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• ### First 230 GHz VLBI Fringes on 3C 279 using the APEX Telescope(1506.03244)

June 10, 2015 astro-ph.IM
We report about a 230 GHz very long baseline interferometry (VLBI) fringe finder observation of blazar 3C 279 with the APEX telescope in Chile, the phased submillimeter array (SMA), and the SMT of the Arizona Radio Observatory (ARO). We installed VLBI equipment and measured the APEX station position to 1 cm accuracy (1 sigma). We then observed 3C 279 on 2012 May 7 in a 5 hour 230 GHz VLBI track with baseline lengths of 2800 M$\lambda$ to 7200 M$\lambda$ and a finest fringe spacing of 28.6 micro-arcseconds. Fringes were detected on all baselines with SNRs of 12 to 55 in 420 s. The correlated flux density on the longest baseline was ~0.3 Jy/beam, out of a total flux density of 19.8 Jy. Visibility data suggest an emission region <38 uas in size, and at least two components, possibly polarized. We find a lower limit of the brightness temperature of the inner jet region of about 10^10 K. Lastly, we find an upper limit of 20% on the linear polarization fraction at a fringe spacing of ~38 uas. With APEX the angular resolution of 230 GHz VLBI improves to 28.6 uas. This allows one to resolve the last-photon ring around the Galactic Center black hole event horizon, expected to be 40 uas in diameter, and probe radio jet launching at unprecedented resolution, down to a few gravitational radii in galaxies like M 87. To probe the structure in the inner parsecs of 3C 279 in detail, follow-up observations with APEX and five other mm-VLBI stations have been conducted (March 2013) and are being analyzed.
• ### Greenland Telescope Project --- Direct Confirmation of Black Hole with Sub-millimeter VLBI(1407.2450)

July 9, 2014 astro-ph.IM, astro-ph.HE
A 12-m diameter radio telescope will be deployed to the Summit Station in Greenland to provide direct confirmation of a Super Massive Black Hole (SMBH) by observing its shadow image in the active galaxy M87. The telescope (Greenland Telescope: GLT) is to become one of the Very Long Baseline Interferometry (VLBI) stations at sub-millimeter (submm) regime, providing the longest baseline > 9,000 km to achieve an exceptional angular resolution of 20 micro arc sec at 350 GHz, which will enable us to resolve the shadow size of ~40 micro arc sec. The triangle with the longest baselines formed by the GLT, the Atacama Large Millimeter/submillimeter Array (ALMA) in Chile, and the Submillimeter Array (SMA) in Hawaii will play a key role for the M87 observations. We have been working on the image simulations based on realistic conditions for a better understanding of the possible observed images. In parallel, retrofitting of the telescope and the site developments are in progress. Based on three years of opacity monitoring at 225 GHz, our measurements indicate that the site is excellent for submm observations, comparable to the ALMA site. The GLT is also expected to make single-dish observations up to 1.5 THz.
• ### Zooming towards the Event Horizon - mm-VLBI today and tomorrow(1305.2811)

Global VLBI imaging at millimeter and sub-millimeter wavelength overcomes the opacity barrier of synchrotron self-absorption in AGN and opens the direct view into sub-pc scale regions not accessible before. Since AGN variability is more pronounced at short millimeter wavelength, mm-VLBI can reveal structural changes in very early stages after outbursts. When combined with observations at longer wavelength, global 3mm and 1mm VLBI adds very detailed information. This helps to determine fundamental physical properties at the jet base, and in the vicinity of super-massive black holes at the center of AGN. Here we present new results from multi-frequency mm-VLBI imaging of OJ287 during a major outburst. We also report on a successful 1.3mm VLBI experiment with the APEX telescope in Chile. This observation sets a new record in angular resolution. It also opens the path towards future mm-VLBI with ALMA, which aims at the mapping of the black hole event horizon in nearby galaxies, and the study of the roots of jets in AGN.
• ### HerMES: Candidate Gravitationally Lensed Galaxies and Lensing Statistics at Submillimeter Wavelengths(1205.3778)

Nov. 9, 2012 astro-ph.CO, astro-ph.GA
We present a list of 13 candidate gravitationally lensed submillimeter galaxies (SMGs) from 95 square degrees of the Herschel Multi-tiered Extragalactic Survey, a surface density of 0.14\pm0.04deg^{-2}. The selected sources have 500um flux densities (S_500) greater than 100mJy. Gravitational lensing is confirmed by follow-up observations in 9 of the 13 systems (70%), and the lensing status of the four remaining sources is undetermined. We also present a supplementary sample of 29 (0.31\pm0.06deg^{-2}) gravitationally lensed SMG candidates with S_500=80--100mJy, which are expected to contain a higher fraction of interlopers than the primary candidates. The number counts of the candidate lensed galaxies are consistent with a simple statistical model of the lensing rate, which uses a foreground matter distribution, the intrinsic SMG number counts, and an assumed SMG redshift distribution. The model predicts that 32--74% of our S_500>100mJy candidates are strongly gravitationally lensed (mu>2), with the brightest sources being the most robust; this is consistent with the observational data. Our statistical model also predicts that, on average, lensed galaxies with S_500=100mJy are magnified by factors of ~9, with apparently brighter galaxies having progressively higher average magnification, due to the shape of the intrinsic number counts. 65% of the sources are expected to have intrinsic 500micron flux densities less than 30mJy. Thus, samples of strongly gravitationally lensed SMGs, such as those presented here, probe below the nominal Herschel detection limit at 500 micron. They are good targets for the detailed study of the physical conditions in distant dusty, star-forming galaxies, due to the lensing magnification, which can lead to spatial resolutions of ~0.01" in the source plane.
• ### Surviving the hole I: Spatially resolved chemistry around Sgr A*(1112.0566)

Dec. 2, 2011 astro-ph.GA
The interstellar region within the few central parsecs around the super-massive black hole, Sgr A* at the very Galactic center is composed by a number of overlapping molecular structures which are subject to one of the most hostile physical environments in the Galaxy. We present high resolution (4"x3"~0.16x0.11 pc) interferometric observations of CN, 13CN, H2CO, SiO, c-C3H2 and HC3N emission at 1.3 mm towards the central ~4 pc of the Galactic center region. Strong differences are observed in the distribution of the different molecules. The UV resistant species CN, the only species tracing all previously identified circumnuclear disk (CND) structures, is mostly concentrated in optically thick clumps in the rotating filaments around Sgr A*. H2CO emission traces a shell-like structure that we interpret as the expansion of Sgr A East against the 50 km/s and 20 km/s giant molecular clouds (GMCs). We derive isotopic ratios 12C/13C~15-45 across most of the CND region. The densest molecular material, traced by SiO and HC3N, is located in the southern CND. The observed c-C3H2/HC3N ratio observed in the region is more than an order of magnitude lower than in Galactic PDRs. Toward the central region only CN was detected in absorption. Apart from the known narrow line-of-sight absorptions, a 90 km/s wide optically thick spectral feature is observed. We find evidences of an even wider (>100 km/s) absorption feature. Around 70-75% of the gas mass, concentrated in just the 27% densest molecular clumps, is associated with rotating structures and show evidences of association with each of the arcs of ionized gas in the mini-spiral structure. Chemical differentiation has been proven to be a powerful tool to disentangle the many overlapping molecular components in this crowded and heavily obscured region.
• ### Gas and dust in a submillimeter galaxy at z = 4.24 from the Herschel ATLAS(1107.2924)

