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• ### Intrinsic properties of a complete sample of HETE-2 gamma-ray bursts. A measure of the GRB rate in the Local Universe(0811.3304)

Nov. 20, 2008 astro-ph
Aims: Taking advantage of the forthcoming Catalog of the HETE-2 mission, the aim of this paper is to evaluate the main properties of HETE-2 GRBs - the E_peak, the T_90 and the E_iso - in their source frames and to derive their unbiased distribution. Methods: We first construct a complete sample containing all the bursts localized by the WXM on-board HETE-2, which are selected with a uniform criterion and whose observed parameters can be constrained. We then derive the intrinsic E_peak, T_90 and E_iso distributions using their redshift when it is available, or their pseudo-redshift otherwise. We finally compute the number of GRB (N_Vmax) within the visibility volume (V_max) of each GRB, in order to derive a weight for each detected burst accounting both for the detection significance and the star formation history of the universe. Results: The unbiased distributions obtained clearly show the predominence of X-ray flashes (XRFs) in the global GRB population. We also derive the rate of local GRBs: R0^H2 > 11 Gpc-3 yr-1, which is intermediate between the local rate obtained by considering only the high-luminosity bursts (~1 Gpc-3 yr-1) and that obtained by including the low-luminosity bursts (>200 Gpc-3 yr-1).
• ### X-ray Flashes or soft Gamma-ray Bursts? The case of the likely distant XRF 040912(astro-ph/0610157)

Oct. 5, 2006 astro-ph
In this work, we present a multi-wavelength study of XRF 040912, aimed at measuring its distance scale and the intrinsic burst properties. We performed a detailed spectral and temporal analysis of both the prompt and the afterglow emission and we estimated the distance scale of the likely host galaxy. We then used the currently available sample of XRFs with known distance to discuss the connection between XRFs and classical Gamma-ray Bursts (GRBs). We found that the prompt emission properties unambiguously identify this burst as an XRF, with an observed peak energy of E_p=17+/-13 keV and a burst fluence ratio S(2-30keV)/S(30-400keV)>1. A non-fading optical source with R~24 mag and with an apparently extended morphology is spatially consistent with the X-ray afterglow, likely the host galaxy. XRF 040912 is a very dark burst since no afterglow optical counterpart is detected down to R>25 mag (3 sigma limiting magnitude) at 13.6 hours after the burst. The host galaxy spectrum detected from 3800A to 10000A, shows a single emission line at 9552A. The lack of any other strong emission lines blue-ward of the detected one and the absence of the Ly alpha cut-off down to 3800A are consistent with the hypothesis of the [OII] line at redshift z=1.563+/-0.001. The intrinsic spectral properties rank this XRF among the soft GRBs in the E_peak-E_iso diagram. Similar results were obtained for most XRFs at known redshift. Only XRF 060218 and XRF 020903 represent a good example of instrinsic XRF(i-XRF) and are possibly associated with a different progenitor population. This scenario may calls for a new definition of XRFs.
• ### Galaxy clusters in the CFHTLS. First matched filter candidate catalogue of the Deep fields(astro-ph/0609815)

Sept. 29, 2006 astro-ph
We apply a matched-filter cluster detection algorithm to the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) i-band data for the Deep-1, Deep-2, Deep-3 and Deep-4 fields covering a total of 4square degrees. To test the implemented procedure we carry out simulations for assessing the frequency of noise peaks as well as estimate the recovery efficiency. We estimate that up to z~0.7 the catalogue is essentially complete for clusters of richness class R>~1. The recovered redshifts are in general overestimated by dz=0.1 with a scatter of sigma_dz~0.1, except at redshifts z>~1 where the estimated redshifts are systematically underestimated. The constructed cluster candidate catalogue contains 162 detections over an effective area of 3.112 square degrees corresponding to a density of ~52.1 per square degree. The median estimated redshift of the candidates is z=0.6. The estimated noise frequency is 16.9+-5.4 detections per square degree. From visual inspection we identify systems that show a clear concentration of galaxies with similar colour. These systems have a density of ~20 per square degree.
• ### HETE-2 Localizations and Observations of Four Short Gamma-Ray Bursts: GRBs 010326B, 040802, 051211 and 060121(astro-ph/0605570)

June 3, 2006 astro-ph
Here we report the localizations and properties of four short-duration GRBs localized by the High Energy Transient Explorer 2 satellite (HETE-2): GRBs 010326B, 040802, 051211 and 060121, all of which were detected by the French Gamma Telescope (Fregate) and localized with the Wide-field X-ray Monitor (WXM) and/or Soft X-ray Camera (SXC) instruments. We discuss eight possible criteria for determining whether these GRBs are "short population bursts" (SPBs) or "long population bursts" (LPBs). These criteria are (1) duration, (2) pulse widths, (3) spectral hardness, (4) spectral lag, (5) energy Egamma radiated in gamma rays (or equivalently, the kinetic energy E_KE of the GRB jet), (6) existence of a long, soft bump following the burst, (7) location of the burst in the host galaxy, and (8) type of host galaxy. In particular, we have developed a likelihood method for determining the probability that a burst is an SPB or a LPB on the basis of its T90 duration alone. A striking feature of the resulting probability distribution is that the T90 duration at which a burst has an equal probability of being a SPB or a LPB is T90 = 5 s, not T90 = 2 s, as is often used. All four short-duration bursts discussed in detail in this paper have T90 durations in the Fregate 30-400 keV energy band of 1.90, 2.31, 4.25, and 1.97 sec, respectively, yielding probabilities P(S|T90) = 0.97, 0.91, 0.60, and 0.95 that these bursts are SPBs on the basis of their T90 durations alone. All four bursts also have spectral lags consistent with zero. These results provide strong evidence that all four GRBs are SPBs (abstract continues).
• ### Discovery of the short gamma-ray burst GRB 050709(astro-ph/0510190)

Oct. 6, 2005 astro-ph
Gamma-Ray Bursts (GRBs) fall into two classes: short-hard and long-soft bursts. The latter are now known to have X-ray and optical afterglows, to occur at cosmological distances in star-forming galaxies, and to be associated with the explosion of massive stars. In contrast, the distance scale, the energy scale, and the progenitors of short bursts have remained a mystery. Here we report the discovery of a short-hard burst whose accurate localization has led to follow-up observations that have identified the X-ray afterglow and (for the first time) the optical afterglow of a short-hard burst. These, in turn, have led to identification of the host galaxy of the burst as a late-type galaxy at z=0.16 showing that at least some short-hard bursts occur at cosmological distances in the outskirts of galaxies, and are likely to be due to the merging of compact binaries.
• ### HETE-2 Observation of two gamma-ray bursts at z > 3(astro-ph/0502494)

Feb. 23, 2005 astro-ph
GRB 020124 and GRB 030323 constitute half the sample of gamma-ray bursts with a measured redshift greater than 3. This paper presents the temporal and spectral properties of these two gamma-ray bursts detected and localized with HETE-2. While they have nearly identical redshifts (z=3.20 for GRB 020124, and z=3.37 for GRB 030323), these two GRBs span about an order of magnitude in fluence, thus sampling distinct regions of the GRB luminosity function. The properties of these two bursts are compared with those of the bulk of the GRB population detected by HETE-2. We also discuss the energetics of GRB 020124 and GRB 030323 and show that they are compatible with the Epeak - Eiso relation discovered by Amati et al. (2002). Finally, we compute the maximum redshifts at which these bursts could have been detected by HETE-2 and we address various issues connected with the detection and localization of high-z GRBs.
• ### High Energy Observations of XRF 030723: Evidence for an Off-axis Gamma-Ray Burst?(astro-ph/0408453)

Jan. 13, 2005 astro-ph
We report High Energy Transient Explorer 2 (HETE-2) Wide Field X-ray Monitor/French Gamma Telescope observations of XRF030723 along with observations of the XRF afterglow made using the 6.5m Magellan Clay telescope and the Chandra X-ray Observatory. The observed peak energy E_pk_obs of the nu F_nu burst spectrum is found to lie within (or below) the WXM 2-25 keV passband at 98.5% confidence, and no counts are detected above 30 keV. Our best fit value is E_pk_obs=8.4+3.5/-3.4 keV. The ratio of X-ray to Gamma-ray flux for the burst follows a correlation found for GRBs observed with HETE-2, and the duration of the burst is similar to that typical of long-duration GRBs. If we require that the burst isotropic equivalent energy E_iso and E_pk_rest satisfy the relation discovered by Amati et al. (2002), a redshift of z=0.38+0.36/-0.18 can be determined, in agreement with constraints determined from optical observations. We are able to fit the X-ray afterglow spectrum and to measure its temporal fade. Although the best-fit fade is shallower than the concurrent fade in the optical, the spectral similarity between the two bands indicates that the X-ray fade may actually trace the optical fade. If this is the case, the late time rebrightening observed in the optical cannot be due to a supernova bump. We interpret the prompt and afterglow X-ray emission as arising from a jetted GRB observed off-axis and possibly viewed through a complex circumburst medium due to a progenitor wind.
• ### Global Characteristics of X-Ray Flashes and X-Ray-Rich GRBs Observed by HETE-2(astro-ph/0409128)

