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"Chebyshev polynomials and CR mappings"

Rift-Effex Glass Distortion, unSharp, Color Polynomial, Gamma Adj

Algebra 2 students work on a collaborative graphing polynomials investigation

polynomial functions.

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 1080 / length of data axis 1

NAXIS2 = 1920 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

PROGRAM = 'Siril 1.4.0-beta2' / Software that created this HDU

DATE = '2025-05-31T22:28:35' / UTC date that FITS file was created

DATE-OBS= '2025-05-31T09:45:01.686455' / YYYY-MM-DDThh🇲🇲ss observation start,

IMAGETYP= 'Light ' / Type of image

EXPTIME = 20. / [s] Exposure time duration

TELESCOP= 'S50_123896fd' / Telescope used to acquire this image

FILTER = 'LP ' / Active filter name

APERTURE= 5. / Aperture of the instrument

FOCALLEN= 252.304 / [mm] Focal length

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 2.9 / [um] Pixel X axis size

YPIXSZ = 2.9 / [um] Pixel Y axis size

INSTRUME= 'Seestar S50' / Instrument name

CCD-TEMP= 19.625 / [degC] CCD temperature

GAIN = 80 / Sensor gain

BAYERPAT= 'GRBG ' / Bayer color pattern

FOCPOS = 1755 / [step] Focuser position

STACKCNT= 421 / Stack frames

OBJECT = 'M 27 ' / Name of the object of interest

SITELAT = 35.7786 / [deg] Observation site latitude

SITELONG= -86.8795 / [deg] Observation site longitude

OBJCTRA = '19 59 32.780' / [H M S] Image center Right Ascension

OBJCTDEC= '+22 41 41.197' / [D M S] Image center Declination

RA = 299.886583530977 / [deg] Image center Right Ascension

DEC = 22.6947769216981 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 2000. / Equatorial equinox

CRPIX1 = 540.5 / Axis1 reference pixel

CRPIX2 = 960.5 / Axis2 reference pixel

CRVAL1 = 299.886583530977 / [deg] Axis1 reference value

CRVAL2 = 22.6947769216981 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000658450437231544 / [deg] X pixel size

CDELT2 = 0.000658675224173079 / [deg] Y pixel size

PC1_1 = 0.999999621326472 / Linear transformation matrix (1, 1)

PC1_2 = 0.000870256808540633 / Linear transformation matrix (1, 2)

PC2_1 = 0.000927272136576244 / Linear transformation matrix (2, 1)

PC2_2 = -0.9999995700831 / Linear transformation matrix (2, 2)

A_ORDER = 3 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = -3.75551074224103E-07

A_1_1 = 4.70621980131655E-08

A_0_2 = -6.46199740980993E-08

A_3_0 = -2.06414460548833E-10

A_2_1 = 2.28072188194465E-12

A_1_2 = -1.18426127608931E-10

A_0_3 = -4.75443001579988E-12

B_ORDER = 3 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = 2.33528382502947E-08

B_1_1 = -2.7699444791217E-07

B_0_2 = 1.11543105135615E-08

B_3_0 = -3.0817154867509E-10

B_2_1 = -3.15067124969027E-11

B_1_2 = -1.35065308183837E-11

B_0_3 = -3.8943227939089E-11

AP_ORDER= 3 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = -1.2935141754458E-05

AP_1_0 = 7.47422390645625E-09

AP_0_1 = -1.83851387227512E-09

AP_2_0 = 3.75737040541087E-07

AP_1_1 = -4.70688526910942E-08

AP_0_2 = 6.46542348013858E-08

AP_3_0 = 2.06714915740501E-10

AP_2_1 = -2.32252533027838E-12

AP_1_2 = 1.18522521237245E-10

AP_0_3 = 4.75228000034824E-12

BP_ORDER= 3 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = -5.74380370311076E-06

BP_1_0 = -1.00036284898605E-08

BP_0_1 = 2.44613751476663E-09

BP_2_0 = -2.32079759435774E-08

BP_1_1 = 2.77052406094745E-07

BP_0_2 = -1.11498937265249E-08

BP_3_0 = 3.08253605705153E-10

BP_2_1 = 3.16804835338791E-11

BP_1_2 = 1.35037926197768E-11

BP_0_3 = 3.89635372178474E-11

PLTSOLVD= T / Siril internal solver

HISTORY Plate Solve

HISTORY Photometric CC (algorithm: SPCC)

