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Optimal condition for non-simultaneous blow-up in a reaction-diffusion system
April, 2004 Optimal condition for non-simultaneous blow-up in a reaction-diffusion system
Philippe SOUPLET, Slim TAYACHI
We study the positive blowing-up solutions of the semilinear parabolic system:
{u}_{t}-\Delta u={v}^{p}+{u}^{r},
{v}_{t}-\Delta v={u}^{q}+{v}^{s}
t∊\left(0,T\right),
x∊{R}^{N}
p,
q,
r,
s>1
r>q+1
s>p+1
then one component of a blowing-up solution may stay bounded until the blow-up time, while if
r<q+1
s<p+1
this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters
p,
q,
r
s
, solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.
Philippe SOUPLET. Slim TAYACHI. "Optimal condition for non-simultaneous blow-up in a reaction-diffusion system." J. Math. Soc. Japan 56 (2) 571 - 584, April, 2004. https://doi.org/10.2969/jmsj/1191418646
Keywords: blow-up rate , nonsimultaneous blow-up , reaction-diffusion systems , semilinear parabolic systems , simultaneous
Philippe SOUPLET, Slim TAYACHI "Optimal condition for non-simultaneous blow-up in a reaction-diffusion system," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 56(2), 571-584, (April, 2004)
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LHPDE : ReducedForm
compute reduced form of differential expressions modulo a LHPDE
ReducedForm(expr, obj)
a list or set of differential expressions
a rif-reduced LHPDE object (see IsRifReduced)
The ReducedForm method reduces a list (or set) of differential expressions modulo a rif-reduced LHPDE object. It returns the reduced form as a list (or set) of differential expressions.
Essentially the method substitutes the LHPDE equations into the differential expressions expr, until no more substitutions can be done.
To perform the reduction, the method ultimately calls a version of the Maple command dsubs.
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):
\mathrm{Typesetting}:-\mathrm{Suppress}\left([\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)]\right)
\mathrm{E2}≔\mathrm{LHPDE}\left([\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0],\mathrm{indep}=[x,y],\mathrm{dep}=[\mathrm{\xi },\mathrm{\eta }]\right)
\textcolor[rgb]{0,0,1}{\mathrm{E2}}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
\mathrm{expr}≔[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x,x\right),\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)+\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)]
\textcolor[rgb]{0,0,1}{\mathrm{expr}}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}]
\mathrm{ReducedForm}\left(\mathrm{expr},\mathrm{E2}\right)
[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
\mathrm{ReducedForm}\left(\mathrm{convert}\left(\mathrm{expr},'\mathrm{set}'\right),\mathrm{E2}\right)
{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}}
The ReducedForm command was introduced in Maple 2020.
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(→Spectral radiance/reflectance)
* Transforming a Landsat ETM+ image from DN values to radiance values
* Conducting an atmospheric correction of a Landsat image using the {{cmd|i.atcorr}} module in GRASS GIS
* Conducting an atmospheric correction of a Landsat image using the {{cmd|i.atcorr}} module in GRASS GIS and convert radiance to reflectance values
# Converting '''Digital Numbers''' (DN) to '''Top-of-Atmosphere Radiances''' (ToAR)
#* for Landsat imagery, apply the equation below or use the {{cmd|i.landsat.toar|version=65}} module
#* DNs -- which are actually the pixel values -- are the result of the quantified amount of energy observed and measured at the sensor
#* the {{cmd|i.landsat.toar|version=65}} module supports all Landsat sensors icluding MSS, TM and ETM+
#* for '''Landsat''' imagery, apply the equation below or use the {{cmd|i.landsat.toar}} module
#* the {{cmd|i.landsat.toar}} module supports all Landsat sensors including MSS, TM and ETM+
#* for '''WorldView''' imagery, apply this equation with {{cmd|r.mapcalc}}:<br> <tt>L = Gain * DN * (abscalfactor/effective bandwidth) + Offset</tt><br> with the band specific "abscalfactor" and "effectiveBandwidth" being delivered with the imagery in the metadata file and the Gain and Offset are the absolute radiometric calibration band-dependent adjustment factors that are given in e.g. [https://dg-cms-uploads-production.s3.amazonaws.com/uploads/document/file/209/ABSRADCAL_FLEET_2016v0_Rel20170606.pdf this DigitalGlobe document]. Note that these are not necessarily stagnant values and they are revisited annually.
# Converting '''ToA Radiances''' to '''Top-of-Canopy Reflectances''' (ToCR) → use the {{cmd|i.atcorr}} module
#* the conversion to ''Reflectances'' attempts to assess for and remove atmospheric effects, a process known as ''atmospheric correction''
'''Remember''' to check if water areas are represented by values <code>>0</code>, since reflectance is always positive. If you encounter negative reflectance values, you have a "cornichon" (pickle in English ;-0). This means that the transformation equations used, do not correspond to the image processed.
<nowiki>+------------------------------------------------------------------------------+
| Digital Numbers |
| +-----v-----+ |
| | i.*.toar | ---> Reflectance |
| +-----+-----+ (uncorrected)~~~(DOS methods)--+ |
| (-r flag) (-r flag) +--> "Corrected" |
| | | | Reflectance |
| v +-----v----+ | |
| Radiance ------> i.atcorr +-------------------+ |
+------------------------------------------------------------------------------+</nowiki>
* GRASS 6.4.0 or higher
* North Carolina sample dataset (location): http://grass.osgeo.org/download/data.php
* North Carolina sample dataset (location): https://grass.osgeo.org/download/sample-data/
== Calculating radiance values ==
The NC data set mapset contains, amongst others, a Landsat ETM+ image of 24th of May 2002. Every pixel of this image contains a DN or grey value. In order to be able to make calculations with satellite imagery, or compare values amongst different sensors, these values have to be converted to radiances or reflectances. The formulas to do this conversion are described here for Landsat images (or use [[GRASS_AddOns#i.landsat.toar|i.landsat.toar]]).
The NC data set mapset contains, amongst others, a Landsat ETM+ image of 24th of May 2002. Every pixel of this image contains a DN or grey value. In order to be able to make calculations with satellite imagery, or compare values amongst different sensors, these values have to be converted to radiances or reflectances. The formulas to do this conversion are described here for Landsat images (or use {{cmd|i.landsat.toar}}).
<math>L\lambda = \frac{L_{MAX}λ - L_{MIN}λ}{QCAL_{MAX} - QCAL_{MIN}} * (QCAL - QCAL_{MIN}) + L_{MIN}\lambda</math>
<ul>where:
{{cmd|r.univar}} <source lang="bash" enclose=none>elev_int</source>
== Estimating the overpass time from the sun position ==
== Estimating the overpass time ==
This value is needed for the control file. For details, refer to Wikipedia's entry on [http://en.wikipedia.org/wiki/Decimal_time#Scientific_decimal_time Scientific decimal time].
In case this parameter is not reported in the metadata file, we have two ways to obtain it.
=== From sun position ===
The satellite overpass time can be estimated rather precisely from the sun position reported in metadata using {{cmd|r.sunmask}}.
The [ftp://ftp.glcf.umiacs.umd.edu/glcf/Landsat/WRS2/p016/r035/p016r035_7x20020524.ETM-EarthSat-Orthorectified/p016r035_7x20020524.met metadata file] for the following example contains: <source lang="bash" enclose="none">SUN_AZIMUTH = 120.8810347, SUN_ELEVATION = 64.7730999</source>.
The [http://www.grassbook.org/wp-content/uploads/ncexternal/landsat/2002/p016r035_7x20020524.met.gz metadata file] for the following example contains: <source lang="bash" enclose="none">SUN_AZIMUTH = 120.8810347, SUN_ELEVATION = 64.7730999</source>.
The resulting overpass local time <code>10:42:07</code> corresponds to <code>15:42</code> in GMT which corresponds to <code>15.70</code> in decimal GMT hours (decimal minutes: <math>42 * 100 / 60</math>).
=== From NASA web tool ===
The [http://cloudsgate2.larc.nasa.gov/cgi-bin/predict/predict.cgi NASA LaRC Satellite Overpass Predictor] can compute satellite overpass time from d''ate of acquisition'' and ''scene center coordinates''. The [http://www.grassbook.org/wp-content/uploads/ncexternal/landsat/2002/p016r035_7x20020524.met.gz metadata file] for the following example contains: <source lang="bash" enclose="none">ACQUISITION_DATE = 2002-05-24, SCENE_CENTER_LAT = +36.0512847, SCENE_CENTER_LON = -79.3280820</source>.
# Select the World zone in the bottom-right side of the page and click <code>Go >></code>
# Insert Lat/Long coordinates (decimal degrees, without ''plus'' sign but with ''minus'' sign)
# Select the proper satellite
# Insert the date of acquisition
# ''Optional:'' select "Day" or "Night" to restrict computation to (local) day/night overpass of the satellite
''Example:'' Calculation for NC Landsat scenes: see [http://www.grassbook.org/wp-content/uploads/ncexternal/landsat/landsat_overpass_time_list_NC.txt here]
--- For LANDSAT only ---
USGS provide the [http://landsat.usgs.gov/consumer.php Landsat Bulk Metadata Service] web page where is possible to extract metadata (included overpass time) for all the Landsat missions.
This radiance image can be used for the atmospheric correction with the '''6S algorithm'''. The algorithm will transform the top-of-the-atmosphere radiance values to top-of-canopy reflectance values using predefined information on the aerosol content and atmospheric composition of the time the image was taken. What follows describes the method to use this algorithm in GRASS GIS. Again, only the calculations for band 1 are shown, and the numbers to be changed for the other bands are indicated in red. The 'icnd_lsat1.txt' control file consists of the following parameters, and is written with a text editor:
Radiance values can be converted to reflectance values in undergoing an atmospheric correction by applying the <math>6S</math> algorithm available in {{cmd|i.atcorr}}. The algorithm will transform the Top-of-Atmosphere radiance values to Top-of-Canopy reflectance values using predefined information on the aerosol content and atmospheric composition of the time the image was taken. What follows describes the method to use this algorithm in GRASS GIS. Again, only calculations for band 1 are demonstrated here, and the number(s) to be changed for indicating other spectral bands, are <span style="color:#FF0000">red</span>. The <code>icnd_lsat1.txt</code> control file consists of the following parameters, and is written with a text editor:
8 # indicates that it is an ETM+ image
2 # the midlatitude summer atmospheric model
{{cmd|i.atcorr}} <source lang="bash" enclose=none>-a -o iimg=lsat7_2002_10_rad ialt=elev_int icnd=icdn_lsat1.txt oimg=lsat7_2002_10_atcorr</source>
or, using GRASS 7.x:
{{cmd|i.atcorr}} <source lang="bash" enclose=none>-a input=lsat7_2002_10_rad elevation=elev_int parameters=icdn_lsat1.txt output=lsat7_2002_10_atcorr</source>
* <tt>'''-a'''</tt> refers to a Landsat image taken after July 2000
* <tt>'''-o'''</tt> activates the cache acceleration
* <tt>'''-o'''</tt> activates the cache acceleration (GRASS 6.x only)
* <tt>'''iimg'''</tt> is the image to be corrected
* <tt>'''iimg/input'''</tt> is the image to be corrected
* <tt>'''ialt'''</tt> is the altitude map which overrides the initialization value of 110 meters
* <tt>'''ialt/elevation'''</tt> is the altitude map which overrides the initialization value of 110 meters
* <tt>'''icnd'''</tt> is the path to the icnd.txt file
* <tt>'''icnd/parameters'''</tt> is the path to the icnd.txt file
* <tt>'''oimg'''</tt> is the name of the output image
* <tt>'''oimg/output'''</tt> is the name of the output image
== Sources for aerosol optical depth (AOD) estimations ==
* [http://aeronet.gsfc.nasa.gov AERONET] - provides globally distributed observations of spectral aerosol optical depth (AOD), inversion products, and precipitable water in diverse aerosol regimes.
* [http://www.cesbio.ups-tlse.fr/multitemp/?p=1710 How to estimate Aerosol Optical Thickness]
* Cross-check with [http://atmcorr.gsfc.nasa.gov Landsat Atmospheric Correction Parameter Calculator]
=== 6S algorithm ===
* [http://6s.ltdri.org 6S Web site]
* [http://modis-sr.ltdri.org/ Land Surface Reflectance Science Computing Facility website - 6S]
=== Spectral radiance/reflectance ===
* [http://landsat.usgs.gov/how_is_radiance_calculated.php How is radiance calculated?], question on USGS' [http://landsat.usgs.gov/tools_faq.php Landsat Missions - Frequently Asked Questions]
* (1) http://landsathandbook.gsfc.nasa.gov/handbook/handbook_htmls/chapter11/chapter11.html
* WorldView: https://dg-cms-uploads-production.s3.amazonaws.com/uploads/document/file/209/ABSRADCAL_FLEET_2016v0_Rel20170606.pdf
=== Whitepapers ===
=== Related on-line tools ===
* [http://cloudsgate2.larc.nasa.gov/cgi-bin/predict/predict.cgi NASA LaRC Satellite Overpass Predictor]
* [http://landsat.usgs.gov/consumer.php Landsat Bulk Metadata Service]
* NASA's [http://atmcorr.gsfc.nasa.gov/ Atmospheric Correction Parameter Calculator]
=== How to add new sensors to i.atcorr ===
How to add new sensors to {{cmd|i.atcorr}}:
* see http://trac.osgeo.org/grass/browser/grass/trunk/imagery/i.atcorr/README
* Terrain correction: {{cmd|i.topo.corr}}
4 Estimating the overpass time
4.1 From sun position
4.2 From NASA web tool
the i.landsat.toar module supports all Landsat sensors including MSS, TM and ETM+
for WorldView imagery, apply this equation with r.mapcalc:
L = Gain * DN * (abscalfactor/effective bandwidth) + Offset
with the band specific "abscalfactor" and "effectiveBandwidth" being delivered with the imagery in the metadata file and the Gain and Offset are the absolute radiometric calibration band-dependent adjustment factors that are given in e.g. this DigitalGlobe document. Note that these are not necessarily stagnant values and they are revisited annually.
North Carolina sample dataset (location): https://grass.osgeo.org/download/sample-data/
{\displaystyle QCAL}
{\displaystyle DN}
{\displaystyle QCALMIN}
{\displaystyle QCALMAX}
{\displaystyle QCALMINM}
{\displaystyle DN}
{\displaystyle QCALMAX}
{\displaystyle DN=255}
{\displaystyle DN}
{\displaystyle DN}
{\displaystyle QCALMIN}
{\displaystyle DN}
{\displaystyle QCALMAX}
{\displaystyle QCAL}
{\displaystyle DN}
Estimating the overpass time
This value is needed for the control file. For details, refer to Wikipedia's entry on Scientific decimal time. In case this parameter is not reported in the metadata file, we have two ways to obtain it.
From sun position
{\displaystyle 42*100/60}
From NASA web tool
The NASA LaRC Satellite Overpass Predictor can compute satellite overpass time from date of acquisition and scene center coordinates. The metadata file for the following example contains: ACQUISITION_DATE = 2002-05-24, SCENE_CENTER_LAT = +36.0512847, SCENE_CENTER_LON = -79.3280820.
Select the World zone in the bottom-right side of the page and click Go >>
Insert Lat/Long coordinates (decimal degrees, without plus sign but with minus sign)
Select the proper satellite
Insert the date of acquisition
Optional: select "Day" or "Night" to restrict computation to (local) day/night overpass of the satellite
Example: Calculation for NC Landsat scenes: see here
USGS provide the Landsat Bulk Metadata Service web page where is possible to extract metadata (included overpass time) for all the Landsat missions.
{\displaystyle 6S}
algorithm available in i.atcorr. The algorithm will transform the Top-of-Atmosphere radiance values to Top-of-Canopy reflectance values using predefined information on the aerosol content and atmospheric composition of the time the image was taken. What follows describes the method to use this algorithm in GRASS GIS. Again, only calculations for band 1 are demonstrated here, and the number(s) to be changed for indicating other spectral bands, are red. The icnd_lsat1.txt control file consists of the following parameters, and is written with a text editor:
i.atcorr -a input=lsat7_2002_10_rad elevation=elev_int parameters=icdn_lsat1.txt output=lsat7_2002_10_atcorr
-o activates the cache acceleration (GRASS 6.x only)
iimg/input is the image to be corrected
ialt/elevation is the altitude map which overrides the initialization value of 110 meters
icnd/parameters is the path to the icnd.txt file
oimg/output is the name of the output image
How to estimate Aerosol Optical Thickness
Cross-check with Landsat Atmospheric Correction Parameter Calculator
WorldView: https://dg-cms-uploads-production.s3.amazonaws.com/uploads/document/file/209/ABSRADCAL_FLEET_2016v0_Rel20170606.pdf
Landsat Bulk Metadata Service
Terrain correction: i.topo.corr
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Anti-unification (computer science) - Wikipedia
Anti-unification is the process of constructing a generalization common to two given symbolic expressions. As in unification, several frameworks are distinguished depending on which expressions (also called terms) are allowed, and which expressions are considered equal. If variables representing functions are allowed in an expression, the process is called "higher-order anti-unification", otherwise "first-order anti-unification". If the generalization is required to have an instance literally equal to each input expression, the process is called "syntactical anti-unification", otherwise "E-anti-unification", or "anti-unification modulo theory".
An anti-unification algorithm should compute for given expressions a complete, and minimal generalization set, that is, a set covering all generalizations, and containing no redundant members, respectively. Depending on the framework, a complete and minimal generalization set may have one, finitely many, or possibly infinitely many members, or may not exist at all;[note 1] it cannot be empty, since a trivial generalization exists in any case. For first-order syntactical anti-unification, Gordon Plotkin[1][2] gave an algorithm that computes a complete and minimal singleton generalization set containing the so-called "least general generalization" (lgg).
1.5 Anti-unification problem, generalization set
2 First-order syntactical anti-unification
3 First-order anti-unification modulo theory
3.1 Equational theories
3.2 First-order sorted anti-unification
3.3 Nominal anti-unification
4 Higher-order anti-unification
{\displaystyle \equiv }
{\displaystyle T}
{\displaystyle t\equiv u}
{\displaystyle t}
{\displaystyle u}
{\displaystyle \equiv }
{\displaystyle \oplus }
{\displaystyle t\equiv u}
{\displaystyle u}
{\displaystyle t}
{\displaystyle \oplus }
{\displaystyle V}
{\displaystyle C}
{\displaystyle F_{n}}
{\displaystyle n}
-ary function symbols, also called operator symbols, for each natural number
{\displaystyle n\geq 1}
{\displaystyle T}
from every n terms t1,...,tn, and every n-ary function symbol f ∈ Fn, a larger term
{\displaystyle f(t_{1},\ldots ,t_{n})}
For example, if x ∈ V is a variable symbol, 1 ∈ C is a constant symbol, and add ∈ F2 is a binary function symbol, then x ∈ T, 1 ∈ T, and (hence) add(x,1) ∈ T by the first, second, and third term building rule, respectively. The latter term is usually written as x+1, using Infix notation and the more common operator symbol + for convenience.
{\displaystyle \sigma :V\longrightarrow T}
{\displaystyle \{x_{1}\mapsto t_{1},\ldots ,x_{k}\mapsto t_{k}\}}
{\displaystyle x_{i}}
{\displaystyle t_{i}}
{\displaystyle i=1,\ldots ,k}
, and every other variable to itself. Applying that substitution to a term t is written in postfix notation as
{\displaystyle t\{x_{1}\mapsto t_{1},\ldots ,x_{k}\mapsto t_{k}\}}
{\displaystyle x_{i}}
in the term t by
{\displaystyle t_{i}}
. The result tσ of applying a substitution σ to a term t is called an instance of that term t. As a first-order example, applying the substitution
{\displaystyle \{x\mapsto h(a,y),z\mapsto b\}}
f( x , a, g( z ), y) yields
f( h(a,y) , a, g( b ), y) .
{\displaystyle t}
{\displaystyle u}
{\displaystyle t\sigma \equiv u}
{\displaystyle \sigma }
{\displaystyle t}
{\displaystyle u}
{\displaystyle u}
{\displaystyle t}
{\displaystyle x\oplus a}
{\displaystyle a\oplus b}
{\displaystyle \oplus }
is commutative, since then
{\displaystyle (x\oplus a)\{x\mapsto b\}=b\oplus a\equiv a\oplus b}
{\displaystyle \equiv }
is literal (syntactic) identity of terms, a term may be both more general and more special than another one only if both terms differ just in their variable names, not in their syntactic structure; such terms are called variants, or renamings of each other. For example,
{\displaystyle f(x_{1},a,g(z_{1}),y_{1})}
{\displaystyle f(x_{2},a,g(z_{2}),y_{2})}
{\displaystyle f(x_{1},a,g(z_{1}),y_{1})\{x_{1}\mapsto x_{2},y_{1}\mapsto y_{2},z_{1}\mapsto z_{2}\}=f(x_{2},a,g(z_{2}),y_{2})}
{\displaystyle f(x_{2},a,g(z_{2}),y_{2})\{x_{2}\mapsto x_{1},y_{2}\mapsto y_{1},z_{2}\mapsto z_{1}\}=f(x_{1},a,g(z_{1}),y_{1})}
{\displaystyle f(x_{1},a,g(z_{1}),y_{1})}
{\displaystyle f(x_{2},a,g(x_{2}),x_{2})}
, since no substitution can transform the latter term into the former one, although
{\displaystyle \{x_{1}\mapsto x_{2},z_{1}\mapsto x_{2},y_{1}\mapsto x_{2}\}}
achieves the reverse direction. The latter term is hence properly more special than the former one.
{\displaystyle \sigma }
{\displaystyle \tau }
{\displaystyle x\sigma }
{\displaystyle x\tau }
{\displaystyle x}
{\displaystyle \{x\mapsto f(u),y\mapsto f(f(u))\}}
{\displaystyle \{x\mapsto z,y\mapsto f(z)\}}
{\displaystyle f(u)}
{\displaystyle f(f(u))}
{\displaystyle z}
{\displaystyle f(z)}
Anti-unification problem, generalization set[edit]
An anti-unification problem is a pair
{\displaystyle \langle t_{1},t_{2}\rangle }
of terms. A term
{\displaystyle t}
is a common generalization, or anti-unifier, of
{\displaystyle t_{1}}
{\displaystyle t_{2}}
{\displaystyle t\sigma _{1}\equiv t_{1}}
{\displaystyle t\sigma _{2}\equiv t_{2}}
for some substitutions
{\displaystyle \sigma _{1},\sigma _{2}}
. For a given anti-unification problem, a set
{\displaystyle S}
of anti-unifiers is called complete if each generalization subsumes some term
{\displaystyle t\in S}
; the set
{\displaystyle S}
is called minimal if none of its members subsumes another one.
First-order syntactical anti-unification[edit]
The framework of first-order syntactical anti-unification is based on
{\displaystyle T}
being the set of first-order terms (over some given set
{\displaystyle V}
of variables,
{\displaystyle C}
of constants and
{\displaystyle F_{n}}
{\displaystyle n}
-ary function symbols) and on
{\displaystyle \equiv }
being syntactic equality. In this framework, each anti-unification problem
{\displaystyle \langle t_{1},t_{2}\rangle }
has a complete, and obviously minimal, singleton solution set
{\displaystyle \{t\}}
. Its member
{\displaystyle t}
is called the least general generalization (lgg) of the problem, it has an instance syntactically equal to
{\displaystyle t_{1}}
and another one syntactically equal to
{\displaystyle t_{2}}
. Any common generalization of
{\displaystyle t_{1}}
{\displaystyle t_{2}}
subsumes
{\displaystyle t}
. The lgg is unique up to variants: if
{\displaystyle S_{1}}
{\displaystyle S_{2}}
are both complete and minimal solution sets of the same syntactical anti-unification problem, then
{\displaystyle S_{1}=\{s_{1}\}}
{\displaystyle S_{2}=\{s_{2}\}}
for some terms
{\displaystyle s_{1}}
{\displaystyle s_{2}}
, that are renamings of each other.
Plotkin[1][2] has given an algorithm to compute the lgg of two given terms. It presupposes an injective mapping
{\displaystyle \phi :T\times T\longrightarrow V}
, that is, a mapping assigning each pair
{\displaystyle s,t}
of terms an own variable
{\displaystyle \phi (s,t)}
, such that no two pairs share the same variable. [note 4] The algorithm consists of two rules:
{\displaystyle f(s_{1},\dots ,s_{n})\sqcup f(t_{1},\ldots ,t_{n})}
{\displaystyle \rightsquigarrow }
{\displaystyle f(s_{1}\sqcup t_{1},\ldots ,s_{n}\sqcup t_{n})}
{\displaystyle s\sqcup t}
{\displaystyle \rightsquigarrow }
{\displaystyle \phi (s,t)}
if previous rule not applicable
{\displaystyle (0*0)\sqcup (4*4)\rightsquigarrow (0\sqcup 4)*(0\sqcup 4)\rightsquigarrow \phi (0,4)*\phi (0,4)\rightsquigarrow x*x}
; this least general generalization reflects the common property of both inputs of being square numbers.
Plotkin used his algorithm to compute the "relative least general generalization (rlgg)" of two clause sets in first-order logic, which was the basis of the Golem approach to inductive logic programming.
First-order anti-unification modulo theory[edit]
This section needs expansion with: explain main results from papers below, relate their approaches to each other. You can help by adding to it. (June 2020)
Jacobsen, Erik (Jun 1991), Unification and Anti-Unification (PDF), Technical Report
Østvold, Bjarte M. (Apr 2004), A Functional Reconstruction of Anti-Unification (PDF), NR Note, vol. DART/04/04, Norwegian Computing Center
Boytcheva, Svetla; Markov, Zdravko (2002). "An Algorithm for Inducing Least Generalization Under Relative Implication" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
Kutsia, Temur; Levy, Jordi; Villaret, Mateu (2014). "Anti-Unification for Unranked Terms and Hedges" (PDF). Journal of Automated Reasoning. 52 (2): 155–190. doi:10.1007/s10817-013-9285-6. Software.
Equational theories[edit]
One associative and commutative operation: Pottier, Loic (Feb 1989), Algorithms des completion et generalisation en logic du premier ordre ; Pottier, Loic (1989), Generalisation de termes en theorie equationelle – Cas associatif-commutatif, INRIA Report, vol. 1056, INRIA
Commutative theories: Baader, Franz (1991). "Unification, Weak Unification, Upper Bound, Lower Bound, and Generalization Problems". Proc. 4th Conf. on Rewriting Techniques and Applications (RTA). LNCS. Vol. 488. Springer. pp. 86–91.
Free monoids: Biere, A. (1993), Normalisierung, Unifikation und Antiunifikation in Freien Monoiden (PDF), Univ. Karlsruhe, Germany
Regular congruence classes: Heinz, Birgit (Dec 1995), Anti-Unifikation modulo Gleichungstheorie und deren Anwendung zur Lemmagenerierung, GMD Berichte, vol. 261, TU Berlin, ISBN 978-3-486-23873-0 ; Burghardt, Jochen (2005). "E-Generalization Using Grammars". Artificial Intelligence. 165 (1): 1–35. arXiv:1403.8118. doi:10.1016/j.artint.2005.01.008.
A-, C-, AC-, ACU-theories with ordered sorts: Alpuente, Maria; Escobar, Santiago; Espert, Javier; Meseguer, Jose (2014). "A modular order-sorted equational generalization algorithm" (PDF). Information and Computation. 235: 98–136. doi:10.1016/j.ic.2014.01.006. hdl:2142/25871.
Purely idempontent theories: Cerna, David; Kutsia, Temur (2019). "Idempotent Anti-Unification". ACM Transactions in Computational Logic. 21 (2). hdl:10.1145/3359060.
First-order sorted anti-unification[edit]
Taxonomic sorts: Frisch, Alan M.; Page, David (1990). "Generalisation with Taxonomic Information". AAAI: 755–761. ; Frisch, Alan M.; Page Jr., C. David (1991). "Generalizing Atoms in Constraint Logic". Proc. Conf. on Knowledge Representation. ; Frisch, A.M.; Page, C.D. (1995). "Building Theories into Instantiation". In Mellish, C.S. (ed.). Proc. 14th IJCAI. Morgan Kaufmann. pp. 1210–1216.
Feature terms: Plaza, E. (1995). "Cases as Terms: A Feature Term Approach to the Structured Representation of Cases". Proc. 1st International Conference on Case-Based Reasoning (ICCBR). LNCS. Vol. 1010. Springer. pp. 265–276. ISSN 0302-9743.
Idestam-Almquist, Peter (Jun 1993). "Generalization under Implication by Recursive Anti-Unification". Proc. 10th Conf. on Machine Learning. Morgan Kaufmann. pp. 151–158.
Fischer, Cornelia (May 1994), PAntUDE – An Anti-Unification Algorithm for Expressing Refined Generalizations (PDF), Research Report, vol. TM-94-04, DFKI
A-, C-, AC-, ACU-theories with ordered sorts: see above
Nominal anti-unification[edit]
Baumgartner, Alexander; Kutsia, Temur; Levy, Jordi; Villaret, Mateu (Jun 2013). Nominal Anti-Unification. Proc. RTA 2015. Vol. 36 of LIPIcs. Schloss Dagstuhl, 57-73. Software.
Program analysis: Bulychev, Peter; Minea, Marius (2008). "Duplicate Code Detection Using Anti-Unification". {{cite journal}}: Cite journal requires |journal= (help); Bulychev, Peter E.; Kostylev, Egor V.; Zakharov, Vladimir A. (2009). "Anti-Unification Algorithms and their Applications in Program Analysis". {{cite journal}}: Cite journal requires |journal= (help)
Code factoring: Cottrell, Rylan (Sep 2008), Semi-automating Small-Scale Source Code Reuse via Structural Correspondence (PDF), Univ. Calgary
Induction proving: Heinz, Birgit (1994), Lemma Discovery by Anti-Unification of Regular Sorts, Technical Report, vol. 94–21, TU Berlin
Information Extraction: Thomas, Bernd (1999). "Anti-Unification Based Learning of T-Wrappers for Information Extraction" (PDF). AAAI Technical Report. WS-99-11: 15–20.
Case-based reasoning: Armengol, Eva; Plaza, Enric (2005). "Using Symbolic Descriptions to Explain Similarity on CBR" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
Program synthesis: The idea of generalizing terms with respect to an equational theory can be traced back to Manna and Waldinger (1978, 1980) who desired to apply it in program synthesis. In section "Generalization", they suggest (on p. 119 of the 1980 article) to generalize reverse(l) and reverse(tail(l))<>[head(l)] to obtain reverse(l')<>m' . This generalization is only possible if the background equation u<>[]=u is considered.
Zohar Manna; Richard Waldinger (Dec 1978). A Deductive Approach to Program Synthesis (PDF) (Technical Note). SRI International. — preprint of the 1980 article
Zohar Manna and Richard Waldinger (Jan 1980). "A Deductive Approach to Program Synthesis". ACM Transactions on Programming Languages and Systems. 2: 90–121. doi:10.1145/357084.357090.
Natural language processing: Amiridze, Nino; Kutsia, Temur (2018). "Anti-Unification and Natural Language Processing". Fifth Workshop on Natural Language and Computer Science, NLCS'18. EasyChair Report No. 203.
Higher-order anti-unification[edit]
This section needs expansion with: (as above). You can help by adding to it. (June 2020)
Calculus of constructions: Pfenning, Frank (Jul 1991). "Unification and Anti-Unification in the Calculus of Constructions" (PDF). Proc. 6th LICS. Springer. pp. 74–85.
Simply-typed lambda calculus (Input: Terms in the eta-long beta-normal form. Output: higher-order patterns): Baumgartner, Alexander; Kutsia, Temur; Levy, Jordi; Villaret, Mateu (Jun 2013). A Variant of Higher-Order Anti-Unification. Proc. RTA 2013. Vol. 21 of LIPIcs. Schloss Dagstuhl, 113-127. Software.
Simply-typed lambda calculus (Input: Terms in the eta-long beta-normal form. Output: Various fragments of the simply-typed lambda calculus including patterns): Cerna, David; Kutsia, Temur (June 2019). "A Generic Framework for Higher-Order Generalizations" (PDF). 4th International Conference on Formal Structures for Computation and Deduction, FSCD, June 24–30, 2019, Dortmund, Germany. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. pp. 74–85.
Restricted Higher-Order Substitutions: Wagner, Ulrich (Apr 2002), Combinatorically Restricted Higher Order Anti-Unification, TU Berlin ; Schmidt, Martin (Sep 2010), Restricted Higher-Order Anti-Unification for Heuristic-Driven Theory Projection (PDF), PICS-Report, vol. 31–2010, Univ. Osnabrück, Germany, ISSN 1610-5389
^ Complete generalization sets always exist, but it may be the case that every complete generalization set is non-minimal.
^ Comon referred in 1986 to inequation-solving as "anti-unification", which nowadays has become quite unusual. Comon, Hubert (1986). "Sufficient Completeness, Term Rewriting Systems and 'Anti-Unification'". Proc. 8th International Conference on Automated Deduction. LNCS. Vol. 230. Springer. pp. 128–140.
{\displaystyle a\oplus (b\oplus f(x))\equiv a\oplus (f(x)\oplus b)\equiv (b\oplus f(x))\oplus a\equiv (f(x)\oplus b)\oplus a}
^ From a theoretical viewpoint, such a mapping exists, since both
{\displaystyle V}
{\displaystyle T\times T}
are countably infinite sets; for practical purposes,
{\displaystyle \phi }
can be built up as needed, remembering assigned mappings
{\displaystyle \langle s,t,\phi (s,t)\rangle }
in a hash table.
^ a b Plotkin, Gordon D. (1970). Meltzer, B.; Michie, D. (eds.). "A Note on Inductive Generalization". Machine Intelligence. 5: 153–163.
^ a b Plotkin, Gordon D. (1971). Meltzer, B.; Michie, D. (eds.). "A Further Note on Inductive Generalization". Machine Intelligence. 6: 101–124.
^ C.C. Chang; H. Jerome Keisler (1977). A. Heyting; H.J. Keisler; A. Mostowski; A. Robinson; P. Suppes (eds.). Model Theory. Studies in Logic and the Foundation of Mathematics. Vol. 73. North Holland. ; here: Sect.1.3
Retrieved from "https://en.wikipedia.org/w/index.php?title=Anti-unification_(computer_science)&oldid=1059124591"
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Spherical linear interpolation - MATLAB slerp - MathWorks Benelux
Interpolate Between Two Quaternions
SLERP Minimizes Great Circle Path
Show Interpolated Quaternions on Sphere
q0 = slerp(q1,q2,T)
q0 = slerp(q1,q2,T) spherically interpolates between q1 and q2 by the interpolation coefficient T. The function always chooses the shorter interpolation path between q1 and q2.
Create two quaternions with the following interpretation:
a = 45 degree rotation around the z-axis
c = -45 degree rotation around the z-axis
a = quaternion([45,0,0],'eulerd','ZYX','frame');
c = quaternion([-45,0,0],'eulerd','ZYX','frame');
Call slerp with the quaternions a and c and specify an interpolation coefficient of 0.5.
interpolationCoefficient = 0.5;
b = slerp(a,c,interpolationCoefficient);
The output of slerp, b, represents an average rotation of a and c. To verify, convert b to Euler angles in degrees.
averageRotation = eulerd(b,'ZYX','frame')
averageRotation = 1×3
The interpolation coefficient is specified as a normalized value between 0 and 1, inclusive. An interpolation coefficient of 0 corresponds to the a quaternion, and an interpolation coefficient of 1 corresponds to the c quaternion. Call slerp with coefficients 0 and 1 to confirm.
b = slerp(a,c,[0,1]);
eulerd(b,'ZYX','frame')
-45.0000 0 0
You can create smooth paths between quaternions by specifying arrays of equally spaced interpolation coefficients.
path = 0:0.1:1;
interpolatedQuaternions = slerp(a,c,path);
For quaternions that represent rotation only about a single axis, specifying interpolation coefficients as equally spaced results in quaternions equally spaced in Euler angles. Convert interpolatedQuaternions to Euler angles and verify that the difference between the angles in the path is constant.
k = eulerd(interpolatedQuaternions,'ZYX','frame');
abc = abs(diff(k))
abc = 10×3
Alternatively, you can use the dist function to verify that the distance between the interpolated quaternions is consistent. The dist function returns angular distance in radians; convert to degrees for easy comparison.
def = rad2deg(dist(interpolatedQuaternions(2:end),interpolatedQuaternions(1:end-1)))
def = 1×10
The SLERP algorithm interpolates along a great circle path connecting two quaternions. This example shows how the SLERP algorithm minimizes the great circle path.
Define three quaternions:
q0 - quaternion indicating no rotation from the global frame
q179 - quaternion indicating a 179 degree rotation about the z-axis
q0 = ones(1,'quaternion');
q179 = quaternion([179,0,0],'eulerd','ZYX','frame');
Use slerp to interpolate between q0 and the three quaternion rotations. Specify that the paths are traveled in 10 steps.
q179path = slerp(q0,q179,T);
Plot each path in terms of Euler angles in degrees.
q179pathEuler = eulerd(q179path,'ZYX','frame');
plot(T,q179pathEuler(:,1),'bo', ...
T,q180pathEuler(:,1),'r*', ...
T,q181pathEuler(:,1),'gd');
legend('Path to 179 degrees', ...
'Path to 180 degrees', ...
'Path to 181 degrees')
xlabel('Interpolation Coefficient')
ylabel('Z-Axis Rotation (Degrees)')
The path between q0 and q179 is clockwise to minimize the great circle distance. The path between q0 and q181 is counterclockwise to minimize the great circle distance. The path between q0 and q180 can be either clockwise or counterclockwise, depending on numerical rounding.
Create two quaternions.
q1 = quaternion([75,-20,-10],'eulerd','ZYX','frame');
q2 = quaternion([-45,20,30],'eulerd','ZYX','frame');
Define the interpolation coefficient.
Obtain the interpolated quaternions.
quats = slerp(q1,q2,T);
Obtain the corresponding rotate points.
pts = rotatepoint(quats,[1 0 0]);
Show the interpolated quaternions on a unit sphere.
surf(X,Y,Z,'FaceColor',[0.57 0.57 0.57])
scatter3(pts(:,1),pts(:,2),pts(:,3))
Note that the interpolated quaternions follow the shorter path from q1 to q2.
q1 — Quaternion
Quaternion to interpolate, specified as a scalar, vector, matrix, or multidimensional array of quaternions.
q1, q2, and T must have compatible sizes. In the simplest cases, they can be the same size or any one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are either the same or one of them is 1.
q1, q2, and T must have compatible sizes. In the simplest cases, they can be the same size or any one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are either the same or one of the dimension sizes is 1.
T — Interpolation coefficient
Interpolation coefficient, specified as a scalar, vector, matrix, or multidimensional array of numbers with each element in the range [0,1].
q0 — Interpolated quaternion
Interpolated quaternion, returned as a scalar, vector, matrix, or multidimensional array.
Quaternion spherical linear interpolation (SLERP) is an extension of linear interpolation along a plane to spherical interpolation in three dimensions. The algorithm was first proposed in [1]. Given two quaternions, q1 and q2, SLERP interpolates a new quaternion, q0, along the great circle that connects q1 and q2. The interpolation coefficient, T, determines how close the output quaternion is to either q1 and q2.
The SLERP algorithm can be described in terms of sinusoids:
{q}_{0}=\frac{\mathrm{sin}\left(\left(1-T\right)\theta \right)}{\mathrm{sin}\left(\theta \right)}{q}_{1}+\frac{\mathrm{sin}\left(T\theta \right)}{\mathrm{sin}\left(\theta \right)}{q}_{2}
where q1 and q2 are normalized quaternions, and θ is half the angular distance between q1 and q2.
[1] Shoemake, Ken. "Animating Rotation with Quaternion Curves." ACM SIGGRAPH Computer Graphics Vol. 19, Issue 3, 1985, pp. 345–354.
dist | meanrot
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Domestic Solar Hot Water Financial Analysis
We have covered methods to account for the costs and savings for a generic SECS in the previous pages. In those readings, we introduced the time value of money. So, let's think about the "time value of money" using a spreadsheet. The questions below are to be leading topics that will dig into the coupled meanings of Life Cycle Savings, Solar Savings, Fuel Savings, time value of money, systems payback, and paying back a loan. Some of the questions may be easier than others, but there are not necessarily clear answers to all of them. Also some people in class may have more experience with this type of analysis than others, so it would be beneficial to work together as a group through this discussion.
An example spreadsheet for solar hot water systems in a residential home (Domestic Solar Hot Water, or DSHW) is published as a shared Google spreadsheet. The direct link to access the file is in the middle of this page. This spreadsheet is set up in many columns: each column is representing a separate sequence of years for discrete financial analysis. There are accompanying graphs to link with the data, presenting loan payments and annualized Solar Savings increasing each year. Because the spreadsheet is dynamic, it would be better if you download a copy of the file and try changing things like the discount rate, fuel cost, loan size, and systems size (solar fraction) and see what the response will be.
There are two example systems analyzed in the spreadsheet. The first system has a solar fraction F = 0.65, costing \$16k with a 20% Down Payment and the remainder paid through a back loan at 7% interest. The second system has a solar fraction F = 0.85, costing \$26k with a 20% Down Payment, and the remainder paid through a back loan. Both systems have a potential resale value of 30% of initial investment ($16k), framed in Present Value (a different kind of "PV"). This is a detailed spreadsheet presenting you with an example of discrete financial analysis where we consider the time value of money over 20 year span. Half the battle in developing a useful spreadsheet is figuring out where everything is. Later, we will also dig into the financial output in SAM simulations.
NOTE: You must be logged into Google in order to view this spreadsheet.
Learning Activity 7.1 Google Spreadsheet
Study the spreadsheet and then discuss the following questions in the “Learning Activity 7.1” Discussion Forum.
Why is there Time "Zero?" What years do the two systems "pay back?" Why is there an additional financial increase for Year 20 at the end?
Look at columns
B
I
and identify the role that each of the columns serves leading up to Solar Savings and the Cumulative Solar Savings (framed in present worth).
Where does one find the market discount rates to estimate present values (seen in Column
Q
R
of the first sheet), and why is it that we need to consider future values in present worth when we are accounting for the project finance of SECS? What would the special meaning of the rate be if we raised that value from 8% in Column
R
, to a value high enough to drive the LCS to \$0?
In the red colored "loan" columns, do you see the connection to the reading regarding the rate of the loan and the annual loan payments? Why is the interest rate listed as a "discount rate"?
Which system would seem to be a reasonable investment for a middle-class family of 4 (two incomes, <\$80k annual gross income) living in Michigan, USA? Why?
A comment: Columns
N
O
P
are tied to the use of fuel to heat water (annual loads:
L
), the annual Solar Fraction for the installed system (annual solar fraction:
F
), and the annual cost of the fuel (
{C}_{F}
as electricity in \$/MWh, or \$0.8/kWh). We are initially guessing a system size, and that 65% of the annual energy will be covered by this array. In mid-continental USA, each person consumes ~8 MWh of energy to heat water per year. Here, we are estimating for a residential family of four.
‹ 7.4 Solar Fractions: Gains and Loads up 7.6 Economic Figures of Merit ›
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!! used as default html header if there is none in the selected theme. Primpoly
P\left(x\right)
n
{p}^{n}-1
p
p
Description: search for primitive polynomials over a finite field. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, cryptology, finite_field, order, coding, cyclic_code
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EUDML | The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one. EuDML | The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one.
The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one.
Furushima, Mikio. "The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one.." Mathematische Annalen 297.4 (1993): 627-662. <http://eudml.org/doc/165150>.
@article{Furushima1993,
author = {Furushima, Mikio},
keywords = {second Betti number one; projective compactification of },
title = {The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one.},
AU - Furushima, Mikio
TI - The complete classification of compactifications of C³ which are projective manifolds with the second Betti number one.
KW - second Betti number one; projective compactification of
second Betti number one, projective compactification of
{ℂ}^{3}
Compactification of analytic spaces
Articles by Mikio Furushima
|
EUDML | On Shanks' algorithm for computing the continued fraction of . EuDML | On Shanks' algorithm for computing the continued fraction of .
On Shanks' algorithm for computing the continued fraction of
{log}_{b}a
Jackson, Terence; Matthews, Keith
Jackson, Terence, and Matthews, Keith. "On Shanks' algorithm for computing the continued fraction of .." Journal of Integer Sequences [electronic only] 5.2 (2002): Art. 02.2.7, 9 p., electronic only-Art. 02.2.7, 9 p., electronic only. <http://eudml.org/doc/50287>.
author = {Jackson, Terence, Matthews, Keith},
keywords = {Shanks' algorithm; continued fraction; log; heuristic algorithm},
title = {On Shanks' algorithm for computing the continued fraction of .},
TI - On Shanks' algorithm for computing the continued fraction of .
KW - Shanks' algorithm; continued fraction; log; heuristic algorithm
Shanks' algorithm, continued fraction, log, heuristic algorithm
|
7.4 Solar Fractions: Gains and Loads | EME 810: Solar Resource Assessment and Economics
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 10 (focus on the Solar Fraction and Gains, Loads, Losses)
This is a fairly short portion of the chapter, but it offers a simple way to think about the size of a system relative to a local demand.
Loads, Costs, and Fractions
We know that a local SECS like a Solar Hot Water system will have a certain quantity of demand from a residential family.
Annual Loads (L) Example: In the Midwest of the USA, a residential family will consume 7-8 MWh of energy to heat water per person, per year. That's an energetic load.
Annual Costs (C) Example: In the USA, an average retail electricity cost is \$0.08/kWh, or \$80/MWh.
Annual Solar Fraction (F): The fraction of energy provided by a SECS relative to the total energy demanded for the periodic step size (here, annual). In the solar field, we call the supplemental energy required beyond the SECS: "auxiliary" energy (even if it is a primary energy source in society).
We often design a domestic solar hot water (DSHW) system to provide an annual fraction F = 0.4-0.7 (40-70% of the total annual demand), sized for the summer loads, because the heat would be wasted/dumped in the summer. That would mean the client would be buying a bigger system that does not have utility in the summer. Better to have a less sufficient system for hot water in the winter, than for the client to pay for something they cannot use part of the year.
In our reading, we made the distinction between the annual solar fraction (uppercase F) and the monthly solar fraction (lowercase f). We can use the solar fraction as a factor in project finance to estimate an ideal array size for our client in his/her locale. Consider that a large solar fraction will entail more modules or panels, and will increase the cost for the client in the system investment (according to the unit cost). It will also increase the time to payback the investment. Our clients will no doubt have finite cash on hand to put a down payment into a SECS, and to acquire a loan for the rest of the investment. They may also require a fast payback that will influence the sizing of the system.
An annual solar fraction of zero (F=0) is where the client opts for no installation of a new SECS.
F=0
will have the highest energy costs (fuel costs;
FC
) of any alternative SECS
An annual solar fraction of one
\left(F=1\right)
is where client opts for a SECS that covers all energy Loads for the entire year.
F=1
will have the highest solar investment costs (
{C}_{S}
), with the lowest associated annual energy costs
We are to work with the client to find a strong solution between those two trivial extremes (a maximum return on investment), specifically a return with net positive in cumulative solar savings (Life Cycle Savings: LCS).
L\cdot F\cdot {C}_{fuel}=
annual fuel savings (considered before discounting or fuel inflation rates)
‹ 7.3 Solar Savings and Avoided Fuel Costs up 7.5 Discussion Activity ›
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EUDML | Complete holomorphic vector fields on the second dual of Banach space. EuDML | Complete holomorphic vector fields on the second dual of Banach space.
Dineen, Seán. "Complete holomorphic vector fields on the second dual of Banach space.." Mathematica Scandinavica 59 (1986): 131-142. <http://eudml.org/doc/166995>.
author = {Dineen, Seán},
keywords = {any uniformly bounded collection of complete holomorphic vector fields on the unit ball of Banach spaces can be combined to define such a vector field on the unit ball of any ultraproduct of the Banach spaces; ultraproduct of triple systems; bidual; biholomorphic automorphisms},
title = {Complete holomorphic vector fields on the second dual of Banach space.},
AU - Dineen, Seán
TI - Complete holomorphic vector fields on the second dual of Banach space.
KW - any uniformly bounded collection of complete holomorphic vector fields on the unit ball of Banach spaces can be combined to define such a vector field on the unit ball of any ultraproduct of the Banach spaces; ultraproduct of triple systems; bidual; biholomorphic automorphisms
any uniformly bounded collection of complete holomorphic vector fields on the unit ball of Banach spaces can be combined to define such a vector field on the unit ball of any ultraproduct of the Banach spaces, ultraproduct of
J{B}^{*}
triple systems, bidual, biholomorphic automorphisms
Nonassociative topological algebras
Articles by Seán Dineen
|
Borcherds products and arithmetic intersection theory on Hilbert modular surfaces
15 July 2007 Borcherds products and arithmetic intersection theory on Hilbert modular surfaces
Jan H. Bruinier, José I. Burgos Gil, Ulf Kühn
Jan H. Bruinier,1 José I. Burgos Gil,2 Ulf Kühn3
2Facultat de Matemàtiques, Universitat de Barcelona
We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight
2
. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors
Jan H. Bruinier. José I. Burgos Gil. Ulf Kühn. "Borcherds products and arithmetic intersection theory on Hilbert modular surfaces." Duke Math. J. 139 (1) 1 - 88, 15 July 2007. https://doi.org/10.1215/S0012-7094-07-13911-5
Secondary: 11F41 , 14C17 , 14C20 , 14G40
Jan H. Bruinier, José I. Burgos Gil, Ulf Kühn "Borcherds products and arithmetic intersection theory on Hilbert modular surfaces," Duke Mathematical Journal, Duke Math. J. 139(1), 1-88, (15 July 2007)
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syntax - zxc.wiki
The title of this article is ambiguous. Further meanings are listed under Syntax (disambiguation) .
Among syntax ( ancient Greek σύνταξις syntaxis from σύν sy n , together 'and τάξις taxis , order, order') is generally understood as a control system for the combination of elementary mark on composite characters in natural or artificial sign systems. The assembly rules of the syntax are opposed to the interpretation rules of the semantics .
In particular is understood to mean syntax , the syntax , a subsection of the grammar of natural language , which the joining of words or word groups to sets ( sentence structure (such as a certain prescribed) on the basis of grammatical laws set position ) treated or sets underlying regular pattern ( sentence structure ) describes. The syntax is usually distinguished from the linguistic morphology , which deals with the internal structure of the words, although the transitions between the two areas can be fluid.
The term syntax is used for natural and formal languages. The relationship between natural and formal syntax is seen differently. For the logician Richard Montague ( Universal Grammar , 1970) there was no difference in principle.
Just like the term grammar , the term syntax can refer to the structural properties of sign systems themselves or to the theoretical-scientific description of these structural properties.
1 natural language syntax (natural syntax)
1.1 Position of syntax in grammar
1.2 Sentence syntax, word syntax, text syntax
1.3 Theories of sentence syntax
2 syntax of formal languages
Natural language syntax (natural syntax)
Position of syntax in grammar
In terms of natural languages , syntax is a division of grammar and closely related to morphology . The demarcation between them refers to the complexity levels of the grammatical structure of linguistic expressions. For example: From the minimal language sign ( morpheme ) ask as a word stem to the extended form by adding the prefix be to the word form questioned , the morphology is responsible. From then on the levels of complexity upwards, i.e. from the syntagmatic questioning of the candidate to the simple sentence (if) you question the candidate to the compound sentence, hold back, if you question the candidate , the syntax is responsible. For syntax, the word form is a whole, with the internal structure of which syntactic rules have nothing to do with; they just have to “know” which syntactically relevant morphological categories the word form actually belongs to. So determined z. B. a syntactic rule that the predicate verb in when you ask the candidate is in congruence with his subject in the second person singular. But how this form is (in this verb), so the morphology care (. When the verb, for example, lets in would, it meadow - unlike befragst - umlaut on).
Problems of demarcation between syntax and morphology can u. a. measured by phrasal compound words such as going down (one or two words?) or riding artillery troop (the attribute riding belongs to artillery , which in turn is part of another word). The Derivation , as part of the word education is part of the morphology, has a syntactic aspect.
Sentence syntax, word syntax, text syntax
In the traditional sense, syntax means the theory of the sentence (i.e. the theory of the correct sentence structure ) or the sentence structure itself. Syntax as part of grammar deals with the patterns and rules according to which words are combined into larger functional units, such as the sentence just mentioned and relationships such as part-whole, dependency, etc. are formulated between these clauses .
Apart from this set-centered perspective ( sentence syntax , sentence syntax ) occurs in a broader sense of an intra verbal syntax or word syntax (also: word syntax or morphotactics ), the combinatorial rules examined in morphology, and by a text syntax ( text syntax ), which deals with the rules of combining sentences into texts. The use of the word syntax , in which syntax is coextensive with grammar (i.e. either including morphology or adding to phonology), is found primarily in English-language linguistics and in the theory of formal languages (in which morphology does not play a role).
Sentence Syntax Theories
In general linguistics, there is a variety and competition of syntax models , theories, and schools. "Each of the models presented has its strengths and weaknesses." In addition to the models of traditional school grammar, the syntax is based on hypothetical, universal, innate form principles ( Noam Chomsky ) or their communicative purpose ( functional syntax ) or their role in the construction of complex meanings (logical semantics , Montague or categorical grammar). Many such models are listed in the Syntax Theory article . The more important ones include:
the dependency grammar
the government and binding theory (Chomsky 1981), a variant of generative grammar
the head-driven phrase structure grammar .
