View allAll Photos Tagged polynomials

"The zeros of random harmonic polynomials"

The Mandelbrot Set M is defined by a family of complex quadratic polynomials. Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by coloring all the points c which belong to M black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to how quickly or slowly the sequence diverges to infinity.

 

Learn more about the Mandelbrot Set

Watch a video of a computer zooming into the Mandelbrot Set

"Extensions of the notions of polynomial and rational hull"

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 1689 / length of data axis 1

NAXIS2 = 3166 / 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

MIPS-HI = 65535 / Lower visualization cutoff

MIPS-LO = 21711 / Upper visualization cutoff

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

FILENAME= 'Light_mosaic_NGC 7000_20.0s_LP_20250526-014127.fit'

DATE = '2025-05-26T16:24:15' / UTC date that FITS file was created

DATE-OBS= '2025-05-26T06:41:02.150919' / YYYY-MM-DDThh🇲🇲ss observation start,

IMAGETYP= 'Light ' / Type of image

EXPTIME = 20. / [s] Exposure time duration

TELESCOP= 'S50_f1c10930' / Telescope used to acquire this image

FILTER = 'mixed ' / Active filter name

APERTURE= 5. / Aperture of the instrument

FOCALLEN= 251.83 / [mm] Focal length

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 2.90000009536743 / [um] Pixel X axis size

YPIXSZ = 2.90000009536743 / [um] Pixel Y axis size

INSTRUME= 'Seestar S50' / Instrument name

CCD-TEMP= 17.0625 / [degC] CCD temperature

GAIN = 80 / Sensor gain

FOCPOS = 1620 / [step] Focuser position

STACKCNT= 197 / Stack frames

LIVETIME= 3940. / [s] Exposure time after deadtime correction

EXPSTART= 2460821.77849711 / [JD] Exposure start time (standard Julian date)

EXPEND = 2460821.87932888 / [JD] Exposure end time (standard Julian date)

OBJECT = 'NGC 7000' / Name of the object of interest

SITELAT = 41.756 / [deg] Observation site latitude

SITELONG= -88.1819 / [deg] Observation site longitude

OBJCTRA = '20 58 7.346' / [H M S] Image center Right Ascension

OBJCTDEC= '+44 19 23.624' / [D M S] Image center Declination

RA = 314.530606961891 / [deg] Image center Right Ascension

DEC = 44.3232288329815 / [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 = 895.5 / Axis1 reference pixel

CRPIX2 = 1141.5 / Axis2 reference pixel

CRVAL1 = 314.592106580487 / [deg] Axis1 reference value

CRVAL2 = 44.0330294379358 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000659802746654457 / [deg] X pixel size

CDELT2 = 0.000659800551565207 / [deg] Y pixel size

PC1_1 = 0.965297511245702 / Linear transformation matrix (1, 1)

PC1_2 = 0.261152665662929 / Linear transformation matrix (1, 2)

PC2_1 = -0.261115398307382 / Linear transformation matrix (2, 1)

PC2_2 = 0.965307592825612 / 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 = -9.74063439851947E-09

A_1_1 = -3.25366176150845E-10

A_0_2 = 2.66465106494466E-10

A_3_0 = -1.20153434446517E-11

A_2_1 = 1.80289254965373E-12

A_1_2 = 1.52021593952749E-11

A_0_3 = 2.52566304775469E-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 = -2.7859943045886E-09

B_1_1 = -8.30877626644875E-09

B_0_2 = 2.54422083612868E-08

B_3_0 = 1.02803637639594E-11

B_2_1 = 2.06197952424513E-11

B_1_2 = -4.20223998544153E-12

B_0_3 = 5.35717259136597E-12

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

AP_0_0 = -8.83407116017409E-07

AP_1_0 = -2.11171234523969E-09

AP_0_1 = -1.8693611876812E-09

AP_2_0 = 9.74152551922136E-09

AP_1_1 = 3.28176321156856E-10

AP_0_2 = -2.65501037894396E-10

AP_3_0 = 1.2017170739467E-11

AP_2_1 = -1.80181631366394E-12

AP_1_2 = -1.52001458940546E-11

AP_0_3 = -2.52483359393872E-12

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

BP_0_0 = -4.50129317174452E-06

BP_1_0 = -8.89802043462381E-10

BP_0_1 = -2.38290078088729E-09

BP_2_0 = 2.78816732715415E-09

BP_1_1 = 8.305737175103E-09

BP_0_2 = -2.54374095652426E-08

BP_3_0 = -1.0279738664855E-11

BP_2_1 = -2.0616406271359E-11

BP_1_2 = 4.20177218037046E-12

BP_0_3 = -5.35485004051757E-12

HISTORY SCNR (type=average neutral, amount=1.00, preserve=true)

