text:
sm80s1-4285

Planet Ocean
J.L. Pelegrí and D. Vaqué (eds)

From field experiments to salinity products: a tribute to the contributions of Jordi Font to the SMOS mission

Adriano Camps 1,2,3, Carolina Gabarró 3,4, Mercè Vall-llossera 1,2,3, Sebastià Blanch 1, Albert Aguasca 1, Francesc Torres 1,3, Ignasi Corbella 1,3, Nuria Duffo 1,3, Antonio Turiel 3,5, Marcos Portabella 3,4, Joaquim Ballabrera-Poy 3,4, Verónica González-Gambau 3,4, Justino Martínez 3,4, Ramón Villarino 5, Luís Enrique 6, Alessandra Monerris 7, Xavier Bosch 8, Roberto Sabia 9, Marco Talone 10, Maria Piles 1,2,3, Míriam Pablos 1,2,3, Enric Valencia 1

1 Dept. Teoria del Senyal i Comunicacions, Universitat Politècnica de Catalunya, c/ Jordi Girona 1-3, Campus Nord, 08034 Barcelona, Spain.
2 IEEC/CTE-UPC, C/ Gran Capità, 2-4, desp 201, 08034 Barcelona, Spain.
3 SMOS Barcelona Expert Centre, CSIC-UPC, Pg. Marítim Barceloneta, 37-49, 08003 Barcelona, Spain.
4 Institut de Ciències del Mar, CSIC, Pg. Marítim Barceloneta 37-49, 08003 Barcelona, Spain.
5 Dept. d’Eng. Electrònica, Elèctrica i Automàtica, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain.
6 ICFO, Mediterranean Technology Park, Av. Carl Friedrich Gauss, 3, 08860 Castelldefels, Barcelona, Spain.
7 Faculty of Engineering, Monash University, Clayton Campus, Victoria 3168, Australia.
8 Microwave Systems Lab., Dept. of Electrical and Computer Eng., Colorado State Univ., Fort Collins, CO 80523 USA.
9 Telespazio-Vega UK Ltd for ESA/ESRIN, Via Galileo Galilei, Casella Postale 64, 00044 Frascati (Roma), Italy.
10 Institute for Environment and Sustainability (IES), EC Joint Research Centre, via Enrico Fermi, 2749, Ispra, Italy.

Summary: This article summarizes some of the activities in which Jordi Font, research professor and head of the Department of Physical and Technological Oceanography, Institut de Ciències del Mar (CSIC, Spanish National Research Council) in Barcelona, has been involved as co-Principal Investigator for Ocean Salinity of the European Space Agency Soil Moisture and Ocean Salinity (SMOS) Earth Explorer Mission from the perspective of the Remote Sensing Lab at the Universitat Politècnica de Catalunya. We have probably left out some of his many contributions to salinity remote sensing, but we hope that this review will give an idea of the importance of his work. We focus on the following issues: 1) the new accurate measurements of the sea water dielectric constant, 2) the WISE and EuroSTARRS field experiments that helped to define the geophysical model function relating brightness temperature to sea state, 3) the FROG 2003 field experiment that helped to understand the emission of sea foam, 4) GNSS-R techniques for improving sea surface salinity retrieval, 5) instrument characterization campaigns, and 6) the operational implementation of the Processing Centre of Levels 3 and 4 at the SMOS Barcelona Expert Centre.

Keywords: SMOS; radiometry; interferometry; calibration; validation; salinity; soil moisture; sea ice; GNSS-R.

De campañas de medidas a productos de salinidad: un tributo a las contribuciones de Jordi Font a la mision SMOS

Resumen: Este artículo resume algunas de las actividades en las que Jordi Font, profesor de investigación y jefe del Departamento de Física y Tecnología Oceanográfica, del Institut de Ciències del Mar (CSIC) en Barcelona, ha estado desarrollando como co-Investigador Principal de la parte de la misión SMOS de la ESA, una misión Earth Explorer, desde la perspectiva del Remote Sensing Lab, de la Universitat Politècnica de Catalunya. Seguramente, estamos olvidando algunas de sus muchas contribuciones a la teledetección de la salinidad, pero esperamos que esta revisión dé una idea de la importancia de su trabajo. Este artículo se focaliza en los siguientes puntos: 1) las medidas de alta calidad de la constante dieléctrica del agua marina, 2) las campañas de medidas WISE y EuroSTARRS que ayudaron a la definición del modelo geofísico relacionando la temperatura de brillo con el estado del mar, 3) la campaña de medidas FROG 2003 que ayudó a entender la emisión de la espuma marina 4) presentación de las técnicas de GNSS-R para la mejora de la recuperación de la salinidad superficial 5) campañas para la caracterización del instrumento y 6) la implantación del centro de procesado operacional de niveles 3 y 4 en el SMOS Barcelona Expert Centre.

Citation/Como citar este artículo: Camps A., Gabarró C., Vall-llossera M., Blanch S., Aguasca A., Torres F., Corbella I., Duffo N., Turiel A., Portabella M., Ballabrera-Poy J., González-Gambau V., Martínez J., Villarino R., Enrique L., Monerris A., Bosch X., Sabia R., Talone M., Piles M., Pablos M., Valencia E. 2016. From field experiments to salinity products: a tribute to the contributions of Jordi Font to the SMOS mission. Sci. Mar. 80S1: 159-172. doi: http://dx.doi.org/10.3989/scimar.04285.04A

Editor: J.L. Pelegrí.

Received: June 9, 2015. Accepted: December 10, 2015. Published: September 30, 2016.

Contents

INTRODUCTIONTop

The Remote Sensing Laboratory of the Department of Signal Theory and Communications at the Universitat Politècnica de Catalunya – BarcelonaTech, has been involved since 1993 in the technological aspects of the Microwave Imaging Radiometer with Aperture Synthesis (MIRAS), the single payload of the European Space Agency Soil Moisture and Ocean Salinity (SMOS) Earth Explorer Opportunity mission. We first met Professor Jordi Font in 1998, during the preparation of what later became the SMOS Mission. Since then, we have enjoyed a fruitful scientific collaboration, which has also led to a longstanding friendship. This paper is a tribute to Professor Font’s achievements in remote sensing of the ocean over the last 17 years.

Sea surface salinity (SSS) can be remotely measured using microwave radiometry at L-band (). During the late 1940s and early 1950s, laboratory experiments were conducted to quantify the relationship between salinity and brightness temperature. Until SMOS, only two (experimental) space-borne radiometers at L-band had been launched, in Cosmos 243 (1968) and in Skylab S-194 (1973), but the first evidence from space came only from the Skylab observations (Fig. 1).

The link between salinity and brightness temperature is through the dielectric constant of sea water. Though the “protected” 1.400-1.427 MHz band is the best choice, these measurements are far from trivial due to the low sensitivity of brightness temperature to SSS, 0.2-0.8 K/psu (psu = practical salinity unit, roughly 1 g of salt per 1 kg of water), the dependence on the physical temperature (Fig. 2), incidence angle, polarization, surface roughness (induced by local wind or by far waves), the technical difficulties of achieving a precise calibration and stability, and especially the presence of radio-frequency interference (RFI).

The first SSS airborne map was published in 1996 by Miller and Zaitzeff () using the scanning low frequency microwave radiometer (SLFMR), a six-beam real aperture radiometer (Fig. 3A). In 2000, LeVine et al. obtained an SSS map using the electronically steered thinned array radiometer (ESTAR), the first in-orbit 1D synthetic aperture radiometer () (Fig. 3B).

In 1995, the ESA organized the Soil Moisture and Ocean Salinity Workshop to define the roadmap to remotely measure these two geophysical variables, and it was concluded that the most promising technique was aperture synthesis radiometry, which had been successfully demonstrated a few years earlier with the ESTAR instrument (). In 1998, an international team of scientists proposed the SMOS mission to ESA. It was selected as the second Earth Explorer Opportunity Mission () within the ESA Living Planet Programme and was launched on 2 November 2009 from Plesetsk, Russia (Figs 4 and 5).

TOWARDS A BETTER UNDERSTANDING OF SEA SURFACE EMISSIONTop

In 1998, when the SMOS proposal was submitted, as in any mission, there were many uncertainties and many lessons to be learnt. For ocean salinity retrieval, three key goals related to the geophysical model function had to be met:

– to more accurately characterize the impact of salinity and temperature on the dielectric constant,

– to account for the impact of the sea state (wind speed, significant wave height, etc.) on the brightness temperatures, and

– to account for the impact of sea foam on the brightness temperatures.

Towards a more accurate determination of the sea water dielectric constant

Two of the most widely used dielectric constant models are those of and . Although they show similar results (see Fig. 6), their differences are too great to meet the accuracy requirements, i.e. for an absolute accuracy of 0.2 psu, the brightness temperature must be in the order of 0.1 K for cold waters (Fig. 2).

Most of the methods used to determine the dielectric constant are based on reflectivity measurements in a transmission line, such as the open-ended coaxial sensor. However, these methods require well-known calibrators and they are not accurate enough for this application. Other techniques based on the change in the resonant properties of a cavity are very accurate but provide more limited information (single frequency, temperature and salinity). Finally, transmission methods provide accurate and more comprehensive information from the measurement of the propagation changes of the electromagnetic signal (S-parameters measured with a vector network analyser). This was the technique selected in to derive the dielectric constant of sea water. The test device is shown in Figure 7. It consists of a metallic waveguide through which temperature controlled salty water can circulate by means of a pump. The dimensions were calculated so that a single mode propagates even when the device is filled with salt water.

The main results are summarized in Figure 6. The Klein and Swift (KS) and Blanch and Aguasca (BA) models show similar dielectric constant results for medium temperatures, whereas the Ellison model is slightly different, especially in the real part of the permittivity. Also, the KS and BA models give similar results on brightness temperature, while the Ellison model tends to overestimate it. However, it is difficult to ensure absolute accuracy because of the lack of absolute references.

Towards a better understanding of the impact of sea state

Since 1971, no systematic measurements had been performed to assess the impact of the sea state (parameterized in terms of the wind speed, the significant wave height or other descriptors). Additionally, only a few airborne experiments had been carried out, mostly with the SLFMR real aperture radiometer (, ), and one with the ESTAR synthetic aperture radiometer ().

In 1999 we proposed to ESA the possibility to carry out the WInd and Salinity Experiment (WISE) on the REPSOL Casablanca oil rig (40.72°N 1.36°E), ~50 km offshore of the coast of Tarragona (Spain), and ~115 km south of Barcelona (see Fig. 8). Two WISE campaigns took place (November 2000 to January 2001) and October to November 2001), and were the result of teamwork by the Universitat Politècnica de Catalunya (Barcelona, Spain), the Institut de Ciències del Mar/CSIC (Barcelona, Spain), the Universitat de València (València, Spain), LODYC (later LOCEAN, Paris, France), the Centre d’Etudes des Environnements Terrestre et Planétaires (CETP, later LATMOS, Vélizy, France), and in 2000 the Microwave Remote Sensing Laboratory of the University of Massachusetts (Ahmerst, USA).

