Recently I ran across a link to some fun plotting with Wolfram Alpha (WA). It uses the following syntax more-or-less: “Graph a x curve”, where “x” is some object --- tree, curious george, dragon, etc. It’s fun to play with; at least I think so!
What I thought would be straight-forward turned out not to be (for me).
I thought about copying the equatons given by WA and using them in a Mathematica (v. 8) notebook for further experimentation. Problem is, I’m not able to get the plot to show up without many errors. I’m still in the rudimentary stages with the software (I don’t get the time with it due to other priorities). The ‘help’ is usually pretty good, but I’m not ‘getting it’ I guess.
How do you port the WA output into a Mathematica Notebook to reproduce the plot?
You should make the syntax compareble. Use Auto-Replace of your favourite plain-text editor (AkelPad is mine).
Do the folowing replaces:
“(” → “[”
“)” → “]”
"sin " ->“Sin”
and define the functions as f := <definition>
Take your tree =)
x[t_] := -2/3 Sin[11/8 - 256 t] - 4/11 Sin[1/4 - 235 t] -
2/7 Sin[1/3 - 234 t] - 1/7 Sin[5/7 - 232 t] -
1/2 Sin[11/12 - 227 t] - 3/4 Sin[1/5 - 220 t] - Sin[2/3 - 216 t] -
10/7 Sin[1/11 - 215 t] - 1/2 Sin[3/7 - 212 t] -
1/3 Sin[3/4 - 206 t] - 11/7 Sin[2/7 - 187 t] -
5/6 Sin[1/4 - 183 t] - 7/5 Sin[5/6 - 180 t] -
4/3 Sin[11/9 - 176 t] - 3/8 Sin[3/4 - 174 t] -
20/13 Sin[1/5 - 173 t] - 2 Sin[4/7 - 171 t] -
28/27 Sin[5/4 - 167 t] - 17/11 Sin[5/4 - 163 t] -
20/13 Sin[3/5 - 161 t] - 5/3 Sin[1 - 156 t] -
11/8 Sin[16/11 - 154 t] - 19/6 Sin[5/7 - 152 t] -
13/8 Sin[6/7 - 145 t] - 22/7 Sin[7/6 - 142 t] -
21/10 Sin[1/17 - 141 t] - 11/12 Sin[13/12 - 137 t] -
18/7 Sin[13/10 - 136 t] - 5/4 Sin[2/3 - 135 t] -
29/8 Sin[8/7 - 123 t] - 3/8 Sin[1/6 - 120 t] -
17/16 Sin[3/7 - 110 t] - 2 Sin[13/12 - 109 t] -
8/5 Sin[14/9 - 104 t] - 11/4 Sin[10/11 - 88 t] -
5/7 Sin[11/8 - 79 t] - 7/6 Sin[14/13 - 78 t] -
23/7 Sin[11/8 - 77 t] - 12/7 Sin[21/20 - 74 t] -
14/9 Sin[1/7 - 73 t] - 30/31 Sin[14/15 - 63 t] -
19/6 Sin[23/24 - 60 t] - 22/5 Sin[1/11 - 59 t] -
9/2 Sin[3/10 - 52 t] - 26/3 Sin[7/6 - 50 t] -
76/17 Sin[2/5 - 47 t] - 41/9 Sin[13/9 - 40 t] -
90/13 Sin[1/17 - 33 t] - 607/38 Sin[2/7 - 31 t] -
151/10 Sin[3/5 - 25 t] - 18 Sin[1 - 24 t] -
383/12 Sin[7/13 - 12 t] - 29 Sin[7/6 - 11 t] -