July 14, 2011 astro-ph.CO
We report ground-based follow-up observations of the exceptional source, ID141, one the brightest sources detected so far in the H-ATLAS cosmological survey. ID141 was observed using the IRAM 30-meter telescope and Plateau de Bure interferometer (PdBI), the Submillimeter Array (SMA) and the Atacama Pathfinder Experiment (APEX) submillimeter telescope to measure the dust continuum and emission lines of the main isotope of carbon monoxide and carbon ([C I] and [C II]). The detection of strong CO emission lines with the PdBI confirms that ID141 is at high redshift (z=4.243 +/- 0.001). The strength of the continuum and emission lines suggests that ID141 is gravitationally lensed. The width (Delta V (FWHM) ~ 800 km/s}) and asymmetric profiles of the CO and carbon lines indicate orbital motion in a disc or a merger. The properties derived for ID141 are compatible with a ultraluminous (L_FIR ~ 8.5 +/- 0.3 x 10^13/mu_L Lsun, where mu_L is the amplification factor, dense (n ~ 10^4 cm^-3) and warm (T_kin ~ 40K) starburst galaxy, with an estimated star-formation rate of (0.7 to 1.7) x 10^4/mu_L Msun/yr. The carbon emission lines indicate a dense (n ~ 10^4 cm^-3) Photo-Dominated Region, illuminated by a far-UV radiation field a few thousand times more intense than that in our Galaxy. In conclusion, the physical properties of the high-z galaxy, ID141, are remarkably similar to those of local ultraluminous infrared galaxies.
• ### Discovery of a Multiply-Lensed Submillimeter Galaxy in Early HerMES Herschel/SPIRE Data(1104.4113)

April 20, 2011 astro-ph.CO
We report the discovery of a bright ($f(250\mum) > 400$ mJy), multiply-lensed submillimeter galaxy \obj\ in {\it Herschel}/SPIRE Science Demonstration Phase data from the HerMES project. Interferometric 880\mum\ Submillimeter Array observations resolve at least four images with a large separation of $\sim 9\arcsec$. A high-resolution adaptive optics $K_p$ image with Keck/NIRC2 clearly shows strong lensing arcs. Follow-up spectroscopy gives a redshift of $z=2.9575$, and the lensing model gives a total magnification of $\mu \sim 11 \pm 1$. The large image separation allows us to study the multi-wavelength spectral energy distribution (SED) of the lensed source unobscured by the central lensing mass. The far-IR/millimeter-wave SED is well described by a modified blackbody fit with an unusually warm dust temperature, $88 \pm 3$ K. We derive a lensing-corrected total IR luminosity of $(1.43 \pm 0.09) \times 10^{13}\, \mathrm{L}_{\odot}$, implying a star formation rate of $\sim 2500\, \mathrm{M}_{\odot}\, \mathrm{yr}^{-1}$. However, models primarily developed from brighter galaxies selected at longer wavelengths are a poor fit to the full optical-to-millimeter SED. A number of other strongly lensed systems have already been discovered in early {\it Herschel} data, and many more are expected as additional data are collected.
• ### Dynamical Structure of the Molecular Interstellar Medium in an Extremely Bright, Multiply Lensed z~3 Submillimeter Galaxy Discovered with Herschel(1104.4116)

April 20, 2011 astro-ph.CO
We report the detection of CO 5-4, 3-2, and 1-0 emission in the strongly lensed, Herschel/SPIRE-selected submillimeter galaxy (SMG) HLSW-01 at z=2.9574+/-0.0001, using the Plateau de Bure Interferometer, the Combined Array for Research in Millimeter-wave Astronomy, and the Green Bank Telescope. The observations spatially resolve the molecular gas into four lensed images with a maximum separation of ~9", and reveal the internal gas dynamics in this system. We derive lensing-corrected CO line luminosities of L'(CO 1-0) = (4.17+/-0.41), L'(CO 3-2) = (3.96+/-0.20) and L'(CO 5-4) = (3.45+/-0.20) x 10^10 (mu_L/10.9)^-1 Kkm/s pc^2, corresponding to luminosity ratios of r_31 = 0.95+/-0.10, r_53 = 0.87+/-0.06, and r_51 = 0.83+/-0.09. This suggests a total molecular gas mass of Mgas = 3.3 x 10^10 (alpha_CO/0.8) (mu_L/10.9)^-1 Msun. The gas mass, gas mass fraction, gas depletion timescale, star formation efficiency, and specific star formation rate are typical for an SMG. The velocity structure of the gas reservoir suggests that the brightest two lensed images are dynamically resolved projections of the same dust-obscured region in the galaxy that are kinematically offset from the unresolved fainter images. The resolved kinematics appear consistent with the complex velocity structure observed in major, `wet' (i.e., gas-rich) mergers. Major mergers are commonly observed in SMGs, and are likely to be responsible for fueling their intense starbursts at high gas consumption rates. This study demonstrates the level of detail to which galaxies in the early universe can be studied by utilizing the increase in effective spatial resolution and sensitivity provided by gravitational lensing.
• ### Modeling of the HERMES J105751.1+573027 submillimeter source lensed by a dark matter dominated foreground group of galaxies(1104.4119)

April 20, 2011 astro-ph.CO
We present the results of a gravitational lensing analysis of the bright $\zs=2.957$ sub-millimeter galaxy (SMG), HERMES J105751.1+573027 found in {\it Herschel}/SPIRE Science Demonstration Phase data from the Herschel Multi-tiered Extragalactic Survey (HerMES) project. The high resolution imaging available in optical and Near-IR channels, along with CO emission obtained with the Plateau de Bure Interferometer, allow us to precisely estimate the intrinsic source extension and hence estimate the total lensing magnification to be $\mu=10.9\pm 0.7$. We measure the half-light radius $R_{\rm eff}$ of the source in the rest-frame Near-UV and $V$ bands that characterize the unobscured light coming from stars and find $R_{\rm eff,*}= [2.0 \pm 0.1]$ kpc, in good agreement with recent studies on the Submillimeter Galaxy population. This lens model is also used to estimate the size of the gas distribution ($R_{\rm eff,gas}= [1.1\pm0.5]$) kpc by mapping back in the source plane the CO (J=5-4) transition line emission. The lens modeling yields a relatively large Einstein radius $R_{\rm Ein}= 4\farcs10 \pm 0\farcs02$, corresponding to a deflector velocity dispersion of [$483\pm 16] \,\kms$. This shows that HERMES J105751.1+573027 is lensed by a {\it galaxy group-size} dark matter halo at redshift $\zl\sim 0.6$. The projected dark matter contribution largely dominates the mass budget within the Einstein radius with $f_{\rm dm}(<R_{\rm Ein})\sim 80%$. This fraction reduces to $f_{\rm dm}(<R_{\rm eff,G1}\simeq 4.5\kpc)\sim 47%$ within the effective radius of the main deflecting galaxy of stellar mass $M_{\rm *,G1}=[8.5\pm 1.6] \times 10^{11}\msun$. At this smaller scale the dark matter fraction is consistent with results already found for massive lensing ellipticals at $z\sim0.2$ from the SLACS survey.
• ### Spitzer Imaging of Herschel-ATLAS Gravitationally Lensed Submillimeter Sources(1011.4940)

Jan. 13, 2011 astro-ph.CO
We present physical properties of two submillimeter selected gravitationally lensed sources, identified in the Herschel Astrophysical Terahertz Large Area Survey. These submillimeter galaxies (SMGs) have flux densities > 100 mJy at 500 um, but are not visible in existing optical imaging. We fit light profiles to each component of the lensing systems in Spitzer IRAC 3.6 and 4.5 um data and successfully disentangle the foreground lens from the background source in each case, providing important constraints on the spectral energy distributions (SEDs) of the background SMG at rest-frame optical-near-infrared wavelengths. The SED fits show that these two SMGs have high dust obscuration with Av ~4 to 5 and star formation rates of ~100 M_sun/yr. They have low gas fractions and low dynamical masses compared to 850 um selected galaxies.
• ### The Detection of a Population of Submillimeter-Bright, Strongly-Lensed Galaxies(1011.1255)