The syntactic structure of a natural language sentence is represented differently in these models. They represent the variants of the phrase structure grammar in the form of a structure tree, which graphically reproduces the part-whole relationships of the constituents of the sentence. The dependency grammar represents it in the form of a stem, which reflects the dependencies between the words.
Formal language syntax
The syntax of a formal language (formal syntax) - such as calculi in logic and mathematics or programming languages in computer science - is a system of rules according to which well-formed (“syntactically correct”) expressions, formulas, program texts or other texts can be formed from a basic set of characters (the alphabet ). The rules can take the form of derivation rules for a formal grammar or they can be formulated in natural language.
If it is only about the well-formedness or correctness, the meaning of the characters can be disregarded. If, however, semantics are to be defined on the well-formed expressions, this is usually done inductively using the same rules that also define the syntax, so that the meaning of a complex expression results from the meaning of its components and the rule for the composition ( Frege Principle ). For example, in the language definition of programming languages, the priority of the operators is reflected in the formal grammar of the language, so that, according to their syntactic rules, an expression can only be read as a sum, but not as a product. That would have played no role for the sheer well-formedness.
{\ displaystyle a + b * c}
The programming language Algol 60 was the first to be described with a formal syntax that was written in the Backus-Naur form (BNF; named after two of the authors of the language definition). Since then, formal syntax descriptions have become generally accepted for programming languages, namely with the help of different versions and extensions of the BNF or syntax diagrams , not least because analysis programs ( parsers ) can be automatically generated from the formal rules under certain conditions . As a result, the syntax of a programming language is often only understood to mean these rules, but not those syntax rules that cannot be expressed using context-free grammars , such as the obligation to declare occurring names.
The XML markup language has a syntax that is valid for all documents, which is further restricted by additional syntax rules depending on the area of application. The agreement with the general syntax is called well-formedness , and that with the additional rules is called validity .
Karl-Dieter Bünting, Henning Bergenholtz: Introduction to Syntax. Basic concepts for reading a grammar. (= Athenäums study books. Linguistics. Study book linguistics ). 2nd, revised edition. Athenaeum, Frankfurt am Main 1989, ISBN 3-610-02194-2 .
Christa Dürscheid : Syntax. Basics and theories (= UTB. Linguistics 3319). 5th revised edition. Vandenhoeck & Ruprecht, Göttingen 2010, ISBN 978-3-8252-3319-8 .
Bernhard Engelen: Introduction to the syntax of the German language. 2 volumes (Vol. 1: Preliminary questions and basics. Vol. 2: Sentences and sentence structure plans. ). Pedagogical publisher Burgbücherei Schneider, Baltmannsweiler 1984–1986, ISBN 3-87116-154-3 (vol. 1), ISBN 3-87116-160-8 (vol. 2).
Hans-Werner Eroms: Syntax of the German language. W. de Gruyter, Berlin et al. 2000, ISBN 3-11-015666-0 .
Joachim Jacobs, Arnim von Stechow, Wolfgang Sternefeld, Theo Vennemann , Herbert Ernst Wiegand (eds.): Syntax (= handbooks for language and communication science 9, 1–2). 2 volumes. de Gruyter, Berlin et al. 1993–1995, ISBN 3-11-009586-6 (vol. 1), ISBN 3-11-014263-5 (vol. 2).
Robert D. Van Valin, Jr .: An introduction to syntax. Cambridge University Press, Cambridge et al. 2001, ISBN 0-521-63566-7 .
Robert D. Van Valin, Jr., Randy J. LaPolla: Syntax. Structure, meaning and function. Cambridge University Press, Cambridge et al. 1997, ISBN 0-521-49565-2 .
Wiktionary: Syntax - explanations of meanings, word origins, synonyms, translations
Wiktionary: Sentence theory - explanations of meanings, word origins, synonyms, translations
Introduction to German Linguistics (10): Syntax
Basic linguistic course in morphology and syntax
Syntax theory. University of Jena
↑ See Christa Dürscheid: Syntax. Basics and theories (= UTB. Linguistics 3319). 5th revised edition. Vandenhoeck & Ruprecht, Göttingen 2010, ISBN 978-3-8252-3319-8 , p. 11.
↑ According to Angelika Linke, Markus Nussbaumer, Paul R. Portmann: Study book Linguistics (= series Germanistische Linguistik. Kollegbuch 121). 5th enlarged edition. Niemeyer, Tübingen 2004, ISBN 3-484-31121-5 , p. 84: "in a more or less metaphorical expansion of the core meaning."
↑ See dtv-Lexikon / Syntax
^ Danièle Clément: Basic linguistic knowledge. An introduction for future German teachers (= WV-Studium. Vol. 173 Linguistics ). 2nd Edition. Westdeutscher Verlag, Wiesbaden 2000, ISBN 3-531-23173-1 , p. 44.
↑ Angelika Linke, Markus Nussbaumer, Paul R. Portmann: Study book linguistics (= series Germanistic linguistics. College book 121). 5th enlarged edition. Niemeyer, Tübingen 2004, ISBN 3-484-31121-5 , p. 84.
↑ Ulrike Pospiech: Syntax. In: Johannes Volmert (Hrsg.): Grundkurs Sprachwissenschaft (= UTB for science. Uni-Taschenbücher. Linguistics 1879). 5th, corrected and supplemented edition. Fink, Munich 2005, ISBN 3-8252-1879-1 , pp. 115–150, here p. 149.
↑ See Helmut Glück (Hrsg.): Metzler-Lexikon Sprache. 3rd, revised edition. JB Metzler Verlagsbuchhandlung, Stuttgart et al. 2005, pp. 645 and 651–652.
↑ See syntax. In: Friedrich Kirchner , Carl Michaëlis: Dictionary of philosophical terms (= Philosophical Library 500). Continued by Johannes Hoffmeister . Completely re-edited by Arnim Regenbogen and Uwe Meyer. Meiner, Hamburg 2005, ISBN 3-7873-1325-7 .
↑ Peter Naur (ed.), Revised Report on the Algorithmic Language Algol 60, published in Numerische Mathematik , Vol.4 (1) (1962), pp.420–453, in Comm. ACM , Vol.6 (1) (1963), pp.1-17, and in Computer Journal , Vol.5 (4) (1963), pp.349-367; PDF
This page is based on the copyrighted Wikipedia article "Syntax" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Implement Euler angle representation of six-degrees-of-freedom equations of motion - Simulink - MathWorks Benelux
Implement Euler angle representation of six-degrees-of-freedom equations of motion
The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze). For more information about these reference points, see Algorithms.
The Quaternion selection conforms the equations of motion in Algorithms.
Assign unique name to each state. You can use state names instead of block paths during linearization.
If a parameter is empty (' '), no name assignment occurs.
The 6DOF (Euler Angles) block uses these reference frame concepts.
The origin of the body-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass.
The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth motion relative to the "fixed stars" to be neglected.
Translational motion of the body-fixed coordinate frame, where the applied forces [Fx Fy Fz]T are in the body-fixed frame, and the mass of the body m is assumed constant.
\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)\\ {A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{w}}_{b}\end{array}\right]=\frac{1}{m}{\overline{F}}_{b}-\overline{\omega }×{\overline{V}}_{b}\\ {A}_{be}=\frac{1}{m}{F}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{c}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega }=\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\end{array}
The rotational dynamics of the body-fixed frame, where the applied moments are [L M N]T, and the inertia tensor I is with respect to the origin O.
\begin{array}{l}{\overline{M}}_{B}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\stackrel{˙}{\overline{\omega }}+\overline{\omega }×\left(I\overline{\omega }\right)\\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}
\left[\begin{array}{ccc}\stackrel{˙}{\varphi }\text{ }\text{\hspace{0.17em}}& \stackrel{˙}{\theta }\text{\hspace{0.17em}}\text{ }\text{ }& \stackrel{˙}{\psi }\end{array}{\right]}^{T}
, are determined by resolving the Euler rates into the body-fixed coordinate frame.
\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \stackrel{˙}{\theta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \stackrel{˙}{\psi }\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]
\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)& \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\\ 0& \mathrm{cos}\varphi & -\mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }& \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]
[1] Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation. Hoboken, NJ: Second Edition, John Wiley & Sons, 2003.
[2] Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Reston, Va: Second Edition, AIAA Education Series, 2007.
6DOF (Quaternion) | 6DOF ECEF (Quaternion) | 6DOF Wind (Quaternion) | 6DOF Wind (Wind Angles) | Custom Variable Mass 6DOF (Euler Angles) | Custom Variable Mass 6DOF (Quaternion) | Custom Variable Mass 6DOF ECEF (Quaternion) | Custom Variable Mass 6DOF Wind (Quaternion) | Custom Variable Mass 6DOF Wind (Wind Angles) | Simple Variable Mass 6DOF (Euler Angles) | Simple Variable Mass 6DOF (Quaternion) | Simple Variable Mass 6DOF ECEF (Quaternion) | Simple Variable Mass 6DOF Wind (Quaternion) | Simple Variable Mass 6DOF Wind (Wind Angles)
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EUDML | Some properties of -projecting. EuDML | Some properties of -projecting.
Some properties of
H
-projecting.
Janić, Milan
Janić, Milan. "Some properties of -projecting.." Novi Sad Journal of Mathematics 33.1 (2003): 129-132. <http://eudml.org/doc/124125>.
@article{Janić2003,
author = {Janić, Milan},
keywords = {projective space; projection},
title = {Some properties of -projecting.},
AU - Janić, Milan
TI - Some properties of -projecting.
KW - projective space; projection
projective space, projection
Articles by Janić
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EUDML | Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base. EuDML | Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base.
Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base.
Vâjâitu, Viorel. "Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base.." Mathematica Scandinavica 79.2 (1996): 161-175. <http://eudml.org/doc/167399>.
@article{Vâjâitu1996,
author = {Vâjâitu, Viorel},
keywords = {-complete space; -convexity; cohomological -completeness; -complete holomorphic map; -convex holomorphic map; separated cohomology group; -Runge domain; plurisubharmonic map; complex Hessian; cohomological -convexity},
title = {Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base.},
AU - Vâjâitu, Viorel
TI - Cohomology Groups of Locally q-Complete Morphisms with r-Complete Base.
KW - -complete space; -convexity; cohomological -completeness; -complete holomorphic map; -convex holomorphic map; separated cohomology group; -Runge domain; plurisubharmonic map; complex Hessian; cohomological -convexity
q
-complete space,
q
-convexity, cohomological
q
-completeness,
q
-complete holomorphic map,
q
-convex holomorphic map, separated cohomology group,
q
-Runge domain, plurisubharmonic map, complex Hessian, cohomological
q
q
q
Articles by Viorel Vâjâitu
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Implementation of capacitor banks - Electrical Installation Guide
HomePower Factor CorrectionImplementation of capacitor banks
1 Capacitor elements
2 Choice of protection, control devices and connecting cables
Capacitors at low voltage are dry-type units (i.e. are not impregnated by liquid dielectric) comprising metallised polypropylene self-healing film in the form of a two-film roll.
Self-healing is a process by which the capacitor restores itself in the event of a fault in the dielectric which can happen during high overloads, voltage transients, etc.
When insulation breaks down, a short duration arc is formed (Figure L35 - top). The intense heat generated by this arc causes the metallization in the vicinity of the arc to vaporise (Figure L35 - middle).
Simultaneously it re-insulates the electrodes and maintains the operation and integrity of the capacitor (Figure L35 - bottom).
Fig. L35 – Illustration of self-healing phenomena
Capacitors must be associated with overload protection devices (fuses, or circuit breaker, or overload relay + contactor), in order to limit the consequences of overcurrents. This may occur in case of overvoltage or high harmonic distortion.
In addition to external protection devices, capacitors are protected by a high-quality system (Pressure Sensitive Disconnector, also called ‘tear-off fuse’) which switches off the capacitors if an internal fault occurs. This enables safe disconnection and electrical isolation at the end of the life of the capacitor.
The protection system operates as follows:
Current levels greater than normal, but insufficient to trigger the over-current protection sometimes occur, e.g. due to a microscopic flow in the dielectric film. Such faults are cleared by self-healing.
If the leakage current persists (and self-healing repeats), the defect may produce gas by vaporizing of the metallisation at the faulty location. This will gradually build up a pressure within the container. Pressure can only lead to vertical expansion by bending lid outwards. Connecting wires break at intended spots. Capacitor is disconnected irreversibly.
Fig. L36 – Cross-section view of a three-phase capacitor after Pressure Sensitive Device operated: bended lid and disconnected wires
Main electrical characteristics, according to IEC standard 60831-1/2: "Shunt power capacitors of the self-healing type for a.c. systems having a rated voltage up to and including 1000 V".
Fig. L37 – Main characteristics of capacitors according to IEC 60831-1/2
Capacitance tolerance –5 % to +10 % for units and banks up to 100 kvar
–5 % to +5 % for units and banks above 100 kvar
Temperature range Min: from -50 to +5°C
Max: from +40 to +55°C
Permissible current overload 1.3 x IN
Permissible voltage overload 1.1 x UN , 8 h every 24 h
1.15 x UN , 30 min every 24 h
1.2 x UN , 5min
2.15 x UN for 10 s (type test)
Discharging unit to 75 V in 3 min or less
Choice of protection, control devices and connecting cables
The choice of upstream cables, protection and control devices depends on the current loading.
For capacitors, the current is a function of:
The system voltage (fundamental and harmonics),
The power rating.
The rated current IN of a 3-phase capacitor bank is equal to:
{\displaystyle I_{N}={\frac {Q}{{\sqrt {3}}.U}}}
Q: power rating (kvar)
U: phase-to-phase voltage (kV)
Overload protection devices have to be implemented and set according to the expected harmonic distortion. The following table summarizes the harmonic voltages to be considered in the different configurations, and the corresponding maximum overload factor IMP/IN. (IMP = maximum permissible current)
Fig. L38 – Typical permissible overload currents
THDu max (%)
IMP/IN
Standard capacitors 5 1.5
Heavy Duty capacitors 7 1.8
Capacitors + 5.7% reactor 0.5 5 4 3.5 3 10 1.31
Capacitors + 7% reactor 0.5 6 4 3.5 3 8 1.19
Capacitors + 14% reactor 3 8 7 3.5 3 6 1.12
Short time delay setting of circuit breakers (short-circuit protection) should be set at 10 x IN in order to be insensitive to inrush current.
50 kvar – 400V – 50 Hz – Standard capacitors
{\displaystyle I_{N}={\frac {50}{{\sqrt {3}}\times 0.4}}=72A}
Long time delay setting: 1.5 x 72 = 108 A
Short time delay setting: 10 x 72 = 720 A
50 kvar – 400V – 50 Hz – Capacitors + 5.7% detuned reactor
Long time delay setting: 1.31 x 72 = 94 A
Upstream cables
Figure L39 gives the minimum recommended cross section area of the upstream cable for capacitor banks.
Cables for control
The minimum cross section area of these cables will be 1.5 mm2 for 230 V. For the secondary side of the current transformer, the recommended cross section area is ≥ 2.5 mm2.
Fig. L39 – Cross-section of cables connecting medium and high power capacitor banks[1]
Bank power (kvar)
Copper cross- section
Aluminium cross- section
120 240 185 2 x 95
150 250 240 2 x 120
300 2 x 95 2 x 150
180 - 210 360 2 x 120 2 x 185
High-frequency voltage and current transients occur when switching a capacitor bank into service. The maximum voltage peak does not exceed (in the absence of harmonics) twice the peak value of the rated voltage when switching uncharged capacitors.
In the case of a capacitor being already charged at the instant of switch closure, however, the voltage transient can reach a maximum value approaching 3 times the normal rated peak value.
This maximum condition occurs only if:
The existing voltage at the capacitor is equal to the peak value of rated voltage, and
The switch contacts close at the instant of peak supply voltage, and
The polarity of the power-supply voltage is opposite to that of the charged capacitor
In such a situation, the current transient will be at its maximum possible value, viz: Twice that of its maximum when closing on to an initially uncharged capacitor, as previously noted.
For any other values of voltage and polarity on the pre-charged capacitor, the transient peaks of voltage and current will be less than those mentioned above. In the particular case of peak rated voltage on the capacitor having the same polarity as that of the supply voltage, and closing the switch at the instant of supply-voltage peak, there would be no voltage or current transients.
Where automatic switching of stepped banks of capacitors is considered, therefore, care must be taken to ensure that a section of capacitors about to be energized is fully discharged.
The discharge delay time may be shortened, if necessary, by using discharge resistors of a lower resistance value.
^ Minimum cross-section not allowing for any correction factors (installation mode, temperature, etc.). The calculations were made for single-pole cables laid in open air at 30°C.
Retrieved from "http://www.electrical-installation.org/enw/index.php?title=Implementation_of_capacitor_banks&oldid=26828"
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Three-phase exterior permanent magnet synchronous motor with sinusoidal back electromotive force - Simulink - MathWorks Italia
Mechanical input configuration
Number of pole pairs (P)
Stator phase resistance per phase (Rs)
Stator d-axis inductance (Ldq_)
Permanent flux linkage constant (lambda_pm)
Back-emf constant (Ke)
Torque constant (Kt)
Physical inertia, viscous damping, and static friction (mechanical)
Initial d-axis and q-axis current (idq0)
Initial mechanical position (theta_init)
Initial mechanical speed (omega_init)
Three-phase exterior permanent magnet synchronous motor with sinusoidal back electromotive force
The Surface Mount PMSM block implements a three-phase exterior permanent magnet synchronous motor (PMSM) with sinusoidal back electromotive force. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.
On the Parameters tab, if you select Back-emf or Torque constant, the block implements one of these equations to calculate the permanent flux linkage constant.
{\lambda }_{pm}=\frac{1}{\sqrt{3}}\cdot \frac{{K}_{e}}{1000P}\cdot \frac{60}{2\pi }
{\lambda }_{pm}=\frac{2}{3}\cdot \frac{{K}_{t}}{P}
This figure shows the motor construction with a single pole pair on the motor.
The motor magnetic field due to the permanent magnets creates a sinusoidal rate of change of flux with motor angle.
For the axes convention, the a-phase and permanent magnet fluxes are aligned when motor angle θr is zero.
The block implements these equations, expressed in the motor flux reference frame (dq frame). All quantities in the motor reference frame are referred to the stator.
\begin{array}{l}{\omega }_{e}=P{\omega }_{m}\\ \frac{d}{dt}{i}_{d}=\frac{1}{{L}_{d}}{v}_{d}-\frac{R}{{L}_{d}}{i}_{d}+\frac{{L}_{q}}{{L}_{d}}P{\omega }_{m}{i}_{q}\end{array}
\frac{d}{dt}{i}_{q}=\frac{1}{{L}_{q}}{v}_{q}-\frac{R}{{L}_{q}}{i}_{q}-\frac{{L}_{d}}{{L}_{q}}P{\omega }_{m}{i}_{d}-\frac{{\lambda }_{pm}P{\omega }_{m}}{{L}_{q}}
{T}_{e}=1.5P\left[{\lambda }_{pm}{i}_{q}+\left({L}_{d}-{L}_{q}\right){i}_{d}{i}_{q}\right]
The Lq and Ld inductances represent the relation between the phase inductance and the motor position due to the saliency of the motor magnets. For the surface mount PMSM,
{L}_{d}={L}_{q}
iq, id
q- and d-axis currents (A)
vq, vd
q- and d-axis voltages (V)
Angular electrical velocity of the motor (rad/s)
Back electromotive force (EMF) (Vpk_LL/krpm, where Vpk_LL is the peak voltage line-to-line measurement)
Torque constant (N·m/A)
Electrical angle (rad)
\begin{array}{c}\frac{d}{dt}{\omega }_{m}=\frac{1}{J}\left({T}_{e}-{T}_{f}-F{\omega }_{m}-{T}_{m}\right)\\ \frac{d{\theta }_{m}}{dt}={\omega }_{m}\end{array}
{P}_{mot}= -{\omega }_{m}{T}_{e}
{P}_{bus}= {v}_{an}{i}_{a}+ {v}_{bn}{i}_{b}+{v}_{cn}{i}_{c}
{P}_{elec}= -\frac{3}{2}\left({R}_{s}{i}_{sd}^{2}+{R}_{s}{i}_{sq}^{2}\right)
{P}_{mech}= -\left({\omega }_{m}^{2}F+ |{\omega }_{m}|{T}_{f}\right)
{P}_{mech}= 0
{P}_{str}= {P}_{bus}+ {P}_{mot}+ {P}_{elec} + {P}_{mech}
Combined motor and load viscous damping N·m/(rad/s)
Spd — Motor shaft speed
Angular velocity of the motor, ωm, in rad/s.
Angular mechanical velocity of the motor
Motor mechanical angular position
To create this port, select Speed for the Mechanical input configuration parameter.
To create this port, select Torque for the Mechanical input configuration parameter.
Mechanical input configuration — Select port configuration
Sample Time (Ts) — Sample time for discrete integration
Number of pole pairs (P) — Pole pairs
Stator phase resistance per phase (Rs) — Resistance
Stator phase resistance per phase, Rs, in ohm.
Stator d-axis inductance (Ldq_) — Inductance
Stator inductance, Ldq, in H.
Permanent flux linkage constant (lambda_pm) — Flux
Permanent flux linkage constant, λpm, in Wb.
Back-emf constant (Ke) — Back electromotive force
Back electromotive force, EMF, Ke, in peak Vpk_LL/krpm. Vpk_LL is the peak voltage line-to-line measurement.
To calculate the permanent flux linkage constant, the block implements this equation.
{\lambda }_{pm}=\frac{1}{\sqrt{3}}\cdot \frac{{K}_{e}}{1000P}\cdot \frac{60}{2\pi }
Torque constant (Kt) — Torque constant
Torque constant, Kt, in N·m/A.
{\lambda }_{pm}=\frac{2}{3}\cdot \frac{{K}_{t}}{P}
Physical inertia, viscous damping, and static friction (mechanical) — Inertia, damping, friction
Inertia, J, in kg.m^2
To enable this parameter, select the Torque configuration parameter.
Initial d-axis and q-axis current (idq0) — Current
Initial q- and d-axis currents, iq, id, in A.
Initial mechanical position (theta_init) — Angle
Initial motor angular position, θm0, in rad.
Initial mechanical speed (omega_init) — Speed
Initial angular velocity of the motor, ωm0, in rad/s.
Surface Mount PM Controller | Flux-Based PMSM | Induction Motor | Interior PMSM | Mapped Motor
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Convert digital filter state-space parameters to second-order sections form - MATLAB ss2sos - MathWorks India
Second-Order Section Form of Filter
Convert digital filter state-space parameters to second-order sections form
[sos,g] = ss2sos(A,B,C,D) returns second-order section form sos with gain g that is equivalent to the state-space system represented by input arguments A, B, C, and D. The input state-space system must be single-output and real.
[sos,g] = ss2sos(A,B,C,D,iu) specifies index iu that indicates which input of the state-space system A, B, C, D the function uses in the conversion.
[sos,g] = ss2sos(A,B,C,D,order) specifies the order of the rows in sos with order.
[sos,g] = ss2sos(A,B,C,D,iu,order) specifies both the index ui and the order of the rows order.
[sos,g] = ss2sos(A,B,C,D,iu,order,scale) specifies the desired scaling of the gain and the numerator coefficients of all second-order sections.
sos = ss2sos(___) embeds the overall system gain g in the first section. You can specify an input combination from any of the previous syntaxes.
Design a fifth-order Butterworth lowpass filter, specifying a cutoff frequency of
0.2\pi
rad/sample and expressing the output in state-space form. Convert the state-space result to second-order sections. Visualize the frequency response of the filter.
m
, attached to a wall by a spring of unit elastic constant. A sensor measures the acceleration,
a
, of the mass.
The system is sampled at
{F}_{s}=5
Hz. Generate 50 time samples. Define the sampling interval
\Delta t=1/{F}_{s}
The oscillator can be described by the state-space equations
\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right),\end{array}
x={\left(\begin{array}{cc}r& v\end{array}\right)}^{T}
r
v
are respectively the position and velocity of the mass, and the matrices
A=\left(\begin{array}{cc}\mathrm{cos}\Delta t& \mathrm{sin}\Delta t\\ -\mathrm{sin}\Delta t& \mathrm{cos}\Delta t\end{array}\right),\phantom{\rule{1em}{0ex}}B=\left(\begin{array}{c}1-\mathrm{cos}\Delta t\\ \mathrm{sin}\Delta t\end{array}\right),\phantom{\rule{1em}{0ex}}C=\left(\begin{array}{cc}-1& 0\end{array}\right),\phantom{\rule{1em}{0ex}}D=\left(\begin{array}{c}1\end{array}\right).
The system is excited with a unit impulse in the positive direction. Use the state-space model to compute the time evolution of the system starting from an all-zero initial state.
Plot the acceleration of the mass as a function of time.
Compute the time-dependent acceleration using the transfer function to filter the input. Express the transfer function as second-order sections. Plot the result.
State matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then A is of size n-by-n.
Input-to-state matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then B is of size n-by-p.
C — Output-to-state matrix
Output-to-state matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then C is of size q-by-n.
Feedthrough matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then D is of size q-by-p.
iu — Index
Index, specified as an integer.
Row order in sos, specified as one of these values:
'down' — Order the sections so that the first row of sos contains the poles that are closest to the unit circle.
'up' — Order the sections so that the first row of sos contains the poles that are farthest from the unit circle.
The zeros are paired with the poles that are closest to them.
'none' (default) | 'inf'
Scaling of the gain and numerator coefficients, specified as one of these values:
Using infinity-norm scaling in conjunction with up-ordering minimizes the probability of overflow in the realization. Using 2-norm scaling in conjunction with down-ordering minimizes the peak round-off noise.
Infinity-norm and 2-norm scaling are appropriate for only direct-form II implementations.
Second-order section representation, returned as a matrix. sos is an L-by-6 matrix of the form
\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right]
whose rows contain the numerator and denominator coefficients bik and aik of the second-order sections of H(z), which is given by
H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right)=g\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}
Overall system gain, returned as a real-valued scalar.
If you call the function with one output argument, the function embeds the gain in the first section, H1(z), so that
H\left(z\right)=\prod _{k=1}^{L}{H}_{k}\left(z\right)
Embedding the gain in the first section when scaling a direct-form II structure is not recommended and can result in erratic scaling. To avoid embedding the gain, use the function with two outputs: sos and g.
The ss2sos function uses this four-step algorithm to determine the second-order section representation for an input state-space system.
Find the poles and zeros of the system given by A, B, C, and D.
Use the function zp2sos, which first groups the zeros and poles into complex conjugate pairs using the cplxpair function. zp2sos then forms the second-order sections by matching the pole and zero pairs according to these rules:
Match the poles that are closest to the unit circle with the zeros that are closest to those poles.
Match the poles that are next closest to the unit circle with the zeros that are closest to those poles.
Continue this process until all of the poles and zeros are matched.
The ss2sos function groups real poles into sections with the real poles that are closest to them in absolute value. The same rule holds for real zeros.
Order the sections according to the proximity of the pole pairs to the unit circle. The ss2sos function normally orders the sections with poles that are closest to the unit circle last in the cascade. You can specify for ss2sos to order the sections in the reverse order by setting the order input to 'down'.
Scale the sections by the norm specified by the scale input. For arbitrary H(ω), the scaling is defined by
{‖H‖}_{p}={\left[\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{|H\left(\omega \right)|}^{p}d\omega \right]}^{1/p}
where p can be either ∞ or 2. For details, see the references. This scaling is an attempt to minimize overflow or peak round-off noise in fixed-point filter implementations.
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Generate univariate autoregressive integrated moving average (ARIMA) model impulse response function (IRF) - MATLAB impulse - MathWorks 日本
{y}_{t}=0.5{y}_{t-1}-0.7{y}_{t-2}+{\mathrm{ε}}_{t},
{\mathrm{ε}}_{\mathit{t}}
is a standard Gaussian process.
Plot the IRF of
{\mathit{y}}_{\mathit{t}}
{y}_{t}=0.004+1.372{y}_{t-1}-0.807{y}_{t-2}+{\mathrm{ε}}_{t}-1.143{\mathrm{ε}}_{t-1}+0.674{\mathrm{ε}}_{t-2},
{\mathrm{ε}}_{\mathit{t}}
is a Gaussian series with mean 0 and variance 0.00008.
{y}_{t}=0.7{y}_{t-1}+{\mathrm{ε}}_{t}+0.2{\mathrm{ε}}_{t-1}.
irf=15×2 table
y(j) is the impulse response of yt at period j – 1. y(0) represents the impulse response during the period in which impulse applies the unit shock to the innovation (ε0 = 1).
{y}_{t}={m}_{t}+\mathrm{Ï}\left(L\right){\mathrm{ε}}_{t},
ψ(L) is the infinite-degree MA lag operator polynomial
{\mathrm{Ï}}_{0}+{\mathrm{Ï}}_{1}L+{\mathrm{Ï}}_{2}{L}^{2}+â¦
The IRF measures the change to the response j periods in the future due to a change in the innovation at time t, for j = 0,1,2,…. Symbolically, the IRF at period j is
\frac{â{y}_{t+j}}{â{\mathrm{ε}}_{t}}={\mathrm{Ï}}_{j}.
\left\{{\mathrm{Ï}}_{j}\right\}
[4] Lütkepohl, Helmut. New Introduction to Multiple Time Series Analysis. New York, NY: Springer-Verlag, 2007.
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Real Analysis - Wikibooks, open books for an open world
The subject of real analysis is concerned with studying the behavior and properties of functions, sequences, and sets on the real number line, which we denote as the mathematically familiar R. Concepts that we wish to examine through real analysis include properties like Limits, Continuity, Derivatives (rates of change), and Integration (amount of change over time). Many of these ideas are, on a conceptual or practical level, dealt with at lower levels of mathematics, including a regular First-Year Calculus course, and so, to the uninitiated reader, the subject of Real Analysis may seem rather senseless and trivial. However, Real Analysis is at a depth, complexity, and arguably beauty, that it is because under the surface of everyday mathematics, there is an assurance of correctness, that we call rigor, that permeates the whole of mathematics. Thus, Real Analysis can, to some degree, be viewed as a development of a rigorous, well-proven framework to support the intuitive ideas that we frequently take for granted.
Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of mathematical ideas you have come across throughout your story of mathematics. However, instead of relying on sometimes uncertain intuition (which we have all felt when we were solving a problem we did not understand), we will anchor it to a rigorous set of mathematical theorems. Throughout this book, we will begin to see that we do not need intuition to understand mathematics - we need a manual.
The overarching thesis of this book is how to define the real numbers axiomatically. How would that work? This book will read in this manner: we set down the properties which we think define the real numbers. We then prove from these properties - and these properties only - that the real numbers behave in the way which we have always imagined them to behave. We will then rework all our elementary theorems and facts we collected over our mathematical lives so that it all comes together, almost as if it always has been true before we analyzed it; that it was in fact rigorous all along - except that now we will know how it came to be.
Do not believe that once you have completed this book, mathematics is over. In other fields of academic study, there are glimpses of a strange realm of mathematics increasingly brought to the forefront of standard thought. After understanding this book, mathematics will now seem as though it is incomplete and lacking in concepts that maybe you have wondered before. In this book, we will provide glimpses of something more to mathematics than the real numbers and real analysis. After all, the mathematics we talk about here always seems to only involve one variable in a sea of numbers and operations and comparisons.
Note: A table of the math symbols used below and their definitions is available in the Appendix.
Manual of Style – How to read this wikibook
A select list of chapters curated from other books are listed below. They should help develop your mathematical rigor that is a necessary mode of thought you will need in this book as well as in higher mathematics.
The set theory notation and mathematical proofs, from the book Mathematical Proof
The experience of working with calculus concepts, from the book Calculus
The real numbers Edit
This part of the book formalizes the various types of numbers we use in mathematics, up to the real numbers. This part focuses on the axiomatic properties (what we have defined to be true for the sake of analysis) of not just the numbers themselves but the arithmetic operations and the inequality comparators as well.
{\displaystyle \mathbb {N} }
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Q} }
Axioms of The Real Numbers
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} }
Functions, Trigonometry, and Graphical Analysis Edit
This part of the book formalizes the definition and usage of graphs, functions, as well as trigonometry. The most curious aspect of this section is its usage of graphics as a method of proof for certain properties, such as trigonometry. These methods of proof are mostly frowned upon (due to the inaccuracy and lack of rigorous definition when it comes to graphical proofs), but they are essential to derive the trigonometric relationships, as the analytical definition of the trigonometric functions will make using trigonometry too difficult—especially if they are described early on.
The theorems described on the Inverse Functions chapter requires knowledge of Derivatives.
Trigonometric Functions, as Axioms
Trigonometric Theorems, as Axioms
The following chapters will rigorously define the trigonometric functions. They should only be read after you have a good understanding of derivatives, integrals, and inverse functions.
Trigonometric Functions, Defined
Trigonometric Theorems, Defined
Sequences and series (May 25, 2008)Edit
This part of the book formalizes sequences of numbers bound by arithmetic, set, or logical relationships. This part focuses on concepts such as mathematical induction and the properties associated with sets that are enumerable with natural numbers as well as a limit set of integers.
{\displaystyle \mathbb {R} }
Metric Spaces Edit
This part of the book formalizes the concept of distance in mathematics, and provides an introduction to the analysis of metric space.
Basic Topology of
{\displaystyle \mathbb {R} ^{n}}
(March 29, 2009)Edit
This part of the book formalizes the concept of intervals in mathematics, and provides an introduction to topology.
Limit Points (Accumulation Points)
Limits and Continuity Edit
This part of the book formalizes the concept of limits and continuity and how they form a logical relationship between elementary and higher mathematics. This part focuses on the epsilon-delta definition, how proofs following epsilon-delta operate on, and the implications of limits. It also discusses other topics such as continuity, a special case of limits.
Topological Continuity
This part of the book formalizes differentiation and how they are used to describe the nature of functions. This part focuses on proving how derivatives study the nature of change of a function and how derivatives can provide properties to functions.
Multivariable Derivatives
Neither constructions of the Riemann Integral or the Darboux Integral definition need epsilon-delta limits.
This part of the book formalizes integration and how imagining what area means can yield many different forms of integration. This part focuses on proving how derivatives study the nature of change of a function and how derivatives can provide properties to functions.
Darboux Integral – The method of integration this book defaults to.
Riemann integration – The popular form of integration.
Applications of Integration – The theorems and algebra that use integration
Generalized Integration
Sequences of Functions Edit
Power Series Edit
Multivariate analysis Edit
Differentiation in Rn
Appendices Edit
Here, you will find a list of unsorted chapters. Some of them listed here are highly advanced topics, while others are tools to aid you on your mathematical journey. Since this is the last heading for the wikibook, the necessary book endings are also located here.
Dedekind's construction
Retrieved from "https://en.wikibooks.org/w/index.php?title=Real_Analysis&oldid=3693116"
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Ameena’s boat travels
35
miles per hour. The best fishing spot in the lake is
27
miles away from her starting point.
If she drives her boat for
\frac { 2 } { 3 }
of an hour, will she make it to the best fishing spot on the lake?
Find how far Ameena's boat will have traveled at two thirds of an hour.
(35\text{ miles per hour})\left(\frac{2}{3}\text{ of an hour}\right) = 23.33\text{ miles}
Ameena's boat will have traveled just over
23
miles. She will not have reached the best fishing spot which will still be approximately another
4
miles away.
How long will Ameena need to drive to get to the best fishing spot on the lake? Express your answer in both a portion of an hour and in minutes.
To find how far Ameena's boat has traveled you multiplied the speed by the time, but since time is now the unknown variable, divide the distance by the speed.
\frac{27\text{ miles}}{35\text{ miles per hour}}
0.77
hours. What is the conversion for hours to minutes?
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(Redirected from Trillion (short scale))
{\displaystyle e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots ,}
Literature: 11,206,310 words in Devta by Mohiuddin Nawab, the longest continuously published story known in the history of literature.
{\displaystyle 2^{2^{n}}+1}
{\displaystyle 19683^{3}}
{\displaystyle 27^{9}}
{\displaystyle 3^{27}}
{\displaystyle 3^{3^{3}}}
{\displaystyle 3\uparrow \uparrow 3}
{\displaystyle 3\uparrow \uparrow \uparrow 2}
{\displaystyle 10^{10^{100}}}
{\displaystyle 10^{10^{1,834,102}}}
{\displaystyle 10^{10^{10,000,000}}}
{\displaystyle 10^{\,\!10^{10^{34}}}}
{\displaystyle 10^{10^{10^{56}}}}
{\displaystyle 10^{\,\!10^{10^{100}}}}
{\displaystyle 10^{\,\!10^{10^{963}}}}
{\displaystyle 10^{\,\!10^{10^{...}}}}
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Use-Before-Define: Applying Reaching Definitions Analysis to Workflow Validation | elliott.dev
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Use-Before-Define: Applying Reaching Definitions Analysis to Workflow Validation
This post was adapted from the original post on blog.yolk.dev.
As programmers, we're familiar with linter errors like the one above. When a variable (or a constant, or any named value) is referenced before it's defined, there is a good chance that the program will not work as intended when executed. In many programming languages, a use-before-define will in fact result in a run time error and crash the program:
This raises a question: How does ESLint do this? How can we detect use-before-define without actually executing the program?
How does ESLint detect use-before-define?
ESLint is fundamentally a static code analysis tool. It works by parsing a JavaScript source file into an abstract syntax tree (AST), then applying a sequence of hand-written rules to the AST in order to find various issues. The AST is never actually executed as a program; instead, it's treated as a static structure to be manually traversed, much like how a table of contents can be extracted from an HTML document without rendering the full document in a browser.
Let's look at the source of ESLint's no-use-before-define rule:
eslint/lib/rules/no-use-before-define.js
The implementation is quite simple because ESLint relies on the structured nature of JavaScript to make assumptions about the control flow of the program, in particular sequential ordering and lexical scope. Given the lexical scopes and their variable definitions & references within are already computed (see scope, scope.references, scope.childScopes), finding use-before-define references boils down to checking whether the reference comes after the definition in the source code:
variable.identifiers[0].range[1] < reference.identifier.range[1];
Statements are always executed sequentially within a block, so ESLint can assume that any variable reference which comes before its variable definition yields a use-before-define error.
Why do we use static analysis at all? If ReferenceError is raised at run time, why not simply execute the program to find all such errors? In other words, why not exclusively use dynamic analysis?
Most programs have branching control flow based on external input. In order for a dynamic analysis tool to work, it would need to generate sufficient input data in order to traverse all branches in the program. This presents a challenge because we would need to either automatically generate valid test inputs, or provide a way for the author to manually provide test inputs. This is more akin to unit tests which, while useful for their own reasons, cannot provide the same quick feedback during editing when compared to static analysis.
The Bot Studio programming model and control-flow graphs
Before we get into this, a quick glossary of some Yolk-specific terms:
Bot: a collection of Decision Trees used to power a chat session, with a specific Decision Tree as an entry point.
Bot Studio: the web application where users can create and configure Bots and Decision Trees.
Decision Tree: an automation flow which powers a Bot's chat session, configured by a user in Bot Studio. Not actually a decision tree data structure, but a directed graph.
Validation: the process of analyzing a Bot and its Decision Trees for common configuration mistakes, such as use-before-define instances.
In contrast to JavaScript, the Bot Studio programming model is non-structured. Statements (nodes) are not constrained to sequential execution; the author can link a node to any other node in the Decision Tree as its successor. There are no explicit looping structures (e.g. for / while) to create repetition; a cycle may be arbitrarily created by linking a node to a predecessor.
│node-1│
│node-2│───────────────────────────┐
└──────┘ │
┌──────┐ ┌──────┐ ┌──────┐ ┌──────┐
┌──────│node-3│◀─│node-8│◀──│node-7│◀─│node-6│──┐
│ └──────┘ └──────┘ └──────┘ └──────┘ │
│ │ │ ▲ │
▼ ▼ │ │ │
┌──────┐ ┌──────┐ │ │ │
│node-4│──▶│node-5│─────────────────┼─────────┘ │
└──────┘ └──────┘ ▼ │
│ ┌──────┐ │
└──────────────────────────▶│node-9│◀───────────┘
This model of programming is essentially creating a control-flow graph with no structured constraints. In particular, a Decision Tree is an irreducible control-flow graph. This is a key fact that underlies the types of static analysis and optimizations which are possible on a Decision Tree.
There is no lexical scope or sequential ordering in Bot Studio to help us make assumptions about the control flow, so we need to look to more low-level compiler theory in order to perform analysis on a control-flow graph with looser constraints.
Data-flow analysis applies here; it tells about how values assigned to variables might propagate while executing a program.
A specific type of data-flow analysis, reaching definitions, tells us about which variable definitions can reach a particular instruction in a program. For Bot Studio, we can apply this theory to find out which variables (as defined by SaveValueNode, an input node result, a DT parameter, or a global) can "reach" a given node. If a node references a variable which does not reach, then a use-before-define error should be raised.
There is more formal computer science background on this topic which you can read more about by reading "Additional resources" below, but the following will illustrate the most important concepts as they apply to Bot Studio.
Take this pseudocode example:
node-1: bar = true
node-2: if (bar === true)
node-3: foo = true
goto node-7
node-4: foo = false
node-5: if (bar === false)
node-6: botMessage(bar)
node-7: botMessage(foo)
node-8: return
bar "reaches" all other nodes in the program, because node-1 is executed before any other node. However, foo does not reach node-1 or node-2, because there is no control flow path from node-3 or node-4 to node-1 or node-2.
The above program does not raise any use-before-define errors because all references to bar or foo have corresponding definitions which "reach" the reference.
This time, there is no definition of foo which reaches node-5. The only definition of foo is node-3, and there is no control flow path from node-3 to node-5, so a use-before-define error should be raised at node-5.
Let's look at a final example:
node-3: qux = true
node-9: botMessage(qux)
node-10: return
There are two control flow paths to node-8:
node-1, node-2, node-3, node-8.
node-1, node-2, node-4, node-5, node-8.
Only path (2) includes a definition for foo: node-4: foo = false.
However, even though a path exists along which foo is not defined, foo still "reaches" node-8 according to reaching definitions analysis.
A definition reaches a node if any path to that node includes that definition. It is not a requirement that all paths include that definition. This is a limitation of reaching definitions analysis, because we might expect that a use-before-define error is raised since path (1) exists. However, it turns out that determining all possibly executed paths is not possible in polynomial time. In other words, it is undecidable; an NP-Hard problem. See also:
page 11 of the Carnegie Mellon slides
the Monotone data flow analysis frameworks paper
http://pages.cs.wisc.edu/~horwitz/CS704-NOTES/2.DATAFLOW.html#MOP
Formally, a solution which includes all paths is called the "Meet Over All Paths" (MOP) solution. Achieving a MOP solution in polynomial time is only possible when the framework is distributive. Reaching definitions analysis is a framework which is monotone but not distributive, so achieving a MOP solution is undecidable. By contrast, constant propagation is a framework which is distributive, so achieving a MOP solution is possible.
The worklist algorithm
To perform reaching definitions analysis, the "iterative worklist algorithm" is used. This University of Texas lecture is a great resource describing this algorithm.
Some key points about this algorithm:
The algorithm computes a Maximal Fixed Point (MFP) solution. An MFP solution is the largest of all Fixed Point (FP) solutions, and is unique regardless of the order of iteration. MFP ≤ MOP, so MFP is not guaranteed to reflect all paths in the program.
Yolk uses a custom JavaScript implementation of this algorithm to detect use-before-define instances, which may be open-sourced soon.
https://en.wikipedia.org/wiki/Reaching_definition#Worklist_algorithm
https://www.youtube.com/watch?v=OROXJ9-wUQE
https://suif.stanford.edu/~courses/cs243-winter07.bak/lectures/l2.pdf
http://www.cs.toronto.edu/~chechik/courses16/csc410/dataflowReadings.pdf
https://greg4cr.github.io/courses/spring17csce747/Lectures/Spring17-Lecture9DataFlowAnalysis.pdf
http://www.cs.toronto.edu/~pekhimenko/courses/cscd70-w19/docs/Lecture%202%20[Dataflow]%2001.17.2019.pdf
https://www.cs.utexas.edu/users/mckinley/380C/lecs/04.pdf
https://www.cs.cmu.edu/afs/cs/academic/class/15745-s13/public/lectures/L5-Foundations-of-Dataflow-1up.pdf
https://www.cs.cmu.edu/afs/cs/academic/class/15745-s16/www/lectures/L6-Foundations-of-Dataflow.pdf
http://pages.cs.wisc.edu/~horwitz/CS704-NOTES/2.DATAFLOW.html
https://stackoverflow.com/questions/9535819/find-all-paths-between-two-graph-nodes
A concern was raised that reaching definitions analysis' underreporting of use-before-define errors may be problematic, since an actual use-before-define issue might occur at run time and be missed during validation.
This is a valid concern which may prompt us to create a stricter data-flow analysis framework which raises errors in more cases. However, such a framework might raise errors for a program which is actually valid when executed.
node-1: foo = true
node-2: if (foo === true)
node-3: bar = true
node-4: botMessage("Hello")
node-5: botMessage(bar)
There is a control flow path to node-5 along which bar is not defined: node-1, node-2, node-4, node-5. We want to raise a use-before-define error on node-5 because this path exists.
A data-flow analysis framework which achieves this behaviour is a slight modification on reaching definitions analysis. Instead of using the union meet operator
\wedge = \cup
, use the intersection meet operator
\wedge = \cap
\textrm{IN}[n] = \cup\textrm{ OUT}[p], p \in\textrm{predecessors}(n)
\textrm{IN}[n] = \cap\textrm{ OUT}[p], p \in\textrm{predecessors}(n)
When the intersection is performed, elements must be compared by their name and not their identity, so that two definitions introduced on separate paths are treated as equivalent when merged.
The biggest caveat with this approach is that errors may be raised when the program is actually correct. For example:
node-6: botMessage(bar)
node-7: botMessage(foo)
There is no actual code execution path to node-6 along which bar is not defined. The only case in which node-6 is executed is when foo === true, which also means that bar = true must have been executed.
However, with the modified analysis, a use-before-define error would be raised on node-6 because bar is not defined along the path node-1, node-2, node-4, node-5, node-6. To fix this, the author could add a bar = true statement after node-4:
node-4a: bar = true
A product decision must be made to determine whether this is desired. With the modified analysis, the author would need to create a definition for bar along all paths which reach node-6, even paths which would never be executed.
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!! used as default html header if there is none in the selected theme. Q-Puzzle
This is a mathematical jigsaw puzzle! You have a photo cut into
q\times q
pieces, where
q
is a power of a prime
p
. Therefore it represents the vector plane over the finite field with
q
q\times q
pieces will be presented to you in disorder, result of a (hidden) affine transformation of the plane. And your goal is to recover the image by finding the inverse affine transformation. You have the right to multiple tries in order to do it.
Recall that an affine transformation is a linear transformation followed by a translation. All are operations over .
You may play this game even without understanding the mathematics behind. In this case we suggest you don't go further than than
q=5
otherwise the game risks to be very frustrating.
Choose :
q
= 2 3 4 5 7 8 9 11 (Bigger usually means more difficult puzzle.)
Description: puzzle based on affine transformations on a finite field. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, affine_geometry, finite_field, vector_space, jigsaw
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100
days of school. The numbers of books are
12
17
10
24
18
31
17
21
20
14
30
9
25
. Help Jerome present the data to his teacher.
How many pieces of data, or observations, does Jerome have?
Refer to the data.
\begin{array}{l|l} \text{Note: 4} & \ 1 \text{ means }41 \quad \end{array}
\begin{array}{r|l} 0 & 9 \\ 1 & 0, 2, 4, 7, 7, 8\\ 2 & 0, 1, 4, 5\\ 3 & 0, 1 \end{array}
Jerome wants to present the data with a plot that makes it possible to calculate the mean and the median. Can he do this with a stem-and-leaf plot? He is not asking you to calculate them, but he wants you to tell him if it is possible and why.
Does the stem-and-leaf plot show all of the data points?
Use the stem-and-leaf plot to describe how the data is spread. That is, is it spread out, or is it concentrated mostly in a narrow range?
Where are most of the values concentrated?
Would it be helpful for Jerome to create a dot plot to display and analyze the data from problem 8-66? Why or why not?
Would it be practical to put this information into a dot plot?
No, there are too many data points that do not repeat.
Make a stem-and-leaf plot using the eTool below.
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Research:New active editor - Meta
Research:New active editor
New active editor
A new active editor is a newly registered user completing
{\displaystyle 5}
edits to the contents of a Wikimedia project within
{\displaystyle 30}
days of registration.
This page proposes a canonical definition of a "new active editor", a user class describing the activation of new registered users.
1 WMF standardization
WMF standardization[edit]
The definition of this metric is based on the same assumptions as the new editor definition.
The following plots represent the activation rate for cohorts of monthly registered users on the English Wikipedia since 2007.[1] We compute how many new registered users reach a specified level of activity in the main namespace (1, 5, 10, 100 edits) within 30 days of registering a new account.
Activation of new registered users by month of registration (absolute number).
Activation of new registered users by month of registration (proportion).
This plot compares the e5 and e10 series above with monthly New Wikipedians (based on the canonical definition).
Activation of new registered users by month of registration compared to New Wikipedians (absolute number).
↑ We limit our analysis to users registered after 2007 because of inconsistencies in the logging table with 2005-2006 data.
Retrieved from "https://meta.wikimedia.org/w/index.php?title=Research:New_active_editor&oldid=7883461"
Metrics with no measured phenomena specified
Standardized metric draft
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2020 Optimal Controls of the Highly Active Antiretroviral Therapy
Ellina V. Grigorieva, Evgenii N. Khailov, Andrei Korobeinikov
In this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the
3
-dimensional Nowak–May within-host HIV dynamics model. Taking into consideration that precise forms of functional responses are usually unknown, we introduce into this model a nonlinear incidence rate of a rather general form given by an unspecified function of the susceptible cells and free virus particles. We also add a term responsible to the loss of free virions due to infection of the target cells. To mirror the idea of highly active anti-HIV therapy, in this model we assume six controls that can act simultaneously. These six controls affecting different stage of virus life cycle comprise all controls possible for this model and account for all feasible actions of the existing anti-HIV drugs. With this control model, we consider an optimal control problem of minimizing the infection level at the end of a given time interval. Using an analytical mathematical technique, we prove that the optimal controls are bang-bang, find accurate estimates for the maximal possible number of switchings of these controls and establish qualitative types of the optimal controls as well as mutual relationships between them. Having the estimate for the number of switchings found, we can reduce the two-point boundary value problem for Pontryagin Maximum Principle to a considerably simpler problem of the finite-dimensional optimization, which can be solved numerically. Despite this possibility, the obtained theoretical results are illustrated by numerical calculations using BOCOP–2.0.5 software package, and the corresponding conclusions are made.
Ellina V. Grigorieva. Evgenii N. Khailov. Andrei Korobeinikov. "Optimal Controls of the Highly Active Antiretroviral Therapy." Abstr. Appl. Anal. 2020 1 - 23, 2020. https://doi.org/10.1155/2020/8107106
Received: 16 April 2019; Accepted: 8 August 2019; Published: 2020
Ellina V. Grigorieva, Evgenii N. Khailov, Andrei Korobeinikov "Optimal Controls of the Highly Active Antiretroviral Therapy," Abstract and Applied Analysis, Abstr. Appl. Anal. 2020(none), 1-23, (2020)
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Econometrics - Wikipedia
Basic models: linear regression[edit]
For example, consider Okun's law, which relates GDP growth to the unemployment rate. This relationship is represented in a linear regression where the change in unemployment rate (
{\displaystyle \Delta \ {\text{Unemployment}}}
) is a function of an intercept (
{\displaystyle \beta _{0}}
), a given value of GDP growth multiplied by a slope coefficient
{\displaystyle \beta _{1}}
and an error term,
{\displaystyle \varepsilon }
{\displaystyle \Delta \ {\text{Unemployment}}=\beta _{0}+\beta _{1}{\text{Growth}}+\varepsilon .}
The unknown parameters
{\displaystyle \beta _{0}}
{\displaystyle \beta _{1}}
can be estimated. Here
{\displaystyle \beta _{0}}
is estimated to be 0.83 and
{\displaystyle \beta _{1}}
is estimated to be -1.77. This means that if GDP growth increased by one percentage point, the unemployment rate would be predicted to drop by 1.77 * 1 points, other things held constant. The model could then be tested for statistical significance as to whether an increase in GDP growth is associated with a decrease in the unemployment, as hypothesized. If the estimate of
{\displaystyle \beta _{1}}
were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related. The variance in a prediction of the dependent variable (unemployment) as a function of the independent variable (GDP growth) is given in polynomial least squares.
{\displaystyle \ln({\text{wage}})=\beta _{0}+\beta _{1}({\text{years of education}})+\varepsilon .}
This example assumes that the natural logarithm of a person's wage is a linear function of the number of years of education that person has acquired. The parameter
{\displaystyle \beta _{1}}
measures the increase in the natural log of the wage attributable to one more year of education. The term
{\displaystyle \varepsilon }
is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters,
{\displaystyle \beta _{0}{\mbox{ and }}\beta _{1}}
under specific assumptions about the random variable
{\displaystyle \varepsilon }
{\displaystyle \varepsilon }
is uncorrelated with years of education, then the equation can be estimated with ordinary least squares.
The most obvious way to control for birthplace is to include a measure of the effect of birthplace in the equation above. Exclusion of birthplace, together with the assumption that
{\displaystyle \epsilon }
is uncorrelated with education produces a misspecified model. Another technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render
{\displaystyle \beta _{1}}
identifiable.[17] An overview of econometric methods used to study this problem were provided by Card (1999).[18]
Wikimedia Commons has media related to Econometrics.
