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1. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa ( April 2022 , Mathematische Annalen)

   Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$$a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$$a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$$a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m\times m$minor of the$$m\times k$$$m\times k$matrix$$(a_{j,i})_{j,i}$$${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$$\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$$x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

    

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2. Photo-response of the $$N=Z$$ nucleus $$^{24}$$Mg

   https://doi.org/10.1140/epja/s10050-023-01111-7  

   Deary, J ; Scheck, M ; Schwengner, R ; O’Donnell, D ; Bemmerer, D ; Beyer, R ; Hensel, Th ; Junghans, A R ; Kögler, T ; Müller, S E ; et al ( September 2023 , The European Physical Journal A)

   The electricE1 and magneticM1 dipole responses of the$$N=Z$$$N=Z$nucleus$$^{24}$$${}^{24}$Mg were investigated in an inelastic photon scattering experiment. The 13.0 MeV electrons, which were used to produce the unpolarised bremsstrahlung in the entrance channel of the$$^{24}$$${}^{24}$Mg($$\gamma ,\gamma ^{\prime }$$$\gamma ,{\gamma }^{\prime }$) reaction, were delivered by the ELBE accelerator of the Helmholtz-Zentrum Dresden-Rossendorf. The collimated bremsstrahlung photons excited one$$J^{\pi }=1^-$$$J^{\pi }=1^{-}$, four$$J^{\pi }=1^+$$$J^{\pi }=1^{+}$, and six$$J^{\pi }=2^+$$$J^{\pi }=2^{+}$states in$$^{24}$$${}^{24}$Mg. De-excitation$$\gamma $$$\gamma$rays were detected using the four high-purity germanium detectors of the$$\gamma $$$\gamma$ELBE setup, which is dedicated to nuclear resonance fluorescence experiments. In the energy region up to 13.0 MeV a total$$B(M1)\uparrow = 2.7(3)~\mu _N^2$$$B\left(M1\right)\uparrow =2.7\left(3\right){\mu }_N^2$is observed, but this$$N=Z$$$N=Z$nucleus exhibits only marginalE1 strength of less than$$\sum B(E1)\uparrow \le 0.61 \times 10^{-3}$$$\sum B\left(E1\right)\uparrow \le 0.61\times {10}^{-3}$ e$$^2 \, $$${}^2$fm$$^2$$${}^2$. The$$B(\varPi 1, 1^{\pi }_i \rightarrow 2^+_1)/B(\varPi 1, 1^{\pi }_i \rightarrow 0^+_{gs})$$$B(\Pi 1,1_i^{\pi }\to 2_1^{+})/B(\Pi 1,1_i^{\pi }\to 0_{\mathrm{gs}}^{+})$branching ratios in combination with the expected results from the Alaga rules demonstrate thatKis a good approximative quantum number for$$^{24}$$${}^{24}$Mg. The use of the known$$\rho ^2(E0, 0^+_2 \rightarrow 0^+_{gs})$$${\rho }^2(E0,0_2^{+}\to 0_{\mathrm{gs}}^{+})$strength and the measured$$B(M1, 1^+ \rightarrow 0^+_2)/B(M1, 1^+ \rightarrow 0^+_{gs})$$$B(M1,1^{+}\to 0_2^{+})/B(M1,1^{+}\to 0_{\mathrm{gs}}^{+})$branching ratio of the 10.712 MeV$$1^+$$$1^{+}$level allows, in a two-state mixing model, an extraction of the difference$$\varDelta \beta _2^2$$$\Delta {\beta }_2^2$between the prolate ground-state structure and shape-coexisting superdeformed structure built upon the 6432-keV$$0^+_2$$$0_2^{+}$level.

    

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3. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

   https://doi.org/10.1007/s10240-024-00144-y  

   De Lellis, Camillo ; Nardulli, Stefano ; Steinbrüchel, Simone ( February 2024 , Publications mathématiques de l'IHÉS)

   We consider integral area-minimizing 2-dimensional currents$T$$T$in$U\subset \mathbf {R}^{2+n}$$U\subset R^{2+n}$with$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$\partial T=Q〚\Gamma 〛$, where$Q\in \mathbf {N} \setminus \{0\}$$Q\in N\setminus \left\{0\right\}$and$\Gamma $$\Gamma$is sufficiently smooth. We prove that, if$q\in \Gamma $$q\in \Gamma$is a point where the density of$T$$T$is strictly below$\frac{Q+1}{2}$$\frac{Q+1}{2}$, then the current is regular at$q$$q$. The regularity is understood in the following sense: there is a neighborhood of$q$$q$in which$T$$T$consists of a finite number of regular minimal submanifolds meeting transversally at$\Gamma $$\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$Q=1$$Q=1$. As a corollary, if$\Omega \subset \mathbf {R}^{2+n}$$\Omega \subset R^{2+n}$is a bounded uniformly convex set and$\Gamma \subset \partial \Omega $$\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$T$$T$with$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $\Gamma $$\Gamma$.

    

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4. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    

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5. Spin density matrix elements in exclusive $$\rho ^0$$ meson muoproduction

   https://doi.org/10.1140/epjc/s10052-023-11359-4  

   Alexeev, G D ; Alexeev, M G ; Alice, C ; Amoroso, A ; Andrieux, V ; Anosov, V ; Augsten, K ; Augustyniak, W ; Azevedo, C_D R ; Badelek, B ; et al ( October 2023 , The European Physical Journal C)

   We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$${\rho }^0$meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$${\mu }^{+}$and$$ \mu ^{-}$$${\mu }^{-}$beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$$c^2$$$< W<$$$<W<$17.0 GeV/$$c^2$$$c^2$, 1.0 (GeV/c)$$^2$$${}^2$$$< Q^2<$$$<Q^2<$10.0 (GeV/c)$$^2$$${}^2$and 0.01 (GeV/c)$$^2$$${}^2$$$< p_{\textrm{T}}^2<$$$<p_{\text{T}}^2<$0.5 (GeV/c)$$^2$$${}^2$. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$$Q^2$the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$$p_{\text{T}}$the transverse momentum of the$$\rho ^0$$${\rho }^0$meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$${\gamma }_T^{\ast }\to V_L^{}$) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$${\rho }^0$production.

    

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