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sr s t r P t rs s
❢❛❝✉❧t②✳✈❛❧♣♦✳❡❞✉✴❧♣✉❞✇❡❧❧ ❱❛❧♣❛r❛✐s♦ ❙②♠♣♦s✐✉♠ ♦♥ ❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❛♥❞ ❈r❡❛t✐✈❡ ❊①♣r❡ss✐♦♥ ▼❛② ✶✹✱ ✷✵✷✵
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ■♠❛❣❡ ❝r❡❞✐t✿ ❤tt♣s✿✴✴♣❤✐❧✳❝❞❝✳❣♦✈✴❞❡t❛✐❧s✳❛s♣①❄♣✐❞❂✷✸✸✶✷ ❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ❖✉ts✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ■♥s✐❞❡
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ■♥s✐❞❡
I’ve discovered just how much nature there is to explore and appreciate even though we are in the middle of a giant city. I’m better at baking than I thought, and I’m only getting better while this goes on.
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ■♥s✐❞❡
I am excited that I may be able to attend more conferences than I ever thought I would because they are going virtual. I’ve come to realize just how much I appreciate my office, my campus, and my face-to-face colleagues and students.
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❚❤❡ ❚✐♠❡ ♦❢ ❈♦r♦♥❛ ❉✐s❝♦✈❡r②✱ ■♥s✐❞❡
I got to teach my 4 year old to read! Watching someone learn to read is AWESOME. I’ve realized anew how much I really do enjoy my kids, and also how beautiful the inquiring mind is.
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❛t ❱❛❧♣♦
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❛t ❱❛❧♣♦
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❛t ❱❛❧♣♦
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❛t ❱❛❧♣♦
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❲❤✐❝❤ ♣❡r♠✉t❛t✐♦♥s ❤❛✈❡ ♥♦ ♦r ❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❲❤✐❝❤ ♣❡r♠✉t❛t✐♦♥s ❤❛✈❡ ♥♦ ♦r ❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❲❤✐❝❤ ♣❡r♠✉t❛t✐♦♥s ❤❛✈❡ ♥♦ ♦r ❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
❲❤✐❝❤ ♣❡r♠✉t❛t✐♦♥s ❤❛✈❡ ♥♦ ♦r ❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r
❚❤❡♦r❡♠ ✭❙✐♠❡♦♥ ❛♥❞ ❙❝❤♠✐❞t✱ ✶✾✽✺✮
❚❤❡r❡ ❛r❡ ✷n−✶ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ n ✇✐t❤ ♥♦ ♦r ✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r
❚❤❡♦r❡♠ ✭❙✐♠❡♦♥ ❛♥❞ ❙❝❤♠✐❞t✱ ✶✾✽✺✮
❚❤❡r❡ ❛r❡ ✷n−✶ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ n ✇✐t❤ ♥♦ ♦r ✳ ❍♦✇ ♠❛♥② ❡①♣❡❝t❡❞✳✳✳ ❛s❝❡♥ts❄ ❞❡s❝❡♥ts❄ ♣❡❛❦s❄ ✈❛❧❧❡②s❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r
