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<DIV></DIV>
<DIV> This subject is not
exactly signature sequences, but is somewhat related.</DIV>
<DIV> </DIV>
<DIV> Below I append the union of the
multiples m of the reciprocal Gauss-Kuzmin frequencies of
integers k, numbered in ascending order by n.</DIV>
<DIV> </DIV>
<DIV>Here I denote Gauss-Kuzmin frequency of k by GK(k),
where</DIV>
<DIV> GK(k) = log(1 + 1/(k*(k+2)) / log(2).</DIV>
<DIV> </DIV>
<DIV>ANY PATTERNS TO THE SEQUENCES?</DIV>
<DIV>Notice one pattern of the values of n when m=1:</DIV>
<DIV>n=1, k=1, m/GK(k)= 2.409420839653463384</DIV>
<DIV>n=3, k=2, m/GK(k)= 5.884949192362146710</DIV>
<DIV>n=6, k=3, m/GK(k)=10.740053666281483132</DIV>
<DIV>n=11, k=4, m/GK(k)=16.865945877574243693</DIV>
<DIV>n=18, k=5, m/GK(k)=24.605097717735793855</DIV>
<DIV>n=24, k=6, m/GK(k)=33.616447248866527729</DIV>
<DIV>n=34, k=7, m/GK(k)=44.013936308554193692</DIV>
<DIV>n=45, k=8, m/GK(k)=55.797630484035824061</DIV>
<DIV>n=56, k=9, m/GK(k)=68.967563936505075182</DIV>
<DIV>n=68, k=10,m/GK(k)=83.523755900347857243</DIV>
<DIV>...</DIV>
<DIV>In each case, the occurrence of the next 'm=1' is predicted closely by
the value of floor(1/GK(k)). Is this due to the fact that
sum(k>0,GK(k))=1 ?</DIV>
<DIV> </DIV>
<DIV>WHAT IS 'BEST-FIT' of GK?</DIV>
<DIV>
<DIV>Note that this construct causes the frequency of the values of {k} to
obey the Gauss-Kuzmin law for continued fractions. </DIV>
<DIV> </DIV>
<DIV>My initial idea was to obtain the partial quotients {k} that would be the
'best-fit' of the Gauss-Kuzmin frequency, and this is what I proposed:<BR>x=
0.5815803358828329856145...<BR> =
[0;1,1,2,1,1,3,2,1,1,1,4,2,1,3,1,2,1,5,1,1,2,1,3,6,1,4,...]</DIV>
<DIV>
<DIV>By construction, this constant would obey Khintchine's
theorem.</DIV>
<DIV> </DIV>
<DIV>But I wonder if the constant is truly the 'best-fit' to
Gauss-Kuzmin, and if such a constant could be defined.</DIV>
<DIV>See: <A
href="http://www.math.niu.edu/~rusin/known-math/99/kusmin">http://www.math.niu.edu/~rusin/known-math/99/kusmin</A></DIV></DIV></DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>At any rate, I find these sequences, {k} and {m}, interesting, as
well as the gaps between m/GK(k): which integers are not included in
{floor(m/GK(k))}?</DIV>
<DIV> </DIV>
<DIV>Sequence {k}:</DIV>
<DIV> 1,1,2,1,1,3,2,1,1,1,4,2,1,3,1,2,1,5,1,1,2,<BR> 1,3,6,1,4,2,1,1,1,2,3,1,7,1,2,1,5,1,4,2,1,<BR> 3,1,8,1,2,1,1,3,2,1,6,1,4,9,1,2,1,5,1,3,2,<BR> 1,1,1,2,10,1,4,3,1,7,2,1,1,1,2,1,3,5,1,11,<BR> 2,6,1,4,1,2,1,3,1,1,8,2,1,1,12,2,1,3,4,1,<BR> 1,5,2,1,1,3,2,1,7,1,6,1,2,13,4,1,9,3,1,2,<BR> 1,1,1,2,5,1,3,1,4,2,1,14,1,2,1,3,1,1,2,1,<BR> 10,8,6,1,4,2,1,3,5,1,1,7,2,15,1,1,2,3,1,1,<BR> 4,1,2,1,1,3,2,1,5,1,11,16,1,2,6,1,4,3,1,2,<BR> 9,1,1,2,1,1,3,1,2,1,7,4,5,1,8,2,1,17,3,1,<BR> 1,2,1,12,1,6,2,1,3,4,1,1,2,1,1,5,3,2,1,18,<BR> 10,1,1,2,4,1,3,1,2,1,1,7,2,1,1,3,6,1,5,2,<BR> 13,4,1,1,9,2,19,1,8,3,1,1,2,1,1,2,4,1,3,1,<BR> 1,2,5,1,11,1,2,3,1,6,1,20,4,1,2,7,1,1,14,3,<BR> 2,1,1,2,1,5,1,3,4,1,2,1,1,2,1,1,3,10,8,1,</DIV>