335/11 Sin[10/7 - 10 t] - 140/3 Sin[5/6 - 9 t] -
755/8 Sin[7/5 - 2 t] - 2675/6 Sin[5/6 - t] + 27/7 Sin[132 t] +
5/11 Sin[245 t] + 345/7 Sin[3 t + 14/3] + 2753/32 Sin[4 t + 27/8] +
533/6 Sin[5 t + 10/7] + 1177/21 Sin[6 t + 5/8] +
319/13 Sin[7 t + 5/8] + 81/4 Sin[8 t + 7/5] +
101/5 Sin[13 t + 17/5] + 149/5 Sin[14 t + 25/6] +
37/7 Sin[15 t + 31/8] + 1059/23 Sin[16 t + 19/5] +
97/6 Sin[17 t + 109/36] + 89/6 Sin[18 t + 8/5] +
436/29 Sin[19 t + 18/7] + 149/8 Sin[20 t + 7/11] +
33/2 Sin[21 t + 12/7] + 113/6 Sin[22 t + 1/9] +
80/3 Sin[23 t + 2/7] + 181/11 Sin[26 t + 16/7] +
151/8 Sin[27 t + 17/11] + 41/7 Sin[28 t + 17/6] +
44/9 Sin[29 t + 20/7] + 39/4 Sin[30 t + 8/5] +
39/10 Sin[32 t + 40/11] + 53/10 Sin[34 t + 17/4] +
63/5 Sin[35 t + 29/8] + 16/5 Sin[36 t + 6/5] +
111/10 Sin[37 t + 80/27] + 67/17 Sin[38 t + 19/7] +
29/15 Sin[39 t + 4] + 11/3 Sin[41 t + 23/8] +
109/9 Sin[42 t + 20/9] + 13/3 Sin[43 t + 31/9] +
21/11 Sin[44 t + 31/9] + 23/6 Sin[45 t + 17/7] +
23/3 Sin[46 t + 5/4] + 19/13 Sin[48 t + 9/7] +
21/8 Sin[49 t + 2/5] + 13/5 Sin[51 t + 21/5] +
9/5 Sin[53 t + 34/9] + 25/6 Sin[54 t + 13/7] +
7/8 Sin[55 t + 51/26] + 15/8 Sin[56 t + 24/7] +
7/2 Sin[57 t + 7/6] + 7/5 Sin[58 t + 23/8] +
11/5 Sin[61 t + 33/7] + 22/5 Sin[62 t + 17/7] + 7/3 Sin[64 t + 3] +
17/5 Sin[65 t + 11/5] + 7/3 Sin[66 t + 70/23] +
11/3 Sin[67 t + 15/16] + 17/7 Sin[68 t + 6/7] +
35/9 Sin[69 t + 23/8] + 129/32 Sin[70 t + 12/5] +
13/5 Sin[71 t + 30/7] + 10/7 Sin[72 t + 19/9] +
13/6 Sin[75 t + 11/9] + 16/11 Sin[76 t + 18/7] +
9/5 Sin[80 t + 6/5] + 4/9 Sin[81 t + 8/5] + 13/6 Sin[82 t + 1/14] +
35/8 Sin[83 t + 11/9] + 13/4 Sin[84 t + 2/9] +
32/11 Sin[85 t + 29/10] + Sin[86 t + 5/4] + 18/7 Sin[87 t + 1/5] +
17/10 Sin[89 t + 11/6] + 9/10 Sin[90 t + 17/7] +
19/7 Sin[91 t + 16/15] + 4/7 Sin[92 t + 29/7] +
57/28 Sin[93 t + 12/11] + 26/7 Sin[94 t + 3/2] +
11/8 Sin[95 t + 4/3] + 10/3 Sin[96 t + 19/10] +
8/5 Sin[97 t + 5/3] + 22/9 Sin[98 t + 39/10] +
27/14 Sin[99 t + 9/7] + 12/7 Sin[100 t + 27/8] +
40/9 Sin[101 t + 97/32] + 3/7 Sin[102 t + 53/27] +
8/5 Sin[103 t + 11/9] + 34/11 Sin[105 t + 4] +
1/2 Sin[106 t + 11/6] + 5/9 Sin[107 t + 25/12] +
19/9 Sin[108 t + 11/6] + 13/8 Sin[111 t + 7/2] +
5/2 Sin[112 t + 1/5] + 14/5 Sin[113 t + 5/2] +
21/5 Sin[114 t + 2/5] + 49/9 Sin[115 t + 44/15] +
13/5 Sin[116 t + 11/4] + 5/8 Sin[117 t + 11/3] +