HISTORY Saturation enhancement (amount=0.23)

HISTORY Saturation enhancement (amount=0.42)

HISTORY Saturation enhancement (amount=0.30)

HISTORY Saturation enhancement (amount=0.42)

HISTORY Saturation enhancement (amount=0.64)

BG-EXTR = 'GraXpert'

CBG-1 = 0.0050184443

CBG-2 = 0.0050184443

CBG-3 = 0.0050184443

INTP-OPT= 'RBF '

HIERARCH SMOOTHING = 0.9

HIERARCH CORR-TYPE = 'Subtraction'

HIERARCH SAMPLE-SIZE = 15

HIERARCH RBF-KERNEL = 'thin_plate'

HIERARCH SPLINE-ORDER = 3

BG-PTS = '[[77, 77, 1], [262, 107, 1], [447, 77, 1], [632, 77, 1], [817, &'

CONTINUE '107, 1], [1017, 92, 1], [1202, 92, 1], [1387, 92, 1], [1587, 77, &'

CONTINUE '1], [1742, 107, 1], [1957, 77, 1], [2142, 77, 1], [2297, 107, 1], &'

CONTINUE '[2497, 92, 1], [2682, 92, 1], [107, 262, 1], [292, 262, 1], [477, &'

CONTINUE '262, 1], [632, 262, 1], [847, 292, 1], [1032, 262, 1], [1217, 262, &'

CONTINUE '1], [1402, 292, 1], [1557, 292, 1], [1742, 262, 1], [1927, 262, &'

CONTINUE '1], [2142, 292, 1], [2327, 292, 1], [2497, 277, 1], [2682, 277, &'

CONTINUE '1], [107, 447, 1], [292, 477, 1], [462, 462, 1], [662, 477, 1], &'

CONTINUE '[832, 462, 1], [1002, 447, 1], [1187, 477, 1], [1372, 477, 1], &'

CONTINUE '[1587, 447, 1], [1742, 447, 1], [1957, 477, 1], [2112, 477, 1], &'

CONTINUE '[2327, 477, 1], [2497, 462, 1], [2682, 462, 1], [77, 632, 1], &'

CONTINUE '[262, 662, 1], [447, 632, 1], [632, 662, 1], [847, 632, 1], [1002, &'

CONTINUE '632, 1], [1402, 662, 1], [1557, 662, 1], [1742, 632, 1], [1927, &'

CONTINUE '662, 1], [2127, 647, 1], [2312, 647, 1], [2497, 647, 1], [2682, &'

CONTINUE '647, 1], [107, 847, 1], [277, 832, 1], [477, 847, 1], [662, 817, &'

CONTINUE '1], [817, 847, 1], [1032, 847, 1], [1217, 847, 1], [1402, 847, 1], &'

CONTINUE '[1557, 847, 1], [1742, 847, 1], [1927, 847, 1], [2142, 847, 1], &'

CONTINUE '[2297, 817, 1], [2512, 817, 1], [2682, 832, 1], [107, 1002, 1], &'

CONTINUE '[292, 1002, 1], [477, 1002, 1], [632, 1002, 1], [832, 1017, 1], &'

CONTINUE '[1002, 1002, 1], [1217, 1002, 1], [1402, 1002, 1], [1587, 1002, &'

CONTINUE '1], [1772, 1002, 1], [1957, 1032, 1], [2142, 1032, 1], [2327, &'

CONTINUE '1002, 1], [2512, 1002, 1], [2682, 1017, 1], [107, 1217, 1], [292, &'

CONTINUE '1217, 1], [447, 1217, 1], [632, 1217, 1], [847, 1217, 1], [1002, &'

CONTINUE '1187, 1], [1187, 1217, 1], [1372, 1217, 1], [1587, 1217, 1], &'

CONTINUE '[1772, 1187, 1], [1927, 1217, 1], [2112, 1187, 1], [2297, 1187, &'

CONTINUE '1], [2482, 1187, 1], [2682, 1202, 1], [107, 1372, 1], [262, 1372, &'

CONTINUE '1], [447, 1372, 1], [632, 1402, 1], [817, 1372, 1], [1002, 1372, &'

CONTINUE '1], [1187, 1372, 1], [1587, 1372, 1], [1772, 1402, 1], [1927, &'