<DIV> </DIV>
<DIV>Sequence {m}:</DIV>
<DIV> 1,2,1,3,4,1,2,5,6,7,1,3,8,2,9,4,10,1,11,12,5,13,<BR> 3,1,14,2,6,15,16,17,7,4,18,1,19,8,20,2,21,3,9,22,5,<BR> 23,1,24,10,25,26,6,11,27,2,28,4,1,29,12,30,3,31,7,<BR> 13,32,33,34,14,1,35,5,8,36,2,15,37,38,39,16,40,9,4,<BR> 41,1,17,3,42,6,43,18,44,10,45,46,2,19,47,48,1,20,49,<BR> 11,7,50,51,5,21,52,53,12,22,54,3,55,4,56,23,1,8,57,<BR> 2,13,58,24,59,60,61,25,6,62,14,63,9,26,64,1,65,27,66,<BR> 15,67,68,28,69,2,3,5,70,10,29,71,16,7,72,73,4,30,1,<BR> 74,75,31,17,76,77,11,78,32,79,80,18,33,81,8,82,2,1,<BR> 83,34,6,84,12,19,85,35,3,86,87,36,88,89,20,90,37,91,<BR> 5,13,9,92,4,38,93,1,21,94,95,39,96,2,97,7,40,98,22,<BR> 14,99,100,41,101,102,10,23,42,103,1,3,104,105,43,15,<BR> 106,24,107,44,108,109,6,45,110,111,25,8,112,11,46,2,16,<BR> 113,114,4,47,1,115,5,26,116,117,48,118,119,49,17,120,<BR> 27,121,122,50,12,123,3,124,51,28,125,9,126,1,18,127,52,<BR> 7,128,129,2,29,53,130,131,54,132,13,133,30,19,134,55,</DIV>
<DIV> </DIV>
<DIV>Sequence {floor(m/GK(k))}:</DIV>
<DIV> 2,4,5,7,9,10,11,12,14,16,16,17,19,21,21,23,24,24,26,<BR> 28,29,31,32,33,33,33,35,36,38,40,41,42,43,44,45,47,48,<BR> 49,50,50,52,53,53,55,55,57,58,60,62,64,64,65,67,67,67,<BR> 68,69,70,72,73,74,75,76,77,79,81,82,83,84,84,85,86,88,<BR> 88,89,91,93,94,96,96,98,98,99,100,100,101,101,103,105,<BR> 106,107,108,110,111,111,113,115,116,117,118,118,118,120,<BR> 122,123,123,125,127,128,129,130,132,132,134,134,135,135,<BR> 135,137,137,139,139,141,142,144,146,147,147,149,150,151,<BR> 152,153,154,155,156,158,159,161,161,163,164,166,167,167,<BR> 168,168,169,170,171,171,172,173,175,176,176,177,178,180,<BR> 182,182,183,185,186,187,188,190,192,193,194,195,196,197,<BR> 198,199,199,200,201,202,203,204,204,205,206,207,209,211,<BR> 212,214,214,216,217,219,220,220,221,221,223,223,224,224,<BR> 225,226,228,229,231,233,233,235,235,236,236,237,238,240,<BR> 241,243,245,246,247,247,248,249,250,250,252,253,254,255,</DIV>
<DIV> </DIV>
<DIV>Missing integers in {floor(m/GK(k))}:</DIV>
<DIV> 1,3,6,8,13,15,18,20,22,25,27,30,34,37,39,46,51,54,56,59,</DIV>
<DIV> 61,63,66,71,78,80,87,90,92,95,97,102,104,109,112,114,119,</DIV>
<DIV> 121,124,126,131,133,136,138,140,143,145,148,157,160,162,</DIV>
<DIV> 165,174,179,181,184,189,191,208,210,213,215,218,222,...</DIV>
<DIV> </DIV>
<DIV>Any comments would be welcomed.</DIV>
<DIV> </DIV>
<DIV> Thanks,</DIV>
<DIV>
Paul</DIV>
<DIV>------------------------------------------------</DIV>
<DIV>. n
k
m m / GK(k)<BR>.