23/8 Sin[118 t + 21/5] + 16/5 Sin[119 t + 35/9] +
13/9 Sin[121 t + 2/7] + 7/13 Sin[122 t + 5/3] +
11/4 Sin[124 t + 3/2] + 11/7 Sin[125 t + 17/5] +
3/5 Sin[126 t + 2/3] + 10/9 Sin[127 t + 88/29] +
9/5 Sin[128 t + 11/3] + 15/8 Sin[129 t + 5/6] +
17/10 Sin[130 t + 46/13] + 9/4 Sin[131 t + 15/4] +
5/2 Sin[133 t + 2] + 43/14 Sin[134 t + 13/7] +
9/7 Sin[138 t + 16/7] + 5/3 Sin[139 t + 5/3] +
4/7 Sin[140 t + 10/3] + 25/12 Sin[143 t + 4/5] +
12/5 Sin[144 t + 7/5] + 3/2 Sin[146 t + 13/10] +
19/10 Sin[147 t + 2/3] + 23/8 Sin[148 t + 3] +
8/3 Sin[149 t + 1/8] + 43/22 Sin[150 t + 32/9] +
12/7 Sin[151 t + 21/8] + 5/6 Sin[153 t + 9/2] +
11/7 Sin[155 t + 6/5] + 4/3 Sin[157 t + 6/5] + 5/8 Sin[158 t + 4] +
8/9 Sin[159 t + 10/7] + 15/16 Sin[160 t + 5/2] +
6/7 Sin[162 t + 11/7] + 2/3 Sin[164 t + 11/5] +
5/11 Sin[165 t + 17/10] + 11/3 Sin[166 t + 26/11] +
6/5 Sin[168 t + 67/15] + 3/4 Sin[169 t + 1/7] +
38/13 Sin[170 t + 19/7] + Sin[172 t + 137/34] +
17/9 Sin[175 t + 13/4] + 7/4 Sin[177 t + 3/10] +
5/6 Sin[178 t + 22/21] + 10/9 Sin[179 t + 9/2] +
12/7 Sin[181 t + 4/7] + 10/9 Sin[182 t + 9/5] +
5/6 Sin[184 t + 13/6] + 4/9 Sin[185 t + 18/7] +
1/2 Sin[186 t + 19/8] + 1/4 Sin[188 t + 28/11] +
3/5 Sin[189 t + 43/10] + 9/7 Sin[190 t + 7/3] +
2/7 Sin[191 t + 19/7] + 5/7 Sin[192 t + 7/5] +
3/5 Sin[193 t + 23/7] + 7/13 Sin[194 t + 1/5] +
11/8 Sin[195 t + 18/7] + 6/7 Sin[196 t + 11/3] +
1/2 Sin[197 t + 16/17] + 12/11 Sin[198 t + 3/7] +
3/5 Sin[199 t + 13/6] + 3/5 Sin[200 t + 23/6] +
5/7 Sin[201 t + 5/6] + 19/20 Sin[202 t + 40/11] +
3/4 Sin[203 t + 19/8] + 11/12 Sin[204 t + 9/7] +
22/21 Sin[205 t + 9/10] + 2/7 Sin[207 t + 52/17] +
3/7 Sin[208 t + 20/7] + 7/6 Sin[209 t + 1/6] +
6/13 Sin[210 t + 7/3] + 4/3 Sin[211 t + 12/5] +
7/9 Sin[213 t + 10/3] + 5/4 Sin[214 t + 41/10] +
2/3 Sin[217 t + 16/9] + 7/8 Sin[218 t + 12/5] +
2/3 Sin[219 t + 23/7] + 5/4 Sin[221 t + 53/18] +
33/32 Sin[222 t + 4/5] + 5/4 Sin[223 t + 35/8] +
5/8 Sin[224 t + 3] + 5/8 Sin[225 t + 3] + 2/7 Sin[226 t + 21/5] +
1/9 Sin[228 t + 24/7] + 1/7 Sin[229 t + 53/26] +
9/8 Sin[230 t + 18/5] + 6/5 Sin[231 t + 17/5] +
17/16 Sin[233 t + 9/8] + 3/10 Sin[236 t + 16/9] +
7/10 Sin[237 t + 25/7] + 6/5 Sin[238 t + 17/7] +
5/11 Sin[239 t + 15/8] + 2/9 Sin[240 t + 3/8] +
3/7 Sin[241 t + 11/3] + 1/3 Sin[242 t + 154/51] +