The instruments deployed were (Fig. 9): LAURA (L-band AUtomatic RAdiometer), the first ever fully polarimetric (4 Stokes elements) L-band radiometer (UPC), a Ka-band polarimetric radiometer (UMass), a stereo-camera to determine surface topography and root mean square slopes of the sea surface (CETP), four oceanographic and climatological buoys for near-surface salinity, temperature, wind speed and direction, wave height and period, etc. (ICM-CSIC and LODYC), a portable meteorological station with atmospheric pressure, air temperature, relative humidity and rain rate (UPC), a video camera mounted on the L-band radiometer pedestal to determine sea surface foam coverage (UPC), and an IR radiometer to determine sea surface temperature estimates (UV). Additionally, during WISE 2000, temperature and salinity were recorded from the platform at 5 m below sea level, and simultaneously ocean colour, wind vector and sea surface temperature were acquired by different satellites.

Professor Font and Agustí Julià designed an ad hoc buoy for WISE 2000, to collect conductivity and temperature data near the sea surface close to the radiometer field of view and send them to a data logging station installed on the platform using a real-time link (Figs 9E and 9F). The buoy was built by EMS Environmental Monitoring Systems S.L. and modified for WISE 2001 mainly to host extra power batteries. It was a toroidal body with a stainless steel structure to house the signalization elements (flash, radar reflector and satellite ARGOS beacon) and the measuring and transmitting instruments. The main instrument was a SeaBird MicroCAT system (model SBE37-SM) used to record water temperature and conductivity for further salinity determination. An RS-232 interface allowed real-time data transmission by an external UHF link. An additional submersible pump was added to ensure a constant water flow through the conductivity cell. The water inlet was situated at 20 cm below sea level in the central part of the toroid, to minimize the effect of waves that might cause air bubbles to be introduced into the measuring cell. Salinity was computed according to established standards with 0.003 psu/month stability, and 0.0002 psu resolution. The conductivity cell was equipped with a chemical poison device to avoid biofouling and degradation of the conductivity measurement.

The main accomplishment of the WISE campaigns was probably the determination of the brightness temperature dependence with respect to the 10 m height wind speed and the significant wave height required to perform the necessary corrections to retrieve SSS (, , ), including a weak azimuthal signature, which was more clearly seen during a very intense storm that occurred in November 2001:

 $Δ T h ≈0.25( 1+ θ i 118° ) U 10 ; R=0.79$ (1a)

 $Δ T v ≈0.25( 1+ θ i 45° ) U 10 ; R=0.90$ (1b)

 $Δ T h ≈1.09( 1+ θ i 142° )SWH ; R=0.88$ (2a)

 $Δ T v ≈0.92( 1+ θ i 51° ) U 10 ; R=0.78$ (2b)

The processing of the data revealed the following:

– A sensitivity to wind speed extrapolated at nadir of ~0.23 K/(m s–1), or a little bit higher, ~0.25 K/(m s–1), when the atmospheric instability or only the measurements corresponding to U10>2 m s–1 are accounted for. This sensitivity increases at H-polarization up to ~0.5 K/(m s–1) at 65°, and decreases at V-polarization down to ~–0.2 K/(m s–1) at 65°, with a zero-crossing around 50°-55°, in agreement with the small slope approximation method using Durden-Vesecky times 2 and Elfouhaily et al. sea spectra. It is very likely that the computed wind speed sensitivities below 2 m s–1 are erroneous.

– A modulation of the instantaneous brightness temperatures due to wave slopes (and also foam), which makes the standard deviation of this modulation increase with wind speed at a rate of ~0.1-0.15 K/(m s–1), depending on polarization, and very weakly on incidence angle.

– A sensitivity to significant wave height extrapolated to nadir of ~1 K/m, increasing at H-polarization up to ~1.5 K/m at 65°, and decreasing at V-polarization down to –0.5 K/m at 65°.

– A small azimuthal modulation of ~0.2-0.3 K for low-to-moderate wind speeds, in reasonable agreement with the numerical models, but very large (peak-to-peak modulations of 4-5 K) during a strong storm recorded on 10 November 2001 (Fig. 10).

The brightness temperature sensitivity to wind speed and significant wave height compared satisfactorily with numerical models; these sensitivities were used until calibrated SMOS data were available, from which new roughness models were formulated (, ).

EuroSTARRS was an ESA campaign in support of the SMOS mission that took place from 17-24 November 2001, while the WISE 2001 campaign was taking place. The US STARRS (Salinity Temperature and Roughness Remote Scanner) L-band radiometer was mounted on a DLR plane and overflew several regions of Europe (see Fig. 11A). The main objective of EuroSTARRS 2001 was to acquire SMOS-like observations to address a range of critical issues relevant to the SMOS mission. Several flights were dedicated to ocean salinity experiments and were coordinated by Jordi Font. The experiment was defined, among other objectives, to allow the data acquired to estimate multi-angular L-band microwave signatures from low-altitude flights; to compare data acquired at low altitude with data acquired at high altitude; to investigate emission from urban areas and the level of RFI, to validate ocean salinity retrievals along salinity gradients from the coast out to the sea; and to investigate azimuthal effects on the emission over sea.

One of the sites was around the Casablanca oil rig, near the mouth of the Ebro River (Tarragona), mainly to study the effect of wind speed on the salinity measurements and to investigate azimuth dependence. In the Casablanca area, the flight acquisition plan consisted of three phases (see Fig. 11B, C). The first one was a straight transect from the coast towards the platform, which was selected to avoid interferences from the platform. The second consisted in performing ten circles at a constant bank angle of 22° to obtain a large range of incidence angles. The last was the reverse course along the first transect flown at low altitude. The measurements were made after sunshine, in order to avoid interference from sun glint.

The Institut de Ciències del Mar (ICM) participated actively during the flight over the Casablanca oil rig on 21 November. Simultaneously to the airborne flight, two types of measurement were made over the area. The CSIC’s R/V Garcia del Cid carried out a survey in a rectangular area around the Casablanca rig coincident with the plane overflight, collecting oceanographic data, such as underway near-surface temperature and salinity records, CTD profiles spread across the site, acoustic Doppler current profiles and meteorological measurements on-board in a continuous mode. The results were published (, , ).

Understanding the impact of sea foam

One of the open questions from WISE 2000 and 2001 was the evident modulation of the brightness temperatures when a patch of foam appeared in the antenna beam (e.g. Fig. 10B). This was confirmed by the infrared images, in which the foam patches appeared as hot spots. To quantify this effect, as well as the impact of rain and oil spills, and to validate the existing numerical models, the FROG 2003 (Foam, Rain, Oil spills and GPS-reflections) experiment was performed at the IRTA facilities at Poble Nou del Delta (Tarragona, Spain) to study specifically the impact of foam, rain, and oil spills on sea surface emissivity (). Figure 12A shows the experimental set-up in which the LAURA radiometer was installed on the side of a 3×7 m pool filled with a mixture of fresh and sea water. A total of 104 air diffusers distributed over the pool produced a fairly uniform foam coverage. Foam structure was analysed using video images (e.g. Fig. 12B) that were processed to infer the bubble size and distribution, and an array of conductivity sensors was used to infer the air/water fraction as a function of depth (). Results indicated that foam impact is not negligible and should be included in the emission model at L-band. Rain (Fig. 12C) and a thin layer of oil on the surface produce a modification of the water surface roughness that is sufficient to be detectable at L-band.

Ultimately, the impact of sea foam can be parameterized as an emissivity increment, multiplied by the foam fraction F(U10):

 $e p total ( θ )=F( U 10 ) e p foam ( θ )+[ 1−F( U 10 ) ] e p sea ( θ )= = e p sea ( θ )+F( U 10 )Δ e p foam ( θ )$ (3)

 $F\left({U}_{10}\right)=2.32\cdot {10}^{-6}{U}_{10}^{3.4988}$ , during WISE 2000, and $F\left({U}_{10}\right)=0.43\cdot {10}^{-6}{U}_{10}^{3.6824}$ , during WISE 2001. (4)

Actually, F(U10) is also dependent on the air-sea stability and salinity, among others, and can be quite variable (see Fig. 13).

The WISE and FROG campaigns were very useful before the SMOS launch to confirm the general behaviour of the models as a function of the wind speed and foam, but once the satellite had been launched the models needed to be recomputed using the new SMOS data, and the BEC group worked hard to improve the emissivity model (), which is still being used as one of the three parameterizations in the official processing chain.

GNSS-R TECHNIQUES TO IMPROVE SEA SURFACE SALINITY RETRIEVALS FOR SMOS FOLLOW-ON MISSIONSTop

As explained, the capability to properly correct for the sea state impact on the brightness temperature is limited; that is, the remaining uncertainty is too large to meet the scientific requirements. In this Section the main outcome of a project conducted to assess the reliance of GNSS-R (Global Navigation Satellite Systems Reflectometry) techniques to perform a better sea state correction is presented. We take advantage of the closeness in frequency of the protected passive microwave band (1400-1427 MHz) to the radio-navigation bands (L1, L2, and L5, mainly), since both emission and scattering should be affected by the same range of mechanical wavelengths ().

The underlying concept in GNSS-R is quite simple: when the electromagnetic wave is scattered over the sea surface, the scattered signal changes its polarization (from right-hand circular to mostly left-hand circular) and arrives at the receiver mainly from the specular reflection point, determined by the shortest distance between the transmitting GPS satellite (Glonass, Beidou or Galileo) and the receiver. However, as the sea gets roughened, the scattered signals come from a wider region (known as the “glistening zone”) that increases in size with increasing roughness, in a similar manner as the Sun reflecting over the sea.

In classical GNSS-R, observables are derived by cross-correlating the received scattered signal with a local replica of the pseudo-random noise signals (i.e. the open C/A code in the case of GPS) transmitted by the navigation satellites, generated at different delays and Doppler frequency shifts. Actually, the values of this two-dimensional cross-correlation are called the Delay-Doppler Map (DDM). When the sea gets rougher, the volume below the normalized DDM (peak equal to 1) down to a predetermined threshold larger than the noise level increases.

The field experiments ALBATROSS 2008 and 2009 performed in the Gran Canaria island already demonstrated that the changes in the brightness temperatures at different polarizations and incidence angles were correlated to the changes in the volume of the DDM, and that this information could be used synergetically to correct for the sea state. Actually, it was also found that the length of the tail of the waveform (the cut of the DDM for Doppler shift equal to 0 Hz) at 1/e was also highly correlated to the sea state (, , ).

This last result was also used in the COSMOS-OS experimental data to improve the retrieval of the salinity. The experiment conditions were a number of flights perpendicular to the coast in the Gulf of Finland passing from fresh water in a lake close to the coast to (a slightly) salty water. The results are shown in Figure 14. In Figure 14A the peaks of the brightness temperature correlate well with the peaks in the normalized waveform area, suggesting a sea state correction without other auxiliary data. Figure 14B shows the improvement produced by these corrections (green dots), as compared with the retrievals using raw data.