CONTINUE '1372, 1], [2142, 1402, 1], [2297, 1402, 1], [2482, 1402, 1], &'

CONTINUE '[2682, 1387, 1], [77, 1557, 1], [262, 1557, 1], [447, 1587, 1], &'

CONTINUE '[662, 1587, 1], [817, 1557, 1], [1032, 1557, 1], [1372, 1557, 1], &'

CONTINUE '[1557, 1557, 1], [1742, 1587, 1], [1927, 1557, 1], [2112, 1587, &'

CONTINUE '1], [2297, 1557, 1], [2482, 1587, 1], [2682, 1572, 1], [77, 1772, &'

CONTINUE '1], [262, 1772, 1], [462, 1757, 1], [662, 1772, 1], [1187, 1742, &'

CONTINUE '1], [1400.5570032573287, 1763.0032573289905, 1.0], [1587, 1772, &'

CONTINUE '1], [1742, 1742, 1], [1927, 1742, 1], [2142, 1742, 1], [2297, &'

CONTINUE '1742, 1], [2482, 1742, 1], [2682, 1757, 1], [77, 1957, 1], [292, &'

CONTINUE '1927, 1], [447, 1927, 1], [632, 1927, 1], [1217, 1957, 1], [1402, &'

CONTINUE '1927, 1], [1587, 1957, 1], [1772, 1957, 1], [1957, 1957, 1], &'

CONTINUE '[2142, 1957, 1], [2297, 1927, 1], [2482, 1957, 1], [2682, 1942, &'

CONTINUE '1], [77, 2142, 1], [277, 2127, 1], [447, 2112, 1], [662, 2112, 1], &'

CONTINUE '[817, 2142, 1], [1217, 2112, 1], [1372, 2142, 1], [1557, 2112, 1], &'

CONTINUE '[1742, 2142, 1], [1957, 2142, 1], [2112, 2142, 1], [2297, 2142, &'

CONTINUE '1], [2482, 2112, 1], [2682, 2127, 1], [77, 2297, 1], [292, 2327, &'

CONTINUE '1], [447, 2297, 1], [662, 2297, 1], [847, 2327, 1], [1032, 2297, &'

CONTINUE '1], [1402, 2327, 1], [1557, 2327, 1], [1742, 2327, 1], [1957, &'

CONTINUE '2297, 1], [2112, 2297, 1], [2327, 2297, 1], [2482, 2297, 1], &'

CONTINUE '[2682, 2312, 1], [92, 2497, 1], [292, 2482, 1], [477, 2482, 1], &'

CONTINUE '[662, 2512, 1], [847, 2512, 1], [1032, 2512, 1], [1402, 2482, 1], &'

CONTINUE '[1557, 2482, 1], [1772, 2512, 1], [1927, 2482, 1], [2112, 2482, &'

CONTINUE '1], [2312, 2497, 1], [2512, 2482, 1], [2667, 2512, 1], [92, 2682, &'

CONTINUE '1], [292, 2697, 1], [477, 2667, 1], [632, 2697, 1], [847, 2667, &'

CONTINUE '1], [1002, 2697, 1], [1202, 2682, 1], [1372, 2667, 1], [1587, &'

CONTINUE '2697, 1], [1957, 2697, 1], [2142, 2667, 1], [2327, 2667, 1], &'

CONTINUE '[2497, 2682, 1], [2697, 2667, 1], [92, 2867, 1], [292, 2852, 1], &'

CONTINUE '[462, 2867, 1], [647, 2867, 1], [817, 2852, 1], [1002, 2882, 1], &'

CONTINUE '[1187, 2882, 1], [1372, 2852, 1], [2112, 2882, 1], [2312, 2867, &'

CONTINUE '1], [2482, 2852, 1], [2697, 2882, 1], [92, 3052, 1], [262, 3037, &'

CONTINUE '1], [477, 3067, 1], [662, 3067, 1], [817, 3037, 1], [1032, 3067, &'

CONTINUE '1], [1772, 3067, 1], [1942, 3052, 1], [2142, 3037, 1], [2297, &'

CONTINUE '3067, 1], [2497, 3052, 1], [2697, 3067, 1], [92, 3237, 1], [262, &'

CONTINUE '3252, 1], [477, 3252, 1], [662, 3222, 1], [847, 3252, 1], [1017, &'

CONTINUE '3237, 1], [1217, 3252, 1], [1772, 3222, 1], [1957, 3252, 1], &'

CONTINUE '[2142, 3252, 1], [2327, 3222, 1], [2482, 3222, 1], [2667, 3222, &'