4/7 Sin[243 t + 8/3] + Sin[244 t + 17/6] + 7/6 Sin[246 t + 32/9] +
1/7 Sin[247 t + 58/19] + 4/5 Sin[248 t + 7/8] +
1/23 Sin[249 t + 12/5] + 5/7 Sin[250 t + 10/3] +
1/5 Sin[251 t + 10/3] + 2/3 Sin[252 t + 21/5] +
1/3 Sin[253 t + 1] + 3/5 Sin[254 t + 14/5] + 1/3 Sin[255 t + 35/18]
y[t_] := -4/7 Sin[19/13 - 256 t] - 6/7 Sin[7/6 - 250 t] -
1/4 Sin[4/3 - 248 t] - 1/9 Sin[10/9 - 247 t] -
8/15 Sin[1/2 - 236 t] - 2/5 Sin[4/3 - 235 t] -
7/8 Sin[10/7 - 233 t] - 1/10 Sin[5/7 - 230 t] -
1/6 Sin[5/6 - 226 t] - 10/9 Sin[5/7 - 222 t] -
9/8 Sin[8/7 - 220 t] - 2/7 Sin[2/5 - 211 t] - 3/8 Sin[5/4 - 208 t] -
Sin[1/4 - 203 t] - 9/10 Sin[10/7 - 198 t] - 4/5 Sin[5/6 - 197 t] -
3/5 Sin[5/6 - 194 t] - 7/9 Sin[6/5 - 193 t] -
3/5 Sin[1/16 - 186 t] - 19/9 Sin[3/10 - 181 t] -
5/9 Sin[7/5 - 178 t] - 4/3 Sin[1/9 - 177 t] -
8/9 Sin[1/10 - 164 t] - 9/5 Sin[18/17 - 161 t] -
19/8 Sin[1/2 - 157 t] - 1/4 Sin[1/5 - 155 t] -
13/6 Sin[6/11 - 153 t] - 11/7 Sin[4/3 - 147 t] -
11/4 Sin[6/7 - 144 t] - 23/7 Sin[7/8 - 143 t] -
29/15 Sin[1/6 - 134 t] - 11/7 Sin[6/5 - 131 t] -
7/6 Sin[1 - 128 t] - 9/5 Sin[1/4 - 118 t] -
23/24 Sin[11/10 - 113 t] - 3/4 Sin[7/8 - 112 t] -
19/4 Sin[10/11 - 105 t] - 17/8 Sin[2/5 - 98 t] -
19/7 Sin[5/9 - 85 t] - 9/17 Sin[10/7 - 80 t] -
16/7 Sin[7/9 - 76 t] - 18/17 Sin[5/4 - 70 t] -
13/5 Sin[7/5 - 69 t] - 11/5 Sin[1/2 - 66 t] - 8/3 Sin[3/2 - 64 t] -
11/3 Sin[6/7 - 60 t] - 23/6 Sin[2/7 - 59 t] -
43/8 Sin[7/6 - 43 t] - 43/22 Sin[17/11 - 41 t] -
9/2 Sin[4/9 - 37 t] - 305/34 Sin[10/7 - 35 t] - 3 Sin[6/7 - 32 t] -
21/5 Sin[1/2 - 28 t] - 47/6 Sin[6/7 - 27 t] -
173/7 Sin[3/2 - 16 t] - 19/3 Sin[15/14 - 10 t] -
58/3 Sin[1 - 5 t] - 221/11 Sin[3/4 - 3 t] + 16/9 Sin[45 t] +
11/5 Sin[79 t] + 1626/5 Sin[t + 43/11] + 1462/13 Sin[2 t + 28/9] +
674/9 Sin[4 t + 20/9] + 191/7 Sin[6 t + 3/5] +
65/9 Sin[7 t + 1/18] + 224/5 Sin[8 t + 1/23] +
163/4 Sin[9 t + 1/11] + 112/3 Sin[11 t + 50/11] +
178/7 Sin[12 t + 51/13] + 118/5 Sin[13 t + 38/9] +
94/5 Sin[14 t + 18/5] + 59/4 Sin[15 t + 10/7] +
37/6 Sin[17 t + 31/9] + 299/20 Sin[18 t + 9/5] +
143/6 Sin[19 t + 30/7] + 17/5 Sin[20 t + 7/6] +
110/9 Sin[21 t + 14/5] + 31/8 Sin[22 t + 5/8] +
11 Sin[23 t + 13/3] + 139/6 Sin[24 t + 13/14] +
32/3 Sin[25 t + 4/3] + 89/6 Sin[26 t + 26/9] +
65/7 Sin[29 t + 57/14] + 113/8 Sin[30 t + 3/2] +
