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1. Lieb–Thirring Inequality for the 2D Pauli Operator

   https://doi.org/10.1007/s00220-024-05177-2  

   Frank, Rupert_L; Kovařík, Hynek (January 2025, Communications in Mathematical Physics)

   Abstract By the Aharonov–Casher theorem, the Pauli operatorPhas no zero eigenvalue when the normalized magnetic flux$$\alpha $$ $\alpha$satisfies$$|\alpha |<1$$ $\left|\alpha \right|<1$, but it does have a zero energy resonance. We prove that in this case a Lieb–Thirring inequality for the$$\gamma $$ $\gamma$-th moment of the eigenvalues of$$P+V$$ $P+V$is valid under the optimal restrictions$$\gamma \ge |\alpha |$$ $\gamma \ge \left|\alpha \right|$and$$\gamma >0$$ $\gamma >0$. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order$$\gamma \ge 1$$ $\gamma \ge 1$. 

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2. On a Bohr set analogue of Chowla’s conjecture

   https://doi.org/10.1007/s00209-025-03770-2  

   Teräväinen, Joni; Walker, Aled (May 2025, Mathematische Zeitschrift)

   Abstract Let$$\lambda $$ $\lambda$denote the Liouville function. We show that the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\lambda \left(\lfloor {\alpha }_{2}n\rfloor \right)$is 0 whenever$$\alpha _1,\alpha _2$$ ${\alpha }_1,{\alpha }_2$are positive reals with$$\alpha _1/\alpha _2$$ ${\alpha }_1/{\alpha }_2$irrational. We also show that for$$k\geqslant 3$$ $k⩾3$the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\cdots \lambda \left(\lfloor {\alpha }_{k}n\rfloor \right)$has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers$$\alpha _i.$$ ${\alpha }_i.$Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets. 

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3. Integer Optimal Control with Fractional Perimeter Regularization

   https://doi.org/10.1007/s00245-024-10157-y  

   Antil, Harbir; Manns, Paul (July 2024, Applied Mathematics & Optimization)

   Abstract Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension$$d - 1$$ $d-1$for a domain of dimensiond. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent$$0<\alpha <1$$ $0<\alpha <1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a$$\Gamma $$ $\Gamma$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent$$\alpha $$ $\alpha$tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for$$\alpha \in (0.5,1)$$ $\alpha \in (0.5,1)$under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional. 

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4. The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

   https://doi.org/10.1007/s00526-024-02660-5  

   Bauer, Martin; Le Brigant, Alice; Lu, Yuxiu; Maor, Cy (February 2024, Calculus of Variations and Partial Differential Equations)

   Abstract We introduce a family of Finsler metrics, called the$$L^p$$ $L^p$-Fisher–Rao metrics$$F_p$$ $F_p$, for$$p\in (1,\infty )$$ $p\in (1,\infty )$, which generalizes the classical Fisher–Rao metric$$F_2$$ $F_2$, both on the space of densities$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$and probability densities$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$. We then study their relations to the Amari–C̆encov$$\alpha $$ $\alpha$-connections$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$from information geometry: on$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$, the geodesic equations of$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$coincide, for$$p = 2/(1-\alpha )$$ $p=2/(1-\alpha )$. Both are pullbacks of canonical constructions on$$L^p(M)$$ $L^p\left(M\right)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$ $\alpha$-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$, the$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$ $L^p\left(M\right)$, but in this case they no longer coincide unless$$p=2$$ $p=2$. Using this transformation, we solve the geodesic equation of the$$\alpha $$ $\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$ $F_p$, and study their relation to$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$. 

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5. Towards a Schinzel–Wójcik theorem for number fields

   https://doi.org/10.1007/s40879-025-00819-8  

   Pollack, Paul (April 2025, European Journal of Mathematics)

   Abstract Schinzel and Wójcik have shown that for every$$\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}$$ $\alpha ,\beta \in Q^{\times }\backslash \{\pm 1\}$, there are infinitely many primespwhere$$v_p(\alpha )=v_p(\beta )=0$$ $v_p\left(\alpha \right)=v_p\left(\beta \right)=0$and where$$\alpha $$ $\alpha$and$$\beta $$ $\beta$generate the same multiplicative group modp. We prove a weaker result in the same direction for algebraic numbers$$\alpha , \beta $$ $\alpha ,\beta$. Let$$\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }$$ $\alpha ,\beta \in {\overline{Q}}^{\times }$, and suppose$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\alpha \right)|\ne 1$and$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\beta \right)|\ne 1$. Then for some positive integer$$C = C(\alpha ,\beta )$$ $C=C(\alpha ,\beta )$, there are infinitely many prime idealsPof Equation missing<#comment/>where$$v_P(\alpha )=v_P(\beta )=0$$ $v_P\left(\alpha \right)=v_P\left(\beta \right)=0$and where the group$$\langle \beta \bmod {P}\rangle $$ $⟨\beta modP⟩$is a subgroup of$$\langle \alpha \bmod {P}\rangle $$ $⟨\alpha modP⟩$with$$[\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]$$ $[⟨\alpha modP⟩:⟨\beta modP⟩]$dividingC. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shiftedS-units. 

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