HISTORY Histogram Transf. (mid=0.105, lo=0.008, hi=1.000)

HISTORY Histogram Transf. (mid=0.060, lo=0.014, hi=1.000)

HISTORY Histogram Transf. (mid=0.129, lo=0.024, hi=1.000)

HISTORY Histogram Transf. (mid=0.436, lo=0.000, hi=1.000)

HISTORY Saturation enhancement (amount=0.25)

HISTORY NL-Bayes denoise (mod=1.000, CC enabled)

CREATOR = 'ZWO Seestar S50' / Capture software

PRODUCER= 'ZWO ' / Powered by ZWO

XORGSUBF= 0 / Subframe X position in binned pixels

YORGSUBF= 0 / Subframe Y position in binned pixels

EQMODE = 1 / Equatorial mode

CCDXBIN = 1 / Camera X Bin

CCDYBIN = 1 / Camera Y Bin

TOTALEXP= 8420. / Total Exposure Time in seconds

IMAGEW = 1080 / Image width, in pixels.

IMAGEH = 1920 / Image height, in pixels.

BG-EXTR = 'GraXpert'

CBG-1 = 0.0095570935

CBG-2 = 0.0095570935

CBG-3 = 0.0095570935

INTP-OPT= 'AI '

HIERARCH SMOOTHING = 0.5

HIERARCH CORR-TYPE = 'Subtraction'

HIERARCH BGE-AI-VER = '1.0.1 '

END

 

"Isolated hypersurface singularities and polynomial realizations of affine quadrics"

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 3233 / length of data axis 1

NAXIS2 = 2381 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

PROGRAM = 'Siril 1.4.0-beta2' / Software that created this HDU

DATE = '2025-05-30T15:34:34' / UTC date that FITS file was created

DATE-OBS= '2025-05-29T22:51:11.440000' / YYYY-MM-DDThh🇲🇲ss observation start,

IMAGETYP= 'LIGHT ' / Type of image

ROWORDER= 'BOTTOM-UP' / Order of the rows in image array

EXPTIME = 180. / [s] Exposure time duration

TELESCOP= 'Evolux62ED + red 0,9X' / Telescope used to acquire this image

FOCALLEN= 362.746 / [mm] Focal length

CENTALT = 46.5351886177 / [deg] Altitude of telescope

CENTAZ = 328.529153054 / [deg] Azimuth of telescope

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 4.80000019073486 / [um] Pixel X axis size

YPIXSZ = 4.80000019073486 / [um] Pixel Y axis size

INSTRUME= 'QHY168C ' / Instrument name

CCD-TEMP= -10.1 / [degC] CCD temperature

SET-TEMP= -10. / [degC] CCD temperature setpoint

GAIN = 10 / Sensor gain

OFFSET = 50 / Sensor gain offset

STACKCNT= 50 / Stack frames

LIVETIME= 9000. / [s] Exposure time after deadtime correction

EXPSTART= 2460825.44591554 / [JD] Exposure start time (standard Julian date)

EXPEND = 2460825.58988351 / [JD] Exposure end time (standard Julian date)

OBJECT = 'M 81 ' / Name of the object of interest

AIRMASS = 1.37656926256564 / Airmass at frame center (Gueymard 1993)

SITELAT = 47.8461111111111 / [deg] Observation site latitude

SITELONG= 2.03333333333333 / [deg] Observation site longitude

SITEELEV= 100. / [m] Observation site elevation

OBJCTRA = '09 56 28.991' / [H M S] Image center Right Ascension

OBJCTDEC= '+69 13 39.535' / [D M S] Image center Declination

RA = 149.120797829765 / [deg] Image center Right Ascension

DEC = 69.2276487128759 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 2000. / Equatorial equinox

CRPIX1 = 1398.5 / Axis1 reference pixel

CRPIX2 = 1324.5 / Axis2 reference pixel

CRVAL1 = 149.232400331936 / [deg] Axis1 reference value

CRVAL2 = 69.4176747680329 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.00075846636598721 / [deg] X pixel size

CDELT2 = 0.000757856678281084 / [deg] Y pixel size

PC1_1 = 0.684319883532855 / Linear transformation matrix (1, 1)

