Almost-periodic Functions in Abstract Spaces (Research Notes by S. Zaidman

By S. Zaidman

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Extra info for Almost-periodic Functions in Abstract Spaces (Research Notes Inmathematics Series)

Example text

HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡s s✐♥❞ ❞✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t❡♥ ❡✐♥❡r ❇✐♥♦♠✐❛❧✈❡rt❡✐❧✉♥❣ ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥   ! ;$ " %;?? H " %<%1 " %$H$ " %$$H " %%1! @ ❆❜❜✳ ✶✳✼✿ ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ♠✐t ✸ P❡r✐♦❞❡♥ ❢ür ❞✐❡ ❆❞✐❞❛s✲❆❦t✐❡ ③✉r ✒Pr♦❣♥♦s❡✏ ❞❡r ❦ü♥❢t✐❣❡♥ ❆❦t✐❡♥❦✉rs❡♥t✇✐❝❦❧✉♥❣ ▲❛✉t ✉♥s❡r❡s ▼♦❞❡❧❧s ❦❛♥♥ ❞❡r ❆❦t✐❡♥❦✉rs ❞❡r ❆❞✐❞❛s✲❆❦t✐❡ ❛♠ ✵✾✳✵✻✳✵✽ ✈✐❡r ❲❡r✲ t❡ ❛♥♥❡❤♠❡♥✿ ✺✵✱✷✶ ❜③✇✳ ✹✸✱✸✷ ♠✐t ❥❡✇❡✐❧s ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥ ✽✶ ✉♥❞ ✹✼✱✽✵ ❜③✇✳ ✹✺✱✺✵ ♠✐t ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥ ❥❡✇❡✐❧s ✸✽ ✳ ❉❡r r❡❛❧❡ ❑✉rs ❞❡r ❆❞✐❞❛s✲❆❦t✐❡ ❧❛❣ ❛♠ ✷✻✳✵✺✳✵✽ ❜❡✐ ✹✺✱✶✺✱ ❛♠ ✵✷✳✵✻✳✵✽ ❜❡✐ ✹✺✱✵✺ ✉♥❞ ❛♠ ✵✾✳✵✻✳✵✽ ❜❡✐ ✹✹✱✸✷✳ ✶✳✶✵✳ ◆❖❘▼❆▲❱❊❘❚❊■▲❯◆● ❯◆❉ ❆❑❚■❊◆❑❯❘❙❊ ✷✼ ❉❛s ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ✐st ✈❡r❣❧❡✐❝❤❜❛r ♠✐t ❞❡r ❇❡s❝❤r❡✐❜✉♥❣ ❡✐♥❡r ❆❦t✐❡♥❦✉rs✲ ❡♥t✇✐❝❦❧✉♥❣ ❞✉r❝❤ ❞❛s ❲❡r❢❡♥ ❡✐♥❡r ▼ü♥③❡ ❢ür ❥❡❞❡ ❡✐♥③❡❧♥❡ P❡r✐♦❞❡✳ ❊rs❝❤❡✐♥t ❑♦♣❢✱ st❡✐❣t ❞❡r ❆❦t✐❡♥❦✉rs ✐♥ ❞❡r ❜❡tr❡✛❡♥❞❡♥ P❡r✐♦❞❡ ❛✉❢ (  ✱ ❢ä❧❧t ❤✐♥❣❡❣❡♥ ❩❛❤❧✱ s✐♥❦t ❞❡r ❑✉rs ❛✉❢   ✳ ❉✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ❢ür ❑♦♣❢ ❜③✇✳ ❩❛❤❧ ❜❡trä❣t ❥❡  ✱ ❞❛♠✐t st❡✐❣t ❜③✇✳ s✐♥❦t ❞❡r ❆❦t✐❡♥❦✉rs ❡❜❡♥❢❛❧❧s ♠✐t ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥  ✳ ❉❛ ❞✐❡ ▼ü♥③❡ ❞❛rü❜❡r ❤✐♥❛✉s ❣❡❞ä❝❤t♥✐s❧♦s ✐st✱ tr❡t❡♥ ❩❛❤❧ ✉♥❞ ❑♦♣❢ ✐♠♠❡r ♠✐t ❞❡r✲ s❡❧❜❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ❛✉❢✱ ✉♥❛❜❤ä♥❣✐❣ ❞❛✈♦♥✱ ✇❛s ❞✐❡ ▼ü♥③❡ ❡✐♥❡♥ ❲✉r❢ ✈♦r❤❡r ❛♥③❡✐❣t❡✳ ➘❤♥❧✐❝❤ ✈❡r❤ä❧t ❡s s✐❝❤ ♠✐t ❞❡♠ ❆❦t✐❡♥❦✉rs✳ ❯♥❛❜❤ä♥❣✐❣ ❞❛✈♦♥✱ ♦❜ ❞❡r ❆❦t✐❡♥❦✉rs ✐♥ ❞❡r ✈♦r❤❡r✐❣❡♥ P❡r✐♦❞❡ st✐❡❣ ♦❞❡r s❛♥❦✱ st❡✐❣t ❜③✇✳ s✐♥❦t ❞❡r ❆❦t✐✲ ❡♥❦✉rs ✐♥ ❞❡r ❞❛r❛✉✛♦❧❣❡♥❞❡♥ P❡r✐♦❞❡ ♠✐t ❞❡rs❡❧❜❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥  ✳ ▼✐t ❞❡♠ ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ❤❛❜❡♥ ✇✐r ❡✐♥ ❡rst❡s ❡✐♥❢❛❝❤❡s ▼♦❞❡❧❧ ❦❡♥♥❡♥ ❣❡❧❡r♥t✱ ♠✐t ❞❡♠ ❡s ♠ö❣❧✐❝❤ ✐st✱ ✐♠ ❘❛❤♠❡♥ ❞❡r ❡r❧ä✉t❡rt❡♥ ▼♦❞❡❧❧❛♥♥❛❤♠❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤✲ ❦❡✐ts❛✉ss❛❣❡♥ ü❜❡r ❦ü♥❢t✐❣❡ ❆❦t✐❡♥❦✉rs❡ ③✉ tr❡✛❡♥✳ ■♠ ❘❛❤♠❡♥ ❞✐❡s❡s ▼♦❞❡❧❧s ✐st ❡s ❞❡♥♥♦❝❤ ✇✐❝❤t✐❣✱ ❞✐❡ ▼♦❞❡❧❧❛♥♥❛❤♠❡♥ ③✉ ❤✐♥t❡r❢r❛❣❡♥✳ ❑r✐t✐s❝❤ ③✉ s❡❤❡♥ ✐st ③✉♠ ❇❡✐s♣✐❡❧✱ ❞❛ss ❛✉s ❉❛t❡♥ ❞❡r ❱❡r❣❛♥❣❡♥❤❡✐t Pr♦❣♥♦s❡♥ ❢ür ❞✐❡ ❩✉❦✉♥❢t ❣❡tät✐❣t ✇❡r✲ ❞❡♥✳ ❉❛rü❜❡r ❤✐♥❛✉s ❜❧❡✐❜❡♥ ( ✉♥❞  ü❜❡r sä♠t❧✐❝❤❡ Pr♦❣♥♦s❡♥ ❦♦♥st❛♥t✱ ❞✳ ❤✳✱ ❞✐❡ ▼♦❞❡❧❧♣❛r❛♠❡t❡r s✐♥❞ st❛t✐s❝❤✳ ❉✐❡s ✐st ✐♥s♦❢❡r♥ ♣r♦❜❧❡♠❛t✐s❝❤✱ ❞❛ss ✐♥ Pr♦❣♥♦s❡♥ ü❜❡r ❡✐♥❡♥ ❧ä♥❣❡r❡♥ ❩❡✐tr❛✉♠ ♥❡✉❡ ■♥❢♦r♠❛t✐♦♥❡♥ ♥✐❝❤t ✐♥ ❞✐❡ ❇❡✇❡rt✉♥❣ ❡✐♥✢✐❡✲ ÿ❡♥✳ ❙♦ ❜❧❡✐❜❡♥ ❜❡✐s♣✐❡❧s✇❡✐s❡ ✉♥✈♦r❤❡rs❡❤❜❛r❡ ❊r❡✐❣♥✐ss❡ ✭③✳ ❇✳ ❆♥❧❡❣❡r♠❡♥t❛❧✐tät✱ ✇✐rts❝❤❛❢t❧✐❝❤❡ ➘♥❞❡r✉♥❣❡♥ ✐♥ ❞❡r ❆●✮ ✉♥❜❡rü❝❦s✐❝❤t✐❣t✳ ●❡♥❡r❡❧❧ ✇✐r❞ ❞❛s ❆❦t✐❡♥✲ ❦✉rs❣❡s❝❤❡❤❡♥ st❛r❦ ✈❡r❡✐♥❢❛❝❤t ♠♦❞❡❧❧✐❡rt✳ ❉✐❡ ❊✐❣❡♥s❝❤❛❢t ❞❡s ▼♦❞❡❧❧s✱ ❞❛ss ❞❡r ❆❦t✐❡♥❦✉rs ♥❛❝❤ ❡✐♥❡r P❡r✐♦❞❡ ♥✉r ③✇❡✐ ❲❡rt❡ ❛♥♥❡❤♠❡♥ ❦❛♥♥✱ ✐st ✉♥r❡❛❧✐st✐s❝❤✳ ✶✳✶✵ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ ✉♥❞ ❆❦t✐❡♥❦✉rs❡ ✶✳✶✵✳✶ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ ❉✐❡ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ st❡❧❧t ❡✐♥❡ ❣r✉♥❞❧❡❣❡♥❞❡ ❱❡rt❡✐❧✉♥❣ ❞❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐ts✲ r❡❝❤♥✉♥❣ ❞❛r✳ ❙✐❡ ✜♥❞❡t ❜❡✐ ③❛❤❧r❡✐❝❤❡♥ ♣r❛❦t✐s❝❤❡♥ Pr♦❜❧❡♠❡♥ ❆♥✇❡♥❞✉♥❣✳ ❇❡✲ tr❛❝❤t❡♥ ✇✐r ③✉♥ä❝❤st ❞✐❡ ❉❡✜♥✐t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✶ ✭◆♦r♠❛❧✈❡rt❡✐❧t❡ ❩✉❢❛❧❧s❣röÿ❡✮✳ ❊✐♥❡ st❡t✐❣❡ ❩✉❢❛❧❧s❣röÿ❡  ❤❡✐ÿt ♥♦r♠❛❧✈❡rt❡✐❧t ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥ ❢ür ❛❧❧❡ *  ❢♦❧❣❡♥❞❡ ❉✐❝❤t❡ ❜❡s✐t③t✿   ▼❛♥ s❝❤r❡✐❜t✿    &   ✉♥❞   *      & "!