Look up econometrics in Wiktionary, the free dictionary.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Econometrics&oldid=1049722590"
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Synthesis and Crystal Structure of Alkynylplatinum(IV) Complex Containing the Terpyridine Ligand
Tsugiko Takase, Dai Oyama, "Synthesis and Crystal Structure of Alkynylplatinum(IV) Complex Containing the Terpyridine Ligand", Journal of Crystallography, vol. 2014, Article ID 280247, 5 pages, 2014. https://doi.org/10.1155/2014/280247
Tsugiko Takase1 and Dai Oyama 2
1Institute of Environmental Radioactivity, Fukushima University, 1 Kanayagawa, Fukushima 960-1296, Japan
2Department of Industrial Systems Engineering, Cluster of Science and Engineering, Fukushima University, 1 Kanayagawa, Fukushima 960-1296, Japan
Reaction of square planar [(C≡CPh)(tpy)]+ (tpy = 2,2′:6′,2′′-terpyridine) with bromine at low temperature provides a general route for the synthesis of octahedral alkynyl(terpyridine)platinum(IV) complex. In this first example of alkynyl(terpyridine)platinum(IV) complex, the alkynyl group is situated in trans position relative to the central nitrogen atom of the terpyridine ligand, and the two bromido ligands are situated in trans positions; an X-ray structural analysis has been completed for trans(Br)-[Br2(C≡CPh)(tpy)]+.
Platinum(II) terpyridyl complexes with alkynyl ligands have attracted great interest in recent years due to their unique photophysics [1, 2] and their potential applications as photocatalysts for hydrogen evolution [3, 4] and molecular frameworks for light-to-chemical energy conversion [5]. On the other hand, there is a limited range of reported alkynylplatinum(IV) complexes [6–8]: there is no report on alkynylplatinum(IV) complex with terpyridine ligands. Although general oxidizing agents such as halogens and hydrogen peroxide are utilized for oxidation reactions of to centers, halogens are particularly the most useful reagents [9, 10]. However, only one example for oxidation of alkynylplatinum(II) by halogens has been known in [(Me2bpy)(C≡C-4-tol)2] by iodine to form [I2(Me2bpy)(C≡C-4-tol)2] [6], due to instability of the -alkynyl bond to halogen-containing oxidants [11]. Therefore, other procedures using 4-nitrophenyl azide and alkynyliodine(III) reagents as oxidants have been explored for oxidation of to [7, 8, 12]. In order to generalize about routes to alkynylplatinum(IV) compounds by halogen-oxidation reaction without Pt-alkynyl bond breaking, we have optimized reaction conditions (the kind of oxidants and temperature) using alkynylplatinum(II) containing the terpyridine ligand as a prototype complex.
All chemicals employed here were used without further purification unless otherwise stated. All solvents purchased for organic synthesis were anhydrous and used without further purification. A Br2–CHCl3 solution was prepared by addition of 12 drops of Br2 to 10 mL of CHCl3 [10]. A Cl2–CHCl3 solution was prepared by a saturated Cl2 gas through 10 mL of CHCl3 for approximately 15 sec [9]. The platinum(II) precursor ([(C≡CPh)(tpy)]PF6) was prepared in accordance with the published method [2]. 1H NMR spectra were recorded on a JEOL JMN-AL300 spectrometer (25°C) operating at 1H frequency of 300 MHz. ESI-MS data were measured on a Bruker Daltonics micrOTOF equipped with electrospray ionization (ESI) source and CH3CN was used as the solvent. The instrument was operated at positive ion mode using m/z range of 100–1000. IR spectra were obtained using the KBr pellet method with a JASCO FT-IR 4100 spectrometer.
2.1. X-Ray Crystal Structure Determination
Single crystals of 1Br were obtained from a solution of the complex in DMF/diethyl ether. A brown crystal of 1Br with dimensions mm was mounted on a glass fiber. All measurements were performed on a Rigaku R-AXIS RAPID diffractometer with graphite monochromated Mo K radiation . All calculations were conducted using the CrystalStructure program package [13] except for refinement, which was performed using SHELXL97 [14]. The structure was solved by direct methods using SIR92 program [15]. A numerical absorption correction (ABSCOR) [16] was applied to data. Aromatic hydrogen atoms were fixed at C–H lengths of 0.95 Å refined as riding, with . Both the highest residual electron peak and the deepest hole are located within 1 Å from atom Pt1. The crystallographic data are summarized in Table 1 and the geometrical parameters are summarized in Table 2. The crystal data for 1Br has been deposited with Cambridge Crystallographic Data Centre as supplementary publication number CCDC-1014039.
Empirical formula C23H16Br3N3Pt
(Å) 0.71075
Space group -1
(Å) 7.2090(4)
(Å) 12.9921(6)
(deg) 67.0477(14)
(Å3) 1089.25(9)
(Mg/m3) 2.345
(Mo K) (mm−1) 11.950
GOF () 1.095
1a () 0.0486
2b (all data) 0.1210
Crystallographic details for trans(Br)-[PtBr2(C≡CPh)(tpy)]Br.
(a) Selected bond lengths and angles (Å, °)
Pt1–Br1 2.4498(12) C1–C2 1.059(17)
Pt1–Br2 2.4644(12) N1–Pt1–N2 80.8(4)
Pt1–N1 2.041(10) N2–Pt1–N3 80.2(4)
Pt1–N2 1.976(9) Pt1–C1–C2 175.3(12)
Pt1–N3 2.030(11) C1–C2–C3 174.4(14)
Pt1–C1 2.054(10) Br1–Pt1–Br2 178.46(5)
(b) Interatomic distances (Å) for intermolecular - interactions
C2⋯C22i 3.299(15)
C4⋯C23ii 3.366(17)
Symmetry operators: i, , ; ii, , .
(c) Hydrogen-bond geometry (Å, °)
D–H⋯A D–H H⋯A D⋯A D–H⋯A
C4–H1⋯Br1i 0.95 2.84 3.688(14) 149
C11–H8⋯Br1ii 0.95 2.86 3.761(13) 160
C22–H15⋯Br1i 0.95 3.34 3.836(14) 115
C4–H1⋯Br2iii 0.95 3.49 4.072(13) 122
C16–H11⋯Br2iv 0.95 2.87 3.702(14) 147
C10–H7⋯Br2v 0.95 2.94 3.733(13) 142
C8–H5⋯Br3ii 0.95 3.12 3.758(13) 126
C22–H15⋯Br3vi 0.95 3.20 3.872(12) 129
C12–H9⋯Br3 0.95 3.19 3.945(13) 138
C15–H10⋯Br3 0.95 2.89 3.597(13) 132
C6–H3⋯Br3vii 0.95 3.32 3.943(13) 125
C17–H12⋯Br3viii 0.95 2.89 3.819(15) 166
Symmetry operators: i, , ; ii, , ; iii, , ; iv, , ; v, , ; vi, , ; vii, , ; viii, , .
2.2. Preparation of trans(Br)-[PtBr2(C≡CPh)(tpy)]Br (1Br)
An orange suspension of [Pt(C≡CPh)(tpy)]PF6 (30 mg, 0.044 mmol) in acetonitrile (10 mL) was stirred at −20°C for 30 min. Addition of Br2–CHCl3 (70 drops) to the suspension resulted in a homogeneous solution. The mixture was further stirred at −20°C for 2 h; during this time some light brown solids gradually appeared out of the solution. The product was collected by filtration, washed with diethyl ether, and then dried in vacuo (25 mg, 73%). ESI+-MS: m/z = 690 ([M]+), 529 ([M–2Br]+). 1H NMR (DMSO-): 9.29 ( with broad 195Pt satellites, 2H), 9.13–8.88 (, 5H), 8.70 (, 2H), 8.20 (, 2H), 7.65 (, 2H), 7.47–7.35 (, 3H). IR: (C≡C) 2165 cm−1.
3.1. Synthesis of trans(Br)-[PtBr2(C≡CPh)(tpy)]Br
The synthetic route for the formation of platinum(IV) terpyridyl complexes is summarized in Scheme 1. Addition of Br2–CHCl3 solution to [(C≡CPh)(tpy)]+ in acetonitrile at −20°C gave [Br2(C≡CPh)(tpy)]+ (1+) as light brown solid in 73% yield. When this reaction was performed at room temperature, 1+ was produced but a small quantity of the alkynyl dissociated species was detected. In the same condition, however, addition of Cl2–CHCl3 solution gave some impurities in addition to [Cl3(tpy)]+ (m/z = 534, major product) and [Cl2(C≡CPh)(tpy)]+ (m/z = 600, minor product) (Scheme 1). These results indicate that the Pt–C bond in [Pt(C≡CPh)(tpy)]+ is influenced by the strength of oxidants and reaction temperatures: milder conditions (Br2 as an oxidant and low temperature) are required for bond retention of the Pt-alkynyl moiety.
Synthetic route for platinum(IV) complexes in this study.
3.2. Characterization of trans(Br)-[PtBr2(C≡CPh)(tpy)]+
The identities of the complex 1+ have been confirmed by IR spectroscopy, 1H NMR spectroscopy, and ESI-mass spectroscopy. The molecular structure of 1Br has also been determined. To the best of our knowledge, this study demonstrates the first crystal structure of an alkynylplatinum(IV) with the terpyridine ligand.
The IR spectrum of 1Br exhibited (C≡C) absorption at 2165 cm−1, which is 40 cm−1 higher than that of [(C≡CPh)(tpy)]+ [17]. The 1H NMR spectra of 1+ showed resonances at more downfield regions compared to the corresponding complex [17]. For example, the doublet signals of 6,6′′-positions in terpyridine ligand of 1+ appeared at 9.29 ppm, whereas those of [(C≡CPh)(tpy)]+ appeared at 9.10 ppm, indicating a lowering of the electron density on a Pt center.
The molecular structure of 1Br is shown in Figure 1 with atom-labeling scheme. The complex consists of a distorted octahedral geometry around the center with three nitrogen atoms of the terpyridine ligand, two bromido ligands in trans positions, and an alkynyl group in trans position relative to the central nitrogen atom (1′-position) of the terpyridine ligand. The terpyridine ligand is coordinated in a planar tridentate fashion with the central nitrogen atom closest to the atom. As shown in Table 2(a), the bond lengths of Pt–N are comparable to the precursor platinum(II) terpyridine complex [2]. Additionally, the Pt1–C1–C2 and C1–C2–C3 fragments are nearly linear with bond angles of 175.3(12)° and 174.4(14)°, respectively. However, both the Pt–C and the C≡C bond lengths (2.054(10) and 1.059(17) Å, resp.) are different from those of the corresponding complex (1.98(1) and 1.19(1) Å, resp.) [2]. In the crystal, the cation of 1Br features intermolecular - stacking between the phenyl group of the alkynyl ligand and the terpyridine ligand, as represented by the shortest contact of 3.256(16) Å for C5⋯C23 (Table 2(b)). The phenyl ring of the alkynyl ligand is almost parallel to the plane of Pt-terpyridine with a dihedral angle of 5.438°, indicating the formation of a face-to-face - interaction (Figure 2). In addition, there are a number of intermolecular weak C–H⋯Br interactions in the crystal as illustrated in Figure 2. These interactions help to stabilize the structure.
The molecular structure of trans(Br)-[PtBr2(C≡CPh)(tpy)]Br (1Br), showing the atom-labeling scheme and displacement ellipsoids drawn at the 50% probability level.
Packing diagram. The green lines denote intermolecular hydrogen bonds.
In summary, we have prepared an alkynylplatinum(IV) complex by simple oxidation of the corresponding alkynylplatinum(II) one. It is anticipated that the successful route described here provides a useful methodology to obtain a variety of platinum(IV) complexes with a wide range of alkynyl ligands.
Complete lists of positional and isotropic displacement coefficients for hydrogen atoms and anisotropic displacement coefficients for the nonhydrogen atoms, bond lengths and angles, and torsion angles are listed as supplementary material available on line at http://dx.doi.org/10.1155/2014/280247.
H. Watanabe is thanked for experimental assistance at an early stage of the project.
Complete lists of positional and isotropic displacement coefficients for hydrogen atoms and anisotropic displacement coefficients for the nonhydrogen atoms, bond lengths and angles, and torsion angles.
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Copyright © 2014 Tsugiko Takase and Dai Oyama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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!! used as default html header if there is none in the selected theme. SQRT shoot
n
n
n\ge 1
z
z
Description: locate (square, cubic, ...) roots of a complex number by clicking on the complex plane. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, complex_number, complex_plane, roots
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\mathrm{ω}
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{\mathrm{ω}}_{2}
\textcolor[rgb]{0,0,1}{\mathrm{ω2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{\mathrm{ω}}_{3}
{\mathrm{ω}}_{3}
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{J}^{3}\left(E\right)
E
\left(x,y, u, v\right)→ \left(x,y\right)
{\mathrm{ω}}_{4}.
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\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{d}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{f}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{e}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}]]]\right)
{\mathrm{ω}}_{5}.
\textcolor[rgb]{0,0,1}{\mathrm{ω5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]]]\right)
{\mathrm{ω}}_{6}
\mathrm{η}.
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\textcolor[rgb]{0,0,1}{\mathrm{ω6}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]]]\right)
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Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks | J. Turbomach. | ASME Digital Collection
Tony Shardlow,
Manuscript received June 26, 2015; final manuscript received July 29, 2015; published online September 23, 2015. Editor: Kenneth C. Hall.
Tang, H., Shardlow, T., and Michael Owen, J. (September 23, 2015). "Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks." ASME. J. Turbomach. December 2015; 137(12): 121003. https://doi.org/10.1115/1.4031355
Conduction in thin disks can be modeled using the fin equation, and there are analytical solutions of this equation for a circular disk with a constant heat-transfer coefficient. However, convection (particularly free convection) in rotating-disk systems is a conjugate problem: the heat transfer in the fluid and the solid are coupled, and the relative effects of conduction and convection are related to the Biot number,
Bi,
which in turn is related to the Nusselt number. In principle, if the radial distribution of the disk temperature is known then
Bi
can be determined numerically. But the determination of heat flux from temperature measurements is an example of an inverse problem where small uncertainties in the temperatures can create large uncertainties in the computed heat flux. In this paper, Bayesian statistics are applied to the inverse solution of the circular fin equation to produce reliable estimates of Bi for rotating disks, and numerical experiments using simulated noisy temperature measurements are used to demonstrate the effectiveness of the Bayesian method. Using published experimental temperature measurements, the method is also applied to the conjugate problem of buoyancy-induced flow in the cavity between corotating compressor disks.
Buoyancy, Cavities, Disks, Flow (Dynamics), Heat transfer, Inverse problems, Rotating disks, Temperature, Temperature measurement, Compressors, Statistics as topic, Temperature distribution
A Bayesian Inference Approach to the Inverse Heat Conduction Problem
A Bayesian Approach for the Simultaneous Estimation of Surface Heat Transfer Coefficient and Thermal Conductivity From Steady State Experiments on Fins
Flow and Heat Transfer in Rotating Disc Systems—Rotating Cavities
Research Studies Press, Taunton, UK/Wiley
Vortex Breakdown in a Rotating Cylindrical Cavity
Rotating Cavity With Axial Throughflow of Cooling Air: Heat Transfer
Spatial Variation. Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations
Medd. Statens Skogsforskningsinstitut
Heat Transfer in High-Pressure Compressor Gas Turbine Internal Air Systems: A Rotating Disc-Cone Cavity With Axial Throughflow
,” D.Phil.,
Flow and Heat Transfer in Rotating-Disc Systems—Rotor-Stator Systems
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There are five students of different ages. Their median age is
13
What are two possibilities for the ages of the three oldest students?
The median is the middle number of a data set ordered from least to greatest.
13
is the median age in a set of five students, the two oldest students must be older than
13
If each student is a different age, how many students must be younger than
13
\text{ __ __ } \ \color{blue}{13}\ \ \text{__ __}
2
In complete sentences, describe what a median is to another student in the class. Pretend that the student was absent the day you learned about medians
What do you do when two numbers share the middle?
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Liebig's_law_of_the_minimum Knowpia
Liebig's law of the minimum, often simply called Liebig's law or the law of the minimum, is a principle developed in agricultural science by Carl Sprengel (1840) and later popularized by Justus von Liebig. It states that growth is dictated not by total resources available, but by the scarcest resource (limiting factor). The law has also been applied to biological populations and ecosystem models for factors such as sunlight or mineral nutrients.
This was originally applied to plant or crop growth, where it was found that increasing the amount of plentiful nutrients did not increase plant growth. Only by increasing the amount of the limiting nutrient (the one most scarce in relation to "need") was the growth of a plant or crop improved. This principle can be summed up in the aphorism, "The availability of the most abundant nutrient in the soil is only as good as the availability of the least abundant nutrient in the soil." Or, to put it more plainly, "A chain is only as strong as its weakest link." Though diagnosis of limiting factors to crop yields is a common study, the approach has been criticized.[1]
Scientific applicationsEdit
Liebig's law has been extended to biological populations (and is commonly used in ecosystem modelling). For example, the growth of an organism such as a plant may be dependent on a number of different factors, such as sunlight or mineral nutrients (e.g., nitrate or phosphate). The availability of these may vary, such that at any given time one is more limiting than the others. Liebig's law states that growth only occurs at the rate permitted by the most limiting factor.[2]
For instance, in the equation below, the growth of population
{\displaystyle O}
is a function of the minimum of three Michaelis-Menten terms representing limitation by factors
{\displaystyle I}
{\displaystyle N}
{\displaystyle P}
{\displaystyle {\frac {dO}{dt}}=O\left(\min \left({\frac {\mu _{I}I}{k_{I}+I}},{\frac {\mu _{N}N}{k_{N}+N}},{\frac {\mu _{P}P}{k_{P}+P}}\right)-m\right)}
The use of the equation is limited to a situation where there are steady state ceteris paribus conditions, and factor interactions are tightly controlled.
Protein nutritionEdit
In human nutrition, the law of the minimum was used by William Cumming Rose to determine the essential amino acids. In 1931 he published his study "Feeding experiments with mixtures of highly refined amino acids".[3] Knowledge of the essential amino acids has enabled vegetarians to enhance their protein nutrition by protein combining from various vegetable sources. One practitioner was Nevin S. Scrimshaw fighting protein deficiency in India and Guatemala. Francis Moore Lappe published Diet for a Small Planet in 1971 which popularized protein combining using grains, legumes, and dairy products.
More recently Liebig's law is starting to find an application in natural resource management where it surmises that growth in markets dependent upon natural resource inputs is restricted by the most limited input. As the natural capital upon which growth depends is limited in supply due to the finite nature of the planet, Liebig's law encourages scientists and natural resource managers to calculate the scarcity of essential resources in order to allow for a multi-generational approach to resource consumption.
Neoclassical economic theory has sought to refute the issue of resource scarcity by application of the law of substitutability and technological innovation. The substitutability "law" states that as one resource is exhausted—and prices rise due to a lack of surplus—new markets based on alternative resources appear at certain prices in order to satisfy demand. Technological innovation implies that humans are able to use technology to fill the gaps in situations where resources are imperfectly substitutable.
A market-based theory depends on proper pricing. Where resources such as clean air and water are not accounted for, there will be a "market failure". These failures may be addressed with Pigovian taxes and subsidies, such as a carbon tax. While the theory of the law of substitutability is a useful rule of thumb, some resources may be so fundamental that there exist no substitutes. For example, Isaac Asimov noted, "We may be able to substitute nuclear power for coal power, and plastics for wood ... but for phosphorus there is neither substitute nor replacement."[4]
Where no substitutes exist, such as phosphorus, recycling will be necessary. This may require careful long-term planning and governmental intervention, in part to create Pigovian taxes to allow efficient market allocation of resources, in part to address other market failures such as excessive time discounting.
Liebig's barrelEdit
Dobenecks[5] used the image of a barrel—often called "Liebig's barrel"—to explain Liebig's law. Just as the capacity of a barrel with staves of unequal length is limited by the shortest stave, so a plant's growth is limited by the nutrient in shortest supply.
If a system satisfies the law of the minimum then adaptation will equalize the load of different factors because the adaptation resource will be allocated for compensation of limitation.[6] Adaptation systems act as the cooper of Liebig's barrel and lengthens the shortest stave to improve barrel capacity. Indeed, in well-adapted systems the limiting factor should be compensated as far as possible. This observation follows the concept of resource competition and fitness maximization.[7]
Due to the law of the minimum paradoxes, if we observe the Law of the Minimum in artificial systems, then under natural conditions adaptation will equalize the load of different factors and we can expect a violation of the law of the minimum. Inversely, if artificial systems demonstrate significant violation of the law of the minimum, then we can expect that under natural conditions adaptation will compensate this violation. In a limited system life will adjust as an evolution of what came before.[6]
One example of technological innovation is in plant genetics whereby the biological characteristics of species can be changed by employing genetic modification to alter biological dependence on the most limiting resource. Biotechnological innovations are thus able to extend the limits for growth in species by an increment until a new limiting factor is established, which can then be challenged through technological innovation.
Theoretically there is no limit to the number of possible increments towards an unknown productivity limit.[8] This would be either the point where the increment to be advanced is so small it cannot be justified economically or where technology meets an invulnerable natural barrier. It may be worth adding that biotechnology itself is totally dependent on external sources of natural capital.
Bottleneck (disambiguation)
^ Thomas R. Sinclair and Wayne R. Park (1993) "Inadequacy of the Liebig limiting-factor paradigm for explaining varying crop yields", Agronomy Journal 85(3): 472–6 doi:10.2134/agronj1993.00021962008500030040x
^ Sinclair, Thomas R. (1999). "Limits to Crop Yield". Plants and Population: is there time?. Colloquium. Washington DC: National Academy of Sciences. doi:10.17226/9619. ISBN 978-0-309-06427-9. Archived from the original on 2011-07-03.
^ W.C. Rose (1931) Feeding Experiments, Journal of Biological Chemistry 94: 155–65
^ Asimov, Issac (1962). "Life's Bottleneck". Fact and Fancy. Doubleday.
^ Whitson, A.R.; Walster, H.L. (1912). Soils and soil fertility. St. Paul, MN: Webb. p. 73. OCLC 1593332. 100. Illustration of Limiting Factors. The accompanying illustration devised by Dr. Dobenecks is intended to illustrate this principle of limiting factors.
^ a b A.N. Gorban, L.I. Pokidysheva, E.V. Smirnova, T.A. Tyukina. Law of the Minimum Paradoxes, Bull Math Biol 73(9) (2011), 2013–2044
^ D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, NJ (1982).
^ Reilly, J.M.; Fuglie, K.O. (6 July 1998). "Future yield growth in field crops: what evidence exists?". Soil and Tillage Research. 47 (3–4): 275–290. doi:10.1016/S0167-1987(98)00116-0.
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Home : Support : Online Help : Graphics : Packages : Plot Tools : cylinder
generate 3-D plot object for a cylinder
cylinder(c, r, h, capped=boolean, strips=n, options)
center of the base circle of the cylinder
(optional) radius of the cylinder; default is 1
(optional) height of the cylinder; default is 1
(optional) specifies whether the resulting object should be capped; default is true
strips=n
(optional) number of sides; default is 24
The cylinder command creates a three-dimensional plot data object, which when displayed is a cylinder of height h. The base circle of the cylinder is centered at c, of radius r, and parallel to the xy-plane.
The plot data object produced by the cylinder command can be used in a PLOT3D data structure, or displayed using the plots[display] command.
The cylinder is capped on both ends unless you specify the capped = false option.
The cylinder is constructed of n vertical strips, which can be specified with the strips = n option. The default number of strips is 24.
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{display}\left(\mathrm{cylinder}\left([1,1,1],1,3\right),\mathrm{orientation}=[45,70],\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{display}\left(\mathrm{cylinder}\left([1,1,1],1,3,\mathrm{capped}=\mathrm{false},\mathrm{strips}=40\right),\mathrm{orientation}=[45,70],\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{display}\left(\mathrm{cylinder}\left([1,1,1],3,1\right),\mathrm{orientation}=[45,70],\mathrm{scaling}=\mathrm{constrained}\right)
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Description: a computing exercise on Mean Value Theorem. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
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Effect of confinement pressure on bearing capacity of two samples of square and strip footing (numerical study) | SpringerPlus | Full Text
Aarash Hosseini1
This paper presents the results of modeling tests of the Effect of Confinement Pressure on footing bearing capacity of two kinds of square and strip footing. Footings bearing capacity depends upon many factors including soil kind, depth, form and kind of loading. Soil behavior is variable regarding the kind of loading and the kind of deformations in that can have great importance in the amount of bearing capacity. The kind of deformations depends on the amount of pressure on soil in the past and present. Therefore, studying the role of stress way, which is subject to the amount of confinement pressure on soil, will have an important role in identifying soil behavior. In this study, primarily the effect of confinement pressure on the cohesion and friction angle is studied. Then the effect of both on the bearing capacity with the Meyerhof and Terzaghi methods is evaluated. By using Plaxis software, changes of shearing resistance parameters of both samples different Confinement pressures are studied and bearing capacity of two kinds of square and strip footing has been computed and compared. This study indicated that the amount of bearing capacity by increasing lateral pressure increased, and this increasing is more in grain soil than cohesion one.
The practices of improvement of soil by different techniques have been received as of late by Civil Engineering experts. Utilization of sites with marginal soil properties has been expanded due to the need accessibility of good construction sites. Because of this, improve foundation soil bearing capacity has risen noticeably. One method of improving soil capacity is soil confinement. Using metalcell, geocell are the current improvement this field to supply confinement to the soil. Civil Engineering professionals have applied these novel approaches efficiently in several fields of Geotechnical engineering; however they have not obtained much attention in foundation applications. Over the last few decades, via consideration of soil and structure interaction great strides in the modification of existing forms of foundations along with the development of new and unconventional types of foundation systems have occurred. This results in a system utilizing the form and material strength that is more realistic in performance. One of these novel methods is the lateral confinement of cohesion less soil. The effect of lateral confinement on bearing capacity, especially on sandy soil has been studied by many researchers.
Confining the soil is reductions in the settlement have been concluded by these researchers, and hence an increase is in the bearing capacity of the soil.
Arrive at optimum dimensions of the cell have been concluded by a series of model plate loading tests on circular footings supported over sand-filled square-shaped paper grid cells to identify different modes of failure (Rea and Mitchell (1978)).
experimental study concerning a method of improving the bearing capacity of the strip footing resting on sand sub-grades utilizing vertical non extensible reinforcement were presented by Mahmoud and Abdrabbo (1989). The test results indicate that the bearing capacity of sub-grades and modifies the load– displacement behavior of the footing is increased with this type of reinforcement.
The laboratory-model test results for the bearing capacity of a strip foundation supported by a sand layer reinforced with layers of geogrid were investigated by Khing et al. (1993).
The ultimate bearing capacity of strip and square foundations supported by sand reinforced with Geogrid were studied by Puri et al. (1993). The ultimate bearing capacity of surface strip foundations on geo grid-reinforced sand and unreinforced sand were presented by Omar et al. (1993a, [b]). the use of vertical reinforcement along with horizontal reinforcement consisted of a series of interlocking cells, constructed from polymer Geogrids, which contain and confine the soil within its pockets were investigated by Dash et al. (2001b). Mandal and Manjunath (1995) used geo grid and bamboo sticks as vertical reinforcement elements, also they studied their effect on the bearing capacity of a strip footing. Rajagopal et al. (1999) have studied the strength of confined sand, the influence of geo cell confinement on the strength and stiffness behavior of granular soils. An experimental study on the bearing capacity of a strip footing supported by a sand bed reinforced with a geo cell mattress was performed by Dash et al. (2001a). Strip foundations but reinforced with different materials such as steel bars also was studied by Several authors (Milovic, 1977; Bassett and Last, 1978; Verma and Char 1986), steel grids (Dawson and Lee, 1988; Abdel-baki et al. 1993), geotextile (Das 1987), and geogrids (Milligan and Love, 1984; Ismail and Raymond, 1995). The results of laboratory model tests on the effect of soil confinement on the behavior of a model footing resting on Ganga sand under eccentric – inclined load were presented by Vinod Kumar Singh et al. Confining cells with different heights and widths have been used to confine the sand.
In this research, Plaxis software has been used for numerical modeling. PLAXIS is a three-dimensional finite element program especially developed for the analysis of foundation structures, including off-shore foundations. It combines simple graphical input procedures, which allow the user to automatically generate complex finite element models, with advanced output facilities and robust calculation procedures. The program is designed such that a user can analyze complex constructions after only a few hours of training. This program can model the soil behavior under loading as well as it happens in the nature.
In order to simulating soil behavior, hardening soil model has been used. Used parameters for samples are presented in Table 1.
Table 1 The specified parameters of soil samples used in numeral analyses
The boundary condition is modified in one of vertical sides of the model as grid along × direction and transferable along y direction and beneath the model is grid along y direction and transferable along × direction. So, in addition to preservation of balance of the entire model in horizontal side, it’s move along with vertical will also be released that are the direction of weight power and enforcing load.
The following assumptions have been considerable for simpler analysis.
The issue has been analyzed as an axisymmetry model.
Considering long term behavior of soil, the sample has been studied in drained condition.
The study has been carried out parametrical.
3 Methodology and the results of analysis
Several different approaches in determination of the bearing capacity of shallow foundations have been generally employed in the past decades. The famous triple-N formula of them is Terzaghi, and can be written as given in equation (1)
{q}_{\mathrm{u}\mathrm{l}\mathrm{t}}=c{N}_{c}+q{N}_{q}+0.5\mathrm{\gamma }B{N}_{\gamma }
Where, qult is the ultimate bearing capacity of soil mass, c is the cohesion, q is the surcharge pressure, B is the foundation width and γ is the unit weight of soil mass. Similarly N c , N q , N γ are bearing capacity factors, which are functions of the soil friction angle. The second and third terms in equation (1) have been known as the main contributor to the bearing capacity of shallow foundations on non-cohesive soils. Different investigators such as Terzaghi (1943), Meyerhof (1963), Hansen (1970), Vesic (1973), Bolton and Lau (1989) suggested values for the third factor.
Although all these methods are generally based on a limit equilibrium solution, there are differences between their assumptions for boundary conditions and consideration of the soil weight effect. taking several assumptions in to account comp uted the third bearing capacity factor i.e. N γ . Terzaghi (1943) assumed that, the components of bearing capacity equation can be safely superposed. Meyerhof (1951, 1963), proposed a bearing-capacity equation similar to that of Terzaghi but included a shape factor s-q with the depth term Nq. He also included depth factors and inclination factors.
Beside these assumptions, almost all conventional methods assume a constant value of soil friction angle to compute the bearing capacity factors. Generally, in calculating the footing bearing capacity, the condition of lateral pressure on soil is not considered. The amount of friction angle and cohesion are counted based on resulted average of some experimental tests while the structural value of foundation is ignored. However, foundation can have major effect on the amount of stress in the soil [Abdel-baki et al. (1993), Das and Omar (1994), Das et al. (1996), De Beer (1970), Fragaszy and Lawton (1984), Meyerhof (1953, 1965)].
In this research, by using Plaxis software, changes of shearing resistance parameters of both samples confinement pressure of 100,300,600 ,1000 ,1500 and 2000 kN/m2 are studied and bearing capacity of two kinds of square and strip footing based on Terzaghi and Meyerhof methods has been computed and compared.
These numerals analysis attempt to provide a better understanding of the effect of confinement pressure on the bearing capacity.
In order to study the effect of confinement pressure on bearing capacity and its parameters, at first, the changes of coherency and the friction angle has been studied and represented in Table 2.
Table 2 Amount of coherency and friction angle
After that, a square footing and a strip one in dimensions of 2 × 2 m and 2 × 10 m has been and the coefficients of bearing capacity and ultimate bearing capacity based on Terzaghi and Meyerhof methods, which have the most applications, by considering the amount of friction angle and coherency obtained in confinement pressure have calculated. The obtained result has been presented in Tables 3and 4 and Figures 1 and 2.
Table 3 The coefficients and the ultimate bearing capacity in sample 1
Bearing capacity changes in sample 1.
In order to Study The effect of confinement pressure on bearing capacity, The ultimate bearing capacity related to each Sample has been presented at to pressure of 100 and 2000 kN/m2 in Table 5 and are compared by applying bearing capacity ratio coefficient (BCR).
Table 5 Comparing the bearing capacity of samples
Based on results obtained, it is observed that bearing capacity in sample 1 in strip footing with Terzaghi method increases 6.59 fold this increase in square footing is with Terzaghi and Meyerhof method 7.84 and 6.79 fold, respectively. Also in sample2 in strip footing with Terzaghi method increases 1.85 fold and with Meyerhof method increases 1.89 fold this increases in square footing is 1.87 Fold with Terzaghi and Meyerhof methods. By comparing obtained results, it way resulted that increasing confinement pressure in grain soils have more effect on increasing bearing capacity.
Abdel-baki S, Raymond GP, Johnson P: Improvement of the Bearing Capacity of Footing by a Single Layer of Reinforcement. Proceedings, Vol. 2, Geosynthetics 93 Conference, Vancouver, Canada; 1993:407-416.
Bassett RH, Last NC: Reinforcing Earth Below Footing and Embankments. Symposium on Earth Reinforcement, ASCE, Pittsburgh; 1978:202-231.
Bolton MD, Lau CK: Scale Effect in the Bearing Capacity of Granular Soils. International Proceeding of the 12th International conference on Soil Mechanics and Foundations Engineering, Rio De Janeiro, Brazil; 1989:895-898.
Das BM: Shallow Foundation in Clay with Geotextile Layers. Proceedings of the 8th Pan American Conference on Soil Mechanics and Foundation Engineering 1987, 2: 497-506.
Das BM, Omar MT: The effects of foundation width on model tests for the bearing capacity of sand with geo grid reinforcement. Geotech Geol Eng 1994, 12: 133-141. 10.1007/BF00429771
Das BM, Puri VK, Omar MT, Evigin E: Bearing capacity of strip foundation on geogrid reinforced sand-scale effects in model tests. Proc, 6th International Conference on Offshore and Polar Engineering 1996, 12: 527-530.
Dash S, Rajagopal K, Krishnaswamy N: Strip footing on geocell reinforced sand beds with additional planar reinforcement. Geotextile and Geomembrane 2001, 19: 529-538. 10.1016/S0266-1144(01)00022-X
Dash S, Krishnaswamy N, Rajagopal K: “Bearing capacity of strip footing supported on geocell-reinforced sand”. Geotextile and Geomembrane 2001, 19: 535-256.
Dawson A, Lee R: “Full scale foundation trials on grid reinforced clay”. Geosynthetics for Soil Improvement 1988, 127-147.
De Beer EE: Experimental determination of the shape factor and the bearing capacity f actors of sand. Geotechnique 1970, 20: 387-411. 10.1680/geot.1970.20.4.387
Fragaszy RJ, Lawton E: Bearing capacity of reinforced sand subgrades. J Geotech Eng Div 1984, 110(10):1500-1507. 10.1061/(ASCE)0733-9410(1984)110:10(1500)
Hansen JB: Revised and extended formula for bearing capacity. Danish Geotechnical Institute, Copenhagen, Bulletin 1970, 28: 5-11.
Ismail I, Raymond GP: Geosynthetic reinforcement of granular layered soils. Proceedings, 1, Geosynthetics 1995, 95: 317-330.
Khing KH, Das BM, Puri VK, Cook EE, Yen SC: “The bearing capacity of a strip foundation on geogrid-reinforced sand”. Geotextiles and Geomembranes 1993, 12: 351-361. 10.1016/0266-1144(93)90009-D
Mahmoud MA, Abdrabbo FM: Bearing capacity tests on strip footing on reinforced sand sub grade. Can Geotech J 1989, 26: 154159.
Mandal JM, Manjunath VR: Bearing capacity of strip footing resting on reinforced sand sub grades. Construction and Building Material 1995, 9(1):35-38. 10.1016/0950-0618(95)92858-E
Meyerhof GG: “The Ultimate Bearing Capacity of Foundations,”. Geotechnique 1951, 2(4):301-332. 10.1680/geot.1951.2.4.301
Meyerhof GG: The bearing capacity of foundations under eccentric-3rd inclined loads. ICSMFE Zurich 1953, 1: 1-19.
Meyerhof GG: Some recent research on the bearing capacity of foundations. Can Geotech J 1963, 1(1):16-26. 10.1139/t63-003
Meyerhof GG: Shallow foundations. J of S MFD, ASC E 1965, 91: SM2. 21–31
Milligan GWE, Love JP Proceedings Symposium on polymer Grid Reinforcement in Civil Engineering. In Model Testing of Geogrids Under and Aggregate Layer in Soft Ground. ICI, London, England; 1984:4.2.1-4.2.11.
Milovic D Proc. Of the 9thInternational conf. on Soil Mechanics and Foundation Engineering. In Bearing Capacity Tests on Reinforced Sand. ᅟ, Tokyo, Japan; 1977:1. 651–654
Omar MT, Das BM, Puri VK, Yen SC: Ultimate bearing capacity of shallow foundations on sand with geogrid reinforcement. Can Geotech J 1993, 30: 545-549. 10.1139/t93-046
Omar MT, Das BM, Yen SC, Puri VK, Cook EE: Ultimate bearing capacity of rectangular foundations on geo grid reinforced sand. Geotech Test J 1993, 16: 246-252. 10.1520/GTJ10041J
Puri VK, Khing KH, Das BM, Cook EE, Yen SC: The bearing capacity of a strip foundation on geo grid reinforced sand. Geotextile and Geomembrane 1993, 12: 351-361. 10.1016/0266-1144(93)90009-D
Rajagopal K, Krishnaswamy N, Latha G: Behavior of sand confined with single and multiple geocells. Geotextile and Geomembrane 1999, 17: 171-184. 10.1016/S0266-1144(98)00034-X
Rea C, Mitchell JK Proc. Symposium on Earth Reinforcement. In Sand Reinforcement Using Paper Grid Cells. ASCE, Pittsburg; 1978:644-663.
Terzaghi K: Theoretical Soil Mechanics. John Wiley and sons, New York, USA; 1943.
Verma BP, Char ANR: Bearing capacity tests on reinforced sand subgrades. J Geotech Eng 1986, 112(7):701-706. 10.1061/(ASCE)0733-9410(1986)112:7(701)
Vesic AS: Analysis of ultimate loads of shallow foundations. J Soil Mech Foundat Div, ASCE 1973, 99: 45-73.
Aarash Hosseini
Correspondence to Aarash Hosseini.
Hosseini, A. Effect of confinement pressure on bearing capacity of two samples of square and strip footing (numerical study). SpringerPlus 3, 593 (2014). https://doi.org/10.1186/2193-1801-3-593
Footing bearing capacity
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EUDML | Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2. EuDML | Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2.
Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2.
Shaw, Mei-Chi. "Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2.." Mathematische Annalen 279.4 (1987/88): 635-652. <http://eudml.org/doc/164344>.
@article{Shaw1987/88,
author = {Shaw, Mei-Chi},
keywords = {pseudoconvex domain; tangential Cauchy-Riemann equation; Hölder and estimates},
title = {Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2.},
TI - Hölder and LP Estimates for ...b on Weakly Pseudo-Convex Boundaries in C2.
KW - pseudoconvex domain; tangential Cauchy-Riemann equation; Hölder and estimates
Joachim Michel, Regularity of the tangential Cauchy-Riemann complex and applications
pseudoconvex domain, tangential Cauchy-Riemann equation, Hölder and
{L}^{p}
\overline{\partial }
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Articles by Mei-Chi Shaw
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Common-mode impedance coupling - Electrical Installation Guide
HomeElectroMagnetic Compatibility (EMC)EMC - Coupling mechanisms and counter-measuresCommon-mode impedance coupling
Two or more devices are interconnected by the power supply and communication cables (see Fig. R30). When external currents (lightning, fault currents, disturbances) flow via these common-mode impedances, an undesirable voltage appears between points A and B which are supposed to be equipotential. This stray voltage can disturb low-level or fast electronic circuits.
All cables, including the protective conductors, have an impedance, particularly at high frequencies.
The exposed conductive parts (ECP) of devices 1 and 2 are connected to a common earthing terminal via connections with impedances Z1 and Z2.
The stray overvoltage flows to the earth via Z1. The potential of device 1 increases to Z1 I1. The difference in potential with device 2 (initial potential = 0) results in the appearance of current I2.
{\displaystyle Z1\,I1=\left(Zsign\,+Z2\right)I2\Rightarrow {\frac {I2}{I1}}={\frac {Z1}{\left(Zsign\,+Z2\right)}}}
Current I2, present on the signal line, disturbs device 2.
Fig. R30 – Definition of common-mode impedance coupling
Devices linked by a common reference conductor (e.g. PEN, PE) affected by fast or intense (di/dt) current variations (fault current, lightning strike, short-circuit, load changes, chopping circuits, harmonic currents, power factor correction capacitor banks, etc.)
A common return path for a number of electrical sources
Fig. R31 – Example of common-mode impedance coupling
If they cannot be eliminated, common-mode impedances must at least be as low as possible. To reduce the effects of common-mode impedances, it is necessary to:
Reduce impedances:
Mesh the common references,
Use short cables or flat braids which, for equal sizes, have a lower impedance than round cables,
Install functional equipotential bonding between devices.
Reduce the level of the disturbing currents by adding common-mode filtering and differential-mode inductors
If the impedance of the parallel earthing conductor PEC (Z sup) is very low compared to Z sign, most of the disturbing current flows via the PEC, i.e. not via the signal line as in the previous case.
The difference in potential between devices 1 and 2 becomes very low and the disturbance acceptable.
Fig. R32 – Counter-measures of common-mode impedance coupling
Retrieved from "http://www.electrical-installation.org/enw/index.php?title=Common-mode_impedance_coupling&oldid=27050"
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!! used as default html header if there is none in the selected theme. Sequence plot
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Description: plot a numerical sequence or series. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, calculus, sequence, series, graphing, convergence
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Understanding the Payback Period
Payback Period FAQs
The term payback period refers to the amount of time it takes to recover the cost of an investment. Simply put, it is the length of time an investment reaches a breakeven point. People and corporations mainly invest their money to get paid back, which is why the payback period is so important. In essence, the shorter payback an investment has, the more attractive it becomes. Determining the payback period is useful for anyone and can be done by dividing the initial investment by the average cash flows.
The payback period is the length of time it takes to recover the cost of an investment or the length of time an investor needs to reach a breakeven point.
Shorter paybacks mean more attractive investments, while longer payback periods are less desirable.
The payback period is calculated by dividing the amount of the investment by the annual cash flow.
Account and fund managers use the payback period to determine whether to go through with an investment.
One of the downsides of the payback period is that it disregards the time value of money.
The payback period is a method commonly used by investors, financial professionals, and corporations to calculate investment returns. It helps determine how long it takes to recover the initial costs associated with an investment. This metric is useful before making any decisions, especially when an investor needs to make a snap judgment about an investment venture.
You can figure out the payback period by using the following formula:
Payback Period = Cost of Investment ÷ Average Annual Cash Flow
PaybackPeriod=CostofInvestment÷AverageAnnualCashFlow
The shorter the payback, the more desirable the investment. Conversely, the longer the payback, the less desirable it becomes. For example, if solar panels cost $5,000 to install and the savings are $100 each month, it would take 4.2 years to reach the payback period. In most cases, this is a pretty good payback period as experts say it can take as much as eight years for residential homeowners in the United States to break even on their investment.
Capital budgeting is a key activity in corporate finance. One of the most important concepts every corporate financial analyst must learn is how to value different investments or operational projects to determine the most profitable project or investment to undertake. One way corporate financial analysts do this is with the payback period.
Although calculating the payback period is useful in financial and capital budgeting, this metric has applications in other industries. It can be used by homeowners and businesses to calculate the return on energy-efficient technologies such as solar panels and insulation, including maintenance and upgrades.
Average cash flows represent the money going into and out of the investment. Inflows are any items that go into the investment, such as deposits, dividends, or earnings. Cash outflows include any fees or charges that are subtracted from the balance.
There is one problem with the payback period calculation. Unlike other methods of capital budgeting, the payback period ignores the time value of money (TVM). This is the idea that money is worth more today than the same amount in the future because of the earning potential of the present money.
Most capital budgeting formulas, such as net present value (NPV), internal rate of return (IRR), and discounted cash flow, consider the TVM. So if you pay an investor tomorrow, it must include an opportunity cost. The TVM is a concept that assigns a value to this opportunity cost.
The payback period disregards the time value of money and is determined by counting the number of years it takes to recover the funds invested. For example, if it takes five years to recover the cost of an investment, the payback period is five years.
This period does not account for what happens after payback occurs. Therefore, it ignores an investment's overall profitability. Many managers and investors thus prefer to use NPV as a tool for making investment decisions. The NPV is the difference between the present value of cash coming in and the current value of cash going out over a period of time.
Some analysts favor the payback method for its simplicity. Others like to use it as an additional point of reference in a capital budgeting decision framework.
Here's a hypothetical example to show how the payback period works. Assume Company A invests $1 million in a project that is expected to save the company $250,000 each year. If we divide $1 million by $250,000, we arrive at the payback period of four years for this investment.
Consider another project that costs $200,000 with no associated cash savings that will make the company an incremental $100,000 each year for the next 20 years at $2 million. Clearly, the second project can make the company twice as much money, but how long will it take to pay the investment back?
The answer is found by dividing $200,000 by $100,000, which is two years. The second project will take less time to pay back, and the company's earnings potential is greater. Based solely on the payback period method, the second project is a better investment.
The best payback period is the shortest one possible. Getting repaid or recovering the initial cost of a project or investment should be achieved as quickly as it allows. However, not all projects and investments have the same time horizon, so the shortest possible payback period needs to be nested within the larger context of that time horizon. For example, the payback period on a home improvement project can be decades while the payback period on a construction project may be five years or less.
Is the Payback Period the Same Thing as the Break-Even Point?
While the two terms are related, they are not the same. The breakeven point is the price or value that an investment or project must rise to cover the initial costs or outlay. The payback period refers to how long it takes to reach that breakeven.
Payback Period = Initial Investment ÷ Annual Cash Flow
What Are Some of the Downsides of Using the Payback Period?
As the equation above shows, the payback period calculation is a simple one. It does not account for the time value of money, the effects of inflation, or the complexity of investments that may have unequal cash flow over time.
The discounted payback period is often used to better account for some of the shortcomings, such as using the present value of future cash flows. For this reason, the simple payback period may be favorable, while the discounted payback period might indicate an unfavorable investment.
When Would a Company Use the Payback Period for Capital Budgeting?
The payback period is favored when a company is under liquidity constraints because it can show how long it should take to recover the money laid out for the project. If short-term cash flows are a concern, a short payback period may be more attractive than a longer-term investment that has a higher NPV.
EcoWatch. "How Long Will Your Solar Panel Payback Period Be?"
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Description: for one-variable real functions: limits, integrals, roots... interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
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Which Satellite Messenger Should I Get – Steve Corino
Comparing the location X, Identify Gen3, Garmin inReach Mini and Garmin inReach Explorer+
With the recent start of two new satellite messengers, the Garmin inReach Mini and the SPOT X, backcountry enthusiasts have significantly more options than ever for staying linked when venturing off the grid. You might be pondering, ‘Which satellite messenger is best for me?’
We’re here to help you produce that decision with a good rundown of the key differences around four of the very most popular satellite messengers we hold in REI-the Garmin inReach Mini, Garmin inReach Explorer+, Place X and Location Gen3.
All four of the devices have numerous basic functions in common. When activated with a satellite registration, all of them allow you do the following in (most) spots without cellphone reception:
Track your voyage (default of 10 min. intervals; could be customized as needed)
Send text-based text messages to your individual contacts
Create shareable over the internet maps of your adventure so others can carry out along in (near) real time
Automatically post improvements, including GPS site, to your Facebook or Twitter accounts
In case of a non-life-threatening emergency, alert your individual contacts that you’ll require help
In case of a life-threatening emergency, activate an SOS button (protected against accidental activation in your pack) that directly notifies emergency responders of your distress signal, plus your GPS coordinates
Notably, all devices will be impact-resistant and rated IPX7 waterproof, signifying they’re protected from normal water submersion up to at least one 1 meter deep for at least thirty minutes (i.e., They’ll definitely have the desired effect hanging off the trunk of your pack in a rainstorm.) Additionally, Place X can be considered dustproof (IP67 rating).
Beyond having these baseline capabilities in keeping, each device offers a definite set of features. Read on to understand the differences between these essential features:
Device and Satellite Registration Plan Costs
Battery Lifestyle, Type and Performance
Comparing Messaging Capabilities
Three of the four devices-the Garmin inReach Mini, inReach Explorer+ and Location X-possess 2-way messaging capabilities, meaning you can both send and receive messages-to cellphones, email addresses and, regarding Garmin, to other inReach equipment. In case of a life-threatening emergency, you’ll also have the ability to text immediately with rescue staff to make clear your emergency and receive confirmation if so when help is along the way.
The Location Gen3 can only just send messages. (And it’s priced accordingly; additional on that shortly.) Aside from the device’s SOS function, you have three message options to select from-a check-in message, an added preprogrammed custom communication (composed in the home, not on the device) and a obtain help in non-life-threatening situations. Messages can be sent to up to 10 contacts predetermined before your trip. A green light confirms that your message has been sent.
The two Garmin devices allow you to send a multitude of preset texts, including messages customized ahead of time in the home, or custom texts typed on these devices itself. Neither includes a built-in keyboard; typing messages is moderately time-consuming on the inReach Explorer+ (using predictive text message and arrow buttons to get around an on-display screen keyboard), and remarkably time-eating on the inReach Mini (employing arrow buttons and a vertically scrolling alphabet to choose letters one at a time). In both circumstances, if you’re likely to message frequently, you might be best away pairing your inReach with a Bluetooth-in a position smartphone and instead composing texts in Garmin’s Earthmate software (free with the order of an inReach device).
The SPOT X will not integrate directly with smartphones, but instead, has its built-in, lit QWERTY keyboard, letting you easily and quickly compose messages from these devices itself. The compact key pad is similar to certain early on smartphone keyboards; typing onto it might take some practice. Much like both Garmin gadgets, you can even compose preset communications at home (e.g., “I’m feeling incredibly wonderful!” or “I love you, honey” or “Are we there however?”, and then send them with the press of a good button once you’re found in the backcountry, stopping you from needing to type long text messages on the device itself.
Navigation Helps and Other Unique Features
The Garmin inReach Explorer+ has a panoply of features. It’s the only one of the four devices that also acts as a handheld Navigation unit with built-in topo maps and sensors, incorporating a barometric altimeter and accelerometer offering specific metrics on your journey. Also you can create a breadcrumb trail as you go, and make make use of it to navigate the right path back the same manner you came.
When paired with a smartphone and the Earthmate app, the inReach Mini offers most of the same operation. On its own, with out a paired mobile, it still lets you follow routes (pre-uploaded to the device) and drop waypoints as you go. Additionally, one unique characteristic is that it really is synced up applying ANT+ cellular technology to compatible wearables just like the Garmin Forerunner 935 or Fenix 5 series watches, in order to receive and send inReach messages, as well as trigger an SOS, on your own watch.
Also you can request basic or perhaps premium weather updates anytime on both Garmin gadgets. Based on your subscription method, additional costs may apply.
Though the SPOT X does not include maps, sensors or weather updates, it has several built-in navigation tools, including a compass and programmable waypoints. Each SPOT X as well comes with a unique U.S. mobile amount to make it possible for friends and liked ones to send your machine text messages from their cellphones. (In comparison, messages are delivered to the Garmin units with a customized inReach email or through the Earthmate app.)
The SPOT Gen3 doesn’t have any navigational features.
The Garmin inReach Mini may be the tiniest of all the unit, with the SPOT Gen3 not too much behind.
Both SPOT X and Garmin inReach Explorer+ are larger and heavier compared to the other devices by a few ounces. Compared to one another, they’re in the same ballpark-though the location X is slightly lighter.
Listed below are each device’s major dimensions, starting with the tiniest and lightest weight.
Garmin inReach Mini: 3.9 x 2 x 1 in. (3.5 oz.*)
SPOT Gen3: 3.4 x 2.5 x 1 in. (4 oz.*)
SPOT X: 6.5 x 2.9 x 0.9 in. (6.7 oz.*)
Garmin inReach Explorer+: 6.5 x 2.7 x 1.5 in. (7.5 oz.*)
*All given merchandise weights include battery excess weight, whether batteries are included or removable.
Choosing what you want to spend is determined by which features matter many to you. Each gadget is costed to reflect the level of features it offers.
SPOT Gen3: $149.95
SPOT X: $249.95
Garmin inReach Mini: $350
Garmin inReach Explorer+: $450
However, evaluating the price tag on a satellite messenger is not as simple as looking at the device price tag. Each one as well requires investing in a satellite membership from Garmin or SPOT.
SPOT plans require a 12-month contract, billed every month or annually, and also a 19.99activationfee.TheessentialSPOTplan(19.99activationfee.TheessentialSPOTplan(
19.99 activation fee. The essential SPOT plan (
199.99/time) provides tracking alternatives of 5-, 10-, 30- or perhaps 60-minute intervals. Upgrading to 2.5-minute tracking intervals costs yet another $99.99/season. Notably, unlike Garmin, SPOT also offers a Gen3 system rental program for one-period uses, or if you wish to try the merchandise out before buying.
The Garmin devices operate on the Iridium satellite network, which provides completely pole-to-pole coverage, without gaps or fringe areas. (Yes, this implies that if you want to venture south Pole, or the center of the Pacific Ocean, your inReach device will keep you connected.)
The SPOT devices operate on the Globalstar satellite network, which gives SPOT messenger equipment with coverage for almost each of the continental United States, Canada, Mexico, Europe, Australia, portions of SOUTH USA, portions of North and South Africa, Northeast Asia, and a huge selection of miles offshore of those areas. An in depth coverage map can be found at that moment website.
