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1. Lieb's most useful contribution to density functional theory?

   https://doi.org/10.4171/90-1/7  

   Burke, Kieron (February 2022, ArXivorg)

   The importance of the Lieb-Simon proof of the relative exactness of Thomas-Fermi theory in the large-Z limit to modern density functional theory (DFT) is explored. The principle, that there is a specific semiclassical limit in which functionals become local, implies that there exist well-defined leading functional corrections to local approximations that become relatively exact for the error in local approximations in this limit. It is argued that this principle might be used to greatly improve the accuracy of the thousand or so DFT calculations that are now published each week. A key question is how to find the leading corrections to any local density approximation as this limit is approached. These corrections have been explicitly derived in ridiculously simple model systems to ridiculously high order, yielding ridiculously accurate energies. Much analytic work is needed to use this principle to improve realistic calculations of molecules and solids. 

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2. A Duality Principle for Groups II: Multi-frames Meet Super-Frames

   https://doi.org/10.1007/s00041-020-09792-0  

   Balan, R.; Dutkay, D.; Han, D.; Larson, D.; Luef, F. (December 2020, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

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3. Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

   https://doi.org/10.1093/imrn/rny022  

   Bhattacharya, Bhaswar B; Ganguly, Shirshendu; Shao, Xuancheng; Zhao, Yufei (March 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract Let Xk denote the number of k-term arithmetic progressions in a random subset of $$\mathbb{Z}/N\mathbb{Z}$$ or $$\{1, \dots , N\}$$ where every element is included independently with probability p. We determine the asymptotics of $$\log \mathbb{P}\big (X_{k} \ge \big (1+\delta \big ) \mathbb{E} X_{k}\big )$$ (also known as the large deviation rate) where p → 0 with $$p \ge N^{-c_{k}}$$ for some constant ck > 0, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of p, the large deviation rate up to a constant factor. 

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4. Local-global principles in circle packings

   https://doi.org/10.1112/S0010437X19007139  

   Fuchs, Elena; Stange, Katherine E.; Zhang, Xin (June 2019, Compositio Mathematica)

   We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$$ satisfying certain conditions, where $$K$$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that $${\mathcal{A}}$$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$$ containing a Zariski dense subgroup of $$\operatorname{PSL}_{2}(\mathbb{Z})$$ . 

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5. Neutrino oscillation constraints on U(1)′ models: from non-standard interactions to long-range forces

   https://doi.org/10.1007/JHEP01(2021)114  

   Coloma, Pilar; Gonzalez-Garcia, M. C.; Maltoni, Michele (January 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We quantify the effect of gauge bosons from a weakly coupled lepton flavor dependent U(1) ′ interaction on the matter background in the evolution of solar, atmospheric, reactor and long-baseline accelerator neutrinos in the global analysis of oscillation data. The analysis is performed for interaction lengths ranging from the Sun-Earth distance to effective contact neutrino interactions. We survey ∼ 10000 set of models characterized by the six relevant fermion U(1) ′ charges and find that in all cases, constraints on the coupling and mass of the Z′ can be derived. We also find that about 5% of the U(1) ′ model charges lead to a viable LMA-D solution but this is only possible in the contact interaction limit. We explicitly quantify the constraints for a variety of models including $$ \mathrm{U}{(1)}_{B-3{L}_e} $$ U 1 B − 3 L e , $$ \mathrm{U}{(1)}_{B-3{L}_{\mu }} $$ U 1 B − 3 L μ , $$ \mathrm{U}{(1)}_{B-3{L}_{\tau }} $$ U 1 B − 3 L τ , $$ \mathrm{U}{(1)}_{B-\frac{3}{2}\left({L}_{\mu }+{L}_{\tau}\right)} $$ U 1 B − 3 2 L μ + L τ , $$ \mathrm{U}{(1)}_{L_e-{L}_{\mu }} $$ U 1 L e − L μ , $$ \mathrm{U}{(1)}_{L_e-{L}_{\tau }} $$ U 1 L e − L τ , $$ \mathrm{U}{(1)}_{L_e-\frac{1}{2}\left({L}_{\mu }+{L}_{\tau}\right)} $$ U 1 L e − 1 2 L μ + L τ . We compare the constraints imposed by our oscillation analysis with the strongest bounds from fifth force searches, violation of equivalence principle as well as bounds from scattering experiments and white dwarf cooling. Our results show that generically, the oscillation analysis improves over the existing bounds from gravity tests for Z′ lighter than ∼ 10 − 8 → 10 − 11 eV depending on the specific couplings. In the contact interaction limit, we find that for most models listed above there are values of g′ and M Z′ for which the oscillation analysis provides constraints beyond those imposed by laboratory experiments. Finally we illustrate the range of Z′ and couplings leading to a viable LMA-D solution for two sets of models. 

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