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Sweep completed on December 14, 2016.

## Sweep data ($\delta\lambda\approx\pm 0.1$) for lowest 164,300+ eigenvalues  Unit-edged Regular Pentagon, both Dirichlet and Neumann boundary conditions

Bob Jones
Summer 2016

This page simply lists the eigenvalues corresponding to each boundary condition (Dirichlet D or Neumann N) within each symmetry class (A,B,C,S) for the regular pentagon, i.e., the values of $\lambda$ which are solutions to $$(\Delta+\lambda)\Psi(\mathbf{r}) = 0\qquad\mbox{in \Omega, with} \qquad\mbox{D: } \Psi=0\quad\mbox{or}\quad \qquad\mbox{N: } \frac{\partial\Psi}{\partial n}=0 \qquad \mbox{on \partial\Omega}$$ where $\Omega$ is the unit-edged regular pentagon, $\partial\Omega$ is its boundary, and $n$ the outward pointing normal.

I also include Weyl counting function plots which provide some numerical evidence that this set of numbers is complete.

Since the edge-length of the regular pentagon is one, the lowest Dirichlet eigenvalue is about 11.00. Also of note, for this tally, I do include the trivial NS $\lambda=0$ eigenvalue.

The total number of distinct eigenvalues with $\lambda<10^6$ is $$\#(10^6) = \#_\mathrm{D}(10^6) + \#_\mathrm{N}(10^6) = 81,913 + 82,360 = 164,303$$ where $\#(\lambda)$ counts the number of distinct eigenvalues less than $\lambda$ (here a continuous parameter).

In addition, for each of the eight towers, I include the next one up beyond $\lambda=10^6$, making a total of 164,311 eigenvalues displayed on this page.

For a glimpse of what the four symmetry classes (A,B,C,S) look like, the plots of the eigenfunctions for the lowest 802 Dirichlet eigenvalues here.

I used the GSVD method to sweep over the interval $\lambda=0$ to just over 1,000,000. I used various but related methods to identify the eigenvalues by watching the smallest singular value as I "swept" over the interval from $\lambda=0$ to $10^6$.

Actual number of distinct Dirichlet eigenvalues with $\lambda< 1,000,000$: $$13531+27302+27304+13776= 81,913$$

Actual number of Dirichlet eigenfunctions with $\lambda< 1,000,000$: $$13531+2(27302+27304)+13776= 136,519$$

Weyl's two term counting formula (disregarding constant term) for Dirichlet eigenvalues: $$N_w(10^6) = \frac{1}{4\pi}\,\left\{ A \times (10^6) - 5 \times (10^3)\right\} = 136,513.4$$ where $A=10\tan(3\pi/10)/8\approx 1.7204774005$ and 5 are the area and perimeter of the pentagon.

Actual number of distinct Neumann eigenvalues with $\lambda< 1,000,000$: $$13610+ 27460 + 27464 + 13856 = 82,390$$

Actual number of Neumann eigenfunctions with $\lambda< 1,000,000$: $$13610+ 2(27460 + 27464) + 13856 = 137,314$$

Weyl's two term counting formula (disregarding constant term) for Neumann eigenvalues: $$N_w(10^6) = \frac{1}{4\pi}\,\left\{ A \times (10^6) + 5 \times (10^3)\right\} = 137,309.2$$ where $A=10\tan(3\pi/10)/8\approx 1.7204774005$ and 5 are the area and perimeter of the pentagon.

