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1. Thermal convection over fractal surfaces

   https://doi.org/10.1017/jfm.2020.826  

   Toppaladoddi, Srikanth; Wells, Andrew J.; Doering, Charles R.; Wettlaufer, John S. (January 2021, Journal of Fluid Mechanics)

   null (Ed.)

   We use well resolved numerical simulations with the lattice Boltzmann method to study Rayleigh–Bénard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $$Ra \in \left [10^7, 10^{10}\right ]$$ , where Pr and Ra are the Prandtl and Rayleigh numbers. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $$k$$ , as $$S(k) \sim k^{p}$$ ( $p < 0$ ). The degree of roughness is quantified by the exponent $$p$$ with $p < -3$ for smooth (differentiable) surfaces and $$-3 \le p < -1$$ for rough surfaces with Hausdorff dimension $$D_f=\frac {1}{2}(p+5)$$ . By computing the exponent $$\beta$$ using power law fits of $$Nu \sim Ra^{\beta }$$ , where $Nu$ is the Nusselt number, we find that the heat transport scaling increases with roughness through the top two decades of $$Ra \in \left [10^8, 10^{10}\right ]$$ . For $$p$$ $= -3.0$ , $-2.0$ and $-1.5$ we find $$\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$$ and $$0.352 \pm 0.011$$ , respectively. We also find that the Reynolds number, $Re$ , scales as $$Re \sim Ra^{\xi }$$ , where $$\xi \approx 0.57$$ over $$Ra \in \left [10^7, 10^{10}\right ]$$ , for all $$p$$ used in the study. For a given value of $$p$$ , the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness. 

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2. Small Fractional Parts of Polynomials and Mean Values of Exponential Sums

   https://doi.org/10.1093/imrn/rnad082  

   Yeon, Kiseok (May 2023, International Mathematics Research Notices)

   Abstract Let $$k_i\ (i=1,2,\ldots ,t)$$ be natural numbers with $$k_1>k_2>\cdots >k_t>0$$, $$k_1\geq 2$$ and $$t<k_1.$$ Given real numbers $$\alpha _{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$$, we consider polynomials of the shape $$\begin{align*} &\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},\end{align*}$$and derive upper bounds for fractional parts of polynomials in the shape $$\begin{align*} &\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),\end{align*}$$by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017). 

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3. Limiting Betti distributions of Hilbert schemes on n points

   https://doi.org/10.4153/S0008439522000261  

   Griffin, Michael; Ono, Ken; Rolen, Larry; Tsai, Wei-Lun (March 2023, Canadian Mathematical Bulletin)

   Abstract Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $$(\mathbb {C}^{2})^{[n]}$$ on $$n$$ points, as $$n\rightarrow +\infty ,$$ is a Gumbel distribution . In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $$((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $$A\geq 2.$$ Furthermore, if $$p_{k}(A;n)$$ denotes the number of partitions of $$n$$ with exactly $$k$$ parts that are multiples of $$A$$ , then we obtain the asymptotic $$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$ a result which is of independent interest. 

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4. On lower bounds for Erdős-Szekeres products

   https://doi.org/10.1090/proc/15503  

   Billsborough, C.; Freedman, M.; Hart, S.; Kowalsky, G.; Lubinsky, D.; Pomeranz, A.; Sammel, A. (October 2021, Proceedings of the American Mathematical Society)

   Let { s j } j = 1 n \left \{ s_{j}\right \} _{j=1}^{n} be positive integers. We show that for any 1 ≤ L ≤ n , 1\leq L\leq n, ‖ ∏ j = 1 n ( 1 − z s j ) ‖ L ∞ ( | z | = 1 ) ≥ exp ⁡ ( 1 2 e L ( s 1 s 2 … s L ) 1 / L ) . \begin{equation*} \left \Vert \prod _{j=1}^{n}\left ( 1-z^{s_{j}}\right ) \right \Vert _{L_{\infty }\left ( \left \vert z\right \vert =1\right ) }\geq \exp \left ( \frac {1}{2e}\frac {L}{\left ( s_{1}s_{2}\ldots s_{L}\right ) ^{1/L}}\right ) . \end{equation*} In particular, this gives geometric growth if a positive proportion of the { s j } \left \{ s_{j}\right \} are bounded. We also show that when the { s j } \left \{ s_{j}\right \} grow regularly and faster than j ( log ⁡ j ) 2 + ε j\left ( \log j\right ) ^{2+\varepsilon } , some ε > 0 \varepsilon >0 , then the norms grow faster than exp ⁡ ( ( log ⁡ n ) 1 + δ ) \exp \left ( \left ( \log n\right ) ^{1+\delta }\right ) for some δ > 0 \delta >0 . 

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5. Universal features of higher-form symmetries at phase transitions

   https://doi.org/10.21468/SciPostPhys.11.2.033  

   Wu, Xiao-Chuan; Jian, Chao-Ming; Xu, Cenke (January 2021, SciPost Physics)

   We investigate the behavior of higher-form symmetries at variousquantum phase transitions. We consider discrete 1-form symmetries, whichcan be either part of the generalized concept “categorical symmetry”(labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) )introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. We demonstrate that for many quantum phase transitionsinvolving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O 𝒞 ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P ⟨ ( log O 𝒞 ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 𝒞 is an operator defined associated with loop \mathcal{C} 𝒞 (or its interior \mathcal{A} 𝒜 ),which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. P P is the perimeter of \mathcal{C} 𝒞 ,and the b \log P b log P term arises from the sharp corners of the loop \mathcal{C} 𝒞 ,which is consistent with recent numerics on a particular example. b b is a universal microscopic-independent number, which in (2+1)d ( 2 + 1 ) d is related to the universal conductivity at the quantum phasetransition. b b can be computed exactly for certain transitions using the dualitiesbetween (2+1)d ( 2 + 1 ) d conformal field theories developed in recent years. We also compute the"strange correlator" of O_\mathcal{C} O 𝒞 : S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle S 𝒞 = ⟨ 0 | O 𝒞 | 1 ⟩ / ⟨ 0 | 1 ⟩ where |0\rangle | 0 ⟩ and |1\rangle | 1 ⟩ are many-body states with different topological nature. 

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