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1. Primitivity rank for random elements in free groups

   https://doi.org/10.1515/jgth-2021-0150  

   Kapovich, Ilya (March 2022, Journal of Group Theory)

   Abstract For a free group F r F_{r} of finite rank r ≥ 2 r\geq 2 and a non-trivial element w ∈ F r w\in F_{r} , the primitivity rank π ⁢ ( w ) \pi(w) is the smallest rank of a subgroup H ≤ F r H\leq F_{r} such that w ∈ H w\in H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ \pi(w)=\infty ).The set of all subgroups of F r F_{r} of rank π ⁢ ( w ) \pi(w) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) \operatorname{Crit}(w) .These notions were introduced by Puder (2014).We prove that there exists an exponentially generic subset V ⊆ F r V\subseteq F_{r} such that, for every w ∈ V w\in V , we have π ⁢ ( w ) = r \pi(w)=r and Crit ⁡ ( w ) = { F r } \operatorname{Crit}(w)=\{F_{r}\} . 

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2. Symmetry of Narayana Numbers and Rowvacuation of Root Posets

   https://doi.org/10.1017/fms.2021.47  

   Defant, Colin; Hopkins, Sam (January 2021, Forum of Mathematics, Sigma)

   Abstract For a Weyl group W of rank r , the W -Catalan number is the number of antichains of the poset of positive roots, and the W -Narayana numbers refine the W -Catalan number by keeping track of the cardinalities of these antichains. The W -Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W -Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution. 

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3. Energy Harvesting Performance of a Flapping Foil with Leading Edge Motion

   Prier, M. (April 2019, Dissertation)

   F or c e d at a f or a fl a p pi n g f oil e n er g y h ar v e st er wit h a cti v e l e a di n g e d g e m oti o n o p er ati n g i n t h e l o w r e d u c e d fr e q u e n c y r a n g e i s c oll e ct e d t o d et er mi n e h o w l e a di n g e d g e m oti o n aff e ct s e n er g y h ar v e sti n g p erf or m a n c e. T h e f oil pi v ot s a b o ut t h e mi dc h or d a n d o p er at e s i n t h e l o w r e d u c e d fr e q u e n c y r a n g e of 𝑓𝑓 𝑓𝑓 / 𝑈𝑈 ∞ = 0. 0 6 , 0. 0 8, a n d 0. 1 0 wit h 𝑅𝑅 𝑅𝑅 = 2 0 ,0 0 0 − 3 0 ,0 0 0 , wit h a pit c hi n g a m plit u d e of 𝜃𝜃 0 = 7 0 ∘ , a n d a h e a vi n g a m plit u d e of ℎ 0 = 0. 5 𝑓𝑓 . It i s f o u n d t h at l e a di n g e d g e m oti o n s t h at r e d u c e t h e eff e cti v e a n gl e of att a c k e arl y t h e str o k e w or k t o b ot h i n cr e a s e t h e lift f or c e s a s w ell a s s hift t h e p e a k lift f or c e l at er i n t h e fl a p pi n g str o k e. L e a di n g e d g e m oti o n s i n w hi c h t h e eff e cti v e a n gl e of att a c k i s i n cr e a s e d e arl y i n t h e str o k e s h o w d e cr e a s e d p erf or m a n c e. I n a d diti o n a di s cr et e v ort e x m o d el wit h v ort e x s h e d di n g at t h e l e a di n g e d g e i s i m pl e m e nt f or t h e m oti o n s st u di e d; it i s f o u n d t h at t h e m e c h a ni s m f or s h e d di n g at t h e l e a di n g e d g e i s n ot a d e q u at e f or t hi s p ar a m et er r a n g e a n d t h e m o d el c o n si st e ntl y o v er pr e di ct s t h e a er o d y n a mi c f or c e s. 

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4. Toward Personalizing Students' Education with Crowdsourced Tutoring

   https://doi.org/10.1145/3430895.3460130  

   Prihar, Ethan; Patikorn, Thanaporn; Botelho, Anthony; Sales, Adam; Heffernan, Neil (June 2021, Lerning @ Scale 2021)

   null (Ed.)