CONTINUE '1], [92, 3422, 1], [277, 3422, 1], [462, 3422, 1], [662, 3437, 1], &'

CONTINUE '[817, 3437, 1], [1002, 3437, 1], [1187, 3407, 1], [1957, 3407, 1], &'

CONTINUE '[2112, 3407, 1], [2327, 3437, 1], [2482, 3437, 1], [2682, 3422, &'

CONTINUE '1], [92, 3607, 1], [262, 3622, 1], [447, 3622, 1], [632, 3622, 1], &'

CONTINUE '[817, 3592, 1], [1032, 3622, 1], [1372, 3592, 1], [1957, 3622, 1], &'

CONTINUE '[2112, 3592, 1], [2327, 3592, 1], [2512, 3592, 1], [2667, 3592, &'

CONTINUE '1], [92, 3792, 1], [262, 3777, 1], [477, 3777, 1], [662, 3777, 1], &'

CONTINUE '[832, 3792, 1], [1002, 3807, 1], [1217, 3777, 1], [1372, 3807, 1], &'

CONTINUE '[1772, 3777, 1], [1927, 3807, 1], [2112, 3777, 1], [2297, 3777, &'

CONTINUE '1], [2497, 3792, 1], [2667, 3807, 1], [92, 3977, 1], [277, 3977, &'

CONTINUE '1], [447, 3962, 1], [647, 3977, 1], [817, 3992, 1], [1002, 3992, &'

CONTINUE '1], [1187, 3992, 1], [1387, 3977, 1], [1572, 3977, 1], [1772, &'

CONTINUE '3962, 1], [1957, 3962, 1], [2112, 3992, 1], [2297, 3962, 1], &'

CONTINUE '[2482, 3962, 1], [2697, 3992, 1], [92, 4162, 1], [277, 4162, 1], &'

CONTINUE '[447, 4177, 1], [662, 4177, 1], [847, 4177, 1], [1017, 4162, 1], &'

CONTINUE '[1187, 4147, 1], [1402, 4147, 1], [1587, 4177, 1], [1772, 4177, &'

CONTINUE '1], [1957, 4177, 1], [2112, 4177, 1], [2297, 4177, 1], [2512, &'

CONTINUE '4177, 1], [2667, 4177, 1], [92, 4347, 1], [277, 4347, 1], [447, &'

CONTINUE '4362, 1], [632, 4362, 1], [817, 4362, 1], [1017, 4347, 1], [1217, &'

CONTINUE '4332, 1], [1402, 4362, 1], [1557, 4362, 1], [1772, 4362, 1], &'

CONTINUE '[1927, 4332, 1], [2112, 4362, 1], [2297, 4362, 1], [2497, 4347, &'

CONTINUE '1], [2667, 4332, 1], [92, 4532, 1], [277, 4532, 1], [477, 4547, &'

CONTINUE '1], [662, 4547, 1], [817, 4547, 1], [1017, 4532, 1], [1187, 4517, &'

CONTINUE '1], [1402, 4517, 1], [1587, 4547, 1], [1772, 4517, 1], [1957, &'

CONTINUE '4547, 1], [2112, 4547, 1], [2327, 4547, 1], [2497, 4532, 1], &'

CONTINUE '[2697, 4517, 1], [1134, 1700, 1], [1433, 1760, 1]]'

END

 

"Root statistics of random polynomials with bounded Mahler measure"

Homework for 12/4/09

"Polynomially convex embeddings of compact real manifolds"

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 1689 / length of data axis 1

NAXIS2 = 3166 / 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

MIPS-HI = 65535 / Lower visualization cutoff

MIPS-LO = 21711 / Upper visualization cutoff

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

FILENAME= 'Light_mosaic_NGC 7000_20.0s_LP_20250526-014127.fit'

DATE = '2025-05-26T16:24:15' / UTC date that FITS file was created

DATE-OBS= '2025-05-26T06:41:02.150919' / YYYY-MM-DDThh🇲🇲ss observation start,

IMAGETYP= 'Light ' / Type of image

EXPTIME = 20. / [s] Exposure time duration

TELESCOP= 'S50_f1c10930' / Telescope used to acquire this image

FILTER = 'mixed ' / Active filter name

APERTURE= 5. / Aperture of the instrument

FOCALLEN= 251.83 / [mm] Focal length

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 2.90000009536743 / [um] Pixel X axis size

