Ries Via the Generalized Weighted-Averages Method

Ries Via the Generalized Weighted-Averages Method

<p>Progress In Electromagnetics Research M, Vol. 14, 233–245, 2010 </p><p>Received 7 October 2010, Accepted 28 October 2010, Scheduled 8 November 2010 </p><p>Corresponding author: Athanasios G. Polimeridis (athanasios.polymeridis@epfl.ch). </p><p></p><ul style="display: flex;"><li style="flex:1">234 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">235 </li></ul><p></p><p>∞</p><p>=0 =0 </p><p>→ ∞ </p><p>→∞ </p><p>−</p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p></p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li><li style="flex:1">{</li><li style="flex:1">}</li></ul><p></p><p>0</p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">+1 </li><li style="flex:1">+1 </li></ul><p></p><p>−</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">236 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li></ul><p></p><p>→ ∞ | |&nbsp;∈ </p><p>→∞ </p><p>•••</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p></p><p>=0 </p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">+ +1 </li><li style="flex:1">+</li></ul><p></p><p>4</p><p>+</p><p>4</p><p>+<br>+<br>+<br>+</p><p></p><ul style="display: flex;"><li style="flex:1">+ −1 </li><li style="flex:1">+2 −1 </li></ul><p>( ) <br>+1 <br>( +1) <br>−1 </p><p>≥</p><p>( ) <br>( ) −1 <br>( ) 0</p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">( ) </li></ul><p>+1 </p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">(0) </li></ul><p>2</p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">2</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">237 </li></ul><p></p><p>(1) 2</p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">2 +1 </li></ul><p></p><p>−</p><p>−1 </p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p></p><p>=0 </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p>{ } </p><p>+1 +1 <br>+1 </p><p>−</p><p>0<br>0</p><p></p><ul style="display: flex;"><li style="flex:1">238 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li></ul><p></p><p>0</p><p>+1 <br>+1 <br>+1 <br>+1 </p><p>0</p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>+1 +1 <br>+1 </p><p>0</p><p>0</p><p>0</p><p>+1 </p><p>−</p><p>+1 </p><p>∞<br>−</p><p></p><ul style="display: flex;"><li style="flex:1">∼</li><li style="flex:1">→ ∞ </li></ul><p></p><p>=0 <br>−2 </p><p>→ ∞ </p><p></p><ul style="display: flex;"><li style="flex:1">+1 </li><li style="flex:1">+1 </li></ul><p></p><p>−<br>∼</p><p>+1 </p><p>−<br>0</p><p>||<br>||</p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">239 </li></ul><p></p><p>−</p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">( )&nbsp;( ) </li></ul><p>+1 <br>( ) <br>( +1) </p><p>≥</p><p>( ) </p><p>−</p><p>+1 <br>( ) </p><p>≈ − </p><p>+1 <br>(0) 0<br>(1) 0<br>( ) 0</p><p>(0) 1<br>(1) 1<br>( −1) 1<br>(0) 2<br>( −2) 2</p><p>(0) </p><ul style="display: flex;"><li style="flex:1">(0) </li><li style="flex:1">( ) </li></ul><p>0</p><p>· · · </p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">240 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">II </li><li style="flex:1">III </li><li style="flex:1">IV </li></ul><p>( ) </p><p>→<br>→</p><p></p><ul style="display: flex;"><li style="flex:1">III </li><li style="flex:1">IV </li></ul><p></p><p>−</p><p>∞</p><p>√<br>−</p><p></p><ul style="display: flex;"><li style="flex:1">√</li><li style="flex:1">−</li></ul><p></p><p>=0 </p><p>∞</p><p>−</p><p>=0 </p><p>·</p><p>( ) </p><p></p><ul style="display: flex;"><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">241 </li></ul><p></p><p>16 14 12 10 8<br>16 14 12 10 8</p><ul style="display: flex;"><li style="flex:1">6</li><li style="flex:1">6</li></ul><p></p><p>WA <sub style="top: 0.178em;">IV </sub>(asymptotic) </p><p></p><ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">4</li></ul><p></p><p>WA <sub style="top: 0.178em;">I </sub>(asymptotic) u</p><p>ε</p><p>u</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><p>ελλ</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><ul style="display: flex;"><li style="flex:1">(a) Test example (29) </li><li style="flex:1">(b) Test example (30) </li></ul><p></p><p>∞</p><p>2<br>2<br>=0 </p><p>∞</p><p>·</p><p>=0 </p><p></p><ul style="display: flex;"><li style="flex:1">242 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">16 </li><li style="flex:1">16 </li></ul><p></p><p>WA<sub style="top: 0.1031em;">III </sub>(asymptotic) <br>WA <sub style="top: 0.1031em;">III </sub>(asymptotic) </p><p></p><ul style="display: flex;"><li style="flex:1">u</li><li style="flex:1">u</li></ul><p></p><p>14 12 10 8<br>14 12 10 8</p><ul style="display: flex;"><li style="flex:1">6</li><li style="flex:1">6</li></ul><p>4<br>4</p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><ul style="display: flex;"><li style="flex:1">(a) Test example (31) </li><li style="flex:1">(b) Test example (32) </li></ul><p></p><p>16 14 12 10 8<br>16 14 12 10 8</p><p>WA<sub style="top: 0.1031em;">IV </sub>(asymptotic) <br>WA<sub style="top: 0.1032em;">IV </sub>(asymptotic) vv</p><p>ε</p><p></p><ul style="display: flex;"><li style="flex:1">6</li><li style="flex:1">6</li></ul><p>4<br>4</p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p>0<br>0</p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li><li style="flex:1">6</li><li style="flex:1">8</li><li style="flex:1">10 12 14 16 18 20 </li></ul><p></p><p>k</p><p></p><ul style="display: flex;"><li style="flex:1">(a) Test example (33) </li><li style="flex:1">(b) Test example (34) </li></ul><p></p><p>∞</p><p>+1 <br>=0 </p><p>∞</p><p>=0 </p><p></p><ul style="display: flex;"><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">243 </li><li style="flex:1">244 </li><li style="flex:1">Polimeridis, Golubovi´c Ni´ciforovi´c, and Mosig </li><li style="flex:1">Progress In Electromagnetics Research M, Vol. 14, 2010 </li><li style="flex:1">245 </li></ul><p></p>

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