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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. On the construction of tame supercuspidal representations

   https://doi.org/10.1112/S0010437X21007636  

   Fintzen, Jessica ( December 2021 , Compositio Mathematica)

   Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$ . Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$ . We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [ Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference. 

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3. Weighted estimates for one-sided martingale transforms

   https://doi.org/10.1090/proc/14665  

   Chen, Wei ; Han, Rui ; Lacey, Michael T. ( October 2019 , Proceedings of the American Mathematical Society)

   Let $ Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert =1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, $\displaystyle \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} .$ Above, we use the one-sided $ A_2$ characteristic for the weight $ w$. This is an instance of a one-sided $ A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $ A_2$ theorem do not seem to apply in the one-sided setting. 

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4. The factorisation property of l ∞ ( Xk )

   https://doi.org/10.1017/s0305004120000304  

   LECHNER, RICHARD ; MOTAKIS, PAVLOS ; MÜLLER, PAUL F.X. ; SCHLUMPRECHT, THOMAS ( September 2021 , Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by ${(e_{(k,j)}^*)_j}$ , for $k\in\N$ , let $Z=\ell^\infty(X_k:k\kin\N)$ be their l ∞ -sum, and let $T:Z\to Z$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 

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5. On the perturbation effect and LET dependence of beam quality correction factors in carbon ion beams

   https://doi.org/10.1002/mp.16089  

   Khan, Ahtesham Ullah ; Nelson, Nicholas P. ; Culberson, Wesley S. ; DeWerd, Larry A. ( December 2022 , Medical Physics)

   In a recent study, we reported beam quality correction factors,f_Q, in carbon ion beams using Monte Carlo (MC) methods for a cylindrical and a parallel‐plate ionization chamber (IC). A non‐negligible perturbation effect was observed; however, the magnitude of the perturbation correction due to the specific IC subcomponents was not included. Furthermore, the stopping power data presented in the International Commission on Radiation Units and Measurements (ICRU) report 73 were used, whereas the latest stopping power data have been reported in the ICRU report 90.

   The aim of this study was to extend our previous work by computingf_Qcorrection factors using the ICRU 90 stopping power data and by reporting IC‐specific perturbation correction factors. Possible energy or linear energy transfer (LET) dependence of thef_Qcorrection factor was investigated by simulating both pristine beams and spread‐out Bragg peaks (SOBPs).

   The TOol for PArticle Simulation (TOPAS)/GEANT4 MC code was used in this study. A 30 × 30 × 50 cm³water phantom was simulated with a uniform 10 × 10 cm²parallel beam incident on the surface. A Farmer‐type cylindrical IC (Exradin A12) and two parallel‐plate ICs (Exradin P11 and A11) were simulated in TOPAS using the manufacturer‐provided geometrical drawings. Thef_Qcorrection factor was calculated in pristine carbon ion beams in the 150–450 MeV/u energy range at 2 cm depth and in the middle of the flat region of four SOBPs. Thek_Qcorrection factor was calculated by simulating thef_Qocorrection factor in a⁶⁰Co beam at 5 cm depth. The perturbation correction factors due to the presence of the individual IC subcomponents, such as the displacement effect in the air cavity, collecting electrode, chamber wall, and chamber stem, were calculated at 2 cm depth for monoenergetic beams only. Additionally, the mean dose‐averaged and track‐averaged LET was calculated at the depths at which thef_Qwas calculated.

   The ICRU 90f_Qcorrection factors were reported. Thep_discorrection factor was found to be significant for the cylindrical IC with magnitudes up to 1.70%. The individual perturbation corrections for the parallel‐plate ICs were <1.0% except for the A11p_celcorrection at the lowest energy. Thef_Qcorrection for the P11 IC exhibited an energy dependence of >1.00% and displayed differences up to 0.87% between pristine beams and SOBPs. Conversely, thef_Qfor A11 and A12 displayed a minimal energy dependence of <0.50%. The energy dependence was found to manifest in the LET dependence for the P11 IC. A statistically significant LET dependence was found only for the P11 IC in pristine beams only with a magnitude of <1.10%.

   The perturbation andk_Qcorrection factor should be calculated for the specific IC to be used in carbon ion beam reference dosimetry as a function of beam quality.

    

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