Battery Existence, Type and Performance
Battery life of the devices will depend on many factors, including:
Frequency of monitoring intervals-the more often your machine logs or reports GPS navigation coordinates, the more quickly it’ll eat through its battery life
Temperature-extreme hot or chilly can decrease battery life
Whether the device has a clear line of view to the sky-the electric battery may drain quicker in remarkably dense tree cover, or if you make an effort to send text messages from indoors or in the cave
Whether it’s connected to ANT+ or Bluetooth cellular technology-using each one can decrease battery life
Frequency of message-examining intervals-a higher regularity can decrease battery life
For comparison, here’s the vendor-provided information on battery life of each device, assuming a full charge in optimal operating circumstances, with continuous 10-minute tracking intervals:
Garmin inReach Mini: 50 hrs. (2 days)
Garmin inReach Explorer+: 100 hrs. (4 days)
SPOT X: 240 hrs. (10 days)
SPOT Gen3: 17 days
With less frequent tracking intervals, battery life capacity boosts significantly. For instance, both Garmin gadgets have a protracted tracking option-30-minute monitoring intervals, with messaging, detailed monitor lines and Bluetooth disabled-that lengthens their battery existence by weeks. (Total battery life as high as 20 days for the inReach Mini, or more to thirty days for the inReach Explorer.)
Similarly, the location devices can last practically three times as long when they’re set to 60-minute tracking intervals (vs. the default 10-minute intervals).
Three of the four devices-the Garmin inReach Mini, Garmin inReach Explorer+ and Area X-operate on a, rechargeable lithium ion battery that recharges via an included micro USB charger.
The SPOT Gen3, on the other hand, operates on 4 AAA lithium batteries, 4 AAA NiMH rechargeable batteries or 5-volt USB series power. This enables you to very easily carry external power packs or back up batteries to swap out on the go if need be. Be aware: AAA alkaline batteries will work, too, but aren’t recommended for optimal performance.
The Bottom Series: Which Satellite Messenger Must I Get?
As you’ve probably determined, your answer will be based upon your unique needs. Consider the following problems, and then read on for our high-level summary of each device:
What exactly are your major priorities? Easy, rapid messaging? Battery lifestyle? Cost benefits? Compact size and excess weight? Navigational tools?
Do you desire a device first or two major adventures each year, or are you looking for something to use on a regular basis year-round?
How important to you is 2-method messaging, and how sometimes do you intend to use this feature?
Will primarily preset messages (incorporating custom kinds you create in the home before leaving for your trip) suffice, or would you like the flexibility to easily type messages in the backcountry?
Do you intend to also bring a smartphone and make usage of it together with your satellite messenger, or would you like a standalone device?
Pick the Garmin inReach Explorer+ if you’re seeking a full-featured 2-way messenger, handheld Navigation tool, tracker and SOS system that can be utilised anywhere on the planet. It’s great if you’re preparing deep backcountry explorations and benefit robust navigational assistance, info tracking and a user-friendly interface. Though it’s the heaviest & most expensive of most four devices reviewed right here, it provides plenty of value for your money as a standalone, do-it-all unit or when paired with a Bluetooth-capable smartphone.
Pick the SPOT X if you adventure year-round and want a good value not merely for baseline monitoring and SOS capability, also for receiving and sending custom messages easily. Its battery lifestyle is two-and-a-half to five instances as long as the various other 2-method messengers discussed below, and it doesn’t require pairing with a smartphone for relatively fast composition of text messages, making it a great standalone messaging gadget. Plus, its built-in compass and programmable waypoints happen to be ideal for pursuits like hiking and backpacking.
Choose the Garmin inReach Mini if you’re trying to find an impressively compact, forget-it’s-right now there satellite television lifeline with 2-approach messaging capabilities. It’s ideal for gram-counting adventurers (backpackers and thru-hikers, trail runners, etc.). Though it accomplishes each of the basic capabilities of a standalone unit, typing custom text messages on these devices itself is time-eating. Its operation and user-friendliness soar when paired with a Bluetooth-capable smartphone-thus if you’re already likely to carry your cellphone, it’s a particularly great choice. But also without, it provides sound tracking and SOS capacities all over the globe. Plus, if you private a compatible Garmin wearable, the check out integration is a huge boon.
Choose the SPOT Gen3 if you wish just the essentials of a satellite communication device: GPS monitoring, SOS capacities and 1-way satellite messaging to let your friends and loved types know whether you’re carrying out OK or need support. It’s compact, compact and by far the most budget-friendly alternative for individuals who adventure regularly but don’t need fancy bells and whistles. It offers set-it-and-forget-it tracking and satisfaction that you’ll be in a position to call for assist in case of emergency.
Comparing the location X, Identify Gen3, Garmin inReach Mini and Garmin inReach Explorer+ With the recent start of two new satellite messengers, the Garmin inReach Mini and the SPOT X, backcountry enthusiasts have significantly more options than ever for staying linked when venturing off the grid. You might be pondering, ‘Which satellite messenger is best…
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EUDML | Periodic solutions of quasilinear equations with discontinuous perturbations. EuDML | Periodic solutions of quasilinear equations with discontinuous perturbations.
Periodic solutions of quasilinear equations with discontinuous perturbations.
Crema, Janete; Boldrini, José Luiz
Crema, Janete, and Boldrini, José Luiz. "Periodic solutions of quasilinear equations with discontinuous perturbations.." Southwest Journal of Pure and Applied Mathematics [electronic only] 2000.1 (2000): 55-73. <http://eudml.org/doc/233145>.
author = {Crema, Janete, Boldrini, José Luiz},
keywords = {set-valued right hand side; weak periodic solutions; -Laplacian; discontinuous perturbations; -Laplacian},
title = {Periodic solutions of quasilinear equations with discontinuous perturbations.},
TI - Periodic solutions of quasilinear equations with discontinuous perturbations.
KW - set-valued right hand side; weak periodic solutions; -Laplacian; discontinuous perturbations; -Laplacian
set-valued right hand side, weak periodic solutions,
p
-Laplacian, discontinuous perturbations,
p
Linear parabolic unilateral problems and linear parabolic variational inequalities
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Some New Classes of Generalized Difference Strongly Summable -Normed Sequence Spaces Defined by Ideal Convergence and Orlicz Function
Adem Kılıçman, Stuti Borgohain, "Some New Classes of Generalized Difference Strongly Summable -Normed Sequence Spaces Defined by Ideal Convergence and Orlicz Function", Abstract and Applied Analysis, vol. 2014, Article ID 621383, 7 pages, 2014. https://doi.org/10.1155/2014/621383
Adem Kılıçman1 and Stuti Borgohain 2
1Department of Mathematics and Institute for Mathematical Research, University of Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra 400076, India
We study some new generalized difference strongly summable n-normed sequence spaces using ideal convergence and an Orlicz function in connection with de la Vallèe Poussin mean. We give some relations related to these sequence spaces also.
A sequence is said to be almost convergent if all of its Banach limits coincide.
Let denote the space of all almost convergent sequences.
Lorentz in [1] proved that where The following space of strongly almost convergent sequence was introduced by Maddox in [2]: where .
Let be a one-to-one mapping from the set of positive integers into itself such that , , where denotes the th iterative of the mapping in .
A continuous linear functional on is said to be an invariant mean or a -mean, if and only if it satisfies the following conditions:(1) , when the sequence is such that for all ;(2) , where ;(3) , for all .For a certain kind of mapping , we get that every invariant mean extends the functional limit on the space , such that for all . Consequently, we get that , where is the set of bounded sequences with equal -means.
Schaefer in [3] proved that where Thus we say that a bounded sequence is -convergent, if and only if such that for all , .
Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the -convergence leads naturally to the concept of strong -convergence.
A sequence is said to be strongly -convergent (Mursaleen [4]), if there exists a number such that We write to denote the set of all strong -convergent sequences and when (6) holds, we write .
Taking , we obtain . Then the strong -convergence generalizes the concept of strong almost convergence.
We also note that The notion of ideal convergence was first introduced by Kostyrko et al. [5] as a generalization of statistical convergence which was later studied by many other authors.
An Orlicz function is a function , which is continuous, nondecreasing, and convex with , , for , and , as .
Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to construct the sequence space: The space with the norm becomes a Banach space which is called an Orlicz sequence space.
Kızmaz [7] studied the difference sequence spaces , , and of crisp sets. The notion is defined as follows: for and , where , for all .
The above spaces are Banach spaces, normed by Later the idea of Kızmaz [7] was applied to introduce different types of difference sequence spaces and study their different properties by many others later on.
The generalized difference notion is defined as follows. For and , This generalized difference has the following binomial representation: The concept of 2-normed space was initially introduced by Gähler [8], in the mid of 1960s, while that of -normed spaces can be found in Misiak [9]. Since then, many other authors have used this concept and obtained various results. Recently, several various activities have been initiated to study summability, sequence spaces, and related topics in these spaces. The notion of ideal convergence in 2-normed spaces was initially introduced by Gurdal [10]. Later on, it was extended to -normed spaces by Gurdal and Sahiner in [11].
Let and be a real vector space. A real valued function on satisfies the following four properties:(1) if and only if are linearly dependent;(2) is invariant under permutation;(3) , for all ;(4) is called an -norm on and the pair is called an -normed space.
Let be a nonempty set. Then a family of sets (power sets of ) is said to be an if is additive that is and hereditary that is .
A sequence in a normed space is said to be -convergent to with respect to -norm, if for each , the set The generalized de la Vallée Poussin mean is defined by where for .
Then a sequence is said to be -summable to a number , if as , and we write for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by de la Vallée Poussin method.
Maddox introduced and studied the special case, where , for ; the sets , , and reduce to the sets , , and .
In this paper, we define some new sequence spaces in -normed spaces by using Orlicz function with notion of generalized de la Vallèe Poussin mean, generalized difference sequences, and ideals. We will also introduce and examine certain new sequence spaces using the above tools as also the -norm.
Let be an admissible ideal of , let be an Orlicz function, and let be a -normed space. Further, let be a bounded sequence of positive real numbers. By , we denote the space of all sequences defined over .
In this paper, we have introduced the following sequence spaces: In particular, if we take , for all , we have Similarly, when , then , , and are reduced to In particular, if we put , for all , then we have the spaces Further when , for , the sets and are reduced to and , respectively.
Now, if we consider , then we can easily obtain If , with as uniformly in , then we write .
The following well-known inequality will be used later.
Lemma 1. Let and . Then , if and only if , where .
Note that no other relation between and is needed in Lemma 1.
Theorem 2. Let . Then, implies . Let . If , then is unique.
By the definition of Orlicz function, we have, for all , Since , it follows that And consequently, .
Let . Suppose that , , and .
Now, from (22) and the definition of Orlicz, we have since Hence, Further, as , and therefore From (27) and (28), it follows that and by the definition of an Orlicz function, we have .
Hence, and this completes the proof.
Theorem 3. (i) Let . Then, (ii) Let . Then,
Theorem 4. Let stand for , , or and . Then the inclusion is strict. In general, for all and the inclusion is strict.
Proof. Let us take .
Let . Then for given , we have Since is nondecreasing and convex, it follows that Hence we have Since the set on the right hand side belongs to , so does the left hand side. The inclusion is strict as the sequence , for example, belongs to but does not belong to for and for all .
Theorem 5. and are complete linear topological spaces, with paranorm , where is defined by where .
The first author acknowledges that this research was part of the research project and partially supported by the Universiti Putra Malaysia under Grant no. ERGS 1-2013/5527179. The work of second author was carried out under the Postdoctoral Fellow under National Board of Higher Mathematics, DAE, Project no. NBHM/PDF.50/2011/64.
I. J. Maddox, “Spaces of strongly summable sequences,” The Quarterly Journal of Mathematics, vol. 18, pp. 345–355, 1967. View at: Publisher Site | Google Scholar | MathSciNet
P. Schaefer, “Infinite matrices and invariant means,” Proceedings of the American Mathematical Society, vol. 36, pp. 104–110, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Mursaleen, “Matrix transformations between some new sequence spaces,” Houston Journal of Mathematics, vol. 9, no. 4, pp. 505–509, 1983. View at: Google Scholar | MathSciNet
P. Kostyrko, T. Salat, and W. Wilczynski, “On I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2001. View at: Google Scholar
S. Gähler, “Lineare 2-normierte Räume,” Mathematische Nachrichten, vol. 28, pp. 1–43, 1964. View at: Publisher Site | Google Scholar | MathSciNet
n
M. Gurdal, “On ideal convergent sequences in 2-normed spaces,” Thai Journal of Mathematics, vol. 4, no. 1, pp. 85–91, 2006. View at: Google Scholar | MathSciNet
M. Gurdal and A. Sahiner, “Ideal convergence in n-normed spaces and some new sequence spaces via n-norm,” Journal of Fundamental Sciences, vol. 4, no. 1, pp. 233–244, 2008. View at: Google Scholar
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Incenter | Brilliant Math & Science Wiki
Xuming Liang, Alexander Katz, Harsh Poonia, and
The incenter of a triangle is the center of its inscribed circle. It has several important properties and relations with other parts of the triangle, including its circumcenter, orthocenter, area, and more.
The incenter is typically represented by the letter
I
All triangles have an incenter, and it always lies inside the triangle. One way to find the incenter makes use of the property that the incenter is the intersection of the three angle bisectors, using coordinate geometry to determine the incenter's location. Unfortunately, this is often computationally tedious.
Generally, the easiest way to find the incenter is by first determining the inradius, or radius of the incircle, usually denoted by the letter
(the letter
R
is reserved for the circumradius). This can be done in a number of ways, detailed in the 'Basic properties' section below. Once the inradius is known, each side of the triangle can be translated by the length of the inradius, and the intersection of the resulting three lines will be the incenter. This, again, can be done using coordinate geometry.
Alternatively, the following formula can be used. For a triangle with side lengths
a,b,c
, with vertices at the points
(x_1, y_1), (x_2, y_2), (x_3, y_3)
, the incenter lies at
\left(\dfrac{ax_1+bx_2+cx_3}{a+b+c}, \dfrac{ay_1+by_2+cy_3}{a+b+c}\right).
A triangle has vertices at
A=(0,0), B=(14,0)
C=(5,12)
. What are the coordinates of the incenter?
The lengths of the sides (using the distance formula) are
a=\sqrt{(14-5)^2+(12-0)^2}=15, b=\sqrt{(5-0)^2+(12-0)^2}=13, c=\sqrt{(14-0)^2+(0-0)^2}=14.
Now the above formula can be used:
I = \left(\dfrac{15 \cdot 0+13 \cdot 14+14 \cdot 5}{13+14+15}, \dfrac{15 \cdot 0+13 \cdot 0+14 \cdot 12}{13+14+15}\right)=\left(6, 4\right).\ _\square
The simplest proof is a consequence of the trigonometric version of Ceva's theorem, which states that
AD, BE, CF
\frac{\sin\angle BAD}{\sin\angle ABE} \cdot \frac{\sin \angle CBE}{\sin \angle BCF} \cdot \frac{\sin\angle ACF}{\sin \angle CAD} = 1.
D,E,F
are the feet of the angle bisectors, so
\angle BAD=\angle CAD
\angle ABE=\angle CBE
\angle ACF=\angle BCF
. As a result,
\frac{\sin\angle BAD}{\sin\angle CAD} \cdot \frac{\sin\angle ABE}{\sin\angle CBE} \cdot \frac{\sin\angle ACF}{\sin\angle BCF} = 1 \cdot 1 \cdot 1 = 1
and rearranging the left hand side gives
\frac{\sin\angle BAD}{\sin\angle ABE} \cdot \frac{\sin \angle CBE}{\sin \angle BCF} \cdot \frac{\sin\angle ACF}{\sin \angle CAD} = 1.
Therefore, the three angle bisectors intersect at a single point,
I
I
lies on the angle bisector of
\angle BAC
I
AB
is equal to the distance from
I
AC
. Similarly, this is also equal to the distance from
I
BC
I
is the center of the inscribed circle, proving the existence of the incenter.
An alternate proof involves the length version of Ceva's theorem and the angle bisector theorem.
It's been noted above that the incenter is the intersection of the three angle bisectors.
For a triangle with semiperimeter (half the perimeter)
s
and inradius
r
The area of the triangle is equal to
sr
This is particularly useful for finding the length of the inradius given the side lengths, since the area can be calculated in another way (e.g. Heron's formula), and the semiperimeter is easily calculable.
ABC
AB = 13, BC = 14
CA = 15
. What is the length of the inradius of
\triangle ABC
In a right triangle with integer side lengths, the inradius is always an integer.
D
is the point where the incircle touches
BC
E,F
are where the incircle touches
AC
AB
AE=AF=s-a, BD=BF=s-b, CD=CE=s-c
. As a corollary,
AE+BF+CD=s
r = \sqrt{\dfrac{AE \cdot BF \cdot CD}{AE+BF+CD}}.
AD, BE,
CF
intersect at a single point, called the Gergonne point.
If the altitudes of a triangle have lengths
h_1, h_2, h_3
\dfrac{1}{h_1}+\dfrac{1}{h_2}+\dfrac{1}{h_3}=\dfrac{1}{r}.
a,b,c
\frac{1}{2} a h_1= \Delta
, and similarly for other sides,
\sum \frac{1}{h_1} = \sum \frac{a}{2 \Delta} = \frac{2s}{2 \Delta} = \frac{1}{ \Delta/s}= \frac{1}{r}.
ABC
has area 15 and perimeter 20. Furthermore, the product of the 3 side lengths is 255. If the three altitudes of the triangle have lengths
d, e
f
de+ef+fd
\frac{m}{n}
m
n
m+n
The incircle and circumcircle are also intimately related. According to Euler's theorem,
(R-r)^2 = d^2+r^2,
R
is the circumradius,
r
the inradius, and
the distance between the incenter and the circumcenter. Equivalently,
d=\sqrt{R(R-2r)}
. This also proves Euler's inequality:
R \geq 2r
. Equality holds only for equilateral triangles.
rR=\frac{abc}{2(a+b+c)}, ~\text{ and }~ IA \cdot IB \cdot IC = 4Rr^2.
Incircles also relate well with themselves. If
r_1, r_2, r_3
are the radii of the three circles tangent to the incircle and two sides of the triangle, then
r=\sqrt{r_1r_2}+\sqrt{r_2r_3}+\sqrt{r_3r_1}.
On a different note, if the circumcircle of
ABC
is drawn, and
M
is the midpoint of minor arc
BC
M
is also the circumcenter of
\triangle BIC
MB=MI=MC
. This is known as "Fact 5" in the Olympiad community.
Similarly, if point
E
lies on the circumcircle of
BCI
BC=EC
\angle BCE=\angle BAC
EC
CO
O
ABC
All triangles have an incircle, and thus an incenter, but not all other polygons do. When one exists, the polygon is called tangential. As in a triangle, the incenter (if it exists) is the intersection of the polygon's angle bisectors.
In the case of quadrilaterals, an incircle exists if and only if the sum of the lengths of opposite sides are equal: Both pairs of opposite sides sum to
a+b+c+d
Cite as: Incenter. Brilliant.org. Retrieved from https://brilliant.org/wiki/triangles-incenter/
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Description: parametrize a function to make it continue or differentiable on 2 points. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, algebra, continuity, derivative, limit
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Black Holes | Brilliant Math & Science Wiki
Sravanth C., Muhammad Arifur Rahman, Matt DeCross, and
A black hole is a region of spacetime in which the attractive force of gravity is so strong that not even light escapes. As a result, black holes are not visible to the eye, although they can be detected from the behavior of light and matter nearby.
The most well-studied black holes are formed from stars collapsing under the gravitational attraction of their own mass, but black holes of any mass can theoretically exist even down to sizes as small as a single atom.
Supermassive black holes, such as the one at the center of the Milky Way, are the most important for studying the development of galaxies and the universe as a whole. Supermassive black holes are defined as black holes with a mass on the scale of hundreds of times the mass of the Sun and greater. It is believed that every galaxy is centered around one such supermassive black hole.
Since the equations of general relativity that govern Einstein's gravity break down at the center of a black hole, a region of enormous energy density, black holes are intensely studied for clues about how quantum mechanics and general relativity can be combined to form a unified theory of "quantum gravity" such as string theory.
Escape Velocity and Event Horizons
Black holes are a natural prediction of Einstein's theory of general relativity. General relativity describes both how spacetime bends in response to mass, and how mass moves in response to bent spacetime. When spacetime is completely flat because no mass is nearby, an object moving through it stays at constant velocity. But when other mass is present, spacetime curves, and the object accelerates toward the mass. The escape velocity is the velocity the object would need to move away far enough away from the mass that it would no longer be affected by its gravitational pull, i.e., to infinity.
The escape velocity is given by
V_{e} = \sqrt{\frac{2 G M}{R}}
G
M
is the mass of the object to be escaped from, and
R
is the radius of the object.
Escape velocity for an object:
The escape velocity of an object in Newtonian gravity can be derived as follows:
\begin{aligned} \dfrac12mV_e^2&=\dfrac{GMm}{R}\\ V_e^2&=\dfrac{2GM}{R}\\ \implies V_e&=\sqrt{\dfrac{2GM}{R}}. \end{aligned}
The event horizon of a black hole is equivalent to the set of points surrounding the black hole at which the escape velocity is equal to the speed of light:
299,000 \text{ km}/\text{s}
. (For scale, the escape velocity at Earth's surface is
11.2\text{ km}/\text{s}
.) Because escape velocity increases as you get closer to a spherically symmetric mass distribution, objects closer to the black hole than the event horizon would need to move faster than the speed of light in order to escape. But since it is impossible to move faster than the speed of light, nothing can escape. The event horizon can therefore be thought of as a boundary that screens off everything happening inside from the rest of the world.
Note that using the Newtonian escape velocity to describe the motion of masses near an inherently relativistic object is not technically correct. Fortunately, the answer works out the same in the full relativistic computation.
Black holes are massive objects that have become so dense that they collapse in on themselves under their own gravitational attraction. The Schwarzschild radius
R_{S}
is defined as the distance from a spherically symmetric mass distribution at which the escape velocity from the sphere is equal to the speed of light, i.e., it is the location of the event horizon. If this distance is greater than the radius of the sphere of mass itself, the event horizon is outside the sphere of mass, and the sphere cannot be seen: it is a black hole.
The Schwarzschild radius can be calculated by substituting
v_{e} = c
into the equation for the escape velocity above:
c = \sqrt{\frac{2 G M}{R_{S}}}.
Solving in terms of
R_{S}
R_{S} = \frac{2 G M}{c^{2}}.
For scale,
R_{S}
for the sun is about 3 km, and
R_{S}
for the Earth is a mere 9 mm. Since the radii of both the Sun and the Earth are much larger than either of these numbers, neither is a black hole, as one would hope and expect.
Consider the volume of a black hole to be the volume of a sphere with radius equal to the Schwarzschild radius
R_{S} = \frac{2 G M}{c^{2}}
How does the average density of a black hole change as the mass increases?
As a point of historical interest, the Schwarzschild radius is named after Karl Schwarzschild, a German physicist who formulated the first nontrivial solution to the Einstein field equations in 1915 while fighting in World War I on the Russian front. The name "Schwarzschild" also happens to literally translate to "black shield," ironically.
A singularity is a point at which the curvature of spacetime is undefined or divergent. The center of black holes in general relativity may contain singularities at a single point of infinite mass density. The gravitational force becomes so strong that no other forces (including electrostatic repulsion and the strong or weak nuclear forces) can prevent the mass from collapsing further and further in on itself, resulting in a point of infinite density.
However, the existence of an event horizon does not necessarily imply the existence of a point of infinite density. An object with finite density that was compressed within its Schwarzschild radius would still have an event horizon, but no singularity. While black holes are observed astronomically to definitely exist, it is not yet understood what happens near the singularity of a black hole, or even whether true singularities exist. Since the energy density near the center of a black hole is so high, there may be effects from theories of high energy physics / quantum gravity such as string theory that prevent singularities from forming.
Even without knowing what happens at the center of a black hole, it is still possible describe what happens around it. A theorem in classical (non-quantum) general relativity known as the "no-hair theorem" states that the only variables that matter in terms of the physics outside the event horizon are the total mass, total angular momentum, and total electric charge of the black hole. (The "hair" in "no-hair" refers to details more specific than these general qualities.) The specific distribution of mass inside the event horizon doesn't matter, nor do other details like whether the mass/energy in the black hole consists primarily of matter or antimatter.
It is also mathematically possible that a singularity could exist without an event horizon, though most physicists reject the notion that such a "naked" singularity exists in the universe. Based on mathematical observations that any process one could design to expose the singularity of a black hole seems to fail, Roger Penrose formulated the "cosmic censorship hypothesis." This hypothesis states that all singularities in the universe are contained inside event horizons and therefore are in principle not observable (because no information about the singularity can make it past the event horizon to the outside world). However, this hypothesis is unproven: it is possible that so-called "naked singularities" might exist, and indeed many physicists in recent years have shown that in at least some spacetimes (though not the physical universe, yet) naked singularities are possible.
"Spaghettification" is a whimsical term for how objects falling into a black hole get stretched out due to massive tidal forces.
Tendency of a distribution of mass to flatten in one dimension and stretch in the other due to tidal forces [2].
Spaghettification in principle occurs anywhere there is a difference in the strength of gravity between one end of an object and another. For instance, if you stand up, the force of gravity on your head is smaller than the force of gravity on your feet because your feet are closer to the center of the Earth. Near a black hole, the same is true, but the difference in gravitational strength between your feet and your head is much larger. Because your feet accelerate towards the center of the hole faster than your head, you would be stretched out like a piece of spaghetti.
Depending on the mass of the black hole, spaghettification may in fact occur outside the event horizon and therefore be observable to a faraway observer, especially if the infalling body is a large source of light such as a star. This is due to the fact that the gravitational strength of the black hole is immense whether or not one is inside the event horizon.
One mechanism of black hole formation is star death. When a star runs out of its fuel storage it explodes into fragments, burning the remainder of its fuel and giving rise to a supernova, a sudden outburst of energy by a star as it dies. A supernova forms when there is a sudden disruption in the ongoing nuclear reactions in the core, which causes an explosion that sends the particles of the star away from it rapidly in a cosmic shockwave. The energy released by a supernova is approximately the amount of energy released by a medium sized star over its entire lifetime. A supernova is highly unstable and exists for about a month before the remaining mass collapses under its own gravity, forming a neutron star. Neutron stars are the densest stars known to exist (though not the theoretically densest possible stars). The eventual fate of this star depends upon its mass. If the star has a mass greater than about 3 times the mass of the sun, it will eventually collapse into a black hole. This limit on how large a neutron star can be is known as the Tolman-Oppenheimer-Volkoff limit.
Black Hole Accretion Credit: ROSAT/MPE/NASA A compact star in an binary system, specifically, an accreting binary system, may also form a black hole. Accreting binary systems generally form when a lower mass star expands into a more compact star, and is said to accrete more particles, often forming a astronomical structure called an accretion disk. Accretion disks can form around black holes, where they look similar to accretion onto a neutron star or white dwarf.
To the right is a ROAST X-ray image of LMC X-1 (the bright cluster on the left). This bright cluster is an accretion disk in the Large Magellanic Cloud, giving off x-ray emissions in the vicinity of a massive star that can be detected by astronomers. This star isn't visible, but is estimated to have a mass of 5 solar masses or more, and is a candidate for a black hole. On the right side of the image, normal stars with 1 solar mass or lower are visible.
Bipolar Mass Ejection
One way to detect a black hole is to look for rapidly ejected mass from a local region of spacetime. Some of the material accreting onto a black hole may gain a large amount of angular momentum that propels it above the escape velocity. It then may be thrown off at a high velocity in the direction defined by the black hole's rotation. This process is called bipolar flow. However, ejected mass could also come from neutron stars that are accreting mass. The key difference between a neutron star and a black hole is the magnitude of the mass of the object, and estimating the mass of the unseen object that is ejecting particles is a key means of proving that bipolar mass is associated with a black hole.
Binary Black Hole Systems
The LIGO experiment, which announced the first-ever detection of gravitational waves on February 11, 2016 [1], observed gravitational waves coming from a binary black hole system consisting of two black holes of about 30 solar masses each. In a binary black hole system, two black holes orbit each other. In this case, the two black holes inspiraled towards each other until they collided, forming a single black hole. This merger of the two black holes generated waves of expanding and contracting spacetime that spread out from the new black hole, that is, gravitational waves. As they passed through the earth, they caused the earth to expand and contract by about the width of an atomic nucleus, sufficiently large to be detected in an interferometry experiment.
A binary black hole system emitting graviational waves, Credit: T. Carnahan NASA GSFC
Despite the fact that nothing can escape from within the event horizon, black holes still give off a form of radiation from the event horizon called Hawking radiation, via which they lose energy to the surrounding space. Heuristically, this process occurs as particle/anti-particle pairs are created near the event horizon of the black hole, and one particle escapes from the black hole as the other one falls in. This explanation is not quite mathematically or physically precise, however.
A black hole created from a collapsing star would take at least 57 orders of magnitude longer than the current age of the universe for the hole to completely evaporate due to the energy lost in Hawking radiation. However, extremely small black holes, such as the ones that some people worried could be created in the Large Hadron Collider, can exist for extremely short periods of time before evaporating due to Hawking radiation.
Hawking radiation is central to the black hole information paradox, a subject of intense recent study. If an object with finite entropy (and therefore some finite amount of information, in the statistical sense) falls into a black hole, but the black hole evaporates due to Hawking radiation, it seems as though the information has been forever destroyed in violation of the second law of thermodynamics. Thus it would seem that information must be destroyed on entering a black hole, which is in contradiction with the idea that in general relativity nothing special occurs to an observer falling into a black hole at the instant he/she crosses the event horizon. Recent resolutions of the paradox suggest among other things that the Hawking radiation does in fact contain the information (i.e., correlations in entropy) of whatever fell into the black hole.
Black holes are regions of space from which nothing can escape. If you consider a spherical object of mass M and radius R and set the escape velocity from the object to be
c
, the speed of light, you can determine a relationship between R and M,
R=2 GM/c^2
G
is Newton's gravitational constant. This radius is called the Schwarzschild radius, denoted by
R_s
. If mass M is concentrated into a region with a radius smaller than
R_s
then you have a black hole, and if not, there is no black hole.
From the above relation you can determine the minimum mass of a black hole, as roughly speaking the Schwarzschild radius must be larger or equal to the Compton wavelength - the minimum size of the region in which an object at rest can be localized.
Find the minimum mass of a black hole in
\mu g
Finally, a bonus thing to think about. What does this result mean for the masses of the particles that we see in nature?
The value of the gravitational constant is
G=6.67 \times 10^{-11} \text{ m}^3\text{/kg s}^2
c=3 \times 10^8\text{ m/s}
Planck's constant is
h=6.63 \times 10^{-34} \text{ kgm}^2\text{/s}
1 ~\mu g = 10^{-6} \text{ g} = 10^{-9} \text{ kg}
[1] Science Magazine. Gravitational Waves, Einstein's Ripples in Spacetime, Spotted For First Time. Retrieved February 11, 2016 from http://www.sciencemag.org/news/2016/02/gravitational-waves-einstein-s-ripples-spacetime-spotted-first-time
[2] By Krishnavedala - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=32941013
Cite as: Black Holes. Brilliant.org. Retrieved from https://brilliant.org/wiki/black-hole/
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Represent sensor configuration for tracking - MATLAB - MathWorks España
trackingSensorConfiguration
SensorTransformFcn
MaxNumDetsPerObject
MinDetectionProbability
Create Radar Sensor Configuration
Create Tracking Sensor Configuration for fusionRadarSensor
Create a Tracking Sensor Configuration
Represent sensor configuration for tracking
The trackingSensorConfiguration object creates the configuration for a sensor used with a trackerPHD System object™. It allows you to specify the sensor parameters such as clutter density, sensor limits, sensor resolution. You can also specify how a tracker perceives the detections from the sensor using properties such as FilterInitializationFcn, SensorTransformFcn, and SensorTransformParameters. See Create a Tracking Sensor Configuration for more details. The trackingSensorConfiguration object enables the tracker to perform three main routine operations:
Evaluate the probability of detection at points in state-space.
Initiate components in the probability hypothesis density.
Obtain the clutter density of the sensor.
config = trackingSensorConfiguration(sensor)
config = trackingSensorConfiguration(SensorIndex)
config = trackingSensorConfiguration(___,Name,Value)
config = trackingSensorConfiguration(sensor) creates a trackingSensorConfiguration object based on a fusionRadarSensor object. You must specify the SensorIndex property of the fusionRadarSensor object.
config = trackingSensorConfiguration(SensorIndex) creates a trackingSensorConfiguration object with a specified sensor index, SensorIndex, and default property values.
config = trackingSensorConfiguration(___,Name,Value) set properties using one or more name-value pairs.
Unique sensor identifier, specified as a positive integer. This property distinguishes detections that come from different sensors in a multi-sensor system. When creating a trackingSensorConfiguration object using a fusionRadarSensor object, the SensorIndex property of the fusionRadarSensor object specifies the value of the SensorIndex property of the trackingSensorConfiguration object. Otherwise, you must specify this property using the SensorIndex argument.
IsValidTime — Indicate detection reporting status
Indicate the detection reporting status of the sensor, specified as false or true. Set this property to true when the sensor must report detections within its sensor limits to the tracker. If a track or target was supposed to be detected by a sensor but the sensor reported no detections, then this information is used to count against the probability of existence of the track when the isValidTime property is set to true.
@initcvggiwphd (default) | function handle | character vector
Filter initialization function, specified as a function handle or as a character vector containing the name of a valid filter initialization function. The function initializes the PHD filter used by trackerPHD. The function must support the following syntaxes:
filter = filterInitializationFcn()
filter = filterInitializationFcn(detections)
filter is a valid PHD filter with components for new-born targets, and detections is a cell array of objectDetection objects. The first syntax allows you to specify the predictive birth density in the PHD filter without using detections. The second syntax allows the filter to initialize the adaptive birth density using detection information. See the BirthRate property of trackerPHD for more details. If you create your own FilterInitilizationFcn, you must also provide a transform function using the SensorTransformFcn property. Other than the default filter initialization function initcvggiwphd, Sensor Fusion and Tracking Toolbox™ also provides other initialization functions, such as initctrectgmphd, initctgmphd, initcvgmphd, initcagmphd, initctggiwphd and initcaggiwphd.
SensorTransformFcn — Sensor transform function
@cvmeas | function handle | character vector
Sensor transform function, specified as a function handle or as a character vector containing the name of a valid sensor transform function. The function transforms a track's state into the sensor's detection state. For example, the function transforms the track's state in the scenario Cartesian frame to the sensor's spherical frame. You can create your own sensor transform function, but it must support the following syntax:
detStates = SensorTransformFcn(trackStates,params)
params are the parameters stored in the SensorTransformParameters property. Notice that the signature of the function is similar to a measurement function. Therefore, you can use a measurement function (such as cvmeas, ctmeas, or cameas) as the SensorTransformFcn.
Depending on the filter type and the target type, the output, detStates, needs to return differently.
When used with gmphd for non-extended targets or with ggiwphd, detStates is a N-by-M matrix, where N is the number of rows in the SensorLimits property and M is the number of input states in trackStates. For gmphd, non-extended targets refer to point targets and extended targets whose MeasurementOrigin is 'center'.
When used with gmphd for extended targets, the SensorTransformFcn allows you to specify multiple detStates per trackState. In this case, detStates is a N-by-M-by-S matrix, where S is the number of detectable sources on the extended target. For example, if the target is described by a rectangular state, the detectable sources can be the corners of the rectangle.
If any of the source falls inside the SensorLimits, the target is declared detectable. The functions uses the spread (maximum coordinate − minimum coordinate) of each detStates and the ratio between the spread and sensor resolution on each sensor limit to calculate the expected number of detections from each extended target. You can override this default setting by providing an optional output in the SensorTransformFcn as:
[..., Nexp] = SensorTransformFcn(trackStates, params)
where Nexp is the expected number of detections from each extended track state.
Note that the default SensorTransformFcn is the sensor transform function of the filter returned by FilterInitilizationFcn. For example, the initicvggiwphd function returns the default cvmeas, whereas initictggiwphd and initicaggiwphd functions return ctmeas and cameas, respectively.
SensorTransformParameters — Parameters for sensor transform function
Parameters for the sensor transform function, returned as a structure or an array of structures. If you only need to transform the state once, specify it as a structure. If you need to transform the state n times, specify it as an n-by-1 array of structures. For example, to transform a state from the scenario frame to the sensor frame, you usually need to first transform the state from the scenario rectangular frame to the platform rectangular frame, and then transform the state from the platform rectangular frame to the sensor spherical frame. The fields of the structure are:
Child coordinate frame type, specified as 'Rectangular' or 'Spherical'.
Child frame origin position expressed in the Parent frame, specified as a 3-by-1 vector.
Child frame origin velocity expressed in the parent frame, specified as a 3-by-1 vector.
Relative orientation between frames, specified as a 3-by-3 rotation matrix. If the IsParentToChild property is set to false, then specify Orientation as the rotation from the child frame to the parent frame. If the IsParentToChild property is set to true, then specify Orientation as the rotation from the parent frame to the child frame.
Flag to indicate the direction of rotation between parent and child frame, specified as true or false. The default is false. See description of the Orientation field for details.
HasAzimuth
Indicates whether outputs contain azimuth components, specified as true or false.
Indicates whether outputs contain elevation components, specified as true or false.
Indicates whether outputs contain range components, specified as true or false.
Indicates whether outputs contains velocity components, specified as true or false.
Note that here the scenario frame is the parent frame of the platform frame, and the platform frame is the parent frame of the sensor frame.
The default values for SensorTransformParameters are a 2-by-1 array of structures as:
Fields Struct 1 Struct 2
Frame 'Spherical' 'Rectangular'
OriginPosition [0;0;0] [0;0;0]
OriginVelocity [0;0;0] [0;0;0]
Orientation eye(3) eye(3)
IsParentToChild false false
HasAzimuth true true
HasElevation true true
HasRange true true
HasVelocity false true
In this table, Struct 2 accounts for the transformation from the scenario rectangular frame to the platform rectangular frame, and Struct 1 accounts for the transformation from the platform rectangular frame to the sensor spherical frame, given the isParentToChild property is set to false.
SensorLimits — Sensor's detection limits
Sensor's detection limits, specified as an N-by-2 matrix, where N is the output dimension of the sensor transform function. The matrix must describe the lower and upper detection limits of the sensor in the same order as the outputs of the sensor transform function.
If you use cvmeas, cameas, or ctmeas as the sensor transform function, then you need to provide the sensor limits in order as:
\text{SensorLimits = }\left[\begin{array}{cc}\text{minAz}& \text{maxAz}\\ \text{minEl}& \text{maxEl}\\ \text{minRng}& \text{maxRng}\\ \text{minRr}& \text{maxRr}\end{array}\right]
The description of these limits and their default values are given in the following table. Note that the default values for SensorLimits are a 3-by-2 matrix including the top six elements in the table. Moreover, if you use these three functions, you can specify the matrix to be in other sizes (1-by-2, 2-by-2, or 3-by-4), but you have to specify these limits in the sequence shown in the SensorLimits matrix.
Limits Description Default values
Minimum detectable azimuth in degrees.
Maximum detectable azimuth in degrees.
Minimum detectable elevation in degrees.
Maximum detectable elevation in degrees.
minRng
Minimum detectable range in meters.
Maximum detectable range in meters.
minRr
Minimum detectable range rate in meters per second.
Maximum detectable range rate in meters per second.
SensorResolution — Resolution of sensor
[4;2;10] (default) | N-element positive-valued vector
Resolution of a sensor, specified as a N-element positive-valued vector, where N is the number of parameters specified in the SensorLimits property. If you want to assign only one resolution cell for a parameter, simply specify its resolution as the difference between the maximum limit and the minimum limit of the parameter.
MaxNumDetsPerObject — Maximum number of detections per object
Maximum number of detections the sensor can report per object, specified as a positive integer.
ClutterDensity — Expected number of false alarms per unit volume
Expected number of false alarms per unit volume from the sensor, specified as a positive scalar.
MinDetectionProbability — Probability of detecting track estimated to be outside of sensor limits
Probability of detecting a target estimated to be outside of the sensor limits, specified as a positive scalar. This property allows a trackerPHD object to consider that the estimated target, which is outside the sensor limits, may be detectable.
Consider a radar with the following sensor limits and sensor resolution.
elLimits = [-2.5 2.5];
rangeLimits = [0 500];
rangeRateLimits = [-50 50];
sensorLimits = [azLimits;elLimits;rangeLimits;rangeRateLimits];
sensorResolution = [5 2 10 3];
Specifying the sensor transform function that transforms the Cartesian coordinates [x;y;vx;vy] in the scenario frame to the spherical coordinates [az;el;range;rr] in the sensor's frame. You can use the measurement function cvmeas as the sensor transform function.
transformFcn = @cvmeas;
To specify the parameters required for cvmeas, use the SensorTransformParameters property. Here, you assume the sensor is mounted at the center of the platform and the platform located at [100;30;20] is moving with a velocity of [-5;4;2] units per second in the scenario frame.
The first structure defines the sensor's location, velocity, and orientation in the platform frame.
params(1) = struct('Frame','Spherical','OriginPosition',[0;0;0],...
'OriginVelocity',[0;0;0],'Orientation',eye(3),'HasRange',true,...
'HasVelocity',true);
The second structure defines the platform's location, velocity, and orientation in the scenario frame.
params(2) = struct('Frame','Rectangular','OriginPosition',[100;30;20],...
'OriginVelocity',[-5;4;2],'Orientation',eye(3),'HasRange',true,...
config = trackingSensorConfiguration('SensorIndex',3,'SensorLimits',sensorLimits,...
'SensorTransformParameters',params,...
'FilterInitializationFcn',@initcvggiwphd)
trackingSensorConfiguration with properties:
SensorLimits: [4x2 double]
SensorResolution: [4x1 double]
SensorTransformFcn: @cvmeas
SensorTransformParameters: [1x2 struct]
FilterInitializationFcn: @initcvggiwphd
MaxNumDetsPerObject: Inf
MinDetectionProbability: 0.0500
Create a fusionRadarSensor object and specify its properties.
sensor = fusionRadarSensor(1, ...
'FieldOfView',[20 5], ...
'RangeLimits',[0 500], ...
'HasRangeRate',true, ...
'RangeRateLimits',[-50 50], ...
'AzimuthResolution',5, ...
'RangeResolution',10, ...
'ElevationResolution',2, ...
'RangeRateResolution',3);
Specify the cvmeas function as the sensor transform function.
Create a trackingSensorConfiguration object.
config = trackingSensorConfiguration(sensor,'SensorTransformFcn',transformFcn)
FilterInitializationFcn: []
MaxNumDetsPerObject: 1
To create the configuration for a sensor, you first need to specify the sensor transform function, which is usually given as:
Y=g\left(x,p\right)
where x denotes the tracking state, Y denotes detection states, and p denotes the required parameters. For object tracking applications, you mainly focus on obtaining an object's tracking state. For example, a radar sensor can measure an object's azimuth, elevation, range, and possibly range-rate. Using a trackingSensorConfiguration object, you can specify a radar's transform function using the SensorTransformFcn property and specify the radar's mounting location, orientation, and velocity using corresponding fields in the SensorTransformParameters property. If the object is moving at a constant velocity, constant acceleration, or constant turning, you can use the built-in measurement function – cvmeas, cameas, or ctmeas, respectively – as the SensorTransformFcn. To set up the exact outputs of these three functions, specify the hasAzimuth, hasElevation, hasRange, and hasVelocity fields as true or false in the SensorTransformParameters property.
To set up the configuration of a sensor, you also need to specify the sensor's detection ability. Primarily, you need to specify the sensor's detection limits. For all the outputs of the sensor transform function, you need to provide the detection limits in the same order of these outputs using the SensorLimits property. For example, for a radar sensor, you might need to provide its azimuth, elevation, range, and range-rate limits. You can also specify the radar's SensorResolution and MaxNumDetsPerObject properties if you want to consider extended object detection. You might also want to specify other properties, such as ClutterDensity, IsValidTime, and MinDetectionProbability to further clarify the sensor's detection ability.
trackerPHD | ggiwphd | cvmeas | cameas | ctmeas
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(Redirected from Three-dimensional space (mathematics))
For a broader, less mathematical treatment related to this topic, see Space.
"Three-dimensional" redirects here. For other uses, see 3D (disambiguation).
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension.
A representation of a three-dimensional Cartesian coordinate system with the x-axis pointing towards the observer.
In mathematics, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted
{\displaystyle \mathbb {R} ^{n},}
and can be identified to the n-dimensional Euclidean space. When n = 3, this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear).[1] It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced,[2] it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width/breadth, height/depth, and length.
1.3 Spheres and balls
2 In linear algebra
2.1 Dot product, angle, and length
3.1 Gradient, divergence and curl
3.2 Line integrals, surface integrals, and volume integrals
3.3 Fundamental theorem of line integrals
5 In finite geometry
Coordinate systemsEdit
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.[3]
Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. For more, see Euclidean space.
Below are images of the above-mentioned systems.
Lines and planesEdit
Two distinct points always determine a (straight) line. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.
Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.
A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.
Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, and hence are coplanar.
Spheres and ballsEdit
Main article: Sphere
A perspective projection of a sphere onto two dimensions
A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by
{\displaystyle V={\frac {4}{3}}\pi r^{3}}
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P(x, y, z, w), then x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.
Regular polytopes in three dimensions
Surfaces of revolutionEdit
Main article: Surface of revolution
A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.
Quadric surfacesEdit
In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
{\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,}
where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero, is called a quadric surface.[4]
There are six types of non-degenerate quadric surfaces:
The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of ℝ3 through that conic that are normal to π).[4] Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family.[5] Each family is called a regulus.
In linear algebraEdit
Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors.
Dot product, angle, and lengthEdit
A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in ℝ3 can be represented by an ordered triple of real numbers. These numbers are called the components of the vector.
The dot product of two vectors A = [A1, A2, A3] and B = [B1, B2, B3] is defined as:[6]
{\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}.}
The magnitude of a vector A is denoted by ||A||. The dot product of a vector A = [A1, A2, A3] with itself is
{\displaystyle \mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2},}
{\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }}={\sqrt {A_{1}^{2}+A_{2}^{2}+A_{3}^{2}}},}
Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by[7]
{\displaystyle \mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}
One can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[8]
In calculusEdit
Gradient, divergence and curlEdit
{\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }
The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:
{\displaystyle \operatorname {div} \,\mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}.}
Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [Fx, Fy, Fz]:
{\displaystyle {\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\\\F_{x}&F_{y}&F_{z}\end{vmatrix}}}
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[9]
{\displaystyle \left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} }
Line integrals, surface integrals, and volume integralsEdit
For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as
{\displaystyle \int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.}
where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and
{\displaystyle a<b}
For a vector field F : U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
{\displaystyle \int \limits _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.}
A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
{\displaystyle \iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {x} (s,t))\left\|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right\|\mathrm {d} s\,\mathrm {d} t}
where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. Given a vector field v on S, that is a function that assigns to each x in S a vector v(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
A volume integral refers to an integral over a 3-dimensional domain.
It can also mean a triple integral within a region D in R3 of a function
{\displaystyle f(x,y,z),}
{\displaystyle \iiint \limits _{D}f(x,y,z)\,dx\,dy\,dz.}
Fundamental theorem of line integralsEdit
{\displaystyle \varphi :U\subseteq \mathbb {R} ^{n}\to \mathbb {R} }
{\displaystyle \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{\gamma [\mathbf {p} ,\,\mathbf {q} ]}\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .}
Stokes' theoremEdit
Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:
{\displaystyle \iint _{\Sigma }\nabla \times \mathbf {F} \cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {r} .}
Divergence theoremEdit
Suppose V is a subset of
{\displaystyle \mathbb {R} ^{n}}
(in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂V = S ). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:[10]
{\displaystyle \iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {F} \right)\,dV=}
{\displaystyle \scriptstyle S}
{\displaystyle (\mathbf {F} \cdot \mathbf {n} )\,dS.}
The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.)
In topologyEdit
Wikipedia's globe logo in 3-D
Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a knot in a piece of string.[11]
In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble
{\displaystyle {\mathbb {R} }^{3}}
In finite geometryEdit
Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one regulus.[12]
Skew lines § Distance
^ "Euclidean space - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-12.
^ "Euclidean space | geometry". Encyclopedia Britannica. Retrieved 2020-08-12.
^ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). Calculus : Single and Multivariable (6 ed.). John wiley. ISBN 978-0470-88861-2.
^ a b Brannan, Esplen & Gray 1999, pp. 34–5
^ Brannan, Esplen & Gray 1999, pp. 41–2
^ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
^ M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum’s Outlines (2nd ed.). USA: McGraw Hill. ISBN 978-0-07-161545-7.
^ Rolfsen, Dale (1976). Knots and Links. Berkeley, California: Publish or Perish. ISBN 0-914098-16-0.
^ Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry, page 72, Cambridge University Press ISBN 0-521-48277-1
Anton, Howard (1994), Elementary Linear Algebra (7th ed.), John Wiley & Sons, ISBN 978-0-471-58742-2
Wikiquote has quotations related to Three-dimensional space.
Wikimedia Commons has media related to 3D.
The dictionary definition of three-dimensional at Wiktionary
Weisstein, Eric W. "Four-Dimensional Geometry". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Three-dimensional_space&oldid=1064034037"
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Sylvester's sequence - Wikipedia
Integer sequence in number theory
Graphical demonstration of the convergence of the sum 1/2 + 1/3 + 1/7 + 1/43 + ... to 1. Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. Squares with side lengths 1/1807 or smaller are too small to see in the figure and are not shown.
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms.
2 Closed form formula and asymptotics
3 Connection with Egyptian fractions
4 Uniqueness of quickly growing series with rational sums
5 Divisibility and factorizations
Formally, Sylvester's sequence can be defined by the formula
{\displaystyle s_{n}=1+\prod _{i=0}^{n-1}s_{i}.}
The product of the empty set is 1, so s0 = 2.
Alternatively, one may define the sequence by the recurrence
{\displaystyle \displaystyle s_{i}=s_{i-1}(s_{i-1}-1)+1,}
with s0 = 2.
It is straightforward to show by induction that this is equivalent to the other definition.
Closed form formula and asymptotics[edit]
The Sylvester numbers grow doubly exponentially as a function of n. Specifically, it can be shown that
{\displaystyle s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor ,}
for a number E that is approximately 1.26408473530530...[1] (sequence A076393 in the OEIS). This formula has the effect of the following algorithm:
s0 is the nearest integer to E2; s1 is the nearest integer to E4; s2 is the nearest integer to E8; for sn, take E2, square it n more times, and take the nearest integer.
This would only be a practical algorithm if we had a better way of calculating E to the requisite number of places than calculating sn and taking its repeated square root.
The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of Fermat numbers Fn; the Fermat numbers are usually defined by a doubly exponential formula,
{\displaystyle 2^{2^{n}}+1}
, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:
{\displaystyle F_{n}=2+\prod _{i=0}^{n-1}F_{i}.}
Connection with Egyptian fractions[edit]
The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series:
{\displaystyle \sum _{i=0}^{\infty }{\frac {1}{s_{i}}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+{\frac {1}{1807}}+\cdots .}
The partial sums of this series have a simple form,
{\displaystyle \sum _{i=0}^{j-1}{\frac {1}{s_{i}}}=1-{\frac {1}{s_{j}-1}}={\frac {s_{j}-2}{s_{j}-1}}.}
This may be proved by induction, or more directly by noting that the recursion implies that
{\displaystyle {\frac {1}{s_{i}-1}}-{\frac {1}{s_{i+1}-1}}={\frac {1}{s_{i}}},}
so the sum telescopes
{\displaystyle \sum _{i=0}^{j-1}{\frac {1}{s_{i}}}=\sum _{i=0}^{j-1}\left({\frac {1}{s_{i}-1}}-{\frac {1}{s_{i+1}-1}}\right)={\frac {1}{s_{0}-1}}-{\frac {1}{s_{j}-1}}=1-{\frac {1}{s_{j}-1}}.}
Since this sequence of partial sums (sj − 2)/(sj − 1) converges to one, the overall series forms an infinite Egyptian fraction representation of the number one:
{\displaystyle 1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+{\frac {1}{1807}}+\cdots .}
One can find finite Egyptian fraction representations of one, of any length, by truncating this series and subtracting one from the last denominator:
{\displaystyle 1={\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{6}},\quad 1={\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{7}}+{\tfrac {1}{42}},\quad 1={\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{7}}+{\tfrac {1}{43}}+{\tfrac {1}{1806}},\quad \dots .}
The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any k-term Egyptian fraction.[2] For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806, 1) requires at least five terms.
It is possible to interpret the Sylvester sequence as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one. Alternatively, the terms of the sequence after the first can be viewed as the denominators of the odd greedy expansion of 1/2.
Uniqueness of quickly growing series with rational sums[edit]
As Sylvester himself observed, Sylvester's sequence seems to be unique in having such quickly growing values, while simultaneously having a series of reciprocals that converges to a rational number. This sequence provides an example showing that double-exponential growth is not enough to cause an integer sequence to be an irrationality sequence.[3]
To make this more precise, it follows from results of Badea (1993) that, if a sequence of integers
{\displaystyle a_{n}}
grows quickly enough that
{\displaystyle a_{n}\geq a_{n-1}^{2}-a_{n-1}+1,}
and if the series
{\displaystyle A=\sum {\frac {1}{a_{i}}}}
converges to a rational number A, then, for all n after some point, this sequence must be defined by the same recurrence
{\displaystyle a_{n}=a_{n-1}^{2}-a_{n-1}+1}
that can be used to define Sylvester's sequence.