73/8 Sin[31 t + 17/8] + 62/9 Sin[33 t + 11/4] +
26/5 Sin[34 t + 7/3] + 20/3 Sin[36 t + 25/6] +
19/3 Sin[38 t + 11/12] + 36/7 Sin[39 t + 3/2] + 4 Sin[40 t + 7/2] +
28/5 Sin[42 t + 15/4] + 2 Sin[44 t + 18/7] +
11/6 Sin[46 t + 16/9] + 31/5 Sin[47 t + 8/5] +
51/8 Sin[48 t + 15/7] + 34/11 Sin[49 t + 7/3] +
289/48 Sin[50 t + 14/5] + 32/9 Sin[51 t + 3] +
67/11 Sin[52 t + 15/16] + 12/7 Sin[53 t + 4/3] +
43/8 Sin[54 t + 7/5] + 25/8 Sin[55 t + 23/12] +
11/10 Sin[56 t + 6/7] + 4/5 Sin[57 t + 19/5] +
26/7 Sin[58 t + 29/8] + 16/7 Sin[61 t + 21/5] +
39/11 Sin[62 t + 45/11] + 70/23 Sin[63 t + 8/5] +
8/3 Sin[65 t + 29/9] + 20/9 Sin[67 t + 6/5] +
16/5 Sin[68 t + 11/5] + 23/6 Sin[71 t + 5/7] +
7/4 Sin[72 t + 47/16] + 11/7 Sin[73 t + 13/5] +
8/5 Sin[74 t + 7/2] + 8/5 Sin[75 t + 8/7] + 17/7 Sin[77 t + 16/7] +
15/8 Sin[78 t + 13/7] + Sin[81 t + 10/7] + 29/7 Sin[82 t + 11/9] +
65/16 Sin[83 t + 14/5] + 69/17 Sin[84 t + 3/4] +
5/8 Sin[86 t + 23/5] + 34/11 Sin[87 t + 2] + 5/3 Sin[88 t + 2] +
10/3 Sin[89 t + 17/6] + 97/32 Sin[90 t + 5/3] +
53/18 Sin[91 t + 3/2] + 61/20 Sin[92 t + 19/18] +
7/8 Sin[93 t + 19/5] + 31/9 Sin[94 t + 55/18] +
19/7 Sin[95 t + 5/2] + 16/7 Sin[96 t + 19/7] + 15/4 Sin[97 t + 2] +
1/11 Sin[99 t + 1/7] + 35/12 Sin[100 t + 2/9] +
40/9 Sin[101 t + 13/3] + 33/10 Sin[102 t + 99/49] +
34/9 Sin[103 t + 7/5] + 21/11 Sin[104 t + 27/13] +
11/7 Sin[106 t + 19/5] + 10/9 Sin[107 t + 15/8] +
13/6 Sin[108 t + 17/11] + 5 Sin[109 t + 3/4] +
27/10 Sin[110 t + 17/10] + 8/9 Sin[111 t + 16/5] +
34/5 Sin[114 t + 13/7] + 41/9 Sin[115 t + 31/7] +
5/4 Sin[116 t + 67/17] + 3/4 Sin[117 t + 16/5] +
13/5 Sin[119 t + 1/4] + 14/5 Sin[120 t + 11/6] +
20/19 Sin[121 t + 3/7] + 72/13 Sin[122 t + 59/15] +
3 Sin[123 t + 1/22] + 6/7 Sin[124 t + 21/5] +
11/8 Sin[125 t + 37/8] + 23/7 Sin[126 t + 3/7] +
13/7 Sin[127 t + 29/8] + 16/15 Sin[129 t + 47/12] +
21/8 Sin[130 t + 61/15] + 19/13 Sin[132 t + 1/2] +
29/14 Sin[133 t + 10/3] + 29/8 Sin[135 t + 9/2] +
5/11 Sin[136 t + 13/8] + 71/14 Sin[137 t + 23/9] +
4/9 Sin[138 t + 13/9] + 23/10 Sin[139 t + 34/9] +
17/9 Sin[140 t + 31/8] + 21/5 Sin[141 t + 29/7] +
28/11 Sin[142 t + 7/2] + 3/5 Sin[145 t + 16/9] +
4/3 Sin[146 t + 13/6] + 6/5 Sin[148 t + 7/6] +
9/7 Sin[149 t + 28/9] + 4/3 Sin[150 t + 5/2] +
12/7 Sin[151 t + 1] + 7/4 Sin[152 t + 17/4] +