Nov. 4, 2010 astro-ph.CO
Gravitational lensing is a powerful astrophysical and cosmological probe and is particularly valuable at submillimeter wavelengths for the study of the statistical and individual properties of dusty starforming galaxies. However the identification of gravitational lenses is often time-intensive, involving the sifting of large volumes of imaging or spectroscopic data to find few candidates. We used early data from the Herschel Astrophysical Terahertz Large Area Survey to demonstrate that wide-area submillimeter surveys can simply and easily detect strong gravitational lensing events, with close to 100% efficiency.
• ### Phase Closure at 691 GHz using the Submillimeter Array(0704.2641)

April 20, 2007 astro-ph
Phase closure at 682 GHz and 691 GHz was first achieved using three antennas of the Submillimeter Array (SMA) interferometer located on Mauna Kea, Hawaii. Initially, phase closure was demonstrated at 682.5 GHz on Sept. 19, 2002 using an artificial ground-based "beacon" signal. Subsequently, astronomical detections of both Saturn and Uranus were made at the frequency of the CO(6-5) transition (691.473 GHz) on all three baselines on Sept. 22, 2002. While the larger planets such as Saturn are heavily resolved even on these short baselines (25.2m, 25.2m and 16.4m), phase closure was achieved on Uranus and Callisto. This was the first successful experiment to obtain phase closure in this frequency band. The CO(6-5) line was also detected towards Orion BN/KL and other Galactic sources, as was the vibrationally-excited 658 GHz water maser line toward evolved stars. We present these historic detections, as well as the first arcsecond-scale images obtained in this frequency band.
• ### The Submillimeter Array Antennas and Receivers(astro-ph/0508492)

Aug. 23, 2005 astro-ph
The Submillimeter Array (SMA) was conceived at the Smithsonian Astrophysical Observatory in 1984 as a six element interferometer to operate in the major atmospheric windows from about 200 to 900 GHz. In 1996, the Academica Sinica Institute of Astronomy and Astrophysics of Taiwan joined the project and agreed to provide additional hardware to expand the interferometer to eight elements. All eight antennas are now operating at the observatory site on Mauna Kea, and astronomical observations have been made in the 230, 345, and 650 GHz bands. The SMA antennas have a diameter of 6 m, a surface accuracy of better than 25 micron rms, and can be reconfigured to provide spatial resolutions down to about 0.5" at 200 GHz and, eventually, 0.1" at 850 GHz. Coupling to the receiver package within each antenna is achieved via a beam waveguide, in a bent Nasmyth configuration, comprised of a flat tertiary mirror and two ellipsoidal mirrors that form a secondary pupil used for receiver calibration. An additional fixed mirror and a rotating wire grid polarizer are then used for receiver selection. Each antenna houses a single cryostat, with an integrated cryocooler capable of cooling up to eight receivers to 4 K. In the current configuration only three receiver bands are available: 175-255 GHz, 250-350 GHz, and 600-720 GHz, and simultaneous operation of the 650 GHz receiver with either of the lower frequency receivers is possible. Eventually dual polarization will be available from 325-350 GHz, and dual frequency operation will be possible, pairing either of the lower frequency receivers with any of the high frequency units: 325-425 GHz, 425-510 GHz, 600-720 GHz, and 800-900 GHz. (Abridged)
• ### A map of OMC-1 in CO 9-8(astro-ph/0405530)

Sept. 6, 2004 astro-ph
The distribution of 12C16O J=9-8 (1.037 THz) emission has been mapped in OMC-1 at 35 points with 84" resolution. This is the first map of this source in this transition and only the second velocity-resolved ground-based observation of a line in the terahertz frequency band. There is emission present at all points in the map, a region roughly 4' by 6' in size, with peak antenna temperature dropping only near the edges. Away from the Orion KL outflow, the velocity structure suggests that most of the emission comes from the OMC-1 photon-dominated region, with a typical linewidthof 3-6 km/s. Large velocity gradient modeling of the emission in J=9-8 and six lower transitions suggests that the lines originate in regions with temperatures around 120 K and densities of at least 10^(3.5) cm^(-3) near theta^(1) C Ori and at the Orion Bar, and from 70 K gas at around 10^(4) cm^(-3) southeast and west of the bar. These observations are among the first made with the 0.8 m Smithsonian Astrophysical Observatory Receiver Lab Telescope, a new instrument designed to observe at frequencies above 1 THz from an extremely high and dry site in northern Chile.


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# Automatically Finding Useful Augmentations with Minimal Use of Compute

Categories computer vision

Cristian Duguet, Adrian Hutter, Nandika Kalra, Arshak Navruzyan

## Overview

Data augmentation of images is a widely adopted technique to improve models generalization [3] by applying transformations (or sets of combined transformations) during training and optionally at test time. Often practitioners look for the best set of transformations manually, either by relying on domain expertise or making assumptions about generally useful transformations. This process of manual search for optimal augmentations can be time-consuming and compute intensive.

The fast.ai library does a great job in providing smart defaults in other areas like setting hyperparameters, model architecture (e.g. custom head) and augmentations are no exception. After a lot of experimentation, the fast.ai team has found a standard set of augmentations which is applied indiscriminately to every dataset, and has proven to work effectively most of the time. However, there may be room for improvement if the appropriate augmentations can be found for a given dataset in a fast way.

Over the last months, our team in fellowship.ai has been working to create a method to automate the search for the best augmentation set, in a computationally efficient manner, and with as little domain-specific input as possible. The purpose of this research aligns with the goals of platform.ai, which seeks to offer deep learning to a wider non-engineer audience.

### Current approaches

There are a few approaches to automatically finding an optimal set of augmentations, but the most common paradigm is to generate various augmented datasets and fully or partially train “child models” with each set to determine impact on performance. The central focus of these approaches is to make the search as efficient as possible through gaussian process, reinforcement learning and other search methods.

One obvious disadvantage of this formulation is that they are not computationally cheap. In order to achieve even a modest improvement, thousands or tens of thousands of child models have to be trained and evaluated.

A more novel approach is from Ratner [5] where they use a GAN to find the set of transformation parameters to create augmented data which lies within a defined distribution of interest, representative of the training set. By treating the augmentation search as a sequence modeling problem, this approach attempts to find not only the right augmentations but also their parameterization, which can be non-differentiable and hence requires a reinforcement learning loop in addition to the generative and discriminative model training.

We are looking for a more computationally efficient way (at the sacrifice of any performance due to composability) for automatically finding a set of parameters for the image transform functions provided by the fast.ai library.

## Methodology

The fastai library provides different types of image transformations which are used for data augmentation. These transformations can be grouped in affine, pixel, cropping, lighting and coordinate transformations. The default augmentation set, which is obtained by calling get_transforms() has the following parameters:

Table 1: List of transformations available in fast.ai. There are more transforms available in their documentation, but we focused on the ones listed

 transform type parameters prob crop_pad TfmCrop 'row_pct': (0, 1), 'col_pct': (0, 1) 1.0 flip_affine TfmAffine   0.5 symmetric_warp TfmCoord 'magnitude': (-0.2, 0.2) 0.75 rotate TfmAffine 'degrees': (-10.0, 10.0) 0.75 zoom TfmAffine 'row_pct': (0, 1), 'col_pct': (0, 1), 'scale': (1.0, 1.1) 0.75 brightness TfmLighting 'change': (0.4, 0.6) 0.75 contrast TfmLighting 'scale': (0.8, 1.25) 0.75

### Method Validation

The performance of the training set augmented with the chosen transformations will be compared to the performance of the training set augmented with the default fast.ai augmentation set, by training both networks for a determined routine. This will be done for different datasets. The performance metric is the error rate on the validation set.

