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1. A New Family of Algebraically Defined Graphs with Small Automorphism Group

   https://doi.org/10.37236/10707  

   Lazebnik, Felix; Taranchuk, Vladislav (January 2022, The Electronic Journal of Combinatorics)

   Let $$p$$ be an odd  prime, $q=p^e$, $$e \geq 1$$, and $$\mathbb{F} = \mathbb{F}_q$$ denote the finite field of $$q$$ elements.  Let $$f: \mathbb{F}^2\to \mathbb{F}$$ and  $$g: \mathbb{F}^3\to \mathbb{F}$$  be functions, and  let $$P$$ and $$L$$ be two copies of the 3-dimensional vector space $$\mathbb{F}^3$$. Consider a bipartite graph $$\Gamma_\mathbb{F} (f, g)$$ with vertex partitions $$P$$ and $$L$$ and with edges defined as follows: for every $$(p)=(p_1,p_2,p_3)\in P$$ and every $$[l]= [l_1,l_2,l_3]\in L$$, $$\{(p), [l]\} = (p)[l]$$ is an edge in $$\Gamma_\mathbb{F} (f, g)$$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question  appeared in Nassau: Given $$\Gamma_\mathbb{F} (f, g)$$,  is it always possible to find a function $$h:\mathbb{F}^2\to \mathbb{F}$$ such that the graph $$\Gamma_\mathbb{F} (f, h)$$  with the same vertex set as $$\Gamma_\mathbb{F} (f, g)$$ and with edges $(p)[l]$  defined in a similar way  by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $$\Gamma_\mathbb{F} (f, g)$$ for infinitely many $$q$$?  In this paper we show that the  answer to the question is negative and the graphs $$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$$ provide such an example for $$p \equiv 1 \pmod{3}$$. Our argument is based on proving that the automorphism group of these graphs has order $$p$$, which is the smallest possible order of the automorphism group of graphs of the form $$\Gamma_{\mathbb{F}}(f, g)$$. 

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2. New germanate and mixed cobalt germanate salt inclusion materials: [(Rb ₆ F)(Rb ₄ F)][Ge ₁₄ O ₃₂ ] and [(Rb ₆ F)(Rb _3.1 Co _0.9 F _0.96 )][Co _3.8 Ge _10.2 O ₃₀ F ₂ ]

   https://doi.org/10.1039/D0CE01099E  

   Carone, Darren; Usman, Mohammad; Klepov, Vladislav V.; Smith, Mark D.; Kocevski, Vancho; Besmann, Theodore M.; zur Loye, Hans-Conrad (December 2020, CrystEngComm)

   null (Ed.)

   Single crystals of two new germanates, [(Rb 6 F)(Rb 4 F)][Ge 14 O 32 ] and [(Rb 6 F)(Rb 3.1 Co 0.9 F 0.96 )][Co 3.8 Ge 10.2 O 30 F 2 ], were synthesized via high temperature RbCl/RbF flux growth. Both compounds crystallize in the cubic space group F 4̄3 m and possess the germanium framework of the previously reported salt inclusion material (SIM), [(Cs 6 F)(Cs 3 AgF)][Ge 14 O 32 ], related to the Ge 7 O 16 zeolitic family. These materials demonstrate the ability to accommodate a variety of salt-inclusions, and exhibit chemical flexibility enabling modifications of the framework through incorporation of Co. Alteration of the salt-inclusion led to intrinsic luminescence of [(Rb 6 F)(Rb 4 F)][Ge 14 O 32 ] while modification of the framework resulted in an unanticipated Rb/Co salt/inclusion in [(Rb 6 F)(Rb 3.1 Co 0.9 F 0.96 )][Co 3.8 Ge 10.2 O 30 F 2 ]. Fluorescence measurements were performed on [(Rb 6 F)(Rb 4 F)][Ge 14 O 32 ]. First-principles calculations in the form of density functional theory (DFT) were performed for [(Rb 6 F)(Rb 3.1 Co 0.9 F 0.96 )][Co 3.8 Ge 10.2 O 30 F 2 ] to elucidate its electronic and magnetic properties, and stability at 0 K. 

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3. pH-dependent 11° F1FO ATP synthase sub-steps reveal insight into the FO torque generating mechanism

   https://doi.org/10.7554/eLife.70016  

   Yanagisawa, Seiga; Frasch, Wayne D (December 2021, eLife)

   Most cellular ATP is made by rotary F 1 F O ATP synthases using proton translocation-generated clockwise torque on the F O c-ring rotor, while F 1 -ATP hydrolysis can force counterclockwise rotation and proton pumping. The F O torque-generating mechanism remains elusive even though the F O interface of stator subunit-a, which contains the transmembrane proton half-channels, and the c-ring is known from recent F 1 F O structures. Here, single-molecule F 1 F O rotation studies determined that the pKa values of the half-channels differ, show that mutations of residues in these channels change the pKa values of both half-channels, and reveal the ability of F O to undergo single c-subunit rotational stepping. These experiments provide evidence to support the hypothesis that proton translocation through F O operates via a Grotthuss mechanism involving a column of single water molecules in each half-channel linked by proton translocation-dependent c-ring rotation. We also observed pH-dependent 11° ATP synthase-direction sub-steps of the Escherichia coli c 10 -ring of F 1 F O against the torque of F 1 -ATPase-dependent rotation that result from H + transfer events from F O subunit-a groups with a low pKa to one c-subunit in the c-ring, and from an adjacent c-subunit to stator groups with a high pKa. These results support a mechanism in which alternating proton translocation-dependent 11° and 25° synthase-direction rotational sub-steps of the c 10 -ring occur to sustain F 1 F O ATP synthesis. 

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4. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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5. Odoni’s conjecture on arboreal Galois representations is false

   https://doi.org/10.1090/proc/15920  

   Dittmann, Philip; Kadets, Borys (August 2022, Proceedings of the American Mathematical Society)

   Suppose f ∈ K [ x ] f \in K[x] is a polynomial. The absolute Galois group of K K acts on the preimage tree T \mathrm {T} of 0 0 under f f . The resulting homomorphism ϕ f : Gal K → Aut ⁡ T \phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T} is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K K there exists a polynomial f f for which ϕ f \phi _f is surjective. We show that this conjecture is false. 

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