YPIXSZ = 2.90000009536743 / [um] Pixel Y axis size

INSTRUME= 'Seestar S50' / Instrument name

CCD-TEMP= 17.0625 / [degC] CCD temperature

GAIN = 80 / Sensor gain

FOCPOS = 1620 / [step] Focuser position

STACKCNT= 197 / Stack frames

LIVETIME= 3940. / [s] Exposure time after deadtime correction

EXPSTART= 2460821.77849711 / [JD] Exposure start time (standard Julian date)

EXPEND = 2460821.87932888 / [JD] Exposure end time (standard Julian date)

OBJECT = 'NGC 7000' / Name of the object of interest

SITELAT = 41.756 / [deg] Observation site latitude

SITELONG= -88.1819 / [deg] Observation site longitude

OBJCTRA = '20 58 7.346' / [H M S] Image center Right Ascension

OBJCTDEC= '+44 19 23.624' / [D M S] Image center Declination

RA = 314.530606961891 / [deg] Image center Right Ascension

DEC = 44.3232288329815 / [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 = 895.5 / Axis1 reference pixel

CRPIX2 = 1141.5 / Axis2 reference pixel

CRVAL1 = 314.592106580487 / [deg] Axis1 reference value

CRVAL2 = 44.0330294379358 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000659802746654457 / [deg] X pixel size

CDELT2 = 0.000659800551565207 / [deg] Y pixel size

PC1_1 = 0.965297511245702 / Linear transformation matrix (1, 1)

PC1_2 = 0.261152665662929 / Linear transformation matrix (1, 2)

PC2_1 = -0.261115398307382 / Linear transformation matrix (2, 1)

PC2_2 = 0.965307592825612 / 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 = -9.74063439851947E-09

A_1_1 = -3.25366176150845E-10

A_0_2 = 2.66465106494466E-10

A_3_0 = -1.20153434446517E-11

A_2_1 = 1.80289254965373E-12

A_1_2 = 1.52021593952749E-11

A_0_3 = 2.52566304775469E-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 = -2.7859943045886E-09

B_1_1 = -8.30877626644875E-09

B_0_2 = 2.54422083612868E-08

B_3_0 = 1.02803637639594E-11

B_2_1 = 2.06197952424513E-11

B_1_2 = -4.20223998544153E-12

B_0_3 = 5.35717259136597E-12

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

AP_0_0 = -8.83407116017409E-07

AP_1_0 = -2.11171234523969E-09

AP_0_1 = -1.8693611876812E-09

AP_2_0 = 9.74152551922136E-09

AP_1_1 = 3.28176321156856E-10

AP_0_2 = -2.65501037894396E-10

AP_3_0 = 1.2017170739467E-11

AP_2_1 = -1.80181631366394E-12

AP_1_2 = -1.52001458940546E-11

AP_0_3 = -2.52483359393872E-12

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

BP_0_0 = -4.50129317174452E-06

BP_1_0 = -8.89802043462381E-10

BP_0_1 = -2.38290078088729E-09

BP_2_0 = 2.78816732715415E-09

BP_1_1 = 8.305737175103E-09

BP_0_2 = -2.54374095652426E-08

BP_3_0 = -1.0279738664855E-11

BP_2_1 = -2.0616406271359E-11

BP_1_2 = 4.20177218037046E-12

BP_0_3 = -5.35485004051757E-12

HISTORY SCNR (type=average neutral, amount=1.00, preserve=true)

HISTORY Saturation enhancement (amount=0.23)

HISTORY Saturation enhancement (amount=0.42)

HISTORY Saturation enhancement (amount=0.30)

HISTORY Saturation enhancement (amount=0.42)

HISTORY Saturation enhancement (amount=0.64)

BG-EXTR = 'GraXpert'

CBG-1 = 0.0050184443

CBG-2 = 0.0050184443

CBG-3 = 0.0050184443

INTP-OPT= 'RBF '

HIERARCH SMOOTHING = 0.9

HIERARCH CORR-TYPE = 'Subtraction'

HIERARCH SAMPLE-SIZE = 15

HIERARCH RBF-KERNEL = 'thin_plate'

HIERARCH SPLINE-ORDER = 3

BG-PTS = '[[77, 77, 1], [262, 107, 1], [447, 77, 1], [632, 77, 1], [817, &'

CONTINUE '107, 1], [1017, 92, 1], [1202, 92, 1], [1387, 92, 1], [1587, 77, &'

CONTINUE '1], [1742, 107, 1], [1957, 77, 1], [2142, 77, 1], [2297, 107, 1], &'

CONTINUE '[2497, 92, 1], [2682, 92, 1], [107, 262, 1], [292, 262, 1], [477, &'