Erdős & Graham (1980) conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,
{\displaystyle \lim _{n\rightarrow \infty }{\frac {a_{n}}{a_{n-1}^{2}}}=1.}
Badea (1995) surveys progress related to this conjecture; see also Brown (1979).
Divisibility and factorizations[edit]
If i < j, it follows from the definition that sj ≡ 1 (mod si). Therefore, every two numbers in Sylvester's sequence are relatively prime. The sequence can be used to prove that there are infinitely many prime numbers, as any prime can divide at most one number in the sequence. More strongly, no prime factor of a number in the sequence can be congruent to 5 modulo 6, and the sequence can be used to prove that there are infinitely many primes congruent to 7 modulo 12.[4]
Are all the terms in Sylvester's sequence squarefree?
Much remains unknown about the factorization of the numbers in the Sylvester's sequence. For instance, it is not known if all numbers in the sequence are squarefree, although all the known terms are.
As Vardi (1991) describes, it is easy to determine which Sylvester number (if any) a given prime p divides: simply compute the recurrence defining the numbers modulo p until finding either a number that is congruent to zero (mod p) or finding a repeated modulus. Using this technique he found that 1166 out of the first three million primes are divisors of Sylvester numbers,[5] and that none of these primes has a square that divides a Sylvester number. The set of primes which can occur as factors of Sylvester numbers is of density zero in the set of all primes:[6] indeed, the number of such primes less than x is
{\displaystyle O(\pi (x)/\log \log \log x)}
The following table shows known factorizations of these numbers (except the first four, which are all prime):[8]
Factors of sn
5 3263443, which is prime
9 181 × 1987 × 112374829138729 × 114152531605972711 × 35874380272246624152764569191134894955972560447869169859142453622851
16 128551 × C13335
22 91798039513 × C853750
As is customary, Pn and Cn denote prime numbers and unfactored composite numbers n digits long.
Boyer, Galicki & Kollár (2005) use the properties of Sylvester's sequence to define large numbers of Sasakian Einstein manifolds having the differential topology of odd-dimensional spheres or exotic spheres. They show that the number of distinct Sasakian Einstein metrics on a topological sphere of dimension 2n − 1 is at least proportional to sn and hence has double exponential growth with n.
As Galambos & Woeginger (1995) describe, Brown (1979) and Liang (1980) used values derived from Sylvester's sequence to construct lower bound examples for online bin packing algorithms. Seiden & Woeginger (2005) similarly use the sequence to lower bound the performance of a two-dimensional cutting stock algorithm.[9]
Znám's problem concerns sets of numbers such that each number in the set divides but is not equal to the product of all the other numbers, plus one. Without the inequality requirement, the values in Sylvester's sequence would solve the problem; with that requirement, it has other solutions derived from recurrences similar to the one defining Sylvester's sequence. Solutions to Znám's problem have applications to the classification of surface singularities (Brenton and Hill 1988) and to the theory of nondeterministic finite automata.[10]
D. R. Curtiss (1922) describes an application of the closest approximations to one by k-term sums of unit fractions, in lower-bounding the number of divisors of any perfect number, and Miller (1919) uses the same property to upper bound the size of certain groups.
^ Graham, Knuth & Patashnik (1989) set this as an exercise; see also Golomb (1963).
^ This claim is commonly attributed to Curtiss (1922), but Miller (1919) appears to be making the same statement in an earlier paper. See also Rosenman & Underwood (1933), Salzer (1947), and Soundararajan (2005).
^ Guy & Nowakowski (1975).
^ This appears to be a typo, as Andersen finds 1167 prime divisors in this range.
^ Odoni (1985).
^ All prime factors p of Sylvester numbers sn with p < 5×107 and n ≤ 200 are listed by Vardi. Ken Takusagawa lists the factorizations up to s9 and the factorization of s10. The remaining factorizations are from a list of factorizations of Sylvester's sequence maintained by Jens Kruse Andersen. Retrieved 2014-06-13.
^ In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of Salzer (1947) on closest approximation.
^ Domaratzki et al. (2005).
Badea, Catalin (1993). "A theorem on irrationality of infinite series and applications". Acta Arithmetica. 63 (4): 313–323. doi:10.4064/aa-63-4-313-323. MR 1218459.
Badea, Catalin (1995). "On some criteria for irrationality for series of positive rationals: a survey" (PDF). Archived from the original (PDF) on 2008-09-11.
Boyer, Charles P.; Galicki, Krzysztof; Kollár, János (2005). "Einstein metrics on spheres". Annals of Mathematics. 162 (1): 557–580. arXiv:math.DG/0309408. doi:10.4007/annals.2005.162.557. MR 2178969. S2CID 13945306.
Brenton, Lawrence; Hill, Richard (1988). "On the Diophantine equation 1=Σ1/ni + 1/Πni and a class of homologically trivial complex surface singularities". Pacific Journal of Mathematics. 133 (1): 41–67. doi:10.2140/pjm.1988.133.41. MR 0936356.
Brown, D. J. (1979). "A lower bound for on-line one-dimensional bin packing algorithms". Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign. {{cite journal}}: Cite journal requires |journal= (help)
Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023.
Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005). "Non-uniqueness and radius of cyclic unary NFAs". International Journal of Foundations of Computer Science. 16 (5): 883–896. doi:10.1142/S0129054105003352. MR 2174328.
Erdős, Paul; Graham, Ronald L. (1980). Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique, No. 28, Univ. de Genève. MR 0592420.
Galambos, Gábor; Woeginger, Gerhard J. (1995). "On-line bin packing — A restricted survey". Mathematical Methods of Operations Research. 42 (1): 25. doi:10.1007/BF01415672. MR 1346486. S2CID 26692460.
Golomb, Solomon W. (1963). "On certain nonlinear recurring sequences". American Mathematical Monthly. 70 (4): 403–405. doi:10.2307/2311857. JSTOR 2311857. MR 0148605.
Graham, R.; Knuth, D. E.; Patashnik, O. (1989). Concrete Mathematics (2nd ed.). Addison-Wesley. Exercise 4.37. ISBN 0-201-55802-5.
Guy, Richard K. (2004). "E24 Irrationality sequences". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 346. ISBN 0-387-20860-7. Zbl 1058.11001.
Guy, Richard; Nowakowski, Richard (1975). "Discovering primes with Euclid". Delta (Waukesha). 5 (2): 49–63. MR 0384675.
Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math.NT/0612415. Bibcode:2006math.....12415J. doi:10.1112/jlms/jdn034. S2CID 15310955.
Liang, Frank M. (1980). "A lower bound for on-line bin packing". Information Processing Letters. 10 (2): 76–79. doi:10.1016/S0020-0190(80)90077-0. MR 0564503.
Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867.
Odoni, R. W. K. (1985). "On the prime divisors of the sequence wn+1 =1+w1⋯wn". Journal of the London Mathematical Society. Series II. 32: 1–11. doi:10.1112/jlms/s2-32.1.1. Zbl 0574.10020.
Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036.
Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339.
Seiden, Steven S.; Woeginger, Gerhard J. (2005). "The two-dimensional cutting stock problem revisited". Mathematical Programming. 102 (3): 519–530. doi:10.1007/s10107-004-0548-1. MR 2136225. S2CID 35815524.
Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247. {{cite journal}}: Cite journal requires |journal= (help)
Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. ISBN 0-201-52989-0.
Irrationality of Quadratic Sums, from K. S. Brown's MathPages.
Weisstein, Eric W. "Sylvester's Sequence". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sylvester%27s_sequence&oldid=1041880324"
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Revision as of 11:22, 29 August 2019 by Munich (talk | contribs) (→Data sets for download)
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle S3}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u_{i}'u_{j}'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle x_{\mathrm {adj} }=-1.5}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle C=-{\frac {\partial k}{\partial t}}-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle w'>0}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {p} }=1.08}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle z_{\mathrm {V1} }=0.06D}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle 0.3}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }}
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Plz solve q1 and q2: Solve The following Inequalities : 1 1 3x2-7x+8x2+1≤21 1,6 - Maths - Linear Inequalities - 12655636 | Meritnation.com
Plz solve q1 and q2:
Solve The following Inequalities :
1. 1<\frac{3{\mathrm{x}}^{2}-7\mathrm{x}+8}{{\mathrm{x}}^{2}+1}\le 2\phantom{\rule{0ex}{0ex}}\left(1\right) \left[1,6\right] \left(2\right) \left[-1,6\right]\phantom{\rule{0ex}{0ex}}\left(3\right) \left(-1,6\right] \left(4\right) \mathrm{R}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}2. 1-{\mathrm{e}}^{\left(\frac{1}{\mathrm{x}}-1\right)} >0\phantom{\rule{0ex}{0ex}}\left(1\right) \mathrm{x}\in \left(-\infty ,0 \right)\cup \left(1,\infty \right)\phantom{\rule{0ex}{0ex}}\left(2\right) \mathrm{x}\in \left(-\infty ,0 \right)\cup \left[1,\infty \right)\phantom{\rule{0ex}{0ex}}
@ Priyanshu Raj good answer for the first part.
Here is the answer for second part
1-{\mathrm{e}}^{\left(\frac{1}{\mathrm{x}}-1\right)}>0 \phantom{\rule{0ex}{0ex}}{\mathrm{e}}^{\left(\frac{1}{\mathrm{x}}-1\right)}<1\phantom{\rule{0ex}{0ex}}{\mathrm{e}}^{\left(\frac{1}{\mathrm{x}}-1\right)}<{\mathrm{e}}^{0}\phantom{\rule{0ex}{0ex}}\mathrm{since} \mathrm{base} \mathrm{is} \mathrm{same} \mathrm{and} \mathrm{greater} \mathrm{than} 1 \mathrm{so} \phantom{\rule{0ex}{0ex}}\left(\frac{1}{\mathrm{x}}-1\right)<0\phantom{\rule{0ex}{0ex}}\frac{1-\mathrm{x}}{\mathrm{x}}<0\phantom{\rule{0ex}{0ex}}\frac{\mathrm{x}-1}{\mathrm{x}}>0\phantom{\rule{0ex}{0ex}}-\infty <+++++++\left(0\right)-----\left(1\right)++++++>\infty \phantom{\rule{0ex}{0ex}}\mathrm{we} \mathrm{need} \mathrm{positive} \mathrm{region} \mathrm{so} \phantom{\rule{0ex}{0ex}}\mathrm{x} \in \left(-\infty ,0\right)\cup \left(1,\infty \right)\phantom{\rule{0ex}{0ex}}
Yatharth answered this
hi yatharth
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A Story of Basis and Kernel - Part I: Function Basis
1. Review of Basis Concept
We know that everything in the world can be decomposed into the combination of the basic elements. For example, water is the combination of hydrogen and oxygen. Similarly, in mathematics, basis is used to represent various things in a simple and unified way.
\mathcal{R}^n
space, we can use
n
independent vectors to represent any vector by linear combination. The
n
independent vectors can be viewed as a set of basis. There are infinite basis sets in
\mathcal{R}^n
space. Among them, basis vectors that are orthogonal to each other are of special interests. For example,
\{\mathbf{e}_i\}_{i=1}^n
is a special basis set with mutually orthogonal basis vectors in the same length, where
\mathbf{e}_i
is a vector that has all zero entries except the
i
th entry which equals 1.
The inner product operator measures the similarity between vectors. For two vectors
\mathbf{x}
\mathbf{y}
, the inner product is the projection of one vector to the other.
<\mathbf{x},\mathbf{y}>=|\mathbf{x}||\mathbf{y}|\cos\theta
\mathbf{x}=(x_1,\cdots,x_n)
\mathbf{y}=(y_1, \cdots, y_n)
<\mathbf{x},\mathbf{y}>=\sum_{i=1}^n x_i y_i
Until now it is a review of vector basis. These knowledge can also be extended to functions and function space.
A function is an infinite vector. As the following figure shows
For a function defined on the interval
[a,b]
, we take samples by an interval
\Delta x
. If we sample the function
f(x)
a, x_1,\cdots,x_n,b
, then we can transform the function into a vector
(f(a),f(x_1),\cdots,f(x_n),f(b))^T
\Delta x\rightarrow 0
, the vector should be more and more close to the function and at last, it becomes infinite.
The above analysis assumes
x
to be a real number. But when
x
is a vector, it still holds. In this article, we use bold font such as
\mathbf{x}
to denote a vector in
\mathcal{R}^n
space; use
f
to denote the function itself, namely the infinite vector; use
f(\mathbf{x})
to denote the evaluation of the function at point
\mathbf{x}
. And the evaluation of a function should be a real number.
Since functions are so close to vectors, we can also define the inner product of functions similarly. For two functions
f
g
sampling by interval
\Delta x
, the inner product may be defined as
<f,g>=\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i) g(x_i)\Delta x=\int f(x)g(x)dx
For a vector, the dimension is discrete. We only have the first, second,… dimension. But we don't have the 0.5, 1.5,… dimension. However, the dimension is not discrete for functions, but continuous. Thus we use the difference between adjacent dimensions (i.e.,
\Delta x
) for normalization.
The expression of function inner product is seen everywhere. It has various meanings in various context. For example, if
X
is a continuous random variable with probability density function
f(x)
f(x)>0
\int f(x) dx = 1
, then the expectation
E[g(x)]=\int f(x) g(x) dx=<f,g>
Similar to vector basis, we can use a set of functions to represent other functions. The difference is that in a vector space, we only need finite vectors to construct a complete basis set, but in function space, we may need infinite basis functions. Two functions can be regarded as orthogonal if their inner product is zero. In function space, we can also have a set of function basis that are mutually orthogonal.
3. Example: Fourier Series
Let the basis functions
\{h_p\}_{p=-\infty}^{+\infty}, (p
is integer) be
h_p(x)=e^{i2\pi p x/T}
defined on interval
[0, T]
is the imaginary number. These functions construct a function space and any function defined on interval
[0, T]
can be represented as linear combination of the basis functions. We can prove that any two basis functions are orthogonal (for complex numbers, the latter term should take conjugate transpose when calculating the inner product).
<h_p, h_q>=\int_0^T h_p (x) \bar{h}_q(x) dx=\int_0^T e^{i2\pi p x/T} e^{-i2\pi q x/T} dx=0
p \neq q
\overline{a+bi}=a-bi
. The "length" of basis is
|h_p|^2=<h_p,h_p>=T
f
[0,T]
is within the space, it can be written as
f(x)=\sum_p c_p h_p(x)=\sum_p c_p e^{i2\pi px/T}
<f,h_p>=<\sum_q c_q h_q,h_p>=\sum_q c_q < h_q,h_p>=c_p < h_p,h_p>=c_p T
the coefficient can be calculated as
c_p=\frac{1}{T} <f,h_p>=\frac{1}{T} \int f(x) \bar{h}_p(x) dx=\frac{1}{T} \int_0^T f(x) e^{-i2\pi p x} dx
which is the Fourier series.
4. Example: Wavelet Analysis
Define function
\psi(x)
\mathbf{R}
\psi(x)= \begin{cases} 1 \quad & 0 \leq x < \frac{1}{2},\\ -1 & \frac{1}{2} \leq x < 1\\ 0 & \text{otherwise} \end{cases}
Define a series of functions for every pair
n,k
of intergers
\psi_{n,k}(x) = 2^{n / 2} \psi(2^n x-k), \quad x \in \mathbf{R}
That is, we rescaled
\psi(x)
2^n
and shift by
2^{-n} k
<\psi_{n_1,k_1},\psi_{n_2,k_2} > = \int \psi_{n_1,k_1} (x),\psi_{n_2,k_2} (x) dx = \begin{cases} 1 \quad & n_1=n_2, k_1=k_2 \\ 0 \quad & \text{otherwise} \end{cases}
Any function can be represented as the linear combination of
\{ \psi_{n,k} \}_{n,k}
. By the same technique as the former example, we can also obtain the analytical expression of the coefficient for each basis function, which is the Haar wavelet analysis.
In the next part, fundermentals about kernel functions and kernel method will be discussed.
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Juan loves to ride his bicycle. Today he is cruising along Exeter Street, which is on flat ground, at
20
miles per hour (
29.33
feet per second). On his front wheel is a reflector
11
inches from the center of the wheel. Juan’s bike has
26
inch wheels. Assume that the reflector is farthest from the ground when the time
t=0
h
be the height of the reflector above the ground (in inches),
t
be the elapsed time (in seconds) and
x
the distance (in inches) Juan has traveled in
t
How far does Juan travel (in inches) when the wheel makes one complete revolution?
If the wheel has a diameter of
26
inches, what is the circumference of the wheel?
h(x)
, the height of the reflector, in terms of the distance traveled.
The center of the wheel is
13
inches off the ground. The reflector oscillates between
11
inches above and below the reflector.
These hints should tell you the amplitude and vertical shift.
Help More:
Since the reflector starts at the top, use the cosine function.
Since this function is in terms of distance traveled, one revolution of the wheel is the same as the circumference you found in part (a), or the period.
What is the height of the reflector after Juan has ridden
20
h(20 \text{ feet})
using your equation from part (b).
Careful! The equation in part (b) uses inches.
h(t)
, the height of the reflector, in terms of the time
t
This will be similar to your equation from part (b), but the period will be the time in seconds to complete one revolution.
Use dimensional analysis to determine "seconds per revolution" for this situation.
What is the height of the reflector after Juan has ridden for
5
h(5 \text{ minutes})
using your equation from part (d).
Careful! The equation in part (d) uses time in seconds.
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Defuzzification Methods - MATLAB & Simulink
Middle, Smallest, and Largest of Maximum
Choosing Defuzzification Method
This example describes the built-in methods for defuzzifying the output fuzzy set of a type-1 Mamdani fuzzy inference system.
Consider the following output fuzzy set, which is an aggregation of three scaled trapezoidal membership functions.
mf1 = trapmf(x,[0 2 8 12]);
mf2 = trapmf(x,[5 7 12 14]);
mf3 = trapmf(x,[12 13 18 19]);
mf = max(0.5*mf2,max(0.9*mf1,0.1*mf3));
figure('Tag','defuzz')
plot(x,mf,'LineWidth',3)
h_gca.YTick = [0 .5 1] ;
Fuzzy Logic Toolbox™ software supports five built-in methods for computing a single crisp output value for such a fuzzy set.
Middle of maximum
Smallest of maximum
Largest of maximum
You can also define your own custom defuzzification method. For more information, see Build Fuzzy Systems Using Custom Functions.
Centroid defuzzification returns the center of gravity of the fuzzy set along the x-axis. If you think of the area as a plate with uniform thickness and density, the centroid is the point along the x-axis about which the fuzzy set would balance. The centroid is computed using the following formula, where
\mu \left({\mathit{x}}_{\mathit{i}}\right)
is the membership value for point
{\mathit{x}}_{\mathit{i}}
in the universe of discourse.
\mathrm{xCentroid}=\frac{\sum _{\mathit{i}}\mu \left({\mathit{x}}_{\mathit{i}}\right){\mathit{x}}_{\mathit{i}}}{\sum _{\mathit{i}}\mu \left({\mathit{x}}_{\mathit{i}}\right)}
Compute the centroid of the fuzzy set.
xCentroid = defuzz(x,mf,'centroid');
Indicate the centroid defuzzification result on the original plot.
hCentroid = line([xCentroid xCentroid],[-0.2 1.2],'Color','k');
tCentroid = text(xCentroid,-0.2,' centroid','FontWeight','bold');
The bisector method finds the vertical line that divides the fuzzy set into two sub-regions of equal area. It is sometimes, but not always, coincident with the centroid line.
xBisector = defuzz(x,mf,'bisector');
Indicate the bisector result on the original plot, and gray out the centroid result.
hBisector = line([xBisector xBisector],[-0.4 1.2],'Color','k');
tBisector = text(xBisector,-0.4,' bisector','FontWeight','bold');
gray = 0.7*[1 1 1];
hCentroid.Color = gray;
tCentroid.Color = gray;
MOM, SOM, and LOM stand for middle, smallest, and largest of maximum, respectively. In this example, since the aggregate fuzzy set has a plateau at its maximum value, the MOM, SOM, and LOM defuzzification results have distinct values. If the aggregate fuzzy set has a unique maximum, then MOM, SOM, and LOM all produce the same value.
xMOM = defuzz(x,mf,'mom');
xSOM = defuzz(x,mf,'som');
xLOM = defuzz(x,mf,'lom');
Indicate the MOM, SOM, and LOM results on the original plot, and gray out the bisector result.
hMOM = line([xMOM xMOM],[-0.7 1.2],'Color','k');
tMOM = text(xMOM,-0.7,' MOM','FontWeight','bold');
hSOM = line([xSOM xSOM],[-0.7 1.2],'Color','k');
tSOM = text(xSOM,-0.7,' SOM','FontWeight','bold');
hLOM = line([xLOM xLOM],[-0.7 1.2],'Color','k');
tLOM = text(xLOM,-0.7,' LOM','FontWeight','bold');
hBisector.Color = gray;
tBisector.Color = gray;
In general, using the default centroid method is good enough for most applications. Once you have created your initial fuzzy inference system, you can try other defuzzification methods to see if any improve your inference results.
Highlight the centroid result, and gray out the MOM, SOM, and LOM results.
hCentroid.Color = 'red';
tCentroid.Color = 'red';
hMOM.Color = gray;
tMOM.Color = gray;
hSOM.Color = gray;
tSOM.Color = gray;
hLOM.Color = gray;
tLOM.Color = gray;
|
Distance Formula - Course Hero
College Algebra/Coordinates, Distance, and Midpoint/Distance Formula
The distance between two points with the same
x
y
-coordinate is the length of a horizontal or vertical line segment connecting those points. This can be found by counting grid squares or subtracting the coordinates that are not equal.
The distance between two points on a number line is the absolute value of the difference between the endpoints.
The distance between two points on a number line with coordinates
x
y
\lvert y-x\rvert
\lvert x-y\rvert
. The two expressions are equivalent because of the absolute value symbols.
x
y
The distance between pairs of points can be found by subtracting the coordinates or by counting squares on the grid.
The Distance between Points with an Identical Coordinate
A (5, 11)
B (5, -4)
A (-3, 3)
B (5, 3)
x
-values are equal. So, the distance between the points along the
y
|-4-11|=15
units. The
y
x
|5-(-3)|=8
There are two ways to determine the distance between two points with the same coordinate: subtracting the coordinates that are not the same or counting the number of units from the first point to the second point.
Using the Pythagorean Theorem to Find Distance
The distance between two points that do not have the same
x
y
-coordinate can be found by drawing a right triangle with the line segment between the points as the hypotenuse and using the Pythagorean theorem.
For two points in a plane that do not have equal
x
y
-coordinates, the distance between the points can be found by using the Pythagorean theorem: If
and
b
c
a^2+b^2=c^2
First, sketch a right triangle. The line segment joining the two points is the hypotenuse, while horizontal and vertical line segments form the legs. Then apply the Pythagorean theorem to find the length of the hypotenuse.
Determine the distance between point
A
(-3, 13)
B
(5, -2)
using the Pythagorean theorem.
Plot point
A
B
. Then draw a right triangle to locate point
C
(-3, -2)
to draw a right triangle.
Determine the length of the horizontal leg, or line segment
a
, of the triangle.
B
(5,-2)
C
(-3, -2)
y
-coordinates. So, calculate the distance between the
x
-coordinates or count the number of squares between the points.
\begin{aligned}{\text{Line segment}}\;a &= |x_{2}-x_{1}|\\&=|5-(-3)|\\&=|5+3|\\&=|8|\end{aligned}
Determine the length of the vertical leg, or line segment
b
A
(-3,13)
C
(-3, -2)
x
y
\begin{aligned}{\text{Line segment}}\;b &= |y_{2}-y_{1}|\\&=|13-(-2)|\\&=|13+2|\\&=|15|\end{aligned}
a^2+b^2=c^2
a
is 8 units. Line segment
b
The hypotenuse of the triangle, line segment
c
, is the distance between the points.
Substitute the lengths of each leg in the theorem:
\begin{aligned}a^{2}+b^{2}&=c^{2}\\ 8^{2}+15^{2}&=c^{2} \\ 64+225&=c^{2} \\ 289&=c^{2}\\ \sqrt{289}&=c\\17&=c\end{aligned}
A
(-3, 13)
B
(5, -2)
Deriving and Applying the Distance Formula
Using a right triangle, a formula can be derived for the distance between any two points in the coordinate plane.
The distance formula is used to calculate the distance between the points
A
(x_2, y_2)
B
(x_1, y_1)
in the coordinate plane. The distance formula is derived from the Pythagorean theorem. For a right triangle
\Delta ABC
with hypotenuse
\overline{AB}
, the lengths of the horizontal and vertical legs are:
\begin{aligned} |x_{2}-x_{1}|\\ |y_{2}-y_{1}|\end{aligned}
To find the distance between point
A
B
, draw a vertical line segment from point
A
and a horizontal line segment from point
B
to form a right triangle. The point where the line segments meet is point
C
. Then use the coordinates of point
A
B
to determine the lengths of the legs of the right triangle. Lastly, apply the Pythagorean theorem to determine the length of the hypotenuse, or the distance between point
A
B
Substitute the lengths of the legs into the Pythagorean theorem. Let
d
represent the distance between the two points, or the hypotenuse, line segment
c
d^2=|x_{2}-x_{1}|^{2}+|y_{2}-y_{1}|^{2}
Then take the square root of both sides of the equation to get the distance formula.
d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}
Since the squared values are always positive, the absolute-value signs are no longer needed.
A
(-3, 3)
B
(9, -2)
using the distance formula.
Substitute the coordinates of the points in the distance formula.
\begin{aligned}d&=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} \\ &=\sqrt{[(9-(-3)]^{2}+(-2-3)^{2}} \end{aligned}
\begin{aligned}d&=\sqrt{(9+3)^{2}+(-5)^{2}}\\ &=\sqrt{12^{2}+25}\\&=\sqrt{144+25}\\&=\sqrt{169}\\&=13\end{aligned}
A
(-3, 3)
B
(9, -2)
<Coordinate Plane>Midpoint Formula
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Precision and Range - MATLAB & Simulink - MathWorks Benelux
You must pay attention to the precision and range of the fixed-point data types and scalings you choose in order to know whether rounding methods will be invoked or if overflows or underflows will occur.
The range is the span of numbers that a fixed-point data type and scaling can represent. The range of representable numbers for a two's complement fixed-point number of word length
wl
S
B
For example, in two's complement, negative numbers must be represented as well as zero, so the maximum value is 2wl -1 – 1. Because there is only one representation for zero, there are an unequal number of positive and negative numbers. This means there is a representation for
-{2}^{wl-1}
{2}^{wl-1}
Because a fixed-point data type represents numbers within a finite range, overflows and underflows can occur if the result of an operation is larger or smaller than the numbers in that range.
Fixed-Point Designer™ software allows you to either saturate or wrap overflows. Saturation represents positive overflows as the largest positive number in the range being used, and negative overflows as the largest negative number in the range being used. Wrapping uses modulo arithmetic to cast an overflow back into the representable range of the data type.
When you create a fi object, any overflows are saturated. The OverflowAction property of the default fimath is saturate. You can log overflows and underflows by setting the LoggingMode property of the fipref object to on.
When you represent numbers with finite precision, not every number in the available range can be represented exactly. If a number cannot be represented exactly by the specified data type and scaling, a rounding method is used to cast the value to a representable number. Although precision is always lost in the rounding operation, the cost of the operation and the amount of bias that is introduced depends on the rounding method itself. To provide you with greater flexibility in the trade-off between cost and bias, Fixed-Point Designer software currently supports the following rounding methods:
Ceiling rounds to the closest representable number in the direction of positive infinity.
Convergent rounds to the closest representable number. In the case of a tie, convergent rounds to the nearest even number. This is the least biased rounding method provided by the toolbox.
Zero rounds to the closest representable number in the direction of zero.
Floor, which is equivalent to two's complement truncation, rounds to the closest representable number in the direction of negative infinity.
Nearest rounds to the closest representable number. In the case of a tie, nearest rounds to the closest representable number in the direction of positive infinity. This rounding method is the default for fi object creation and fi arithmetic.
Round rounds to the closest representable number. In the case of a tie, the round method rounds:
Positive numbers to the closest representable number in the direction of positive infinity.
Negative numbers to the closest representable number in the direction of negative infinity.
Choosing a Rounding Method. Each rounding method has a set of inherent properties. Depending on the requirements of your design, these properties could make the rounding method more or less desirable to you. By knowing the requirements of your design and understanding the properties of each rounding method, you can determine which is the best fit for your needs. The most important properties to consider are:
Ε\left(\stackrel{^}{\theta }-\theta \right)
Ε\left(\stackrel{^}{\theta }-\theta \right)<0
Ε\left(\stackrel{^}{\theta }-\theta \right)=0
Ε\left(\stackrel{^}{\theta }-\theta \right)>0
The following table shows a comparison of the different rounding methods available in the Fixed-Point Designer product.
Ceiling Low Large positive
Convergent High Unbiased
Floor Low Large negative
Nearest Moderate Small positive
(Simulink® only) Low Depends on the operation
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Existence of Solutions for a Baby-Skyrme Model
Hongan Hu, Kunlin Hu, "Existence of Solutions for a Baby-Skyrme Model", Journal of Applied Mathematics, vol. 2014, Article ID 530682, 4 pages, 2014. https://doi.org/10.1155/2014/530682
Hongan Hu1 and Kunlin Hu2
Academic Editor: Senlin Guo
The existence of the energy-minimizing solutions for a baby-Skyrme model on the sphere is proved using variational method. Some properties of the solutions are also established.
Half a century ago, Skyrme [1] firstly suggested that the soliton in the nonlinear -model [2] may be explained by the baryon number, which is corresponding to the winding number of soliton.
The Skyrmions were originally introduced to describe baryons in three spatial dimensions [1]. In a nonlinear scalar field theory, a Skyrmion is a classical static field configuration of minimal energy. The scalar field is the pion field, and the Skyrmion represents a baryon. The Skyrmion has a topological charge which prevents continuously deforming to the vacuum field configuration. This charge is identified with the conserved baryon number which prevents a baryon from decaying into pions [1, 3].
Skyrmions have been shown to exist for a very wide class of geometries [4], which are now playing an increasing role in other areas of physics as well. For example, in certain condense matter systems, Skyrmions are used to model the bubbles that appear in the presence of an external magnetic field in two dimensions; they could provide a mechanism associated with the disappearance of antiferromagnetism, the onset of HTc superconductivity, and so on. In condensed matter physics [5], the model [6] has direct applications which may give an effective description in quantum Hall systems. In the context of condensed matter physics [7, 8], direct experimental observations can be made. In three spatial dimensions [6], baby Skyrmions have been studied in the context of strong interactions as a toy-model in order to understand the more complicated dynamics of usual Skyrmions which live.
In the present paper we consider a baby-Skyrme model, that is, Skyrmional model in two spatial dimensions, which was introduced in [9]. Our purpose of this paper is to establish the existence of the energy-minimizing solutions for this baby-Skyrme model rigorously by the variational method. In Section 2, we will present the mathematical structure of the model and the main existence theorem. In Section 3, we will show the existence of the energy-minimizing solutions by the variational method and establish some properties of the solutions.
2. The Mathematical Structure and Existence Theorem
Baby Skyrmions are obtained as the nontrivial solutions of the well-known nonlinear model. The model consists of three real scalars subject to the constraint The equation of motion admits solutions with finite energy which represents a mapping of into . They are characterized by the density , and the winding number , The energy functional of this model is as follows: with where is a unit vector in the third derivation in internal space and is a parameter that is assumed positive.
By using the inequality we may find the Bogomol’nyi bound
We are to extend the model above by going from to where is the radius of the two-sphere. By the polar coordinates , and , And the Jacobian of the transformation and the metric associated with the polar coordinates are
In order to obtain explicit static solutions in the winding number sector, we introduce the hedgehog parameterization where is subject to the boundary conditions
The energy functional is as follows: while the winding number density results in
It is not difficult to show that the Euler-Lagrange equation of (13) is
Next we are to find a solution of the boundary problem (15) and (12). We will establish the existence of solutions by the indirect variational method.
Here is our main existence theorem, which solves the above problem.
Theorem 1. The boundary value problem (15) and (12) has a solution such that and there hold the sharp asymptotic estimates
In this section, we will divide the proof of Theorem 1 into two lemmas.
Lemma 2. The boundary value problem (15) and (12) has a solution such that
Proof. In order to get a solution of (15) with the boundary condition (12), we may look for the minimizers of the functional (13).
We first introduce the admissible space Obviously the set is not empty.
We intend to find a solution of (15) and (12) by solving the minimization problem: Let be a minimizing sequence of (20). Without loss of generality, we may assume that Otherwise, we may modify the sequence to fulfill (21) meanwhile without enlarging the energy. From the inequality we may see that uniformly as .
Similarly, we have Then, we may find that uniformly as .
In view of (22) and (23), letting , we have
We may get that the sequence is bounded in for any Using weak compactness, we may assume that (in fact, a subsequence in it) is weakly convergent in . Applying a diagonal subsequence argument, we may assume there is an such that weakly in . In view of the compact embedding theorem, we may get That is, can be compactly embedded into . So we see that the convergence (27) is strong in . Consequently, we know that is absolutely continuous in any compact subinterval of and continuous on .
Let Using the weak lower semicontinuity property of the functional, we obtain the inequality for any Letting we have
Thus we see that fulfills the complete boundary conditions (12). Therefore and (30) allows us to obtain That is, is found to be a solution of (20). As a consequence, is a finite-energy solution of (12) and (15).
Next we will establish some properties of the energy-minimizing solutions.
Lemma 3. Let be the energy-minimizing solution obtained in Lemma 2. Then
Proof. Evidently, is an equilibrium point of (15). We assume that there is such that Hence, attains its global minimum, so Using the uniqueness theorem for the initial value problem of ordinary differential equations, we can get which contradicts so Similarly, we may find that
Combining Lemmas 2 and 3, we complete the proof of Theorem 1.
This work was supported by the Natural Science Foundation of Henan Province under Grant 122300410188. The authors thank the referees for their valuable suggestions which improve the quality of this paper. They also thank Professor S.-X. Chen for his guidance and assistance throughout the work.
T. H. R. Skyrme, “A non-linear field theory,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 260, pp. 127–138, 1961. View at: Publisher Site | Google Scholar | MathSciNet
Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, NY, USA, 2001. View at: Publisher Site | MathSciNet
T. H. R. Skyrme, “A unified field theory of mesons and baryons,” Nuclear Physics, vol. 31, pp. 556–569, 1962. View at: Publisher Site | Google Scholar | MathSciNet
N. S. Manton, “Geometry of skyrmions,” Communications in Mathematical Physics, vol. 111, no. 3, pp. 469–478, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. D. Jackson, N. S. Manton, and A. Wirzba, “New skyrmion solutions on a
3
-sphere,” Nuclear Physics A, vol. 495, no. 3-4, pp. 499–522, 1989. View at: Publisher Site | Google Scholar | MathSciNet
B. M. A. Piette, B. J. Schroers, and W. J. Zakrzewski, “Dynamics of baby skyrmions,” Nuclear Physics B, vol. 439, no. 1-2, pp. 205–235, 1995. View at: Publisher Site | Google Scholar | MathSciNet
S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, “Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies,” Physical Review B, vol. 47, no. 24, pp. 16419–16426, 1993. View at: Publisher Site | Google Scholar
X. Z. Yu, Y. Onose, N. Kanazawa et al., “Real-space observation of a two-dimensional skyrmion crystal,” Nature, vol. 465, no. 7300, pp. 901–904, 2010. View at: Publisher Site | Google Scholar
N. N. Scoccola and D. R. Bes, “Two-dimensional skyrmions on the sphere,” Journal of High Energy Physics, vol. 1998, no. 9, 9 pages, 1998. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 Hongan Hu and Kunlin Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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m (→Modules overview: some re-structuring)
[http://en.wikipedia.org/wiki/Pansharpened_image Pan-Sharpening] / [http://en.wikipedia.org/wiki/Image_fusion Fusion] is the process of merging high-resolution panchromatic and lower resolution multi-spectral imagery. [http://grass.osgeo.org/grass70/ GRASS 7] holds a dedicated pan-sharpening module, {{cmd|i.pansharpen}} which features three techniques for sharpening, namely the [http://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Brovey_Transformation Brovey transformation], the classical IHS method and one that is based on [[Principal Components Analysis]] (PCA). Another algorithm deriving excellent detail and a realistic representation of original multispectral scene colors, is the High-Pass Filter Addition (HPFA) technique.
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L\lambda ={\frac {10^{4}*DN\lambda }{CalCoef\lambda *Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
{\displaystyle [0,255]}
{\displaystyle [0,2047]}
All of the above steps can be replicated in GRASS GIS. A script that implements the algorithm is currently developed and hopefully published as a grass add-on.
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : AlgebraicGeometryTools Subpackage : TangentPlane
compute the plane plane of an hypersurface at a point given by a regular chain
TangentPlane(rc, f, R)
The command TangentPlane(rc, f, R) returns the tangent plane of the hypersurface defined by f at every point of F defined by rc.
The result is a list of pairs [g,ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that g, and g a polynomial the zero set of which defines the tangent plane of f at ts.
This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form TangentPlane(..) only after executing the command with(RegularChains[AlgebraicGeometryTools]). However, it can always be accessed through the long form of the command by using RegularChains[AlgebraicGeometryTools][TangentPlane](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right):
R≔\mathrm{PolynomialRing}\left([x,y,z]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
\mathrm{rc}≔\mathrm{Empty}\left(R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{rc}≔\mathrm{Chain}\left([z-1,y,x],\mathrm{rc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{Equations}\left(\mathrm{rc},R\right)
[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}]
f≔xz+zy+yx
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}
\mathrm{tp}≔\mathrm{TangentPlane}\left(\mathrm{rc},f,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{tp}}\textcolor[rgb]{0,0,1}{≔}[[\textcolor[rgb]{0,0,1}{\mathrm{_x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rc}}]]
The RegularChains[AlgebraicGeometryTools][TangentPlane] command was introduced in Maple 2020.
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Special relativity - Simple English Wikipedia, the free encyclopedia
(Redirected from Special Relativity)
Special relativity (or the special theory of relativity) is a theory in physics that was developed and explained by Albert Einstein in 1905. It applies to all physical phenomena, so long as gravitation is not significant. Special relativity applies to Minkowski space, or "flat spacetime" (phenomena which are not influenced by gravitation).
Einstein knew that some weaknesses had been discovered in older physics. For example, older physics thought light moved in luminiferous aether. Various tiny effects were expected if this theory were true. Gradually it seemed these predictions were not going to work out.
Eventually, Einstein (1905) drew the conclusion that the concepts of space and time needed a fundamental revision. The result was special relativity theory, which brought together a new principle "the constancy of the speed of light" and the previously established "principle of relativity".
Galileo had already established the principle of relativity, which said that physical events must look the same to all observers, and no observer has the "right" way to look at the things studied by physics. For example, the Earth is moving very fast around the Sun, but we do not notice it because we are moving with the Earth at the same speed; therefore, from our point of view, the Earth is at rest. However, Galileo's math could not explain some things, such as the speed of light. According to him, the measured speed of light should be different for different speeds of the observer in comparison with its source. However, the Michelson-Morley experiment showed that this is not true, at least not for all cases. Einstein's theory of special relativity explained this among other things.
1 Basics of special relativity
2 The Lorentz transformations
3 Mass, energy and momentum
5 Experimental confirmations
Basics of special relativity[change | change source]
Suppose that you are moving toward something that is moving toward you. If you measure its speed, it will seem to be moving faster than if you were not moving. Now suppose you are moving away from something that is moving toward you. If you measure its speed again, it will seem to be moving more slowly. This is the idea of "relative speed"—the speed of the object relative to you.
Before Albert Einstein, scientists were trying to measure the "relative speed" of light. They were doing this by measuring the speed of star light reaching the Earth. They expected that if the Earth was moving toward a star, the light from that star should seem faster than if the Earth was moving away from that star. However, they noticed that no matter who performed the experiments, where the experiments were performed, or what star light was used, the measured speed of light in a vacuum was always the same.[1]
Einstein said this happens because there is something unexpected about length and duration, or how long something lasts. He thought that as Earth moves through space, all measurable durations change very slightly. Any clock used to measure a duration will be wrong by exactly the right amount so that the speed of light remains the same. Imagining a "light clock" allows us to better understand this remarkable fact for the case of a single light wave.
Also, Einstein said that as Earth moves through space, all measurable lengths change (ever so slightly). Any device measuring length will give a length off by exactly the right amount so that the speed of light remains the same.
The most difficult thing to understand is that events that appear to be simultaneous in one frame may not be simultaneous in another. This has many effects that are not easy to perceive or understand. Since the length of an object is the distance from head to tail at one simultaneous moment, it follows that if two observers disagree about what events are simultaneous then this will affect (sometimes dramatically) their measurements of the length of objects. Furthermore, if a line of clocks appear synchronized to a stationary observer and appear to be out of sync to that same observer after accelerating to a certain velocity then it follows that during the acceleration the clocks ran at different speeds. Some may even run backwards. This line of reasoning leads to general relativity.
Other scientists before Einstein had written about light seeming to go the same speed no matter how it was observed. What made Einstein's theory so revolutionary is that it considers the measurement of the speed of light to be constant by definition, in other words it is a law of nature. This has the remarkable implications that speed-related measurements, length and duration, change in order to accommodate this.
The Lorentz transformations[change | change source]
The mathematical bases of special relativity are the Lorentz transformations, which mathematically describe the views of space and time for two observers who are moving relative to each other but are not experiencing acceleration.
To define the transformations we use a Cartesian coordinate system to mathematically describe the time and space of "events".
Each observer can describe an event as the position of something in space at a certain time, using coordinates (x,y,z,t).
The location of the event is defined in the first three coordinates (x,y,z) in relation to an arbitrary center (0,0,0) so that (3,3,3) is a diagonal going 3 units of distance (like meters or miles) out in each direction.
The time of the event is described with the fourth coordinate t in relation to an arbitrary (0) point in time in some unit of time (like seconds or hours or years).
Let there be an observer K who describes when events occur with a time coordinate t, and who describes where events occur with spatial coordinates x, y, and z. This is mathematically defining the first observer whose "point of view" will be our first reference.
Let us specify that the time of an event is given: by the time that it is observed t(observed) (say today, at 12 o'clock) minus the time that it took for the observation to reach the observer.
This can be calculated as the distance from the observer to the event d(observed) (say the event is on a star which is 1 light year away, so it takes the light 1 year to reach the observer) divided by c, the speed of light (several million miles per hour), which we define as being the same for all observers.
This is correct because distance, divided by speed gives the time it takes to go that distance at that speed (e.g. 30 miles divided by 10 mph: give us 3 hours, because if you go at 10 mph for 3 hours, you reach 30 miles). So we have:
{\displaystyle t=d/c}
This is mathematically defining what any "time" means for any observer.
Now with these definitions in place, let there be another observer K' who is
moving along the x axis of K at a rate of v,
has a spatial coordinate system of x' , y' , and z' ,
where x' axis is coincident with the x axis, and with the y' and z' axes - "always being parallel" to the y and z axes.
This means that when K' gives a location like (3,1,2), the x (which is 3 in this example) is the same place that K, the first observer would be talking about, but the 1 on the y axis or the 2 on the z axis are only parallel to some location on the K' observer's coordinate system, and
where K and K' are coincident at t = t' = 0
This means that the coordinate (0,0,0,0) is the same event for both observers.
In other words, both observers have (at least) one time and location that they both agree on, which is location and time zero.
The Lorentz Transformations then are
{\displaystyle t'=(t-vx/c^{2})/{\sqrt {1-v^{2}/c^{2}}}}
{\displaystyle x'=(x-vt)/{\sqrt {1-v^{2}/c^{2}}}}
{\displaystyle y'=y}
{\displaystyle z'=z}
Define an event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in a reference frame moving at a velocity v with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way: is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′ is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.
Solving the above four transformation equations for the unprimed coordinates yields the inverse Lorentz transformation:
{\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}
There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction, which can be done by directions parallel to the motion (which are warped by the γ factor) and perpendicular; see the article Lorentz transformation for details.
{\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}
{\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}
{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }
{\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}
{\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }
{\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}
{\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }
{\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}
{\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }
{\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}
Mass, energy and momentum[change | change source]
In special relativity, the momentum
{\displaystyle p}
and the total energy
{\displaystyle E}
of an object as a function of its mass
{\displaystyle m}
{\displaystyle p={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
{\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
A frequently made error (also in some books) is to rewrite this equation using a "relativistic mass" (in the direction of motion) of
{\displaystyle m_{r}={\frac {m}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
. The reason why this is incorrect is that light, for example, has no mass, but has energy. If we use this formula, the photon (particle of light) has a mass, which is according to experiments incorrect.
In special relativity, an object's mass, total energy and momentum are related by the equation
{\displaystyle E^{2}=p^{2}c^{2}+m^{2}c^{4}}
For an object at rest,
{\displaystyle p=0}
so the above equation simplifies to
{\displaystyle E=mc^{2}}
. Hence, a massive object at rest still has energy. We call this rest energy and denote it by
{\displaystyle E_{0}}
{\displaystyle E_{0}=mc^{2}}
The need for special relativity arose from Maxwell's equations of electromagnetism, which were published in 1865. It was later found that they call for electromagnetic waves (such as light) to move at a constant speed (i.e., the speed of light).
To have James Clerk Maxwell's equations be consistent with both astronomical observations[1] and Newtonian physics,[2] Maxwell proposed in 1877 that light travels through an ether which is everywhere in the universe.
In 1887, the famous Michelson-Morley experiment tried to detect the "ether wind" generated by the movement of the Earth.[3] The persistent null results of this experiment puzzled physicists, and called the ether theory into question.
In 1895, Lorentz and Fitzgerald noted that the null result of the Michelson-Morley experiment could be explained by the ether wind contracting the experiment in the direction of motion of the ether. This effect is called the Lorentz contraction, and (without ether) is a consequence of special relativity.
In 1899, Lorentz first published the Lorentz equations. Although this was not the first time they had been published, this was the first time that they were used as an explanation of the Michelson-Morley's null result, since the Lorentz contraction is a result of them.
In 1900, Poincaré gave a famous speech in which he considered the possibility that some "new physics" was needed to explain the Michelson-Morley experiment.
In 1904, Lorentz showed that electrical and magnetic fields can be modified into each other through the Lorentz transformations.
In 1905, Einstein published his article introducing special relativity, "On the Electrodynamics of Moving Bodies", in Annalen der Physik. In this article, he presented the postulates of relativity, derived the Lorentz transformations from them, and (unaware of Lorentz's 1904 article) also showed how the Lorentz Transformations affect electric and magnetic fields.
Later in 1905, Einstein published another article presenting E = mc2.
In 1908, Max Planck endorsed Einstein's theory and named it "relativity". In that same year, Hermann Minkowski gave a famous speech on Space and Time in which he showed that relativity is self-consistent and further developed the theory. These events forced the physics community to take relativity seriously. Relativity came to be more and more accepted after that.
In 1912, Einstein and Lorentz were nominated for the Nobel prize in physics due to their pioneering work on relativity. Unfortunately, relativity was so controversial then, and remained controversial for such a long time that a Nobel prize was never awarded for it.
Experimental confirmations[change | change source]
The Michelson-Morley experiment, which failed to detect any difference in the speed of light based on the direction of the light's movement.
Fizeau's experiment, in which the index of refraction for light in moving water cannot be made to be less than 1. The observed results are explained by the relativistic rule for adding velocities.
The energy and momentum of light obey the equation
{\displaystyle E=pc}
. (In Newtonian physics, this is expected to be
{\displaystyle E={\begin{matrix}{\frac {1}{2}}\end{matrix}}pc}
The transverse doppler effect, which is where the light emitted by a quickly moving object is red-shifted due to time dilation.
The presence of muons created in the upper atmosphere at the surface of Earth. The issue is that it takes much longer than the half-life of the muons to get down to Earth surface even at nearly the speed of light. Their presence can be seen as either being due to time dilation (in our view) or length contraction of the distance to the earth surface (in the muon's view).
Particle accelerators cannot be constructed without accounting for relativistic physics.
[1] Observations of binary stars show that light takes the same amount of time to reach the Earth over the same distance for both stars in such systems. If the speed of light was constant with respect to its source, the light from the approaching star would arrive sooner than the light from the receding star. This would cause binary stars to appear to move in ways that violate Kepler's Laws, but this is not seen.
[2] The second postulate of special relativity (that the speed of light is the same for all observers) contradicts Newtonian physics.
[3] Since the Earth is constantly being accelerated as it orbits the Sun, the initial null result was not a concern. However, that did mean that a strong ether wind should have been present 6 months later, but none was observed.
↑ Light in different media (water,air..) may travel at different speeds.
↑ Okun, L. B. (July 1998), "Note on the meaning and terminology of Special Relativity", European Journal of Physics, 19 (4): 403–406, doi:10.1088/0143-0807/19/4/015
W. Rindler, Introduction to Special Relativity, 2nd edition, Oxford Science Publications, 1991, ISBN 0-19-853952-5.
Web article on the history of special relativity Archived 2013-12-09 at the Wayback Machine
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html Archived 2015-10-13 at the Wayback Machine
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Special_relativity&oldid=7943348"
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I know that there is the addition rule of probability,
I suppose that you question is this:
A
B
cannot occur simultaneously, then why is the probability that
A
occurs or
B
occurs equal to the probability that
A
occurs plus the probability that
B
occurs?
Suppose that there are, say, 100 possibilities, that
A
takes place in 50 of them and that
B
takes place in 20 of them. Then the probability that
A
occurs is
\frac{1}{2}\left(=\frac{50}{100}\right)
and the probability that
B
\frac{1}{5}\left(=\frac{20}{100}\right)
. What is the probability that
A
B
occurs? Well, out of those 100 possibilities,
A
B
occurs exactly in 70 of them (this is where I use the fact that
A
B
cannot occur simultaneously). So, the probability that
A
B
\begin{array}{rl}\frac{70}{100}& =\frac{50}{100}+\frac{20}{100}\\ & =\text{probability that }A\text{ occurs}+\text{probability that }B\text{ occurs.}\end{array}
Tyler Velasquez
Because the probability is the number of favorable draws over the total number of draws.
And the number of favorable draws are additive: the number of [red or green] balls is the number of red plus the number of green.
The additive rule is only valid for disjoint categories. For example, if you have black/white balls and dice, it is not necessarily true that
\text{#(black or ball)}=\text{#black}+\text{#balls}.
P\left(A\cup \left(B\cap C\right)\right)
P\left(A\cap \left(B\cup C\right)\right)=P\left(A\cap B\right)+P\left(A\cap C\right).
P\left(A\cup \left(B\cap C\right)\right)=P\left(A\cup B\right)+P\left(A\cup C\right)?
P\left(2\right)+P\left(5\right)
P\left(2\right)=1/6,P\left(5\right)=1/6
3-2-1
3\oplus 2\oplus 1
|
Math - Balancer
The Balancer whitepaper describes a set of formulas derived from the value function for interacting with the protocol. The formulas in the Theory section are sufficient to describe the functional specification, but they are not straightforward to implement for the EVM, in part due to a lack of mature fixed-point math libraries.
Our implementation uses a combination of a few algebraic transformations, approximation functions, and numerical hacks to compute these formulas with bounded maximum error and reasonable gas cost.
Exponentiation Approximation
SP^o_i = \frac{ \frac{B_i}{W_i} }{ \frac{B_o}{W_o} }
Bi is the balance of token i, the token being sold by the trader which is going into the pool.
Bo is the balance of token o, the token being bought by the trader which is coming out of the pool.
Wi is the weight of token i
Wo is the weight of token o
When we consider swap fees, we do exactly the same calculations as without fees, but using
A_i \cdot (1-swapFee)
A_i
. This strategy is referred to as charging fees "on the way in." With the swap fee, the spot price increases. It then becomes:
SP^o_i = \frac{ \frac{B_i}{W_i} }{ \frac{B_o}{W_o} } \cdot \frac{1}{(1-swapFee)}
In the Whitepaper, we derive the following formula to calculate the amount of tokens out –
A_o
– a trader gets in return for a given amount of tokens in –
A_i
, considering a Balancer pool without any swap fees:
A_{o} = B_{o} \cdot \left(1 - \left(\frac{B_{i}}{B_{i}+A_{i}}\right)^{\frac{W_{i}}{W_{o}}}\right)
To take into account the swap fees charged by the Balancer pool, we replace
A_i
A_i \cdot (1-swapFee)
. This is known as charging the fees "on the way in"
A_{o} = B_{o} \cdot \left(1 - \left(\frac{B_{i}}{B_{i}+A_{i} \cdot (1-swapFee)}\right)^{\frac{W_{i}}{W_{o}}}\right)
In the Whitepaper, we derive the following formula for the amount of tokens in –
A_i
– a trader needs to swap to get a desired amount
A_o
of tokens out in return, considering a Balancer pool without any swap fees:
A_{i} = B_{i} \cdot \left(\left(\frac{B_{o}}{B_{o}-A_{o}}\right)^{\frac{W_{o}}{W_{i}}}-1\right)
A_i
is the amount the user has to swap to get a desired amount out
A_o
, all we have to do to include swap fees is divide the formula above by
(1-swapFee)
. This is because we know the fee charged on the way in will multiply that amount
A_i
(1-swapFee)
. This will cross out both terms
(1-swapFee)
and the amount out will be
A_o
A_{i} = B_{i} \cdot \left(\left(\frac{B_{o}}{B_{o}-A_{o}}\right)^{\frac{W_{o}}{W_{i}}}-1\right) \cdot \frac{1}{(1-swapFee)}
All-Asset Deposit/Withdrawal
Anyone can be issued Balancer pool tokens (provided the pool is finalized) by depositing proportional amounts of each of the assets contained in the pool. So, for each token k in the pool, the amounts of token k –
D_k
– that need to be deposited for someone to get
P_{issued}
pool tokens are:
D_k = \left(\frac{P_{supply}+P_{issued}}{P_{supply}}-1\right) \cdot B_k
Conversely, if a user wants to redeem their pool tokens to get their proportional share of each of the underlying tokens in the pool, the amounts of token k –
A_k
– a user gets for redeeming
P_{redeemed}
pool tokens will be:
A_k = \left(1-\frac{P_{supply}-P_{redeemed}}{P_{supply}}\right) \cdot B_k
All Balancer Protocol smart contracts were coded supporting a protocol-level exit fee to be charged that goes to Balancer Labs for supporting the development of the protocol. However, after careful consideration the Balancer Labs team decided to launch the first version of Balancer without any protocol fees whatsoever. (For technical reasons, this is unlikely to change.)