Sept. 6, 2004 astro-ph
We describe and discuss the global properties of 45 gamma-ray bursts (GRBs) observed by HETE-2 during the first three years of its mission, focusing on the properties of X-Ray Flashes (XRFs) and X-ray-rich GRBs (XRRs). We find that the numbers of XRFs, XRRs, and GRBs are comparable. We find that the durations and the sky distributions of XRFs and XRRs are similar to those of GRBs. We also find that the spectral properties of XRFs and XRRs are similar to those of GRBs, except that the values of the peak energy $E^{\rm obs}_{\rm peak}$ of the burst spectrum in $\nu F_\nu$, the peak energy flux $\Fp$, and the energy fluence $S_E$ of XRFs are much smaller -- and those of XRRs are smaller -- than those of GRBs. Finally, we find that the distributions of all three kinds of bursts form a continuum in the [$S_E$(2-30 keV),$S_E$(30-400) keV]-plane, the [$S_E$(2-400 keV), $E_{\rm peak}$]-plane, and the [$F_{\rm peak}$(50-300 keV), $E_{\rm peak}$]-plane. These results provide strong evidence that all three kinds of bursts arise from the same phenomenon.
• ### Highlights of the HETE-2 Mission(astro-ph/0310414)

Dec. 18, 2003 astro-ph
The HETE-2 mission has been highly productive. It has observed more than 250 GRBs so far. It is currently localizing 25 - 30 GRBs per year, and has localized 43 GRBs to date. Twenty-one of these localizations have led to the detection of X-ray, optical, or radio afterglows, and as of now, 11 of the bursts with afterglows have redshift determinations. HETE-2 has also observed more than 45 bursts from soft gamma-ray repeaters, and more than 700 X-ray bursts. HETE-2 has confirmed the connection between GRBs and Type Ic supernovae, a singular achievement and certainly one of the scientific highlights of the mission so far. It has provided evidence that the isotropic-equivalent energies and luminosities of GRBs may be correlated with redshift; such a correlation would imply that GRBs and their progenitors evolve strongly with redshift. Both of these results have profound implications for the nature of GRB progenitors and for the use of GRBs as a probe of cosmology and the early universe. HETE-2 has placed severe constraints on any X-ray or optical afterglow of a short GRB. It has made it possible to explore the previously unknown behavior optical afterglows at very early times, and has opened up the era of high-resolution spectroscopy of GRB optical afterglows. It is also solving the mystery of "optically dark" GRBs, and revealing the nature of X-ray flashes (XRFs).
• ### Scientific Highlights of the HETE-2 Mission(astro-ph/0309462)

Dec. 18, 2003 astro-ph
The HETE-2 mission has been highly productive. It has observed more than 250 GRBs so far. It is currently localizing 25 - 30 GRBs per year, and has localized 43 GRBs to date. Twenty-one of these localizations have led to the detection of X-ray, optical, or radio afterglows, and as of now, 11 of the bursts with afterglows have known redshifts. HETE-2 has confirmed the connection between GRBs and Type Ic supernovae, a singular achievement and certainly one of the scientific highlights of the mission so far. It has provided evidence that the isotropic-equivalent energies and luminosities of GRBs are correlated with redshift, implying that GRBs and their progenitors evolve strongly with redshift. Both of these results have profound implications for the nature of GRB progenitors and for the use of GRBs as a probe of cosmology and the early universe. HETE-2 has placed severe constraints on any X-ray or optical afterglow of a short GRB. It is also solving the mystery of "optically dark' GRBs, and revealing the nature of X-ray flashes.
• ### HETE-2 Observations of the Extremely Soft X-Ray Flash XRF 020903(astro-ph/0309455)

Nov. 10, 2003 astro-ph
We report HETE-2 WXM/FREGATE observations of the X-ray flash, XRF 020903. This event was extremely soft: the ratio log(S_X/S_gamma) = 0.7, where S_X and S_gamma are the fluences in the 2-30 and 30-400 keV energy bands, is the most extreme value observed so far by HETE-2. In addition, the spectrum has an observed peak energy E^{obs}_{peak} < 5.0 keV (99.7 % probability upper limit) and no photons were detected above ~10 keV. The burst is shorter at higher energies, which is similar to the behavior of long GRBs. We consider the possibility that the burst lies at very high redshift and that the low value of E^{obs}_{peak} is due to the cosmological redshift, and show that this is very unlikely. We find that the properties of XRF 020903 are consistent with the relation between the fluences S(7-30 keV) and S(30-400 keV) found by Barraud et al. for GRBs and X-ray-rich GRBs, and are consistent with the extension by a decade of the hardness-intensity correlation (Mallozzi et al. 1995) found by the same authors. Assuming that XRF 020903 lies at a redshift z = 0.25 as implied by the host galaxy of the candidate optical and radio afterglows of this burst, we find that the properties of XRF 020903 are consistent with an extension by a factor ~300 of the relation between the isotropic-equivalent energy E_iso and the peak E_peak of the nu F_nu spectrum (in the source frame of the burst) found by Amati et al. for GRBs. The results presented in this paper therefore provide evidence that XRFs, X-ray-rich GRBs, and GRBs form a continuum and are a single phenomenon. The results also impose strong constraints on models of XRFs and X-ray-rich GRBs.
• ### HETE-2 Localization and Observation of the Bright, X-Ray-Rich Gamma-Ray Burst GRB021211(astro-ph/0303470)

Sept. 2, 2003 astro-ph
A bright, x-ray-rich GRB was detected by HETE-2 at 11:18:34.03 UT on 11 Dec 2002. The WXM localization was to 14' and relayed to the GCN 22 s after the start of the burst. The ground SXC localization was within 2' of R.A. 08h 09m 00s, Dec 06d 44' 20" (J2000). GRB021211 consists of a single, FRED-like pulse with t90s of 2.3 s (85-400 keV) and 8.5 s (2-10 keV). The peak photon number and photon energy fluxes in the 2-400 keV band, are 34.0 +/- 1.8 ph cm^-2 s^-1 and 1.68 +/- 0.11 x 10^-6 erg cm^-2 s^-1, respectively. The energy fluences in the 2-30 keV and 30-400 kev energy bands are S_X = 1.36 +/- 0.05 x 10^-6 erg cm^-2 and S_gamma 2.17 +/- 0.15 x 10^-6 erg cm^-2, respectively. Thus GRB021211 is an X-ray-rich GRB (S_X/S_gamma = 0.63 > 0.32). The spectrum is well-fit by a Band function (alpha = -0.805, beta = -2.37, E_peak = 46.8 keV). The prompt localization allowed the detection of an optical afterglow for what would otherwise have been an optically dark'' GRB. GRB 021211 demonstrates that some fraction of burst afterglows are optically dark'' because their optical afterglows at times > 1 hr after the burst are very faint, and thus have often escaped detection. GRB 021211 shows that such optically dim'' bursts can have very bright afterglows at times < 20 min after the burst.
• ### HETE-2 Localization and Observations of the Short, Hard Gamma-Ray Burst GRB020531(astro-ph/0206151)

June 11, 2002 astro-ph
The {\it HETE-2} (hereafter \HETE) French Gamma Telescope (FREGATE) and the Wide-field X-ray Monitor (WXM) instruments detected a short ($t_{50} = 360$ msec in the FREGATE 85-300 keV energy band), hard gamma-ray burst (GRB) that occurred at 1578.72 SOD (00:26:18.72 UT) on 31 May 2002. The WXM flight localization software produced a valid location in spacecraft (relative) coordinates. However, since no on-board real-time star camera aspect was available, an absolute localization could not be disseminated. A preliminary localization was reported as a GCN Position Notice at 01:54:22 UT, 88 min after the burst. Further ground analysis produced a refined localization, which can be expressed as a 90% confidence rectangle that is 67 arcminutes in RA and 43 arcminutes in Dec (90% confidence region), centered at RA = +15$^{\rm h}$ 14$^{\rm m}$ 45$^{\rm s}$, Dec = -19$^\circ$ 21\arcmin 35\arcsec (J2000). An IPN localization of the burst was disseminated 18 hours after the GRB (Hurley et al. 2002b). A refined IPN localization was disseminated $\approx$ 5 days after the burst. This hexagonal-shaped localization error region is centered on RA = 15$^{\rm h}$ 15$^{\rm m}$ 03.57$^{\rm s}$, -19$^\circ$ 24\arcmin 51.00\arcsec (J2000), and has an area of $\approx$ 22 square arcminutes (99.7% confidence region). The prompt localization of this short, hard GRB by \HETE and the anti-Sun pointing of the \HETE instruments, coupled with the refinement of the localization by the IPN, has made possible rapid follow-up observations of the burst at radio, optical, and X-ray wavelengths.
• ### GRB010921: Localization and Observations by the HETE Satellite(astro-ph/0201461)