1 1:
1.
2.409420839653463384<BR>. 2
1: 2.
4.818841679306926769<BR>. 3
2: 1.
5.884949192362146710<BR>. 4
1: 3.
7.228262518960390154<BR>. 5
1: 4.
9.637683358613853539<BR>. 6
3: 1.
10.740053666281483132<BR>. 7
2: 2.
11.769898384724293421<BR>. 8
1: 5.
12.047104198267316923<BR>. 9
1: 6.
14.456525037920780308<BR>. 10
1: 7.
16.865945877574243693<BR>. 11
4: 1.
16.979748018338516689<BR>. 12
2: 3.
17.654847577086440132<BR>. 13
1: 8.
19.275366717227707078<BR>. 14
3: 2.
21.480107332562966265<BR>. 15
1: 9.
21.684787556881170462<BR>. 16
2: 4.
23.539796769448586843<BR>. 17
1: 10.
24.094208396534633847<BR>. 18
5: 1.
24.605097717735793855<BR>. 19
1: 11.
26.503629236188097232<BR>. 20
1: 12.
28.913050075841560617<BR>. 21
2: 5.
29.424745961810733554<BR>. 22
1: 13.
31.322470915495024002<BR>. 23
3: 3.
32.220160998844449398<BR>. 24
6: 1.
33.616447248866527729<BR>. 25
1: 14.
33.731891755148487386<BR>. 26
4: 2.
33.959496036677033379<BR>. 27
2: 6.
35.309695154172880265<BR>. 28
1: 15.
36.141312594801950771<BR>. 29
1: 16.
38.550733434455414156<BR>. 30
1: 17.
40.960154274108877541<BR>. 31
2: 7.
41.194644346535026976<BR>. 32
3: 4.
42.960214665125932530<BR>. 33
1: 18.
43.369575113762340925<BR>. 34
7: 1.
44.013936308554193692<BR>. 35
1: 19.
45.778995953415804310<BR>. 36
2: 8.
47.079593538897173687<BR>. 37
1: 20.
48.188416793069267695<BR>. 38
5: 2.
49.210195435471587710<BR>. 39
1: 21.
50.597837632722731080<BR>. 40
4: 3.
50.939244055015550069<BR>. 41
2: 9.
52.964542731259320398<BR>. 42
1: 22.
53.007258472376194465<BR>. 43
3: 5.
53.700268331407415663<BR>. 44
1: 23.
55.416679312029657849<BR>. 45
8: 1.
55.797630484035824061<BR>. 46
1: 24.
57.826100151683121234<BR>. 47
2: 10.
58.849491923621467109<BR>. 48
1: 25.
60.235520991336584619<BR>. 49
1: 26.
62.644941830990048004<BR>. 50
3: 6.
64.440321997688898796<BR>. 51
2: 11.
64.734441115983613820<BR>. 52
1: 27.
65.054362670643511388<BR>. 53
6: 2.
67.232894497733055458<BR>. 54
1: 28.
67.463783510296974773<BR>. 55
4: 4.
67.918992073354066759<BR>. 56
9: 1.
68.967563936505075182<BR>. 57
1: 29.
69.873204349950438158<BR>. 58
2: 12.
70.619390308345760531<BR>. 59
1: 30.
72.282625189603901543<BR>. 60
5: 3.
73.815293153207381565<BR>. 61
1: 31.
74.692046029257364927<BR>. 62
3: 7.
75.180375663970381929<BR>. 63
2: 13.
76.504339500707907242<BR>. 64
1: 32.
77.101466868910828312<BR>. 65
1: 33.
79.510887708564291697<BR>. 66
1: 34.
81.920308548217755082<BR>. 67
2: 14.
