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1. Short Proof of the Asymptotic Confirmation of the Faudree-Lehel Conjecture

   https://doi.org/10.37236/11413  

   Przybyło, Jakub; Wei, Fan (October 2023, The Electronic Journal of Combinatorics)

   Given a simple graph $$G$$, the irregularity strength of $$G$$, denoted $s(G)$, is the least positive integer $$k$$ such that there is a weight assignment on edges $$f: E(G) \to \{1,2,\dots, k\}$$ for which each vertex weight $$f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$$ is unique amongst all $$v\in V(G)$$. In 1987, Faudree and Lehel conjectured that there is a constant $$c$$ such that $$s(G) \leq n/d + c$$ for all $$d$$-regular graphs $$G$$ on $$n$$ vertices with $d>1$, whereas it is trivial that $$s(G) \geq n/d$$. In this short note we prove that the Faudree-Lehel Conjecture holds when $$d \geq n^{0.8+\epsilon}$$ for any fixed $$\epsilon >0$$, with a small additive constant $c=28$ for $$n$$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $$\beta\in(0,1/4)$$ there is a constant $$C$$ such that for all $$d$$-regular graphs $$G$$, $$s(G) \leq \frac{n}{d}(1+\frac{C}{d^{\beta}})+28$$, extending and improving a recent result of Przybyło that $$s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{\epsilon/19}n})$$ whenever $$d\in [\ln^{1+\epsilon} n, n/\ln^{\epsilon}n]$$ and $$n$$ is large enough. 

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2. Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

   https://doi.org/10.1007/s10955-020-02500-8  

   Boßmann, Lea; Pavlović, Nataša; Pickl, Peter; Soffer, Avy (March 2020, Journal of Statistical Physics)

   Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N $$d$$ d -dimensional bosons for large N . The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form $$(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )$$ ( N - 1 ) - 1 N d β v ( N β · ) for $$\beta \in [0,\frac{1}{4d})$$ β ∈ [ 0 , 1 4 d ) . We derive a sequence of N -body functions which approximate the true many-body dynamics in $$L^2({\mathbb {R}}^{dN})$$ L 2 ( R dN ) -norm to arbitrary precision in powers of $$N^{-1}$$ N - 1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution. 

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3. The effects of changing vegetative composition on the abundance, species diversity and activity of birds at the Jornada Basin LTER site, 1997

   https://doi.org/10.6073/pasta/9ce9a7139153fa22636bdfef63c4cdb8  

   Bogner, Heidi; Huenneke, Laura F (January 2021, Environmental Data Initiative)

   This data package contains bird abundance data collected in plots that have had various plant functional groups or species experimentally removed at the Jornada Basin LTER site in southern New Mexico, USA. This data was collected in an effort to distinguish the differential effects of plant community biomass, plant community functional groups, and biodiversity within functional groups on plant community function, including effects on animals. To make these distinctions, treatments were established by the selective removal of plant species or functional groups within experimental plots. There are eight treatments: control (C, no removals); four functional group removal treatments (PG, perennial grass removed; S, shrubs removed; SSh, subshrubs removed; Succ, succulents removed), and three species richness manipulation treatments. Richness manipulations included a simplified treatment (Simp), where only the single most abundant species of each growth form is preserved and all other species in the growth form are removed, a reduced‐Larrea treatment (rL), where the Larrea is assumed to be the dominant and is removed while minority components remain, and a reduced-Prosopsis treatment (rP), where Prosopis rather than Larrea is removed as the shrub dominant. Following treatments, bird abundance and habitat preference data was collected in 1997. This data set consists of plot number, treatment type, and time of bird presence by taxa and by habitat and behavior. This study is complete. 

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4. Sums of proper divisors follow the Erdős–Kac law

   https://doi.org/10.1090/proc/16167  

   Pollack, Paul; Troupe, Lee (March 2023, Proceedings of the American Mathematical Society)

   Let s ( n ) = ∑ d ∣ n ,   d > n d s(n)=\sum _{d\mid n,~d>n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log ⁡ log ⁡ n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [ The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log ⁡ log ⁡ n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function). 

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5. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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