Jan. 29, 2002 astro-ph
On September 21 at 18950.56 SOD (05:15:50.56) UT the FREGATE gamma-ray instrument on the High Energy Transient Explorer (HETE) detected a bright gamma-ray burst (GRB). The burst was also seen by the X-detector on the WXM X-ray instrument and was therefore well-localized in the X direction; however, the burst was outside the fully-coded field-of-view of the WXM Y-detector, and therefore information on the Y direction of the burst was limited. Cross-correlation of the HETE and Ulysses time histories yielded an Interplanetary Network (IPN) annulus that crosses the HETE error strip at a ~45 degree angle. The intersection of the HETE error strip and the IPN annulus produces a diamond-shaped error region for the location of the burst having an area of 310 square arcminutes. Based on the FREGATE and WXM light curves, the duration of the burst is characterized by a t90 = 18.4 s in the WXM 4 - 25 keV energy range, and 23.8 s and 21.8 s in the FREGATE 6 - 40 and 32 - 400 keV energy ranges, respectively. The fluence of the burst in these same energy ranges is 4.8 10^{-6}, 5.5 10^{-6}, and 11.4 10^{-6} erg cm^{-2}, respectively. Subsequent optical and radio observations by ground-based observers have identified the afterglow of GRB010921 and determined an apparent redshift of z = 0.450.


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0

Exercise 13.7

Assume π = ${{22} \over 7}$ unless stated otherwise.

1. Find the volume of the right circular cone with
(i) radius 6 cm, height 7 cm                  (ii) radius 3.5 cm, height 12 cm.

Sal. (i) Radius (r) = 6 cm
Height (h) = 7 cm

Volume = ${1 \over 3}\pi {r^2}h$

Volume =  $= {1 \over 3} \times {{22} \over 7} \times {(6)^2} \times 7c{m^3} = 264c{m^3}$

(ii) Radius (r) = 3.5 ern
Height (h) = 12 cm
Volume = $= {1 \over 3}\pi {r^2}h$
$= {1 \over 3} \times {{22} \over 7} \times {(3.5)^2} \times 12c{m^3} = 154c{m^3}$

2. Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm.

Sol. (i) r = 7 cm, l = 25 cm.

∴    $h = \sqrt {{l^2} - {r^2}} = \sqrt {{{(125)}^2} - {{(7)}^2}} cm$

$= \sqrt {625 - 49} cm = \sqrt {576} cm = 24cm$

∴  Capacity = 1 22 x x 7 x 7 x 24 cm3
= 1232 cm³ = 1.232 l.

(ii) h = 12 cm, l = 13 cm.

∴      $r = \sqrt {{l^2} - {h^2}} = \sqrt {{{(13)}^2} - {{(12)}^2}} cm$

$= \sqrt {169 - 144} cm = \sqrt {25} cm = 5cm$

Capacity = $= {1 \over 3} \times {{22} \over 7} \times 5 \times 5 \times 12c{m^3}$

$= {{2200} \over 7}c{m^3} = {{11} \over {35}}litre$

3. The height of a cone is 15 cm. If its volume is 1570 cm³, find the radius of the base. (Use π = 3.14).

Sol. Let r be the required radius.

Height (h) = 15 cm

Volume = 1570 ⇒    ${1 \over 3} \times 3.14 \times {r^2} \times 15 = 1570$

$\Rightarrow {r^2} = {{1570 \times 3} \over {3.14 \times 15}} = 100 \Rightarrow r = 10cm$

4. If the volume of a right circular cone of height 9 cm is 48 π cm³, find the diameter of its base.

Sol. Height (h) = 9 cm, radius (r) = ?

Volume = 48π cm³ ⇒ ${1 \over 3} \times \pi \times {r^2} \times 9 = 48\pi$

⇒    r2  =  16 ⇒    r  = 4  cm.

Diameter = 2r = 8 cm.

2. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Sol. Diameter (d) = 3.5 m

$Radius(r) = {{diameter} \over 2} = {{3.5} \over 2}m = {{35} \over {20}}m$

Deep, i.e., height (h) = 12 m

Capacity of the pit = ${1 \over 3}$πr²h

$= {1 \over 3} \times {{22} \over 7} \times {{35} \over {20}} \times {{35} \over {20}} \times 12{m^3}$

= 38.5 m³ = 38.5 kilolitres.  [1 m³ = 1 kl]

6. The volume of a right circular cone is 9856 cm³. If the diameter of the base is 28 cm, find
(i) height of the cone
(ii) slant height of the cone
(iii) curved surface area of the cone.

Sol. Diameter = 28 cm, radius =  ${{28} \over 2} = 14cm$

(i) Volume of cone = 9856 cm³

${1 \over 3} \times {{22} \over 7} \times 14 \times 14 \times h = 9856$

$\Rightarrow h = {{4856 \times 3} \over {44 \times 14}} \Rightarrow h = 48$

∴   Height (h) of the cone = 48 cm.

(ii) Slant height of cone (l)

$= \sqrt {{{(14)}^2} + {{(48)}^2}} cm$

$= \sqrt {196 + 2304} cm = \sqrt {2500} cm = 50cm.$

(iii) Curved surface area of the cone = πrl

$= {{22} \over 7} \times 14 \times 50 = 2200c{m^2}$

7. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.

Sol. As sides are 5 cm, 12 cm, 13 cm. Hence, triangle is right angled.

We have r = 5 cm, h = 12 cm.

∴   $Volume = {1 \over 3}\pi {r^2}h = {1 \over 3} \times \pi \times 5$

× 5  ×  12  cm3

= 100π cm3

8. If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.

Sol. In this case, r = 12 cm, h = 5 cm.

∴     $Volume = {1 \over 3}\pi {r^2}h = {1 \over 3} \times \pi \times 12 \times 12 \times 5c{m^3}$

= 240 π cm3

Ratio of volumes obtained in Question 7 and Question 8:

${{{V_1}} \over {{V_2}}} = {{100\pi } \over {240\pi }} = {5 \over {12}},i.e,{V_1}:{V_2} = 5:12$

9. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.

Sol. Diameter (d) = 10.5 m  ⇒   radius (r) = ${{10.5} \over 2} = 5.25m$
Height (h) = 3 m
Volume of the heap of wheat = ${1 \over 3}\pi {r^2}h$

$= {1 \over 3} \times {{22} \over 7} \times 5.25 \times 5.25 \times 3{m^3}$

= 86.625 m3

Slant height of the cone = $\sqrt {{{(5.25)}^2} + {{(3)}^2}} m$

$= \sqrt {27.5625 + 9} m$

$= \sqrt {36.5625} m \approx 6.05m.$

Area of canvas required = ${{22} \over 7} \times 5.25 \times 6.05{m^2}$

= 99.825 m²

format: plain


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bray_curtis {abdiv} R Documentation

## Bray-Curtis distance

### Description

The Bray-Curtis distance is the Manhattan distance divided by the sum of both vectors.

### Usage

bray_curtis(x, y)

### Arguments

 x, y Numeric vectors

### Details

For two vectors x and y, the Bray-Curtis distance is defined as

d(x, y) = \frac{∑_i |x_i - y_i|}{∑_i x_i + y_i}.

The Bray-Curtis distance is connected to many other distance measures in this package; we try to list some of the more important connections here. Relation to other definitions:

• Equivalent to vegdist() with method = "bray".

• Equivalent to the braycurtis() function in scipy.spatial.distance for positive vectors. They take the absolute value of x_i + y_i in the denominator.

• Equivalent to the braycurtis and odum calculators in Mothur.

• Equivalent to D_14 = 1 - S_17 in Legendre & Legendre.

• The Bray-Curtis distance on proportions is equal to half the Manhattan distance.

• The Bray-Curtis distance on presence/absence vectors is equal to the Sorenson index of dissimilarity.

### Value

The Bray-Curtis distance between x and y. The Bray-Curtis distance is undefined if the sum of all elements in x and y is zero, in which case we return NaN.

### Examples

x <- c(15, 6, 4, 0, 3, 0)
y <- c(10, 2, 0, 1, 1, 0)
bray_curtis(x, y)

# For proportions, equal to half the Manhattan distance
bray_curtis(x / sum(x), y / sum(y))
manhattan(x / sum(x), y / sum(y)) / 2

[Package abdiv version 0.2.0 Index]


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# Importance of rigor

I always have a hard time explaining the importance of rigor to my friends who are not mathematically minded. A lot of past mathematicians develop the foundations of today's mathematics without going through all the rigor but their work is still relevant now, e.g. Newton and Leibniz's calculus and Euler's work on infinite series.