CONTINUE '262, 1], [632, 262, 1], [847, 292, 1], [1032, 262, 1], [1217, 262, &'

CONTINUE '1], [1402, 292, 1], [1557, 292, 1], [1742, 262, 1], [1927, 262, &'

CONTINUE '1], [2142, 292, 1], [2327, 292, 1], [2497, 277, 1], [2682, 277, &'

CONTINUE '1], [107, 447, 1], [292, 477, 1], [462, 462, 1], [662, 477, 1], &'

CONTINUE '[832, 462, 1], [1002, 447, 1], [1187, 477, 1], [1372, 477, 1], &'

CONTINUE '[1587, 447, 1], [1742, 447, 1], [1957, 477, 1], [2112, 477, 1], &'

CONTINUE '[2327, 477, 1], [2497, 462, 1], [2682, 462, 1], [77, 632, 1], &'

CONTINUE '[262, 662, 1], [447, 632, 1], [632, 662, 1], [847, 632, 1], [1002, &'

CONTINUE '632, 1], [1402, 662, 1], [1557, 662, 1], [1742, 632, 1], [1927, &'

CONTINUE '662, 1], [2127, 647, 1], [2312, 647, 1], [2497, 647, 1], [2682, &'

CONTINUE '647, 1], [107, 847, 1], [277, 832, 1], [477, 847, 1], [662, 817, &'

CONTINUE '1], [817, 847, 1], [1032, 847, 1], [1217, 847, 1], [1402, 847, 1], &'

CONTINUE '[1557, 847, 1], [1742, 847, 1], [1927, 847, 1], [2142, 847, 1], &'

CONTINUE '[2297, 817, 1], [2512, 817, 1], [2682, 832, 1], [107, 1002, 1], &'

CONTINUE '[292, 1002, 1], [477, 1002, 1], [632, 1002, 1], [832, 1017, 1], &'

CONTINUE '[1002, 1002, 1], [1217, 1002, 1], [1402, 1002, 1], [1587, 1002, &'

CONTINUE '1], [1772, 1002, 1], [1957, 1032, 1], [2142, 1032, 1], [2327, &'

CONTINUE '1002, 1], [2512, 1002, 1], [2682, 1017, 1], [107, 1217, 1], [292, &'

CONTINUE '1217, 1], [447, 1217, 1], [632, 1217, 1], [847, 1217, 1], [1002, &'

CONTINUE '1187, 1], [1187, 1217, 1], [1372, 1217, 1], [1587, 1217, 1], &'

CONTINUE '[1772, 1187, 1], [1927, 1217, 1], [2112, 1187, 1], [2297, 1187, &'

CONTINUE '1], [2482, 1187, 1], [2682, 1202, 1], [107, 1372, 1], [262, 1372, &'

CONTINUE '1], [447, 1372, 1], [632, 1402, 1], [817, 1372, 1], [1002, 1372, &'

CONTINUE '1], [1187, 1372, 1], [1587, 1372, 1], [1772, 1402, 1], [1927, &'

CONTINUE '1372, 1], [2142, 1402, 1], [2297, 1402, 1], [2482, 1402, 1], &'

CONTINUE '[2682, 1387, 1], [77, 1557, 1], [262, 1557, 1], [447, 1587, 1], &'

CONTINUE '[662, 1587, 1], [817, 1557, 1], [1032, 1557, 1], [1372, 1557, 1], &'

CONTINUE '[1557, 1557, 1], [1742, 1587, 1], [1927, 1557, 1], [2112, 1587, &'

CONTINUE '1], [2297, 1557, 1], [2482, 1587, 1], [2682, 1572, 1], [77, 1772, &'

CONTINUE '1], [262, 1772, 1], [462, 1757, 1], [662, 1772, 1], [1187, 1742, &'

CONTINUE '1], [1400.5570032573287, 1763.0032573289905, 1.0], [1587, 1772, &'

CONTINUE '1], [1742, 1742, 1], [1927, 1742, 1], [2142, 1742, 1], [2297, &'

CONTINUE '1742, 1], [2482, 1742, 1], [2682, 1757, 1], [77, 1957, 1], [292, &'

CONTINUE '1927, 1], [447, 1927, 1], [632, 1927, 1], [1217, 1957, 1], [1402, &'

CONTINUE '1927, 1], [1587, 1957, 1], [1772, 1957, 1], [1957, 1957, 1], &'

CONTINUE '[2142, 1957, 1], [2297, 1927, 1], [2482, 1957, 1], [2682, 1942, &'

