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Abstract The physics of recombination lines in the Heisinglet system is expected to be relatively simple, supported by accurate atomic models. We examine the intensities of Heisingletsλ3614, λ3965, λ5016, λ6678, and λ7281 and the triplet Heiλ5876 in various types of ionized nebulae and compare them with theoretical predictions to test the validity of the “Case B” recombination scenario and the assumption of thermal homogeneity. Our analysis includes 85 spectra from Galactic and extragalactic Hiiregions, 90 from star-forming galaxies, and 218 from planetary nebulae, all compiled by the Deep Spectra of Ionized Regions Database Extended (DESIRED-E) project. By evaluating the ratios Heiλ7281/λ6678 and Heiλ7281/λ5876, we determineT_e(Hei) and compare it with direct measurements ofT_e([Oiii]λ4363/λ5007). We find thatT_e(Hei) is systematically lower thanT_e([Oiii]) across most objects and nebula types. Additionally, we identify a correlation between the abundance discrepancy factor (ADF(O²⁺)) and the differenceT_e([Oiii]) –T_e(Hei) for planetary nebulae. We explore two potential explanations: photon loss fromn¹P → 1¹Stransitions and temperature inhomogeneities. Deviations from “Case B” may indicate photon absorption by Hirather than Heiand/or generalized ionizing photon escape, highlighting the need for detailed consideration of radiative transfer effects. If temperature inhomogeneities are widespread, identifying a common physical phenomenon affecting all ionized nebulae is crucial. Our results suggest that both scenarios can contribute to the observed discrepancies. 

Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH ). We also discuss a number- theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. 

We consider the sensitivity of real zeros of structured polynomial systems to pertubations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems and extend these estimates to the smoothed analysis setting. 

Suppose $$F:=(f_1,\ldots,f_n)$$ is a system of random $$n$$-variate polynomials with $$f_i$$ having degree $$\leq\!d_i$$ and the coefficient of $$x^{a_1}_1\cdots x^{a_n}_n$$ in $$f_i$$ being an independent complex Gaussian of mean $$0$$ and variance $$\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$$. Recent progress on Smale's 17$$\thth$$ Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, B\"{u}rgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $$F$$ using just $$N^{O(1)}$$ arithmetic operations on average, where $$N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$$ ($$=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$$) is the maximum possible total number of monomial terms for such an $$F$$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $$F$$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $$O(n^3\log^2(n\max_i d_i))$$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $$n\log \max_i d_i$$ when $$F$$ has more terms. 

Abstract Multi‐locus sequence data are widely used in fungal systematic and taxonomic studies to delimit species and infer evolutionary relationships. We developed and assessed the efficacy of a multi‐locus pooled sequencing method using PacBio long‐read high‐throughput sequencing. Samples included fresh and dried voucher specimens, cultures and archival DNA extracts of Agaricomycetes with an emphasis on the order Cantharellales. Of the 283 specimens sequenced, 93.6% successfully amplified at one or more loci with a mean of 3.3 loci amplified. Our method recovered multiple sequence variants representing alleles of rDNA loci and single copy protein‐coding genesrpb1,rpb2 andtef1. Within‐sample genetic variation differed by locus and taxonomic group, with the greatest genetic divergence observed among sequence variants ofrpb2 andtef1 from corticioid Cantharellales. Our method is a cost‐effective approach for generating accurate multi‐locus sequence data coupled with recovery of alleles from polymorphic samples and multi‐organism specimens. These results have important implications for understanding intra‐individual genomic variation among genetic loci commonly used in species delimitation of fungi.