PC1_2 = 0.729181936831666 / Linear transformation matrix (1, 2)

PC2_1 = -0.72912939960989 / Linear transformation matrix (2, 1)

PC2_2 = 0.684375860638379 / Linear transformation matrix (2, 2)

A_ORDER = 3 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = 4.88282691352465E-07

A_1_1 = -4.84143767300149E-08

A_0_2 = 2.47902930496684E-07

A_3_0 = -9.38902752301354E-10

A_2_1 = -2.60908941548892E-11

A_1_2 = -9.59812714947483E-10

A_0_3 = -2.43623482638473E-10

B_ORDER = 3 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = -5.55579050606427E-08

B_1_1 = 2.85841832834249E-07

B_0_2 = -1.71345239101971E-07

B_3_0 = -5.37968508068464E-11

B_2_1 = -1.06440014406305E-09

B_1_2 = -2.85233836657898E-12

B_0_3 = -8.98180265970671E-10

AP_ORDER= 3 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = 0.0128303089794321

AP_1_0 = -5.52758456159319E-05

AP_0_1 = -1.03930010898326E-05

AP_2_0 = -5.0466446282257E-07

AP_1_1 = 5.04489272149756E-08

AP_0_2 = -2.54046104525162E-07

AP_3_0 = 9.65790534423701E-10

AP_2_1 = 3.06235131352588E-11

AP_1_2 = 9.91309560570671E-10

AP_0_3 = 2.5020950408452E-10

BP_ORDER= 3 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = -0.0028162775575758

BP_1_0 = -5.58511609164564E-06

BP_0_1 = -4.20740732082203E-05

BP_2_0 = 5.68390105826947E-08

BP_1_1 = -2.9435324378723E-07

BP_0_2 = 1.74875324548103E-07

BP_3_0 = 5.59034993100432E-11

BP_2_1 = 1.09476749115569E-09

BP_1_2 = 6.72664307213777E-12

BP_0_3 = 9.19144264237707E-10

DATE-LOC= '2025-05-30T00:51:11.440' / Time of observation (local)

MJD-OBS = 60824.9522157409 / Modified Julian Date of observation

DATE-AVG= '2025-05-29T22:52:40.986' / Averaged midpoint time (UTC)

MJD-AVG = 60824.953252153 / Modified Julian Date of averaged midpoint

CAMERAID= 'QHY168C-f3c29947a8d19c337' / Imaging instrument identifier

USBLIMIT= 0 / Camera-specific USB setting

FOCRATIO= 58.0 / Focal ratio

PIERSIDE= 'East' / Telescope pointing state

OBJCTROT= 312.29 / [deg] planned rotation of imaged object

ROTNAME = 'Manual Rotator' / Rotator equipment name

ROTATOR = 312.086853027344 / [deg] Mechanical rotator angle

ROTATANG= 312.086853027344 / [deg] Mechanical rotator angle

ROTSTPSZ= 0.0 / [deg] Rotator step size

END

 

"The protean chromatic polynomial"

"The protean chromatic polynomial"

"The protean chromatic polynomial"

"Ruling polynomials and augmentations over finite fields"

"Ruling polynomials and augmentations over finite fields"

"Rational and polynomial density on compact real manifolds"

"Rational and polynomial density on compact real manifolds"

Histograms demonstrating the calculation of the R iso, sky (E, L) function. Top left: amount of time Fermi spent while at each L bin; top

right: number of events in the

θ< 50◦subset of the isotropic-component data set detected while Fermi was inside each of the L bins; bottom: ratio

of these two histograms showing the detection rate of such events while at some

L bin. We fit the rate histogram with a fifth degree polynomial,

shown with the black curve. The histograms shown here correspond to log-energy bin #1 (energy≃100 MeV).

Polynomial Diffusion Models for Life Insurance Liabilities. Biagini, Zhang arxiv.org/abs/1602.07910 #q-fin

"Chebyshev polynomials and CR mappings"

"Chebyshev polynomials and CR mappings"

We did this lame shit in Algebra II with graphing of polynomials, reducing them to prime factors, and then graphing them. This was my response.