### Datasets

A variety of datasets and domains were selected to assess the robustness of this method. A group of datasets, which could be representative of very different use cases was gathered and the following 6 were selected.

### TTA Search

The Test Time Augmentation is a technique to leverage the use of augmentation for prediction instead of training. With TTA, the probability of prediction of an image would comprise 2 different predictions:

$p_{final} = \beta p_{original} + (1-\beta) p_{augmented}$

Where $p_{original}$ and $p_{augmented}$ are the predicted values for the original and the augmented image, respectively, and $\beta$ is a weight factor.

We have observed that the Test Time Augmentation can be a good indicator of when a certain augmentation is a good candidate for training. This means that if the average of all TTA predictions on a set using certain augmentation had higher accuracy than the average of normal predictions, then training the network with that augmentation will most likely improve the performance of the network.

Figure 1: Normalized error rate of a trained dataset with a certain augmentation vs normalized TTA for that augmentation as well. The normalization is the relative change to the baseline case. Each point is a different augmentation tried on the network, each color is an experiment on a different dataset.

We identified that very harmful transformations for training, which would be characterized by a high err/err_none and TTA_err/TTA_err_none value. For the Pets and Dogs dataset, for example, it was the dihedral transform, while for Planet it was resize_crop. Most transforms of our set were detrimental for CIFAR-10.

Based on the observed behaviour, we search for a certain augmentation set using TTA, in a procedure which works as follows.

1. Split the training set into two subsets of size 80% and 20%, respectively.
2. Train the last layer group on 80% of the training set for EPOCHS_HEAD epochs, without any data augmentation.
3. Calculate the error rate ERR_NONE on the remaining 20% of the training set.
4. For each kind of transformation, for each possible magnitude, calculate the TTA error rate on the remaining 20% of the training set. For TTA, we base predictions on WEIGHT_UNTRANSFORMED * LOGITS_UNTRANSFORMED + (1 - WEIGHT_UNTRANSFORMED) * LOGITS_TRANSFORMED. Where WEIGHT_UNTRANSFORMED describes the amount of influence the augmentation has on the prediction.
5. For each kind of transformation, choose the magnitude which leads to the lowest TTA error rate, if that error rate is lower than THRESHOLD * ERR_NONE; otherwise, don’t include that kind of transformation in the final set of augmentations.
6. With the chosen set of augmentations, train the head for EPOCHS_HEAD epochs and the full network for EPOCHS_FULL.
7. As a baseline, train the network for the same number of epochs using the transforms provided by get_transforms().

Out of the transformations available in the fast.ai library, we have tested our method with the following transforms/parameters list

Table 2: List of tested augmentations using our search method, including a list of the tested parameters.

 Transform Type Tested parameters Prob* flip TfmAffine Either flip_affine, or dihedral_affine 0.5 symmetric_warp TfmCoord 'magnitude': (-m, m) for m in [0.1,0.2,0.3,0.4] 0.75 rotate TfmAffine 'degrees': (-m, m) for m in [10., 20., 30., 40.] 0.75 zoom TfmAffine 'Scale': (1.,max_zoom) for max_zoom in [1.1, 1.2, 1.3, 1.4] 0.75 brightness TfmLighting 'change': (0.5*(1-max_lighting), 0.5*(1+max_lighting)) for max_lighting in [0.1, 0.2, 0.3, 0.4] 0.75 contrast TfmLighting 'scale': (1-max_lighting, 1/(1-max_lighting)) for max_lighting in [0.1, 0.2, 0.3, 0.4] 0.75 skew TfmCoord 'direction': (0,7), 'magnitude': max_skew for max_skew in [0.2, 0.4, 0.6, 0.8] 0.75 squish TfmAffine 'scale': (1/max_scale, max_scale) for max_scale in [1.2, 1.8, 2.4, 3.] 0.75 rand_pad+crop TfPixel 'padding': p for p in [size/16, size/8, size/4] 0.75

* that is the probability with which each transform was used for final training. For the TTA search method the probability was 1.0.

# Findings

We have found that the Test Time Augmentation for transformation delivers information about the performance improvement with a particular augmentation, and helps rapidly deciding on image transformations which are constructive for better generalizing the network.

Table 3 shows the performance improvement by this method, in comparison to the baseline case. The list of augmentations picked out for each dataset are detailed in Table 4.

Table 3: Top-1 error rates for the found augmentation sets.

 Dataset Baseline error rate TTA Search rel. improvement Oxford-IIIT Pets 5.07% 4.74% 7.14% Stanford Dogs 15.41% 15.17% 1.61% Planet: Understanding the Amazon from Space 3.29% 3.30% -0.13% CIFAR-10 2.99% 3.11% -3.86% Kuzushiji-MNIST 1.55% 1.48% 4.73% Food-101 19.39% 18.65% 3.93%

UPDATE: The platform.ai team has just reached SOTA performance in the dataset Food-101 using our augmentation search technique.

Table 4 : List of selected augmentations and its parameters, for each dataset. The values between brackets represent uniform distribution of the RandTransform class.

 transform pets dogs planet cifar k-mnist food flip flip_lr flip_lr dihedral flip_affine flip_affine symmetric_warp(magnitude) [-0.1, 0.1] [-0.1, 0.1] [-0.2, 0.2] [-0.2, 0.2] rotate(degrees) [-20, 20] [-20, 20] [-10, 10] [-10, 10] [-40, 40] zoom(scale) [1.0, 1.3] [1.0, 1.2] [1.0, 1.2] [1.0, 1.2] [1.0, 1.4] brightness(change) [0.45, 0.55] [0.3, 0.7] [0.45, 0.55] [0.35, 0.65] contrast(scale) [0.7, 1.43] [0.9, 1.11] [0.9, 1.11] [0.8, 1.25] [0.7, 1.43] skew(scale, magnitude) [0, 7], 0.4 [0,7], 0.2 [0, 7], 0.4 [0, 7], 0.2 [0, 7], 0.2 squish(scale) [0.83, 1.2] [0.83, 1.2] [0.83, 1.2] [0.42, 2.4]

It is worth noting that the augmentations picked by our method seemed qualitatively reasonable. For example, for the Planet dataset it chooses dihedral flips (which might include upside-down flips), while for Kuzushiji-MNIST it chooses neither left-right nor dihedral flips, since any of these flips would be damaging for the transformation.

With more time, it may be worth investigating how consistent the differences between the error rates are. The differences between the TTA-based selection of augmentations and get_transforms() might be smaller than the differences between different runs for the same set of augmentations.

### Acknowledgements

We would like to thank Jeremy Howard for his insightful guidance and David Tedaldi for mentorship during this project.

# References

1. Cubuk, Ekin D., et al. AutoAugment: Learning Augmentation Policies from Data. arXiv preprint arXiv:1805.09501 (2018).
2. Geng, Mingyang, et al. Learning data augmentation policies using augmented random search. arXiv preprint arXiv:1811.04768 (2018).
3. Perez, Luis, et al. The Effectiveness of Data Augmentation in Image Classification using Deep Learning. arXiv preprint arXiv:1712.04621 (2017).
4. Krizhevsky, Alex, et al. ImageNet Classification with Deep Convolutional Neural Networks. Neural Information Processing Systems. 25. 10.1145/3065386 (2012).
5. Ratner, Alexander J., et al. Learning to compose domain-specific transformations for data augmentation. Advances in neural information processing systems. 2017.