Single-Asset Deposit / Withdrawal
In the Whitepaper, we derive the following formula for the amount of pool tokens –
P_{issued}
– a liquidity provider gets in return for depositing an amount
A_t
of a single token t present in the pool:
P_{issued} = P_{supply} \cdot \left(\left(1+\frac{A_t}{B_t}\right)^{W_t} -1\right)
Since Balancer allows for depositing and withdrawing liquidity to Balancer pools using only one of the tokens present in the pool, this could be used to do the equivalent of a swap: provide liquidity depositing token A, and immediately withdraw that liquidity in token B. Therefore a swap fee has to be charged, proportional to the tokens that would need to be swapped for an all-asset deposit.
Another justification for charging a swap fee when a liquidity provider does a single-asset deposit is that they are getting a share of a pool that contains a basket of different assets. So what they are really doing is trading one of the pool assets (the token t being deposited) for proportional shares of all the pool assets.
Since the pool already has a share of its value in token t, represented by the weight
W_t
, it only makes sense to charge a swap fee for the remaining portion of the deposit
A_t \cdot(1 - W_t)
P_{issued} = P_{supply} \cdot \left(\left(1+\frac{\left(A_t-A_t\cdot(1 - W_t)\cdot swapFee\right)}{B_t}\right)^{W_t} -1\right)
The formula above calculates the amount of pool tokens one receives in return for a deposit of a given amount of a single asset. We also allow for users to define a given amount of pool tokens they desire to get –
P_{issued}
– and calculate what amount of tokens t is needed –
A_t
A_t = B_t \cdot \left(\left(1+\frac{P_{issued}}{P_{supply}}\right)^{\frac{1}{W_t}} -1\right)
Taking into account the swap fees, we have:
A_t = B_t \cdot \frac{\left(\left(1+\frac{P_{issued}}{P_{supply}}\right)^{\frac{1}{W_t}} -1\right)}{(1 - W_t)\cdot swapFee}
Without considering swap fees, each withdrawal formula is simply the inverse of the corresponding deposit formula. In other words, if you deposit a given amount of token t for pool tokens and then immediately redeem these pool tokens for token t, you should receive exactly what you started off with.
The formula without considering swap fees is then:
A_t = B_t \cdot \left(1-\left(1-\frac{P_{redeemed}}{P_{supply}}\right)^\frac{1}{W_t}\right)
A_t
is the amount of token t one receives when redeeming
P_{redeemed}
pool tokens.
Considering swap fees, we have the following:
A_t = B_t \cdot \left(1-\left(1-\frac{P_{redeemed}}{P_{supply}}\right)^\frac{1}{W_t}\right)\cdot \left(1-(1 - W_t)\cdot swapFee\right)
If there were an exit fee, it would be taken from the amount of tokens redeemed
P_{redeemed}
but as mentioned above this fee is zero in the first version of Balancer.
Balancer also allows for a liquidity provider to choose a desired amount of token t,
A_t
, they would like to withdraw from the pool, and calculates the necessary amount of pool tokens required for that,
P_{redeemed}
. The formula without considering swap fees is:
P_{redeemed} = P_{supply} \cdot \left(1-\left(1-\frac{A_t}{B_t}\right)^{W_t} \right)
A_t
P_{redeemed}
P_{redeemed} = P_{supply} \cdot \left(1-\left(1-\frac{\frac{A_t}{\left(1-(1 - W_t)\cdot swapFee\right)}}{B_t}\right)^{W_t} \right)
|
Improved Infra-Chromatic Bound for Exact Maximum Clique Search
Pablo San Segundo Alexey Nikolaev Mikhail Batsyn Panos M. Pardalos
This paper improves an infra-chromatic bound which is used by the exact branch-and-bound maximum clique solver BBMCX (San Segundo et al., 2015) as an upper bound on the clique number for every subproblem. The infra-chromatic bound looks for triplets of colour subsets which cannot contain a 3-clique. As a result, it is tighter than the bound obtained by widely used approximate-colouring algorithms because it can be lower than the chromatic number. The reported results show that our algorithm with the new bound significantly outperforms the state-of-the-art algorithms in a number of structured and uniform random graphs.
\mathit{n}\ge 3
|
Pound (force) — Wikipedia Republished // WIKI 2
Earth's gravitational pull on a one-pound mass
Look up pound-force or pound in Wiktionary, the free dictionary.
"lbf" redirects here. For the airport in North Platte, Nebraska, assigned the IATA code LBF, see North Platte Regional Airport.
For the unit of mass, see Pound (mass). For the basis weight of paper, see Paper density. For the monetary unit, see Pound (currency).
English Engineering units, British Gravitational System
1 lbf in ... ... is equal to ...
CGS units
444,822.2 dyn
Absolute English System
32.17405 pdl
The pound of force or pound-force (symbol: lbf,[1] sometimes lbf,[2]) is a unit of force used in some systems of measurement, including English Engineering units[a] and the foot–pound–second system.[3]
Pound-force should not be confused with pound-mass (lb), often simply called pound, which is a unit of mass, nor should these be confused with foot-pound (ft⋅lbf), a unit of energy, or pound-foot (lbf⋅ft), a unit of torque.
POUND MASS vs. POUND FORCE (lbm vs. lbf)
Pound-force (lbf) vs Pound-mass (lbm) vs Slugs
Pound Force, Kilogram Force and Gravitational Constant - Applied Fluid Dynamics - Class 006
Pounds Force vs Pounds Mass (LBf vs LBm)
Pound Mass versus Pound Force
1.1 Product of avoirdupois pound and standard gravity
2 Conversion to other units
3 Foot–pound–second (FPS) systems of units
4 Pound of thrust
The pound-force is equal to the gravitational force exerted on a mass of one avoirdupois pound on the surface of Earth. Since the 18th century, the unit has been used in low-precision measurements, for which small changes in Earth's gravity (which varies from equator to pole by up to half a percent) can safely be neglected.[4]
The 20th century, however, brought the need for a more precise definition, requiring a standardized value for acceleration due to gravity.
Product of avoirdupois pound and standard gravity
The pound-force is the product of one avoirdupois pound (exactly 0.45359237 kg) and the standard acceleration due to gravity, 9.80665 m/s2 (32.174049 ft/s2).[5][6][7]
The standard values of acceleration of the standard gravitational field (gn) and the international avoirdupois pound (lb) result in a pound-force equal to 4.4482216152605 N.[b]
{\displaystyle {\begin{aligned}1\,{\text{lbf}}&=1\,{\text{lb}}\times g_{\text{n}}\\&=1\,{\text{lb}}\times 9.80665\,{\tfrac {\text{m}}{{\text{s}}^{2}}}/0.3048\,{\tfrac {\text{m}}{\text{ft}}}\\&\approx 1\,{\text{lb}}\times 32.174049\,\mathrm {\tfrac {ft}{s^{2}}} \\&\approx 32.174049\,\mathrm {\tfrac {ft{\cdot }lb}{s^{2}}} \\1\,{\text{lbf}}&=1\,{\text{lb}}\times 0.45359237\,{\tfrac {\text{kg}}{\text{lb}}}\times g_{\text{n}}\\&=0.45359237\,{\text{kg}}\times 9.80665\,{\tfrac {\text{m}}{{\text{s}}^{2}}}\\&=4.4482216152605\,{\text{N}}\end{aligned}}}
This definition can be rephrased in terms of the slug. A slug has a mass of 32.174049 lb. A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s2, so:
{\displaystyle {\begin{aligned}1\,{\text{lbf}}&=1\,{\text{slug}}\times 1\,{\tfrac {\text{ft}}{{\text{s}}^{2}}}\\&=1\,{\tfrac {{\text{slug}}\cdot {\text{ft}}}{{\text{s}}^{2}}}\end{aligned}}}
Foot–pound–second (FPS) systems of units
Main article: Foot–pound–second system
In some contexts, the term "pound" is used almost exclusively to refer to the unit of force and not the unit of mass. In those applications, the preferred unit of mass is the slug, i.e. lbf⋅s2/ft. In other contexts, the unit "pound" refers to a unit of mass. The international standard symbol for the pound as a unit of mass is lb.[8]
Three approaches to units of mass and force or weight[9][10]
In the "engineering" systems (middle column), the weight of the mass unit (pound-mass) on Earth's surface is approximately equal to the force unit (pound-force). This is convenient because one pound mass exerts one pound force due to gravity. Note, however, unlike the other systems the force unit is not equal to the mass unit multiplied by the acceleration unit[11]—the use of Newton's second law, F = m ⋅ a, requires another factor, gc, usually taken to be 32.174049 (lb⋅ft)/(lbf⋅s2). "Absolute" systems are coherent systems of units: by using the slug as the unit of mass, the "gravitational" FPS system (left column) avoids the need for such a constant. The SI is an "absolute" metric system with kilogram and meter as base units.
Pound of thrust
Further information: thrust
The term pound of thrust is an alternative name for pound-force in specific contexts. It is frequently seen in US sources on jet engines and rocketry, some of which continue to use the FPS notation. For example, the thrust produced by each of the Space Shuttle's two Solid Rocket Boosters was 3,300,000 pounds-force (14.7 MN), together 6,600,000 pounds-force (29.36 MN).[12][13]
Foot-pound (energy)
Mass in general relativity
Mass in special relativity
Mass versus weight for the difference between the two physical properties
Pounds per square inch, a unit of pressure
^ Despite its name, this system is based on United States customary units and is only used in the US.
^ The international avoirdupois pound is defined to be exactly 0.45359237 kg.
^ IEEE Standard Letter Symbols for Units of Measurement (SI Units, Customary Inch-Pound Units, and Certain Other Units), IEEE Std 260.1™-2004 (Revision of IEEE Std 260.1-1993)
^ Fletcher, Leroy S.; Shoup, Terry E. (1978), Introduction to Engineering, Prentice-Hall, ISBN 978-0135018583, LCCN 77024142, archived from the original on 2019-12-06, retrieved 2017-08-03. : 257
^ "Mass and Weight". engineeringtoolbox.com. Archived from the original on 2010-08-18. Retrieved 2010-08-03.
^ Acceleration due to gravity varies over the surface of the Earth, generally increasing from about 9.78 m/s2 (32.1 ft/s2) at the equator to about 9.83 m/s2 (32.3 ft/s2) at the poles.
^ BS 350 : Part 1: 1974 Conversion factors and tables, Part 1. Basis of tables. Conversion factors. British Standards Institution. 1974. p. 43.
^ In 1901 the third CGPM Archived 2012-02-07 at the Wayback Machine declared (second resolution) that:
The value adopted in the International Service of Weights and Measures for the standard acceleration due to Earth's gravity is 980.665 cm/s2, value already stated in the laws of some countries.
This value was the conventional reference for calculating the kilogram-force, a unit of force whose use has been deprecated since the introduction of SI.
^ Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24
^ IEEE Std 260.1™-2004, IEEE Standard Letter Symbols for Units of Measurement (SI Units, Customary Inch-Pound Units, and Certain Other Units)
^ The acceleration unit is the distance unit divided by the time unit squared.
^ "Space Launchers - Space Shuttle". www.braeunig.us. Archived from the original on 6 April 2018. Retrieved 16 February 2018. Thrust: combined thrust 29.36 MN SL (maximum thrust at launch reducing by 1/3 after 50 s)
^ Richard Martin (12 January 2001). "From Russia, With 1 Million Pounds of Thrust". wired.com. Archived from the original on 25 September 2019. Retrieved 25 November 2019.
Obert, Edward F. (1948). Thermodynamics. New York: D. J. Leggett Book Company. Chapter I "Survey of Dimensions and Units", pp. 1-24.
Comparison with imperial unit system
Board foot
Face cord
Feet per second
Inch of mercury
Kilopounds per square inch
Bolt (cloth)
Degree (angle)
Pound-foot (torque)
Ton of refrigeration
American wire gauge
Body jewelry sizes
|
Distance from Bloch-Type Functions to the Analytic Space
Cheng Yuan, Cezhong Tong, "Distance from Bloch-Type Functions to the Analytic Space ", Abstract and Applied Analysis, vol. 2014, Article ID 610237, 7 pages, 2014. https://doi.org/10.1155/2014/610237
Cheng Yuan1 and Cezhong Tong2
1Institute of Mathematics, School of Science, Tianjin University of Technology and Education, Tianjin 300222, China
2Department of Mathematics, School of Science, Hebei University of Technology, Tianjin 300401, China
The analytic space can be embedded into a Bloch-type space. We establish a distance formula from Bloch-type functions to , which generalizes the distance formula from Bloch functions to BMOA by Peter Jones, and to by Zhao.
Let denote the unit disc of the complex plane and let be its boundary. As usual, denotes the space of all analytic functions on .
Recall that, for , the Bloch-type space is the space of analytic functions on satisfying The little Bloch-type space is the subspace of all with It is well known that is a Banach space under the norm In particular, when , becomes the classic Bloch space , which is the maximal Möbius invariant Banach space that has a decent linear functional; see [1, 2] for more details on the Bloch spaces.
For , the involution of the unit disk is denoted by . It is well known and easy to check that
Let , , , and . The space , introduced by Zhao in [3] and known as the general family of function spaces, is defined as the set of for which where is the normalized area measure on . The space consists of all such that For appropriate parameter values , , and , coincides with several classical function spaces. For instance, if . The space is the classical Bergman space , and is the classical Besov space . The spaces are the spaces, in particular, , and the function space of bounded mean oscillation. See [3–9] for these basic facts.
For , we say that a nonnegative Borel measure defined on is an -Carleson measure if where the supremum ranges over all subarcs of , denotes the arc length of , and is the Carleson square based on a subarc . We write for the class of all -Carleson measures. Moreover, is said to be a vanishing -Carleson measure if
For an analytic function on , we define It was proved in [3] that if and only if is an -Carleson measure and if and only if is a vanishing -Carleson measure.
Let be an analytic function space. The distance from a Bloch-type function to is defined by
The following result is obtained by Zhao in [9].
Theorem 1. Suppose , , and . The following two quantities are equivalent: (1);(2), where and denotes the characteristic function of a set.
When and , the above characterization is Peter Jone’s distance formula from a Bloch function to BMOA (Peter Jone never published his result but a proof was provided in [10]). Also, similar type results can be found in [11–13]. Specifically, distance from Bloch function to -type space is given in [11]; to the little Bloch space is obtained in [12], and to the space of the ball is characterized in [13]. All these spaces are Möbius invariant.
This paper is dedicated to characterize the distance from to , which extends Zhao’s result. The main result is following.
Theorem 2. Suppose , , , and . Then where
The strategy in this paper follows from Theorem in [14]. The distance from a function to Campanato-Morrey space was given in [15] with similar idea.
Notation. Throughout this paper, we only write (or ) for for a positive constant , and moreover for both and .
We begin with a lemma quoted from Lemma in [14].
Lemma 3. Let , , and be nonnegative Radon measures on . Then, if and only if
According to Lemma 3 and the fact that if and only if is an -Carleson measure, we can easily get the following corollary.
Corollary 4. Let be an analytic function on . if and only if there exists an such that
We will also need the following standard result from [16].
Lemma 5. Suppose and . Then, for all .
The following lemma, quoted from Lemma 1 in [9], is an extension of Lemma 5. See also [17].
Lemma 6. Suppose and , . If , then
Next, we see that is contained in . We thank Zhao for pointing out that the following result is firstly proved in [3]. Here, we give another proof with a different approach.
Lemma 7. For , , and , . In particular, if , then .
Proof. We can use the reproducing formula for to get that for some constant , where is a real number greater than ; see, for example, [14, page 55].
Let . If , denote ; it follows from the Hölder’s inequality and (15) that Apparently, we have used Lemma 5 in the last inequality. This gives that when .
If , then Recall that and . We can easily use (4) to check that Thus, when .
Now, suppose and let , then for all . It follows that Again, the above inequality follows from Lemma 5. This completes the proof.
Our strategy relies on an integral operator preserving the -Carleson measures. For , we define the integral operator as
The following lemma is similar to Theorem 2.5 in [18]. Indeed, Qiu and Wu proved the case . Specially, the case is just Lemma in [14].
Lemma 8. Assume , , and . Let , let , and let be Lebesgue measurable on . If belongs to , then also belongs to .
Proof. We firstly prove the case and then sketch the outline argument of the case modified from [18] for the completeness.
When , according to Lemma 3, it is sufficient to show that for some . That is to show is finite. By Fubini’s theorem, it is enough to verify that is finite.
Choosing such that , we can use Lemma 6 to control the last integral by Since is an -Carleson measure, we can complete the proof by using Lemma 3 again.
When , we need to verify that
holds for any arc . In order to make this estimate, let , be the biggest integer satisfying , and let , , denotes the arcs on with the same center as and length , and is just . We can control and decompose the integral as
In order to estimate Int1, we define the linear operator as where If we choose a test function , then Schur’s lemma combines with Lemma 5 implying that Hence, is a bounded operator. Letting , then with Thus,
To handle , first note that, for , if and , then . Further, it is easy to check that, for any fixed , Now, splitting as we have Recall that . It follows from Hölder’s inequality that
Now, an easy computation gives that since and . This completes the proof.
Proof of Theorem 2. For , it is easy to establish the following formula (see, e.g., [19, (1.1)] or [14, page 55]. Notice that it is a special case of the -order derivative of , as in [14], which holds for all holomorphic on ). Consider Define, for each , Then, Write Then,
So, if is in , Lemma 8 implies that By Corollary 4, . Meanwhile, recall that, for and , we can use Lemma 5 to obtain This means that To summarize the above argument, we have , (by (47)), and (by (49)), and is an -Carleson measure for each . Thus,
In order to prove the other direction of the inequality, we assume that equals the right-hand quantity of the last inequality and We only consider the case . Then, there exists an such that Hence, by definition, we can find a function such that Now, for any , we have that is not in . But, according to (53), we get and so This implies that does not belong to . But, it follows from (13) that . Therefore, Since , Corollary 4 implies that is in . This means that is in , and so is . This contradicts (57). Thus, we must have as required.
Remark 9. Theorem 2 characterizes the closure of in the norm. That is, for , is in the closure of in the norm if and only if, for every , for any Carleson square .
The authors would like to thank the referee for her/his helpful comments and suggestions which improved this paper. Cheng Yuan is supported by NSFC 11226086 of China and Tianjin Advanced Education Development Fund 20111005; Cezhong Tong is supported by the National Natural Science Foundation of China (Grant nos. 11301132 and 11171087) and Natural Science Foundation of Hebei Province (Grant no. A2013202265).
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Copyright © 2014 Cheng Yuan and Cezhong Tong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Question in population dynamics using exponential growth rate equation Given population doubles
\frac{dN}{dt}=2N
N\left(t\right)={N}_{0}{e}^{2t}
r=2
Since the population doubles in 20 minutes, assuming your t variable is in minutes, you should have that:
N\left(20\right)=2N\left(0\right)
If you have an exponential growth rate by assumption,
\frac{dN}{dt}=\lambda N
N={N}_{0}{e}^{\lambda t}
If you evaluate this at times
t=0\text{ }\text{and}\text{ }t=20
, you'll find that
N\left(0\right)={N}_{0}\text{ }\text{and}\text{ }N\left(20\right)={N}_{0}{e}^{20\lambda }
. You can then plug in these expressions into the initial equation comparing the population at these two times to solve for
\lambda
, which should not be equal to 2.
\frac{dw}{dt}
\frac{dw}{dt}
w=x\mathrm{sin}y,\text{ }x={e}^{t},\text{ }y=\pi -t
t=0
Removal of absolute signs
An object is dropped from a cliff. The object leaves with zero speed, and t seconds later its speed v metres per second satisfies the differential equation
\frac{dv}{dt}=10-0.1{v}^{2}
So I found t in terms of v
t=\frac{1}{2}\mathrm{ln}|\frac{10+v}{10-v}|
The questions goes on like this: Find the speed of the object after 1 second. Part of the answer key shows this
t=\frac{1}{2}\mathrm{ln}|\frac{10+v}{10-v}|
2t=\mathrm{ln}|\frac{10+v}{10-v}|
{e}^{2t}=\frac{10+v}{10-v}
So here's my question: why is it not like this?
±{e}^{2t}=\frac{10+v}{10-v}
Why can you ignore the absolute sign?
How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on
\left[1,\mathrm{\infty }\right)
{r}^{2}f{ }^{″}\left(r\right)+2r{f}^{\prime }\left(r\right)-l\left(l+1\right)f\left(r\right)=-\frac{\left({f}^{\prime }\left(1\right)+f\left(1\right)\right)}{{r}^{2}}
f\left(1\right)=\alpha
\underset{r\to \mathrm{\infty }}{lim}f\left(r\right)=0
where l is a positive integer and
\alpha
Define the following norm
‖·‖
‖f{‖}^{2}:={\int }_{1}^{\infty }{r}^{2}f\text{'}\left(r{\right)}^{2}dr+l\left(l+1\right){\int }_{1}^{\infty }f\left(r{\right)}^{2}dr
I want to prove the estimate:
‖f‖\le C\sqrt{l\left(l+1\right)}|\alpha |
for some constant C independent of
\alpha
, l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:
\begin{array}{rl}‖f{‖}^{2}={\int }_{1}^{\mathrm{\infty }}{r}^{2}{f}^{2}+{\int }_{1}^{\mathrm{\infty }}l\left(l+1\right){f}^{2}& ={f}^{\prime }\left(1\right)f\left(1\right)+\left({f}^{\prime }\left(1\right)+f\left(1\right)\right){\int }_{1}^{\mathrm{\infty }}\frac{f}{{r}^{2}}\\ & \le {f}^{\prime }\left(1\right)\alpha +\frac{\left({f}^{\prime }\left(1\right)+\alpha \right)}{3}\sqrt{{\int }_{1}^{\mathrm{\infty }}{f}^{2}}\\ & \le {f}^{\prime }\left(1\right)\alpha +\frac{\left({f}^{\prime }\left(1\right)+\alpha \right)}{3}‖f‖\end{array}
where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.
Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.
\mathrm{cos}x\frac{dy}{dx}+\left(\mathrm{sin}x\right)y=1
Solving homogenous system with complex eigenvalues
\frac{dx}{dt}=2x+8y
\frac{dy}{dt}=-x-2y
When I solve the determinant of the matrix, I get
\lambda =±2i
. Then , I plug it in the matrix, and get the for the first eigenvalue,
\lambda =2i
0=\left(2-2i\right)x+8y
0=-x-\left(2+2i\right)y
x=\frac{-8}{2-2i}y
x=-\left(2+2i\right)y
\stackrel{\to }{Y}\text{'}=\left(\begin{array}{cc}-4& 2\\ 2& -1\end{array}\right)\stackrel{\to }{Y}+\left(\begin{array}{c}{x}^{-1}\\ 2{x}^{-1}+4\end{array}\right)
z=\frac{1}{2}\left({e}^{{x}^{2}+{y}^{2}}-{e}^{-{x}^{2}-{y}^{2}}\right)
w=2{z}^{3}y\mathrm{sin}x
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Chebyshev Polynomials - Application to Polynomial Interpolation | Brilliant Math & Science Wiki
Chebyshev Polynomials - Application to Polynomial Interpolation
Kev Du, Calvin Lin, and Jimin Khim contributed
Recall that the Chebyshev polynomials are defined by
T_n (x) = \cos ( n \arccos x ),\quad T_n ( \cos \theta) = \cos n \theta .
T_{n} (x)
\begin{aligned} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^2 - 1 \\ T_3(x) &= 4x^3 - 3x \\ T_4(x) &= 8x^4 - 8x^2 + 1 \\ T_5(x) &= 16x^5 - 20x^3 + 5x \\ T_6(x) &= 32x^6 - 48x^4 + 18x^2 - 1 \\ T_7(x) &= 64x^7 - 112x^5 + 56x^3 - 7x \\ T_8(x) &= 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \\ T_9(x) &= 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \\ T_{10}(x) &= 512x^{10} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1. \\ \end{aligned}
Finding Roots of a Chebyshev Polynomial
Finding Minimal Polynomial of Roots in Trigonometric Form
y
between -1 and 1, the solutions to
T_n (x) = y
\cos \frac{ \theta + 2 \pi k } { n }
k
n
\cos \theta = y
For each of these values, we have
T_n \left( \cos \frac{ \theta + 2 \pi k } { n }\right) = \cos\left( n \times \frac{ \theta + 2 \pi k } { n } \right ) = \cos \theta = y.
n
solutions to a degree
n
polynomial, these are all of the roots.
_\square
\cos A
, what are the roots of
T_3 ( x) = 0?
\cos A
2 T_4 ( x) - 1 = 0?
\cos A
4 T_5 ^2 ( x) - 3 = 0?
The converse of the above theorem is as follows:
The polynomial whose roots are
\cos \frac{ \theta + 2\pi k } { n }
k
n
T_n ( x) = \cos \theta .
T_n
, find a polynomial whose roots are
\cos 10 ^ \circ , \cos 100^\circ, \cos 190^ \circ, \cos 180 ^ \circ
T_n
\cos \left( \theta + \frac{ 2 n } { 5} \pi \right)
i = 1
5.
To answer problems in this section, you should be familiar with Vieta's formula to help you find the sum and product of roots.
\cos 10 ^ \circ \times \cos 50 ^ \circ \times \cos 70 ^ \circ.
\frac{ 1}{ \cos 25 ^ \circ } + \frac{ 1}{ \cos 115 ^ \circ } + \frac{ 1}{ \cos 205 ^ \circ } + \frac{ 1}{ \cos 295 ^ \circ }.
T_n
, find a polynomial whose roots are exactly
\cos^2 1 ^ \circ, \cos^2 3 ^ \circ, \cos^2 5 ^ \circ, \ldots , \cos^2 89 ^ \circ .
Cite as: Chebyshev Polynomials - Application to Polynomial Interpolation. Brilliant.org. Retrieved from https://brilliant.org/wiki/chebyshev-polynomials-application-to-polynomial/
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A bullet of 10 g strikes a sand-bag at a speed of 10^3 m/s and gets embedded after travelling 5 - Physics - Motion in one dimension - 16823901 | Meritnation.com
Mass of the bullet, m = 10 g = 10-2 kg
Speed of the bullet, u = 103 m/s
Distance travelled by the bullet before stopping (v=0), d = 5 cm = 0.05 m
(i)Let the acceleration of the bullet be a.
Using the kinematic equation for motion under uniform acceleration:
{v}^{2}={u}^{2}+2ad\phantom{\rule{0ex}{0ex}}0={\left({10}^{3}\right)}^{2}+2×a×0.05\phantom{\rule{0ex}{0ex}}a=-\frac{{10}^{6}}{2×0.05}=-{10}^{7}m/{s}^{2}
(ii) Let the force exerted by the sand on the bullet be F
Using Newton's 2nd law of motion
F=ma\phantom{\rule{0ex}{0ex}}F={10}^{-2}×\left(-{10}^{7}\right)=-{10}^{5}N\phantom{\rule{0ex}{0ex}}Force is negative which means that it is resistive or retarding in nature.
(iii) Let t be the time taken by the bullet to come to rest.
v=u+at\phantom{\rule{0ex}{0ex}}0=u+at\phantom{\rule{0ex}{0ex}}-{10}^{3}=-{10}^{7}×t\phantom{\rule{0ex}{0ex}}t=\frac{{10}^{3}}{{10}^{7}}={10}^{-4}s
Dhananjay Srivastava answered this
Acrlaration 10000000m/s^2
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H(s)=\frac{s-5}{(s-3)(s-1)} The inverse Laplace transform of H(s) is equal to f*g
Gage Potter 2022-05-01 Answered
H\left(s\right)=\frac{s-5}{\left(s-3\right)\left(s-1\right)}
f\cdot g
icebox2686zsd
F\left(s\right)=\frac{s-5}{s-3}=1-\frac{2}{s-3}
f\left(t\right)=\delta \left(t\right)-2{e}^{3t}u\left(t\right)
h\left(t\right)={\int }_{0}^{t}{e}^{t-{t}^{\prime }}\left(\delta \left({t}^{\prime }\right)-2{e}^{3{t}^{\prime }}\right){dt}^{\prime }
={e}^{t}-{e}^{t}\left({e}^{2t}-1\right)
=2{e}^{t}-{e}^{3t}
Note that partial fraction expansion makes things easier. We simply write
H\left(s\right)=\frac{2}{s-1}-\frac{1}{s-3}
what is the squareroot of 2
Use Laplace transform to solve the following initial value problem:
{y}^{″}+5y=1+t,y\left(0\right)=0,{y}^{\prime }\left(0\right)=4
\frac{7}{25}{e}^{t}\mathrm{cos}\left(2t\right)+\frac{21}{10}{e}^{t}\mathrm{sin}\left(2t\right)
\frac{7}{2}+\frac{t}{5}-\frac{t}{25}{e}^{t}\mathrm{cos}\left(2t\right)+\frac{21}{10}{e}^{t}\mathrm{sin}\left(2t\right)
\frac{7}{25}+\frac{t}{5}
\frac{7}{25}+\frac{t}{5}-\frac{7}{25}{e}^{t}\mathrm{cos}\left(2t\right)+\frac{21}{10}{e}^{t}\mathrm{sin}\left(2t\right)
\frac{7}{25}+\frac{t}{5}-\frac{7}{5}{e}^{t}\mathrm{cos}\left(2t\right)+\frac{21}{10}{e}^{t}\mathrm{sin}\left(2t\right)
The Laplace transform X(s) of a signal x(t) has four poles and an unknown number of zeroes. The signal x(t) is known to have an impulse at t=0. Determine what information, if any , this provides about the number of zeroes and their locations.
Find y which satisfies:
{y}^{\prime }={y}^{a},y\left(a\right)=a-2
a\in N
\frac{dx}{dt}+2x=2{e}^{-3t}
x\left(0\right)=2
Finding eigenvalues by inspection?
I need to solve the following problem,
In this problem, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of the system.
{x}_{1}^{{}^{\prime }}=2{x}_{1}+{x}_{2}-{x}_{3}
{x}_{2}^{{}^{\prime }}=-4{x}_{1}-3{x}_{2}-{x}_{3}
{x}_{3}^{{}^{\prime }}=4{x}_{1}+4{x}_{2}+2{x}_{3}
Now I know how to find the eigenvalues by using the fact that
|A-\lambda I|=0
, but how would I do it by inspection? Inspection is easy for matrices that have the sum of their rows adding up to the same value, but this coefficient matrix doesn't have that property.
EDIT: Originally I didn't understand what inspection meant either. After googling it this is what I found. Imagine you have the matrix,
A=\left(\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right)
By noticing (or inspecting) that each row sums up to the same value, which is 0, we can easily see that [1, 1, 1] is an eigenvector with the associated eigenvalue of 0.
Solution of the following initial value problem using the Laplace transform
y"+4y=4t
y\left(0\right)=1
{y}^{\prime }\left(0\right)=5
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Logarithmic potentials, quasiconformal flows, and Q-curvature
1 April 2008 Logarithmic potentials, quasiconformal flows, and
Q
Mario Bonk, Juha Heinonen, Eero Saksman
Mario Bonk,1 Juha Heinonen,1 Eero Saksman2
By using quasiconformal flows, we establish that exponentials of logarithmic potentials of measures of small mass are comparable to Jacobians of quasiconformal homeomorphisms of
{\mathbb{R}}^{n}
n\ge 2
. As an application, we obtain the fact that certain complete conformal deformations of an even-dimensional Euclidean space
{\mathbb{R}}^{n}
with small total Paneitz or
Q
-curvature are bi-Lipschitz equivalent to standard
{\mathbb{R}}^{n}
Mario Bonk. Juha Heinonen. Eero Saksman. "Logarithmic potentials, quasiconformal flows, and
Q
-curvature." Duke Math. J. 142 (2) 197 - 239, 1 April 2008. https://doi.org/10.1215/00127094-2008-005
Mario Bonk, Juha Heinonen, Eero Saksman "Logarithmic potentials, quasiconformal flows, and
Q
-curvature," Duke Mathematical Journal, Duke Math. J. 142(2), 197-239, (1 April 2008)
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Bose-Einstein condensate - zxc.wiki
The Bose-Einstein condensate (after Satyendranath Bose and Albert Einstein ; abbreviation BEK , English BEC ) is an extreme aggregate state of a system of indistinguishable particles in which the majority of the particles are in the same quantum mechanical state . This is only possible if the particles are bosons and are therefore subject to Bose-Einstein statistics .
Bose-Einstein condensates are macroscopic quantum objects in which the individual bosons are completely de- localized . This is also known as the macroscopic quantum state . The bosons are completely indistinguishable. The state can therefore be described by a single wave function .
The resulting properties are superfluidity , superconductivity , suprasolidity or coherence over macroscopic distances. The latter allows interference experiments with Bose-Einstein condensates and the production of an atomic laser , which can be obtained by controlled decoupling of part of the matter wave from the trap holding the condensate.
2 conditions of existence
Theoretically, Albert Einstein predicted in 1924 - based on a work by Satyendranath Bose on the quantum statistics of photons - that a homogeneous, ideal Bose gas would condense at low temperatures .
The superfluid properties of liquid helium at temperatures below 2.17 K were then attributed to the Bose-Einstein condensation. However, direct observation of the effect in this system is extremely difficult because here the interaction between the atoms cannot be neglected. Therefore, contrary to the Bose-Einstein theory, which has since been proven experimentally in ultracold gases, in superfluid helium not a maximum of 100%, but only 8% of the atoms are in the ground state .
Attempts to achieve a Bose-Einstein condensation in a gas composed of polarized hydrogen atoms were initially unsuccessful.
The first Bose-Einstein condensates were experimentally produced in June and September 1995 by Eric A. Cornell and Carl E. Wieman at JILA and by Wolfgang Ketterle , Kendall Davis and Marc-Oliver Mewes at MIT . In 2001, Cornell, Wiemann, and Ketterle received the Nobel Prize in Physics for this .
The phase transition from a classical atomic gas to a Bose-Einstein condensate takes place when a critical phase space density is reached, that is, when the density of the particles with almost the same momentum is large enough.
You can understand it this way: the atoms are quantum particles , the movement of which is represented by a wave packet . The expansion of this wave packet is the thermal De Broglie wavelength . This becomes greater the further the temperature drops. When the De Broglie wavelength reaches the mean distance between two atoms, the quantum properties come into play. Bose-Einstein condensation now sets in in a three-dimensional ensemble . It is therefore necessary to increase the density of the gas and lower the temperature in order to achieve the phase transition.
In the context of statistical physics , the Bose-Einstein statistics can be used to calculate the critical temperature of an ideal Bose gas , below which the Bose-Einstein condensation begins:
{\ displaystyle T _ {\ mathrm {C}}}
{\ displaystyle T _ {\ mathrm {C}} = {\ frac {h ^ {2}} {2 \ pi \ cdot m \ cdot k _ {\ mathrm {B}}}} \ left ({\ frac {n} {(2S + 1) \ cdot \ zeta (3/2)}} \ right) ^ {2/3}}
{\ displaystyle h}
: Planck's quantum of action
{\ displaystyle m}
: Mass of the particles
{\ displaystyle k _ {\ mathrm {B}}}
: Boltzmann constant
{\ displaystyle n}
: Density of particles
{\ displaystyle S}
: Spin of the particles
{\ displaystyle \ zeta (x)}
: Riemann zeta function ,
{\ displaystyle \ zeta (3/2) \ approx 2 {,} 6124}
“Ideal Bosegas” means that an infinitely extensive, homogeneous, non-interacting gas is considered for the calculation. The inclusion of the atoms in the falling potential and the interactions between them lead to a slight deviation of the actually observed critical temperature from the calculated value, but the formula gives the correct order of magnitude. For typical, experimentally realizable parameters, one finds temperatures of significantly less than 100 nK, so-called ultra - low temperatures .
The usual method of creating Bose-Einstein condensates from atoms consists of two phases:
First, the atoms are caught in a magneto-optical trap and pre-cooled by laser cooling . The laser cooling, however, has a lower limit for temperatures (typically about 100 µK), which is caused by the recoil during the spontaneous emission of photons.
However, the mean speed of the atoms cooled in this way, only a few centimeters per second, is small enough to catch them in a magnetic or optical trap. The temperature of the atomic cloud is further reduced through evaporative cooling , i.e. continuous removal of the most energetic atoms. In this process, mostly over 99.9% of the atoms are deliberately removed. In this way, the remaining atoms achieve the necessary phase space density to complete the phase transition into a Bose-Einstein condensate.
In this way, it was possible until 2004 to achieve Bose-Einstein condensation for many different isotopes at ultra-low temperatures of 100 nK and below ( 7 Li , 23 Na , 41 K , 52 Cr , 85 Rb , 87 Rb, 133 Cs and 174 Yb ). In the end, they were also successful with hydrogen, albeit with slightly different methods.
The fact that the above-mentioned gases show bosonic behavior and not - as solid-state physicists or chemists would expect from alkali atoms - fermionic behavior (for which the Pauli principle would apply) is based on a subtle interplay of electron and nuclear spin at ultra-low temperatures: with correspondingly low excitation energies the half- integer total spin of the electron shell of the atoms and the half- integer nuclear spin are coupled by the weak hyperfine interaction to form an integer total spin of the system. In contrast, the behavior at room temperature (the “chemistry” of the systems) is determined solely by the spin of the electron shell, because here the thermal energies are much greater than the hyperfine field energies.
In 2006 Demokritov and co-workers achieved the Bose-Einstein condensation of magnons (quantized spin waves) at room temperature, but by using optical pumping processes .
In 2009 the Physikalisch-Technische Bundesanstalt succeeded for the first time in producing a Bose-Einstein condensate from calcium atoms. Such alkaline earth metals have - in contrast to the previously used alkali metals - an optical transition that is one million times narrower and are therefore suitable for new types of precision measurements, e.g. B. of gravitational fields , usable.
In November 2010, a research group at the University of Bonn reported on the generation of a Bose-Einstein condensate from photons. The photons were trapped in an optical resonator between two curved mirrors. Since photons cannot be cooled down, dye molecules were placed in the resonator to establish a thermal equilibrium . The condensation that occurred after optical pumping could be detected in the form of a coherent yellow light beam. According to the research group around Martin Weitz, the photonic Bose-Einstein condensate can be used to produce short-wave lasers in the UV or X-ray range .
The first Bose-Einstein condensate in space was created in 2017. For this purpose, the MAIUS rocket was launched with a VSB-30 engine on the European Space and Sounding Rocket Range and brought to a weightless parabolic flight at an altitude of more than 240 km. There have been a previously created in ultra-high vacuum -Kammer rubidium -atoms by diode laser in a magneto-optical trap by evaporative cooling almost up to the absolute zero accommodated. The Bose-Einstein condensate was then generated using an atom chip . It was released from the center of the trap in weightlessness before a harmonic potential was briefly applied by means of a magnetic field and the states were measured using a Mach-Zehnder interferometer . The mission was a cooperation project in which the following institutions were involved under the leadership of the Gottfried Wilhelm Leibniz University of Hanover : Humboldt University of Berlin , Ferdinand Braun Institute, Leibniz Institute for High Frequency Technology , ZARM , Johannes Gutenberg University Mainz , University of Hamburg , University of Ulm , Darmstadt University of technology , simulation and software technology Braunschweig and the Mobile rocket base .
On May 21, 2018, the Cold Atom Laboratory (CAL) experiment was flown to the ISS space station on a Cygnus ferry .
Density distribution of a Bose-Einstein condensate
The proof that a Bose-Einstein condensate was actually generated is usually provided with the help of absorption images after a flight time for atomic gases .
To do this, the trap in which the gas was trapped is switched off suddenly. The gas cloud then expands and is irradiated with resonant laser light after a flight time . The photons of the beam are scattered by the atoms of the gas cloud , so the beam is effectively weakened. The resulting (half) shadow can be recorded with a sensitive CCD camera, and the density distribution of the gas cloud can be reconstructed from its image.
This is anisotropic for Bose-Einstein condensates , while a classic gas always expands isotropically in thermal equilibrium . In many cases the density distribution is parabolic , which can be understood as a consequence of the interaction between the atoms and which distinguishes the Bose-Einstein condensate from an ideal Bosegas .
In the case of the fermion condensate , the effect is also based on bosons. Due to the Pauli principle , it is not possible for fermions to be in the same state. However, this does not apply to fermions which combine in pairs to form bosons, which can then form a condensate as bosons.
Satyendranath Bose : Planck's law and light quantum hypothesis . In: Zeitschrift für Physik No. 26, p. 178, Springer, Berlin / Heidelberg 1924 (English translation published in American Journal of Physics , Vol. 44, No. 11, November 1976).
Albert Einstein: Quantum Theory of the Monatomic Ideal Gas - Second Treatise . In: Meeting reports of the Prussian Academy of Sciences . Berlin, 1925, pp. 3-10.
Kai Bongs, Jakob Reichel, Klaus Sengstock: Bose Einstein Condensation: The Ideal Quantum Laboratory . In: Physics in Our Time . Volume 34, number 4, Wiley-VCH, Weinheim / Berlin 2003, ISSN 0031-9252 , pp. 168–176.
Jan Klaers, Julian Schmitt, Frank Vewinger, Martin Weitz: Bose-Einstein condensate from light . In: Phys. Our time . tape 42 , no. 2 , 2011, p. 58–59 ( uni-bonn.de [PDF; 196 kB ]).
Commons : Bose-Einstein Condensate - collection of images, videos and audio files
Max Planck Institute for Quantum Optics (generally understandable description, history)
What does vacuum mean? , Evaporative cooling , laser cooling , bosons and fermions , what's colder than cold? , At the coldest point in the universe , the wave character of atoms , a cage for atoms , what is a super atom ? , Indistinguishability of similar particles , What does the condensate look like? ' What use is that for? , Laboratory photos and literature
Bose-Einstein condensation in a gas. iap.uni-bonn, archived from the original on July 13, 2013 ; accessed on June 20, 2016 (illustrative explanations).
How cold is cold ( Memento of July 13, 2013 in the Internet Archive ), Bose-Einstein Condensation - What is it? ( Memento from July 13, 2013 in the Internet Archive ), Magnetic trap ( Memento from February 2, 2013 in the Internet Archive ), Evaporative cooling ( Memento from February 2, 2013 in the Internet Archive ), What does a Bose-Einstein condensate look like? ( Memento of December 3, 2012 in the Internet Archive ),
Online script with detailed theoretical derivation
Audio recording of a general lecture given by the German Nobel Prize laureate Wolfgang Ketterle on Bose-Einstein condensates (1998) (English)
Video of Wolfgang Ketterle's reading "Bose-Einstein Condensates: The Coldest Matter in the Universe" - Massachusetts Institute of Technology , October 11, 2001 (English)
Bose-Einstein Condensation of Photons. Institute of Applied Physics at the University of Bonn, accessed on June 20, 2016 .
↑ Albert Einstein: Quantum Theory of the Monatomic Ideal Gas (Handwritten manuscript, discovered in August 2005 at the Lorentz Institute for Theoretical Physics of the Dutch University of Leiden ) 1924. Retrieved on March 21, 2010.
↑ Albert Einstein: Quantum Theory of the Monatomic Ideal Gas - Second Treatise . In: Meeting reports of the Prussian Academy of Sciences . 1925, pp. 3-10.
↑ First Bose-Einstein condensate with strontium atoms. In: iqoqi.at. Austrian Academy of Sciences , November 10, 2009, accessed on September 10, 2016.
↑ Michael Breu: Frozen. 100 atoms at the lowest temperatures: quantum opticians produce one-dimensional Bose-Einstein condensate. In: ethz.ch. ETH Zurich , February 26, 2004, accessed on June 6, 2010.
↑ Demokritov SO, Demidov VE, Dzyapko O, et al. : Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping . In: Nature . 443, No. 7110, September 2006, pp. 430-3. doi : 10.1038 / nature05117 . PMID 17006509 .
↑ Patryk Nowik-Boltyk: Magnon Bose Einstein condensation simply depicted. In: uni-muenster.de. Westfälische Wilhelms-Universität , June 6, 2012, accessed on September 10, 2016.
↑ S. Kraft et al. : Bose-Einstein Condensation of Alkaline Earth Atoms: 40 Approx . In: Phys. Rev. Lett. . 103, No. 13, August, pp. 130401-130404. doi : 10.1103 / PhysRevLett.103.130401 .
↑ Chilled light enters a new phase. In: nature.com. Nature News, November 24, 2010, accessed November 25, 2010 .
↑ Bonn physicists create a new light source. In: handelsblatt.com. Handelsblatt , November 25, 2010, accessed on November 25, 2010 .
↑ MAIUS - Atom-optical experiments on sounding rockets ( Memento of the original from August 1, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.iqo.uni-hannover.de
↑ V. Schkolnik et al .: A compact and robust diode laser system for atom interferometry on a sounding rocket , 2016, arXiv 1606.0027 ( online )
↑ A laboratory for "coldest point in space" orf.at, May 18, 2018, accessed May 18, 2018.
↑ Launch of the space freighter “Cygnus” postponed to the ISS orf.at, May 19, 2018, accessed May 19, 2018.
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EUDML | A Hölder-type inequality for positive functionals on -algebras. EuDML | A Hölder-type inequality for positive functionals on -algebras.
A Hölder-type inequality for positive functionals on
\text{Φ}
Boulabiar, Karim. "A Hölder-type inequality for positive functionals on -algebras.." JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] 3.5 (2002): Paper No. 74, 7 p., electronic only-Paper No. 74, 7 p., electronic only. <http://eudml.org/doc/122917>.
@article{Boulabiar2002,
author = {Boulabiar, Karim},
keywords = {Hölder inequality; positive linear functional; -algebra; uniformly complete -algebra; constructive proof; -algebra; uniformly complete -algebra},
title = {A Hölder-type inequality for positive functionals on -algebras.},
AU - Boulabiar, Karim
TI - A Hölder-type inequality for positive functionals on -algebras.
KW - Hölder inequality; positive linear functional; -algebra; uniformly complete -algebra; constructive proof; -algebra; uniformly complete -algebra
Hölder inequality, positive linear functional,
f
-algebra, uniformly complete
\text{Φ}
-algebra, constructive proof,
f
{\Phi }
Ordered rings, algebras, modules
Articles by Boulabiar
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Convolutionally encode binary data and modulate using PSK method - Simulink - MathWorks France
Specifying the Encoder
Coding Gains
Convolutionally encode binary data and modulate using PSK method
The M-PSK TCM Encoder block implements trellis-coded modulation (TCM) by convolutionally encoding the binary input signal and mapping the result to a PSK signal constellation.
The M-ary number parameter is the number of points in the signal constellation, which also equals the number of possible output symbols from the convolutional encoder. (That is, log2(M-ary number) is equal to n for a rate k/n convolutional code.)
If the convolutional encoder described by the trellis structure represents a rate k/n code, then the block input signal must be a binary column vector with a length of L*k for some positive integer L.
This block accepts a binary-valued input signal. The output signal is a complex column vector of length L.
To define the convolutional encoder, use the Trellis structure parameter. This parameter is a MATLAB® structure whose format is described in Trellis Description of a Convolutional Code. You can use this parameter field in two ways:
If you want to specify the encoder using its constraint length, generator polynomials, and possibly feedback connection polynomials, then use a poly2trellis command within the Trellis structure field. For example, to use an encoder with a constraint length of 7, code generator polynomials of 171 and 133 (in octal numbers), and a feedback connection of 171 (in octal), set the Trellis structure parameter to
poly2trellis(7,[171 133],171)
If you have a variable in the MATLAB workspace that contains the trellis structure, then enter its name as the Trellis structure parameter. This way is faster because it causes Simulink® software to spend less time updating the diagram at the beginning of each simulation, compared to the usage in the previous bulleted item.
The encoder registers begin in the all-zeros state. You can configure the encoder so that it resets its registers to the all-zeros state during the course of the simulation. To do this, set the Operation mode to Reset on nonzero input via port. The block then opens a second input port, labeled Rst. The signal at the Rst port is a scalar signal. When it is nonzero, the encoder resets before processing the data at the first input port.
The trellis-coded modulation technique partitions the constellation into subsets called cosets, so as to maximize the minimum distance between pairs of points in each coset. This block internally forms a valid partition based on the value you choose for the M-ary number parameter.
The figure below shows the labeled set-partitioned signal constellation that the block uses when M-ary number is 8. For constellations of other sizes, see [1].
Coding gains of 3 to 6 decibels, relative to the uncoded case can be achieved in the presence of AWGN with multiphase trellis codes [3].
MATLAB structure that contains the trellis description of the convolutional encoder.
In Continuous mode (default setting), the block retains the encoder states at the end of each frame, for use with the next frame.
In Truncated (reset every frame) mode, the block treats each frame independently. I.e., the encoder states are reset to all-zeros state at the start of each frame.
In Terminate trellis by appending bits mode, the block treats each frame independently. For each input frame, extra bits are used to set the encoder states to all-zeros state at the end of the frame. The output length is given by
y=n\cdot \left(x+s\right)/k
, where x is the number of input bits, and
s=\text{constraint length}-1
(or, in the case of multiple constraint lengths, s =sum(ConstraintLength(i)-1)). The block supports this mode for column vector input signals.
In Reset on nonzero input via port mode, the block has an additional input port, labeled Rst. When the Rst input is nonzero, the encoder resets to the all-zeros state.
The output type of the block can be specified as a single or double. By default, the block sets this to double.
[2] Proakis, John G., Digital Communications, Fourth edition, New York, McGraw-Hill, 2001
[3] Ungerboeck, G., “Channel Coding with Multilevel/Phase Signals”, IEEE Trans. on Information Theory, Vol IT28, Jan. 1982, pp. 55–67.
General TCM Encoder | M-PSK TCM Decoder
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f(x,y)=x\sin(x)+\cos(x)-y\sin(x)+\frac{y^2}{2} My task is to find the critical points of t
f\left(x,y\right)=x\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)-y\mathrm{sin}\left(x\right)+\frac{{y}^{2}}{2}
My task is to find the critical points of this multivariable function (Determine the set of critical points of the function). Now I found the partial derivative to be
{f}_{x}^{\prime }=\left(x-y\right)\mathrm{cos}\left(x\right)
{f}_{y}^{\prime }=y-\mathrm{sin}\left(x\right)
Wolfram Alpha says the critical point(s) is/are at
x=y
? How do I get the above in that form? The best I could do was get it in the form
x-y\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)=y
Brynn Ortiz
Only one critical point is located on the line
y=x
. There are an infinite number of critical points on the lines
y=±1
Rosa Nicholson
How did you compute that?
x=r\mathrm{cos}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}
\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}
P\left(x\right)=-12{x}^{2}+2136x-41000
{f}_{xx}
{f}_{yy}
f\left(x,y\right)=\mathrm{ln}\left({x}^{2}y\right)+{y}^{3}{x}^{2}
A polynomial P is given.
P\left(x\right)={x}^{3}+64
a)Find all zeros of P , real and complex.(Enter your answers as a comma-separated list. Enter your answers as a comma-separated list.)
x=?
b) Factor P completely.
P\left(z\right)=?
Solve for x in the equation a-7x+4=0.
Solve using separation of variables:
{y}^{\prime }-xy+y=2y
Convert the equation into a first-order linear differential equation system with an appropriate transformation of variables.
y"+4{y}^{\prime }+3y={x}^{2}
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Patching and multiplicity $2^k$ for Shimura curves
We use the Taylor–Wiles–Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod
\ell
cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local deformation rings done by Shotton, which we use to compute the Weil class group of various deformation rings. Exploiting the natural self-duality of the cohomology groups, we use these class group computations to precisely determine the structure of a patched module in many new cases in which the patched module is not free (and so multiplicity one fails).
Our main result is a “multiplicity
{2}^{k}
” theorem in the minimal level case (which we prove under some mild technical hypotheses), where
k
is a number that depends only on local Galois theoretic information at the primes dividing the discriminant of the Shimura curve. Our result generalizes Ribet’s classical multiplicity 2 result and the results of Cheng, and provides progress towards the Buzzard–Diamond–Jarvis local-global compatibility conjecture. We also prove a statement about the endomorphism rings of certain modules over the Hecke algebra, which may have applications to the integral Eichler basis problem.
We have not been able to recognize your IP address 3.235.176.80 as that of a subscriber to this journal.
Shimura curves, Taylor–Wiles–Kisin patching, Galois deformation theory, multiplicity
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Kinematic-Motion and time | Brilliant Math & Science Wiki
Kinematic-Motion and time
tanveen dhingra contributed
First of all we will study what exactly is rest and motion.
An object is said to be in motion when it changes it's position with respect to an observer along the passage of time. An object is said to be at rest when it does not change it position wih respect to am observer along the passage of time.
If a person is standing on the Earth then he would be in state of ________ with respect to a person standing on the Moon.