Is there a historical event where a wrong conclusion is derived due to lack of rigor and cause a harmful effect?

Edit: Thanks so much for the lively discussion! I just want to point out that I'm not interested in counterintuitive results or paradoxes, instead I'm looking for instances where wrongly derived conclusion caused a disastrous effect.

-
I am pretty sure the harmonic series was believed to be convergent for a long time. There is even some funny quote on that but I can't find it anymore. –  Simon Markett May 1 '12 at 16:21
The divergence of the harmonic series was proved by Oresme, about $300$ years before Newton was born. I have read the argument, the idea is the same as the one we usually give to students. Bernoulli's argument was more manipulational. The proof by Oresme is buried in a philosophical/theological work, and had zero impact on mathematics. –  André Nicolas May 1 '12 at 16:32
@SimonMarkett If you've ever taught calculus, you know that the harmonic series is still believed (by students) to be convergent. –  Brett Frankel May 1 '12 at 16:42
Basically, rigor marks the frontier between what might or might not be a good intuitive thought. Euler as some say, had a "genius intuition" so he never crossed the line, but most of us need that dose of rigor to guide us to correct ideas, and not slip into nonsense. –  Pedro Tamaroff May 1 '12 at 17:09
@Simon: Godel only showed that CH is consistent with ZF (as is the axiom of choice). Only Cohen proved the undecidability of both from ZF. –  Asaf Karagila May 1 '12 at 17:42

Cauchy and Fourier had a famous dispute about whether Fourier's use of trigonometric series to solve the heat equation was valid. To a large extent, the basic rigorous notions of analysis were formalized in order to resolve this dispute. Along the way, Cauchy published a famous "wrong" theorem. (It wasn't really wrong; it just used a different definition than that which became standard.)

Here's another, more applied, example. I remember going to an joint meetings talk years ago in which the speaker described a situation in which engineers had employed widely-used software to numerically compute solutions to PDE's. But they had neglected to prove that, at the level of precision they were using, their methods would converge to the correct solution. The result was disastrous: a brand new 180 million dollar oil rig collapsed so thunderously that it registered a 3 on the Richter scale.

EDIT: Thanks to Kcd here's the link to more details.

-
It was not multi-billion or 4.0... but close. Details on your second example are at failures.wikispaces.com/…. Here is a quote from the middle of that page: "The sinking was partially caused by the inappropriate use of finite element-code with regards to the global analysis of the finalized design: the finite element mesh used to analyze the tri-cells was too coarse to predict the shear stress accurately. The flawed analysis and post-processing of the tri-cell design led to shear forces being underestimated by some 45%. [contd.]" –  KCd May 3 '12 at 4:29
"Another flaw with the use of the software was the human error involved: the design software involved considerable complexity, which led to a a very high perception of precision by those who used it. One of the features of this software was the way it ran calculations: it would only flag certain sections that were deemed critical by the software presets, and indicate those as sections that the engineers on the design team needed to check. The probable failure point of the SLA-1 GBS was not flagged." –  KCd May 3 '12 at 4:29
@Kcd Thanks. I was trying to remember the details from a talk I was at years ago. Nice to have the reference for the correct details. –  Brett Frankel May 3 '12 at 4:49
Another link for this story on the Sleipner platform accident is at www5.in.tum.de/~huckle/sleipner_Jakobsen.pdf. –  KCd May 4 '12 at 3:06

This was one of the top voted questions on Math Overflow. You'll find many interesting answers there.

Added: One of my favorite examples for regarding the unreliability of numerics is looking at the difference $\operatorname{li}(x)-\pi(x)$ where $\pi(x)=\sum_{p\leq x}1$ is the prime counting function, and $\text{li}(x)=\int_{2}^x \frac{1}{\log t}dt$ is the logarithmic integral. The prime number theorem tells us that $$\lim_{x\rightarrow \infty}\frac{\pi(x)}{\operatorname{li}(x)}=1,$$ but we can still ask what does $\operatorname{li}(x)-\pi(x)$ look like? If we create a plot going all the way to $10^{22}$, which is very far, it appears that $\operatorname{li}(x)-\pi(x)$ is always about $\sqrt{x}$. That is, it is increasing to infinity, and the data seem to suggest that we could find a monotonic function $f(x)$ about the size of $\sqrt{x}$ where $\operatorname{li}(x)-f(x)$ would be an even better approximation to $\pi(x)$.

However, this is not true at all. Littlewood proved that $\text{li(x)}-\pi(x)$ switches sign infinitely often, and we believe that the first time it does so is around $10^{316}$.

I first learned about this while reading the expository paper Prime Number Races by Andrew Granville and Greg Martin. I recommend reading it; it is accessible and interesting. The result and numerics I mentioned above are on pages 5 and 6. (At the start of section 2.)

-
John Derbyshire mentions the same example in his book "Prime Obsession" about the Riemann Hypothesis. –  Fredrik Meyer May 1 '12 at 18:08
Eric, could you provide a link to the graph of $\operatorname{li}(x)-\pi(x)$? –  Pedro Tamaroff May 1 '12 at 21:28
@Look at page 6 of the paper Prime Number Races. There is a convincing chart with the values at $10^i$ for $8\leq i \leq 22$. –  Eric Naslund May 2 '12 at 8:40
It is Littlewood, not Landau, who proved that $\mathrm{li}(x) - \pi(x)$ changes sign infinitely often. The first one to give an upper bound on the range of $x$ when this first happens is Skewes. –  Francesco Sica May 2 '12 at 16:16
@FSica: Thank you for the correction, it was indeed Littlewood. Letting $x_1$ denote the first sign switch, as you mention the first upper bound was due to Skewes. He showed that $$x_1<10^{10^{10^{10^{34}}}},$$ the so called Skewes number. This was proven in 1933 conditionally on the Riemann hypothesis. Skewes later gave a new larger bound which was not dependent on RH, and several improvements over the years have led to Bays and Hudson's 1999 result which shows that $$x_1<1.4\times 10^{316}.$$ –  Eric Naslund May 2 '12 at 18:24

As your friends are non-mathematical, you might try examples they can understand--simple paradoxes. For example if the base and height of the red triangle is 1, then the length of each staircase below is two, so we have a simplistic argument that 2 = $\sqrt{2}$. This error, from a lack of rigor, would cause major troubles with Google maps, measuring to build a building ... (trying to stay non-mathematical here). There are many such easy paradoxes.

(I know this argument is wrong--the points is lack of rigor can be dangerous)

There are times at which such naive dissection do give the correct answer, such as when we dissect a circle of radius $r$ into arcs and rearrange them into an approximate rectangle of (limiting) height $r$ and width circumference/2 = $\pi r$.

Without rigor, we can not know when to trust our 'reasoning'.

-
Maybe it be interesting to include some basic geometrical insight that $ds \neq dx+dy$ but rather "$ds^2 =dx^2+dy^2$" which is not rigorous but rather intuitive. –  Pedro Tamaroff May 1 '12 at 17:07
@PeterTamaroff it might, but then you are bringing in the missing rigor, I thought the point was to show missing rigor causes big problems! My suggestion is just do it on a third grade level instead of using one of the other very impressive answers here that may not convince the non-mathematical. –  Chris K. Caldwell May 1 '12 at 17:09
@Chris I see. ------------ –  Pedro Tamaroff May 1 '12 at 17:12
This example can be usefully combined with that proof that the area of a circle is $\pi r^2$ which cuts the circle up into wedges and rearranges them to form an approximate parallelogram. That argument asserts that the limiting rectangle is $r\times \pi r$, but the naive argument for these dimensions assumes that arclength is preserved under limits, which the above example shows is not the case in general. So why is this idea right in one argument but wrong in the other? Getting to the bottom of this requires getting more rigorous. –  Steven Taschuk May 1 '12 at 17:55

The "era of rigor" in mathematics began about 150 years ago, a time in which a lot of the mathematics before this era was turned rigorous and cleared from many mistakes. For this reason it is not very trivial to come up with good examples without knowing a lot about mathematical history.

One of the reasons that pre-rigorous mathematics is still useful is that it was essentially created to help developing physical theories. There was no real theoretical mathematics as we have today, and in fact if you look closely you will see that many mathematicians from that time were also physicists, chemists and alchemists (not to mention dark wizards and theological nuts).

I can give a similar example about the axiom of choice, which as a mathematically inclined person you can probably relate to. Consider the mathematicians of exactly a century ago. Many refused to accept the axiom of choice and it was not fully clear whether or not it is consistent or preposterous. Only after Gödel's work people began relaxing the constant questioning (although you still see people today which claim that the axiom of choice is false).