82.389288693070053953<BR>. 68
10: 1.
83.523755900347857243<BR>. 69
1: 35.
84.329729387871218467<BR>. 70
4: 5.
84.898740091692583449<BR>. 71
3: 8.
85.920429330251865061<BR>. 72
1: 36.
86.739150227524681851<BR>. 73
7: 2.
88.027872617108387385<BR>. 74
2: 15.
88.274237885432200664<BR>. 75
1: 37.
89.148571067178145236<BR>. 76
1: 38.
91.557991906831608621<BR>. 77
1: 39.
93.967412746485072006<BR>. 78
2: 16.
94.159187077794347375<BR>. 79
1: 40.
96.376833586138535390<BR>. 80
3: 9.
96.660482996533348194<BR>. 81
5: 4.
98.420390870943175420<BR>. 82
1: 41.
98.786254425791998775<BR>. 83
11: 1.
99.466217884583035803<BR>. 84
2: 17.
100.044136270156494085<BR>. 85
6: 3.
100.849341746599583188<BR>. 86
1: 42.
101.195675265445462160<BR>. 87
4: 6.
101.878488110031100139<BR>. 88
1: 43.
103.605096105098925545<BR>. 89
2: 18.
105.929085462518640796<BR>. 90
1: 44.
106.014516944752388930<BR>. 91
3: 10.
107.400536662814831327<BR>. 92
1: 45.
108.423937784405852314<BR>. 93
1: 46.
110.833358624059315699<BR>. 94
8: 2.
111.595260968071648122<BR>. 95
2: 19.
111.814034654880787507<BR>. 96
1: 47.
113.242779463712779084<BR>. 97
1: 48.
115.652200303366242469<BR>. 98
12: 1.
116.794957120786374235<BR>. 99
2: 20.
117.698983847242934218<BR>. 100
1: 49.
118.061621143019705853<BR>. 101
3: 11.
118.140590329096314459<BR>. 102
4: 7.
118.858236128369616829<BR>. 103
1: 50.
120.471041982673169238<BR>. 104
1: 51.
122.880462822326632623<BR>. 105
5: 5.
123.025488588678969275<BR>. 106
2: 21.
123.583933039605080929<BR>. 107
1: 52.
125.289883661980096008<BR>. 108
1: 53.
127.699304501633559393<BR>. 109
3: 12.
128.880643995377797592<BR>. 110
2: 22.
129.468882231967227640<BR>. 111
1: 54.
130.108725341287022777<BR>. 112
7: 3.
132.041808925662581078<BR>. 113
1: 55.
132.518146180940486162<BR>. 114
6: 4.
134.465788995466110917<BR>. 115
1: 56.
134.927567020593949547<BR>. 116
2: 23.
135.353831424329374351<BR>. 117
13: 1.
135.509978339807338529<BR>. 118
4: 8.
135.837984146708133519<BR>. 119
1: 57.
137.336987860247412932<BR>. 120
9: 2.
137.935127873010150364<BR>. 121
3: 13.
139.620697661659280725<BR>. 122
1: 58.
139.746408699900876316<BR>. 123
2: 24.
141.238780616691521062<BR>. 124
1: 59.
142.155829539554339701<BR>. 125
1: 60.
144.565250379207803086<BR>. 126
1: 61.
146.974671218861266471<BR>. 127
2: 25.
147.123729809053667773<BR>. 128
5: 6.
147.630586306414763130<BR>. 129
1: 62.
149.384092058514729855<BR>. 130
3: 14.
150.360751327940763858<BR>. 131
1: 63.
151.793512898168193240<BR>. 132
4: 9.
152.817732165046650209<BR>. 133
2: 26.
153.008679001415814484<BR>. 134
1: 64.
154.202933737821656625<BR>. 135
14: 1.
155.611284742426132435<BR>. 136
1: 65.
156.612354577475120010<BR>. 137
2: 27.
158.893628193777961195<BR>. 138
1: 66.
159.021775417128583395<BR>. 139
3: 15.
161.100804994222246990<BR>. 140
1: 67.
161.431196256782046779<BR>. 141
1: 68.
163.840617096435510164<BR>. 142
2: 28.
164.778577386140107906<BR>. 143
1: 69.
166.250037936088973549<BR>. 144
10: 2.