Solution: Motion. This is because the Moon is revolving around the Earth and so the person will be seen in motion.
In the above example the earth is rotating around it's axis and the moon is revolving around the earth. Then why does the moon appear in motion to us and not in rest?
Solution: The speed of the rotation of the earth and that of the revolution of the moon is different. Hence the resultant speed is not zero and the moon appears to be in motion to us.
Now we will come to what is distance and displacement. Distance-It is the total path length travelled between the initial and final point Displacement-It is the shortest distance between the initial and final point. Both have same S.I. Unit that is m. Distance is a scalar quantity while displacement is a vector quantity.
A man started from point A moved 3m left and then 4m south reaching point B. Find the total distance and the displacement he travelled between point A and B.
Solution: As the distance is the total path length travelled it will be (3+4)m = 7m.Therefore the distance is 7m. Now to find the displacement we will refer to the given figure. As we know that the displacement is the shortest length between 2 points it will be the diagonal distance between A an C. It is shown by the dotted lines. Now to find this distance we will use the Pythagoras Theorem. It states that
{ (Hypotenuse) }^{ 2 }\quad =\quad { Perpendicular }^{ 2 }+\quad { Base }^{ 2 }
.Therefore, we get that
{ AC }^{ 2 }\quad =\quad { 3 }^{ 2 }\quad +\quad { 4 }^{ 2 }\\ therefore\quad AC\quad =\quad \sqrt { 25 } \quad =\quad 5
Hence the displacement is 5m
Further we will know about speed and velocity. Speed is the ratio of the distance and time.
Speed\quad =\quad \frac { Distance }{ Time }
Velocity is the ratio of displacement.
Velocity\quad =\quad \frac { Displacement }{ Time }
Both speed and velocity have S.I. Unit m/s. Speed is a scalar quantity and vector is a vector quantity. Now what if the speed keeps changing throughout a journey. We will simply find the average speed. The average speed is the ratio of the total distance travelled and the total time taken.
Average\quad Speed\quad =\quad \frac { Total\quad distance\quad travelled }{ Total\quad time\quad taken }
A person travels from A to B with a speed of 5m/s. He returns from B to A with a speed of 6 m/s. Find the average speed and average velocity.
Solution: Let the distance from A to B be x meters. So time taken from A to B is
\frac { distance }{ speed }
\frac { x }{ 5 }
. Now time taken from B to A is
\frac { x }{ 6}
So, Average speed is
\frac { total\quad distance\quad }{ total\quad time } \quad
\frac { x+x\quad }{ \frac { x }{ 5 } +\frac { x }{ 6 } } \quad \\ =\quad \frac { 2x }{ \frac { 6x+5x }{ 30 } } \\ =\quad \frac { 2x\quad \times \quad 30 }{ 11x } \\ =\frac { 60 }{ 11 } \\ =5.45
Therefore the average speed is 5.45m/s. Now, we know that Average Velocity =
\frac { Total\quad displacement }{ Total\quad time\quad } \quad
=0, because total displacement is o.
Now acceleration will be studied. Acceleration is a vector quantity that is defined as a rate at which an object changes its velocity. An object is accelerating if it is changing its velocity. So,
Acceleration\quad =\quad \frac { change\quad in\quad velocity }{ time\quad taken } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac { v-u }{ t } \\ where,\quad t\quad =\quad time\quad taken\\ v\quad =\quad final\quad velocity\quad and\\ u\quad =\quad initial\quad velocity
Cite as: Kinematic-Motion and time. Brilliant.org. Retrieved from https://brilliant.org/wiki/kinematic-motion-and-time/
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How many ways can you buy 2 DVDs from a
How many ways can you buy 2 DVDs from a display of 15?
verrainellewtzri
We assume that someone will not buy two of the same DVD.
For the first DVD, there is a choice of 15 , but once the first is bought, there is a choice of 14 for the second one.
15×14
=210
However, this does not take order into consideration .... the 2 DVD's might be the same, just bought in a different order, so we need to divide this number by 2 to avoid the duplicates.
\frac{210}{2}=105
Assumption: choosing
a+b
is classified the same as choosing
b+a
This is the condition of 'combinations'
⇒{.}^{n}{C}_{r}\to \frac{n!}{\left(n-r\right)!r!}
⇒{.}^{n}{C}_{r}\to {.}^{15}{C}_{2}\to \frac{15!}{\left(15-2\right)!2!}
=\frac{15×14×\overline{)13!}}{\overline{)13!}.2!}=\frac{15×14}{2}
=105
Giving a test to a group of students, the grades and gender are summarized below
If one student is chosen at random,
Find the probability that the student was female OR got an "C".
Interest centers around the life of a concrete mixer. Let A be the event that the concrete mixer fails a test, but B be the event that the concrete mixer displays strain wears but does not actually fail. Event A occurs with probability 0.20, and event B occurs with probability 0.35.
a)What is the probability that the concrete mixer does not fail the test?
b)What is the probability that the concrete mixer works perfectly well (i.e., neither displays strain nor fails the test)?
c)What is the probability that the concrete mixer either fails or shows strain in the test?
Two fair dice are rolled. Find the joint probability mass function of X and Y when (a) X is the largest value obtained on any die and Y is the sum of the values; (b) X is the value on the first die and Y is the larger of the two values; (c) X is the smallest and Y is the largest value obtained on the dice.
A box contains 50 balls, exactly ten of them are red and the others are black.
If five balls are chosen at random from the box knowing that the first two balls are red, what is the probability that the third one is black?
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Using MicroCT Imaging Technique to Quantify Heat Generation Distribution Induced by Magnetic Nanoparticles for Cancer Treatments | J. Heat Transfer | ASME Digital Collection
Attaluri, A., Ma, R., and Zhu, L. (September 27, 2010). "Using MicroCT Imaging Technique to Quantify Heat Generation Distribution Induced by Magnetic Nanoparticles for Cancer Treatments." ASME. J. Heat Transfer. January 2011; 133(1): 011003. https://doi.org/10.1115/1.4002225
Magnetic nanoparticles have been used in clinical and animal studies to generate localized heating for tumor treatments when the particles are subject to an external alternating magnetic field. Currently, since most tissue is opaque, the detailed information of the nanoparticle spreading in the tissue after injections cannot be visualized directly and is often quantified by indirect methods, such as temperature measurements, to inversely determine the particle distribution. In this study, we use a high resolution microcomputed tomography (microCT) imaging system to investigate nanoparticle concentration distribution in a tissue-equivalent agarose gel. The local density variations induced by the nanoparticles in the vicinity of the injection site can be detected and analyzed by the microCT system. Heating experiments are performed to measure the initial temperature rise rate to determine the nanoparticle-induced volumetric heat generation rates (or specific absorption rate (SAR
W/m3
)) at various gel locations. A linear relationship between the measured SARs and their corresponding microCT pixel index numbers is established. The results suggest that the microCT pixel index number can be used to represent the nanoparticle concentration in the media since the SAR is proportional to the local nanoparticle concentration. Experiments are also performed to study how the injection amount, gel concentration, and nanoparticle concentration in the nanofluid affect the nanoparticle spreading in the gel. The nanoparticle transport pattern in gels suggests that convection and diffusion are important mechanisms in particle transport in the gel. Although the particle spreading patterns in the gel may not be directly applied to real tissue, we believe that the current study lays the foundation to use microCT imaging systems to quantitatively study nanoparticle distribution in opaque tumor.
biological fluid dynamics, biomedical imaging, cancer, convection, flow visualisation, gels, heating, magnetic particles, nanofluidics, nanoparticles, patient treatment, tumours, magnetic nanoparticles, hyperthermia, cancer, heating, temperature, microCT imaging
Imaging, Nanoparticles, Cancer, Heat, Heating, Temperature, Particulate matter, Tumors, Convection, Ferrofluids, Biological tissues, Agar
Lapatnikov
Maghemite nanoparticles With Very High AC-Losses for Application in RF-Magnetic Hyperthermia
Thermotherapy of Prostate Cancer Using Magnetic Nanoparticles—Feasibility, Imaging, and Three-Dimensional Temperature Distribution
Viroonchatapan
Possibility of Thermosensitive Magnetoliposomes as a New Agent for Electromagnetic Induced Hyperthermia
Z. -S.
Thermoablation of Malignant Kidney Tumors Using Magnetic Nanoparticles: An In-Vivo Feasibility Study in a Rabbit Model
Cardiovasc. Intervent Radiol.
Superparamagnetic Iron Oxide Nanoparticle ‘Theranostics’ for Multimodality Tumor Imaging, Gene Delivery, Targeted Drug and Prodrug Delivery
Intraparenchymal Drug Delivery via Positive-Pressure Infusion: Experimental and Modeling Studies of Poroelasticity in Brain Phantom Gels
The Heating Effect of Magnetic Fluids in an Alternating Magnetic Field
Diffusion From an Injected Volume of a Substance in Brain Tissue With Arbitrary Volume Fraction and Tortuosity
Extracellular Space Revealed by Diffusion Analysis
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On Some Classes of Linear Volterra Integral Equations
Anatoly S. Apartsyn, "On Some Classes of Linear Volterra Integral Equations", Abstract and Applied Analysis, vol. 2014, Article ID 532409, 6 pages, 2014. https://doi.org/10.1155/2014/532409
Anatoly S. Apartsyn1
1Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia
The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.
Volterra integral equations of the first kind with variable upper and lower limits of integration were studied by Volterra himself [1]. The publications on this topic in the first half of the 20th century were reviewed in [2] and later studies were discussed in [3–5].
A noticeable impetus to the development of this area is related to the research [6] which suggested a macroeconomic two-sector integral model. The Glushkov’s models of developing systems were further extended in [7, 8] and used in many applications (see [9] and references therein). In particular, a one-sector version of the Glushkov’s model applied to the power engineering problems was considered in [10–12]. In the recent years the researchers have got attracted by the equation (see [13] and references therein) that in a general case has the following form: where kernels and right-hand side are given, and is an unknown desired solution.
At the problems of the existence and uniqueness of solution to (1) in the space , as well as the numerical methods, are studied in detail in [5]. In this paper we will be interested in the same problems for (1) at . Further, for simplicity, we will consider only the case , since many results are easily generalized for the case .
2. Sufficient Conditions for the Correctness of (1) at in Pair ,
For convenience, present (1) with in operator form (in (3) is assumed with no loss of generality).
Let kernels and be continuous in arguments and continuously differentiable with respect to in regions and , respectively, so that , , , . We will assume that
In particular, (4) holds true for , . is further taken to mean the space of continuously differentiable functions on with the norm and additional condition . If then, as established in [5, page 106], the following estimate is true: where
Estimating (6) makes it possible to obtain the sufficient condition for the existence, uniqueness, and stability of the solution to (3) in pair .
Theorem 1. Let the following inequality hold true: where Then (3) is correct in the sense of Hadamard in pair .
Proof. By virtue of a well-known theorem of functional analysis (see, e.g., [14, page 212]), if then the operator has a bounded inverse, and, consequently, (3) is correct in the sense of Hadamard in pair . We show that under (8)-(9) inequality (10) holds true.
As then and (10) follows from (6) and (12).
Condition (8) was obtained in the assumption that kernel is defined on . If it is possible to expand the domain of definition to , so that , then the sufficient condition for the correctness of (3) is modified in the following way. Represent the first term in (3) in the form Then (3) can be represented as
Since (see [5, page 12]) where then sufficient conditions for the correctness of (14) give the following theorem.
Theorem 2. Let inequality where hold true. Then (14) is correct in the sense of Hadamard in pair .
Proof. With obvious changes, repeat the proof of Theorem 1.
Let us illustrate the obtained results with the following example.
Consider the equation Here by (5)–(7) , , , , , , , and ; therefore based on (8) inequality and based on (17) inequality give the following estimates , which guarantee the existence, uniqueness, and stability of solution to (20) in the space : It is useful to compare (23) with the estimate obtained by shifting from (20) to the equivalent functional equation. Differentiation of (20) gives whence and condition provides convergence of series (25) to continuous function on .
If in (20) then condition (26) is violated. Then it is easy to see that the homogeneous equation has a nontrivial solution , and if, for example, , the solution to the nonhomogeneous equation is a one-parameter family: Let now Then, according to (24), whence so that for the right-hand side of (20) , , from (33) we obtain
In conclusion of this section it should be noted that inequalities (8) and (17) can be interpreted as constraints on the value ,which guarantee at given , and the correct solvability of (3) in . Since all parameters in the left-hand side of (8) and (17) are nondecreasing functions of and the right-hand side of (8) and (17) at ( ), on the contrary, monotonously decreases, then the real positive root of corresponding nonlinear equation that gives a guaranteed lower-bound estimate of exists and is unique if is sufficiently small. In some special cases this root can be found analytically in terms of the Lambert function [15, 16].
In [17–22] the authors studied the characteristic of continuous solution locality and the role of the Lambert function as applied to the polynomial (multilinear) Volterra equations of the first kind. The calculations of the test examples show that the locality feature of the solution to the linear equation (3) is not the result of the inaccuracy of estimates (8) and (17) and reflects the specifics of the considered class of problems. In this paper we do not dwell on the problem of numerically solving (3). It is of independent interest and deserves special consideration.
3. The Volterra Integral Equations of the First Kind with Discontinuous Kernels
Equation (2) can be written in the form of Volterra integral equation of the first kind: with discontinuous kernel
To illustrate the fundamental difference between (35), (36), and classical Volterra equation of the first kind with smooth kernel, we confine ourselves to (20) that has the form of (35) at where , and . In particular, at , ,
For this case the solution to (35) with given in [23] is
For kernel (38) If is continuous in arguments and continuously differentiable with respect to in , then condition (40) means that (35) is Volterra integral equation of the third kind.
The theory (whose foundation was laid by Volterra (see [24, pages 104–106])) of such equations is developed in the research done by Magnitsky [25–28].
In particular, the author of [25–28] studies the structure of one- or many-parameter family of solutions to (35).
If is discontinuous, then the solution to (35) may be nonunique, even if .
For example, if and , the solution to equation is a one-parameter family: but, by (37) , , .
Now we show that there can be a nonunique solution to (35) and (36) even in the case . Let so that condition is true.
We prove that solutions to (35), (37) and (35), (43) coincide. It suffices to show that the equivalent functional equations for (35), (37) and (35), (43) coincide. Recall that for (35), (37) the equivalent functional equation is (24).
Theorem 3. The equivalent functional equations for (35), (37) and (35), (43) coincide.
Proof. Let us represent (43) by where – is a Heaviside function:
Substitution of (44) in (35) gives
Transform the second integral. Let . Then By virtue of (47), differentiation of (46) results in But By virtue of (49) we have from (48), whence finally and (51) coincides with (24).
The solution to (35), (43) in the class of piecewise continuous functions with a jump on line is interesting from the application perspective.
It is easy to see that this solution is
At last consider the concept of -convolution. Volterra integral equations of convolution type are important for application.
Examples (38) and (44) show the usefulness of the -convolution concept:
Give some inversion formulas of the integral equation (1)If , and , then (2)If , , and , then (3)If , , and , then At , (55) is Volterra integral equation of the third kind.(4)If , , and , then (5)If , , and , , then
As is mentioned in the introduction, the main results of this study can be easily applied to the case in (1). The equations of type (1) not only are of theoretical interest, but also play an important role in the mathematical modeling of developing dynamic systems. Moreover, by , we can mean some criterion that characterizes the level of development of the system as a whole, and the th term in (1) represents a contribution of the system components of the th age group, whose operation is reflected by the efficiency coefficient . As a rule, . Such an approach is implemented, for instance, in [29, 30], in the problem of the analysis of strategies for the long-term expansion of the Russian electric power system, with the consideration of aging of the power plants equipment.
The author wishes to thank the reviewers for their helpful notes. The study is supported by the Russian Foundation for Basic Research, Grant no. 12-01-00722a.
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W
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A=P\left(1–i{\right)}^{n}\phantom{\rule{0ex}{0ex}}200000–490740=\left(1–0.15{\right)}^{n}\phantom{\rule{0ex}{0ex}}0.4075=\left(0.85{\right)}^{n}\phantom{\rule{0ex}{0ex}}0.{85}^{5.5}=0.{85}^{n}\phantom{\rule{0ex}{0ex}}n=5.5 or 5 yr 6 months\left(approx\right)
A=P\left(1–i{\right)}^{n}\phantom{\rule{0ex}{0ex}}200000–490740=\left(1–0.15{\right)}^{n}\phantom{\rule{0ex}{0ex}}0.4075=\left(0.85{\right)}^{n}\phantom{\rule{0ex}{0ex}}0.{85}^{5.5}=0.{85}^{n}\phantom{\rule{0ex}{0ex}}n=5.5 or 5 yr 6 months\left(approx\right)
let the annual investment be Rs a\phantom{\rule{0ex}{0ex}}A=a/i\left[\left(1+i{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}500000=a/0.04\left[\left(170.04{\right)}^{25}–1\right]\phantom{\rule{0ex}{0ex}}20000=a x 1.6658\phantom{\rule{0ex}{0ex}}a=Rs 12006\left(approx\right)
let the annual investment be Rs a\phantom{\rule{0ex}{0ex}}A=a/i\left[\left(1+i{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}500000=a/0.04\left[\left(170.04{\right)}^{25}–1\right]\phantom{\rule{0ex}{0ex}}20000=a x 1.6658\phantom{\rule{0ex}{0ex}}a=Rs 12006\left(approx\right)
P\left[\left(1+i{\right)}^{n}–1\right]–Pxixt=11\phantom{\rule{0ex}{0ex}}P\left[\left(1+0.1{\right)}^{2}–1–0.1x2\right]=11\phantom{\rule{0ex}{0ex}}P=Rs 1100
P\left[\left(1+i{\right)}^{n}–1\right]–Pxixt=11\phantom{\rule{0ex}{0ex}}P\left[\left(1+0.1{\right)}^{2}–1–0.1x2\right]=11\phantom{\rule{0ex}{0ex}}P=Rs 1100
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}2P=P\left(1+0.05{\right)}^{n}\phantom{\rule{0ex}{0ex}}2=1.{05}^{n}\phantom{\rule{0ex}{0ex}}1.{05}^{14.3}=1.{05}^{n}\phantom{\rule{0ex}{0ex}}n=14.3 yr
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}2P=P\left(1+0.05{\right)}^{n}\phantom{\rule{0ex}{0ex}}2=1.{05}^{n}\phantom{\rule{0ex}{0ex}}1.{05}^{14.3}=1.{05}^{n}\phantom{\rule{0ex}{0ex}}n=14.3 yr
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}2P=P\left(1+i{\right)}^{15}\phantom{\rule{0ex}{0ex}}2=1+{i}^{15}\phantom{\rule{0ex}{0ex}}\left(1.0473{\right)}^{15}=\left(1+i{\right)}^{15}\phantom{\rule{0ex}{0ex}}1+i=1.0473\phantom{\rule{0ex}{0ex}}i=0.473\phantom{\rule{0ex}{0ex}}now,8P=P\left(1+i{\right)}^{n}= 8=\left(1.0473{\right)}^{n} = n=45
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}2P=P\left(1+i{\right)}^{15}\phantom{\rule{0ex}{0ex}}2=1+{i}^{15}\phantom{\rule{0ex}{0ex}}\left(1.0473{\right)}^{15}=\left(1+i{\right)}^{15}\phantom{\rule{0ex}{0ex}}1+i=1.0473\phantom{\rule{0ex}{0ex}}i=0.473\phantom{\rule{0ex}{0ex}}now,8P=P\left(1+i{\right)}^{n}= 8=\left(1.0473{\right)}^{n} = n=45
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=1000\left(1+0.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=\left(1.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1.{10}^{3}=1.{10}^{n}\phantom{\rule{0ex}{0ex}}n=3\phantom{\rule{0ex}{0ex}}hence,Rs1000 amounts to Rs1331 at 10%p.a. C.I in 3 yr
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=1000\left(1+0.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=\left(1.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1.{10}^{3}=1.{10}^{n}\phantom{\rule{0ex}{0ex}}n=3\phantom{\rule{0ex}{0ex}}hence,Rs1000 amounts to Rs1331 at 10%p.a. C.I in 3 yr
P.V=\frac{A}{i}\left[1–\frac{1}{\left(1+i{\right)}^{n}}\right]\phantom{\rule{0ex}{0ex}}20000=\frac{2000}{0.05}\left[1–\frac{1}{1.{05}^{n}}\right]\phantom{\rule{0ex}{0ex}}0.5=1–\frac{1}{1.{05}^{n}}\phantom{\rule{0ex}{0ex}}1.{05}^{n}=1/0.5\phantom{\rule{0ex}{0ex}}n=15 yr\left(approx\right)
P.V=\frac{A}{i}\left[1–\frac{1}{\left(1+i{\right)}^{n}}\right]\phantom{\rule{0ex}{0ex}}20000=\frac{2000}{0.05}\left[1–\frac{1}{1.{05}^{n}}\right]\phantom{\rule{0ex}{0ex}}0.5=1–\frac{1}{1.{05}^{n}}\phantom{\rule{0ex}{0ex}}1.{05}^{n}=1/0.5\phantom{\rule{0ex}{0ex}}n=15 yr\left(approx\right)
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=1000\left(1+0.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=\left(1.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1.{10}^{3}=1.{10}^{n}\phantom{\rule{0ex}{0ex}}n=3\phantom{\rule{0ex}{0ex}}hence,Rs1000 amounts to Rs1331 at 10%p.a. C.I in 3 yr
A=P\left(1+i{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=1000\left(1+0.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1331=\left(1.10{\right)}^{n}\phantom{\rule{0ex}{0ex}}1.{10}^{3}=1.{10}^{n}\phantom{\rule{0ex}{0ex}}n=3\phantom{\rule{0ex}{0ex}}hence,Rs1000 amounts to Rs1331 at 10%p.a. C.I in 3 yr
C.I = P\left[\left(1+i{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}25=P\left[\left(1+0.05{\right)}^{1}–1\right]\phantom{\rule{0ex}{0ex}}25=P\left[0.05\right]\phantom{\rule{0ex}{0ex}}P=25/0.05 = Rs 500\phantom{\rule{0ex}{0ex}}now, R=5% and T=2 Yr\phantom{\rule{0ex}{0ex}}S.i=P R T/100=500x5/100x2=s 50
C.I = P\left[\left(1+i{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}25=P\left[\left(1+0.05{\right)}^{1}–1\right]\phantom{\rule{0ex}{0ex}}25=P\left[0.05\right]\phantom{\rule{0ex}{0ex}}P=25/0.05 = Rs 500\phantom{\rule{0ex}{0ex}}now, R=5% and T=2 Yr\phantom{\rule{0ex}{0ex}}S.i=P R T/100=500x5/100x2=s 50
A = P\left[1+R/100{\right]}^{n}\phantom{\rule{0ex}{0ex}}16 P = P\left[1+R/100{\right]}^{4}\phantom{\rule{0ex}{0ex}}16=\left[1+R/100{\right]}^{4}={2}^{4}\phantom{\rule{0ex}{0ex}}1+R/100=2\phantom{\rule{0ex}{0ex}}R/100=1\phantom{\rule{0ex}{0ex}}R=100%
A = P\left[1+R/100{\right]}^{n}\phantom{\rule{0ex}{0ex}}16 P = P\left[1+R/100{\right]}^{4}\phantom{\rule{0ex}{0ex}}16=\left[1+R/100{\right]}^{4}={2}^{4}\phantom{\rule{0ex}{0ex}}1+R/100=2\phantom{\rule{0ex}{0ex}}R/100=1\phantom{\rule{0ex}{0ex}}R=100%
\left[given \left(1.06{\right)}^{–10}=0.5584\right]
V=A/r \left[1–\left(1+i{\right)}^{–n}\right]\phantom{\rule{0ex}{0ex}}=1000/0.06\left[1–\left(1.06{\right)}^{–10}\right]\phantom{\rule{0ex}{0ex}}=1000/0.06\left[1–0.5584\right]\phantom{\rule{0ex}{0ex}}=Rs7360
V=A/r \left[1–\left(1+i{\right)}^{–n}\right]\phantom{\rule{0ex}{0ex}}=1000/0.06\left[1–\left(1.06{\right)}^{–10}\right]\phantom{\rule{0ex}{0ex}}=1000/0.06\left[1–0.5584\right]\phantom{\rule{0ex}{0ex}}=Rs7360
CI=P\left[\left(1+R/100{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}=15000\left[\left(1+12/100{\right)}^{2}–1\right]\phantom{\rule{0ex}{0ex}}=Rs 3816
CI=P\left[\left(1+R/100{\right)}^{n}–1\right]\phantom{\rule{0ex}{0ex}}=15000\left[\left(1+12/100{\right)}^{2}–1\right]\phantom{\rule{0ex}{0ex}}=Rs 3816
\left[given \left(1.09{\right)}^{8}=1.99256\right]
FV = P\left[\frac{\left(1+i{\right)}^{n}–1}{i}\right] where P=5000, i=9% and n=8\phantom{\rule{0ex}{0ex}}FV = 5000\left[\frac{\left(1+9/100{\right)}^{8}–1}{0.09}\right]=5000\left[\frac{1.99256–1}{0.09}\right]=55142.22
FV = P\left[\frac{\left(1+i{\right)}^{n}–1}{i}\right] where P=5000, i=9% and n=8\phantom{\rule{0ex}{0ex}}FV = 5000\left[\frac{\left(1+9/100{\right)}^{8}–1}{0.09}\right]=5000\left[\frac{1.99256–1}{0.09}\right]=55142.22
nominal rate=6%p.a\phantom{\rule{0ex}{0ex}}effective rate=\left(1+i{\right)}^{n}–1\phantom{\rule{0ex}{0ex}}i=3%\phantom{\rule{0ex}{0ex}}n=2\phantom{\rule{0ex}{0ex}}effective rate=\left(1+3/100{\right)}^{2}–1\phantom{\rule{0ex}{0ex}}=0.0609\phantom{\rule{0ex}{0ex}}=6.09%
nominal rate=6%p.a\phantom{\rule{0ex}{0ex}}effective rate=\left(1+i{\right)}^{n}–1\phantom{\rule{0ex}{0ex}}i=3%\phantom{\rule{0ex}{0ex}}n=2\phantom{\rule{0ex}{0ex}}effective rate=\left(1+3/100{\right)}^{2}–1\phantom{\rule{0ex}{0ex}}=0.0609\phantom{\rule{0ex}{0ex}}=6.09%
\left[given \left(0.9{\right)}^{20}=0.1215\right]
scrap value=P\left(1–i{\right)}^{n}\phantom{\rule{0ex}{0ex}}P=125000,i=10%,n=20\phantom{\rule{0ex}{0ex}}v=125000\left[1–10/100{\right]}^{20}=125000x0.1215=15187
scrap value=P\left(1–i{\right)}^{n}\phantom{\rule{0ex}{0ex}}P=125000,i=10%,n=20\phantom{\rule{0ex}{0ex}}v=125000\left[1–10/100{\right]}^{20}=125000x0.1215=15187
CI=Q\left[\left(1+5/100{\right)}^{2}–1\right]=0.1025Q\phantom{\rule{0ex}{0ex}}P/5=0.1025Q\phantom{\rule{0ex}{0ex}}P/5=1025/10000Q\phantom{\rule{0ex}{0ex}}P/5=41/400Q\phantom{\rule{0ex}{0ex}}P=41/80Q
CI=Q\left[\left(1+5/100{\right)}^{2}–1\right]=0.1025Q\phantom{\rule{0ex}{0ex}}P/5=0.1025Q\phantom{\rule{0ex}{0ex}}P/5=1025/10000Q\phantom{\rule{0ex}{0ex}}P/5=41/400Q\phantom{\rule{0ex}{0ex}}P=41/80Q
CI for 2 Yrs \phantom{\rule{0ex}{0ex}}CI=P\left[\left(1+R/100{\right)}^{t}–1\right]\phantom{\rule{0ex}{0ex}}=X\left[\left(1+12/100{\right)}^{2}–1\right]\phantom{\rule{0ex}{0ex}}CI=159X/625\phantom{\rule{0ex}{0ex}}difference = 159X/625–24X/100=72\phantom{\rule{0ex}{0ex}}on solving we get X= 5000
CI for 2 Yrs \phantom{\rule{0ex}{0ex}}CI=P\left[\left(1+R/100{\right)}^{t}–1\right]\phantom{\rule{0ex}{0ex}}=X\left[\left(1+12/100{\right)}^{2}–1\right]\phantom{\rule{0ex}{0ex}}CI=159X/625\phantom{\rule{0ex}{0ex}}difference = 159X/625–24X/100=72\phantom{\rule{0ex}{0ex}}on solving we get X= 5000
\left[Given \left[\frac{4033}{4000}{\right]}^{12}=1.1036\left(approx\right)\right]
effective rate = \left(1+i{\right)}^{n}–1\phantom{\rule{0ex}{0ex}}=\left(1+33/4000{\right)}^{12}–1\phantom{\rule{0ex}{0ex}}=\left(4033/4000{\right)}^{12}–1\phantom{\rule{0ex}{0ex}}=1.1036–1\phantom{\rule{0ex}{0ex}}=0.1036=10.36%
effective rate = \left(1+i{\right)}^{n}–1\phantom{\rule{0ex}{0ex}}=\left(1+33/4000{\right)}^{12}–1\phantom{\rule{0ex}{0ex}}=\left(4033/4000{\right)}^{12}–1\phantom{\rule{0ex}{0ex}}=1.1036–1\phantom{\rule{0ex}{0ex}}=0.1036=10.36%
ca foundation mock test-ratio and proportion
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Tool bits used in milling machines
2.3 Ball cutter
2.8 Thread mill
3.3 Cutter location (cutter radius compensation)
Features of a milling cutter[edit]
Flutes / teeth: The flutes of the milling bit are the deep helical grooves running up the cutter, while the sharp blade along the edge of the flute is known as the tooth. The tooth cuts the material, and chips of this material are pulled up the flute by the rotation of the cutter. There is almost always one tooth per flute, but some cutters have two teeth per flute.[1] Often, the words flute and tooth are used interchangeably. Milling cutters may have from one to many teeth, with two, three and four being most common. Typically, the more teeth a cutter has, the more rapidly it can remove material. So, a 4-tooth cutter can remove material at twice the rate of a two-tooth cutter.
Roughing or Finishing: Different types of cutter are available for cutting away large amounts of material, leaving a poor surface finish (roughing), or removing a smaller amount of material, but leaving a good surface finish (finishing). A roughing cutter may have serrated teeth for breaking the chips of material into smaller pieces. These teeth leave a rough surface behind. A finishing cutter may have a large number (four or more) teeth for removing material carefully. However, the large number of flutes leaves little room for efficient swarf removal, so they are less appropriate for removing large amounts of material.
Coatings: The right tool coatings can have a great influence on the cutting process by increasing cutting speed and tool life, and improving the surface finish. Polycrystalline diamond (PCD) is an exceptionally hard coating used on cutters that must withstand high abrasive wear. A PCD coated tool may last up to 100 times longer than an uncoated tool. However, the coating cannot be used at temperatures above 600 degrees C, or on ferrous metals. Tools for machining aluminium are sometimes given a coating of TiAlN. Aluminium is a relatively sticky metal, and can weld itself to the teeth of tools, causing them to appear blunt. However, it tends not to stick to TiAlN, allowing the tool to be used for much longer in aluminium.
Shank: The shank is the cylindrical (non-fluted) part of the tool which is used to hold and locate it in the tool holder. A shank may be perfectly round, and held by friction, or it may have a Weldon Flat, where a set screw, also known as a grub screw, makes contact for increased torque without the tool slipping. The diameter may be different from the diameter of the cutting part of the tool, so that it can be held by a standard tool holder.§ The length of the shank might also be available in different sizes, with relatively short shanks (about 1.5x diameter) called "stub", long (5x diameter), extra long (8x diameter) and extra extra long (12x diameter).
End mill[edit]
Roughing end mill[edit]
Roughing end mills quickly remove large amounts of material. This kind of end mill utilizes a wavy tooth form cut on the periphery. These wavy teeth act as many successive cutting edges producing many small chips. This results in a relatively rough surface finish, but the swarf takes the form of short thin sections and is more manageable than a thicker more ribbon-like section, resulting in smaller chips that are easier to clear. During cutting, multiple teeth are in simultaneous contact with the workpiece, reducing chatter and vibration. Rapid stock removal with heavy milling cuts is sometimes called hogging. Roughing end mills are also sometimes known as "rippa" or "ripper" cutters.
Ball cutter[edit]
Slab mill[edit]
High speed steel slab mill
Side-and-face cutter[edit]
Involute gear cutter[edit]
Hob[edit]
These cutters are a type of form tool and are used in hobbing machines to generate gears. A cross-section of the cutter's tooth will generate the required shape on the workpiece, once set to the appropriate conditions (blank size). A hobbing machine is a specialised milling machine.
Thread mill[edit]
Main article: Threading (manufacturing) § Thread milling
A diagram of a solid single-form thread cutting tool
A solid multiple-form thread milling cutter.
Face mill[edit]
Fly cutter[edit]
Woodruff cutter[edit]
Hollow mill[edit]
4 bladed hollow mill
Hollow milling cutters, more often called simply hollow mills, are essentially "inside-out endmills". They are shaped like a piece of pipe (but with thicker walls), with their cutting edges on the inside surface. They were originally used on turret lathes and screw machines as an alternative to turning with a box tool, or on milling machines or drill presses to finish a cylindrical boss (such as a trunnion). Hollow mills can be used on modern CNC lathes and Swiss style machines. An advantage to using an indexable adjustable hollow mill on a Swiss-style machine is replacing multiple tools. By performing multiple operations in a single pass, the machine does not need as can accommodate other tools in the tool zone and improves productivity.
More advanced hollow mills use indexable carbide inserts for cutting, although traditional high speed steel and carbide-tipped blades are still used.
Hollow milling has an advantage over other ways of cutting because it can perform multiple operations. A hollow mill can reduce the diameter of a part and also perform facing, centering, and chamfering in a single pass.
Hollow mills offer an advantage over single point tooling. Multiple blades allow the feed rate to double and can hold a closer concentricity. The number of blades can be as many as 8 or as few as 3. For significant diameter removal (roughing), more blades are necessary.
Trepanning is also possible with a hollow mill. Special form blades can be used on a hollow mill for trepanning diameters, forms, and ring grooves.
Interpolation is also not necessary when using a hollow mill; this can result in a significant reduction of production time.
Both convex and concave spherical radii are possible with a hollow mill. The multiple blades of a hollow mill allow this radius to be produced while holding a tight tolerance.
A common use of a hollow mill is preparing for threading. The hollow mill can create a consistent pre-thread diameter quickly, improving productivity.
An adjustable hollow mill is a valuable tool for even a small machine shop to have because the blades can be changed out for an almost infinite number of possible geometries.
Dovetail cutter[edit]
A dovetail cutter is an end mill whose form leaves behind a dovetail slot, such as often forms the ways of a machine tool.
Shell mill[edit]
Modular principle[edit]
This modular style of construction is appropriate for large milling cutters for about the same reason that large diesel engines use separate pieces for each cylinder and head whereas a smaller engine would use one integrated casting. Two reasons are that (1) for the maker it is more practical (and thus less expensive) to make the individual pieces as separate endeavors than to machine all their features in relation to each other while the whole unit is integrated (which would require a larger machine tool work envelope); and (2) the user can change some pieces while keeping other pieces the same (rather than changing the whole unit). One arbor (at a hypothetical price of USD100) can serve for various shells at different times. Thus 5 different milling cutters may require only USD100 worth of arbor cost, rather than USD500, as long as the workflow of the shop does not require them all to be set up simultaneously. It is also possible that a crashed tool scraps only the shell rather than both the shell and arbor. To also avoid damage to the shell, many cutters, especially in larger diameters, also have another replaceable part called shim, which is mounted to the shell and the inserts are mounted on the shim. That way, in case of light damage, only the insert and maximum the shim needs replacement. The shell is safe. This would be like crashing a "regular" endmill and being able to reuse the shank rather than losing it along with the flutes.
Mounting methods[edit]
Another type of shell fastening is simply a large-diameter fine thread. The shell then screws onto the arbor just as old-style lathe chuck backplates screw onto the lathe's spindle nose. This method is commonly used on the 2" or 3" boring heads used on knee mills. As with the threaded-spindle-nose lathe chucks, this style of mounting requires that the cutter only make cuts in one rotary direction. Usually (i.e., with right-hand helix orientation) this means only M03, never M04, or in pre-CNC terminology, "only forward, never reverse". One could use a left-hand thread if one needed a mode of use involving the opposite directions (i.e., only M04, never M03).
Using a milling cutter[edit]
Chip formation[edit]
Surface cutting speed (Vc)
This is the speed at which each tooth cuts through the material as the tool spins. This is measured either in metres per minute in metric countries, or surface feet per minute (SFM) in America. Typical values for cutting speed are 10m/min to 60m/min for some steels, and 100m/min and 600m/min for aluminum. This should not be confused with the feed rate. This value is also known as "tangential velocity."
This is the rotation speed of the tool, and is measured in revolutions per minute (rpm). Typical values are from hundreds of rpm, up to tens of thousands of rpm.
Diameter of the tool (D)
Feed per tooth (Fz)
This is the distance the material is fed into the cutter as each tooth rotates. This value is the size of the deepest cut the tooth will make.Typical values could be 0.1 mm/tooth or 1 mm/tooth
This is the speed at which the material is fed into the cutter. Typical values are from 20 mm/min to 5000 mm/min.
This is how deep the tool is under the surface of the material being cut (not shown on the diagram). This will be the height of the chip produced. Typically, the depth of cut will be less than or equal to the diameter of the cutting tool.
{\displaystyle S={\frac {V_{c}}{\pi D}}\,}
{\displaystyle F=zSF_{z}\,}
Conventional milling versus climb milling[edit]
Conventional milling (left): The chip thickness starts at zero thickness, and increases up to the maximum. The cut is so light at the beginning that the tool does not cut, but slides across the surface of the material, until sufficient pressure is built up and the tooth suddenly bites and begins to cut. This deforms the material (at point A on the diagram, left), hardening it, and dulling the tool. The sliding and biting behaviour leaves a poor finish on the material.
Climb milling (right): Each tooth engages the material at a definite point, and the width of the cut starts at the maximum and decreases to zero. The chips are disposed behind the cutter, leading to easier swarf removal. The tooth does not rub on the material, and so tool life may be longer. However, climb milling can apply larger loads to the machine, and so is not recommended for older milling machines or machines that are not in good condition. This type of milling is used predominantly on mills with a backlash eliminator.
Cutter location (cutter radius compensation)[edit]
Swarf removal[edit]
Selecting a milling cutter[edit]
Selecting a milling cutter is not a simple task. There are many variables, opinions and lore to consider, but essentially the machinist is trying to choose a tool that will cut the material to the required specification for the least cost. The cost of the job is a combination of the price of the tool, the time taken by the milling machine, and the time taken by the machinist. Often, for jobs of a large number of parts, and days of machining time, the cost of the tool is lowest of the three costs.
Material: High speed steel (HSS) cutters are the least-expensive and shortest-lived cutters. Cobalt-bearing high speed steels generally can be run 10% faster than regular high speed steel. Cemented carbide tools are more expensive than steel, but last longer, and can be run much faster, so prove more economical in the long run.[citation needed] HSS tools are perfectly adequate for many applications. The progression from regular HSS to cobalt HSS to carbide could be viewed as very good, even better, and the best. Using high speed spindles may preclude use of HSS entirely.
Coating: Coatings, such as titanium nitride, also increase initial cost but reduce wear and increase tool life. TiAlN coating reduces sticking of aluminium to the tool, reducing and sometimes eliminating need for lubrication.
^ Rapid Traverse: More Teeth Per Flute Archived 2007-09-27 at the Wayback Machine
^ J.Ramsey, "Max Diameter for a Flycutter?", PracticalMachinist.com discussion board, retrieved 2011-06-05.
De Vries, D. (1910), Milling machines and milling practice: a practical manual for the use of manufacturers, engineering students and practical men, London: E. & F.N. Spon . Co-edition: New York, Spon & Chamberlain, 1910.
Woodbury, Robert S. (1972) [1960], "History of the Milling Machine", Studies in the History of Machine Tools, Cambridge, Mass., and London: MIT Press, ISBN 978-0-262-73033-4, LCCN 72006354 . First published alone as a monograph in 1960.
Media related to Milling heads at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Milling_cutter&oldid=1062275434"
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Number of Elements of order p in S_{p} An exercise from Herstein asks to prove that the nu
Number of Elements of order p in
{S}_{p}
An exercise from Herstein asks to prove that the number of elements of order p, p a ' in
{S}_{p}
\left(p-1\right)!+1
. I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson's theorem which says
\left(p-1\right)!\equiv -1
(mod p) using this result
Maybe you mean the number of elements of order dividing p (so that you are including the identity)? (Think about the case
p=3
- there are two three cycles, not three of them.) For the general question, think about the possible cycle structure of an element of order p in
{S}_{p}
You can go from the formula in your question to Wilson's theorem by counting the number of p-Sylow subgroups (each contains
p-1
elements of order p), and then appealing to Sylow's theorem. (You will find that there are
\left(p-2\right)!
p-Sylow subgroups, and by Sylow's theorem this number is congruent to
1\text{mod}p
p-1
\left(p-1\right)!
-1\text{mod}p
Aliana Porter
Every element of order p in
{S}_{p}
is a p-cycle. The symmetric group
{S}_{p-1}
acts transitively on these p cycles.
={x}^{k}
f,\text{ }g\in \mathbb{C}\left[x\right]
fg=\alpha {x}^{n+m}
\alpha \in \mathbb{C}
{x}^{2}=e
\mathbb{Z}\left[t\right]\left[\sqrt{{t}^{2}-1}\right]
How does cancellation work in polynomial quotient rings?
\frac{\mathbb{Z}\left[x\right]}{\left(2x-6,6x-15\right)}
Can I just automatically say that
\frac{\mathbb{Z}\left[x\right]}{\left(2x-6,6x-15\right)}\stackrel{\sim }{=}\frac{\mathbb{Z}\left[x\right]}{\left(x-3,6x-15\right)}
by just dividng the first polynomial by 2?
\frac{{z}^{11}-1}{z-1}={z}^{10}+{z}^{9}+{z}^{8}+{z}^{7}+{z}^{6}+{z}^{5}+{z}^{4}+{z}^{3}+{z}^{2}+z+1
{\zeta }^{1},{\zeta }^{2},\dots ,{\zeta }^{10}
\begin{array}{rl}{A}_{0}& ={x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}\\ {A}_{1}& ={x}_{1}+\zeta {x}_{2}+{\zeta }^{2}{x}_{3}+{\zeta }^{3}{x}_{4}+{\zeta }^{4}{x}_{5}\\ {A}_{2}& ={x}_{1}+{\zeta }^{2}{x}_{2}+{\zeta }^{4}{x}_{3}+\zeta {x}_{4}+{\zeta }^{3}{x}_{5}\\ {A}_{3}& ={x}_{1}+{\zeta }^{3}{x}_{2}+\zeta {x}_{3}+{\zeta }^{4}{x}_{4}+{\zeta }^{2}{x}_{5}\\ {A}_{4}& ={x}_{1}+{\zeta }^{4}{x}_{2}+{\zeta }^{3}{x}_{3}+{\zeta }^{2}{x}_{4}+\zeta {x}_{5}\end{array}
{A}_{0},\dots ,{A}_{4}
{x}_{1},\dots ,{x}_{5}
{A}_{0}
\tau
{A}_{j}
{\zeta }^{-j}{A}_{j}
{A}_{j}^{5}
{A}_{j}^{5}
{A}_{j}^{5}
\zeta
\zeta
{A}_{1}^{5}
{A}_{1}
{A}_{j}
{A}_{1}
gcd\left(a,b\right)=|a|⇔a\mid b
|
I have a multivariable function that is composed of six variables; four of these are constants C_
Ahmed Stewart 2022-01-25 Answered
I have a multivariable function that is composed of six variables; four of these are constants
{C}_{1},{C}_{2},{C}_{3},{C}_{4}
and the other two variables are x and y. The function is defined as follows
Xi\left(x,y,{C}_{1},{C}_{2},{C}_{3},{C}_{4}\right)
I would like to find the maximum value of
Xi
x\in \left\{-\mathrm{\infty },\mathrm{\infty }\right\}
y\in \left\{-\mathrm{\infty },\mathrm{\infty }\right\}
. What is the rigorous mathematical notation I should use to describe this. Is this fine:
max\left\{Xi\left(x,y,{C}_{1},{C}_{2},{C}_{3},{C}_{4}\right)\right\}
Joy Compton
If it uses four constants, isn't it just a function of 2 variables?
First, don't include the constants in your function. You can just say
Xi\left(x,y\right)
. I think it's pretty clear what you're saying if you say max, and include the intervals of x and y.
P\left(x\right)=-12{x}^{2}+2136x-41000
x=r\mathrm{cos}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}
\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}
How to solve the ordinary differential equation using separation of variables?
\frac{dy}{dt}=\frac{y+1}{t+1}
f:{\mathbb{R}}^{2}\to \mathbb{R}
f\circ f,\text{or}f\circ \dots \circ f?
Here is partial output from a simple regression analysis.
EAFE = 4.76 + 0.663 S&P
\begin{array}{cccccc}Source& DF& SS& MS& F& P\\ Regression& 1& 3445.9& 3445.9& 9.50& 0.005\\ Residual\text{ }Error& & & & & \\ Total& 29& 13598.3& & & \end{array}
Calculate the values of the following:
The regression standard error,
{s}_{e}
(Round to 3 decimal places)
The coefficient of determination,
{r}^{2}
The correlation coefficient, r (Round to 4 decimal places)
Define Clairaut equation?
|
If \cos 2v=- \frac 19 and v is acute, then
\mathrm{cos}2v=-\frac{19}{}
\mathrm{cos}2v={\mathrm{cos}}^{2}v-{\mathrm{sin}}^{2}v
rakije2v
\mathrm{cos}\left(2x\right)=-\frac{19}{}
For the general case, let the R.H.S constant be k. Well
Calculate the average rate of change for the function
f\left(x\right)={x}^{2}+x-12
, over the interval:
2. [-1,10]
Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t.
\mathrm{sin}t=\frac{3}{4}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}t=\frac{\sqrt{7}}{4}
\frac{\mathrm{cos}x}{1-\mathrm{sin}x}\text{ }\text{ as x approaches }\text{ }\frac{\pi }{2}
y=\sqrt{x}
Only 25% of the intensity of a polarized light wave passes through a polarizing filter. What is the angle between the electric field and the axis of the filter?
Find y as a function of t, if
\frac{dy}{dt}+\left(0.2\right)ty=4t\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y\left(0\right)=9
\mathrm{sin}A=0.5
\mathrm{sin}A=1.2654
\mathrm{sin}A=0.9962
\mathrm{sin}A=\frac{3}{4}
|
Find an answer to any advanced math equations
Advanced math questions and answers
Recent questions in Advanced Math
is this true or false
6,a,b,c,b,c,8=6,a,b,c,b,c,8,\text{⧸}0
{\left({x}^{4}{y}^{5}\right)}^{\frac{1}{4}}{\left({x}^{8}{y}^{5}\right)}^{\frac{1}{5}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}
j-k
Find a square number such that when twice its root is added to it or subtracted from it, one obtained other square numbers. In other words, solve a problem of the type.
{x}^{2}+2x={u}^{2}
{x}^{2}-2x={v}^{2}
Using cardinatility of sets in discrete mathematics the value of N is real numbers Currently using elements of discrete mathematics by Richard Hammack chapter 18 Let A be a collection of sets such that X in A if and only if
X\supset N\text{ }\text{and}|X|=n
for some n in N. Prove that
|A|=|N|
Using the Mathematical Induction to prove that:
{3}^{2n}-1
is divisible by 4, whenever n is a positive integer.
prove or disprove the product of two distinct irrational numbers is irrational
For the following questions you must use the rules of logic (Don’t use truth tables)
a) Show that (
p↔q
\mathrm{¬}p↔\mathrm{¬}q
) are logically equivalent.
b) Show that not (
p\oplus q
p↔q
Approximate y(1,2), with h = 0.1 given that
x{y}^{\prime }=y-2{x}^{2},y\left(1\right)=2
, by RK-4 using 5-digits rounding.
A building code requires one square foot (sq ft) of net-free vent area (NFVA) for every 300 sq ft of attic space. How many square feet of NFVA are required for a
1620-sq-ft
attic?
Suppose n is an integer. Using the definitions of even and odd, prove that n is odd if and only if
3n+1
Tories bookcase holds one algebra book one geometry book and 12 books about advanced engineering mathematics. Books are taken at random from her book case, one after another without replacement until an engineer book is had, at which no more books are taken. The random variable Z represents the total number of books taken during this experiment.
Pr\left[z=3\right]
A\left({A}^{\prime }+{B}^{\prime }\right)\left(B+C\right)\left(B+{C}^{\prime }+D\right)
3{m}^{3}+2{m}^{2}
Role of Mathematics in business and economics
List all the steps used to search for 9 in the sequence
1,3,4,5,6,8,9,11
using linear search
A payment of 1500$ is due in three months.Find the equivalent value at nine months if the interest rate is 4%
a:$1500
d:$1470
Students pursuing advanced Math are constantly dealing with advanced Math equations that are mostly used in space engineering, programming, and construction of AI-based solutions that we can see daily as we are turning to automation that helps us to find the answers to our challenges. If it sounds overly complex with subjects like exponential growth and decay, don’t let advanced math problems frighten you because these must be approached through the lens of advanced Math questions and answers. Regardless if you are dealing with simple equations or more complex ones, just break things down into several chunks as it will help you to find the answers.
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Evaluating the Average Power Delivered by a Wind Turbine - MATLAB & Simulink Example - MathWorks Benelux
Derive Equation for Average Power of Wind Turbine
I. Define piecewise expression for power
II. Define external wind conditions
III. Calculate average power
This example uses Symbolic Math Toolbox™ and the Statistics and Machine Learning Toolbox™ to explore and derive a parametric analytical expression for the average power generated by a wind turbine.
The parametric equation can be used for evaluating various wind turbine configurations and wind farm sites. More information see Wind Resource Assessment.
The total power delivered to a wind turbine can be estimated by taking the derivative of the wind's kinetic energy. This results in the following expression:
{\mathit{P}}_{\mathit{w}}=\frac{\rho {\text{\hspace{0.17em}}\mathit{A}\text{\hspace{0.17em}}\mathit{u}}^{3}}{2}
A is the swept area of turbine blades, in
{\mathit{m}}^{2}
ρ = air density, in
\mathrm{kg}/{m}^{3}
u = wind speed, in
\mathit{m}/\mathit{s}
The process of converting wind power to electrical power results in efficiency losses, as described in the diagram below.
The electrical power output of a practical wind turbine can be described using the following equation:
{\mathit{P}}_{\mathit{e}}=\frac{{\mathit{C}}_{\mathrm{tot}}\text{\hspace{0.17em}\hspace{0.17em}}\rho {\text{\hspace{0.17em}}\mathrm{Au}}^{3}}{2}
(2) where
{\mathit{C}}_{\mathrm{tot}}=\mathrm{overall}\text{\hspace{0.17em}}\mathrm{efficiency}={\mathit{C}}_{\mathit{p}}{\mathit{C}}_{\mathit{t}}{\mathit{C}}_{\mathit{g}}
The overall efficiency is between 0.3 and 0.5, and varies with both wind speed and rotational speed of the turbine. For a fixed rotational speed, there is a rated wind speed at which electrical power generated by the wind turbine is near its maximum (
{\mathit{P}}_{\mathrm{er}}
), and overall efficiency at this point is denoted
{\mathit{C}}_{\mathrm{totR}}
{\mathit{P}}_{\mathrm{er}}=\frac{{\mathit{C}}_{\mathrm{totR}}\text{\hspace{0.17em}\hspace{0.17em}}\rho {\text{\hspace{0.17em}}\mathrm{Au}}^{3}}{2}
Assuming a fixed rotational speed, the electrical power output of the wind turbine can be estimated using the following profile:
{\mathit{u}}_{\mathit{r}}
= rated wind speed
{\mathit{u}}_{\mathit{c}}
= cut-in speed, the speed at which the electrical power output rises above zero and power production starts
{\mathit{u}}_{\mathit{f}}
= furling wind speed, the speed at which the turbine is shut down to prevent structural damage
As seen in the figure, we assume that output power increases between
{u}_{c}
{u}_{r}
, then is at a constant maximum value between
{\mathit{u}}_{\mathit{r}}
{\mathit{u}}_{\mathit{f}}
. Power output is zero for all other conditions.
We define a piecewise function that described turbine power.
\left\{\begin{array}{cl}0& \text{ if }u<{u}_{c}\\ {C}_{1}+{C}_{2} {u}^{k}& \text{ if }{u}_{c}\le u\wedge u\le {u}_{r}\\ \mathrm{Per}& \text{ if }u\le {u}_{f}\wedge {u}_{r}\le u\\ 0& \text{ if }{u}_{f}<u\end{array}
{\mathit{C}}_{1}
{\mathit{C}}_{2}
\frac{\mathrm{Per} {{u}_{c}}^{k}}{{{u}_{c}}^{k}-{{u}_{r}}^{k}}
-\frac{\mathrm{Per}}{{{u}_{c}}^{k}-{{u}_{r}}^{k}}
The rated power output offers a good indication of a how much power a wind turbine is capable of producing, however we'd like to estimate how much power (on average) the wind turbine will actually deliver. To calculate average power, we need to account for external wind conditions. A Weibull distribution does a good job at modeling the variance in wind, therefore the wind profile can be estimated using the following probability density function:
f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)=\frac{\left(\frac{b}{a}\right)\phantom{\rule{0.16666666666666666em}{0ex}}{\left(\frac{u}{a}\right)}^{b-1}}{{\mathrm{e}}^{{\left(\frac{u}{a}\right)}^{b}}}
In general, larger 'a' values indicate a higher median wind speed, and larger 'b' values indicate reduced variability.