DA DB DC DS Total of Dirichlet eigenvalues in these lists $= 81,917$ (13,532) (27,303) (27,305) (13,777) 147.141 285.892 390.512 473.102 596.826 705.524 754.490 856.393 981.796 1027.852 1162.797 1238.429 1304.336 1358.455 1510.329 1578.114 1673.638 1739.163 1841.949 1902.186 1979.495 2088.831 2161.112 2246.138 2341.168 2433.964 2548.717 2566.482 2624.207 2718.069 2826.538 2932.149 3030.148 3058.224 3132.734 3246.540 3356.092 3410.731 3469.268 3571.137 3610.888 3688.320 3821.465 3921.552 3948.210 4048.266 4174.440 4209.673 4288.042 4375.881 4435.297 4531.985 4565.623 4674.280 4826.824 4854.290 4925.566 4971.423 5086.580 5207.186 5235.203 5344.890 5461.405 5519.473 5545.124 5604.486 5651.628 5783.672 5931.356 5954.953 6053.245 6117.086 6245.447 6282.435 6345.969 6394.063 6525.990 6665.504 6696.755 6723.911 6799.150 6860.377 7014.967 7065.841 7138.633 7191.474 7306.178 7382.662 7500.418 7528.346 7605.639 7673.239 7829.892 7895.067 7966.932 7999.415 8040.273 8119.362 8189.238 8351.917 8372.628 8468.071 8544.726 8683.365 8762.694 8774.780 8849.982 8923.290 8989.887 9070.881 9248.404 9262.924 9370.135 9444.177 9471.806 9564.816 9636.926 9692.820 9770.732 9841.612 9921.744 10014.295 10179.924 10202.514 10269.492 10327.178 10434.826 10498.603 10587.103 10660.290 10739.220 10803.458 10896.213 11007.276 11021.310 11131.756 11189.410 11209.420 11313.716 11429.687 11500.333 11600.642 11673.769 11751.003 11807.669 11885.331 11935.369 12060.809 12128.637 12219.004 12234.822 12350.731 12466.846 12547.289 12673.135 12703.084 12733.298 12798.253 12843.568 12892.379 12983.575 13133.587 13201.416 13300.974 13316.359 13432.561 13546.904 13617.588 13673.221 13800.241 13837.813 13873.030 13932.934 13982.271 14088.435 14245.954 14323.718 14436.609 14452.342 14510.403 14556.632 14650.259 14698.699 14779.568 14952.381 14990.552 15018.044 15079.330 15131.447 15240.046 15402.846 15471.128 15535.478 15614.303 15653.506 15713.880 15785.421 15861.718 15952.218 16139.448 16189.179 16221.404 16273.279 16342.762 16424.921 16450.331 16579.539 16627.292 16700.274 16833.864 16887.420 16932.903 16992.054 17079.058 17173.660 17372.726 17431.678 17448.935 17506.969 17531.500 17619.471 17666.558 17772.447 17864.916 17941.032 18101.177 18149.070 18215.316 18257.308 18353.835 18435.535 18495.361 18625.373 18676.075 18724.576 18745.375 18805.747 18927.641 18960.258 19040.442 19152.815 19232.271 19414.220 19457.565 19525.465 19565.752 19630.040 19699.564 19745.008 19880.739 19989.816 20053.810 20060.799 20141.128 20261.568 20315.697 20383.296 20490.609 20567.601 20664.797 20765.603 20796.745 20842.243 20907.980 20916.559 21074.710 21115.640 21210.990 21351.519 21414.565 21442.702 21523.094 21645.891 21697.160 21777.729 21853.838 21899.560 21950.425 22104.434 22184.816 22232.328 22291.029 22310.626 22477.052 22551.368 22622.728 22756.835 22822.381 22870.786 22942.457 22962.305 23059.746 23114.831 23156.481 23211.599 23340.292 23391.660 23499.799 23634.222 23672.009 23720.452 23758.862 23932.488 24007.660 24104.591 24195.668 24229.435 24279.963 24345.150 24412.346 24467.654 24586.590 24603.554 24663.545 24810.016 24909.339 24975.522 25129.942 25151.006 25199.233 25252.414 25367.140 25427.763 25503.344 25551.283 25628.067 25728.957 25786.686 25865.050 25899.891 25959.648 26106.211 26117.653 26159.883 26333.154 26445.915 26536.267 26652.771 26677.557 26692.941 26731.999 26794.139 26915.888 27033.308 27056.955 27156.953 27267.000 27381.297 27430.304 27445.072 27519.893 27663.414 27679.338 27706.186 27885.684 27917.481 28016.043 28072.491 28157.874 28242.557 28264.701 28304.564 28375.805 28450.262 28608.819 28655.663 28723.100 28856.542 29003.394 29044.696 29073.143 29128.633 29242.010 29272.894 29305.554 29308.201 29480.272 29599.653 29641.561 29771.568 29848.151 29900.992 29965.010 30005.989 30059.790 30237.101 30295.915 30334.075 30490.853 30560.634 30648.947 30698.550 30714.074 30789.590 30794.784 30892.625 30934.588 30962.422 31083.168 31232.334 31316.692 31412.605 31504.385 31582.375 31676.486 31692.462 31741.087 31902.399 31973.512 31990.136 32005.986 32159.629 32293.649 32344.905 32409.542 32487.178 32501.825 32563.442 32644.134 32677.477 32756.774 32926.177 33035.070 33094.735 33209.856 33308.270 33332.927 33399.228 33418.064 33487.193 33538.187 33636.818 