Compare the above situation to the common mathematician of 2012. The vast majority of mathematicians just assume the axiom of choice so bluntly that they use it for things which it is not even close to being needed for. In fact I hear set theorists who admit they have absolutely no intuition whatsoever how things work without the axiom of choice. Of course this can be remedied, but people will not really put that much effort into because they prefer to use their time for better things.

The case is similar to the importance of rigor because we are so deep into the age of rigor that we have essentially forgotten how non-rigorous mathematics look like. You can see it under the surface when talking to a mathematician on something that he is currently working on, but even then people are usually tend to be careful with their words as to avoid claiming false claims.

The most important use of rigor, I think, comes from logic and set theory in the form of the various paradoxes of set theory. When set theory was first developed by Cantor there was still not enough rigor in mathematics in order to well-define all the notions needed (Cantor's paradox is an exceptionally good example). One excellent example is König's theorem. König figured out a contradiction in set theory but it was a mistake in his understanding originating in a not-so-rigorous proof of some claim that he read. The result is actually one of the basic theorems in cardinal arithmetics.

It is important to understand that once we began dealing with the transfinite it was no longer "intuitive" to our physical intuition how things should work. This caused many people reject these ideas and belittle them (transfinite related ideas, the axiom of choice, and so on). Nowadays we know that if one does not use intuition coming from the physical world, but rather develops a mathematical intuition based on the definitions and the axioms of mathematics - one can have a good grasp how the transfinite basics work.

Other good examples are the Weierstrass function which is continuous but non-differentiable anywhere. Before that it was commonly believed that all continuous functions were differentiable almost everywhere. Gauss himself used topological arguments to prove the fundamental theorem of algebra, although he did not prove these arguments.

All in all, in mathematics we have "notion" and "definition". Rarely these two concepts coincide but usually we think about objects in one way and we define them to be approximately what we would like them to be. The introduction of rigor made sure that we work by the definition, and step by step while being guided by the notion - and not the other way around.

-
"There was no real theoretical mathematics as we have today, and in fact if you look closely you will see that many mathematicians from that time were also physicists, chemists and alchemists." I would like to read more about this idea, do you know any good sources ? –  Kasper Jan 25 at 1:30
On which part? Generally it's just picking up biographies. Newton was a physicist and an alchemist. Fourier wasn't really a mathematician to begin with, and many others were simply scientists in the broad term of the word. As for rigor, just look at Euler, Cauchy, even as far as early 1900s you can find mistakes rooted in lack of correct rigor. Foe example König's theorem in set theory (not about trees) was an attempt to disprove the axiom of choice, but it was just due to formal misunderstanding (or rather the ambiguity) of definitions in set theory at the time. –  Asaf Karagila Jan 25 at 1:38

It seems that all the proposed answers so far have been about why math might get wrong answers without rigor. While that's been shown to be a nontrivial risk, I would think non-mathematicians might be interested in how rigor gives math a unique position as a human activity.

The basic point would be that 18$^{th}$ century and older writers in all kinds of genres used to throw around the phrase "I shall prove..." about all sorts of notions that the mathematically inclined would never admit as a proof today. Yet at the time, there was absolutely nothing on the scene that worked better. Euler's results were and are astounding but, as they say, to derive Euler's results with Euler's methods one has to actually be Euler.

Since the gift of rigor to mathematics by Cauchy, Weierstrass, Cantor, Hilbert, Noether, and others, we've found ourselves in a field that seems completely unique among human activities in this: suppose the mathematical community disagrees about the validity of some new proposed proof of a mathematical proposition. It's an incredibly safe bet that, come back five years later, and the experts in the field will now be in agreement. That's something essentially nobody else has, though of course certain sciences have a closer or further approximation of it.

I should leave a caveat that none of this applies to opinions, about, say, what math is worthwhile-lots of people objected to set theory in 1900, Bourbaki in 1950, and category theory today. It's about specific proofs of well-defined facts that mathematicians don't have to agree to disagree.

-


text:
# Write the solutions or, if there is no solution, say the system is inconsistent.[ 1 0 -1 1 0 1 2 1 ]

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $$\begin{bmatrix}1 & 0 & -1 & | & 1 \\0 & 1 & 2 & | & 1 \end{bmatrix}$$

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

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Corben Pittman
There are no impossible equations, such as (0=1), the system is consistent,
A variable does not have a leading 1 in its corresponding column, so we take it asa parameter ... the system is consistent and has infinitely many solution.
Parameters: $$\displaystyle{x}{3}∈{R}$$
Interpreting row by row as equations.


text:
Simple Lie algebra

In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.

A direct sum of simple Lie algebras is called a semisimple Lie algebra.

A simple Lie group is a connected Lie group whose Lie algebra is simple.

Complex simple Lie algebras

A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$, ${\displaystyle {\mathfrak {so}}_{n}\mathbb {C} }$, ${\displaystyle {\mathfrak {sp}}_{2n}\mathbb {C} }$ (classical Lie algebras) or one of the five exceptional Lie algebras.[1]

To each finite-dimensional complex semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines depending on the angles between the simple roots and the arrows are put to indicate whether the roots are longer or shorter.[2] The Dynkin diagram of ${\displaystyle {\mathfrak {g}}}$ is connected if and only if ${\displaystyle {\mathfrak {g}}}$ is simple. All possible connected Dynkin diagrams are the following:[3]

where n is the number of the nodes (the simple roots). The correspondence of the diagrams and complex simple Lie algebras is as follows:[2]

(An) ${\displaystyle \quad {\mathfrak {sl}}_{n+1}\mathbb {C} }$
(Bn) ${\displaystyle \quad {\mathfrak {so}}_{2n+1}\mathbb {C} }$
(Cn) ${\displaystyle \quad {\mathfrak {sp}}_{2n}\mathbb {C} }$
(Dn) ${\displaystyle \quad {\mathfrak {so}}_{2n}\mathbb {C} }$
The rest, exceptional Lie algebras.

Real simple Lie algebras

If ${\displaystyle {\mathfrak {g}}_{0}}$ is a finite-dimensional real simple Lie algebra, its complexification is either (1) simple or (2) a product of a simple complex Lie algebra and its conjugate. For example, the complexification of ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$ thought of as a real Lie algebra is ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} \times {\overline {{\mathfrak {sl}}_{n}\mathbb {C} }}}$. Thus, a real simple Lie algebra can be classified by the classification of complex simple Lie algebras and some additional information. This can be done by Satake diagrams that generalize Dynkin diagrams. See also Table of Lie groups#Real Lie algebras for a partial list of real simple Lie algebras.

Notes

1. ^ Fulton & Harris 1991, Theorem 9.26.
2. ^ a b Fulton & Harris 1991, § 21.1.
3. ^ Fulton & Harris 1991, § 21.2.


text:
Topic

# Intensity (heat transfer)

About: Intensity (heat transfer) is a(n) research topic. Over the lifetime, 7684 publication(s) have been published within this topic receiving 85856 citation(s). The topic is also known as: intensity.

##### Papers
More

Journal Article
Abstract: An extension is made to Lighthill's general theory of aerodynamic sound, so as to incorporate the influence of solid boundaries upon the sound field. This influence is twofold, namely (i) reflexion and diffraction of the sound waves at the solid boundaries, and (ii) a resultant dipole field at the solid boundaries which are the limits of Lighthill's quadrupole distribution. It is shown that these effects are exactly equivalent to a distribution of dipoles, each representing the force with which unit area of solid boundary acts upon the fluid. A dimensional analysis shows that the intensity of the sound generated by the dipoles should at large distances x be of the general form I$\propto$ $\rho \_{0}$ U$\_{0}^{6}$a$\_{0}^{-3}$ L$^{2}$x$^{-2}$, where U$\_{0}$ is a typical velocity of the flow, L is a typical length of the body, a$\_{0}$ is the velocity of sound in fluid at rest and $\rho \_{0}$ is the density of the fluid at rest. Accordingly, these dipoles should be more efficient generators of sound than the quadrupoles of Lighthill's theory if the Mach number is small enough. It is shown that the fundamental frequency of the dipole sound is one half of the frequency of the quadrupole sound.

Topics: Speed of sound (52%),

1,628 Citations

Journal Article
S. H. Autler1, Charles H. Townes1Institutions (1)
15 Oct 1955-Physical Review
Abstract: A method is developed for calculating the effects of a strong oscillating field on two states of a quantum-mechanical system which are connected by a matrix element of the field. Explicit approximate solutions are obtained for a variety of special cases, and the results of numerical computations are given for others. The effect of an rf field on the $J=2\ensuremath{\rightarrow}1$ $l$-type doublet microwave absorption lines of OCS has been studied in particular both experimentally and theoretically. Each line was observed to split into two components when the frequency of the rf field was near 12.78 Mc or 38.28 Mc, which are the frequencies separating the $J=1$ and $J=2$ pairs of levels, respectively. By measuring the rf frequency, ${\ensuremath{ u}}_{0}$, at which the microwave lines are split into two equally intense components, one may determine the separation between the energy levels. The measured value of ${\ensuremath{ u}}_{0}$ depends upon the intensity of the rf field and the form of this dependence has been calculated and found to be in good agreement with the experimental results.

