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1. The Flex Divisor of a K3 Surface

   https://doi.org/10.1093/imrn/rnac089  

   Alexeev, Valery; Engel, Philip (May 2022, International Mathematics Research Notices)

   Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces. 

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2. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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3. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan; Smirnov, Ilya (February 2023, Proceedings of the American Mathematical Society)

   Let $(R,\mathfrak{m})$be a Noetherian local ring of dimension $d\ge <\#comment/>2$. We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$, then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$for all $\mathfrak{m}$-primary ideals $I\subseteq <\#comment/>R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$. 

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4. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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5. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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