33684.012 33714.638 33834.895 33984.324 34094.052 34160.177 34218.092 34241.323 34286.811 34402.875 34464.122 34502.944 34664.521 34793.403 34814.612 34835.102 34957.578 35078.735 35112.516 35167.755 35215.171 35329.870 35387.255 35436.878 35497.581 35576.352 35744.093 35889.502 35956.844 35977.740 36030.509 36064.017 36193.193 36231.586 36268.312 36335.787 36404.669 36475.436 36590.291 36597.678 36705.621 36858.599 36905.607 36988.391 37028.985 37137.936 37189.844 37246.118 37360.393 37387.965 37552.607 37723.751 37764.190 37788.147 37801.676 37867.767 37895.307 38016.703 38090.279 38111.684 38259.672 38298.131 38400.303 38447.571 38513.472 38687.936 38770.273 38859.619 38891.763 38976.054 39031.210 39104.481 39240.065 39251.386 39294.059 39388.101 39452.388 39601.350 39621.536 39684.428 39699.709 39777.756 39852.613 39946.402 40005.287 40143.244 40201.831 40259.310 40366.937 40397.195 40567.422 40680.955 40769.240 40791.499 40849.696 40868.329 40926.508 41007.955 41132.106 41170.872 41293.634 41331.206 41493.392 41527.678 41569.352 41625.950 41710.554 41753.499 41873.605 41951.040 42062.115 42129.242 42167.988 42334.950 42363.021 42385.663 42484.439 42564.513 42641.905 42724.450 42748.469 42780.134 42831.965 42945.686 43013.607 43136.618 43242.989 43328.399 43422.346 43497.112 43527.052 43606.709 43689.554 43705.969 43847.264 43939.300 44012.306 44062.396 44099.988 44129.313 44299.379 44348.743 44444.514 44487.829 44642.618 44697.177 44725.125 44754.136 44794.653 44932.571 44991.311 45156.057 45235.888 45335.053 45405.937 45525.738 45569.238 45632.427 45638.443 45696.432 45717.539 45803.214 45890.258 45967.406 46030.165 46074.114 46143.407 46232.083 46385.786 46459.837 46571.327 46682.075 46704.559 46767.535 46778.024 46830.386 46963.819 47029.092 47221.066 47269.860 47364.471 47388.893 47428.577 47577.949 47645.254 47707.173 47717.283 47766.292 47793.255 47940.865 48031.863 48046.097 48105.936 48210.800 48259.473 48478.648 48520.279 48666.784 48764.669 48791.637 48836.902 48894.244 48919.020 49014.185 49037.329 49104.104 49203.242 49328.515 49368.117 49446.312 49492.100 49580.699 49748.158 49803.976 49827.451 49907.633 49925.571 50039.787 50111.284 50158.446 50185.831 50326.158 50353.370 50614.403 50634.274 50772.051 50813.474 50872.599 50930.920 50970.032 51045.568 51078.110 51118.764 51168.778 51268.432 51451.119 51471.109 51568.128 51606.967 51670.225 51918.711 51926.421 51992.930 52074.363 52117.303 52198.619 52258.004 52306.802 52342.343 52481.400 52490.518 52519.432 52677.608 52806.220 52809.937 52923.804 53001.636 53048.754 53127.292 53218.063 53251.468 53284.476 53386.464 53445.266 53581.817 53659.013 53725.870 53777.786 53840.008 54100.728 54138.991 54204.393 54282.006 54300.706 54361.037 54431.267 54449.450 54499.212 54591.467 54631.977 54689.277 54733.362 54968.133 55026.625 55120.286 55169.521 55207.770 55359.227 55452.916 55469.053 55497.801 55673.103 55680.246 55785.660 55894.308 55929.763 55996.223 56061.575 56132.883 56269.096 56373.737 56402.167 56462.293 56522.887 56539.646 56623.327 56657.521 56734.213 56804.281 56886.061 56955.172 56972.791 57172.697 57286.229 57353.872 57397.079 57449.516 57638.229 57701.113 57758.329 57775.852 57938.819 57959.074 58051.438 58094.731 58147.469 58227.454 58248.676 58328.351 58359.777 58598.438 58698.618 58765.698 58784.723 58823.896 58865.200 58903.715 59021.765 59103.395 59163.608 59261.433 59344.343 59441.028 59592.012 59637.223 59667.719 59748.324 59869.729 59950.959 59980.923 60066.384 60101.421 60113.518 60243.487 60274.198 60388.230 60421.346 60506.410 60603.836 60613.158 60663.766 60875.079 61034.800 61072.183 61116.185 61160.664 61195.265 61204.896 61361.857 61434.209 61514.965 61606.002 61715.069 61797.879 61865.541 61940.005 61971.493 61997.614 62093.366 62128.587 62334.940 62353.557 62460.533 62488.281 62570.976 62605.391 62705.142 62740.915 62872.306 62947.443 62990.646 63115.256 63212.527 63403.184 63427.216 63511.860 63525.147 63551.762 63594.000 63718.289 63750.216 63802.621 63901.801 63942.824 64014.061 64085.118 64201.557 64301.857 64329.894 64398.164 64436.772 64488.292 64691.675 64761.117 64877.924 64901.986 64920.539 65015.065 65104.055 65104.524 65277.190 65359.031 65408.935 65586.360 65638.622 65770.331 65813.597 65849.287 65915.074 65953.412 