We use the Statistics and Machine Learning Toolbox to generate a Weibull distribution and illustrate the variability in wind at our wind farm site (a=12.5, b=2.2):
The average power output from a wind turbine can be obtained using the following integral:
P{e}_{average}={\int }_{0}^{\infty }Pe\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}f\left(u\right)du
Power is zero when wind speed is less than the cut in wind speed
{u}_{c}
and greater than furling wind speed
{\mathit{u}}_{\mathit{f}}
. Therefore, the integral can be expressed as follows:
P{e}_{average}={C}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{c}}^{{u}_{r}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du\right)+{C}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{c}}^{{u}_{r}}{u}^{b}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)du\right)+\mathrm{P}\mathrm{e}\mathrm{r}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{r}}^{{u}_{f}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du\right)
There are two distinct integrals in equation (7). We plug equation (4) into these integrals and simplify them using the substitutions:
x={\left(\frac{u}{a}\right)}^{b}
\mathrm{d}\mathrm{x}=\left(\frac{b}{a}\right)\phantom{\rule{0.16666666666666666em}{0ex}}{\left(\frac{u}{a}\right)}^{b-1}\phantom{\rule{0.16666666666666666em}{0ex}}\mathrm{d}\mathrm{u}
. This simplifies our original integrals to the following:
\int f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du=\int \frac{1}{{\mathrm{e}}^{x}}\phantom{\rule{0.16666666666666666em}{0ex}}dx
\int {u}^{b}\phantom{\rule{0.16666666666666666em}{0ex}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du={a}^{b}\phantom{\rule{0.16666666666666666em}{0ex}}\left(\int \frac{x}{{\mathrm{e}}^{x}}\phantom{\rule{0.16666666666666666em}{0ex}}dx\right)
Solving these integrals and then replacing x with
{\left(\frac{u}{a}\right)}^{b}
-{\mathrm{e}}^{-{\left(\frac{u}{a}\right)}^{b}}
-{a}^{b} {\mathrm{e}}^{-{\left(\frac{u}{a}\right)}^{b}} \left({\left(\frac{u}{a}\right)}^{b}+1\right)
Substituting the results into equation (6) yields an equation for average power output of the wind turbine.
\begin{array}{l}\mathrm{Per} {\sigma }_{2}-\mathrm{Per} {\mathrm{e}}^{-{\left(\frac{{u}_{f}}{a}\right)}^{b}}+\frac{\mathrm{Per} {{u}_{c}}^{k} {\mathrm{e}}^{-{\left(\frac{{u}_{c}}{a}\right)}^{b}}}{{\sigma }_{1}}-\frac{\mathrm{Per} {{u}_{c}}^{k} {\sigma }_{2}}{{\sigma }_{1}}-\frac{\mathrm{Per} {a}^{b} {\mathrm{e}}^{-{\left(\frac{{u}_{c}}{a}\right)}^{b}} \left({\left(\frac{{u}_{c}}{a}\right)}^{b}+1\right)}{{\sigma }_{1}}+\frac{\mathrm{Per} {a}^{b} {\sigma }_{2} \left({\left(\frac{{u}_{r}}{a}\right)}^{b}+1\right)}{{\sigma }_{1}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={{u}_{c}}^{k}-{{u}_{r}}^{k}\\ \\ \mathrm{ }{\sigma }_{2}={\mathrm{e}}^{-{\left(\frac{{u}_{r}}{a}\right)}^{b}}\end{array}
We have used the Symbolic Math Toolbox to develop a parametric equation which can be used to perform simulation studies to determine the average power generated for various wind turbine configurations and wind farm sites.
|
nxm
\mathrm{rows}=n
\mathrm{cols}=m+2
\mathrm{rcol}=m
nxn
rows have leading identity, the reduction of the remaining rows can be accomplished by specifying
\mathrm{rows}=r..n
\mathrm{cols}=r..n+1
\mathrm{rcol}=n-r+1
nxm
nxn
i
-i
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Modular}]\right):
p≔2741
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2741}
A≔\mathrm{Mod}\left(p,\mathrm{Matrix}\left(4,5,\left(i,j\right)↦\mathrm{rand}\left(\right)\right),\mathrm{integer}[]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{2543}& \textcolor[rgb]{0,0,1}{1568}& \textcolor[rgb]{0,0,1}{127}& \textcolor[rgb]{0,0,1}{356}& \textcolor[rgb]{0,0,1}{581}\\ \textcolor[rgb]{0,0,1}{430}& \textcolor[rgb]{0,0,1}{1549}& \textcolor[rgb]{0,0,1}{2376}& \textcolor[rgb]{0,0,1}{1511}& \textcolor[rgb]{0,0,1}{1839}\\ \textcolor[rgb]{0,0,1}{164}& \textcolor[rgb]{0,0,1}{1946}& \textcolor[rgb]{0,0,1}{211}& \textcolor[rgb]{0,0,1}{49}& \textcolor[rgb]{0,0,1}{2418}\\ \textcolor[rgb]{0,0,1}{30}& \textcolor[rgb]{0,0,1}{1480}& \textcolor[rgb]{0,0,1}{754}& \textcolor[rgb]{0,0,1}{1049}& \textcolor[rgb]{0,0,1}{423}\end{array}]
B≔\mathrm{Copy}\left(p,A\right):
\mathrm{RowReduce}\left(p,B,4,5,4,'\mathrm{det}',0,'\mathrm{rank}',0,0,\mathrm{true}\right):
B,\mathrm{rank}
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1916}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{659}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{181}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{25}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}
\mathrm{det}
\textcolor[rgb]{0,0,1}{82}
\mathrm{modp}\left(\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(\mathrm{Matrix}\left(A[1..4,1..4],\mathrm{datatype}=\mathrm{integer}\right)\right),p\right)
\textcolor[rgb]{0,0,1}{82}
\mathrm{Multiply}\left(p,A,1..4,1..4,B,1..4,5\right),\mathrm{Copy}\left(p,A,1..4,5\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{581}\\ \textcolor[rgb]{0,0,1}{1839}\\ \textcolor[rgb]{0,0,1}{2418}\\ \textcolor[rgb]{0,0,1}{423}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{581}\\ \textcolor[rgb]{0,0,1}{1839}\\ \textcolor[rgb]{0,0,1}{2418}\\ \textcolor[rgb]{0,0,1}{423}\end{array}]
A≔\mathrm{Mod}\left(p,\mathrm{Matrix}\left(4,5,\left(i,j\right)↦\mathrm{rand}\left(\right)\right),\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1980.}& \textcolor[rgb]{0,0,1}{2533.}& \textcolor[rgb]{0,0,1}{1439.}& \textcolor[rgb]{0,0,1}{67.}& \textcolor[rgb]{0,0,1}{2051.}\\ \textcolor[rgb]{0,0,1}{2635.}& \textcolor[rgb]{0,0,1}{353.}& \textcolor[rgb]{0,0,1}{2657.}& \textcolor[rgb]{0,0,1}{1617.}& \textcolor[rgb]{0,0,1}{2198.}\\ \textcolor[rgb]{0,0,1}{2587.}& \textcolor[rgb]{0,0,1}{1857.}& \textcolor[rgb]{0,0,1}{827.}& \textcolor[rgb]{0,0,1}{1848.}& \textcolor[rgb]{0,0,1}{2338.}\\ \textcolor[rgb]{0,0,1}{1720.}& \textcolor[rgb]{0,0,1}{1181.}& \textcolor[rgb]{0,0,1}{493.}& \textcolor[rgb]{0,0,1}{1731.}& \textcolor[rgb]{0,0,1}{2264.}\end{array}]
B≔\mathrm{Copy}\left(p,A\right):
\mathrm{RowReduce}\left(p,B,1..2,5,4,'\mathrm{det1}',0,0,0,0,\mathrm{true}\right):
B
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{732.}& \textcolor[rgb]{0,0,1}{2653.}& \textcolor[rgb]{0,0,1}{1849.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1097.}& \textcolor[rgb]{0,0,1}{941.}& \textcolor[rgb]{0,0,1}{1501.}\\ \textcolor[rgb]{0,0,1}{2587.}& \textcolor[rgb]{0,0,1}{1857.}& \textcolor[rgb]{0,0,1}{827.}& \textcolor[rgb]{0,0,1}{1848.}& \textcolor[rgb]{0,0,1}{2338.}\\ \textcolor[rgb]{0,0,1}{1720.}& \textcolor[rgb]{0,0,1}{1181.}& \textcolor[rgb]{0,0,1}{493.}& \textcolor[rgb]{0,0,1}{1731.}& \textcolor[rgb]{0,0,1}{2264.}\end{array}]
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{AddMultiple}\left(p,p-B[2+i,j],B,2+i,B,j,B,2+i\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
B
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{732.}& \textcolor[rgb]{0,0,1}{2653.}& \textcolor[rgb]{0,0,1}{1849.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1097.}& \textcolor[rgb]{0,0,1}{941.}& \textcolor[rgb]{0,0,1}{1501.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{608.}& \textcolor[rgb]{0,0,1}{581.}& \textcolor[rgb]{0,0,1}{2260.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{508.}& \textcolor[rgb]{0,0,1}{1120.}& \textcolor[rgb]{0,0,1}{2290.}\end{array}]
\mathrm{RowReduce}\left(p,B,3..4,3..5,2,'\mathrm{det2}',0,0,0,0,\mathrm{true}\right):
B
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{732.}& \textcolor[rgb]{0,0,1}{2653.}& \textcolor[rgb]{0,0,1}{1849.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1097.}& \textcolor[rgb]{0,0,1}{941.}& \textcolor[rgb]{0,0,1}{1501.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{2413.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1536.}\end{array}]
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{AddMultiple}\left(p,p-B[i,2+j],B,i,3..5,B,j+2,3,B,i,3\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
B
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1596.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1377.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{2413.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1536.}\end{array}]
\mathrm{modp}\left(\mathrm{round}\left(\mathrm{det1}\mathrm{det2}\right),p\right)
\textcolor[rgb]{0,0,1}{2603}
\mathrm{modp}\left(\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(\mathrm{Matrix}\left(4,4,\left(i,j\right)↦\mathrm{round}\left(A[i,j]\right),\mathrm{datatype}=\mathrm{integer}\right)\right),p\right)
\textcolor[rgb]{0,0,1}{2603}
\mathrm{Multiply}\left(p,A,1..4,1..4,B,1..4,5\right),\mathrm{Copy}\left(p,A,1..4,5\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{2051.}\\ \textcolor[rgb]{0,0,1}{2198.}\\ \textcolor[rgb]{0,0,1}{2338.}\\ \textcolor[rgb]{0,0,1}{2264.}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2051.}\\ \textcolor[rgb]{0,0,1}{2198.}\\ \textcolor[rgb]{0,0,1}{2338.}\\ \textcolor[rgb]{0,0,1}{2264.}\end{array}]
p≔65521
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{65521}
M≔\mathrm{Create}\left(p,4,5,'\mathrm{random}',\mathrm{integer}[]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{37606}& \textcolor[rgb]{0,0,1}{6440}& \textcolor[rgb]{0,0,1}{30791}& \textcolor[rgb]{0,0,1}{17866}& \textcolor[rgb]{0,0,1}{45834}\\ \textcolor[rgb]{0,0,1}{55381}& \textcolor[rgb]{0,0,1}{6159}& \textcolor[rgb]{0,0,1}{7233}& \textcolor[rgb]{0,0,1}{13465}& \textcolor[rgb]{0,0,1}{42145}\\ \textcolor[rgb]{0,0,1}{60796}& \textcolor[rgb]{0,0,1}{36696}& \textcolor[rgb]{0,0,1}{65135}& \textcolor[rgb]{0,0,1}{63713}& \textcolor[rgb]{0,0,1}{52874}\\ \textcolor[rgb]{0,0,1}{49338}& \textcolor[rgb]{0,0,1}{50505}& \textcolor[rgb]{0,0,1}{50237}& \textcolor[rgb]{0,0,1}{6211}& \textcolor[rgb]{0,0,1}{46483}\end{array}]
\mathrm{RowReduce}\left(p,M,4,5,4,0,0,0,0,0,-1\right):
M
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{17540}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{45645}& \textcolor[rgb]{0,0,1}{14195}& \textcolor[rgb]{0,0,1}{36912}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{64728}& \textcolor[rgb]{0,0,1}{52075}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{41141}\end{array}]
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At t=0 the current to a dc electric
J. P. Morgan Asset Management publishes information about financial investments. Between 2002 and 2011 the expected return for the S&P was with a standard deviation of and the expected return over that same period for a Core Bonds fund was with a standard deviation of (J. P. Morgan Asset Management, Guide to the Markets). The publication also reported that the correlation between the S&P and Core Bonds is . You are considering portfolio investments that are composed of an S&P index fund and a Core Bonds fund. a. Using the information provided, determine the covariance between the S&P and Core Bonds. Round your answer to two decimal places. If required enter negative values as negative numbers.
f \left( x \right) = 7 {x}^{2 }+ 2
g \left( x \right) = 5 {x}^{3 }+ 7 {x}^{2 }+ 6 x
\frac{f \left( x \right) }{g \left( x \right) }
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3-hydroxyisobutyryl-CoA hydrolase - Wikipedia
3-hydroxyisobutyryl-CoA hydrolase monomer, Human
In enzymology, a 3-hydroxyisobutyryl-CoA hydrolase (EC 3.1.2.4) is an enzyme that catalyzes the chemical reaction
3-hydroxy-2-methylpropanoyl-CoA + H2O
{\displaystyle \rightleftharpoons }
CoA + 3-hydroxy-2-methylpropanoate
Thus, the two substrates of this enzyme are 3-hydroxy-2-methylpropanoyl-CoA and H2O, whereas its two products are CoA and 3-hydroxy-2-methylpropanoate.
This enzyme belongs to the family of hydrolases, specifically those acting on thioester bonds. The systematic name of this enzyme class is 3-hydroxy-2-methylpropanoyl-CoA hydrolase. Other names in common use include 3-hydroxy-isobutyryl CoA hydrolase, and HIB CoA deacylase. This enzyme participates in 3 metabolic pathways: valine, leucine and isoleucine degradation, beta-alanine metabolism, and propanoate metabolism. 3-hydroxyisobutyryl-CoA hydrolase is encoded by HIBCH gene.[1]
^ "OMIM Entry - * 610690 - 3-HYDROXYISOBUTYRYL-CoA HYDROLASE; HIBCH". www.omim.org. Retrieved 2017-11-20.
Rendina G, Coon MJ (March 1957). "Enzymatic hydrolysis of the coenzyme a thiol esters of beta-hydroxypropionic and beta-hydroxyisobutyric acids". The Journal of Biological Chemistry. 225 (1): 523–34. PMID 13457352.
Retrieved from "https://en.wikipedia.org/w/index.php?title=3-hydroxyisobutyryl-CoA_hydrolase&oldid=917319586"
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Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix <math> \tilde{A}=A-BK</math> must have an eigenvalue with positive real part, to which corresponds a certain eigenvector <math>v</math>. Writing down the algebraic Riccati equation with <math> \tilde{A}</math>, and pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and <math>v^* Q_v v</math> is zero. Which contradicts the initial assumption.
Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix <math> \tilde{A}=A-BK</math> must have an eigenvalue with positive real part, to which corresponds a certain eigenvector <math>v</math>.
One can rewrite the Algebraic Riccati equation using <math> \tilde{A}</math>, where you should note the changes of signs:
<math>P\tilde{A} + \tilde{A}^T P + PBQ_u^{-1}B^T P + Qx=0 </math>
Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and <math>v^* Q_v v</math> is zero. Which contradicts the initial assumption.
{\displaystyle {\tilde {A}}=A-BK}
{\displaystyle v}
One can rewrite the Algebraic Riccati equation using
{\displaystyle {\tilde {A}}}
, where you should note the changes of signs:
{\displaystyle P{\tilde {A}}+{\tilde {A}}^{T}P+PBQ_{u}^{-1}B^{T}P+Qx=0}
Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and
{\displaystyle v^{*}Q_{v}v}
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x^2y''+(3x^2+4x)y'+2(x^2+3x+1)y=0 I try to solve this equation by finding \mu such
Ali Marshall 2022-05-01 Answered
{x}^{2}y{}^{″}+\left(3{x}^{2}+4x\right){y}^{\prime }+2\left({x}^{2}+3x+1\right)y=0
I try to solve this equation by finding
\mu
such that the equation become exact.( I know there are other ways for solving this equation).
{\left(\mu {y}^{\prime }\right)}^{\prime }={\mu }^{\prime }{y}^{\prime }+\mu y{}^{″}=\mu y{}^{″}+\mu \frac{3{x}^{2}+4x}{{x}^{2}}{y}^{\prime }+\mu \frac{2\left({x}^{2}+3x+1\right)}{{x}^{2}}y⇒
{\mu }^{\prime }{y}^{\prime }=\mu \frac{3{x}^{2}+4x}{{x}^{2}}{y}^{\prime }+\mu \frac{2\left({x}^{2}+3x+1\right)}{{x}^{2}}y⇒\frac{{\mu }^{\prime }}{\mu }=\frac{3{x}^{2}+4x}{{x}^{2}}+\frac{2\left({x}^{2}+3x+1\right)}{{x}^{2}}\frac{y}{{y}^{\prime }}
How am I supposed to solve it?
Jonas Dickerson
There is a method to solve equations like this one that uses a clever substitution. The idea is to use substitution to remove the y' term. Whenever you have an equation in the form of
y{ }^{″}+f\left(x\right){y}^{\prime }+g\left(x\right)y=0
, you substitute
y=\mathrm{exp}\left(-\int \frac{f\left(x\right)}{2}, dx \right)\mu
. Divide through by
{x}^{2}
y{ }^{″}+\underset{=f\left(x\right)}{\underset{⏟}{\frac{3x+4}{x}}}{y}^{\prime }+\underset{=g\left(x\right)}{\underset{⏟}{\frac{2\left({x}^{2}+3x+1\right)}{{x}^{2}}}}y=0.
Then substitute in
y=\mathrm{exp}\left(-\int \frac{3x+4}{2x}, dx \right)\mu =\frac{{e}^{-3\frac{x}{2}}}{{x}^{2}}\mu .
\frac{{e}^{-3\frac{x}{2}}}{{x}^{2}}\mu { }^{″}-\frac{{e}^{-3\frac{x}{2}}}{4{x}^{2}}\mu =0
Divide through by
\frac{{e}^{-3\frac{x}{2}}}{{x}^{2}}
\mu { }^{″}-\frac{1}{4}\mu =0.
Using standard methods in ODEs, we get
\mu ={c}_{1}{e}^{-\frac{x}{2}}+{c}_{2}{e}^{\frac{x}{2}}
y=\frac{{e}^{-3\frac{x}{2}}}{{x}^{2}}\mu =\frac{{e}^{-3\frac{x}{2}}}{{x}^{2}}\left({c}_{1}{e}^{-\frac{x}{2}}+{c}_{2}{e}^{\frac{x}{2}}\right)
\frac{dw}{dt}
\frac{dw}{dt}
w=x\mathrm{sin}y,\text{ }x={e}^{t},\text{ }y=\pi -t
t=0
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have
\frac{dN}{dt}=2N
N\left(t\right)={N}_{0}{e}^{2t}
r=2
, but I have a feeling this is wrong since 20 minutes should be used somewhere around here.
Solving a differential equation in matrix form but adding a constant
\begin{array}{rl}{x}_{1}^{\prime }& =2{x}_{1}+3{x}_{2},\\ {x}_{2}^{\prime }& ={x}_{1}-{x}_{2}.\end{array}
Solving a differential equation connecting slope and derivative
\frac{y\left(x\right)-y\left(a\right)}{x-a}={y}^{\prime }\left(x\right)
Explain what is the difference between implicit and explicit solutions for differential equation initial value problems.
How to solve a complicated ODE
{f}^{{}^{\prime }}\left(x\right)=\gamma \frac{f\left(x\right)+{f}^{2}\left(x\right)}{\mathrm{log}\left(\frac{f\left(x\right)}{1+f\left(x\right)}\right)}
with the initial condition f(0), where
x\ge \text{ }\text{and}\text{ }f\left(0\right)\ge 0
f\left(x\right)=\frac{1}{-1+\mathrm{exp}\sqrt{2\gamma x+{\mathrm{log}\left(\frac{f\left(0\right)+1}{f\left(0\right)}\right)}^{2}}}
I think the ODE can be solved by separating the variables
{\int }_{0}^{f\left(x\right)}\frac{\mathrm{log}\left(\frac{f\left(z\right)}{1+f\left(z\right)}\right)}{f\left(z\right)+{f}^{2}\left(z\right)}df={\int }_{0}^{x}\gamma dz
The right-hand size is easy. I do not know how to solve the integration of the left-hand side.
Infer boundedness from differential inequality
\frac{dx}{dt}\le x{\left(t\right)}^{2}+y\left(t\right)
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A stained glass window consists of nine squares of glass in a 3x3 array. Of the nine squares, k are
I upvoted the first answer but I would like to show how to compute the cycle index
Z\left(G\right)
G
of symmetries of the square and apply the Polya Enumeration Theorem to this problem.
We need to enumerate and factor into cycles the eight permutations that contribute to
Z\left(G\right).
There is the identity, which contributes
{a}_{1}^{9}.
The two 90 degree rotations contribute
2{a}_{1}{a}_{4}^{2}.
The 180 degree rotation contributes
{a}_{1}{a}_{2}^{4}.
The vertical and horizontal reflections contribute
2{a}_{1}^{3}{a}_{2}^{3}.
The reflections in a diagonal contribute
2{a}_{1}^{3}{a}_{2}^{3}.
This yields the cycle index
Z\left(G\right)=\frac{1}{8}\left({a}_{1}^{9}+2{a}_{1}{a}_{4}^{2}+{a}_{1}{a}_{2}^{4}+4{a}_{1}^{3}{a}_{2}^{3}\right).
As we are interested in the red squares we evaluate
Z\left(G\right)\left(1+R\right)
1/8\phantom{\rule{thinmathspace}{0ex}}{\left(1+R\right)}^{9}+1/2\phantom{\rule{thinmathspace}{0ex}}{\left(1+R\right)}^{3}{\left({R}^{2}+1\right)}^{3}+1/8\phantom{\rule{thinmathspace}{0ex}}\left(1+R\right){\left({R}^{2}+1\right)}^{4}\phantom{\rule{0ex}{0ex}}+1/4\phantom{\rule{thinmathspace}{0ex}}\left(1+R\right){\left({R}^{4}+1\right)}^{2}
{R}^{9}+3\phantom{\rule{thinmathspace}{0ex}}{R}^{8}+8\phantom{\rule{thinmathspace}{0ex}}{R}^{7}+16\phantom{\rule{thinmathspace}{0ex}}{R}^{6}+23\phantom{\rule{thinmathspace}{0ex}}{R}^{5}+23\phantom{\rule{thinmathspace}{0ex}}{R}^{4}+16\phantom{\rule{thinmathspace}{0ex}}{R}^{3}+8\phantom{\rule{thinmathspace}{0ex}}{R}^{2}+3\phantom{\rule{thinmathspace}{0ex}}R+1.
This is the classification of the orbits according to the number of red squares. Differentiate and multiply by
R
to obtain the total count of the squares, which yields
9\phantom{\rule{thinmathspace}{0ex}}{R}^{9}+24\phantom{\rule{thinmathspace}{0ex}}{R}^{8}+56\phantom{\rule{thinmathspace}{0ex}}{R}^{7}+96\phantom{\rule{thinmathspace}{0ex}}{R}^{6}+115\phantom{\rule{thinmathspace}{0ex}}{R}^{5}+92\phantom{\rule{thinmathspace}{0ex}}{R}^{4}+48\phantom{\rule{thinmathspace}{0ex}}{R}^{3}+16\phantom{\rule{thinmathspace}{0ex}}{R}^{2}+3\phantom{\rule{thinmathspace}{0ex}}R
which matches the accepted answer.
A
B
S
S=A\cup B
A\cap B=\mathrm{\varnothing }
\mathcal{P}\left(X\right)
X
|Y|
Y
|\mathcal{P}\left(A\right)|+|\mathcal{P}\left(B\right)|=|\mathcal{P}\left(A\right)\cup \mathcal{P}\left(B\right)|
\mathcal{P}\left(A\right)
\mathcal{P}\left(B\right)
\mathcal{P}\left(X\right)
|Y|
f
A\to B
|A|=4,|B|=3
{3}^{4}-\left(\genfrac{}{}{0}{}{3}{1}\right){2}^{4}+\left(\genfrac{}{}{0}{}{3}{2}\right){1}^{4}
\left(\genfrac{}{}{0}{}{3}{2}\right){1}^{4}
\left(\genfrac{}{}{0}{}{3}{1}\right){2}^{4}
\left(1,2,...,n\right)
k
\left(n-k\right)!
k
\left(\genfrac{}{}{0}{}{n}{k}\right)\left(n-k\right)!=\frac{n!}{k!}
n!
n!-\left(\frac{n!}{1!}+\frac{n!}{2!}-\frac{n!}{3!}+...+\left(-1{\right)}^{n+1}\frac{n!}{n!}\right)
k
n-k
i have a homework with 10 question but im stuck with 3 i searched about them everywhere read other colleges lectures but i couldnt solved them finally i desired to ask here
Prove that each positive integer can be written in form of
{2}^{k}\ast q
, where q is odd, and k is a non-negative integer.
Hint: Use induction, and the fact that the product of two odd numbers is odd.
\left(x+y{\right)}^{n}=\sum _{k=0}^{n}C\left(n,k\right)\ast {x}^{n-k}\ast {y}^{k}=\frac{{n}^{2}+n}{2}
Prove the above statement by using induction on n.
{n}_{1},{n}_{2},...,{n}_{t}
be positive integers. Show that if
{n}_{1}+{n}_{2}+...+{n}_{t}-t+1
objects are placed into t boxes, then for some i,
i=1,2,3,...,t
, the ith box contains at least
{n}_{i}
{26}^{2}×{10}^{4}
{}^{6}{P}_{2}
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1 Chemical Engineering Department, State University of Maringa, Maringá, Brazil.
2 Graduate Program of Food Science, Universidade Estadual de Maringá, Maringá, Brazil.
3 Pharmaceutical Sciences Department, State University of Maringa, Maringá, Brazil.
Abstract: The process described in the present work uses air supplementation in a fluidized bed reactor containing Bacillus firmus strain 37 immobilized on active bovine bone charcoal, to produce by batch fermentation the enzyme CGTase (cyclomaltodextrin-glucanotransferase). Three different aeration rates were evaluated. The maximum CGTase activity was achieved after 120 hours of fermentation with aeration rate of 2 vvm and was equal to 2.48 U/mL. When 0.5 and 1 vvm were used the enzymatic activities achieved 1.1 and 0.57 U/mL, respectively. Bovine bone charcoal was characterized in terms of surface area, pore size and volume. To the best of our knowledge, the immobilization of microorganism cells in bovine bone charcoal for CGTase production has not been reported in the literature. Our results showed that fluidized bed reactor allows retaining high concentration of biomass, improving biomass-substrate contact and operation at low residence times, which resulted in improved enzyme production. Therefore, the process as proposed has great potential for industrial development.
Keywords: Bacillus firmus, Bone Charcoal Matrix, Cell Immobilization, Microbial Enzyme, Cyclomaltodextrin Glucanotransferase
{C}_{\beta \text{-CD}}=a\left[1-\frac{ABS}{AB{S}_{o}}\right]\left[1+\frac{AB{S}_{o}}{akABS}\right]
{C}_{\beta \text{-CD}}=0.3000\left(1-0.6750\text{ABS}\right)\left(1+\frac{2.0442}{\text{ABS}}\right)
A=\left(K\cdot {V}_{R}\cdot D\right)/{V}_{E}
Cite this paper: Silva, L. , Bieli, B. , Junior, O. , Matioli, G. , Zanin, G. and Moraes, F. (2018) Bovine Bone Charcoal as Support Material for Immobilization of Bacillus firmus Strain 37 and Production of Cyclomaltodextrin Glucanotransferase by Batch Fermentation in a Fluidized Bed. Advances in Chemical Engineering and Science, 8, 11-25. doi: 10.4236/aces.2018.81002.
[1] Matioli, G., Moriwaki, C., Mazzoni, R.B., Zanin, G.M. and Moraes, F.F. (2000) Study of Parameters That Influence the Production of the Enzyme CGTase from Bacillus firmus, Strain no 37. Acta Scientiarum Biological Sciences, 22, 311-316.
[2] Moriwaki, C., Pelissari, F.M., Goncalves, R.A.C., Goncalves, J.E. and Matioli, G. (2007) Immobilization of Bacillus firmus Strain 37 in Inorganic Matrix for Cyclodextrin Production. Journal of Molecular Catalysis B: Enzymatic, 49, 1-7.
[3] Pinto, F.S.T., Flores, S.H., Ayub, M.A.Z. and Hertz, P.F. (2007) Production of Cyclodextrin Glycosyltransferase by Alkaliphilic Bacillus circulans in Submerged and Solid-State Cultivation. Bioprocess and Biosystems Engineering, 30, 377-382.
[4] Moriwaki, C., Ferreira, L.R., Rodella, L.R.T. and Matioli, G. (2009) A Novel Cyclodextrin Glycosyltransferase from Bacillus sphaericus Strain 41: Production, Characterization and Catalytic Properties. Biochemical Engineering Journal, 48, 124-131.
[5] Burhan, N., Sapundzhiev, T. and Beschkov, V. (2007) Mathematical Modelling of Cyclodextrin-Glucano-Transferase Production by Immobilised Cells of Bacillus circulans ATCC 21783 at Batch Cultivation. Biochemical Engineering Journal, 35, 114-119.
[6] Ai-Noi, S., Abd-Aziz, S., Alitheen, N., Hassan, O. and Karim, M.I.A. (2008) Optimization Cyclodextrin Glycosyltransferase Production by Response Surface Methodology Approach. Biotechnology, 7, 10-18.
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[8] Abdel-Naby, M.A., El-Refai, H.A. and Abdel-Fattah, A.F. (2011) Biosynthesis of Cyclodextrin Glucosyltransferase by the Free and Immobilized Cells of Bacillus cereus NRC7 in Batch and Continuous Cultures. Journal Applied Microbiology, 111, 1129-1137.
[9] Costa, G.L., Pazzetto, R., Brol, F. and Matioli, G. (2007) Metodologia de selecao de cepas para producao da ciclodextrina glicosiltransferase e para purificacao da enzima. Acta Scientiarum Health Sciences, 29, 45-50.
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[11] Martins, R.F., Delgado, O. and Hatti-Kaul, R. (2003) Sequence Analysis of Cyclodextrin Glycosyltransferase from the Alkaliphilic Bacillus agaradhaerens. Biotechnology Letters, 25, 1555-1562.
[12] Mazzer, C., Ferreira, L.R., Rodella, J.R.T., Moriwaki, C. and Matioli, G. (2008) Cyclodextrin Production by Bacillus firmus Strain 37 Immobilized on Inorganic Matrices and Alginate Gel. Biochemical Engineering Journal, 41, 79-86.
[13] Szejtli, J. (1988) Cyclodextrin Technology. Dordrecht, Netherlands.
[14] Kunamneni, A., Prabhakar, T., Jyothi, B. and Ellaiah, P. (2007) Investigation of Continuous Cyclodextrin Glucanotransferase Production by the Alginate-Immobilized Cells of alkalophilic Bacillus sp. in an Airlift Reactor. Enzyme and Microbial Technology, 40, 1538-1542.
[15] Cabello, P.E., Scognamiglio, F.P. and Terán, F.J.C. (2009) Vinasses Treatment in Anaerobic Fluidized Bed Reactor. Journal of Environmental Engineering Course, 6, 321-338.
[16] Carvalho, W., Canilha, L. and Silva, S.S. (2006) Uso de biocatalisadores imobilizados: Uma alternativa para a conducao de bioprocessos. Analytica Magazine, 23, 60-70.
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[18] Postma, J., Nijhuis, E.H. and Someus, E. (2010) Selection of Phosphorus Solubilizing Bacteria with Biocontrol Potential for Growth in Phosphorus Rich Animal Bone Charcoal. Applied Soil Ecology, 46, 464-469.
[19] Ngandwe, N. (2007) Toxicity of Animal Bone Charcoal from Pig and Cattle to Aquatic Bioassays: Vibrio Fischeri, Daphnia Magna and Selenastrum Capricornutum. Ph.D. Dissertation, Leipzig University, Leipzig.
[20] Freire, F.B. and Pires, E.C. (2004) Evaluation of the Material Support for Biomass in Fluidized Bed Reactors: Adhesion and Hydrodynamics. Magazine University Rural, Exact and Earth Sciences Series—Seropédica, 23, 34-43.
[21] Pazzeto, R., Delani, T.C.O., Fenelon, V.C. and Matioli, G. (2011) Cyclodextrin Production by Bacillus firmus Strain 37 Cells Immobilized on Loofa Sponge. Process Biochemistry, 46, 46-51.
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https://doi.org/10.1385/ABAB:84-86:1-9:1003
[26] Tardioli, P.W., Zanin, G.M. and Moraes, F.F. (2006) Characterization of Thermoanaerobacter Cyclomaltodextrin Glucanotransferase Immobilized on Glyoxil-Agarose. Enzyme and Microbial Technology, 39, 1270-1278.
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[32] Atanasova, N., Kitayska, T., Yankov, D., Safarikova, M. and Tonkova, A. (2009) Cyclodextrin Glucanotransferase Production by Cell Biocatalysts of Alkaliphilic Bacilli. Biochemical Engineering Journal, 46, 278-285.
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Second-order ODE involving two functions I am wondering how to find
adiadas8o7 2022-04-30 Answered
Second-order ODE involving two functions
I am wondering how to find a general analytical solution to the following ODE:
\frac{dy}{dt}\frac{{d}^{2}x}{{dt}^{2}}=\frac{dx}{dt}\frac{{d}^{2}y}{{dt}^{2}}
The solution method might be relatively simple; but right now I don't know how to approach this problem.
I find it more convenient to rewrite the equation using Newton's notation. Instead of writing
\frac{dx}{dt}
, it is more helpful to write x'. Thus, the equation is
{x}^{\prime }y{}^{″}=x{}^{″}{y}^{\prime }.
{x}^{\prime }=0
x0
is trivial, so every differentiable function
y\in {\mathbb{R}}^{\mathbb{R}}
satisfies the equation. This holds analogously if
{y}^{\prime }=0
. Otherwise, we can divide by x'y', thus
\frac{y{}^{″}}{{y}^{\prime }}=\frac{x{}^{″}}{{x}^{\prime }}
There are four cases to consider from here:
{x}^{\prime }<0\text{ }\text{and}\text{ }{y}^{\prime }<0;{x}^{\prime }<0\text{ }\text{and}\text{ }{y}^{\prime }>0;{x}^{\prime }>0\text{ }\text{and}\text{ }{y}^{\prime }<0
{x}^{\prime }>0\text{ }\text{and}\text{ }{y}^{\prime }>0
. These cases simplify the equation respectively
\mathrm{ln}\left(-{x}^{\prime }\right)+C=\mathrm{ln}\left(-{y}^{\prime }\right)
\mathrm{ln}\left(-{x}^{\prime }\right)+C=\mathrm{ln}\left({y}^{\prime }\right)
\mathrm{ln}\left({x}^{\prime }\right)+C=\mathrm{ln}\left(-{y}^{\prime }\right)
\mathrm{ln}\left({x}^{\prime }\right)+C=\mathrm{ln}\left({y}^{\prime }\right)
{e}^{C}{x}^{\prime }={y}^{\prime }
-{e}^{C}{x}^{\prime }={y}^{\prime }
-{e}^{C}{x}^{\prime }={y}^{\prime }
{e}^{C}{x}^{\prime }={y}^{\prime }
These cases simply reduce to
{y}^{\prime }=A{x}^{\prime }
A\ne 0
. Therefore, we have that
y\left(t\right)=Ax\left(t\right)+B
A\ne 0
. Remember that this is in the case when neither x nor y is a constant function.
\frac{dw}{dt}
\frac{dw}{dt}
w=x\mathrm{sin}y,\text{ }x={e}^{t},\text{ }y=\pi -t
t=0
Prove instability using Lyapunov function
{x}^{\prime }={x}^{3}+xy
{y}^{\prime }=-y+{y}^{2}+xy-{x}^{3}
I have to solve the following Cauchy's problem:
\left\{\begin{array}{rl}& {x}^{2}{x}^{\prime }={\mathrm{sin}}^{2}\left({x}^{3}-3t\right)\\ & x\left(0\right)=1\end{array}
x,dx+y,dy=\frac{{a}^{2}\left(x,dy-y,dx\right)}{{x}^{2}+{y}^{2}}
The author further proceeds to rearrange above in the form
M,dx+N,dy=0
M=x+\frac{{a}^{2}y}{{x}^{2}+{y}^{2}};N=y-\frac{{a}^{2}x}{{x}^{2}+{y}^{2}}
{M}_{y}={N}_{x}=\frac{{a}^{2}\left({x}^{2}-{y}^{2}\right)}{{\left({x}^{2}+{y}^{2}\right)}^{2}}
This is the required condition for the given equation to be exact and the solution is obtained using standard formula.
But, what I did is as follows:
x,dx+y,dy=\frac{{a}^{2}\left(x,dy-y,dx\right)}{{x}^{2}+{y}^{2}}
\to \left({x}^{2}+{y}^{2}\right)x,dx+\left({x}^{2}+{y}^{2}\right)y,dy={a}^{2}\left(x,dy-y,dx\right)
\left({x}^{3}+x{y}^{2}+{a}^{2}y\right),dx+\left({y}^{3}+y{x}^{2}-{a}^{2}x\right),dy=0
Comparing it with the equation
M,dx+N,dy=0
M={x}^{3}+x{y}^{2}+{a}^{2}y;N={y}^{3}+y{x}^{2}-{a}^{2}x
{M}_{y}=2xy+{a}^{2};{N}_{x}=2xy-{a}^{2}
Use the annihilator method and find the general solution of
y{}^{″}-3{y}^{\prime }+2y=4{\mathrm{sin}}^{3}\left(3x\right)
Trying to understand eigenvalues with respect to differential equations.
I am trying to understand how to find eigenvalues from a matrix consisting of exponential terms, considering a differential equation. The examples I've seen online are ODEs. Without using a vector with exponential terms, here is what I have learned.
\frac{d}{dt}\stackrel{\to }{x}\left(t\right)=\lambda \stackrel{\to }{x}\left(t\right)
\frac{d}{dt}\left[\begin{array}{c} {x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right]=\left[\begin{array}{ccc} {\lambda }_{1}& 0& 0\\ 0& {\lambda }_{2}& 0\\ 0& 0& {\lambda }_{3}\end{array}\right]\left[\begin{array}{c} {x}_{1}\left(t=0\right)\\ {x}_{2}\left(t=0\right)\\ {x}_{3}\left(t=0\right)\end{array}\right]
Here is what I am trying to understand
Section of a paper with imag eigenvalue
In this paper, the following assumption is made.
\stackrel{\to }{J}\left(t\right)=|\stackrel{\to }{J}|{e}^{-i\omega t}=\left[\begin{array}{c}|{J}_{x}|{e}^{-i\omega t}\\ |{J}_{y}|{e}^{-i\omega t}\\ |{J}_{z}|{e}^{-i\omega t}\end{array}\right]
They are using partial derivatives (I believe this can be viewed as an ODE then?). Differentiating with respect to time I believe yields the following. Please correct me if I am wrong.
\frac{\partial }{\partial t}\stackrel{\to }{J}\left(t\right)=\frac{\partial }{\partial t}\left[\begin{array}{c}|{J}_{x}|{e}^{-i\omega t}\\ |{J}_{y}|{e}^{-i\omega t}\\ |{J}_{z}|{e}^{-i\omega t}\end{array}\right]=\left[\begin{array}{ccc} -i\omega & 0& 0\\ 0& -i\omega & 0\\ 0& 0& -i\omega \end{array}\right]\left[\begin{array}{c}|{J}_{x}|{e}^{-i\omega t\left(t=0\right)}\\ |{J}_{y}|{e}^{-i\omega t\left(t=0\right)}\\ |{J}_{z}|{e}^{-i\omega t\left(t=0\right)}\end{array}\right]
\frac{\partial }{\partial t}\left[\begin{array}{c}|{J}_{x}|{e}^{-i\omega t}\\ |{J}_{y}|{e}^{-i\omega t}\\ |{J}_{z}|{e}^{-i\omega t}\end{array}\right]=\left[\begin{array}{ccc} -i\omega & 0& 0\\ 0& -i\omega & 0\\ 0& 0& -i\omega \end{array}\right]\left[\begin{array}{c}|{J}_{x}|\\ |{J}_{y}|\\ |{J}_{z}|\end{array}\right]
Are my assumptions correct? If so, is there a deeper analysis to why this is the case with exponential terms?
Find the number of unique solutions of the following nonlinear ODE
{y}^{\prime }=\frac{10}{3}x{y}^{\frac{25}{,}}\phantom{\rule{2em}{0ex}}y\left(0\right)=-1
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Mock test in proportion in Business Mathematics | Online Exam
Home Quantitative Aptitude proportion
Mock test in proportion in Business Mathematics of CA Foundation exams
two numbers are in the ratio of 2:3 and the difference of their squares is 320.The numbers are:
let numbers be 2x and 3x\phantom{\rule{0ex}{0ex}} \left(3x{\right)}^{2}–\left(2x{\right)}^{2}=320\phantom{\rule{0ex}{0ex}} 9{x}^{2}–4{x}^{2}=320\phantom{\rule{0ex}{0ex}} x=8\phantom{\rule{0ex}{0ex}}numbers are:2x=2*8=16\phantom{\rule{0ex}{0ex}} 3x=3*8=24
let numbers be 2x and 3x\phantom{\rule{0ex}{0ex}} \left(3x{\right)}^{2}–\left(2x{\right)}^{2}=320\phantom{\rule{0ex}{0ex}} 9{x}^{2}–4{x}^{2}=320\phantom{\rule{0ex}{0ex}} x=8\phantom{\rule{0ex}{0ex}}numbers are:2x=2*8=16\phantom{\rule{0ex}{0ex}} 3x=3*8=24
If p:q is the sub–duplicate ratio of p–{x}^{2}:q–{x}^{2},then {x}^{2} is:
qp/p-q
An alloy is to contain copper and zinc in the ratio 9:4.the zinc required to melt 24 kg of copper
here,copper:zinc=9x and 4x
then, 9x=24
x=24/9=8/3kg
so zinc=4x=4*8/3 kg =32/3 kg
two numbers are in the ratio 7:8.if 3 is added to each of them.their ratio becomes 8:9.the numbers are:
let the number be 7x and 8x
so,7x+3/8x+3=8/9
numbers are: 7x=7*3=21
A box contains Rs 56 in the form of coins of the one rupee,50 paise and 25 paise.The number of 50 paise coin is double the number of 25 paise coins and 4 times the number of one rupee coins.The numbers of 50 paise coins in the box is:
let the number of 1 rupee coins be x,50 paise coins be 4x and 25 paise coins be 2x
so,x+4x/2+2x/4=56
4x+8x+2x=56*4
no of 50 paise coins is 4*16=64
8 people are planning to share equally the cost of a rental car.if 1 person withdraws from the arrangement and the others share equally entire cost of the car,then the share of each of the remaining persons increased by:
when no of people=8
then,share of each person=1/8 of the total cost
increase in the share of a person=1/7-1/8=1/56
1/7 0f 1/8,i.e.1/7 of the original share of each person.
A bag contains Rs 187 in the form of 1 rupee,50 paise and 10 paise coins in the ratio 3:4:5.find the number of each type of coins
let the no of coins be 3x,4x and 5x
then,3x+4x/2+5x/10=187
30x+20x+5x=187*10
no of coins: 1 rupee=3x=3*34=102
50 paise=4x=4*34=136
Ratio of earnings of A and B is 4:7.if the earnings of A increase by 50% and those of B decrease by 25%.the new ratio of their earning becomes 8:7.what is A’s earning?
let the earning of A and B=4x and 7x.
new earning of A=4*150%=6x
new earning of B=7*75%=5.25x
then, 6x/5.25x=8/7
so,the data is inadequate
P,Q and R there cities.the ratio of average temperature between P and Q is 11:12 and P and R is 9:8.the ratio between the average temperature of Q and R is:
P/Q=11/12 and P/R=9/8
P/Q=11×9/12×9=99/108 and P/R=9×11/8×11=99/88
then,Q/R=108/88=27/22
so,Q:R=27:22
Rs 407 are to be divided among A,B and C so that their shares are in the ratio 1/4:1/5:1/6.the shares of A,B,c are:
here,A:B:C=1/4:1/5:1/6=15:12:10/60=15:12:10
A’s share =407*15/37=Rs165
B’s share =407*12/37=Rs132
C’s share =407*10/37=Rs110
The income of A and B are in the ratio 3:2 and their expenditure in the ratio 5:3.if each saves Rs1500,then B’s income is
let the income of A and B be 3x and 2x
expenditure of A and B be 5y and 3y
then, 3x-5y=1500……….(i)
2x-3y=1500………..(ii)
by solving i and ii we get
x=3000 and y=1500
hence,B’s income = 2x=2*3000=Rs6000
in 40 liters mixture glycerine and water are in the ratio of 3:1.the quantity of water added in the mixture in in order to make this ratio 2:1 is
Qty of glycerine = 40 x 3/4=30 liters
Qty of water = 40×1/4=10 liters
then,total qty of mixture=40+X liters
total qty of water = 10+X liters
so,30/10+X=2/1
X=5 liters of water must be added to the mixture
log 144=
2log4+2log2
3log2-4log3
log 144 = log(16×9)=log 16+log 9 =4 log 2+2 log 3
in what ratio should tea worth rs 10 per kg mixed with tea worth rs 14 per kg, so, that the average price of the mixture may be rs 11 per kg?
let X be qty of tea worth rs 10 per kg
Y be qty of tea worth 14 rs per kg
total , price of the mixture = 10X+14Y and qty of mixture=X+Y
average price=10X+14Y/X+Y=11
X/Y=3/1=3:1
two presons are in the ratio of 5:7. 18 yr ago there ages were in the ratio of 8:13, there present ages are
let present ages of person be 5X and 7X
18yr ago, there ages =5X-18 and 7X-18
then,5X-18/7X-18=8/13
65X-234=56X-144
there present ages are 5X = 5×10=50 yr and 7X =7×10=70
if A,B and C started a business by investing rs 126000, rs 84000 and rs 210000.if at the end of the year profit is rs 242000 then the share of each is
given, A:B:C=126000:84000:210000=126:84:210=3:2:5 and profit = rs 242000
A’s share = 3/10×242000=72600
B’s share = 2/10×242000=48400
C’s share = 5/10×242000=121000
if p/q=-2/3 then the value of 2p+q/2p-q is
given, p/q=-2/3 where p=-2q/3
now, 2p+q/2p-q = 2(-2q/3)+q/2(-2q/3)-q = -4q/3+q/-4q/3-q = -q/3×3/-7q= 1/7
4th proportional to x,2x,(x+1) is
let 4th proportional be x,2x and (x+1) be t
then,x/2x=x+1/t
1/2=x+t/t
4th proportional to x,2x,(x+1) is (2x+2) i.e,x:2x::(x+1):(2x+2)
log(m+n)=log m + log n,m can be expressed as
m=n/n-1
m=n/n+1
m=n+1/n
m=n+1/n-1
log(m+n)=log m + log n
log(m+n)=log(mn)
taking antilog on both side
antilog [log(m+n)]=antilog [log(mn)]
m+n=mn
mn-m=n
m(n-1)=n
what must be added to each term of the ratio 49:68, so that it becomes 3:4?
let the number added be x
49+x/68+x=3/4
196+4x=204+3x
The students of 2 classes are in the ratio of 5:7.if 10 students left from each class,then the remaining students are in the ratio of 4:6 then the number of students in each class is
let the ratio be 5x:7x
if 10 students left,ratio becomes 4:6
5x-10/7x-10=4/6
no of students in each class is 5x and 7x i.e.50 and 70
The recurring decimal 2.7777…… can be expressed as
2+0.7+0.07+0.007+……
2+[7/10+7/100+7/1000+…….]
2+7[1/10+1/100+1/1000+………]
2+7[1/10 / 1-1/10]
2+7×1/9
18+7/9=25/9
if a:b=2:5,then (10a+3b):(5a+2b) =
a/b=2/5=2k/5k
10a+3b/5a+2b = 20k+15k/10k+10k = 35k/20k = 35/20 = 7/4 = 7:4
which of the numbers are not in proportion?
if say a,b,c,d are in proportion they bear a common ratio that is a/b=c/d
(A) 6/8 is not equal to 5/7
(B) 7/3=14/6
(c)18/27=12/18
(D)8/6=12/9
which of the following is true if 1/ab+1/bc+1/ca=1/abc
log(ab+bc+ca)=abc
log(1/a+1/b+1/c)=abc
log(abc)=0
log(a+b+c)=0
given,1/ab+1/bc+1/ca=1/abc
c+a+b/abc=1/abc
log(a+b+c)=log1
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Design of Novel Catheter Insertion Device | J. Med. Devices | ASME Digital Collection
Erik K. Bassett,
, Cambridge, USA
Bassett, E. K., and Slocum, A. (June 25, 2008). "Design of Novel Catheter Insertion Device." ASME. J. Med. Devices. June 2008; 2(2): 027558. https://doi.org/10.1115/1.2936119
Poor positioning of needles and catheters may result in repeated attempts at correct placement, injury to adjacent structures or infusions into inappropriate spaces. Existing catheter insertion methods do not uniformly provide feedback of the tip location, nor prevent the needle from going beyond the target space. The purpose of this research was to develop a design tool to be used to create a new catheter insertion device. This device would advance a needle in firm tissue but automatically release it upon entrance into the desired space. The system studied consisted of a flexible filament (OD
∼0.9mm
) in compression passing through a tube (ID
1.22mm
) with both straight and curved sections. A mathematical model based on oil drilling methods was developed to predict the compressive force dissipated in the filament for any given tube geometry. A correction factor on one of the two terms in the model was necessary to achieve best results, but proved to be accurate for all
100+
tests completed. With it, this model accounted for the following parameters: Angular displacement of tube bends, radial clearance, coefficient of friction, lengths, tube and filament radii, number of bends, moment of inertia, and modulus of elasticity. Implementation of this model should allow for a more safe and effective catheter insertion device.
biological tissues, biomechanics, catheters, elastic moduli, friction, needles, oil drilling
Biological tissues, Catheters, Design, Elastic moduli, Friction, Needles, Oil well drilling
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Gases - Vocabulary - Course Hero
General Chemistry/Gases/Vocabulary
temperature as compared to absolute zero, the cessation of all motion, even subatomic motion, measured in kelvins (K)
unit of pressure (force per unit area) equal to 760 mm Hg. The average atmospheric pressure at sea level is defined as 1 atmosphere.
law that states that equal volumes of gases at the same temperature and pressure have equal numbers of atoms or molecules, represented by the equation
\frac{V_1}{n_1}=\frac{V_2}{n_2}
unit of pressure equal to 100,000 Pa
open tube filled with liquid such as mercury and sealed at the other end under a vacuum, used to measure atmospheric pressure (of gas, in mm Hg)
law that states that pressure of gas increases as volume decreases at constant temperature and moles of gas, represented by the equation
P_1V_1=P_2V_2
law that states that the volume of gas increases as temperature increases at constant pressure and moles of gas, represented by the equation
\frac{V_1}{T_1}=\frac{V_2}{T_2}
describes how much a gas's behavior differs from ideal
law that states that the total pressure of a mixture of ideal and nonreacting gases is the sum of the partial pressures of the individual gases
process by which gas moves from an area of higher concentration to an area of lower concentration
process by which gases move through small openings in solids, one particle at a time
law that states that for an ideal gas with constant mass and volume, the pressure exerted on the container is proportional to its absolute temperature, represented by the equation
\frac{P_1}{T_1}=\frac{P_2}{T_2}
law that states that the rate of effusion of a gas is inversely proportional to the square root of its mass, represented by the equation
\frac{{\rm{rate}}_1}{{\rm{rate}}_2}=\sqrt{\frac{M_2}{M_1}}
C=kP_{\rm{gas}}
Solubility and Pressure
theoretical gas in which no forces are acting on the gas particles, and the particles do not take up space
law that describes the behavior of ideal gases, represented by the equation
PV=nRT
theory involving the relationship between temperature, pressure, and volume that states that the average kinetic energy of a gas is proportional to its temperature
instrument that measures gas pressure (in mm Hg) via a tube of liquid that is open at both ends
average distance that a gas particle travels before colliding with another gas particle
concentration expressed as the moles of solvent divided by the total number of all moles in a solution
pressure of an ideal gas that contributes to the total pressure of a mixture of gases at constant temperature
SI unit of gas pressure
unit that describes the pounds of force applied to a square inch (area) of a container
force applied perpendicular to a unit area of surface
1 atmosphere (atm) pressure and 0°C (273.15 K) temperature
absolute unit of pressure, defined as 1/760 atmosphere (atm)
equation that accounts for intermolecular (nonideal) interactions between gases. It adjusts the ideal gas law to explain and predict the behavior of real gases.
<Overview>Properties of Gases
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