CONTINUE '1], [77, 2142, 1], [277, 2127, 1], [447, 2112, 1], [662, 2112, 1], &'

CONTINUE '[817, 2142, 1], [1217, 2112, 1], [1372, 2142, 1], [1557, 2112, 1], &'

CONTINUE '[1742, 2142, 1], [1957, 2142, 1], [2112, 2142, 1], [2297, 2142, &'

CONTINUE '1], [2482, 2112, 1], [2682, 2127, 1], [77, 2297, 1], [292, 2327, &'

CONTINUE '1], [447, 2297, 1], [662, 2297, 1], [847, 2327, 1], [1032, 2297, &'

CONTINUE '1], [1402, 2327, 1], [1557, 2327, 1], [1742, 2327, 1], [1957, &'

CONTINUE '2297, 1], [2112, 2297, 1], [2327, 2297, 1], [2482, 2297, 1], &'

CONTINUE '[2682, 2312, 1], [92, 2497, 1], [292, 2482, 1], [477, 2482, 1], &'

CONTINUE '[662, 2512, 1], [847, 2512, 1], [1032, 2512, 1], [1402, 2482, 1], &'

CONTINUE '[1557, 2482, 1], [1772, 2512, 1], [1927, 2482, 1], [2112, 2482, &'

CONTINUE '1], [2312, 2497, 1], [2512, 2482, 1], [2667, 2512, 1], [92, 2682, &'

CONTINUE '1], [292, 2697, 1], [477, 2667, 1], [632, 2697, 1], [847, 2667, &'

CONTINUE '1], [1002, 2697, 1], [1202, 2682, 1], [1372, 2667, 1], [1587, &'

CONTINUE '2697, 1], [1957, 2697, 1], [2142, 2667, 1], [2327, 2667, 1], &'

CONTINUE '[2497, 2682, 1], [2697, 2667, 1], [92, 2867, 1], [292, 2852, 1], &'

CONTINUE '[462, 2867, 1], [647, 2867, 1], [817, 2852, 1], [1002, 2882, 1], &'

CONTINUE '[1187, 2882, 1], [1372, 2852, 1], [2112, 2882, 1], [2312, 2867, &'

CONTINUE '1], [2482, 2852, 1], [2697, 2882, 1], [92, 3052, 1], [262, 3037, &'

CONTINUE '1], [477, 3067, 1], [662, 3067, 1], [817, 3037, 1], [1032, 3067, &'

CONTINUE '1], [1772, 3067, 1], [1942, 3052, 1], [2142, 3037, 1], [2297, &'

CONTINUE '3067, 1], [2497, 3052, 1], [2697, 3067, 1], [92, 3237, 1], [262, &'

CONTINUE '3252, 1], [477, 3252, 1], [662, 3222, 1], [847, 3252, 1], [1017, &'

CONTINUE '3237, 1], [1217, 3252, 1], [1772, 3222, 1], [1957, 3252, 1], &'

CONTINUE '[2142, 3252, 1], [2327, 3222, 1], [2482, 3222, 1], [2667, 3222, &'

CONTINUE '1], [92, 3422, 1], [277, 3422, 1], [462, 3422, 1], [662, 3437, 1], &'

CONTINUE '[817, 3437, 1], [1002, 3437, 1], [1187, 3407, 1], [1957, 3407, 1], &'

CONTINUE '[2112, 3407, 1], [2327, 3437, 1], [2482, 3437, 1], [2682, 3422, &'

CONTINUE '1], [92, 3607, 1], [262, 3622, 1], [447, 3622, 1], [632, 3622, 1], &'

CONTINUE '[817, 3592, 1], [1032, 3622, 1], [1372, 3592, 1], [1957, 3622, 1], &'

CONTINUE '[2112, 3592, 1], [2327, 3592, 1], [2512, 3592, 1], [2667, 3592, &'

CONTINUE '1], [92, 3792, 1], [262, 3777, 1], [477, 3777, 1], [662, 3777, 1], &'

CONTINUE '[832, 3792, 1], [1002, 3807, 1], [1217, 3777, 1], [1372, 3807, 1], &'

CONTINUE '[1772, 3777, 1], [1927, 3807, 1], [2112, 3777, 1], [2297, 3777, &'

CONTINUE '1], [2497, 3792, 1], [2667, 3807, 1], [92, 3977, 1], [277, 3977, &'

CONTINUE '1], [447, 3962, 1], [647, 3977, 1], [817, 3992, 1], [1002, 3992, &'

CONTINUE '1], [1187, 3992, 1], [1387, 3977, 1], [1572, 3977, 1], [1772, &'