167.047511800695714487<BR>. 145
8: 3.
167.392891452107472183<BR>. 146
6: 5.
168.082236244332638646<BR>. 147
1: 70.
168.659458775742436934<BR>. 148
4: 10.
169.797480183385166899<BR>. 149
2: 29.
170.663526578502254617<BR>. 150
1: 71.
171.068879615395900318<BR>. 151
3: 16.
171.840858660503730123<BR>. 152
5: 7.
172.235684024150556985<BR>. 153
1: 72.
173.478300455049363703<BR>. 154
1: 73.
175.887721294702827088<BR>. 155
7: 4.
176.055745234216774771<BR>. 156
2: 30.
176.548475770864401328<BR>. 157
15: 1.
177.098878557435750322<BR>. 158
1: 74.
178.297142134356290473<BR>. 159
1: 75.
180.706562974009753858<BR>. 160
2: 31.
182.433424963226548039<BR>. 161
3: 17.
182.580912326785213256<BR>. 162
1: 76.
183.115983813663217242<BR>. 163
1: 77.
185.525404653316680627<BR>. 164
4: 11.
186.777228201723683589<BR>. 165
1: 78.
187.934825492970144012<BR>. 166
2: 32.
188.318374155588694750<BR>. 167
1: 79.
190.344246332623607397<BR>. 168
1: 80.
192.753667172277070781<BR>. 169
3: 18.
193.320965993066696389<BR>. 170
2: 33.
194.203323347950841460<BR>. 171
1: 81.
195.163088011930534166<BR>. 172
5: 8.
196.840781741886350840<BR>. 173
1: 82.
197.572508851583997551<BR>. 174
11: 2.
198.932435769166071606<BR>. 175
16: 1.
199.972761375538618998<BR>. 176
1: 83.
199.981929691237460936<BR>. 177
2: 34.
200.088272540312988171<BR>. 178
6: 6.
201.698683493199166376<BR>. 179
1: 84.
202.391350530890924321<BR>. 180
4: 12.
203.756976220062200279<BR>. 181
3: 19.
204.061019659348179521<BR>. 182
1: 85.
204.800771370544387705<BR>. 183
2: 35.
205.973221732675134882<BR>. 184
9: 3.
206.902691809515225547<BR>. 185
1: 86.
207.210192210197851090<BR>. 186
1: 87.
209.619613049851314475<BR>. 187
2: 36.
211.858170925037281593<BR>. 188
1: 88.
212.029033889504777860<BR>. 189
1: 89.
214.438454729158241244<BR>. 190
3: 20.
214.801073325629662654<BR>. 191
1: 90.
216.847875568811704629<BR>. 192
2: 37.
217.743120117399428304<BR>. 193
1: 91.
219.257296408465168014<BR>. 194
7: 5.
220.069681542770968463<BR>. 195
4: 13.
220.736724238400716968<BR>. 196
5: 9.
221.445879459622144695<BR>. 197
1: 92.
221.666717248118631399<BR>. 198
8: 4.
223.190521936143296244<BR>. 199
2: 38.
223.628069309761575015<BR>. 200
1: 93.
224.076138087772094783<BR>. 201
17: 1.
224.232934356877362998<BR>. 202
3: 21.
225.541126991911145787<BR>. 203
1: 94.
226.485558927425558168<BR>. 204
1: 95.
228.894979767079021553<BR>. 205
2: 39.
229.513018502123721726<BR>. 206
1: 96.
231.304400606732484938<BR>. 207
12: 2.
233.589914241572748470<BR>. 208
1: 97.
233.713821446385948323<BR>. 209
6: 7.
235.315130742065694105<BR>. 210
2: 40.
235.397967694485868437<BR>. 211
1: 98.
236.123242286039411707<BR>. 212
3: 22.
236.281180658192628919<BR>. 213
4: 14.
237.716472256739233658<BR>. 214
1: 99.
238.532663125692875092<BR>. 215
1: 100.
240.942083965346338477<BR>. 216
2: 41.
241.282916886848015148<BR>. 217
1: 101.
243.351504804999801862<BR>. 218
1: 102.
245.760925644653265246<BR>. 219
5: 10.
246.050977177357938550<BR>. 220
3: 23.