Also published on Medium.

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# Tag Info

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An interferometer selects (filters) certain wave lengths and suppresses the others.

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The radiation belts are thought to be produced through multiple processes. One of the original leading ideas was the concept of radial diffusion. Other ideas included energization due to low frequency waves. I have done work in this field, but only with a higher frequency electromagnetic wave called whistler mode waves. I will focus on these in my ...

3

What caused the transition,[...] As the universe expands, the number of particles per unit volume goes down. But in addition to this, photons suffer a cosmological red-shift. So the density of mass-energy due to nonrelativistic particles goes down, and the density due to photons goes down, the latter goes down faster. By extrapolation, we predict that ...

1

I'll tackle Gamma Ray Bursts (GRBs) since neither of the current answers have addressed that yet. This section of the Wikipedia article on Gamma Ray Bursts state that a GRB from a Wolf-Rayet star within the Milky Way Galaxy could have a devestating effect on Earth. Specifically, A GRB at a distance of 8,000 light-years from Earth could deplete 25% of the ...

0

I will address the direct impact of a supernova on the Earth's temperature, ignoring secondary effects like effects on cloud formation which may also be important. Let's assume that a supernova's direct contribution to Earth's heat budget is entirely due to electromagnetic radiation emitted during the supernova (we'll ignore neutrino emissions which are ...

1

If a supernova occurred within 100 light years of Earth it would wreck the atmosphere. However, on a more mundane level it has been suggested that cosmic rays cause water droplets to form in the atmosphere and affect cloud cover and hence climate. The amount of cosmic rays also depends to some extent on the changes in the magnetic field of the sun. Reference ...

0

If I were in your position I would be interested in doing some experimental work on Axion detection Specifically, and tying in with the University of Leicester results, repeating the experiments mentioned in wiki with high energy photons eg X rays

1

For stars (which have huge amount of mass and density), gravity is taken to be responsible for the heat increase. because heat and volume (thus density) thus gravitation of a (massive) star, are related. This is exactly one of the factors that make nuclear fusion (in stars) possible. The two effects thermodynamics (and kinetic energy) and gravity are ...

2

Consider a satellite in orbit about the Earth and moving at some velocity $v$. The orbital velocity is related to the distance from the centre of the Earth, $r$, by: $$v = \sqrt{\frac{GM}{r}}$$ If we take energy away from the satellite then it descends into a lower orbit, so $r$ decreases and therefore it's orbital velocity $v$ increases. Likewise if we ...

5

So a while ago I did a little project where I grabbed a "standard solar model" from this paper, which gives me some information that's useful for actually making an estimate. (Unsurprisingly the link given to download the data has changed in the last ten years; I haven't sleuthed to see whether the data is still publicly available.) Only about 1.5% of the ...

0

At the end of your question you ask if there are other ways to express the temperature of the Sun in terms of energy. This is probably not exactly what you're looking for, but from Wien's law $\lambda_{\rm max}T=b$ ($\lambda_{\rm max}$ is the peak wavelength of the Sun's $\sim$blackbody spectrum, $b$ is Wien's displacement constant) and ...

27

Jupiters moon Io is heated through the gravitational pull of Jupiter, but when Io is heated because of this, where does that energy come from? How does conservation of energy work for this effect, where is energy "lost"? TL;DR: The energy ultimately comes from Jupiter's rotation. Io is tidally locked; it has the same orbital and rotation rates. If Io ...

9

To add to Dancrumb's answer and deal more specifically with energy, page 21 of the MIT thesis says the following: As a secondary object orbits around a primary body, the gravitational force of the primary causes a distortion in the shape of the secondary, and vice versa. We can refer to one body as the perturber and the other body as the extended ...

5

Io undergoes synchronous rotation as it orbits Jupiter. As a result, the same part of Io faces Jupiter all the time. Since Jupiter is so massive and Io orbits so closely, Io's shape is distorted due to tidal forces, that is to say the difference between the gravitational pull on Io's closest side to Jupiter and its furthest side to Jupiter are different and ...

1

Fun, So you are asking about the thermal energy content of the sun? If we assume that all the hydrogen is dissociated. (single atoms) Then each atom has three degrees of freedom and carriers 3/2 kT of energy. So count up the number of atoms at each temperature.... That will work until the atoms ionize. Then there will be equal energy in all the ...

0

The sun is more than a cloud of hot gas that is radiating energy. The sun also has energy stored in hydrogen "fuel" that will be "burned" through nuclear fusion into helium, releasing a lot of energy. The sun is $2 \times 10^{30}$ kg, and about 70% hydrogen, so around $1.4 \times 10^{30}$ kg of hydrogen or $8 \times 10^{56}$ protons. Let's estimate that ...

3

I actually answered a related question a couple of days ago on Astronomy. Small world! One of the important properties of the Oort Cloud is that objects in it are not strongly influenced by the Sun. After all, its inner edge is roughly 2,000 AU away - 300 billion kilometers from the Sun. The Sun's gravitational influence in that region is rather weak, so ...

11

Estimating the mass of a "single star" can be a very difficult task, though perhaps your question is too pessimistic. There are a number of suggested relationships linking the mass of a star to its luminosity. These can of course come from stellar evolution models, but they can also be empirically calibrated using stars in resolved binary systems of known ...

7

A binary star system, which is quite common, will allow us to determine mass with great accuracy, using Kepler's Third Law of Planetary Motion which is as follows: $$\frac{T^2}{r^3} = \frac{4 \pi^2}{GM_{sum}}$$ Using the orbital period, $T$, we can determine the acceleration and affect that stars have on one another to determine mass. Once we have the ...

1

The core-collapse phase of a supernova commences once the final stages of nuclear burning are complete. This final phase of fusion reactions involving silicon, produce a core composed of iron-peak elements (not just iron). The cessation of nuclear burning leads to the contraction of the core. This happens relatively slowly at first, on a timescale given by ...

2

Your first paragraph is not quite right. Gas pressure does not "stop" upon formation of an iron core, it is merely that the star cannot generate further heat from nuclear reactions and becomes unstable to collapse. i.e. The star does collapse! Perhaps what you mean is what halts the collapse (sometimes) before the star disappears inside its own event horizon ...

2

Elements heavier than iron are produced mainly by neutron-capture inside stars, although there are other more minor contributors (cosmic ray spallation, radioactive decay or even the collision of neutron stars). Neutron capture can occur rapidly (the r-process) and occurs mostly inside supernova explosions. The free neutrons are created by electron capture ...

0

The Chandrasekhar mass is not the dividing line between those stellar remnants that will become black holes and those that will become something else. A compact, cold white dwarf (i.e. one supported by electron degeneracy pressure) may become unstable and collapse at close to the value of 1.44$M_{\odot}$ described by Kyle and derived using simple Newtonian ...

1

The apparent radial accelleration due to moving in a circle is ω2R. For earth, ω = 2π rad/24 h = 73 x 10-6 rad/s. Earth's radius is about 6.37 Mm. Therefore the upwards accelleration at the equator is 34 mm/s2. That's pretty small compared to the 9.8 m/s2 downwards accelleration due to gravity, so we generally ignore it. Then there is ...

0

So first of all, $\Phi(L)dL$ is, as you say, the number density (per unit volume) of galaxies with a luminosity between $L$ and $L+dL$. The constant $\Phi_0$ is a normalisation constant, and defines the overall number of galaxies per unit volume. To calculate the average luminosity we simply add up the luminosity from each galaxy and divide by the number ...

3

I had a quick look at the paper - its mostly nonsense. The intrinsic light from a quasar is completely dominated by its emission line spectrum and a mostly featureless continuum. The emission lines give the true redshift of the quasar. Absorption lines in quasar spectra are predominantly due to foreground gas clouds at lower redshifts than the more distant ...

