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1. On ℓ p -Gaussian–Grothendieck Problem

   https://doi.org/10.1093/imrn/rnab311  

   Chen, Wei-Kuo; Sen, Arnab (November 2021, International Mathematics Research Notices)

   Abstract For $$p\geq 1$$ and $$(g_{ij})_{1\leq i,j\leq n}$$ being a matrix of i.i.d. standard Gaussian entries, we study the $$n$$-limit of the $$\ell _p$$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $$p=\infty $$, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $$1\leq p<2$$ and $$2<p<\infty .$$ For the former, we compute the limit of the $$\ell _p$$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $$n^{-1}$$. 

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2. Resistance scaling on 4 N -carpets

   https://doi.org/10.1515/forum-2020-0330  

   Canner, Claire; Hayes, Christopher; Huang, Robin; Orwin, Michael; Rogers, Luke G. (December 2021, Forum Mathematicum)

   Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F . 

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3. Mesodinium–Dinophysis encounters: temporal and spatial constraints on Dinophysis blooms

   https://doi.org/10.1093/plankt/fbae068  

   Díaz, Patricio A; Baldrich, Ángela M; Rodríguez, Francisco; Díaz, Manuel; Álvarez, Gonzalo; Pérez-Santos, Iván; Schwerter, Camila; Rodríguez-Villegas, Camilo; Carbonell, Pamela; Cantarero, Bárbara; et al (February 2025, Journal of Plankton Research)

   Dolan, John (Ed.)

   Abstract Species of the Dinophysis acuminata complex are the main cause of diarrhetic shellfish poisoning worldwide. These mixotrophs perform photosynthesis with plastids stolen from specific ciliate prey. Current transport models forecast advection of established populations, but modelling bloom development and maintenance also needs to consider the prey (Mesodinium spp.) of Dinophysis. Predator and prey have distinct niches, and Dinophysis bloom success relies on matching prey populations in time and place. During autumn 2019, red tides of Mesodinium rubrum in Reloncaví Fjord, Chile, were not followed by Dinophysis growth. The dynamics of Mesodinium–Dinophysis encounters during this and additional multiscale cases elsewhere are examined. Analogies with some classic predator—prey models (match–mismatch hypothesis; Lasker’s stable ocean hypothesis) are explored. Preceding dense populations of Mesodinium do not guarantee Dinophysis blooms if spatial co-occurrence is not accompanied by water column structure, which leads to thin layer formation, as in Lasker’s stable ocean hypothesis or if the predator growth season is over. Tracking the frequency of vacuolate Dinophysis cells, irrefutable signal of prey acquisition, with advanced in situ fluid-imaging instruments, is envisaged as a next-generation tool to predict rising Dinophysis populations. 

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4. Multifaceted effects on even-skipped transcriptional dynamics upon Krüppel dosage changes

   https://doi.org/10.1242/dev.202132  

   Lin, Shufan; Lim, Bomyi (March 2024, Development)

   ABSTRACT Although fluctuations in transcription factor (TF) dosage are often well tolerated, TF dosage modulation can change the target gene expression dynamics and result in significant non-lethal developmental phenotypes. Using MS2/MCP-mediated quantitative live imaging in early Drosophila embryos, we analyzed how changing levels of the gap gene Krüppel (Kr) affects transcriptional dynamics of the pair-rule gene even-skipped (eve). Halving the Kr dosage leads to a transient posterior expansion of the eve stripe 2 and an anterior shift of stripe 5. Surprisingly, the most significant changes are observed in eve stripes 3 and 4, the enhancers of which do not contain Kr-binding sites. In Kr heterozygous embryos, both stripes 3 and 4 display narrower widths, anteriorly shifted boundaries and reduced mRNA production levels. We show that Kr dosage indirectly affects stripe 3 and 4 dynamics by modulating other gap gene dynamics. We quantitatively correlate moderate body segment phenotypes of Kr heterozygotes with spatiotemporal changes in eve expression. Our results indicate that nonlinear relationships between TF dosage and phenotypes underlie direct TF-DNA and indirect TF-TF interactions. 

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5. On the optimal arrangement of 2 d lines in ℂ d

   https://doi.org/10.1093/imaiai/iaaf008  

   Fallon, Kean; Iverson, Joseph W (March 2025, Information and Inference: A Journal of the IMA)

   Abstract We show the optimal coherence of $2d$ lines in $$\mathbb{C}^{d}$$ is given by the Welch bound whenever a skew Hadamard matrix of order $d+1$ exists. Our proof uses a variant of Hadamard matrix doubling that converts any equiangular tight frame of size $$\tfrac{d-1}{2} \times d$$ into another one of size $$d \times 2d$$. Among $$d \leq 160$$, this produces equiangular tight frames of new sizes when $d = 11$, $35$, $39$, $43$, $47$, $59$, $67$, $71$, $83$, $95$, $103$, $107$, $111$, $119$, $123$, $127$, $131$, $143$, $151$ and $155$. 

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