   A s m or e e d u c at or s i nt e gr at e t h eir c urri c ul a wit h o nli n e l e ar ni n g, it i s e a si er t o cr o w d s o ur c e c o nt e nt fr o m t h e m. Cr o w ds o ur c e d t ut ori n g h a s b e e n pr o v e n t o r eli a bl y i n cr e a s e st u d e nt s’ n e xt pr o bl e m c orr e ct n e s s. I n t hi s w or k, w e c o n fir m e d t h e fi n di n g s of a pr e vi o u s st u d y i n t hi s ar e a, wit h str o n g er c o n fi d e n c e m ar gi n s t h a n pr e vi o u sl y, a n d r e v e al e d t h at o nl y a p orti o n of cr o w d s o ur c e d c o nt e nt cr e at or s h a d a r eli a bl e b e n e fit t o st ud e nt s. F urt h er m or e, t hi s w or k pr o vi d e s a m et h o d t o r a n k c o nt e nt cr e at or s r el ati v e t o e a c h ot h er, w hi c h w a s u s e d t o d et er mi n e w hi c h c o nt e nt cr e at or s w er e m o st eff e cti v e o v er all, a n d w hi c h c o nt e nt cr e at or s w er e m o st eff e cti v e f or s p e ci fi c gr o u p s of st u d e nt s. W h e n e x pl ori n g d at a fr o m Te a c h er A SSI S T, a f e at ur e wit hi n t h e A S SI S T m e nt s l e ar ni n g pl atf or m t h at cr o w d s o ur c e s t ut ori n g fr o m t e a c h er s, w e f o u n d t h at w hil e o v erall t hi s pr o gr a m pr o vi d e s a b e n e fit t o st u d e nt s, s o m e t e a c h er s cr e at e d m or e eff e cti v e c o nt e nt t h a n ot h er s. D e s pit e t hi s fi n di n g, w e di d n ot fi n d e vi d e n c e t h at t h e eff e cti v e n e s s of c o nt e nt r eli a bl y v ari e d b y st u d e nt k n o wl e d g e-l e v el, s u g g e sti n g t h at t h e c o nt e nt i s u nli k el y s uit a bl e f or p er s o n ali zi n g i n str u cti o n b a s e d o n st u d e nt k n o wl e d g e al o n e. T h e s e fi n di n g s ar e pr o mi si n g f or t h e f ut ur e of cr o w d s o ur c e d t ut ori n g a s t h e y h el p pr o vi d e a f o u n d ati o n f or a s s e s si n g t h e q u alit y of cr o w d s o ur c e d c o nt e nt a n d i n v e sti g ati n g c o nt e nt f or o p p ort u niti e s t o p er s o n ali z e st u d e nt s’ e d u c ati o n. 

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5. On Multilinear Forms: Bias, Correlation, and Tensor Rank

   Bhrushundi, Abhishek; Harsha, Prahladh; Hatami, Pooya; Kopparty, Swastik; Kumar, Mrinal (August 2020, Leibniz international proceedings in informatics)

   Byrka, Jaroslaw; Meka, Raghu (Ed.)

   In this work, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over F₂. We show the following results for multilinear forms and tensors. Correlation bounds. We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d = 2^{o(k)}, we show that a random d-linear form f(X₁,X₂, … , X_d) : (F₂^{k}) ^d → F₂ has correlation 2^{-k(1-o(1))} with any polynomial of degree at most d/2 with high probability. This result is proved by giving near-optimal bounds on the bias of a random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a sum of t random d-dimensional rank-1 tensors is identically zero. Tensor rank vs Bias. We show that if a 3-dimensional tensor has small rank then its bias, when viewed as a 3-linear form, is large. More precisely, given any 3-dimensional tensor T: [k]³ → F₂ of rank at most t, the bias of the 3-linear form f_T(X₁, X₂, X₃) : = ∑_{(i₁, i₂, i₃) ∈ [k]³} T(i₁, i₂, i₃)⋅ X_{1,i₁}⋅ X_{2,i₂}⋅ X_{3,i₃} is at least (3/4)^t. This bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds. In particular, we use this approach to give a new proof that the finite field multiplication tensor has tensor rank at least 3.52 k, which is the best known rank lower bound for any explicit tensor in three dimensions over F₂. Moreover, this relation between bias and tensor rank holds for d-dimensional tensors for any fixed d. 

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