16

In astronomy parlance, the Sun has a "metal"$^{1}$ mass fraction of about $0.02$. A solar mass is $\sim2\times10^{30}\;\rm{kg}$, so the sun contains about $4\times10^{28}\;{\rm kg}$ of "metals". That's about $20$ times the mass of Jupiter. A lot of that metal mass will be ${\rm C}$ and ${\rm O}$ and other elements a chemist would call non-metals, but I think ...

5

"Storing" something implies the purpose is to put is somewhere safe so that it can eventually be retrieved. Heavy metals should eventually sink to the center of a star, but how are you going to retrieve it? Even in a science fiction context, it's hard to imagine a plausible means or retreiving a large pile of heavy metal from the center of a star. ...

7

Our primary method of understanding stars is using optical telescopes. We can directly measure the temperature of the "surface" of a star by fitting its spectrum to a blackbody. However, this gives us the temperature only at the "surface" or the optical depth of a visible photon. We know that stars are extremely hot through the hydrodynamic/thermodynamic ...

3

This is due to spinning forces, such as centrifuged force. Remember that galaxies form (at least the regular ones) from the spinning of matter around a galaxy nucleus. So, you're not wrong. Gravity is isotropic, but in this cases the spinning forces are the definition to the galaxy form.

3

The resolution is much smaller than the granulation pattern. i.e. The granules are well resolved and are of order 500-1000 km in diameter, not 50 km. The sunspot is of order the diameter of the Earth. Granulation is caused by convective cells rising and falling such that they just poke up into the bottom of the photosphere - i.e. that region where most of ...

2

According to Wikipedia The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The limit was first published by Wilhelm Anderson and E. C. Stoner, and was named after Subrahmanyan Chandrasekhar, the Indian-American astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 19. White dwarfs with masses greater ...

0

Above the Chandrashekhar limit you would obtain a neutron star: https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit Also see: https://en.wikipedia.org/wiki/Chandrasekhar_limit

1

A "collision course" is a very fuzzy concept: if you are "barely going to hit" you are on a collision course but don't need a lot of deflection. However, let's assume for a moment a stationary earth, a meteorite of mass $m$ at distance $D$, heading for earth of radius $R$ with velocity $v$. The equations you need are conservation of angular momentum and ...

1

Here are the systems I found: 6: ADS 9731 Beta Tucanae Gamma Velorum Kappa Tauri Mu Sagittarii 7: AR Cassiopeiae Nu Scorpii ... no physical multiple stars of greater multiplicity yet found.

Top 50 recent answers are included


text:
SciTech.blog
Formal logics are axiomatic systems that formalise inference. Let’s look at what higher-order logic is and how it can be worked with.


text:
# Algorithm to find the size of a quotient of a free group

Are there any algorithms to find the size of an algebraic quotient of a free group? It would take the generators as input and output the size. For example, an input could be something like {a,b: a^8=b^2=1, ab=ba^3} (a quasidihedral group), and the output would be the integer 16, since there are 16 different ways to write a string formed of a's and b's if you treat aaaaaaaa and bb as the empty string, and the string ab as the same as the string baaa.

Ideally the algorithm would also be able to say if a group was infinite. It's not obvious by looking at the presentation what the size of the group is, and I think it's a fairly interesting question that I've comes across a few times in my algebra class, so I was surprised that I couldn't find a tool to solve it for me or any papers talking about it, it seems that computers should be good at this. Wondering if anyone has any good ideas.

I think you're asking: given a group presentation $$\langle X | R \rangle$$, determine the size of the group.


text:
2013
12-23

# Tell me the area

There are two circles in the plane (shown in the below picture), there is a common area between the two circles. The problem is easy that you just tell me the common area.

There are many cases. In each case, there are two lines. Each line has three numbers: the coordinates (X and Y) of the centre of a circle, and the radius of the circle.

For each case, you just print the common area which is rounded to three digits after the decimal point. For more details, just look at the sample.

0 0 2
2 2 1

0.108

#include <iostream>
#include <cmath>
#include <cstdio>

using namespace std;

int main()
{
double x1,y1,r1,x2,y2,r2,s,PI;
PI=2*asin(1.0);                //PI不能写成3.1415926
while(cin>>x1>>y1>>r1)
{
cin>>x2>>y2>>r2;

double d=sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1));
if(d>=(r1+r2)||!r1||!r2)      //外离
s=0;
else if(d<=abs(r2-r1))        //内含
{
if(r2<r1)
r1=r2;
s=r1*r1*PI;
}
else                                   //相交
{
double a1=acos((r1*r1+d*d-r2*r2)/(2.0*r1*d));       //a1,a2分别为两扇形圆心角的一半
double a2=acos((r2*r2+d*d-r1*r1)/(2.0*r2*d));
s=a1*r1*r1+a2*r2*r2-r1*r1*sin(a1)*cos(a1)-r2*r2*sin(a2)*cos(a2);        //相交面积等于两扇形面积之和减四边形

}
printf("%.3lf\n",s);
}

return 0;
}

1. 第2题，TCP不支持多播，多播和广播仅应用于UDP。所以B选项是不对的。第2题，TCP不支持多播，多播和广播仅应用于UDP。所以B选项是不对的。

2. 站长，你好！
你创办的的网站非常好，为我们学习算法练习编程提供了一个很好的平台，我想给你提个小建议，就是要能把每道题目的难度标出来就好了，这样我们学习起来会有一个循序渐进的过程！

3. int half(int *array,int len,int key)
{
int l=0,r=len;
while(l<r)
{
int m=(l+r)>>1;
if(key>array )l=m+1;
else if(key<array )r=m;
else return m;
}
return -1;
}
这种就能避免一些Bug
l,m,r
左边是l,m;右边就是m+1,r;

4. 约瑟夫也用说这么长……很成熟的一个问题了，分治的方法解起来o(n)就可以了，有兴趣可以看看具体数学的第一章，关于约瑟夫问题推导出了一系列的结论，很漂亮

5. “可以发现,树将是满二叉树,”这句话不对吧，构造的树应该是“完全二叉树”，而非“满二叉树”。

6. Excellent Web-site! I required to ask if I might webpages and use a component of the net web website and use a number of factors for just about any faculty process. Please notify me through email regardless of whether that would be excellent. Many thanks


text:
ResearchWorks Archive

# Browsing Mathematics by Author "Wright, James M., 1960-"

Sort by: Order: Results:

• (1996)
A strong Markov process, $W\sp0$, is constructed by a natural linking together of two independent stable processes of type ($\alpha,\ \beta\sb1$) and ($\alpha,\ \beta\sb2$). The drift for a stable process X of type (\$\alpha,\ ...


text:
# Import PCB outline made from connected lines from Inkscape

I have designed an outline for a PCB in Inkscape and wish to import it into Eagle, but the guides I have found leave me with lots of little lines in Eagle, not a continuous line, what's more is there are gaps between the lines, and as this is for the dimensional layer, not ideal. 1

Is there a way to modify the file so the import is made up of connected lines? Or another way to get this to work?

Also tried this with similar disappointing results 2

-
the links are broken... can you fix it? – Butzke Feb 20 '14 at 13:04
Use a program like Draftsight to make sure that the export to DXF step is actually completing correctly. The export may have a bunch of holes that you can close up with a join command. I would recommend drawing the whole thing in Draftsight anyway, in DXF/DWG format. It may be less like drawing than Illustrator but it's more precise and made to handle tasks just like this. – scld Feb 20 '14 at 20:15

Since you designed the outline yourself, just re-enter the same information into Eagle. Eagle lets you define the outline as line segments and arcs.