247.021234324474112052<BR>. 221
2: 42.
247.167866079210161859<BR>. 222
1: 103.
248.170346484306728631<BR>. 223
18: 1.
249.879398363598296293<BR>. 224
10: 3.
250.571267701043571730<BR>. 225
1: 104.
250.579767323960192016<BR>. 226
1: 105.
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270.707662848658748703<BR>. 244
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271.019956679614677059<BR>. 245
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275.870255746020300729<BR>. 249
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543.080742245495363585</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Wed, 28 May 2003 17:23:56 -0400 (EDT) "N. J. A. Sloane" <<A
href="mailto:njas@research.att.com">njas@research.att.com</A>>
writes:<BR>> <BR>> Kerry Mitchell just posted a message to the math fun
list<BR>> which relates to sequences, and I will append it below.<BR>>
<BR>> The OEIS prsently contains these examples:<BR>> <BR>> signature
sequences (1): A007336 A007337 A023115 A023116 A023117 <BR>> A023118 A023119
A023120 A023121 A023122 A023123 A023124<BR>> signature sequences (2): A023125
A023126 A023127 A023128 A023129 <BR>> A023130 A023131 A023132 A023133 A023134
A035796<BR>> <BR>> But the sig. seq. for phi seems to be missing - maybe
someone<BR>> could send it in.<BR>> <BR>> NJAS<BR>> <BR>>
<BR>> >Date: Wed, 28 May 2003 18:57:04 +0000<BR>> ><BR>>
>Lately, I've been playing with the signature of irrational numbers.
<BR>> My <BR>> >understanding, from what I've read on the web, is
this: For a <BR>> positive <BR>> >irrational number x, form the
numbers y = i + j*x, where i and j <BR>> are both <BR>> >positive
integers. Since x is irrational, no 2 y values will be <BR>> the same
for <BR>> >different i and j. Arrange the ys by size, then the
sequence of i <BR>> values is a <BR>> >fractal sequence, and the
signature of x.<BR>> ><BR>> >For x = phi ~ 1.618034, the first few
entries are:<BR>>
>i j y<BR>>
>1 1 2.618033989<BR>>
>2 1 3.618033989<BR>>
>1 2 4.236067977<BR>>
>3 1 4.618033989<BR>>
>2 2 5.236067977<BR>>
>4 1 5.618033989<BR>>
>1 3 5.854101966<BR>>
>3 2 6.236067977<BR>>
>5 1 6.618033989<BR>>
><BR>> >And the signature begins: 1, 2, 1, 3, 2, 4, 1, 3, 5. If
you strike <BR>> the first <BR>> >occurence of every integer in the
sequence, you get the original <BR>> sequence back, <BR>> >which makes
this a fractal sequence.<BR>> ><BR>> >My questions are:<BR>>
><BR>> >- What about the j sequence? From what I've seen
experimentally, <BR>> it seems to <BR>> >be a fractal sequence,
too. Why is the signature the i sequence as <BR>> opposed to <BR>>
>the j sequence? What's known about the relation of the j sequence
<BR>> to the i <BR>> >sequence?<BR>> ><BR>> >- For a
limited set of integers (both i and j run from 1 to 50), I <BR>> plotted i
<BR>> >vs. j, and the result was very interesting (I thought). You
can <BR>> find the <BR>> >picture here:<BR>> ><BR>>
w.fractalus.com/kerry/sigofphi.html<BR>> ><BR>> >The plot is one
continuous zig-zag line which seems to never cross <BR>> itself.
<BR>> >But, the angle of the line changes slightly, causing some areas to
<BR>> bunch up and <BR>> >appear darker, and others to spread out and
appear lighter. The <BR>> overall effect <BR>> >is of a series of
rectangles drawn in different shades of gray. <BR>> Can anyone <BR>>
>point me to other work that has been done on this?<BR>> ><BR>>
>Thanks,<BR>> >Kerry Mitchell<BR>> >--<BR>> ><A
href="mailto:lkmitch@att.net">lkmitch@att.net</A><BR>> ><A
href="http://www.fractalus.com/kerry">www.fractalus.com/kerry</A><BR>>
><BR>> <BR>> </DIV>
<DIV> </DIV>
<DIV></DIV></BODY></HTML>