HH< HH% HH@ HHH (HH@ (HH% (HH< (HH! (H1H (HH! (HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡s s✐♥❞ ❞✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t❡♥ ❡✐♥❡r ❇✐♥♦♠✐❛❧✈❡rt❡✐❧✉♥❣ ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥   ! ;$ " %;?? H " %<%1 " %$H$ " %$$H " %%1! @ ❆❜❜✳ ✶✳✼✿ ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ♠✐t ✸ P❡r✐♦❞❡♥ ❢ür ❞✐❡ ❆❞✐❞❛s✲❆❦t✐❡ ③✉r ✒Pr♦❣♥♦s❡✏ ❞❡r ❦ü♥❢t✐❣❡♥ ❆❦t✐❡♥❦✉rs❡♥t✇✐❝❦❧✉♥❣ ▲❛✉t ✉♥s❡r❡s ▼♦❞❡❧❧s ❦❛♥♥ ❞❡r ❆❦t✐❡♥❦✉rs ❞❡r ❆❞✐❞❛s✲❆❦t✐❡ ❛♠ ✵✾✳✵✻✳✵✽ ✈✐❡r ❲❡r✲ t❡ ❛♥♥❡❤♠❡♥✿ ✺✵✱✷✶ ❜③✇✳ ✹✸✱✸✷ ♠✐t ❥❡✇❡✐❧s ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥ ✽✶ ✉♥❞ ✹✼✱✽✵ ❜③✇✳ ✹✺✱✺✵ ♠✐t ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥ ❥❡✇❡✐❧s ✸✽ ✳ ❉❡r r❡❛❧❡ ❑✉rs ❞❡r ❆❞✐❞❛s✲❆❦t✐❡ ❧❛❣ ❛♠ ✷✻✳✵✺✳✵✽ ❜❡✐ ✹✺✱✶✺✱ ❛♠ ✵✷✳✵✻✳✵✽ ❜❡✐ ✹✺✱✵✺ ✉♥❞ ❛♠ ✵✾✳✵✻✳✵✽ ❜❡✐ ✹✹✱✸✷✳ ✶✳✶✵✳ ◆❖❘▼❆▲❱❊❘❚❊■▲❯◆● ❯◆❉ ❆❑❚■❊◆❑❯❘❙❊ ✷✼ ❉❛s ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ✐st ✈❡r❣❧❡✐❝❤❜❛r ♠✐t ❞❡r ❇❡s❝❤r❡✐❜✉♥❣ ❡✐♥❡r ❆❦t✐❡♥❦✉rs✲ ❡♥t✇✐❝❦❧✉♥❣ ❞✉r❝❤ ❞❛s ❲❡r❢❡♥ ❡✐♥❡r ▼ü♥③❡ ❢ür ❥❡❞❡ ❡✐♥③❡❧♥❡ P❡r✐♦❞❡✳ ❊rs❝❤❡✐♥t ❑♦♣❢✱ st❡✐❣t ❞❡r ❆❦t✐❡♥❦✉rs ✐♥ ❞❡r ❜❡tr❡✛❡♥❞❡♥ P❡r✐♦❞❡ ❛✉❢ (  ✱ ❢ä❧❧t ❤✐♥❣❡❣❡♥ ❩❛❤❧✱ s✐♥❦t ❞❡r ❑✉rs ❛✉❢   ✳ ❉✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ❢ür ❑♦♣❢ ❜③✇✳ ❩❛❤❧ ❜❡trä❣t ❥❡  ✱ ❞❛♠✐t st❡✐❣t ❜③✇✳ s✐♥❦t ❞❡r ❆❦t✐❡♥❦✉rs ❡❜❡♥❢❛❧❧s ♠✐t ❡✐♥❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥  ✳ ❉❛ ❞✐❡ ▼ü♥③❡ ❞❛rü❜❡r ❤✐♥❛✉s ❣❡❞ä❝❤t♥✐s❧♦s ✐st✱ tr❡t❡♥ ❩❛❤❧ ✉♥❞ ❑♦♣❢ ✐♠♠❡r ♠✐t ❞❡r✲ s❡❧❜❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ❛✉❢✱ ✉♥❛❜❤ä♥❣✐❣ ❞❛✈♦♥✱ ✇❛s ❞✐❡ ▼ü♥③❡ ❡✐♥❡♥ ❲✉r❢ ✈♦r❤❡r ❛♥③❡✐❣t❡✳ ➘❤♥❧✐❝❤ ✈❡r❤ä❧t ❡s s✐❝❤ ♠✐t ❞❡♠ ❆❦t✐❡♥❦✉rs✳ ❯♥❛❜❤ä♥❣✐❣ ❞❛✈♦♥✱ ♦❜ ❞❡r ❆❦t✐❡♥❦✉rs ✐♥ ❞❡r ✈♦r❤❡r✐❣❡♥ P❡r✐♦❞❡ st✐❡❣ ♦❞❡r s❛♥❦✱ st❡✐❣t ❜③✇✳ s✐♥❦t ❞❡r ❆❦t✐✲ ❡♥❦✉rs ✐♥ ❞❡r ❞❛r❛✉✛♦❧❣❡♥❞❡♥ P❡r✐♦❞❡ ♠✐t ❞❡rs❡❧❜❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t ✈♦♥  ✳ ▼✐t ❞❡♠ ❘❛♥❞♦♠✲❲❛❧❦✲▼♦❞❡❧❧ ❤❛❜❡♥ ✇✐r ❡✐♥ ❡rst❡s ❡✐♥❢❛❝❤❡s ▼♦❞❡❧❧ ❦❡♥♥❡♥ ❣❡❧❡r♥t✱ ♠✐t ❞❡♠ ❡s ♠ö❣❧✐❝❤ ✐st✱ ✐♠ ❘❛❤♠❡♥ ❞❡r ❡r❧ä✉t❡rt❡♥ ▼♦❞❡❧❧❛♥♥❛❤♠❡♥ ❲❛❤rs❝❤❡✐♥❧✐❝❤✲ ❦❡✐ts❛✉ss❛❣❡♥ ü❜❡r ❦ü♥❢t✐❣❡ ❆❦t✐❡♥❦✉rs❡ ③✉ tr❡✛❡♥✳ ■♠ ❘❛❤♠❡♥ ❞✐❡s❡s ▼♦❞❡❧❧s ✐st ❡s ❞❡♥♥♦❝❤ ✇✐❝❤t✐❣✱ ❞✐❡ ▼♦❞❡❧❧❛♥♥❛❤♠❡♥ ③✉ ❤✐♥t❡r❢r❛❣❡♥✳ ❑r✐t✐s❝❤ ③✉ s❡❤❡♥ ✐st ③✉♠ ❇❡✐s♣✐❡❧✱ ❞❛ss ❛✉s ❉❛t❡♥ ❞❡r ❱❡r❣❛♥❣❡♥❤❡✐t Pr♦❣♥♦s❡♥ ❢ür ❞✐❡ ❩✉❦✉♥❢t ❣❡tät✐❣t ✇❡r✲ ❞❡♥✳ ❉❛rü❜❡r ❤✐♥❛✉s ❜❧❡✐❜❡♥ ( ✉♥❞  ü❜❡r sä♠t❧✐❝❤❡ Pr♦❣♥♦s❡♥ ❦♦♥st❛♥t✱ ❞✳ ❤✳✱ ❞✐❡ ▼♦❞❡❧❧♣❛r❛♠❡t❡r s✐♥❞ st❛t✐s❝❤✳ ❉✐❡s ✐st ✐♥s♦❢❡r♥ ♣r♦❜❧❡♠❛t✐s❝❤✱ ❞❛ss ✐♥ Pr♦❣♥♦s❡♥ ü❜❡r ❡✐♥❡♥ ❧ä♥❣❡r❡♥ ❩❡✐tr❛✉♠ ♥❡✉❡ ■♥❢♦r♠❛t✐♦♥❡♥ ♥✐❝❤t ✐♥ ❞✐❡ ❇❡✇❡rt✉♥❣ ❡✐♥✢✐❡✲ ÿ❡♥✳ ❙♦ ❜❧❡✐❜❡♥ ❜❡✐s♣✐❡❧s✇❡✐s❡ ✉♥✈♦r❤❡rs❡❤❜❛r❡ ❊r❡✐❣♥✐ss❡ ✭③✳ ❇✳ ❆♥❧❡❣❡r♠❡♥t❛❧✐tät✱ ✇✐rts❝❤❛❢t❧✐❝❤❡ ➘♥❞❡r✉♥❣❡♥ ✐♥ ❞❡r ❆●✮ ✉♥❜❡rü❝❦s✐❝❤t✐❣t✳ ●❡♥❡r❡❧❧ ✇✐r❞ ❞❛s ❆❦t✐❡♥✲ ❦✉rs❣❡s❝❤❡❤❡♥ st❛r❦ ✈❡r❡✐♥❢❛❝❤t ♠♦❞❡❧❧✐❡rt✳ ❉✐❡ ❊✐❣❡♥s❝❤❛❢t ❞❡s ▼♦❞❡❧❧s✱ ❞❛ss ❞❡r ❆❦t✐❡♥❦✉rs ♥❛❝❤ ❡✐♥❡r P❡r✐♦❞❡ ♥✉r ③✇❡✐ ❲❡rt❡ ❛♥♥❡❤♠❡♥ ❦❛♥♥✱ ✐st ✉♥r❡❛❧✐st✐s❝❤✳ ✶✳✶✵ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ ✉♥❞ ❆❦t✐❡♥❦✉rs❡ ✶✳✶✵✳✶ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ ❉✐❡ ◆♦r♠❛❧✈❡rt❡✐❧✉♥❣ st❡❧❧t ❡✐♥❡ ❣r✉♥❞❧❡❣❡♥❞❡ ❱❡rt❡✐❧✉♥❣ ❞❡r ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐ts✲ r❡❝❤♥✉♥❣ ❞❛r✳ ❙✐❡ ✜♥❞❡t ❜❡✐ ③❛❤❧r❡✐❝❤❡♥ ♣r❛❦t✐s❝❤❡♥ Pr♦❜❧❡♠❡♥ ❆♥✇❡♥❞✉♥❣✳ ❇❡✲ tr❛❝❤t❡♥ ✇✐r ③✉♥ä❝❤st ❞✐❡ ❉❡✜♥✐t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✶✳✶✵✳✶ ✭◆♦r♠❛❧✈❡rt❡✐❧t❡ ❩✉❢❛❧❧s❣röÿ❡✮✳ ❊✐♥❡ st❡t✐❣❡ ❩✉❢❛❧❧s❣röÿ❡  ❤❡✐ÿt ♥♦r♠❛❧✈❡rt❡✐❧t ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥ ❢ür ❛❧❧❡ *  ❢♦❧❣❡♥❞❡ ❉✐❝❤t❡ ❜❡s✐t③t✿   ▼❛♥ s❝❤r❡✐❜t✿    &   ✉♥❞   *      & "!

P■❚❊▲ ✶✳ ❆❑❚■❊◆ ✭❛✮ ✭❜✮ ✭❝✮ ❆❜❜✳ ✶✳✸✿ ✭❛✮ ◆❡❣❛t✐✈ ❦♦rr❡❧✐❡rt❡✱ ✭❜✮ ✉♥❦♦rr❡❧✐❡rt❡✱ ✭❝✮ ♣♦s✐t✐✈ ❦♦rr❡❧✐❡rt❡ ❉❛t❡♥♣❛❛r❡ ❇❡✐s♣✐❡❧ ✶✳✽✳✻ ✭❑♦rr❡❧❛t✐♦♥ ③✇✐s❝❤❡♥ ❘❡♥❞✐t❡♥ ✈❡rs❝❤✐❡❞❡♥❡r ❆❦t✐❡♥✮✳ ❆❜❜✐❧❞✉♥❣ ✶✳✹ ③❡✐❣t ❞✐❡ ❘❡♥❞✐t❡♣❛❛r❡ ❣❧❡✐❝❤❡r ❩❡✐trä✉♠❡ ❞❡r ❆❧❧✐❛♥③✲❆❦t✐❡ ✉♥❞ ❞❡r ▼ü♥❝❤❡♥❡r✲ ❘ü❝❦✲❆❦t✐❡ ✈♦♠ ✵✹✳✵✻✳✵✼ ❜✐s ✶✶✳✵✻✳✵✽✳ /&8)>',)9'  -)> -  8 B-' - 8 B*(*>) HH! HH< HH% HH@ HHH (HH@ (HH% (HH< (HH! (H1H (HH! (HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡s s✐♥❞ ❞✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t❡♥ ❡✐♥❡r ❇✐♥♦♠✐❛❧✈❡rt❡✐❧✉♥❣ ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥   !

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