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1. 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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2. The universal acceleration scale from stellar feedback

   https://doi.org/10.1093/mnrasl/slaa103  

   Grudić, Michael Y; Boylan-Kolchin, Michael; Faucher-Giguère, Claude-André; Hopkins, Philip F (July 2020, Monthly Notices of the Royal Astronomical Society: Letters)

   ABSTRACT It has been established for decades that rotation curves deviate from the Newtonian gravity expectation given baryons alone below a characteristic acceleration scale $$g_{\dagger }\sim 10^{-8}\, \rm {cm\, s^{-2}}$$, a scale promoted to a new fundamental constant in MOND. In recent years, theoretical and observational studies have shown that the star formation efficiency (SFE) of dense gas scales with surface density, SFE ∼ Σ/Σcrit with $$\Sigma _{\rm crit} \sim \langle \dot{p}/m_{\ast }\rangle /(\pi \, G)\sim 1000\, \rm {M_{\odot }\, pc^{-2}}$$ (where $$\langle \dot{p}/m_{\ast }\rangle$$ is the momentum flux output by stellar feedback per unit stellar mass in a young stellar population). We argue that the SFE, more generally, should scale with the local gravitational acceleration, i.e. that SFE $${\sim}g_{\rm tot}/g_{\rm crit}\equiv (G\, M_{\rm tot}/R^{2}) / \langle \dot{p}/m_{\ast }\rangle$$, where Mtot is the total gravitating mass and $$g_{\rm crit}=\langle \dot{p}/m_{\ast }\rangle = \pi \, G\, \Sigma _{\rm crit} \approx 10^{-8}\, \rm {cm\, s^{-2}} \approx \mathit{ g}_{\dagger }$$. Hence, the observed g† may correspond to the characteristic acceleration scale above which stellar feedback cannot prevent efficient star formation, and baryons will eventually come to dominate. We further show how this may give rise to the observed acceleration scaling $$g_{\rm obs}\sim (g_{\rm baryon}\, g_{\dagger })^{1/2}$$ (where gbaryon is the acceleration due to baryons alone) and flat rotation curves. The derived characteristic acceleration g† can be expressed in terms of fundamental constants (gravitational constant, proton mass, and Thomson cross-section): $$g_{\dagger }\sim 0.1\, G\, m_{\mathrm{ p}}/\sigma _{\rm T}$$. 

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3. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin; Harper, Alicia; Landsberg, J.M. (January 2023, Forum of Mathematics, Pi)

   Abstract Let$$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$$denote the matrix multiplication tensor (and write$$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$$), and let$$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$$denote the determinant polynomial considered as a tensor. For a tensorT, let$$\underline {\mathbf {R}}(T)$$denote its border rank. We (i) give the first hand-checkable algebraic proof that$$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$$, (ii) prove$$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$$and$$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$$M_{\langle 2\rangle }$$, (iii) prove$$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$$, (iv) prove$$\underline {\mathbf {R}}(\operatorname {det}_3)=17$$, improving the previous lower bound of$$12$$, (v) prove$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$$for all$$\mathbf {n}\geq 25$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$$was known, as well as lower bounds for$$4\leq \mathbf {n}\leq 25$$, and (vi) prove$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$$for all$$\mathbf {n} \ge 18$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 

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4. Hopf algebra actions in tensor categories

   https://doi.org/10.1007/s00031-020-09560-w  

   Bischoff, M.; Davydov, A. (April 2020, Transformation Groups)

   We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group. 

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5. Measuring the photoionization rate, neutral fraction, and mean free path of H  i ionizing photons at 4.9 ≤ z ≤ 6.0 from a large sample of XShooter and ESI spectra

   https://doi.org/10.1093/mnras/stad2566  

   Gaikwad, Prakash; Haehnelt, Martin G.; Davies, Fredrick B.; Bosman, Sarah E. I.; Molaro, Margherita; Kulkarni, Girish; D’Odorico, Valentina; Becker, George D.; Davies, Rebecca L.; Nasir, Fahad; et al (September 2023, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT We measure the mean free path ($$\lambda _{\rm mfp,H\, \small {I}}$$), photoionization rate ($$\langle \Gamma _{\rm H\, \small {I}} \rangle$$), and neutral fraction ($$\langle f_{\rm H\, \small {I}} \rangle$$) of hydrogen in 12 redshift bins at 4.85 < z < 6.05 from a large sample of moderate resolution XShooter and ESI QSO absorption spectra. The fluctuations in ionizing radiation field are modelled by post-processing simulations from the Sherwood suite using our new code ‘EXtended reionization based on the Code for Ionization and Temperature Evolution’ (ex-cite). ex-cite uses efficient Octree summation for computing intergalactic medium attenuation and can generate large number of high resolution $$\Gamma _{\rm H\, \small {I}}$$ fluctuation models. Our simulation with ex-cite shows remarkable agreement with simulations performed with the radiative transfer code Aton and can recover the simulated parameters within 1σ uncertainty. We measure the three parameters by forward-modelling the  Lyα forest and comparing the effective optical depth ($$\tau _{\rm eff, H\, \small {I}}$$) distribution in simulations and observations. The final uncertainties in our measured parameters account for the uncertainties due to thermal parameters, modelling parameters, observational systematics, and cosmic variance. Our best-fitting parameters show significant evolution with redshift such that $$\lambda _{\rm mfp,H\, \small {I}}$$ and $$\langle f_{\rm H\, \small {I}} \rangle$$ decreases and increases by a factor ∼6 and ∼104, respectively from z ∼ 5 to z ∼ 6. By comparing our $$\lambda _{\rm mfp,H\, \small {I}}$$, $$\langle \Gamma _{\rm H\, \small {I}} \rangle$$ and $$\langle f_{\rm H\, \small {I}} \rangle$$ evolution with that in state-of-the-art Aton radiative transfer simulations and the Thesan and CoDa-III simulations, we find that our best-fitting parameter evolution is consistent with a model in which reionization completes by z ∼ 5.2. Our best-fitting model that matches the $$\tau _{\rm eff, H\, \small {I}}$$ distribution also reproduces the dark gap length distribution and transmission spike height distribution suggesting robustness and accuracy of our measured parameters. 

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