65972.271 66007.996 66057.965 66201.771 66232.592 66386.179 66423.119 66500.733 66555.671 66701.955 66762.824 66803.562 66897.980 66996.430 67097.765 67224.827 67320.157 67345.935 67361.707 67470.951 67513.342 67553.991 67702.473 67721.804 67813.229 67883.230 67919.801 68037.869 68133.152 68257.597 68308.688 68327.291 68385.575 68440.661 68443.354 68525.518 68629.931 68705.580 68858.011 68887.338 68958.532 69012.546 69138.564 69239.629 69280.991 69385.477 69560.760 69590.487 69733.206 69776.478 69814.882 69860.083 69863.224 69964.262 69972.364 70024.055 70118.785 70218.485 70346.868 70414.411 70523.050 70559.177 70772.370 70783.601 70784.383 70935.492 70976.799 70993.063 71054.626 71107.398 71233.431 71351.547 71413.829 71473.712 71536.232 71619.665 71743.484 71798.919 71825.830 71936.085 71995.528 72096.654 72183.923 72288.992 72311.456 72372.076 72411.808 72437.239 72481.551 72515.605 72622.712 72721.585 72882.097 72983.940 73010.674 73099.447 73294.660 73304.004 73343.003 73493.352 73557.196 73623.980 73648.995 73666.220 73805.746 73884.727 73964.715 73995.944 74046.389 74101.443 74147.855 74228.564 74332.567 74406.607 74547.124 74648.596 74697.391 74880.877 74896.978 74918.593 74983.829 75022.907 75090.157 75114.174 75197.820 75277.175 75472.925 75558.851 75592.508 75702.575 75847.956 75881.213 75939.423 76038.227 76110.717 76176.678 76191.205 76265.333 76272.355 76357.584 76429.525 76480.874 76571.050 76619.580 76709.780 76714.029 76725.665 76902.680 77009.633 77181.690 77209.103 77314.442 77472.796 77543.802 77552.174 77575.558 77667.360 77724.586 77819.801 77844.066 77903.079 78101.764 78160.380 78246.154 78267.531 78343.401 78446.067 78469.413 78566.770 78607.970 78796.445 78855.047 78878.291 78931.216 78972.129 79099.672 79132.413 79176.480 79289.042 79354.302 79365.857 79372.223 79510.648 79677.321 79803.946 79881.682 79995.047 80120.564 80199.332 80218.525 80241.090 80350.468 80383.860 80409.131 80480.072 80558.660 80601.656 80645.667 80768.148 80823.971 80943.764 80968.172 81031.812 81097.135 81201.598 81267.892 81483.286 81536.175 81575.614 81638.450 81668.349 81767.313 81828.353 81892.248 81971.958 82051.224 82081.698 82146.776 82180.802 82391.991 82473.460 82607.744 82655.300 82717.601 82803.526 82873.371 82919.364 82955.755 82987.989 83108.012 83162.345 83226.801 83298.703 83381.494 83475.565 83536.708 83632.286 83690.240 83756.262 83793.169 83878.444 84002.740 84170.386 84291.526 84339.756 84398.651 84428.494 84483.689 84586.265 84627.383 84675.590 84788.098 84839.338 84852.286 84909.896 85006.842 85045.539 85162.114 85189.921 85369.787 85397.418 85479.660 85590.590 85605.041 85673.717 85770.043 85871.648 85928.930 85981.356 86072.985 86135.927 86191.443 86354.736 86397.397 86481.571 86540.605 86579.988 86608.312 86784.217 86920.565 87087.935 87144.264 87158.636 87218.368 87235.236 87257.643 87379.688 87403.542 87419.652 87472.469 87634.281 87657.530 87710.184 87859.377 87935.996 88002.642 88056.823 88208.817 88265.061 88349.155 88363.777 88470.740 88602.277 88648.680 88736.066 88818.545 88911.939 88958.904 88970.660 89166.169 89213.792 89268.869 89369.406 89402.137 89430.792 89509.371 89579.193 89653.748 89719.125 89907.903 89954.171 90008.593 90053.568 90094.970 90118.149 90209.248 90219.181 90257.878 90430.325 90528.848 90552.080 90650.140 90747.946 90864.410 90931.625 91081.550 91117.548 91153.171 91180.073 91312.146 91472.834 91488.089 91604.875 91689.801 91753.728 91823.425 91836.796 91850.253 92005.707 92043.885 92085.484 92165.975 92248.586 92288.990 92440.990 92511.231 92584.216 92633.708 92844.389 92902.174 92910.569 92940.870 92996.086 93046.991 93082.672 93152.278 93259.068 93425.207 93460.884 93553.798 93593.708 93749.047 93834.498 93979.098 94010.231 94031.932 94117.870 94136.848 94194.886 94318.493 94364.577 94410.722 94518.894 94588.750 94630.750 94737.322 94749.089 94782.522 94893.769 94948.128 95016.525 95152.683 95222.037 95296.594 95431.270 95472.542 95600.538 95786.459 95802.384 95826.858 95860.057 95930.791 95943.478 95984.795 96096.709 96158.713 96352.740 96412.491 96482.946 96531.336 96598.912 96683.157 96744.797 96868.638 96872.893 96966.460 97000.430 97115.307 97207.670 97294.348 97336.919 97388.972 97515.153 97590.255 97641.667 97679.453 97720.961 97826.346 97857.987 97940.726 