Topics: , Stark effect (52%), Field (physics) (51%) ...read more

1,232 Citations

Journal Article
Abstract: Experiments on the angular intensity distribution of x‐rays scattered by porous materials (hole structures) in the range of small angles are described. It is shown that the scattering can be characterized by an exponential correlation function in the case of a distribution of holes of random shape and size in solid; a theoretical derivation of the exponential function is given for this case. When the correlation function is an exponential, the rule holds that the reciprocal square root of the scattered intensity is a linear function of the square of the scattering angle. The specific surface of the material is determined by the slope of this straight line. Specific surfaces of a number of compositions are calculated from their experimental correlation functions and compared to surfaces based on adsorption measurements.

Topics: , Exponential function (57%),  ...read more

1,116 Citations

Journal Article
Howard R. Reiss1, Howard R. Reiss2Institutions (2)
01 Nov 1980-Physical Review A
Abstract: The approximation method introduced by Keldysh is revised and extended. The technique is applicable to the photodetachment by a plane-wave field of an electron bound by a short-range potential. The approximation is to neglect the effect of the binding potential as compared to the field effects on the final state of the detached electron. By choice of a different gauge than that used by Keldysh, the formalism becomes very simple and tractable. A general basis for the formalism is developed, and it is then applied to find transition probabilities for any order of interaction for both linearly and circularly polarized plane-wave fields. The low-intensity, first-order limit and the high-intensity, high-order limit yield the correct results. Two intensity parameters are identified. The fundamental one is $z=\frac{{e}^{2}{a}^{2}}{4m\ensuremath{\omega}}$, where $a$ is the magnitude of the vector potential (in radiation gauge) of the field of circular frequency $\ensuremath{\omega}$. The second parameter is ${z}_{1}=\frac{2z\ensuremath{\omega}}{{E}_{B}}$, where ${E}_{B}$ is binding energy, and it becomes important only in the asymptotic case. With the assumption that the field leaves the neutral atomic core relatively unaffected, the formalism is applied to the example of the negative hydrogen ion irradiated by circularly or linearly polarized 10.6-\ensuremath{\mu}m radiation. Photodetachment angular distributions and total transition probabilities are examined for explicit intensity effects. It is found that total transition probability $W$ is not sensitive to intensity since $\frac{d(logW)}{d}$ ($logz$) retains low-intensity straight-line behavior up to quite high values of $z$. An important intensity effect is the major significance of higher-than-lowest-order terms when $z$ is large, especially for circular polarization. A sensitive indicator of intensity is the ratio of photodetachment probabilities in circularly and linearly polarized fields, which increases sharply with intensity. An investigation of the convergence of perturbation expansions gives the upper limit $zl[\frac{{E}_{B}}{\ensuremath{\omega}}]\ensuremath{-}\frac{{E}_{B}}{\ensuremath{\omega}}$, where the square bracket means "smallest integer containing" the quantity in brackets. This limit is $zl0.59$ for ${\mathrm{H}}^{\ensuremath{-}}$ in 10.6-\ensuremath{\mu}m radiation. The failure of perturbation theory is not necessarily manifest in qualitative ways. For example, it is not apparent in total photoelectron yield as a function of intensity.

Topics: Order (ring theory) (64%),

1,047 Citations

Journal Article
Timothy R. Oke1Institutions (1)
Abstract: A simple hardware scale model is used to simulate nocturnal cooling rates for rural and urban environments under calm and cloudless conditions. Comparison with field observations gathered under similar conditions shows the model capable of reproducing many of the features of the temporal development of urban heat islands and the long-wave radiative exchange in urban canyons. The model is used to investigate the roles played by rural/urban differences in geometry and thermal admittance. The results of the experiments show that canyon geometry in the central portion of a city (as measured by the sky view factor) is a relevant variable in producing nocturnal urban heat islands due to its role in regulating long-wave radiative heat loss. It is also demonstrated that this measure is central to the relationship between city size and heat island intensity. The importance of canyon geometry as a feature of urban design is discussed. Thermal admittance differences can also produce realistic heat island features but their magnitude requires quantification in the field.

Topics: Urban heat island (59%), Canyon (53%),

939 Citations

##### Performance
###### Metrics
No. of papers in the topic in previous years
YearPapers
202215
2021329
2020375
2019429
2018418
2017362

###### Top Attributes

Show by:

Topic's top 5 most impactful authors

S. Sinkunas

8 papers, 6 citations

Viktor I. Terekhov

7 papers, 68 citations

Liang-Bi Wang

6 papers, 153 citations

Barry D. Ganapol

4 papers, 38 citations

Meisam Torab-Mostaedi

4 papers, 112 citations

##### Network Information
###### Related Topics (5)
Heat transfer

181.7K papers, 2.9M citations

78% related
Convective heat transfer

29.2K papers, 613.7K citations

77% related
Natural convection

29.5K papers, 608.7K citations

77% related
Heat flux

46.7K papers, 870.4K citations

76% related
Heat transfer coefficient

48.4K papers, 913.9K citations

76% related


text:
7:44 AM
@JonathanBeardsley it's not working here either

2 hours later…
9:43 AM
@HarryGindi Is this the state-of-the-art in the justification of the Appendix in Gaitsgory-Rozenblyum? Would it get close to filling all the gaps? I am writing a paper using their correspondence machinery (stating carefully that it is conditional on their results!) and I would be very happy if that saga was finally complete.

6 hours later…
3:43 PM
@HarryGindi I'd say mostly for the dualities available in $\Theta_2$-sets, but as far as the monoidal structure itself that was already available from Verity's work

1 hour later…
5:02 PM
@SimonPepinLehalleur As Edoardo said, a lot of this stuff was already developed for complicial sets, and the recent work of Gagna-Harpaz-Lanari more or less gives you a way to apply this in concert with Lurie's (un)straightening theorem by completing the equivalence between 2-trivial complicial sets and scaled simplicial sets
However, Yuki's paper is also excellent and has a bunch of interesting (∞,2)-categorical combinatorics. I personally prefer the Θ-type models to complicial sets, mainly because the representable objects are all fibrant and represent the obvious elementary operations of n-category theory. I was very excited to see this come out, but Gagna-Harpaz-Lanari does give you, at least right now, a better toolbox.

5:56 PM
@HarryGindi what's not working

6:09 PM
@JonathanBeardsley yeah it says I need an invitation.

Did you read the conversation following my initial posting of the link
Others had that problem but seemed to resolve it.

6:22 PM
@JonathanBeardsley Did they resolve it or give up?

@HarryGindi People seemed satisfied with the answer (I did not try it)

It was resolved.
The one that Rune posted. Here: twitter.com/_julesh_/status/1242141831057616896

Weird
ok, worked
Just wanted to check it out, don't know if it's for me.

There's not much \infty-category stuff happening there.
I've been learning a bit about algebraic theories and their relationship to operads.

6:25 PM
There's also some interesting discussions happening around cybernetics and semiotics.
Which paper?

Operator categories in this paper are more or less algebraic theories

I have not read it in detail, but I am pretty familiar with it.

Well, if you're looking into the relationship between algebraic theories and ∞-operads, this basically the one-stop-shop
I read it a long time ago, so I don't know exactly which algebraic theories correspond to ∞-operads, but iirc this gives a framework to change back and forth between them

I see. Well yeah I'm not really looking to create more work for myself in interpreting something. I'm more just having things explained to me by logicians, which is nice.

7:24 PM
@HarryGindi Great. So is someone who understands Verity+Gagna-Harpaz-Lanari well going to write down and publish somewhere "Thm: the results in [GR,Appendix.] are true"? That would be really a public service for the poor end-users like me.

1 hour later…
8:39 PM
@SimonPepinLehalleur It probably depends on what you need from GR. If you just need the Gray tensor product and its basic properties, then check out Yuki Maehara's thesis -- he develops the Gray tensor product in 2-quasicategories (i.e. Theta_2 - sets). I believe this is the first model where this has been done other than complicial.


text:
a particle moves in the x-y plane with constant acceleration. At time t=0s, the position vector for the particle is d=4.8mx+10my. The acceleration is given by the vector a=7.5m/s2x+5.8m/s2y. The velocity at time t=0 is v=7.8m/sx-2.6m/sy.
a) Find the magnitude of the velocity vector at t=3.4s.
b). What's the angle between the velocity vector and the x-axis at t=3.4s
c). What's the magnitude of the position vector at t=3.4s
d). what's the angle between the velocity vector and the x-axis at t=3.4s

can you give me formulas i'm supposed to use and why please?


text:
# How do you simplify sqrt(z^3)sqrt(z^7)?

