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1. Inexact Fixed-Point Proximity Algorithm for the $$\ell _0$$ Sparse Regularization Problem

   https://doi.org/10.1007/s10915-024-02600-7  

   Fang, Ronglong; Xu, Yuesheng; Yan, Mingsong (July 2024, Journal of Scientific Computing)

   Abstract We studyinexactfixed-point proximity algorithms for solving a class of sparse regularization problems involving the$$\ell _0$$ ${\ell }_0$norm. Specifically, the$$\ell _0$$ ${\ell }_0$model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the$$\ell _0$$ ${\ell }_0$norm regularization term. Such an$$\ell _0$$ ${\ell }_0$model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the$$\ell _0$$ ${\ell }_0$models found by the proposed inexact algorithm outperform global minimizers of the corresponding$$\ell _1$$ ${\ell }_1$models, in terms of approximation accuracy and sparsity of the solutions. 

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2. The Curvature Operator of the Second Kind in Dimension Three

   https://doi.org/10.1007/s12220-024-01639-0  

   Fluck, Harry; Li, Xiaolong (April 2024, The Journal of Geometric Analysis)

   Abstract This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that$$\alpha $$ $\alpha$-positive/$$\alpha $$ $\alpha$-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all$$\alpha \in [1,5]$$ $\alpha \in [1,5]$. 

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3. On the Korányi spherical maximal function on Heisenberg groups

   https://doi.org/10.1007/s00208-022-02533-2  

   Srivastava, Rajula (December 2022, Mathematische Annalen)

   Abstract We prove$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in (Ganguly and Thangavelu in J Funct Anal 280(3):108832, 2021) for the associated lacunary maximal operator. 

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4. 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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5. 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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