THE SMALL AIRBORNE MIRAS CAMPAIGNTop

The Small airborne MIRAS (AMIRAS) was a reduced version of the MIRAS: Four elements per arm with a unit spacing of 0.875 λ and a noise injection radiometer in the centre. Most of the hardware units, including front-ends, correlators and antennas, were first-version prototypes in the framework of the SMOS project. AMIRAS had been specifically designed mechanically and thermally to fly on board the Short SC-7 Sky-van of the Helsinki University of Technology (TKK). The demonstrator was first installed in an anechoic chamber at UPC, where it underwent a number of characterization tests, including measurement of the antenna patterns.

A ground-based experiment took place with the instrument placed in the open air pointing to the cold sky (IRTA facilities, Spain). This experiment was used to measure the instrument’s response to a constant target and also to make an image of the Galaxy over-pass (). Figure 15 shows the instrument placed at the IRTA facilities and the first image of the Milky Way obtained by AMIRAS.

Finally, the demonstrator was installed on board the TKK sky-van and was flown three times in order to acquire dual-pol images of coastlines and islands. A sequence of snapshots produced during a flight over an island was presented at IGARSS’06 and is shown in Figure 16. It clearly reproduced the coastal geometry, qualitatively demonstrating the imaging capability of the system.

The limited spatial resolution of the AMIRAS instrument and the small number of snapshots over a single point produced a large amount of dispersion of the results, but they were extremely useful for understanding the specificities of the SMOS field of view ().

ON-GROUND CHARACTERIZATION OF THE SMOS PAYLOADTop

The instrument was successfully assembled by EADS-CASA Espacio (Spain) and it was subjected to an extensive testing process aimed at checking the hardware and software operation as well as validating the calibration procedures and the imaging capability. MIRAS was fully deployed inside the Maxwell anechoic chamber at ESA-ESTEC (Fig. 17) and some tests were performed on it (Fig. 18). They included the following: calibration sequence evaluation, polarization check, stability, inter-element phase retrieval, correlation offset, electromagnetic compatibility and image validation ().

MIRAS was successfully characterized on-ground in terms of basic operation, thermal cycling, image validation internal phase and RFI. The results showed that the instrument’s performance was excellent, having all parameters according to specifications, except for an anomaly detected in the temperature dependence of the phase of correlation and a minor interference of the nominal X-band transmitter. The internal calibration concept and operation was validated, providing highly repeatable and accurate calibration parameters with low temperature sensitivity. Absolute amplitude calibration was demonstrated in the anechoic chamber using the average of all Payload Modules, PMS values, showing that the instrument could measure absolute brightness temperature with an accuracy of 0.1 K. Images of the brightness temperature of the empty chamber showed radiometric accuracies of less than half a kelvin and radiometric sensitivities according to the theoretical expectations as a function of integration time and bandwidth.

BARCELONA EXPERT CENTRE AND PROCESSING CENTRE FOR LEVELS 3 AND 4 Top

Some of the above experimental activities and results were the embryo of the so-called empirical model (Model 3) in the SMOS L2 OS Algorithm Theoretical Baseline Document (ATBD) (), which describes all the steps required to perform the retrieval of the ocean surface salinity from the calibrated and geolocated brightness temperatures (Level 1C).

Professor Font’s group was one of the three Expert Support Laboratories giving advice to ESA on how to retrieve SSS (the “holy grail” of physical oceanography) from the instrument measurements from 2005.

The SMOS Barcelona Expert Centre on Radiometric Calibration and Ocean Salinity (SMOS-BEC) was created in 2007 as a joint initiative of the Spanish National Research Council (CSIC) and the Universitat Politècnica de Catalunya (UPC) to contribute to the ground segment of ESA’s SMOS mission, gathering in a single entity the full spectrum of expertise required within the SMOS data processing, ranging from instrument engineers to oceanographers. The UPC’s participation is made through the Passive Remote Sensing Group/Remote Sensing Lab of the Department of Signal Theory and Communications. The main CSIC actor in SMOS is the Physical and Technological Oceanography Department of the ICM, together with the Earth Observation Group of the Institute of Space Sciences (ICE). The SMOS-BEC is installed at the Institute of Marine Sciences (ICM) building, Barcelona, and Professor Font was the head of the group.

BEC’s missions were to provide assessment to ESA as a Level 2 Ocean Salinity Expert Support Laboratory, to contribute to SMOS radiometric calibration and validation activities, and to develop and validate new algorithms for generating added-value products at Levels 3 and 4, helping the Spanish Processing Centre, devoted to Levels 3 and 4 (CP34), to maintain the operational generation and distribution of SSS and soil moisture (SM) products.

ESA’s mandate is to process instrument data up to Level 2, (geophysical products), while Levels 3 and 4 (spatio-temporal averages and synergistic products) are produced by national agencies. Spain pushed for the CP34 (Processing Centre for Levels 3 and 4), which was initially installed at ESAC (the European Space Astronomy Centre of ESA, at Villafranca del Castillo, Spain). BEC was nominated to define and coordinate the SMOS high level products and distribution system of the CP34. INDRA was the company in charge of developing the operational software to generate the products and the dissemination tools, while the company INSA was in charge of the CP34 maintenance at ESAC. All these activities where supported by several National Plan projects led by Professor Font, called MIDAS.

In 2013, the operations of CP34 at ESAC were discontinued and BEC took over. As BEC started to generate and distribute SSS and SM products, some deep, useful changes with respect to the structure of the former CP34 were introduced: a new, easy to navigate webpage was put in place; the structure of the products was simplified and unified under the netCDF format in MF convention to better meet the needs of the scientific community; and new products and new applications were made available: the new CP34-BEC service was created. BEC’s visibility increased enormously thanks to those changes, and now BEC is a reference in the processing and distribution of marine and soil moisture remote sensing products in Spain.

The data produced at BEC are served in netCDF format and are freely distributed to the scientific community by means of OPENDAP, HTTPServer and NetcdfSubset services through a THREDDS server. Maps produced by BEC can also be visualized online by means of a Web Map Service. All products are served at http://cp34-bec.cmima.csic.es/data/available-products.

BEC creates new salinity global maps as new data reprocessing campaigns are performed by ESA and exhaustive quality control reports are made. The product description and quality reports are published at BEC-CP34 (http://cp34-bec.cmima.csic.es/doc/BEC-SMOS-0001-PD.pdf).

Currently, the BEC-CP34 web portal distributes the following products:

Salinity products according to the direction of the SMOS orbit passes: ascending, descending and both products. These products are created in a variety of averaging periods: 3 days and 9 days generated every 3 days, monthly, quarterly and annually. The spatial averaging is computed by default in a regular lat-lon grid of 0.25×0.25°. Several processes have been applied:

- Binned products are constructed by simple weighted averaging of the filtered L2 SSS values. The weighted average of SSS follows ().

- Optimal interpolation (OI) products: Binned products are optimally interpolated (Objective Analysis) to produce maps of higher consistency and fewer gaps as compared with the L3 binned products. The OI is performed using monthly WOA 2009 data as background field.

- L4 Fused products using singularity analysis techniques are obtained with a singularity-analysis-based fusion technique. A template variable of good quality (sea surface temperature) is used as a template to restore the multifractal structure of singularity fronts in a noisy variable (SSS in our case). Further information on the multifractal structure of ocean scalars can be found in .

- SMOS Mediterranean SSS products: Objectively Analysed 9-day 0.25×0.25° L3 SSS product and Fused 1-day 0.25×0.25° L4 SSS product (Fig. 19).

Land products according to ascending and descending passes

- Global Soil Moisture binned products are constructed by simple spatial averaging taking into account only the last orbit measurements in each cell. They are operationally generated by averaging daily, 3-day, 9-day, monthly and annual data on an EASE 25 km grid.

- Soil Moisture Level 4 products: By combining SMOS Level 1C TBs, with Terra/Aqua MODIS land surface temperature and normalized difference vegetation index data, high resolution soil moisture maps (1 km) can be produced over the Iberian Peninsula with a delay of less than 12 hours. ().

- Forest fires risk assessment and senescence: As shown in (), fire ignition is more likely to occur in the northeast of the Iberian Peninsula under dry soil conditions (as seen by SMOS). Since 2012, BEC SMOS high resolution soil moisture maps in near real time are being routinely used by the Diputacio de Barcelona (DiBa) for fire risk assessment and prevention during the forest fire season.

– Other studies and products in development at BEC are:

- Sea ice concentration maps with SMOS data: At 1.4GHz, the brightness temperature values are very different for sea and sea ice, allowing an index of ice concentration to be established. A new statistical algorithm to retrieve this index with SMOS data is being used to produce daily maps ().

- SSS maps at high latitudes (>50ºN) have been produced. It is well known that the sensitivity of the TB to SSS decreases for cold waters (Fig. 2), but by using empirical bias correction technics, it is possible to retrieve salinities from cold waters with reasonable accuracy.

- A sea surface density product has been implemented based on the new equation of state for sea water, the thermodynamic equation of seawater, TEOS10. The product is ready to go into operation, although validation is still needed. The maps are based in the combination of OI SSS L3 and OSTIA SST.

- TB Intercomparison between SMOS and Aquarius: Several studies on SMOS and Aquarius intercomparison have been performed for several Earth regions: Antarctica, the open ocean, dry and wet lands, the Amazon, etc. ().

- The ice sheet layer in Antarctica with Aquarius and SMOS data: A good agreement was observed between the brightness temperatures and the ice thickness variations over selected transects of eastern Antarctica. The presence of subglacial lakes, which are generally located at ~3 km depth, may influence the physical temperature profile and/or the dielectric properties of the ice layers above, affecting the observed brightness temperatures (, ).

All these products and the research activities conducted at BEC during the last few years are broadly described in . The paper reviews the main scientific activities and achievements of the Barcelona Expert Centre under the direction of Jordi Font, as well as the future directions, highlighting the importance of the oceanographic applications of the mission

For his dedication to the pioneer SMOS mission, Professor Font received the Catalan National Art Prize 2011 in the category of Scientific Thought and Culture (see Fig. 20).

CONCLUSIONSTop

This article has summarized some of the contributions of Jordi Font, co-Principal Investigator for Ocean Salinity of the ESA SMOS Earth Explorer Mission.

Among his many direct contributions, and all those triggered by his work, we have emphasized the ICM/CSIC contributions to the field experiments WISE 2000 and 2001, and FROG 2003, which helped define the geophysical model function relating the brightness temperature to the sea state. We have also stressed his efforts to establish a new ocean salinity dielectric constant. Finally, we have highlighted his drive towards the creation of the Barcelona Expert Centre, the implementation of the CP34, and the development of new products. We will miss his leadership, but his approach to oceanography will always remain.

ACKNOWLEDGEMENTSTop

This work has been performed under research grants TEC2005-06863-C02-01/TCM, ESP2005-06823-C05 and ESP2007-65667-C04, AYA2008-05906-C02-01/ESP and AYA2010-22062-C05 and ESP2015-70014-C2-1-R, and EURYI 2004 award.

REFERENCESTop

Blanch S., Aguasca A. 2004. Seawater dielectric permittivity model from measurements at L band. Proced. IEEE Geosci. Rem. Sens. Symp. 2: 1362-1365.

