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1. Quantum group intertwiner space from quantum curved tetrahedron

   https://doi.org/10.1088/1361-6382/ad5f71  

   Han, Muxin; Hsiao, Chen-Hung; Pan, Qiaoyin (July 2024, Classical and Quantum Gravity)

   Abstract In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in Haggardet al(2016Ann. Henri Poincaré172001–48). Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition $M_4{M}_3{M}_2{M}_1=1$, $M_{\nu }\in \mathrm{SU}\left(2\right)$, for the homogeneously curved tetrahedron. The quantum group $U_q\left(\mathfrak{su}\right(2\left)\right)$emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of $U_q\left(\mathfrak{su}\right(2\left)\right)$. In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory. 

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2. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa; Teräväinen, Joni (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

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3. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo; de la Llave, Rafael (March 2022, Nonlinearity)

   Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation ${\mathcal{H}}_{\epsilon }\left(p,q,I,\phi ,t\right)=h\left(I\right)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i\left(q_i\right)\right)+\epsilon H_1\left(p,q,I,\phi ,t\right),$where $\left(p,q\right)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$, $\left(I,\phi \right)\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}. 

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4. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo; Wang, Jinhuan (June 2024, Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

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5. The metastable Q ³ Δ ₂ state of ThO: a new resource for the ACME electron EDM search

   https://doi.org/10.1088/1367-2630/ab6a3a  

   Wu, X.; Han, Z.; Chow, J.; Ang, D. G.; Meisenhelder, C.; Panda, C. D.; West, E. P.; Gabrielse, G.; Doyle, J. M.; DeMille, D. (February 2020, New Journal of Physics)

   Abstract The best upper limit for the electron electric dipole moment was recently set by the ACME collaboration. This experiment measures an electron spin-precession in a cold beam of ThO molecules in their metastable $H{(}^3{\mathrm{\Delta }}_1)$state. Improvement in the statistical and systematic uncertainties is possible with more efficient use of molecules from the source and better magnetometry in the experiment, respectively. Here, we report measurements of several relevant properties of the long-lived $Q{(}^3{\mathrm{\Delta }}_2)$state of ThO, and show that this state is a very useful resource for both these purposes. TheQstate lifetime is long enough that its decay during the time of flight in the ACME beam experiment is negligible. The large electric dipole moment measured for theQstate, giving rise to a large linear Stark shift, is ideal for an electrostatic lens that increases the fraction of molecules detected downstream. The measured magnetic moment of theQstate is also large enough to be used as a sensitive co-magnetometer in ACME. Finally, we show that theQstate has a large transition dipole moment to the $C{(}^1{\mathrm{\Pi }}_1)$state, which allows for efficient population transfer between the ground state $X{(}^1{\mathrm{\Sigma }}^{+})$and theQstate via $X-C-Q$Stimulated Raman Adiabatic Passage (STIRAP). We demonstrate 90 % STIRAP transfer efficiency. In the course of these measurements, we also determine the magnetic moment ofCstate, the $X\to C$transition dipole moment, and branching ratios of decays from theCstate. 

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