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1. 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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2. 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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3. Internal Sequential Commutation and Single Generation

   https://doi.org/10.1093/imrn/rnaf103  

   Gao, David; Kunnawalkam_Elayavalli, Srivatsav; Patchell, Gregory; Tan, Hui (April 2025, International Mathematics Research Notices)

   Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

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4. Uniqueness of some cylindrical tangent cones to special Lagrangians

   https://doi.org/10.1007/s00039-023-00634-x  

   Collins, Tristan C.; Li, Yang (April 2023, Geometric and Functional Analysis)

   Abstract We show that if an exact special Lagrangian $$N\subset {\mathbb {C}}^n$$ N ⊂ C n has a multiplicity one, cylindrical tangent cone of the form $${\mathbb {R}}^{k}\times {\textbf{C}}$$ R k × C where $${\textbf{C}}$$ C is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $${\textbf{C}}$$ C satisfies an integrability condition. This applies, for example, when $${\textbf{C}}= {\textbf{C}}_{HL}^{m}$$ C = C HL m is the Harvey-Lawson $$T^{m-1}$$ T m - 1 cone for $$m\ne 8,9$$ m ≠ 8 , 9 . 

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5. Determinantal Representations and the Image of the Principal Minor Map

   https://doi.org/10.1093/imrn/rnae038  

   Al_Ahmadieh, Abeer; Vinzant, Cynthia (March 2024, International Mathematics Research Notices)

   Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $$(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $$\mathbb{F}$$, there is no finite set of equations whose orbit under $$(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$$ cuts out the image of $$n\times n$$ matrices over $${\mathbb{F}}$$ under the principal minor map for every $$n$$. 

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