Jul 12, 2016

We use the properties of powers to find $\sqrt{{z}^{3}} \cdot \sqrt{{z}^{7}} = {z}^{5}$

#### Explanation:

We need to write the powers of the factors in a single number so that we can add them. We start by noting that a square root is just a power of $\frac{1}{2}$, i.e.:

$\sqrt{{z}^{3}} \cdot \sqrt{{z}^{7}} = {\left({z}^{3}\right)}^{\frac{1}{2}} \cdot {\left({z}^{7}\right)}^{\frac{1}{2}}$

Now we use the property that raising a power to another power multiplies the exponents:

${\left({z}^{3}\right)}^{\frac{1}{2}} \cdot {\left({z}^{7}\right)}^{\frac{1}{2}} = {z}^{\frac{3}{2}} \cdot {z}^{\frac{7}{2}}$

Then we use the property that multiplying powers with the same base adds the exponents:

${z}^{\frac{3}{2}} \cdot {z}^{\frac{7}{2}} = {z}^{\frac{3}{2} + \frac{7}{2}} = {z}^{\frac{10}{2}} = {z}^{5}$


text:
# Working with QEDIT Assets¶

Nodes, wallets, and rules are essentially the infrastructure for issuing and transfering assets. The QEDIT API makes it simple to issue and transfer assets.

## Issuing an asset¶

Issuance of new assets are done by wallets that are permitted to do so by the issuance rules. Issuance of new assets can be done publicly or privately (if permitted). Public issuance disclose to other QEDIT nodes the issuer, the asset ID and the amount being issued. It does not disclose to whom the asset is issued. Such issuance is suitable in some cases, for example the issuance of stocks or bonds, when it is expected everyone to know that total amount in circulation. Private issuance disclosed to other QEDIT nodes the issuer and which issuance rule is being invoked. It does not disclose the asset ID, the amount being issued, and to whom the asset is issued. An issuance can include a memo which will viewable by the receiving party.

Request

 POST /node/issue_asset
{
"wallet_id": "source_wallet",
"authorization": "PrivacyIsAwesome",
"amount": 4,
"asset_id": "bank-a.loans.1",
"confidential": true,
"memo": ""
}

Response

{
"id": "70a88558-2b8b-4b63-a5b6-2c54b24377f5"
}

## Transferring an asset¶

An asset is transferred from a wallet to an address of another wallet. All transfers are private and do not reveal who is the sender, who is the receiver, what asset they are transferring, and what amount. A transfer can include a memo which will be visible to the receiving party. The sender can specify that the transfer requires confirming by the receiving party. In such a case, the transfer will not be final until the receipient confirms the transfer. Prior to the transfer being final, the sender can cancel the transfer by cancelling the transfer task.

Request

POST /wallet/transfer_asset
{
"wallet_id": "source_wallet",
"authorization": "PrivacyIsAwesome",

{


text:
# What could be said about centralizers of normal subgroups if $G$ contains a simple, non-abelian maximal subgroup

Let $G$ be a finite group containing a maximal simple and non-abelian group, is it true that the centraliser of each normal subgroup is either trivial or a minimal normal subgroups?

EDIT: Maybe to give some context. If $G$ contains a maximal subgroup which is core-free, then this property holds for the centraliser of each normal subgroup. For a proof, let $G$ be a group and $L$ be a maximal core-free group, meaning that the intersection of all conjugates of $L$ is trivial, in particular $L$ contains no normal subgroup of $G$. Now let $N \unlhd G$ be non-trivial, as $L \cap N = 1$ we have $G = LN$. Also $C_G(N) \unlhd G$ and so $C_G(N) \cap L \unlhd L$ and therefore $L \le N_G(C_G(N)\cap L)$. As $N \le C_G(C_G(N))$ we have $$N \le C_G(C_G(N) \cap L) \le N_G(C_G(N)\cap L)$$ which implies $G = LN \le N_G(C_G(N)\cap L)$, i.e. $C_G(N) \cap L) \unlhd G$, and as $L$ is core-free it follows that $C_G(N) \cap L = 1$. Now let $M \le C_G(N)$ be a non-trivial normal subgroup of $G$, then again $M \cap L = 1$ and so $G = LM$. By Dedekind's Identity we have with $M \le C_G(N) \le LM$ $$C_G(N) = M(C_G(N) \cap L) = M$$ hence either $C_G(N) = 1$ or it is a minimal normal subgroup of $G$.

Now I am conjecturing that if $L$ is simple and non-abelian, a similar conclusion could be drawn, but I am stuck in the proof.

Take $G=A_5\times Z_2$ then $C_G(1\times Z_2)=G$


text:
This function is used to quickly create a workflowsets object.

## Usage

ts_wfs_auto_arima(.model_type = "auto_arima", .recipe_list)

## Arguments

.model_type

This is where you will set your engine. It uses modeltime::arima_reg() under the hood and can take one of the following:

• "auto_arima"

.recipe_list

You must supply a list of recipes. list(rec_1, rec_2, ...)

## Value

Returns a workflowsets object.

## Details

This function expects to take in the recipes that you want to use in the modeling process. This is an automated workflow process. There are sensible defaults set for the model specification, but if you choose you can set them yourself if you have a good understanding of what they should be. The mode is set to "regression".

This only uses the option set_engine("auto_arima") and therefore the .model_type is not needed. The parameter is kept because it is possible in the future that this could change, and it keeps with the framework of how other functions are written.

modeltime::arima_reg() arima_reg() is a way to generate a specification of an ARIMA model before fitting and allows the model to be created using different packages. Currently the only package is forecast.

https://workflowsets.tidymodels.org/

Other Auto Workflowsets: ts_wfs_arima_boost(), ts_wfs_ets_reg(), ts_wfs_lin_reg(), ts_wfs_mars(), ts_wfs_nnetar_reg(), ts_wfs_prophet_reg(), ts_wfs_svm_poly(), ts_wfs_svm_rbf(), ts_wfs_xgboost()

## Author

Steven P. Sanderson II, MPH

## Examples

suppressPackageStartupMessages(library(modeltime))
suppressPackageStartupMessages(library(timetk))
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(rsample))

data <- AirPassengers %>%
ts_to_tbl() %>%
select(-index)

splits <- time_series_split(
data
, date_col
, assess = 12
, skip = 3
, cumulative = TRUE
)

rec_objs <- ts_auto_recipe(
.data = training(splits)
, .date_col = date_col
, .pred_col = value
)

wf_sets <- ts_wfs_auto_arima("auto_arima", rec_objs)
wf_sets
#> # A workflow set/tibble: 4 × 4
#>   wflow_id                       info             option    result
#>   <chr>                          <list>           <list>    <list>
#> 1 rec_base_arima_reg             <tibble [1 × 4]> <opts[0]> <list [0]>
#> 2 rec_date_arima_reg             <tibble [1 × 4]> <opts[0]> <list [0]>
#> 3 rec_date_fourier_arima_reg     <tibble [1 × 4]> <opts[0]> <list [0]>
#> 4 rec_date_fourier_nzv_arima_reg <tibble [1 × 4]> <opts[0]> <list [0]>


text:
# Fermat primality test

The Fermat primality test is a probabilistic test to determine whether a number is a probable prime.

## Concept

Fermat's little theorem states that if p is prime and ${\displaystyle 1, then

${\displaystyle a^{p-1}\equiv 1{\pmod {p}}.}$

If we want to test whether p is prime, then we can pick random a's in the interval and see whether the equality holds. If the equality does not hold for a value of a, then p is composite. If the equality does hold for many values of a, then we can say that p is probably prime.

It might be in our tests that we do not pick any value for a such that the equality fails. Any a such that

${\displaystyle a^{n-1}\equiv 1{\pmod {n}}}$

when n is composite is known as a Fermat liar. Vice versa, in this case n is called Fermat pseudoprime to base a.

If we do pick an a such that

${\displaystyle a^{n-1}\not \equiv 1{\pmod {n}}}$

then a is known as a Fermat witness for the compositeness of n.

## Example

Suppose we wish to determine whether n = 221 is prime. Randomly pick 1 < a < 221, say a = 38. We check the above equality and find that it holds:

${\displaystyle a^{n-1}=38^{220}\equiv 1{\pmod {221}}.}$

Either 221 is prime, or 38 is a Fermat liar, so we take another a, say 24:

${\displaystyle a^{n-1}=24^{220}\equiv 81\not \equiv 1{\pmod {221}}.}$

So 221 is composite and 38 was indeed a Fermat liar. Furthermore, 24 is a Fermat witness for the compositeness of 221.