98072.026 98217.956 98262.201 98354.070 98417.921 98621.051 98740.050 98790.608 98796.533 98854.419 98882.453 98921.468 98941.723 98961.391 99071.193 99121.535 99157.789 99321.658 99399.894 99437.424 99517.129 99569.637 99696.618 99706.887 99807.634 99917.348 100017.554 100052.751 100213.997 100242.695 100364.739 100402.149 100523.329 100580.102 100620.407 100651.270 100735.593 100795.584 100812.769 100920.761 101052.620 101248.898 101284.621 101335.373 101421.390 101494.418 101610.842 101697.647 101774.951 101810.175 101840.326 101895.909 101915.741 101979.764 102012.256 102119.043 102129.754 102221.622 102418.224 102446.250 102478.673 102572.418 102696.437 102706.599 102790.427 102886.604 103080.240 103091.169 103206.980 103251.786 103463.770 103507.873 103564.700 103606.065 103678.036 103695.830 103789.044 103801.522 103819.117 103904.140 103962.203 104057.186 104130.320 104291.167 104327.597 104382.957 104483.632 104546.235 104629.756 104779.135 104811.132 104924.562 104933.278 104983.759 105061.544 105104.215 105161.723 105228.185 105254.635 105458.040 105508.141 105560.014 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971125.24 971211.81 971417.89 971502.24 971505.25 971710.75 971725.87 971736.96 971826.68 971976.62 972113.29 972184.68 972219.30 972273.06 972278.97 972384.99 972469.48 972549.52 972680.82 972744.02 972798.25 972863.26 972931.35 972966.29 972977.72 973046.59 973093.75 973160.40 973227.23 973246.24 973263.38 973311.52 973413.86 973494.10 973601.42 973644.56 973691.96 973702.02 973840.62 973857.04 974090.21 974134.87 974203.12 974221.16 974320.59 974389.62 974394.78 974410.08 974474.09 974548.49 974596.09 974654.53 974730.08 974774.03 974970.21 975036.95 975155.87 975163.20 975272.11 975375.95 975385.79 975451.50 975488.61 975611.85 975620.70 975712.08 975938.90 976014.57 976156.09 976205.77 976259.86 976268.62 976359.97 976416.15 976462.22 976518.15 976582.19 976768.75 976808.60 976845.76 976918.99 976930.77 977185.83 977358.41 977400.71 977496.05 977657.50 977737.99 977744.11 977865.28 977894.61 977992.11 978028.82 978047.91 978067.81 978076.96 978173.02 978204.39 978267.01 978294.70 978375.71 978388.89 978537.94 978648.93 978653.89 978670.91 978707.90 978756.24 978834.82 979008.93 979171.19 979231.28 979290.03 979385.40 979417.95 979541.93 979632.05 979770.77 979878.87 979891.24 979950.79 979960.22 980064.84 980147.01 980152.95 980318.25 980371.91 980386.41 980564.58 980607.79 980735.37 980744.80 980803.41 980825.15 981002.77 981091.58 981198.00 981212.45 981217.67 981251.22 981338.23 981378.33 981430.32 981562.65 981658.86 981763.93 981863.35 981983.98 981997.57 982041.38 982100.04 982128.29 982172.01 982260.61 982419.51 982458.29 982469.44 982509.65 982553.56 982615.33 982716.68 982723.65 982861.30 982900.56 982971.85 983038.39 983060.42 983280.52 983318.61 983454.90 983502.38 983642.22 983658.08 983738.09 983768.18 983790.18 983859.19 983931.93 983936.81 984005.40 984011.91 984121.12 984247.79 984298.84 984363.13 984471.55 984506.56 984607.38 984685.27 984688.41 984761.44 984921.79 985065.32 985132.74 985159.92 985281.13 985451.43 985559.81 985587.71 985665.59 985718.61 985749.43 985805.63 985829.48 985977.45 986114.93 986190.86 986261.84 986303.21 986386.19 986430.49 986556.40 986830.58 986853.72 986993.42 987063.77 987178.37 987230.24 987255.25 987310.47 987342.78 987384.22 987454.84 987538.58 987572.71 987584.18 987661.14 987676.18 987788.26 987837.04 987921.23 987949.47 988081.26 988093.91 988177.01 988241.17 988303.26 988350.21 988381.54 988544.85 988570.14 988613.11 988739.50 988802.06 988848.30 988859.46 988938.83 989029.56 989202.29 989307.74 989330.51 989435.96 989485.65 989486.89 989520.08 989638.81 989673.68 989719.08 989756.70 989829.66 990029.72 990092.18 990245.04 990305.35 990363.81 990508.70 990545.58 990563.65 990579.93 990600.04 990722.78 990803.43 990877.29 990946.18 991298.31 991348.84 991370.91 991440.68 991456.87 991516.89 991576.44 991593.96 991675.73 991749.27 991872.74 991921.28 991996.74 992044.27 992083.81 992146.00 992194.62 992247.51 992272.39 992450.24 992512.09 992556.08 992754.26 992812.73 992836.27 992914.11 993072.61 993196.88 993212.33 993289.45 993302.96 993353.50 993371.46 993428.80 993570.23 993590.21 993628.23 993680.75 993760.91 993804.49 