❚❤❡♦r❡♠ ✭❙✐♠❡♦♥ ❛♥❞ ❙❝❤♠✐❞t✱ ✶✾✽✺✮
❚❤❡r❡ ❛r❡ ✷n−✶ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ n ✇✐t❤ ♥♦ ♦r ✳ ❍♦✇ ♠❛♥② ❡①♣❡❝t❡❞✳✳✳ ❛s❝❡♥ts❄ ❞❡s❝❡♥ts❄ ♣❡❛❦s❄ ✈❛❧❧❡②s❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
❙❤✐❢t✐♥❣ ♣❡rs♣❡❝t✐✈❡✳✳✳
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r ▲▲▲ ▲▲❘ ▲❘▲ ▲❘❘ ❘▲▲ ❘▲❘ ❘❘▲ ❘❘❘
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r ▲ ▲▲ ▲ ▲❘ ▲ ❘▲ ▲ ❘❘ ❘▲ ▲ ❘▲ ❘ ❘ ❘▲ ❘❘❘ ◆✉♠❜❡r ♦❢ ❛s❝❡♥ts✿ ✭❝♦✉♥t ✐♥✐t✐❛❧ ▲ ♣❧✉s ❘▲s✮
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
P❡r♠✉t❛t✐♦♥s t❤❛t ❤❛✈❡ ♥♦ ♦r ▲▲▲ ▲ ▲❘ ▲❘ ▲ ▲❘ ❘ ❘▲▲ ❘ ▲❘ ❘❘▲ ❘❘❘ ◆✉♠❜❡r ♦❢ ❛s❝❡♥ts✿ ✭❝♦✉♥t ✐♥✐t✐❛❧ ▲ ♣❧✉s ❘▲s✮ ◆✉♠❜❡r ♦❢ ♣❡❛❦s✿ ✭❝♦✉♥t ▲❘s✮
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ P❡r♠✉t❛t✐♦♥s t♦ ❲♦r❞s
◆✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ n ✇✐t❤ t❤❡ ❣✐✈❡♥ st❛t✐st✐❝ ❡q✉❛❧ t♦ k
P❛tt❡r♥s ❆s❝❡♥ts ❉❡s❝❡♥ts P❡❛❦s ❱❛❧❧❡②s ✶✷✸✱✶✸✷ n ✷k
✷(n − k − ✶)
✷k + ✶
n − ✶ ✷k
n − ✶ k
k
✷k + ✶
✷k + ✶
✶, k = n − ✶ n ✷
k = n − ✷ ✶, k = ✵ n ✷
k = ✶ n, k = ✵ n − ✶ ✷
k = ✶ ✷, k = ✵ n ✷
k = ✶ ✷✶✸✱✷✸✶ n − ✶ k
k
✷k + ✶
✷k + ✶
n − ✶ k
k
k = ✵ ✷n−✶ − ✷, k = ✶ tr✐✈✐❛❧
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❉✐ss❡♠✐♥❛t✐♦♥
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❉✐ss❡♠✐♥❛t✐♦♥
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❉✐ss❡♠✐♥❛t✐♦♥
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤ ❉✐ss❡♠✐♥❛t✐♦♥
✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ ✶ ✶
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥ ❘❡s❡❛r❝❤
❇❡♥❡✜ts ♦❢ ✭❯♥❞❡r❣r❛❞✉❛t❡✮ ❘❡s❡❛r❝❤✿
❊♥❤❛♥❝❡s st✉❞❡♥t ❧❡❛r♥✐♥❣ t❤r♦✉❣❤ ♠❡♥t♦r✐♥❣ r❡❧❛t✐♦♥s❤✐♣s ✇✐t❤ ❢❛❝✉❧t② ■♥❝r❡❛s❡s r❡t❡♥t✐♦♥ ❛♥❞ ❣r❛❞✉❛t✐♦♥ ✐♥ ❛❝❛❞❡♠✐❝ ♣r♦❣r❛♠s ■♥❝r❡❛s❡s ❡♥r♦❧❧♠❡♥t ✐♥ ❣r❛❞✉❛t❡ ❡❞✉❝❛t✐♦♥ ❛♥❞ ♣r♦✈✐❞❡s ❡✛❡❝t✐✈❡ ❝❛r❡❡r ♣r❡♣❛r❛t✐♦♥ ❉❡✈❡❧♦♣s ❝r✐t✐❝❛❧ t❤✐♥❦✐♥❣✱ ❝r❡❛t✐✈✐t②✱ ♣r♦❜❧❡♠ s♦❧✈✐♥❣✱ ❛♥❞ ✐♥t❡❧❧❡❝t✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❉❡✈❡❧♦♣s ❛♥ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ r❡s❡❛r❝❤ ♠❡t❤♦❞♦❧♦❣② Pr♦♠♦t❡s ❛♥ ✐♥♥♦✈❛t✐♦♥✲♦r✐❡♥t❡❞ ❝✉❧t✉r❡
❢r♦♠ t❤❡ ❈♦✉♥❝✐❧ ♦♥ ❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤✿ ❤tt♣s✿✴✴✇✇✇✳❝✉r✳♦r❣✴✇❤♦✴♦r❣❛♥✐③❛t✐♦♥✴♠✐ss✐♦♥✴
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥ ❘❡s❡❛r❝❤
❇❡♥❡✜ts ♦❢ ✭❯♥❞❡r❣r❛❞✉❛t❡✮ ❘❡s❡❛r❝❤✿