I just looked at your first link, and it seems the real problem is that you are trying to do something completely unrealistic:

First, if you are designing a PCB, you should do so in the program for that purpose, which in your case is apparently Eagle. Second, you have to understand how printed circuit boards are made to avoid impossible specifications.

Each board will be cut from a larger panel by computer-controller router. These have straight lines and arcs as primitives. This means most of the time arbitrary curves have to be approximated with lots of small line segments, which is inefficient for the board house and very costly for you. Usually after 6 or so route points, they start charging per route point. Every line segment is one route point and every arc two.

Another thing you have to think about is that there can never be a true interior corner because the router bit will have some minimum diameter. If you specify a sharp interior corner, you don't know what you'll get. The board house might use the smallest router bit they have and charge you extra for having to change bits or for the small size (smaller bits break off more easily and cost them more), and you still won't get a sharp interior corner. It is best to always specify interior corners yourself as arcs of some reasonable radius.

All in all what you are trying to do is unreasonable. At the very least, talk to your board house first and find out what special things they might be able to do. It is also unfair of that web page to say that Eagle has a "glaring omission" in this area when the glaring omission is really in their knowledge of PCB fabrication.

If you really need such a strange board shape and this is a one-off, it is probably better to have the board house cut your board to the minimum enclosing rectangle, then cut the wierd shape yourself. It might require a combination of using a band saw and some manual filing. Make sure there is no copper in the areas you don't want and within some minimal distance of the edge. The copper will dull whatever tool you are using to cut or grind the board. You want to cut just fiberglass, not fiberglass and copper.

-
There are places, such as OSH Park, where you can essentially get a free-form outline and cut-out for no additional charge. While you are definitely correct that the inner curves won't be correct, for a lot of applications getting close is all you need. Also, that link was just a third-party guide, not the OP's exact intentions, so maybe direct some anger away from the OP? – W5VO Feb 20 '14 at 15:05
@W5VO: It wasn't at all clear that the OP didn't link to his own web page. It would make sense for him to show us what he is really trying to do, not what someone else on the internet did. The point about the "glaring omission" comments still stands, but I reworded it to refer to the web page author instead of the OP, even though it's still not clear to me they aren't the same. – Olin Lathrop Feb 20 '14 at 15:18
I read the guides I have found to mean that he did not write the guides, and I would expect that if he had written the guide he would not be asking why his guide didn't work. That being said, I agree with the "glaring omission" because complex curved outlines are uncommon in PCB design. However, it should be able to do something like that - I've seen many PCBs that have been shaped to fit whatever space they had. Inkscape is working as "the poor man's AutoCAD" here. – W5VO Feb 20 '14 at 15:35
I didn't write the guide, but I see you have edited your answer to reflect this realisation, and although the guide shows an example with sharp interior angles, I have designed my outlines to not for this very reason. I know what I am asking is possible, and if you don't know how to do this then it is best to say 'I don't know' or not post an answer, instead of blurting out everything you know, which, although may be helpful to some, was not what I was asking about... – holmeswatson Feb 20 '14 at 16:11
@holme: As I said, it is possible. You approximate arbitrary curves with lots of little line segments. These would be bunch of WIRE commands in the Dimension layer. If you use some tool other than Eagle to produce the curves, then it's up to you to eventually extract line segments from that tool, then convert them to the appropriate WIRE commands. Again, you really want to talk to your board house first though. Find out exactly how they prefer complex curves to be specified, if they are willing to do them at all (for a reasonable price). – Olin Lathrop Feb 20 '14 at 16:29

Why don't you just import it however you need to, and then use regular Eagle drawing tools or scripting to trace it into the dimension layer?

-
because the Eagle drawing tools aren't great and I may need to do this a few times. I'm currently using a ULP script, but not getting the desired results – holmeswatson Feb 20 '14 at 13:38
Sometimes doing things a bit out of the ordinary take a few times to get it right – Scott Seidman Feb 20 '14 at 14:47

pstoedit is a tool which processes a PostScript file into other formats. It was quite easy to extend it to create gEDA "pcb" outlines, and assuming Eagle has a reasonably documented file format, it should be easy for you as well.

To understand how the tool works, you need to understand PostScript as a programming language for graphical objects. (Look for Don Lancaster's writings about this.) pstoedit contains a PostScript interpreter, but lets you replace the very lowest level function, the one that actually draws the lines, with one of your own creation. (You can also hook into functions the next level up, that draw text or primitive objects, otherwise they are drawn as lines through your function.)

So if you make a simple Eagle file with one line in it, open the file and divide into "line", "header", and "footer" parts. Your pstoedit plugin will output the header at the beginning of the file, a large number of "line" parts, and the "footer" part at the end.

Then you just export from whatever drawing tool as PostScript or PDF, and use pstoedit to convert to Eagle format.

Disclaimer: I've never looked at Eagle file format documentation; if it is a binary-blob-mystery-meat format you would do well to choose a less limiting tool.

-

There's a tool called PCBmodE which was created to convert Inkscape into Gerber, which there are other tools to import into eagle.

I haven't used it myself so can't comment further.

-
Are you sure you can import Gerber to Eagle?? – Scott Seidman Feb 21 '14 at 2:07
There are other pages on the web recommending tools to do so, but I've not done so myself. – Pete Kirkham Feb 21 '14 at 9:59


text:
# Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN.

Can we say that MN bisects BH? If so, why?

If MN bisects BH (at point X), do they form right angles? If so, why? Is a segment between two midpoints parallel to the triangle's opposite side?

If we have both of those, then triangle BXN is congruent to triangle HXN. Then angle HBN = angle BHN and BN = HN. Also, triangles BXM = HXM, so angles BHM = HBM and sides BM = HM. Then we can prove using Side-angle-side.

Note: Only theorems about triangle congruence and parallel lines are available (no similar triangles).

• "Can we say that MN bisects BH? If so, why?" Yes. Because BMN and BAC are similar (two proportional sides and a common interior angle). So Triangle BXM and BHA are similar (two angles are equal) so $BX$ is proportional to $BH$ as $BM$ is proportional to $BA$. Sep 7, 2016 at 0:25
• "Is a segment between two midpoints parallel to the triangle's opposite side?" Yes. Because they form similar triangles so the form equal interior angles. "If MN bisects BH (at point X), do they form right angles? If so, why? " Yes, because the lines are parallel. Sep 7, 2016 at 0:28
• Oh. No similar triangles. .... Okay. Let X be midpoint BH. Construct perp biscetor of BH. Have it intersect AB at M' and BC at N'. M'HN' is congruent to M'BN'. M'N' is parallel to AC so interior angles the same. Futz to show AM'H and HN'C are iscoselles. So AM' = M'H = M'B so 2AM' - AB so M' is midpoint. Same with N'. So M = M' and N=N' and the triangles are congruent. In other words go backward. Assume the triangles are congruent and prove the points most be midpoints are mid points. Sep 7, 2016 at 0:42

(Someone draw a picture of this because I don't know how)

To answer your second question, yes, a segment between two midpoints is parallel to the triangle's opposite side. In this case, call the intersection between $\overline{BH}$ and $\overline{MN}$ $P$, and draw altitude $\overline{MH'}$ perpendicular to $\overline{AC}$. Notice that $\overline{MH'} || \overline{BH}$, so $\angle AMH' = \angle ABH$, and therefore, $\angle BAC = \angle BMN$, so $\overline{MN} || \overline{AC}$.

Additionally, in this case $\overline{MN}$ bisects $\overline{BH}$. This is because $\triangle AMH' \cong \triangle MBP$ via Angle-Side-Angle, so $MH' = BP$. We also have $MH' = PH$, so $\overline{MN}$ does indeed bisect $\overline{BH}$, and it forms right angles because $\overline{MN} || \overline{AC}$.

