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1. Malnormal matrices

   https://doi.org/10.1090/proc/15821  

   Mulcahy, Garrett; Sinclair, Thomas (July 2022, Proceedings of the American Mathematical Society)

   We exhibit an operator norm bounded, infinite sequence { A n } \{A_n\} of 3 n × 3 n 3n \times 3n complex matrices for which the commutator map X ↦ X A n − A n X X\mapsto XA_n - A_nX is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert–Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and provide numerical evidence. 

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2. Dynamical generation of parameter laminations

   https://doi.org/10.1090/conm/744  

   Blokh, A; Oversteegen, L; Timorin, V. (January 2020, Contemporary mathematics)

   Abstract. Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of “pullbacks” of minors from the main cardioid 

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3. Operator realizations of non-commutative analytic functions

   https://doi.org/10.1017/fms.2025.10038  

   Augat, Méric L; Martin, Robert T_W; Shamovich, Eli (January 2025, Forum of Mathematics, Sigma)

   A realization is a triple, (A,b,c), consisting of a d−tuple, A=(A1,⋯,Ad), d∈N, of bounded linear operators on a separable, complex Hilbert space, H, and vectors b,c∈H. Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin, 0:=(0,⋯,0), of the NC universe of d−tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula b∗(I−zA)−1c . It is well known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at 0 . Such finite realizations contain valuable information about the NC rational functions they generate. By extending to infinite-dimensional realizations, we construct, study and characterize more general classes of analytic NC functions. In particular, we show that an NC function is (uniformly) entire if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that a formal Taylor series extends globally to an entire or meromorphic function in the complex plane, C, if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This motivates our definition of the field of global (uniformly) meromorphic NC functions as the field of fractions generated by NC rational expressions in the ring of NC functions with jointly compact realizations. This definition recovers the field of meromorphic functions in C when restricted to one variable. 

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4. Convergence rate for a Radau hp collocation method applied to constrained optimal control

   https://doi.org/10.1007/s10589-019-00100-1  

   Hager, William W.; Hou, Hongyan; Mohapatra, Subhashree; Rao, Anil V.; Wang, Xiang-Sheng (May 2019, Computational Optimization and Applications)

   For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory. 

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5. On extremizing sequences for adjoint Fourier restriction to the sphere

   https://doi.org/10.1016/j.aim.2024.109854  

   Flock, Taryn C; Stovall, Betsy (September 2024, Advances in Mathematics)

   In this article, we develop a linear profile decomposition for the $$L^p \to L^q$$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p\max\{p,\tfrac{d+2}d p'\}$$, or if $$q=\tfrac{d+2}d p'$$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator. 

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