Boutin J., Martin N., Yin Y., et al. 2012. First assessment of SMOS data over open ocean: Part II-sea surface salinity. IEEE Trans. Geosci. Rem. Sens. 50 : 1662-1675.
http://dx.doi.org/10.1109/TGRS.2012.2184546

Camps A., Font J., Etcheto J., et al. 2002. L-band sea surface emissivity radiometric observations under high winds: Preliminary results of the Wind and Salinity Experiment WISE-2001. Proceed. IEEE Geosci. Rem. Sens. Symp. 3: 1367-1369.

Camps A., Font J., Vall-llossera M., et al. 2004. The WISE 2000 and 2001 field experiments in support of the SMOS mission: sea surface L-band brightness temperature observations and their application to sea surface salinity retrieval. IEEE Trans. Geosci. Rem. Sens. 42(4): 804-823.
http://dx.doi.org/10.1109/tgrs.2003.819444

Camps A., Vall-llossera M., Villarino R., et al. 2005. The emissivity of foam-covered water surface at L-band: theoretical modeling and experimental results from the FROG 2003 field experiment. IEEE Trans. Geosci. Rem. Sens. 43(5): 925-937.
http://dx.doi.org/10.1109/TGRS.2004.839651

Camps A., Bosch-Lluis X., Ramos-Perez I., et al. 2009. New Passive Instruments Developed for Ocean Monitoring at the Remote Sensing Lab—Universitat Politècnica de Catalunya. Sensors 9: 10171-10189.
http://dx.doi.org/10.3390/s91210171

Chaparro D., Vall-llossera M., Piles M., and the SMOS-BEC Team. 2015. Remotely sensed soil moisture and forestry applications. SMOS Science Workshop ESA-ESAC, Villafranca del Castillo (Madrid), Spain.

Corbella I., Torres F., Duffo N., et al. 2008 Brightness Temperature retrievals from the Small Airborne MIRAS, IGARSS’08, Massachusetts, USA.

Corbella I, Torres F., Duffo N., et al. 2009. On-Ground Characterization of the SMOS Payload, 2009. Trans. Geosci. Rem. Sens. 47: 3123-3132.
http://dx.doi.org/10.1109/TGRS.2009.2016333

Ellison W., Balana A., Delbos G., et al. 1998. New Permittivity Measurements of Sea Water. Radio Sci. 33(3): 639-648.
http://dx.doi.org/10.1029/97RS02223

Emelianov M., Font J., Julià A., et al. 2003. Sea surface fields at Casablanca site (NW Mediterranean) during the EuroSTARRS campaign. In: Proceedings of SMOS Campaigns Workshop, ESA SP. 525: 73-80.

Font J., Gabarró C., Julià A., et al. 2003. Oceanographic conditions during the Wind and Salinity Experiment 2000 and 2001, NW Mediterranean Sea. In: Proceedings of SMOS Campaigns Workshop, ESA SP. 525: 51-59.

Gabarró C. 2004. Study of salinity retrieval errors for the SMOS mission. PhD thesis, Tech. Univ. Catalonia.

Gabarró C., Font J., Camps A., et al. 2003. Retrieved Sea Surface Salinity and Wind Speed from L-Band measurements for WISE and EuroSTARRS campaigns. In: Proceedings of SMOS Campaigns Workshop, ESA SP. 525: 163-171.

Gabarró C., Font J., Camps A., et al. 2004. A new empirical model of sea surface microwave emissivity for salinity remote sensing. Geophys. Res. Lett. 31: L01309.
http://dx.doi.org/10.1029/2003GL018964

Gabarró C., Pla Q., Elosegui P., et al. 2015. Investigating SMOS data for sea ice concentration determination. SMOS Science Workshop, ESAC- Madrid, Spain.

Guimbard S., Gourrion J., Portabella M., et al. 2012. SMOS Semi-Empirical Ocean Forward Model Adjustment. IEEE Trans. Geosci. Rem. Sens. 50: 1676-1687.
http://dx.doi.org/10.1109/TGRS.2012.2188410

Hollinger J.P. 1971. Passive Microwave Measurements of Sea Surface Roughness. IEEE Trans. Geosci. Electronics, GE-9(3): 165-169.
http://dx.doi.org/10.1109/TGE.1971.271489

Klein L.A., Swift C.T. 1977. An improved model for the dielectric constant of sea water at microwave frequencies. IEEE Trans. Anten. Propag. AP 25: 104-111.
http://dx.doi.org/10.1109/TAP.1977.1141539

LeVine D.M., Zaitzeff J.B., D’Sa E.J., et al. 2000. Sea surface salinity: toward an operational remote-sensing system. Satellites, oceanography and society. Elsevier Oceanography Series 63: 321-335.

Marchan J.F., Camps A., Rodríguez N., et al. 2009. An Efficient Algorithm to the Simulation of Delay–Doppler Maps of Reflected Global Navigation Satellite System Signals. IEEE Trans. Geosci. Rem. Sens. 47: 2733-2740.
http://dx.doi.org/10.1109/TGRS.2009.2014465

Martin-Neira M., Cabeza I., Pérez C., et al. 2008. AMIRAS – an airborne MIRAS demonstrator. IEEE Trans. Geosci. Rem. Sens. 46(3): 705-716.
http://dx.doi.org/10.1109/TGRS.2008.916266

Miller J., Goodberlet M.A., Zaitzeff J. 1996. Airborne salinity mapper makes debut in coastal zone. EOS Trans. AGU 79: 173-177.
http://dx.doi.org/10.1029/98EO00126

Pablos M., Piles M., González-Gambau V., et al. 2014. SMOS and Aquarius Radiometers: Inter-Comparison over Selected Targets, IEEE J-STARS 7(9): 3833-3844.
http://dx.doi.org/10.1109/jstars.2014.2321455

Pablos M., Piles M. González-Gambau V., et al. 2015a. Ice Thickness Effects on Aquarius Brightness Temperatures over Antarctica. J Geophys Res C 120(4): 2856-2868.
http://dx.doi.org/10.1002/2014JC010151

Pablos M., Piles M., González-Gambau V., et al. 2015b. Influence of Ice Thickness on SMOS and Aquarius Brightness Temperatures over Antarctica. IEEE IGARSS 2015, 26-31 July 2015, Milan (Italy) pp. 5178-5181.

Piles M., Sánchez N., Vall-llossera M., et al. 2014. A Downscaling Approach for SMOS Land Observations: Evaluation of High-Resolution Soil Moisture Maps Over the Iberian Peninsula. IEEE J-STARS 7(9): 3845-3857.
http://dx.doi.org/10.1109/jstars.2014.2325398

Ruf C.S., Swift C.T., Tanner A.B. et al. 1998. Interferometric synthetic aperture microwave radiometry for the remote sensing of the Earth. IEEE Trans. Geosci. Rem. Sens. 26: 597-611.
http://dx.doi.org/10.1109/36.7685

Silvestrin P., Berger M., Kerr Y., et al. 2001. ESA’s second earth explorer opportunity mission: The soil moisture and ocean salinity mission— SMOS. IEEE Geosci. Rem. Sens. Newslett. 118: 11-14.

SMOS Salinity Expert Support Laboratories. 2014. SMOS L2 OS Algorithm Theoretical Baseline Document, ref SO-TN-ARG-GS-0007.
https://earth.esa.int/web/guest/-/data-processors-7632

Swift C.T. 1980. Passive microwave remote sensing of the ocean - a review. Boundary - Layer Meteorology. 18: 25-54.
http://dx.doi.org/10.1007/BF00117909

Swift C.T., McIntosh R.E. 1983. Considerations for microwave remote sensing of ocean-surface salinity. IEEE Trans. Geosci. Elec. 21: 480-491.
http://dx.doi.org/10.1109/tgrs.1983.350511

Talone M., Camps A., Marchan-Hernandez J.F., et al. 2009. Preliminary Results of the Advanced LBand Transmission and Reflection Observation of the Sea Surface (ALBATROSS) Campaign: Preparing the SMOS Calibration and Validation Activities. Proc. IEEE Int. Geosci. and Rem. Sens. Symp., Cape Town, South Africa.

Turiel A., Nieves V., Garcia-Ladona E., et al. 2009. The multifractal structure of satellite sea surface temperature maps can be used to obtain global maps of streamlines. Ocean Sci. 5(4): 447-460.
http://dx.doi.org/10.5194/os-5-447-2009

Turiel A., Piles M., González-Gambau V., et al. 2016. 2000 days of SMOS at the Barcelona Expert Centre: a tribute to the work of Jordi Font. Sci. Mar. 80S1: 173-193.
http://dx.doi.org/10.3989/scimar.04291.15A

Valencia E., Camps A., Rodriguez-Alvarez N., et al. 2011. Improving the accuracy of sea surface salinity retrieval using GNSS-R data to correct the sea state effect. Radio Sci. 46: RS0C02.
http://dx.doi.org/10.1029/2011RS004688


text:
A NEW BIHARMONIC KERNEL FOR THE UPPER HALF PLANE

Title & Authors
A NEW BIHARMONIC KERNEL FOR THE UPPER HALF PLANE
Abkar, Ali;

Abstract
We introduce a new biharmonic kernel for the upper half plane, and then study the properties of its relevant potentials, such as the convergence in the mean and the boundary behavior. Among other things, we shall see that Fatou's theorem is valid for these potentials, so that the biharmonic Poisson kernel resembles the usual Poisson kernel for the upper half plane.
Keywords
Poisson kernel;biharmonic function;biharmonic Green function;convergence in the mean;Fatou's theorem;
Language
English
Cited by
1.
Biharmonic Green Functions on Homogeneous Trees, Mediterranean Journal of Mathematics, 2009, 6, 3, 249
References
1.
A. Abkar and H. Hedenmalm, A Riesz representation formula for super-bi- harmonic functions, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 305-324

2.
P. R. Garabedian, Partial Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964

3.
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981

4.
H. Hedenmalm, A computation of Green functions for the weighted biharmonic operators ${\Delta}|z|^{-2{\alpha}}{\Delta}$ with ${\alpha}>-1$ Duke Math. J. 75 (1994), no. 1, 51-78

format: plain


text:
# Revision history [back]

Hi, this error is because you got more than 1 network available in your project, you need to specify in the stack's YAML the propertie

network_id: <net_ID>

You can see a complete reference of Heat options in http://docs.openstack.org/developer/heat/template_guide/openstack.html

Hi, this error is because you got more than 1 network available in your project, you need to specify in the stack's YAML the propertie

network_id: <net_ID>
networks:
- network: <Network name or UUID>

You can see a complete reference of Heat options in http://docs.openstack.org/developer/heat/template_guide/openstack.html


text:
# Publication Citation

NIST Authors in Bold

Author(s): Sita Ramamurti; David E. Gilsinn; Bicubic B-Spline Surface Approximation of Invariant Tori October 20, 2010 The invariant torus of a coupled system of Van der Pol oscillators is approximated using bicubic B-splines. The paper considers the case of strong nonlinear coupling. In particular, the shapes of invariant torii for the Van der Pol coupling parameter $\lambda$ are computed in the range [0.1, 2.0]. Comparisons are given with results obtained by the MATLAB differential equation solver ode45. Very good normed residual errors of the determining equations for the approximate tori for the cases $\lambda = 0.1,\ 0.6$ are shown. At the upper limit of $\lambda = 2.0$ memory errors occured during the optimization phase for solving the determining equations so that full optimization for some knot sets was not achieved, but a visual comparison of the resulting invariant torus figure showed a close similarity to the solution using ode45. NIST Interagency/Internal Report (NISTIR) - 7731 32 pp. bicubic B-splines; determining equations; invariant torus; large parameter case; optimization; Van der Pol oscillators Software, Math Click here to retrieve PDF version of paper (2MB)


text:
# No sound after Liquorix Kernel update

I recently installed the Liquorix Kernel after hearing that it had the 200 line patch in the kernel.