21/20 Sin[154 t + 1] + 1/4 Sin[156 t + 1/4] + 17/6 Sin[158 t + 3] +
9/4 Sin[159 t + 3/4] + 16/9 Sin[160 t + 43/21] +
4/9 Sin[162 t + 11/7] + 9/7 Sin[163 t + 46/13] +
13/7 Sin[165 t + 16/7] + 8/5 Sin[166 t + 3/10] +
11/6 Sin[167 t + 25/6] + 5/7 Sin[168 t + 39/11] +
3/4 Sin[169 t + 29/7] + 53/18 Sin[170 t + 20/21] +
24/7 Sin[171 t + 22/5] + 8/9 Sin[172 t + 11/5] +
6/13 Sin[173 t + 40/9] + 16/15 Sin[174 t + 3] +
45/22 Sin[175 t + 17/6] + 6/7 Sin[176 t + 32/7] +
9/7 Sin[179 t + 28/11] + 3/2 Sin[180 t + 29/10] +
20/19 Sin[182 t + 9/4] + 9/7 Sin[183 t + 193/48] +
11/10 Sin[184 t + 6/5] + 4/3 Sin[185 t + 18/7] +
3/5 Sin[187 t + 11/4] + 11/8 Sin[188 t + 16/11] +
3/4 Sin[189 t + 27/10] + 1/2 Sin[190 t + 1/6] +
10/7 Sin[191 t + 38/13] + 3/10 Sin[192 t + 52/15] +
4/5 Sin[195 t + 5/8] + 2/3 Sin[196 t + 5/4] +
6/5 Sin[199 t + 1/7] + 1/4 Sin[200 t + 4/5] +
5/8 Sin[201 t + 2/5] + 5/6 Sin[202 t + 87/22] +
15/14 Sin[204 t + 2/7] + 5/7 Sin[205 t + 5/7] +
2/3 Sin[206 t + 7/10] + Sin[207 t + 33/13] +
6/7 Sin[209 t + 37/8] + 3/8 Sin[210 t + 10/3] +
1/10 Sin[212 t + 6/5] + 2/5 Sin[213 t + 20/7] +
1/2 Sin[214 t + 21/8] + 17/16 Sin[215 t + 37/8] +
12/7 Sin[216 t + 19/5] + 1/5 Sin[217 t + 19/7] +
1/5 Sin[218 t + 15/4] + 1/9 Sin[219 t + 2/5] +
1/3 Sin[221 t + 5/3] + 3/4 Sin[223 t + 17/7] +
1/4 Sin[224 t + 19/13] + 1/7 Sin[225 t + 1/7] +
1/3 Sin[227 t + 16/5] + 5/7 Sin[228 t + 10/3] +
4/9 Sin[229 t + 10/7] + 9/10 Sin[231 t + 8/7] +
1/2 Sin[232 t + 13/14] + 7/10 Sin[234 t + 2] +
7/8 Sin[237 t + 9/7] + 3/5 Sin[238 t + 1/4] +
4/7 Sin[239 t + 11/5] + 1/2 Sin[240 t + 11/8] +
1/28 Sin[241 t + 5/7] + 2/7 Sin[242 t + 11/5] +
1/3 Sin[243 t + 21/5] + 1/5 Sin[244 t + 19/8] +
5/8 Sin[245 t + 18/19] + 3/4 Sin[246 t + 29/10] +
1/5 Sin[249 t + 7/10] + 5/6 Sin[251 t + 4/3] +
1/6 Sin[252 t + 22/7] + 3/4 Sin[253 t + 37/8] +
6/11 Sin[254 t + 3/2] + 5/4 Sin[255 t + 7/5] + 647/5;
ParametricPlot[{x[t], y[t]}, {t, 0, 2 Pi}]
Well…I am quite embarrassed! These are things I should have known, or seen. I looked other things, and how I mised these, well, all I can say is, my mind must be tainted from a class I’m taking that involves using Mathcad.
Everything you said makes sense, and it works perfectly. I appreciate your kind help, thank you so much.
Wishing you all the best.
To Vitaliy Kaurov…I thank you as well for your excellent help!
