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1. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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2. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo ; Wang, Jinhuan ( June 2024 , Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following$L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality$\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}$where parameters$q$and$p$satisfy the conditions$p>d\geq 1$,$q\geq 0$. The best constant$C_{q,\infty ,p}$is given by$\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}$where$u_{c,\infty }$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (x-x_0))$for any real numbers$A$,$\lambda >0$and$x_{0}\in \mathbb {R}^d$. In fact, the generalized Lane-Emden equation in$\mathbb {R}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For$q=0$or$d=1$,$u_{c,\infty }$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that$u_{c,m}\to u_{c,\infty }$and$C_{q,m,p}\to C_{q,\infty ,p}$as$m\to +\infty$for$d=1$, where$u_{c,m}$and$C_{q,m,p}$are the function achieving equality and the best constant of$L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively.

    

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3. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael ; Shalev, Aner ; Tiep, Pham ( February 2024 , Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups$G$have the property that for each$\epsilon > 0$there exists$N > 0$such that, if$|G| \ge N$and$S, T$are normal subsets of$G$with at least$\epsilon |G|$elements each, then every non-trivial element of$G$is the product of an element of$S$and an element of$T$.

   We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form$\mathrm {PSL}_n(q)$where$q$is fixed and$n\to \infty$. However, in the case$S=T$and$G$alternating this holds with an explicit bound on$N$in terms of$\epsilon$.

   Related problems and applications are also discussed. In particular we show that, if$w_1, w_2$are non-trivial words,$G$is a finite simple group of Lie type of bounded rank, and for$g \in G$,$P_{w_1(G),w_2(G)}(g)$denotes the probability that$g_1g_2 = g$where$g_i \in w_i(G)$are chosen uniformly and independently, then, as$|G| \to \infty$, the distribution$P_{w_1(G),w_2(G)}$tends to the uniform distribution on$G$with respect to the$L^{\infty }$norm.

    

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4. Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

   https://doi.org/10.1090/cams/17  

   Carmeli, Shachar ; Yuan, Allen ( March 2023 , Communications of the American Mathematical Society)

   We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the$K(1)$-local sphere$\mathbb {S}_{K(1)}$at the prime$2$, in particular realizing the non-$2$-adic rational element$1+\varepsilon \in \pi _0\mathbb {S}_{K(1)}$as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in$\pi _0\mathbb {S}_{K(1)}$.

    

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5. A variant of Maclaurin’s inequality

   https://doi.org/10.1090/bproc/236  

   Tao, Terence ( January 2025 , Proceedings of the American Mathematical Society, Series B)

   The classical Maclaurin inequality asserts that the elementary symmetric means$\begin{equation*} s_k(y) ≔\frac {1}{\binom {n}{k}} \sum _{1 \leq i_1 > \dots > i_k \leq n} y_{i_1} \dots y_{i_k} \end{equation*}$obey the inequality$s_\ell (y)^{1/\ell } \leq s_k(y)^{1/k}$whenever$1 \leq k \leq \ell \leq n$and$y = (y_1,\dots ,y_n)$consists of nonnegative reals. We establish a variant$\begin{equation*} |s_\ell (y)|^{\frac {1}{\ell }} \ll \frac {\ell ^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac {1}{k}}, |s_{k+1}(y)|^{\frac {1}{k+1}}) \end{equation*}$of this inequality in which the$y_i$are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the$k^{1/2}$factor in the denominator.

    

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