Once I installed it, everything worked great and I had HUGE performance improvements. At that point i was using the 2.6.36-14 version.

The kernel has now updated twice to version 2.6.36.-16 and now I have no sound coming from my system. Alsa mixer shows that nothing is muted.

How can i resolve this issue?

Here is the script output file everyone requested. alsa-info.txt

-
You may need to contact Liquorix Kernel maintainer to get help. I have only one clue: you need to rebuild the sound stack (alsa and pulseaudio) to get it working with Liquorix. Thanks! –  crncosta Nov 23 '10 at 20:52
PulseAudio shouldn't need to be rebuilt if the change was only effected by a kernel update. –  Daniel T Chen Nov 23 '10 at 21:07
Yes it is only affected by the kernel because booting into the official ubuntu kernel, the sound works fine. –  user6465 Nov 23 '10 at 21:28
Have you executed the recommended script as referenced at askubuntu.com/questions/14596/…? –  Daniel T Chen Nov 24 '10 at 11:50
Yes, I executed the script and its posted below. –  user6465 Nov 25 '10 at 0:38

I would argue you should ask in the Liquorix forums. Only a few people here have experience with the patches it uses.

Something to try before you do though, and it'll sound silly, but have you run alsamixer and checked that it isn't muted? Sounds silly, but you wouldn't be the first person to overlook it. We've all done it at least once ;)

-
I check and no, my sound is not muted. Thanks for the tip though. –  user6465 Nov 24 '10 at 5:47

Please see Ubuntu sound not working, thanks!

If you feel odd about posting a bug report in Launchpad, the procedure we normally follow to request necessary information is to ask that a developer's debugging shell script be executed. This bash script is available at http://www.alsa-project.org/alsa-info.sh, and you would execute it using bash ~/Desktop/alsa-info.sh --no-upload in a Terminal provided you saved the shell script to your Desktop. You can then pastebin /tmp/alsa-info.txt and/or link to it from a response here.

-
I wouldn't recommend filing a bug for something that only happens when a 3rd party kernel is installed. –  8128 Nov 23 '10 at 20:45
Normally I wouldn't either, but asking someone to endure a rather laborious information-gathering process is worse than having a bug filed then converting it to a question - or just having it filed as a question directly. The former approach is preferable, because we gather all the necessary information using apport hooks. –  Daniel T Chen Nov 23 '10 at 20:51
ubuntuone.com/p/Qfi This is the line to the file. –  user6465 Nov 24 '10 at 20:37


text:
# How do you factor 16x^3 + 54?

In general ${a}^{3} + {b}^{3} = \left(a + b\right) \left({a}^{2} - a b + {b}^{2}\right)$
$16 {x}^{3} + 54$
$= 2 \left({\left(2 x\right)}^{3} + {\left(3\right)}^{3}\right)$
$= 2 \left(2 x + 3\right) \left(4 {x}^{2} - 6 x + 9\right)$


text:
# How do you sketch theta=(7pi)/4 in standard position?

Jul 29, 2015

$2 \pi = \frac{8 \pi}{4}$ is one full revolution
$\rightarrow \frac{7 \pi}{4}$ is $\frac{\pi}{4}$ less than a full revolution.

#### Explanation:

...to look at it another way:

$\frac{\pi}{2}$ is a right angle

$\frac{7 \pi}{4} = 3 \cdot \left(\frac{\pi}{2}\right) + \frac{1}{2} \cdot \frac{\pi}{2}$
$\textcolor{w h i t e}{\text{XXXX}}$ is 3 right-angles plus half a right angle


text:
# Curvature of a measure

In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the Melnikov curvature or Menger-Melnikov curvature. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel.

## Definition

Let μ be a Borel measure on the Euclidean plane R2. Given three (distinct) points x, y and z in R2, let R(xyz) be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear. The Menger curvature c(xyz) is defined to be

$c(x, y, z) = \frac{1}{R(x, y, z)},$

with the natural convention that c(xyz) = 0 if x, y and z are collinear. It is also conventional to extend this definition by setting c(xyz) = 0 if any of the points x, y and z coincide. The Menger-Melnikov curvature c2(μ) of μ is defined to be

$c^{2} (\mu) = \iiint_{\mathbb{R}^{2}} c(x, y, z)^{2} \, \mathrm{d} \mu (x) \mathrm{d} \mu (y) \mathrm{d} \mu (z).$

More generally, for α ≥ 0, define c2α(μ) by

$c^{2 \alpha} (\mu) = \iiint_{\mathbb{R}^{2}} c(x, y, z)^{2 \alpha} \, \mathrm{d} \mu (x) \mathrm{d} \mu (y) \mathrm{d} \mu (z).$

One may also refer to the curvature of μ at a given point x:

$c^{2} (\mu; x) = \iint_{\mathbb{R}^{2}} c(x, y, z)^{2} \, \mathrm{d} \mu (y) \mathrm{d} \mu (z),$

in which case

$c^{2} (\mu) = \int_{\mathbb{R}^{2}} c^{2} (\mu; x) \, \mathrm{d} \mu (x).$

## Examples

• The trivial measure has zero curvature.
• A Dirac measure δa supported at any point a has zero curvature.
• If μ is any measure whose support is contained within a Euclidean line L, then μ has zero curvature. For example, one-dimensional Lebesgue measure on any line (or line segment) has zero curvature.
• The Lebesgue measure defined on all of R2 has infinite curvature.
• If μ is the uniform one-dimensional Hausdorff measure on a circle Cr or radius r, then μ has curvature 1/r.

## Relationship to the Cauchy kernel

In this section, R2 is thought of as the complex plane C. Melnikov and Verdera (1995) showed the precise relation of the boundedness of the Cauchy kernel to the curvature of measures. They proved that if there is some constant C0 such that

$\mu(B_{r} (x)) \leq C_{0} r$

for all x in C and all r > 0, then there is another constant C, depending only on C0, such that

$\left| 6 \int_{\mathbb{C}} | \mathcal{C}_{\varepsilon} (\mu) (z) |^{2} \, \mathrm{d} \mu (z) - c_{\varepsilon}^{2} (\mu) \right| \leq C \| \mu \|$

for all ε > 0. Here cε denotes a truncated version of the Menger-Melnikov curvature in which the integral is taken only over those points x, y and z such that

$| x - y | > \varepsilon;$
$| y - z | > \varepsilon;$
$| z - x | > \varepsilon.$

Similarly, $\mathcal{C}_{\varepsilon}$ denotes a truncated Cauchy integral operator: for a measure μ on C and a point z in C, define

$\mathcal{C}_{\varepsilon} (\mu) (z) = \int \frac{1}{\xi - z} \, \mathrm{d} \mu (\xi),$

where the integral is taken over those points ξ in C with

$| \xi - z | > \varepsilon.$

## References

• Mel'nikov, Mark S. (1995). "Analytic capacity: a discrete approach and the curvature of measure". Mat. Sb. 186 (6): 57–76. ISSN 0368-8666.
• Melnikov, Mark S. and Verdera, Joan (1995). "A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs". Internat. Math. Res. Notices 1995 (7): 325–331. doi:10.1155/S1073792895000249.
• Tolsa, Xavier (2000). "Principal values for the Cauchy integral and rectifiability". Proc. Amer. Math. Soc. 128 (7): 2111–2119. doi:10.1090/S0002-9939-00-05264-3.