## Algorithm and running time

The algorithm can be written as follows:

Inputs: n: a value to test for primality, n>3; k: a parameter that determines the number of times to test for primality
Output: composite if n is composite, otherwise probably prime
Repeat k times:
Pick a randomly in the range [2, n − 2]
If ${\displaystyle a^{n-1}\not \equiv 1{\pmod {n}}}$, then return composite
If composite is never returned: return probably prime

The a values 1 and n-1 are not used as the equality holds for all n and all odd n respectively, hence testing them adds no value.

Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k × log2n × log log n × log log log n), where k is the number of times we test a random a, and n is the value we want to test for primality.

## Flaw

There are infinitely many values of ${\displaystyle n}$ (known as Carmichael numbers) for which all values of ${\displaystyle a}$ for which ${\displaystyle gcd(a,n)=1}$ are Fermat liars. For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers,[1] there are enough of them that Fermat's primality test is not often used in the above form. Instead, other more powerful extensions of the Fermat test, such as Baillie-PSW, Miller-Rabin, and Solovay-Strassen are more commonly used.

In general, if ${\displaystyle n}$ is a composite number that is not a Carmichael number, then at least half of all

${\displaystyle a\in (\mathbb {Z} /n\mathbb {Z} )^{*}}$

are Fermat witnesses. For proof of this, let ${\displaystyle a}$ be a Fermat witness and ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, ..., ${\displaystyle a_{s}}$ be Fermat liars. Then

${\displaystyle (a\cdot a_{i})^{n-1}\equiv a^{n-1}\cdot a_{i}^{n-1}\equiv a^{n-1}\not \equiv 1{\pmod {n}}}$

and so all ${\displaystyle a\times a_{i}}$ for ${\displaystyle i=1,2,...,s}$ are Fermat witnesses.

## Applications

As mentioned above, most applications use a Miller-Rabin or Baillie-PSW test for primality. Sometimes a Fermat test (along with some trial division by small primes) is performed first to improve performance. GMP since version 3.0 uses a base-210 Fermat test after trial division and before running Miller-Rabin tests. Libgcrypt uses a similar process with base 2 for the Fermat test, but OpenSSL does not.

In practice with most big number libraries such as GMP, the Fermat test is not noticeably faster than a Miller-Rabin test, and can be slower for many inputs.[2]

As an exception, OpenPFGW uses only the Fermat test for probable prime testing. The program is typically used with multi-thousand digit inputs with a goal of maximum speed with very large inputs. Another well known program that relies only on the Fermat test is PGP where it is only used for testing of self-generated large random values (an open source counterpart, GNU Privacy Guard, uses a Fermat pretest followed by Miller-Rabin tests).

## References

1. ^ Erdös' upper bound for the number of Carmichael numbers is lower than the prime number function n/log(n)
2. ^ Joe Hurd (2003), Verification of the Miller-Rabin Probabilistic Primality Test, p. 2


text:
# Trigo is not such easy!!

Algebra Level 5

If $$\lfloor 2 \cos x \rfloor + \lfloor \sin x \rfloor =-3$$ for $$x \in [0, 2 \pi]$$, then the range of the function $$f(x)= \sin x+\sqrt 3 \cos x$$ can be written as $$[a, b]$$. Find $$a^2 + b^2$$.

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

×


text:
## Puzzles: solution to problem 2

How many squares (of any size) are there in an 8 by 8 chessboard? What about an 8000 by 8000 chessboard?

Unfortunately this problem has not been formulated correctly. Oh well. If you try to solve it as stated, the answer is obvious: uncountable infinity, even for a 1 by 1 board. Clearly what should have been said is that we are looking for square shapes that cover completely the little black and white squares of the chess board. In other words, if you place the board on Cartesian coordinates in such a way that if the left bottom corner coincides with the origin of your coordinates and space is scaled so that the vertices of the little squares of the board fall on points with integer coordinates, the question is how many squares of integer side length there are with vertices having integer coordinates and with sides parallel to the axes.

Quite a mouthful all that.

Once we know what it is we are looking for, let us see. If the board is a 1 by 1 one, there is only one square, of side 1. If the board is $2 \times 2$, there are 5 squares, one of side 2 and 4 of side 1. So the MSfMC is 3, where I have just invented the concept of “minimal size for maximal complexity”. In a $3\times 3$ board there clearly are 9 squares of side length 1 and 1 square of side length 3. But how many are there of side length 2? To get such a square, you need to count 2 on the x-axis, starting either from 0 or from 1: so there are two ways to lay the projection of your intended square onto the x-axis. Similarly, there are 2 ways to project an $2 \times 2$ square onto the y-axis. Hence there are 4 $2\times2$ squares: their left lower vertices are at $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. So all together there are $9+4+1=14$ squares. But this argument works for chess boards of any size and squares of any size. For example, think about projecting a $6 \times 6$ square contained in an $8 \times 8$ board onto the x-axis. It can fall on one of the intervals $[0,6]$, $[1,7]$, $[2,8]$. So there are 9 such squares, $(8-6+1)^2$. Continuing this train of thought, we see that in an $8 \times 8$ board there are

$(8-1+1)^2+(8-2+1)^2+ \cdots + (8-8+1)^2 =1^2+2^2+\cdots + 8^2=204$

squares. Once the idea is clear, it is easy to prove that an $n \times n$ board contains $\displaystyle\sum_{k=1}^n k^2$ squares. So if you want to work out the answer for $n=800$, you should either prove (by induction) that

$\displaystyle\sum_{k=1}^n k^2 = \displaystyle\frac{1}{6}[2n^3+3n^2+n]$

and then plug in $n=800$, or let MAPLE do the work for you:

sum(k^2,k=1..800);

170986800.

Note that the problem can be easily made harder (!): How many squares are there in an $8 \times 8$ board having vertices at points with integer coordinates and having integer length side (i.e. removing the conditions that the little squares are to be covered completely and that the sides have to be parallel to the axes)? For an $8 \times 8$ board one can count all such squares, but for an $800 \times 800$ one the problem looks challenging to say the least.

(MG)


text:
# AvgPool3d¶

Applies a 3D average pooling over an input signal composed of several input planes.

In the simplest case, the output value of the layer with input size $(N, C, D, H, W)$, output $(N, C, D_{out}, H_{out}, W_{out})$ and kernel_size $(kD, kH, kW)$ can be precisely described as:

\begin{aligned} \text{out}(N_i, C_j, d, h, w) ={} & \sum_{k=0}^{kD-1} \sum_{m=0}^{kH-1} \sum_{n=0}^{kW-1} \\ & \frac{\text{input}(N_i, C_j, \text{stride}[0] \times d + k, \text{stride}[1] \times h + m, \text{stride}[2] \times w + n)} {kD \times kH \times kW} \end{aligned}

If padding is non-zero, then the input is implicitly zero-padded on all three sides for padding number of points.

Note

When ceil_mode=True, sliding windows are allowed to go off-bounds if they start within the left padding or the input. Sliding windows that would start in the right padded region are ignored.

The parameters kernel_size, stride can either be:

• a single int – in which case the same value is used for the depth, height and width dimension

• a tuple of three ints – in which case, the first int is used for the depth dimension, the second int for the height dimension and the third int for the width dimension

Parameters:
• kernel_size (Union[int, Tuple[int, int, int]]) – the size of the window

• stride (Union[int, Tuple[int, int, int]]) – the stride of the window. Default value is kernel_size

• padding (Union[int, Tuple[int, int, int]]) – implicit zero padding to be added on all three sides

• ceil_mode (bool) – when True, will use ceil instead of floor to compute the output shape

• count_include_pad (bool) – when True, will include the zero-padding in the averaging calculation

• divisor_override (Optional[int]) – if specified, it will be used as divisor, otherwise kernel_size will be used

Shape:
• Input: $(N, C, D_{in}, H_{in}, W_{in})$ or $(C, D_{in}, H_{in}, W_{in})$.

• Output: $(N, C, D_{out}, H_{out}, W_{out})$ or $(C, D_{out}, H_{out}, W_{out})$, where

$D_{out} = \left\lfloor\frac{D_{in} + 2 \times \text{padding}[0] - \text{kernel\_size}[0]}{\text{stride}[0]} + 1\right\rfloor$
$H_{out} = \left\lfloor\frac{H_{in} + 2 \times \text{padding}[1] - \text{kernel\_size}[1]}{\text{stride}[1]} + 1\right\rfloor$
$W_{out} = \left\lfloor\frac{W_{in} + 2 \times \text{padding}[2] - \text{kernel\_size}[2]}{\text{stride}[2]} + 1\right\rfloor$

Examples:

>>> # pool of square window of size=3, stride=2
>>> m = nn.AvgPool3d(3, stride=2)
>>> # pool of non-square window
>>> m = nn.AvgPool3d((3, 2, 2), stride=(2, 1, 2))
>>> input = torch.randn(20, 16, 50,44, 31)
>>> output = m(input)