"The protean chromatic polynomial"

"The protean chromatic polynomial"

"The protean chromatic polynomial"

Polynomial term structure models. Cheng, Tehranchi arxiv.org/abs/1504.03238 #q-fin

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 6036 / length of data axis 1

NAXIS2 = 4020 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

PROGRAM = 'Siril 1.4.0-beta4' / Software that created this HDU

FILENAME= 'img_0011.nef'

DATE = '2025-11-01T10:06:59' / UTC date that FITS file was created

DATE-OBS= '2017-08-20T22:39:36' / YYYY-MM-DDThh🇲🇲ss observation start, UT

ROWORDER= 'BOTTOM-UP' / Order of the rows in image array

EXPTIME = 123. / [s] Exposure time duration

ISOSPEED= 250. / ISO camera setting

FOCALLEN= 1158.489 / [mm] Focal length

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 5. / [um] Pixel X axis size

YPIXSZ = 5. / [um] Pixel Y axis size

INSTRUME= 'Nikon D5200' / Instrument name

STACKCNT= 78 / Stack frames

LIVETIME= 9529.5998916626 / [s] Exposure time after deadtime correction

EXPSTART= 2457986.43081019 / [JD] Exposure start time (standard Julian date)

EXPEND = 2457986.61906943 / [JD] Exposure end time (standard Julian date)

OBJCTRA = '00 42 26.189' / [H M S] Image center Right Ascension

OBJCTDEC= '+41 19 38.552' / [D M S] Image center Declination

RA = 10.6091190843413 / [deg] Image center Right Ascension

DEC = 41.3273755778947 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 2000. / Equatorial equinox

CRPIX1 = 3018.5 / Axis1 reference pixel

CRPIX2 = 2010.5 / Axis2 reference pixel

CRVAL1 = 10.6091190843413 / [deg] Axis1 reference value

CRVAL2 = 41.3273755778947 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000247277488832208 / [deg] X pixel size

CDELT2 = 0.000247295686311149 / [deg] Y pixel size

PC1_1 = 0.723517411000228 / Linear transformation matrix (1, 1)

PC1_2 = 0.690306132074406 / Linear transformation matrix (1, 2)

PC2_1 = -0.690449231186407 / Linear transformation matrix (2, 1)

PC2_2 = 0.723380853461094 / Linear transformation matrix (2, 2)

A_ORDER = 3 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = 2.60771910752349E-08

A_1_1 = -1.86634136428528E-08

A_0_2 = 4.06674834757682E-08

A_3_0 = -6.05931488148592E-10

A_2_1 = 2.56883210469854E-11

A_1_2 = -6.53524530828069E-10

A_0_3 = 7.25083282937301E-11

B_ORDER = 3 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = 1.50276602598993E-07

B_1_1 = -1.6434617742354E-08

B_0_2 = -1.98369209680245E-08

B_3_0 = 3.90050019482281E-11

B_2_1 = -5.74875020546825E-10

B_1_2 = -4.68060214532268E-11

B_0_3 = -5.39125306836144E-10

AP_ORDER= 3 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = 0.00466628178056705

AP_1_0 = -5.77783443298507E-05

AP_0_1 = 2.46590735907731E-06

AP_2_0 = -2.7642869210866E-08

AP_1_1 = 1.86261052482085E-08

AP_0_2 = -4.20836666275435E-08

AP_3_0 = 6.23033563027215E-10

AP_2_1 = -2.72709576667416E-11

AP_1_2 = 6.74726273410097E-10

AP_0_3 = -7.40353685585022E-11

BP_ORDER= 3 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = 0.00238839704565799

BP_1_0 = 7.02212940738368E-08

BP_0_1 = -3.74149093612663E-05

BP_2_0 = -1.52984415511728E-07

BP_1_1 = 1.54068935989443E-08

BP_0_2 = 1.95356304418592E-08

BP_3_0 = -3.98923175670779E-11

BP_2_1 = 5.90706303502153E-10

BP_1_2 = 4.75457974444921E-11

BP_0_3 = 5.5121324303925E-10

PLTSOLVD= T / Siril internal solver

HISTORY Extraction du gradient (Correction : Subtraction)

HISTORY R solution astrom trique

HISTORY Photom trie CC (algorithme : PCC)

HISTORY D convolution

HISTORY Transf. histogramme (mid=0.001, lo=0.100, hi=1.000)

HISTORY SCNR (type=neutre moyen, qt =1.00, pr serve=true)

HISTORY Rehaussement de la saturation (quantit =0.25)

END

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