text:
Java MD5 HackerRank Solution. HackerRank solutions in Java/JS/Python/C++/C#. Beeze Aal 09.Jun.2020. My Hackerrank profile.. An organized, detail-oriented, and conscientious self-starter. For example, computing the hash of a downloaded file and comparing the result to a … (Java Solution) HackerRank Java- MD5 MD5 is one in a series of message digest algorithms designed by Professor Ronald Rivest of MIT (Rivest, 1994); however, the security of MD5 has been severely compromised, most infamously by the Flame malware in 2020. An English text needs to be encrypted using the following encryption scheme. they're used to log you in. Hackerrank Solutions. This is the solution for the Encryption problem in the implementation section of the Algorithm domain. Sample Output 0eval(ez_write_tag([[580,400],'thepoorcoder_com-medrectangle-4','ezslot_6',104,'0','0'])); ,  is between  and .Rewritten with  rows and  columns: eval(ez_write_tag([[300,250],'thepoorcoder_com-box-4','ezslot_3',108,'0','0']));,  is between  and .Rewritten with  columns and  rows ( so we have to use .). Cryptographic hash functions are mathematical operations run on digital data; by comparing the computed hash (i.e., the output produced by executing a hashing algorithm) to a known and expected hash value, a person can determine the data's integrity. Contribute to RyanFehr/HackerRank development by creating an account on GitHub. If multiple grids satisfy the above conditions, choose the one with the minimum area, i.e. We use essential cookies to perform essential website functions, e.g. Here you will get program for caesar cipher in Java for encryption and decryption. Anyway, on HackerRank the tests pass with no timeout issues. Then, characters are written into a grid, whose rows and columns have the following constraints: We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. However, since distance in the worst case is linear, the overall complexity might turn to $$O(N^2)$$. def encrypt_list(some_list): encrpted_list = [] for i in range(c): string = "" for j in range(f): string += my_list[j][i:i + 1] encrpted_list.append(string) return encrpted_list Which looks like a hack. is between and , so it is written in … Here are some common uses for MD5: To store a one-way hash of a password. Now the final step remaining is to print the values separated by a spaceeval(ez_write_tag([[300,250],'thepoorcoder_com-large-leaderboard-2','ezslot_4',110,'0','0'])); © 2020 The Poor Coder | Hackerrank Solutions - This post covers the solutions of certification problems of problem solving. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. Check if incoming character is letter. Hackerrank Problem, encryption python solution is given in this video, its explanation is also provided. We say that a string contains the word hackerrank if a subsequence of its characters spell the word hackerrank.For example, if string it does contain hackerrank, but does not.In the second case, the second r is missing. We consider two strings to be anagrams of each other if the first string's letters can be rearranged to form the second string. You will be given a message to encode and print. It should return a single string composed as described. Create a function for encrypting which takes a cipher shift as parameter as well. Learn more, Cannot retrieve contributors at this time. Let be the length of this text. (This is essentially the Encryption problem from HackerRank.) Caesar cipher or Shift Cipher is a Substitution cipher algorithm in which each letter of the plain text (message) is substituted with another letter. Challenge Name: Super Reduced String Problem: Alice wrote a sequence of words in CamelCase as a string of letters, s, having the following properties: When you select a contiguous block of text in a PDF viewer, the selection is highlighted with a blue rectangle. (Java Solution) Project Euler > Problem 185 > Number Mind (Java Solution) Project Euler > Problem 186 > Connectedness of a network. Approach: Create a frequency array to store the frequency of each character . In a new kind of PDF viewer, the selection of each word is independent of the other words; this means that each rectangular selection area forms independently around each highlighted word. You are given an unordered array consisting of consecutive integers [1, 2, 3, ..., n] without any duplicates. HackerRank Problem Solving Basic Certification Solutions 2020. Complete the encryption function in the editor below. eval(ez_write_tag([[580,400],'thepoorcoder_com-medrectangle-3','ezslot_8',103,'0','0']));Output Format. This post is “For Educational Purposes Only” Also Read: HackerRank JAVA Basic Certification Solutions 2020 . Using default dict we will initialize an empty dictionary. For each test case, print the encoded string. Flexible and analytical with an infectious enthusiasm for technology. Problem:-. The rest of them are getting timed out. We are not providing the only hackerRank solution but we have also a huge collection of List of C Programs With Complete Solution, List of CPP Programs. Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. First, the spaces are removed from the text. they're used to gather information about the pages you visit and how many clicks you need to accomplish a task. Leave a Reply Cancel reply. Print the encoded message on one line as described. In this An English text needs to be encrypted using the following encryption scheme.First, the spaces are removed from the text. The solution uses extra space but it does not change the input. .MathJax_SVG_Display {text-align: center; margin: 1em 0em; position: relative; display: block!important; text-indent: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; width: 100%} .MathJax_SVG .MJX-monospace {font-family: monospace} .MathJax_SVG .MJX-sans-serif {font-family: sans-serif} .MathJax_SVG {display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: Hackerrank Breadth First Search: Shortest Reach Solution. It passes 7 test cases out of 15. Hackerrank - Encryption Solution. Sample Input. Hackerrank is a site where you can test your programming skills and learn something new in many domains.. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. Therefore, if we choose an offset greater or equal to 26, we loop, at least one time, over the entire alphabet. Caesar Cipher Hackerrank Algorithm Solution in Java Input Format. I found the problem from HackerRank Algorithms practice part. Java SHA-256 HackerRank Solution. You signed in with another tab or window. Let be the length of this text. Let  be the length of this text.Then, characters are written into a grid, whose rows and columns have the following constraints: For example, the sentence , after removing spaces is  characters long. I think it's because of … For more information, see our Privacy Statement. Then, characters are written into a grid, whose rows and columns have the following constraints: For example, the sentence , after removing spaces is characters long. Is there a more Pythonic way? The second line... Output Format. Previous post Huffman Decoding – Hackerrank. Sample Output. Home >> Programming Questions >> Minimum Swaps 2 Minimum Swaps 2 Hackerrank Solution In this post, you will learn how to solve Hackerrank's Minimum Swaps 2 Problem and its solution in Java. 30 Days Of Code HackerRank. To provide some assurance that a transferred file has arrived intact. Let L be the length of this text. The Question can be found in the Algorithm domain of Hackerrank. Then we use math.floor and math.ceil function to calculate the required number of rows and columns denoted by r and c respectively. eval(ez_write_tag([[300,250],'thepoorcoder_com-banner-1','ezslot_5',109,'0','0']));We use for loop to divide our string into equal number of columns, Instead of printing the value of our variable sub we add each letter to our default dict. I created almost all solutions in 4 programming languages - Scala, Javascript, Java and Ruby. Published with, Hackerrank Snakes and Ladders: The Quickest Way Up Solution. is between  and , so it is written in the form of a grid with 7 rows and 8 columns. This is not a regular post of mine about a certain mobile topic, this post is about my solution for a general interesting challenge that I found in HackerRank. The encryption problem is solved in python language with full … Here's the solution I came up with (in Java, but anyone with c++ background can read this code) import java.util.Arrays; import java.util.List; public class TickerPurcahseTime { public static int calculateTimeOptimized(List line, int pos) { // Minimum time I have to wait is the number of tickets I want to buy. Get code examples like "diagonal difference hackerrank solution in java 8 using list" instantly right from your google search results with the Grepper Chrome Extension. Problem Statement: Given a string consisting of letters, a, b and c, we can perform the following operation: Take any two adjacent distinct characters and replace them with the third character. The overall complexity seems $$O(N \cdot log(N))$$. You can always update your selection by clicking Cookie Preferences at the bottom of the page. Project Euler > Problem 182 > RSA encryption (Java Solution) Project Euler > Problem 183 > Maximum product of parts (Java Solution) Project Euler > Problem 184 > Triangles containing the origin. Given an unsorted array of n elements, find if the element k is present in the array or not. For example, the encoded message for the above rectangle is: imtgdvs fearwer mayoogo anouuio ntnnlvt wttddes aohghn sseoau. I found this page around 2014 and after then I exercise my brain for FUN. encryption has the following parameter(s):eval(ez_write_tag([[468,60],'thepoorcoder_com-box-3','ezslot_2',102,'0','0'])); is comprised only of characters in the range ascii[a-z]. In this time, I used C++ as an… Find the solution of other programming problems ie, Data Structure and Algorithms, or GeeksforGeeks solution. The first line contains an integer, N, which is the length of the unencrypted string. Next post The Coin Change Problem – Hackerrank Solution. MD5 ( Message-Digest algorithm 5) is a widely-used cryptographic hash function with a -bit hash value. Enter your comment … Tagged Encryption, Encryption Hackerrank, Encryption Solution in Java Post navigation. Make it Anagram Hacker Rank Problem Solution Using JAVA Alice is taking a cryptography class and finding anagrams to be very useful. The encoded message is obtained by displaying the characters in a column, inserting a space, and then displaying the next column and inserting a space, and so on. Hello Friends, in this tutorial we are going to learn CamelCase Hackerrank Algorithm Solution in Java. You are given an unordered array consisting of consecutive integers [1, 2, 3, ..., n] without any duplicates. I have a solution to the HackerRank Circular Array Rotation challenge. \ ) use GitHub.com so we can make them better, e.g with an infectious enthusiasm technology... No timeout issues, Java and Ruby complexity seems \ ( O ( N ) ) )! Hash value encryption hackerrank solution in java clicking Cookie Preferences at the bottom of the Algorithm domain of HackerRank )! Solution in Java for encryption and decryption \ ) Structure and Algorithms, or solution... Is the solution to the solution for the above rectangle is: imtgdvs fearwer mayoogo anouuio ntnnlvt aohghn. Encrypting which takes a cipher shift as parameter as well to be encrypted using the following scheme... … HackerRank solutions in 4 programming languages - Scala, Javascript, Java and.... \ ( O ( N \cdot log ( N ) ) \.! Selection is highlighted with a -bit hash value Javascript, Java and Ruby when select... Tagged encryption, encryption solution in Java store the frequency of each character ( Message-Digest Algorithm )... One-Way hash of a grid with 7 rows and columns denoted by and. Area, i.e your approach on first, before moving on to solution! First line contains an integer, N ] without any duplicates for FUN gather information the... Then i exercise my brain for FUN after then i exercise my for! Encoded message for the encryption problem from HackerRank. Rotation challenge use third-party. Almost all solutions in Java/JS/Python/C++/C # the second string Tagged encryption, encryption HackerRank encryption. Solved in python widely-used cryptographic hash function with a blue rectangle is: imtgdvs fearwer mayoogo anouuio ntnnlvt aohghn... Without any duplicates creating an account on GitHub is the length of Algorithm! Is written in the form of a grid with 7 rows and 8 columns solution uses extra space it! 50 million developers working together to host and review code, manage projects, and build together! Solution uses extra space but it does not Change the input found the problem from HackerRank practice... Together to host and review code, manage projects, and build software together learn something new many. Uses extra space but it does not Change the input clicks you need to a. Return a single string composed as described on HackerRank the tests pass with no timeout issues (! On one line as described and how many clicks you need to accomplish a task for cipher. A cipher shift as parameter as well contiguous block of text in a PDF viewer, selection. How many clicks you need to accomplish a task learn something new in many domains them better,.! Its explanation is also provided which takes a cipher shift as parameter as well solution is in! Array consisting of consecutive integers [ 1, 2, 3,..., N, which the! Integer, N, which is the solution of other programming problems ie, Data Structure and Algorithms or... Contribute to RyanFehr/HackerRank development by creating an account on GitHub consecutive integers [,! The length of the page GitHub is home to over 50 million developers working together to host and code! Can be found in the implementation section of the unencrypted string and, so it is written …! Learn more, can not retrieve contributors at this time array Rotation challenge conditions, choose one! For the encryption problem in the implementation section of the page solution uses extra space but it does Change. Java and Ruby N ) ) \ ) then we use essential cookies to perform essential website functions e.g. Assurance that a transferred file has arrived intact is a site where you can always update your by. 2014 and after then i exercise my brain for FUN, i.e so we can make them better e.g. A widely-used cryptographic hash function with a blue rectangle of HackerRank. clicking Cookie Preferences the. Working together to host and review code, manage projects, and build together. Are given an unordered array consisting of consecutive integers [ 1, 2, 3,..., N without. Some assurance that a transferred file has arrived intact website functions, e.g test programming! I found this page around 2014 and after then i exercise my brain FUN! C++ as an… this is essentially the encryption problem in the form of a password Java for and... Of HackerRank. - Scala, Javascript, Java and Ruby created almost all solutions in Java/JS/Python/C++/C.... Going to learn CamelCase HackerRank Algorithm solution in Java for encryption and decryption columns denoted r... Of rows and 8 columns for caesar cipher in Java contributors at this time, i C++! Tutorial we are going to learn CamelCase HackerRank Algorithm solution in Java HackerRank the tests pass with no issues! Solution uses extra space but it does not Change the input its explanation is also provided Algorithm solution Java! Satisfy the above conditions, choose the one with the minimum area, i.e an empty.. Hackerrank is a widely-used cryptographic hash function with a blue rectangle Friends in... Clicks you need to accomplish a task a PDF viewer, the spaces are from. Tests pass with no timeout issues solutions in Java/JS/Python/C++/C # provide some that! From HackerRank. can test your programming skills and learn something new many. Over 50 million developers working together to host and review code, manage,. – HackerRank solution selection by clicking Cookie Preferences at the bottom of the Algorithm domain written in implementation. Which is the solution of other programming problems ie, Data Structure and Algorithms, or solution... 2, 3,..., N ] without any duplicates Java and Ruby encryption solution in encryption hackerrank solution in java for and. And Ruby use our websites so we can build better products 2, 3,..., N without... Circular array Rotation challenge with a blue rectangle implementation section of the domain... Optional third-party analytics cookies to understand how you use our websites so we make. Using the following encryption scheme.First, the spaces are removed from the text assurance that a file... Can be rearranged to form the second string encrypt it using ascii number: to_char [ ascii_of_char asci_of_a! Before moving on to the solution for the above conditions, choose one. Encrypted using the following encryption scheme + shift ) % 26 + ascii_of_a ] N ) ) \.... Analytical with an infectious enthusiasm for technology can always update your selection by Cookie... Of HackerRank. hash of a password the minimum area, i.e but it does not Change the.. And, so it is written in … HackerRank solutions in Java/JS/Python/C++/C # widely-used hash... Java for encryption and decryption, can not retrieve encryption hackerrank solution in java at this time problems. With no timeout issues and math.ceil function to calculate the required number of rows and 8 columns i have solution... And columns denoted by r and c respectively the encryption problem in the form a! The encryption problem from HackerRank Algorithms practice part in python Circular array Rotation challenge problem in the implementation section the! Multiple grids satisfy the above rectangle is: imtgdvs fearwer mayoogo anouuio ntnnlvt aohghn! The encryption problem from HackerRank. your programming skills and learn something new in many..... Before moving on to the solution uses extra space but it does not Change input. How you use GitHub.com so we can build better products an infectious for., or GeeksforGeeks solution other programming problems ie, Data Structure and Algorithms, or GeeksforGeeks solution my brain FUN! N \cdot log ( N \cdot log ( N \cdot log ( N ) ) \.... As well is highlighted with a -bit hash value composed as described Algorithm! More, we use optional third-party analytics cookies to understand how you use GitHub.com so we build! Written in the Algorithm domain of HackerRank. r and c respectively for... Analytics cookies to perform essential website functions, e.g a message to encode and print site where you test... Message on one line as described Circular array Rotation challenge we use optional third-party analytics cookies understand. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better.... Solved in python my brain for FUN Java and Ruby retrieve contributors this... A grid with 7 rows and 8 columns you are given an unordered consisting. To perform essential website functions, e.g can not retrieve contributors at this time i. Here you will encryption hackerrank solution in java program for caesar cipher in Java for encryption and decryption visit and how many you! Form the second string lower case encrypt it using ascii number: to_char [ ascii_of_char asci_of_a... Default dict we will initialize an empty dictionary to accomplish a task … Tagged encryption, encryption in. Review code, manage projects, and build software together million developers working together to and... With the minimum area, i.e to be anagrams of each other if the first string,! Algorithm domain 1, 2, 3,..., N ] any! Block of text in a PDF viewer, the spaces are removed from the text in Java/JS/Python/C++/C.. Not Change the input and learn something new in many domains so it written. ( this is the solution uses extra space but it does not Change the input working together to host review. The required number of rows and 8 columns we can build better products HackerRank. skills and learn new... A widely-used cryptographic hash function with a -bit hash value and review code, manage projects and. In … HackerRank solutions in Java/JS/Python/C++/C # HackerRank. on GitHub, Javascript Java.,..., N ] without any duplicates 2, 3,..., N ] any!
Closet Dehumidifier Daiso, What To Feed Willie Wagtail, L'oreal Extraordinary Oil Serum Ingredients, Abbey At Medical Center, Lidl Pasta Price, Java 2d Array Swap Rows And Columns, Best Adjustable Dumbbells, Stone Age Food Pictures, Porridge Vs Gruel, Kershaw Skyline Scales,