• I don't see how you established that $\angle BAC = \angle BMN$.
– Jeff
Aug 25, 2014 at 23:20
• $\overline{MH'}$ is perpendicular to $\overline{AC}$, so $\angle MH'A = \angle BHA$. We have $\angle AMH′= \angle ABH$ by parallel lines. Logically, that means $\angle BAC= \angle BMN$ (see triangles $\triangle AMH', \triangle MBP$) Aug 25, 2014 at 23:56
• I think that's only true if you've already established that $MN || AC$. But I've found that can be established using a parallelogram with side $CE$ parallel to $AM$.
– Jeff
Aug 26, 2014 at 3:55

By midpoint theorem, MN//AC.

By intercept theorem, BP = PH

Also angle BPM = angle MPH = 90 degrees.

MP = PM (common side)

Thus ⊿BPM is congruent to ⊿HPM (SAS)

Similarly, ⊿BPN is congruent to ⊿HPN

Hence, ⊿BMN is congruent to ⊿HMN.

Note (1) My P is your X.

Note (2) The way you think is essentially correct. I just add in the details with corresponding reasons as support.

Note (3) We (including others) were confused at first and I finally realized that those questions you are asking are in fact your attempt to the question.

• Can you explain how BP=PH? I don't see how the intercept theorem applys. We can only use congruent triangle parallel line theorems.
– Jeff
Aug 26, 2014 at 14:33
• @Jeff From BM = MA (given); and MP(N) // AH(C) (proved), BP = PH (intercept theorem).
– Mick
Aug 27, 2014 at 3:38
• @Jeff Intercept theorem follows right after, congruent triangles, parallel theorem and the midpoint theorem. It is not that advanced and not that difficult to prove and learn.
– Mick
Aug 27, 2014 at 3:43

The circles with $AB$ and $BC$ as diameters have $BH$ as their radical axis and $M,N$ as the respective centers. Its clear now that $MB= MH, NB = NH$ and congruence follows


text:
cohomotopy groups

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, particularly
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...
, cohomotopy sets are particular contravariant functors from the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
of pointed topological spaces and basepoint-preserving
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
maps to the category of sets and
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. They are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the
homotopy groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, but less studied.

# Overview

The ''p''-th cohomotopy set of a pointed topological space ''X'' is defined by :$\pi^p\left(X\right) = \left[X,S^p\right]$ the set of pointed homotopy classes of continuous mappings from $X$ to the ''p''-hypersphere, sphere $S^p$. For ''p'' = 1 this set has an abelian group structure, and, provided $X$ is a CW-complex, is group isomorphism, isomorphic to the first cohomology group $H^1\left(X\right)$, since the circle $S^1$ is an Eilenberg–MacLane space of type $K\left(\mathbb,1\right)$. In fact, it is a theorem of Heinz Hopf that if $X$ is a CW-complex of dimension at most ''p'', then $\left[X,S^p\right]$ is in bijection with the ''p''-th cohomology group $H^p\left(X\right)$. The set $\left[X,S^p\right]$ also has a natural group (mathematics), group structure if $X$ is a suspension (topology), suspension $\Sigma Y$, such as a sphere $S^q$ for $q \ge 1$. If ''X'' is not homotopy equivalent to a CW-complex, then $H^1\left(X\right)$ might not be isomorphic to $\left[X,S^1\right]$. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to $S^1$ which is not homotopic to a constant map.Polish Circle
Retrieved July 17, 2014.

# Properties

Some basic facts about cohomotopy sets, some more obvious than others: * $\pi^p\left(S^q\right) = \pi_q\left(S^p\right)$ for all ''p'' and ''q''. * For $q= p + 1$ and $p > 2$, the group $\pi^p\left(S^q\right)$ is equal to $\mathbb_2$. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.) * If $f,g\colon X \to S^p$ has $\, f\left(x\right) - g\left(x\right)\, < 2$ for all ''x'', then $\left[f\right] = \left[g\right]$, and the homotopy is smooth if ''f'' and ''g'' are. * For $X$ a compact space, compact smooth manifold, $\pi^p\left(X\right)$ is isomorphic to the set of homotopy classes of smooth function, smooth maps $X \to S^p$; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic. * If $X$ is an $m$-manifold, then $\pi^p\left(X\right)=0$ for $p > m$. * If $X$ is an $m$-manifold#Manifold with boundary, manifold with boundary, the set $\pi^p\left(X,\partial X\right)$ is natural isomorphism, canonically in bijection with the set of cobordism classes of codimension-''p'' framed submanifolds of the Interior (topology), interior $X \setminus \partial X$. * The stable cohomotopy group of $X$ is the colimit :$\pi^p_s\left(X\right) = \varinjlim_k$ :which is an abelian group.

# References

Homotopy theory {{topology-stub


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# Math Help - Totally baffled

1. ## Totally baffled

I have an algebra problem where I need to find X, hopefully someone can help as I cannot see the wood for the trees on this one.

THanks for any help anyone

X - ((0.2*X) + (0.3*X)) = 100

2. "I cannot see the wood for the trees." Nice expression ^^

$X-((0.2X)+(0.3X))=100$

$\implies X(1-(0.2+0.3))=100$

$\implies X=\frac{100}{(1-(0.2+0.3))}$

$\implies X=\frac{100}{0.5}$

$\implies X=200$

3. Originally Posted by Sean12345
"I cannot see the wood for the trees." Nice expression ^^

$X-((0.2X)+(0.3X))=100$

$\implies X(1-(0.2+0.3))=100$

$\implies X=\frac{100}{(1-(0.2+0.3))}$

$\implies X=\frac{100}{0.5}$

$\implies X=200$
just in-regards to your answer, why did u remove the X's and place a 1 before the additions ?

4. Good point actually. I don't really know why i did that come to think of it. I'll just say i was giving an insight into how he would do harder questions of the same principal.


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## anonymous 5 years ago Determine whether the sequence converges or diverges. If it converges, nd the limit n!/2^n

1. anonymous

You can use the ratio test. $\lim_{n->\infty}\left|\frac{a_{n+1}}{a_n} \right|=\lim_{n->\infty}\frac{(n+1)!/2^{n+1}}{n!/2^{n}}=\lim_{n->\infty}\frac{(n+1)n!}{2 \times 2^n}\frac{2^n}{n!}$$=\lim_{n->\infty}\frac{n}{2} \rightarrow \infty$ The sequence does not converge since the ratio in the limit is not less than 1.

2. anonymous

nice!

3. anonymous

Thanks ;)

4. anonymous

$\frac{n+1}{2}$

5. anonymous

A question.. So does it mean that you cant test some values as N... and see whats happening?

6. anonymous

Can you rephrase that? I'm not sure what you mean.

7. anonymous

n!/2^n... Put in some values, as a check.. ..n= 1, 2, 3

8. anonymous

And see if it diverges or converges

9. anonymous

I hope you didnt misunderstand me, it was just a question, if i would in some values, in the funktion , n!/2^n .. could i see right away if it diverges or converges..

10. anonymous

Ah I see. No, because that's not actually testing for ALL n. The tests are derived from the definition of convergence.

11. anonymous

You could get an idea, but it wouldn't prove it. That's all.

12. anonymous

Okey, so u have to, show it for all n. like u did

13. anonymous

Exactly.

14. anonymous

Well, thx.

15. anonymous

No worries.

16. anonymous

Hehe, I thought this was XiaoHong's question.

17. anonymous

So did i ;)

18. anonymous

thanks

19. anonymous

Welcome.