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1. Planar chemical reaction systems with algebraic and non-algebraic limit cycles

   https://doi.org/10.1007/s00285-025-02221-0  

   Craciun, Gheorghe; Erban, Radek (May 2025, Journal of Mathematical Biology)

   Abstract The Hilbert numberH(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most$$n \in {{\mathbb {N}}}$$ $n\in N$. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, wherenis equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number H(n) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to then-th order; (ii) systems with up ton-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve$$h(x,y)=0$$ $h(x,y)=0$of degree$$n_h \in {{\mathbb {N}}}$$ $n_h\in N$and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most$$n=2\,n_h+1.$$ $n=2n_h+1.$Considering$$n_h \ge 4,$$ $n_h\ge 4,$the algebraic curve$$h(x,y)=0$$ $h(x,y)=0$can contain multiple closed components with the maximum number of ovals given by Harnack’s curve theorem as$$1+(n_h-1)(n_h-2)/2$$ $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for$$n_h=4.$$ $n_h=4.$Algebraic curve$$h(x,y)=0$$ $h(x,y)=0$with$$n_h=4$$ $n_h=4$and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles. 

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2. Searching for dark photon dark matter in LIGO O1 data

   https://doi.org/10.1038/s42005-019-0255-0  

   Guo, Huai-Ke; Riles, Keith; Yang, Feng-Wei; Zhao, Yue (December 2019, Communications Physics)

   Abstract Dark matter exists in our Universe, but its nature remains mysterious. The remarkable sensitivity of the Laser Interferometer Gravitational-Wave Observatory (LIGO) may be able to solve this mystery. A good dark matter candidate is the ultralight dark photon. Because of its interaction with ordinary matter, it induces displacements on LIGO mirrors that can lead to an observable signal. In a study that bridges gravitational wave science and particle physics, we perform a direct dark matter search using data from LIGO’s first (O1) data run, as opposed to an indirect search for dark matter via its production of gravitational waves. We demonstrate an achieved sensitivity on squared coupling as$$\sim\! 4\times 1{0}^{-45}$$ $~4\times 10^{-45}$, in a$$U{(1)}_{{\rm{B}}}$$ $U{\left(1\right)}_B$dark photon dark matter mass band around$${m}_{{\rm{A}}} \sim 4\,\times 1{0}^{-13}$$ $m_A~4\times 10^{-13}$eV. Substantially improved search sensitivity is expected during the coming years of continued data taking by LIGO and other gravitational wave detectors in a growing global network. 

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3. Search for dark matter from the center of the Earth with 10 years of IceCube data

   https://doi.org/10.1140/epjc/s10052-025-14144-7  

   Abbasi, R; Ackermann, M; Adams, J; Agarwalla, S K; Aguilar, J A; Ahlers, M; Alameddine, J M; Amin, N M; Andeen, K; Argüelles, C; et al (May 2025, The European Physical Journal C)

   Abstract The nature of dark matter remains unresolved in fundamental physics. Weakly Interacting Massive Particles (WIMPs), which could explain the nature of dark matter, can be captured by celestial bodies like the Sun or Earth, leading to enhanced self-annihilation into Standard Model particles including neutrinos detectable by neutrino telescopes such as the IceCube Neutrino Observatory. This article presents a search for muon neutrinos from the center of the Earth performed with 10 years of IceCube data using a track-like event selection. We considered a number of WIMP annihilation channels ($$\chi \chi \rightarrow \tau ^+\tau ^-$$ $\chi \chi \to {\tau }^{+}{\tau }^{-}$/$$W^+W^-$$ $W^{+}{W}^{-}$/$$b\bar{b}$$ $b\overline{b}$) and masses ranging from 10 GeV to 10 TeV. No significant excess over background due to a dark matter signal was found while the most significant result corresponds to the annihilation channel$$\chi \chi \rightarrow b\bar{b}$$ $\chi \chi \to b\overline{b}$for the mass$$m_{\chi }=250$$ $m_{\chi }=250$ GeV with a post-trial significance of$$1.06\sigma $$ $1.06\sigma$. Our results are competitive with previous such searches and direct detection experiments. Our upper limits on the spin-independent WIMP scattering are world-leading among neutrino telescopes for WIMP masses$$m_{\chi }>100$$ $m_{\chi }>100$ GeV. 

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4. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

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5. An algorithm and computation to verify Legendre’s conjecture up to $$7\cdot 10^{13}$$

   https://doi.org/10.1007/s40993-024-00589-4  

   Sorenson, Jonathan; Webster, Jonathan (March 2025, Research in Number Theory)

   Abstract We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integern, there exists a prime between$$n^2$$ $n^2$and$$(n+1)^2$$ ${(n+1)}^2$. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between$$n^2$$ $n^2$and$$n(n+1)$$ $n(n+1)$and also between$$n(n+1)$$ $n(n+1)$and$$(n+1)^2$$ ${(n+1)}^2$. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all$$n\le N$$ $n\le N$in time$$O( N \log N \log \log N)$$ $O(NlogNloglogN)$and space$$N^{O(1/\log \log N)}$$ $N^{O(1/loglogN)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous$$N = 2\cdot 10^{9}$$ $N=2·{10}^9$up to$$N = 7.05\cdot 10^{13} > 2^{46}$$ $N=7.05·{10}^{13}>2^{46}$, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors. 

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