993872.03 993936.34 994006.43 994056.22 994073.23 994301.02 994431.52 994558.38 994603.40 994742.58 994758.33 994893.99 994915.99 995099.15 995112.40 995183.12 995248.04 995269.34 995315.54 995522.00 995611.81 995667.75 995692.49 995757.38 995822.89 996006.18 996053.18 996191.78 996288.67 996365.75 996386.11 996408.49 996446.97 996573.05 996576.43 996654.49 996752.27 996791.88 996815.86 996939.59 996969.82 997064.17 997074.75 997134.77 997170.58 997182.57 997211.62 997300.12 997383.29 997455.77 997490.22 997559.35 997614.08 997632.21 997778.88 997823.33 997982.99 998036.56 998104.26 998127.23 998253.23 998306.18 998333.50 998586.96 998655.08 998680.01 998766.55 998878.66 998901.58 998986.27 999050.16 999065.09 999107.19 999179.54 999281.31 999380.38 999453.50 999534.76 999697.45 999760.15 999796.05 999810.52 999864.56 999894.86 999944.95 1000125.200 [L.DS.dat] Total number$^\dagger$ of Neumann eigenvalues in these lists $= 82,394$ (13,611) (27,461) (27,465) (13,857)$^\dagger$ 80.619 204.392 261.227 372.232 457.637 557.253 586.528 709.271 808.964 846.277 973.597 1009.427 1125.105 1159.040 1279.892 1346.471 1477.768 1512.617 1545.438 1657.309 1731.077 1862.633 1889.733 1962.161 2092.558 2165.843 2197.145 2295.948 2328.078 2423.580 2556.541 2615.775 2687.031 2786.801 2823.034 2931.672 2974.578 3058.535 3120.040 3231.983 3321.773 3367.327 3464.033 3513.886 3612.511 3680.339 3816.366 3873.500 3901.820 3947.050 4036.120 4146.758 4218.080 4294.664 4431.116 4452.051 4524.213 4570.194 4664.366 4818.239 4862.613 4905.807 4955.281 5071.682 5149.999 5194.921 5243.731 5345.079 5507.205 5521.689 5576.455 5651.517 5772.121 5899.592 5924.287 5960.137 6048.212 6081.471 6213.077 6265.762 6322.610 6397.719 6521.647 6660.748 6694.523 6719.222 6759.020 6852.497 6995.977 7065.770 7127.669 7191.567 7310.889 7328.861 7446.350 7482.058 7531.627 7574.858 7672.083 7822.364 7906.669 7983.942 8031.659 8073.381 8171.417 8297.205 8333.989 8379.564 8451.164 8543.052 8688.870 8704.423 8778.257 8839.585 8923.842 8948.469 9064.524 9216.648 9238.644 9273.663 9380.363 9460.892 9512.877 9608.666 9686.965 9752.546 9841.637 9900.139 10010.550 10164.339 10184.731 10199.377 10233.332 10327.027 10431.194 10439.386 10574.560 10662.066 10731.869 10802.147 10896.288 11006.839 11070.376 11155.341 11171.379 11210.412 11301.329 11423.418 11473.293 11593.246 11686.839 11767.410 11804.354 11846.716 11931.273 12033.389 12060.261 12188.876 12206.413 12242.897 12340.717 12461.771 12540.242 12668.057 12725.945 12775.265 12845.856 12866.681 12978.550 13114.244 13163.602 13273.362 13292.030 13331.599 13442.688 13542.315 13592.709 13649.007 13748.919 13798.849 13866.023 13939.157 13975.425 14078.936 14235.773 14311.874 14404.612 14439.340 14458.600 14554.240 14622.902 14670.590 14779.484 14892.294 14967.197 15016.754 15088.447 15129.547 15242.300 15396.638 15459.904 15489.063 15569.292 15597.481 15647.722 15679.490 15784.578 15850.410 15945.408 16115.828 16192.968 16213.981 16288.296 16339.070 16448.457 16525.663 16600.148 16700.616 16754.410 16810.386 16887.224 16904.165 17001.600 17073.863 17171.114 17364.030 17433.210 17455.030 17465.409 17528.077 17610.094 17636.745 17746.710 17851.911 17936.858 18050.196 18082.420 18148.662 18202.759 18266.417 18349.527 18451.808 18559.274 18645.818 18723.822 18743.149 18786.973 18902.418 18943.210 19036.718 19147.231 19227.911 19381.120 19423.859 19451.914 19541.335 19556.916 19581.596 19690.948 19727.013 19822.104 19972.241 20042.751 20064.348 20123.819 20239.820 20312.543 20381.086 20487.430 20575.742 20707.515 20747.191 20801.143 20813.607 20906.335 20911.106 21041.836 21100.095 21189.283 21344.640 21408.786 21453.462 21530.986 21632.287 21713.548 21770.061 21799.875 21888.621 21944.278 22008.899 22146.284 22183.841 22223.953 22286.601 22305.407 22445.977 22549.593 22610.460 22754.801 22824.738 22883.585 22959.873 23040.390 23079.204 23148.407 23203.223 23310.679 23366.289 23454.710 23596.484 23636.484 23669.341 23716.435 23762.484 23914.491 24016.625 24093.606 24148.508 24214.031 24280.040 24305.618 24381.145 24468.805 24565.919 24603.281 24665.427 24773.709 24904.542 24949.931 25096.080 25145.964 25152.999 25198.330 25273.679 25429.640 25455.190 25517.203 25614.528 25702.656 25743.883 25827.385 25893.882 25973.588 26092.289 26116.968 