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1. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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2. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin; Iyengar, Srikanth (January 2023, Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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3. Improved Lower Bounds for 3-Query Matching Vector Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2025.2  

   Aggarwal, Divesh; Dutta, Pranjal; Li, Zeyong; Obremski, Maciej; Saraogi, Sidhant (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   A Matching Vector (MV) family modulo a positive integer m ≥ 2 is a pair of ordered lists U = (u_1, ⋯, u_K) and V = (v_1, ⋯, v_K) where u_i, v_j ∈ ℤ_m^n with the following property: for any i ∈ [K], the inner product ⟨u_i, v_i⟩ = 0 mod m, and for any i ≠ j, ⟨u_i, v_j⟩ ≠ 0 mod m. An MV family is called r-restricted if inner products ⟨u_i, v_j⟩, for all i,j, take at most r different values. The r-restricted MV families are extremely important since the only known construction of constant-query subexponential locally decodable codes (LDCs) are based on them. Such LDCs constructed via matching vector families are called matching vector codes. Let MV(m,n) (respectively MV(m, n, r)) denote the largest K such that there exists an MV family (respectively r-restricted MV family) of size K in ℤ_m^n. Such a MV family can be transformed in a black-box manner to a good r-query locally decodable code taking messages of length K to codewords of length N = m^n. For small prime m, an almost tight bound MV(m,n) ≤ O(m^{n/2}) was first shown by Dvir, Gopalan, Yekhanin (FOCS'10, SICOMP'11), while for general m, the same paper established an upper bound of O(m^{n-1+o_m(1)}), with o_m(1) denoting a function that goes to zero when m grows. For any arbitrary constant r ≥ 3 and composite m, the best upper bound till date on MV(m,n,r) is O(m^{n/2}), is due to Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14).In a breakthrough work, Alrabiah, Guruswami, Kothari and Manohar (STOC'23) implicitly improve this bound for 3-restricted families to MV(m, n, 3) ≤ O(m^{n/3}). In this work, we present an upper bound for r = 3 where MV(m,n,3) ≤ m^{n/6 +O(log n)}, and as a result, any 3-query matching vector code must have codeword length of N ≥ K^{6-o(1)}. 

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4. A Combinatorial Proof of a Symmetry for a Refinement of the Narayana Numbers

   https://doi.org/10.37236/12681  

   Bóna, Miklós; Dimitrov, Stoyan; Labelle, Gilbert; Li, Yifei; Pappe, Joseph; Vindas-Meléndez, Andrés R; Zhuang, Yan (April 2025, The Electronic Journal of Combinatorics)

   We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $$w_{n,k,m}$$ is the number of Dyck paths of semilength $$n$$ with $$k$$ occurrences of $UD$ and $$m$$ occurrences of $UUD$, then $$w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $$w_{2k+1,k,m}$$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $$w_{n,k,m}$$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $$w_{n,k_{1},k_{2},\ldots, k_{r}}$$ counting the semilength-$$n$$ Dyck paths with $$k_{1}$$ $UD$-factors, $$k_{2}$$ $UUD$-factors, $$\ldots$$, and $$k_{r}$$ $$U^{r}D$$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $$W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$$, including real-rootedness, $$\gamma$$-positivity, and a symmetric decomposition. 

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5. 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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