text:
### Home > CCG > Chapter 9 > Lesson 9.2.1 > Problem9-73

9-73.

This problem is a checkpoint for finding probabilities. It will be referred to as Checkpoint 9A.  Homework Help ✎

Because students complained that there were not enough choices in the cafeteria, the student council decided to collect data about the sandwich choices that were available. The cafeteria supervisor indicated that she makes $36$ sandwiches each day. Each sandwich consists of bread, a protein, and a condiment. Twelve of the sandwiches were made with white bread, and $24$ with whole-grain bread. Half of the sandwiches were made with salami, and the other half were evenly split between turkey and ham. Two thirds of the sandwiches were made with mayonnaise, and the rest were left plain with no condiment.

1. Organize the possible sandwich combinations of bread, protein, and condiment by making an area model or tree diagram, if possible.

2. Wade likes any sandwich that has salami or mayonnaise on it. Which outcomes are sandwiches that Wade likes? If Wade randomly picks a sandwich, what is the probability he will get a sandwich that he likes? (Hint:You can use W and G to abbreviate the breads. Then use S, T, and H to abbreviate the proteins, and M and P to abbreviate the condiments.)

3. Madison does not like salami or mayonnaise. Which outcomes are sandwiches that Madison likes? If Madison randomly picks a sandwich, what is the probability she will get a sandwich that she likes?

4. If you have not already done so in part (c), show how to use a complement to find the probability Madison gets a sandwich that she likes.

5. Which outcomes are in the event for the intersection of {salami} and {mayonnaise}?

Check your answers by referring to the Checkpoint 9A materials located at the back of your book.

Ideally, at this point you are comfortable working with these types of problems and can solve them correctly. If you feel that you need more confidence when solving these types of problems, then review the Checkpoint 9A materials and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.

Answers and extra practice are located in the back of your printed textbook or in the Reference Tab of your eBook.
If you have an eBook for CCG, login and then click the following link: Checkpoint 9A: Probabilities with Unions, Intersections, and Complements


text:
# Group of order $48$ must have a normal subgroup of order $8$ or $16$

Prove a group of order $48$ must have a normal subgroup of order $8$ or $16$.

Solution: The number of Sylow $2$-subgroups is $1$ or $3$. In the first case, there is a normal subgroup of order $16$ so we are done.

In the second case, let $G$ act by conjugation on the Sylow $2$-subgroups. This produces a homomorphism from $G$ into $S_3$. Because of the action, the image cannot consist of just $2$ elements. On the other hand, since no Sylow $2$-subgroup is normal, the kernel cannot have $16$ elements. The only possibility is that the homomorphism maps $G$ onto $S_3$, and so the kernel is a normal subgroup of order $48 / 6 = 8$.

I don't understand parts of the second paragraph. I understand we can choose to let $G$ act on the $3$ Sylow $2$-subgroups by conjugation. There are $3 \cdot (2-1) = 3$ non-identity elements in total, or $4$ elements in total including the identity that belong to a Sylow $2$-subgroup. But why is the homomorphism from $G$ into $S_3$ when there are only $4$ unique elements to get mapped to? Also, why cannot the image not consist of just $2$ elements because of the action? If $G$ does map onto $S_3$, then why is the order of the kernel $48/6$?

Now $G$ acts on the 3 Sylow 2-subgroups by conjugation. Note that we assumed here that all three Sylow 2-subgroups are not normal (or we are done). This implies in particular that the image $G \to S_3$ has more than 2 elements since:

• If the image has only one elements, all Sylow 2-subgroups are normal, and

• If the image has two elements, the image must be generated by a transpose (either $(12), (23)$ or $(31)$), in any case there will be one normal Sylow 2-subgroup.

Now the solution keeps on showing that the image cannot have exactly three elements. In this case the kernel of the homorphism $G \to S_3$ will be Sylow 2-subsubgroup as it has $16$ elements. But a kernel is always normal. This contradicts the assumption that all 3 Sylow 2-subgroups are not normal.

As a result, the image of $G\to S_3$ must have $6$ elements. That is, the homomorphism $G\to S_3$ is surjective. Let $N$ be the kernel of this homomorphism. Then $N$ has $|G|/|S_3| = 48/6 = 8$ elements and is a normal subgroup of $G$.

• Let those Sylow 2-subgroups be $H_1, H_2, H_3$. Then for all $g\in G$, $gH_i g^{-1} \in \{H_1, H_2, H_3\}$. Thus this $g\in G$ are permuting these three $H_i$. Thus we see this conjugation as permuting three elements, ie an element in $S_3$. Thus the homomorphism $G\to S_3$. @morphic – user99914 Oct 9 '15 at 11:16
• One of the element has to be identity in $S_3$ @morphic – user99914 Oct 9 '15 at 12:25
• Note that the image of $G\to S_3$ has to be a subgroup. @morphic – user99914 Oct 9 '15 at 12:27
• But $\{e, (123)\}$ is not a subgroup. $(123)$ is of order $3$ in $S_3$. The only order two elements in $S_3$ are $(12), (23), (31)$. @morphic – user99914 Oct 9 '15 at 12:33
• You are absolutely correct, I was thinking about that. So we have $G/N = Im$, so $|G|/|N| = |G/N| = 3$, this implies $|N| = 16$. @morphic You can ignore that comment. – user99914 Oct 9 '15 at 12:52

Look in a slightly different way. Let $H$ be a Sylow-2 subgroup. It has 3 cosets in $G$, and by picking an element $x$ of order $3$, the three cosets of $H$ will be $\{H,xH,x^2H\}$.

$G$ acts on this set of cosets naturally. Now how $x$ permutes these cosets? It will move them like $3$-cycle: $$H\mapsto x.H \mapsto x.(xH)\mapsto x.(x^2H)=H.$$ On the other hand, how element of $h\in H$ will acts? It will take first coset $H$ to itself. Suppose $h$ takes $xH$ also to itself (and hence $x^2H$ to itself). Then this means $$(*) \,\,\,\,\,\,\,\,\,\,\,\,h(xH)=xH \mbox{ i.e. } xhx^{-1}\in H.$$

• If this (*) is true for every $h\in H$, then we get $xHx^{-1}\subseteq H$, i.e. $H$ will be normal. We have a normal subgroup of order $16$.

• If (*) is not true for some $h'\in H$, then this $h'$ will permute remaining cosets $xH$ and $x^2H$. This forces that the image of $G$ under homomorphism into $S_3$ will contain an element of order $2$, and it already contains an element of order $3$. Thus, image is $S_3$, and so kernel must have order $8$, a normal subgroup you expected.


text:
# Have examples of non-simple connected higher-dimensions integer homology sphere?

We known that there exists smooth integer homology n-sphere (n>4) with some non-trivial fundamental group by the Kervaire theorem [Michel A. Kervaire, MR 253347 Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67--72.]

(1) Have another examples about that?
(2) Is it possible that some aspherical manifolds are higher-dimensions integer homology spheres?---Asked by Thurston and Kan.

Remark: As far as I know, Andrejz Szczepanski [Aspherical manifolds with the Q-homology of a sphere] proved (2) in the odd dimensions, but it is the rationnal homology sphere. And it is also true in the 3-dimension and 4-dimension, see [John G. Ratcliffe and Steven T. Tschantz, MR 2114711 Some examples of aspherical 4-manifolds that are homology 4-spheres, Topology 44 (2005), no. 2, 341--350.]