26163.879 26309.951 26450.082 26521.107 26613.363 26641.022 26672.751 26686.709 26733.571 26777.187 26879.653 27029.884 27036.578 27157.983 27222.572 27359.825 27400.398 27456.634 27541.450 27661.692 27685.403 27712.433 27907.275 27966.819 28023.207 28134.308 28195.771 28237.093 28251.074 28306.293 28347.713 28448.202 28602.229 28649.342 28724.441 28828.056 29003.896 29044.575 29066.794 29148.177 29218.888 29261.071 29300.086 29308.447 29429.120 29576.532 29616.820 29771.511 29776.808 29866.462 29896.514 29953.596 29990.399 30073.764 30241.543 30294.650 30333.375 30490.815 30608.508 30651.916 30721.055 30761.942 30795.041 30893.658 30909.882 30961.767 31050.847 31231.993 31301.840 31410.062 31466.264 31547.200 31597.354 31673.827 31681.518 31751.591 31919.172 31949.801 31973.426 31992.138 32117.550 32220.400 32324.647 32402.817 32464.612 32497.599 32562.288 32614.242 32682.433 32747.838 32927.863 33032.739 33091.669 33200.598 33273.484 33318.361 33391.827 33400.709 33476.005 33492.112 33630.525 33682.850 33715.635 33774.826 33981.262 34062.199 34152.534 34205.432 34237.132 34291.013 34395.811 34463.144 34496.202 34685.718 34766.675 34794.572 34817.385 34928.952 34982.874 35052.767 35113.302 35157.580 35214.339 35324.421 35353.566 35435.326 35496.003 35550.191 35745.118 35883.346 35959.067 35980.304 36035.413 36069.287 36215.664 36263.551 36312.848 36332.907 36477.469 36590.715 36597.750 36618.517 36825.358 36868.929 36901.979 36969.118 37025.623 37137.074 37170.758 37247.938 37353.041 37375.905 37564.914 37716.585 37725.961 37773.180 37804.088 37863.369 37884.063 37961.109 38087.984 38101.218 38249.078 38266.771 38400.916 38435.304 38474.467 38687.723 38733.792 38779.692 38847.454 38897.420 38974.026 39023.386 39123.737 39243.585 39281.937 39314.031 39439.436 39577.413 39601.717 39622.237 39714.484 39773.092 39833.911 39936.459 39999.612 40124.685 40189.696 40263.127 40345.114 40388.704 40572.649 40647.452 40700.617 40773.256 40790.646 40830.428 40847.316 40920.568 40976.355 41082.133 41158.745 41278.969 41305.525 41455.280 41518.135 41529.082 41642.556 41718.698 41753.568 41865.648 41945.881 42044.205 42134.311 42175.291 42322.428 42361.165 42438.482 42517.459 42601.643 42634.804 42721.513 42728.146 42773.616 42836.069 42898.251 43014.761 43121.220 43224.114 43317.301 43406.165 43470.336 43521.158 43624.344 43704.815 43717.445 43873.630 43937.404 43996.545 44014.053 44098.527 44125.011 44214.342 44325.984 44416.698 44471.756 44570.055 44618.285 44696.568 44717.582 44744.665 44812.550 44918.305 44991.938 45141.584 45217.199 45344.051 45401.690 45505.265 45568.266 45636.833 45688.596 45714.339 45764.673 45889.502 45968.092 46030.138 46038.696 46115.357 46203.458 46366.375 46446.314 46545.161 46618.750 46665.671 46729.282 46756.611 46777.208 46855.697 46962.975 47046.801 47215.112 47269.421 47310.758 47380.556 47424.815 47495.130 47596.133 47679.094 47721.138 47764.102 47794.445 47908.540 48032.650 48047.804 48094.582 48193.077 48259.823 48454.059 48507.999 48670.890 48737.908 48759.659 48811.121 48827.119 48894.492 48942.026 49040.135 49082.516 49168.408 49329.127 49364.608 49391.255 49491.258 49506.678 49719.462 49786.353 49831.259 49875.965 49931.726 50009.201 50131.009 50156.994 50182.554 50325.922 50360.900 50597.610 50635.348 50745.636 50779.289 50867.336 50878.582 50910.930 50973.734 50990.639 51075.468 51099.601 51171.649 51257.150 51423.619 51470.882 51548.985 51606.629 51647.412 51878.059 51922.483 52008.107 52082.617 52119.317 52168.929 52283.678 52303.694 52342.659 52488.232 52515.781 52599.951 52803.237 52810.316 52846.101 52946.439 53007.399 53089.655 53135.093 53194.962 53247.436 53290.756 53364.120 53427.754 53575.432 53655.827 53721.360 53782.764 53836.233 54079.127 54137.127 54217.643 54288.815 54299.247 54328.153 54420.792 54445.340 54481.103 54552.030 54591.631 54690.557 54732.735 54926.804 55026.349 55094.801 55130.695 55189.607 55320.663 55357.844 55437.567 55455.214 55518.790 55649.758 55676.698 55786.311 55893.486 55929.464 56019.790 56061.724 56200.810 56339.203 56398.423 56416.317 56498.015 56528.537 56610.532 56650.093 56735.736 56778.990 56887.740 56923.593 56975.566 57141.212 57284.117 57355.837 57391.851 57437.298 57604.255 57651.806 57687.497 57757.096 57795.722 57937.199 57977.169 58007.765 