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

By S. Zaidman

Best mathematics books

Pre-calculus Demystified (2nd Edition)

Your step by step way to studying precalculus

Understanding precalculus frequently opens the door to studying extra complex and useful math topics, and will additionally support fulfill collage requirements. Precalculus Demystified, moment version, is your key to gaining knowledge of this occasionally tough subject.

This self-teaching advisor offers basic precalculus ideas first, so you'll ease into the fundamentals. You'll progressively grasp services, graphs of capabilities, logarithms, exponents, and extra. As you move, you'll additionally overcome subject matters corresponding to absolute worth, nonlinear inequalities, inverses, trigonometric features, and conic sections. transparent, precise examples make it effortless to appreciate the cloth, and end-of-chapter quizzes and a last examination support make stronger key ideas.

It's a no brainer! You'll examine about:

Linear questions
Functions
Polynomial division
The rational 0 theorem
Logarithms
Matrix arithmetic
Basic trigonometry

Simple adequate for a newbie yet tough adequate for a complicated pupil, Precalculus Demystified, moment variation, moment variation, is helping you grasp this crucial topic.

Il matematico curioso. Dalla geometria del calcio all'algoritmo dei tacchi a spillo

Los angeles matematica informa, in modo consapevole e inconsapevole, anche i più semplici e automatici gesti quotidiani. Avreste mai pensato che los angeles matematica ci può aiutare in line with lavorare a maglia? E che esistono numeri fortunati consistent with giocare al lotto, enalotto e superenalotto? E che addirittura esiste una formulation consistent with scegliere correttamente l. a. coda al casello?

Mathematics Education and Subjectivity: Cultures and Cultural Renewal

This ebook rethinks mathematical instructing and studying with view to altering them to satisfy or face up to rising calls for. via contemplating how academics, scholars and researchers make experience in their worlds, the e-book explores how a few linguistic and socio-cultural destinations hyperlink to ordinary conceptions of arithmetic schooling.

Strong Limit Theorems in Noncommutative L2-Spaces

The noncommutative models of basic classical effects at the virtually definite convergence in L2-spaces are mentioned: person ergodic theorems, robust legislation of enormous numbers, theorems on convergence of orthogonal sequence, of martingales of powers of contractions and so forth. The proofs introduce new concepts in von Neumann algebras.

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

Example text

HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡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✿    &   ✉♥❞   *      & "!

HH< HH% HH@ HHH (HH@ (HH% (HH< (HH! (H1H (HH! (HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡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*(*>) HH! HH< HH% HH@ HHH (HH@ (HH% (HH< (HH! (H1H (HH! (HH< (HH% (HH@ HHH HH@ HH% HH< HH!     ❉✐❡s s✐♥❞ ❞✐❡ ❲❛❤rs❝❤❡✐♥❧✐❝❤❦❡✐t❡♥ ❡✐♥❡r ❇✐♥♦♠✐❛❧✈❡rt❡✐❧✉♥❣ ♠✐t ❞❡♥ P❛r❛♠❡t❡r♥   !