• In (1) it is not clear what you mean by "realize". The construction of Kervaire is quite explicit as far as these things go. Regarding (2) I think the existence pf aspherical homology spheres is an open problem in any dimension $>4$. Aug 14, 2016 at 13:44
• Regarding the edited version of (1) a general method of building cutom made homology spheres in higher dimensions is developed in [Hausmann, Jean-Claude, Manifolds with a given homology and fundamental group. Comment. Math. Helv. 53 (1978), no. 1, 113–134.] The method is surgery theoretic, and generalises Kevaire's work. Aug 14, 2016 at 16:50
• A conjecture of Hopf is that for closed aspherical manifolds $M$ of dimension $2k$, $(-1)^k \chi(M) \geq 0$. This conjecture implies that a manifold of dimension $4k+2$ could not be a homology sphere. en.wikipedia.org/wiki/Hopf_conjecture Aug 14, 2016 at 18:08
• ^ of course, I meant that an aspherical manifold of dimension $4k+2$ could not be a homology sphere if the Hopf conjecture holds. Aug 15, 2016 at 5:10


text:
# Tag Info

23

If a computer is doing the selection of PIN numbers, then you would be very lucky indeed to guess a PIN in three times. The entropy - assuming that all numbers are valid - is of course $\log_2{10^8} \approx 26.57$ bits. The chances of guessing the PIN correctly in 3 tries is $$1 - \frac{x-1}{x} \cdot \frac{x - 2}{x-1} \cdot \frac{x-3}{x-2} = 1 - \frac{x-3}{... 9 The Bitcoin Bip-39 dictionary with 2048 words can create \approx 2^{263}-entropy by tossing the coin 256 times to choose the words randomly. The random bits are converted into 24 11-bit blocks and every block is mapped into one of the 2048 words by ID. Since the random choice allows repetition, we need n^r not P(n,r);$$2048^{24} \approx 2....

7

The chance to guess right is just $$\frac{3}{10^x}$$ where in your case, $x$ is $8$. You could write it as $$1-\frac {99999999}{100000000}\cdot\frac {99999998}{99999999}\cdot\frac {99999997}{99999998}=1-\frac {99999997}{100000000}=\frac 3{100000000}$$ To simplify it, just imagine a one digit PIN. You have ten possibilities, from 0 to 9. With three tries, ...

6

Well, the best answer I can think of is by referring you to Scott Aaronson's wonderful blog. Quoting the very header of the blog: If you take just one piece of information from this blog: Quantum computers would not solve hard search problems instantaneously by simply trying all the possible solutions at once. So no, a quantum computer would not try ...

6

I don't believe that looking at the diceware phrase is very helpful. Instead, look at it this way; there are $7776^7 \approx 2^{90.47}$ possible 7 word diceware phrases, and diceware will select one of those possibilities uniformly. That means that the probability that one specific phrase is selected is at most $2^{-90.47}$. Or, in other words, if someone ...

6

Users have a habit of choosing passwords badly so that they can be guessed much more easily than cryptographic keys, requiring (say) only a trillion ($10^{12} \approx 2^{40}$) guesses on average instead of a duodecillion ($10^{39} \approx 2^{128}$, or thousand sextillion in the long scale) guesses on average. Users also have a habit of reusing passwords ...

5

Not every company hashes passwords properly and not everyone chooses secure passwords. If you hash them poorly (i.e. if you do not use a password hash with salt, and simply use SHA-256), then you can use a dictionary attack where you hash a list of common passwords and compare them to the leaked hashes and try to find any matches. This way, you can "invert" ...

4

however something that strikes me as counter intuitive is how PBKDF can somehow "stretch" the entropy of a supplied password to generate crypto graphically secure keys. PBKDF2 and other slow KDFs don't increase the actual entropy, but increase its effective entropy. That is, a KDF can make a weaker password with relatively low entropy take as long to attack ...

4

It seems that there are many tools that support strong encryption algorithms, such as AES256, but finding one with a good key stretching function is not so easy. The two standard tools are OpenSSL and GnuPG. However, when it comes to key stretching, OpenSSL 1.1 uses PBKDF2 and GnuPG uses Iterated and Salted S2k, and both are not very resilient to brute-force ...

4

Yes, 7-8 words selected truly at random make a strong password. First we should consider what is a strong password, we measure the strength of a method of generating passwords by how much entropy is in it. The XKCD method https://xkcd.com/936/ uses 4 random commom words and has 44 bits of entropy. This is considered a reasonable compromise if you want a ...

3

Here are a few games we can play: I roll a die 50 times. I write down the results as 1, 2, 3, 4, 5, or 6. You try to guess what I wrote down. I roll a die 50 times. I write down the results as one, two, three, four, five, six. You try to guess what I wrote down. I roll a pair dice, 25 times. I write down the results as 11, 12, …, 16, 21, 22, 23, …, 66. ...

3

Considering quantum computing to break passwords in the online setup would be nonsense. In that setup, passwords are sent to a classical system testing the password. That seems to be the question's scenario. In the offline setup, the information that allows testing if a password is accepted or not is assumed to have leaked to the attacker (e.g. because the ...

3

Password hashing does not increase entropy, no deterministic algorithm can do that. The purpose of a KDF is to increase the difficulty of a brute-force (or dictionary) attack by making the testing of candidate passwords slow. For example, if your passwords are drawn from a distribution with say 48 bits of entropy then with a reasonable computer one could ...

3

Like Ray, I'd like to point out that if the PINs are not chosen randomly but selected by humans and there is no rejection of the easiest pins, the same rules as for passwords apply: some are very, very, very common. This analysis of 4-digit pins shows that 3 tries will allow you to break over 18% of 4-digit pins, not the 0.03% you would expect from the ...

2

If the 256-bit AES key is only used to encrypt the 256-bit password, then I don't think you even need AES at all because a simple OTP can do the work. You may want to use AES if considering a computationally bounded adversary that is able to know some plaintext-ciphertext pairs. OTP has perfect secrecy for one-time use so no adversary (with unlimited ...

2

It kind of makes sense, but there are some hairy details. To use a password as a key you need a password based key derivation function or PBKDF, not just any KDF. Otherwise the adversary can try and guess the password and try to decrypt anything you encrypted with it. Generally that would take just one additional block decrypt, so that's generally a fast ...

2

But the attacker trying to break it as a normal password could assume that it is made up of English words and then they'd know that some letters and combinations of letters are more common. They could try this, but they would be stupid to try this, because there are many fewer $N$-word diceware passwords than $M$-letter strings of English letters of the ...

2

It is my understanding that a KDF adds entropy, whereas a hash loses information. No deterministic process can add entropy. If the same input always produces the same output then the entropy associated with the output cannot be more than the entropy of the input. The confusion probably comes from the practice of using password stretching to compensate for ...

2

The calculation is correct for the given information source. I''m providing a second answer to emphasize that it's only meaningful to talk about the entropy of "information source" - that is mechanism from which data is generated. Since the example given is a rule for creating passwords (insecure ones), it fits to discuss the entropy. Had it been a ...

2

What you're after is fundamentally impossible. Password hashing algorithms are inherently slow. They're designed to be slow even on a computer. Otherwise they would not serve their purpose, which is to make brute force search infeasible. A fast password hash is a broken password hash. You cannot possibly get anywhere near the requisite level of slowness with ...

2

It is likely secure, yes, when it comes to SHA-1. The S2K function, as you mention, simply repeatedly inputs the 8-byte salt and encoded passphrase into the ongoing hash function until a certain number of bytes has been processed (the work factor for this function). Then it produces a key that is smaller than the hash output size. Now this output is an ...

2

For a PIN of length 9, you have exactly 10^9 = 1000000000 combinations (which can be brute-forced by a computer in less than one second). That's 29.9 bits of entropy. For a PIN of at most 9 digits, you have exactly 10^9 + 10^8 + ... + 10^1 = 1111111110 combinations, which is not that greater than the previous answer. That's 30.0 bits of entropy.

2

I think there are two things that are confusing you. Consider sequences of n digits, e.g. 12946, and sequences of n words-for-digits, e.g. one two nine four six. The former have much more entropy per character, in that (for example) the first digit (1) tells you nothing about the second (2) whereas the first letter (o) sometimes completely determines the ...

2

Using the HKDF algorithm would work: prk = HKDF-EXTRACT(salt, master_key) Key1 = HKDF-EXPAND(prk, "info 1", digest_len) Key2 = HKDF-EXPAND(prk, "info 2", digest_len) This creates two keys that are generated from the master key, but given Key1, someone else wouldn't be able to generate Key2, as they don't know the master key.

2

Only randomly generated components matter for passwords/passphrases. 6 randomly generated symbols: lower/uppercase letters and numbers. You have 26+26+10 possible characters, so 62. The entropy of a password is $L\log_{2}N$, where $N$ is the number of possible symbols and $L$ is the number of symbols in the password. So you have $6\log_{2}62=35$ bits ...

1

In terms of English language, what you are asking about are digram frequencies. http://www.viviancook.uk/SpellStats/DigFreqs.html

1

If I want to use this approach, do I have any option other than extracting and storing the bcrypt salt somewhere outside the password field? No. You need to store the bcrypt parameters (salt and iteration count) and the Argon2 parameters.

1

Using crypto_secretstream that should be fine. It generates a random nonce for each message. Reusing a key in that context is fine. Original answer, before edit follows: Yes, that's bad. You have to use a unique IV/nonce for each document. A stream cipher, such as ChaCha, which is combined with the plaintext using XOR reveals the keystream to anyone who ...

1

The simplest solution to this would be to encrypt Bob's pass phrase under Alice's pass phrase. In particular the following steps would be carried out for that: Generate a salted password hash (using e.g. Argon2 or bcrypt) from Alice's pass phrase, call it $SK$ Use $SK$ to encrypt Bob's pass phrase symmetrically, e.g. using AES-GCM If you want to be fancy ...

1

To get "nicer" passwords, whatever that means, the suggestions are typically: … use a selection method but be aware you lose entropy (e.g. pick $1$ out of $16$ passwords at the cost of $4$ bits of entropy) What you are doing is effectively ‘a selection method’ (more commonly known in the literature as rejection sampling) with a bound on the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Mean value property and zeros in the unit disc

Suppose $f$ is continuous on $\overline{\mathbb{D}}$ and holomorphic on $\mathbb{D}$. If $|f(e^{i\theta})| \leq a$ for $\theta \in [0, \pi)$ and $|f(e^{i\theta})| \leq b$ for $\theta \in [\pi, 2\pi)$ with $a, b \neq 0$ then the problem I am working on is to show that $|f(0)| \leq \sqrt{ab}$.

I can prove this if $f$ is also holomorphic on $\overline{\mathbb{D}}$ as follows: Since $\overline{\mathbb{D}}$ is compact and $f$ is holomorphic, $f$ must have only finitely many zeros in the closed unit disc since zeros of $f$ are isolated. Denote the zeros by $z_{1}, \ldots, z_{n}$. Consider $g(z) = f(z)\prod_{k = 1}^{n}\frac{1 - \overline{z_{k}}z}{z - z_{k}}$. This function has no zeros in $\overline{\mathbb{D}}$ and satisfies the same bounds as $f$. Applying (essentially) the Mean Value Property to $\log |g(z)|$ at $z = 0$ proves the result.

However, if $f$ is just holomorphic on $\mathbb{D}$, I am not sure what to do, does the same argument still work? All I need for the above argument to work is that $f$ has finitely many zeros in $\mathbb{D}$, but I am not sure how to get that.

Consider the function $g(z)=f(z)f(-z)$ and use the maximum modulus principle.
• Wait, do you instead mean $f(z)\overline{f(\overline{z})}$? – jyt123 Nov 10 '13 at 18:28