58058.044 58148.238 58183.539 58227.436 58315.968 58337.802 58556.172 58673.225 58704.125 58783.835 58796.201 58886.369 58902.346 59017.924 59101.453 59153.072 59247.102 59338.700 59422.147 59591.811 59651.500 59668.652 59746.608 59908.463 59939.105 59989.612 60009.231 60101.542 60106.347 60221.870 60250.765 60383.379 60410.431 60454.562 60603.848 60605.358 60658.361 60823.870 61031.902 61067.256 61109.012 61144.880 61196.027 61227.481 61365.834 61435.641 61502.800 61621.393 61719.640 61779.336 61792.308 61943.088 61955.719 61995.133 62033.080 62093.730 62274.548 62315.144 62364.143 62418.447 62489.024 62554.186 62575.995 62715.442 62740.404 62862.716 62921.460 62994.791 63103.311 63181.519 63406.279 63431.028 63521.064 63530.312 63550.546 63618.940 63758.424 63787.502 63837.612 63926.072 64015.268 64063.178 64163.250 64241.024 64310.696 64395.365 64432.528 64484.080 64611.832 64725.901 64777.883 64854.749 64903.676 64932.590 64985.938 65104.094 65119.068 65274.932 65337.657 65422.371 65583.437 65616.263 65707.104 65822.094 65847.476 65877.137 65914.991 65973.483 65982.481 66066.370 66194.377 66222.842 66358.881 66392.381 66486.972 66541.922 66700.867 66741.856 66791.616 66899.118 66974.731 67054.470 67189.989 67248.853 67319.932 67351.397 67385.834 67461.513 67513.215 67583.591 67702.798 67785.535 67847.901 67894.283 68014.898 68089.430 68161.584 68303.982 68310.867 68383.433 68432.071 68449.481 68506.947 68617.474 68709.621 68828.618 68887.234 68943.086 69013.016 69139.227 69236.393 69266.502 69386.317 69543.441 69574.173 69697.963 69732.550 69765.203 69796.315 69859.907 69860.661 69965.826 69970.684 69976.972 70112.140 70199.353 70317.079 70407.892 70477.032 70521.763 70741.685 70776.725 70784.319 70926.288 70947.673 71000.477 71035.413 71116.883 71255.369 71348.444 71409.216 71482.705 71536.860 71619.679 71739.004 71826.626 71885.479 71934.488 72071.679 72140.669 72196.631 72261.422 72322.621 72348.827 72411.404 72447.407 72480.931 72514.477 72605.291 72731.553 72829.635 72985.401 73012.447 73065.092 73292.993 73299.127 73336.180 73482.602 73570.894 73602.553 73651.430 73676.753 73833.008 73888.605 73918.481 73991.036 74054.374 74094.014 74145.716 74162.233 74300.179 74406.449 74522.731 74558.554 74653.969 74841.589 74865.655 74889.941 74924.682 74982.818 75019.218 75090.144 75112.270 75169.571 75299.905 75451.338 75566.990 75597.932 75684.667 75846.404 75880.758 75951.142 76086.431 76106.787 76200.255 76229.791 76273.279 76356.010 76383.429 76480.729 76534.272 76617.184 76707.895 76714.482 76724.508 76905.821 76958.662 77155.318 77200.541 77269.726 77470.255 77510.080 77537.831 77565.646 77574.889 77665.276 77731.529 77793.406 77841.224 77931.875 78098.265 78185.338 78213.256 78246.065 78337.845 78393.409 78446.161 78520.131 78601.766 78725.269 78795.736 78884.928 78894.626 78990.835 79075.654 79136.226 79166.152 79288.670 79337.111 79365.678 79372.590 79511.277 79652.405 79804.905 79880.669 79967.486 80119.729 80199.123 80205.988 80226.316 80261.461 80372.405 80392.743 80468.827 80492.077 80598.550 80629.618 80747.150 80787.308 80940.088 80944.857 81025.614 81095.416 81202.196 81228.725 81388.475 81524.888 81573.692 81609.711 81698.301 81767.686 81835.586 81885.736 81970.391 82026.419 82085.710 82145.611 82184.754 82396.433 82488.405 82597.320 82605.269 82705.417 82767.717 82828.518 82876.154 82933.388 82948.978 82996.352 83087.636 83149.177 83227.035 83241.056 83390.972 83443.317 83516.797 83632.099 83690.738 83755.762 83793.278 83861.130 83983.422 84144.027 84290.685 84346.762 84372.877 84457.957 84483.541 84617.966 84628.809 84677.094 84784.145 84856.416 84890.264 84943.607 85012.304 85163.013 85170.360 85297.129 85381.926 85469.456 85585.079 85606.103 85630.975 85707.050 85744.832 85834.112 85945.590 85975.521 86031.309 86157.577 86192.885 86332.181 86391.919 86481.474 86540.436 86579.619 86592.824 86790.884 86917.920 87083.876 87113.471 87163.961 87188.901 87230.980 87260.459 87330.468 87404.091 87420.725 87452.963 87612.142 87637.644 87698.192 87807.334 87908.953 87983.460 88051.989 88207.835 88265.289 88326.398 88357.468 88456.796 88547.422 88578.081 88649.309 88762.172 88819.128 88878.495 88967.376 88989.814 89167.127 89222.508 89269.164 89361.611 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