CONTINUE '3962, 1], [1957, 3962, 1], [2112, 3992, 1], [2297, 3962, 1], &'

CONTINUE '[2482, 3962, 1], [2697, 3992, 1], [92, 4162, 1], [277, 4162, 1], &'

CONTINUE '[447, 4177, 1], [662, 4177, 1], [847, 4177, 1], [1017, 4162, 1], &'

CONTINUE '[1187, 4147, 1], [1402, 4147, 1], [1587, 4177, 1], [1772, 4177, &'

CONTINUE '1], [1957, 4177, 1], [2112, 4177, 1], [2297, 4177, 1], [2512, &'

CONTINUE '4177, 1], [2667, 4177, 1], [92, 4347, 1], [277, 4347, 1], [447, &'

CONTINUE '4362, 1], [632, 4362, 1], [817, 4362, 1], [1017, 4347, 1], [1217, &'

CONTINUE '4332, 1], [1402, 4362, 1], [1557, 4362, 1], [1772, 4362, 1], &'

CONTINUE '[1927, 4332, 1], [2112, 4362, 1], [2297, 4362, 1], [2497, 4347, &'

CONTINUE '1], [2667, 4332, 1], [92, 4532, 1], [277, 4532, 1], [477, 4547, &'

CONTINUE '1], [662, 4547, 1], [817, 4547, 1], [1017, 4532, 1], [1187, 4517, &'

CONTINUE '1], [1402, 4517, 1], [1587, 4547, 1], [1772, 4517, 1], [1957, &'

CONTINUE '4547, 1], [2112, 4547, 1], [2327, 4547, 1], [2497, 4532, 1], &'

CONTINUE '[2697, 4517, 1], [1134, 1700, 1], [1433, 1760, 1]]'

END

 

"Polynomially convex embeddings of compact real manifolds"

Can you say epsilon-delta? Can you approximate with polynomials? Good boy!

"The zeros of random harmonic polynomials"

"Random matrices and zeros of polynomials"

"Zeros of ultraspherical and pseudo–ultraspherical polynomials"

"Polynomially convex embeddings of compact real manifolds"

degree of a polynomial In case of a polynomial involving in one variable, the highest power of the variable is called the degree of the polynomial.

"Root statistics of random polynomials with bounded Mahler measure"

Polynomials division algorithm in telugu | polynomial by polynomial division | polynomial division telugu full chapter

 

youtu.be/ThflcU9Y1Gg

 

"Polynomially convex embeddings of compact real manifolds"

Light curve of PG 1553+113 during the 2012 observation campaign. Upper panel: nightly light curve in VHE γ-rays observed with the MAGIC telescopes for E>150 GeV. For comparison the flux during the low state of the source in 2007 and the high state in 2008 measured by MAGIC (Aleksi ́c et al. 2012a) are shown by the dotted and dashed lines, respectively. Middle panel: Fermi -LAT light curve computed with a 10-day binning for energies between 0.1 to 100 GeV; the dashed gray lines show a zero order polynomial fit and the mean flux of the source from 2FGL catalog (Nolan et al. 2012). Lower panel: flux in the X-ray band in one night bins observed by Swift-XRT from 2 to 10 keV is represented in gray squares. For comparison previous measurements from 2010 and 2011 with Swift-XRT are plotted as dashed lines. The vertical lines denote the flare intervals observed in the VHE band and X-rays. No hints of flare are observed in the HE band.

Again, my mom keeps a bunch of spray paints she used to decorate things in the house and here a a few. I figured the bottles would be the variables and the little bottles could be the numbers. This Polynomial is a polynomial because it has a degree of 3 which means it's more than a quadratic.

"Zeros of ultraspherical and pseudo–ultraspherical polynomials"

"The zeros of random harmonic polynomials"

"Polynomially convex embeddings of compact real manifolds"

"Polynomially convex embeddings of compact real manifolds"

"Polynomially convex embeddings of compact real manifolds"

"Zeros of ultraspherical and pseudo–ultraspherical polynomials"

"Zeros of ultraspherical and pseudo–ultraspherical polynomials"

This is my Great Great Great Great Grandfather! He is staring at a photo of his inventor Charles Babbage. My grandfather looks big and intimidating but deep down he was just a sweet and simple calculator. He was a smart guy for his time, capable of calculating polynomial functions. Also he is English.

Halley's rational method applied to a fifth degree polynomial, and examined near the origin, where there is a point of inflection. Again, generated using a Python program I wrote @ community college.

"Random matrices and zeros of polynomials"

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