❊♥❤❛♥❝❡s st✉❞❡♥t ❧❡❛r♥✐♥❣ t❤r♦✉❣❤ ♠❡♥t♦r✐♥❣ r❡❧❛t✐♦♥s❤✐♣s ✇✐t❤ ❢❛❝✉❧t② ■♥❝r❡❛s❡s r❡t❡♥t✐♦♥ ❛♥❞ ❣r❛❞✉❛t✐♦♥ ✐♥ ❛❝❛❞❡♠✐❝ ♣r♦❣r❛♠s ■♥❝r❡❛s❡s ❡♥r♦❧❧♠❡♥t ✐♥ ❣r❛❞✉❛t❡ ❡❞✉❝❛t✐♦♥ ❛♥❞ ♣r♦✈✐❞❡s ❡✛❡❝t✐✈❡ ❝❛r❡❡r ♣r❡♣❛r❛t✐♦♥ ❉❡✈❡❧♦♣s ❝r✐t✐❝❛❧ t❤✐♥❦✐♥❣✱ ❝r❡❛t✐✈✐t②✱ ♣r♦❜❧❡♠ s♦❧✈✐♥❣✱ ❛♥❞ ✐♥t❡❧❧❡❝t✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❉❡✈❡❧♦♣s ❛♥ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ r❡s❡❛r❝❤ ♠❡t❤♦❞♦❧♦❣② Pr♦♠♦t❡s ❛♥ ✐♥♥♦✈❛t✐♦♥✲♦r✐❡♥t❡❞ ❝✉❧t✉r❡
❢r♦♠ t❤❡ ❈♦✉♥❝✐❧ ♦♥ ❯♥❞❡r❣r❛❞✉❛t❡ ❘❡s❡❛r❝❤✿ ❤tt♣s✿✴✴✇✇✇✳❝✉r✳♦r❣✴✇❤♦✴♦r❣❛♥✐③❛t✐♦♥✴♠✐ss✐♦♥✴
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥ ❘❡s❡❛r❝❤
✕ ❩♦r❛ ◆❡❛❧❡ ❍✉rst♦♥
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
“Most importantly, I learned that math is collaborative. When doing research, math becomes a collaborative field where mathematicians are constantly pitching ideas to each other in hopes that together they can reach their goal.” “There is something so satisfying and rewarding about solving problems in a research setting. The results you come up with are novel; no one has ever done it before. Once you find a pattern and prove why it works, you have proven something that all mathematicians can use forever.” “I thoroughly enjoyed unleashing a new sense of self-motivation from the experience.”
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
math is collaborative you have proven something that all mathematicians can use forever
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
math is collaborative you have proven something that all mathematicians can use forever unleashing a new sense of self-motivation something so satisfying and rewarding about solving problems
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
❲❤❛t ❞✐s❝♦✈❡r✐❡s ❤❛✈❡ ②♦✉ ♠❛❞❡✳✳✳ ❛❜♦✉t ②♦✉r ♣r♦❥❡❝t ❛r❡❛❄ ❛❜♦✉t ②♦✉r ❞✐s❝✐♣❧✐♥❡❄ ❛❜♦✉t ②♦✉rs❡❧❢❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
❲❤❛t ❞✐s❝♦✈❡r✐❡s ❤❛✈❡ ②♦✉ ♠❛❞❡✳✳✳ ❛❜♦✉t ②♦✉r ♣r♦❥❡❝t ❛r❡❛❄ ❛❜♦✉t ②♦✉r ❞✐s❝✐♣❧✐♥❡❄ ❛❜♦✉t ②♦✉rs❡❧❢❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
❲❤❛t ❞✐s❝♦✈❡r✐❡s ❤❛✈❡ ②♦✉ ♠❛❞❡✳✳✳ ❛❜♦✉t ②♦✉r ♣r♦❥❡❝t ❛r❡❛❄ ❛❜♦✉t ②♦✉r ❞✐s❝✐♣❧✐♥❡❄ ❛❜♦✉t ②♦✉rs❡❧❢❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧
❉✐s❝♦✈❡r② ■♥s✐❞❡ ❛♥❞ ❖✉t
❲❤❛t ❞✐s❝♦✈❡r✐❡s ❤❛✈❡ ②♦✉ ♠❛❞❡✳✳✳ ❛❜♦✉t ②♦✉r ♣r♦❥❡❝t ❛r❡❛❄ ❛❜♦✉t ②♦✉r ❞✐s❝✐♣❧✐♥❡❄ ❛❜♦✉t ②♦✉rs❡❧❢❄
❉✐s❝♦✈❡r②✱ ■♥s✐❞❡ ❛♥❞ ❖✉t ▲❛r❛ P✉❞✇❡❧❧