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1. 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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2. Operator noncommutative functions

   https://doi.org/10.4153/S0008439522000339  

   Augat, Meric; McCarthy, John E. (June 2023, Canadian Mathematical Bulletin)

   Abstract We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g.,$$B({\mathcal H})$$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the$$k{\mathrm {th}}$$directional derivative of any NC function at a scalar point is ak-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball. 

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3. Efficient simplicial replacement of semialgebraic sets

   https://doi.org/10.1017/fms.2023.36  

   Basu, Saugata; Karisani, Negin (January 2023, Forum of Mathematics, Sigma)

   Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$$S \subset \mathbb {R}^k$$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$$\ell \geq 0$$, there exists an algorithm which takes as input a description of a semialgebraic subset$$S \subset \mathbb {R}^k$$given by a quantifier-free first-order formula$$\phi $$in the language of the reals and produces as output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$is$$\ell $$-equivalent toS. The complexity of our algorithm is bounded by$$(sd)^{k^{O(\ell )}}$$, wheresis the number of polynomials appearing in the formula$$\phi $$, andda bound on their degrees. For fixed$$\ell $$, this bound issingly exponentialink. In particular, since$$\ell $$-equivalence implies that thehomotopy groupsup to dimension$$\ell $$of$$|\Delta |$$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$$\ell $$homotopy groups ofSto the combinatorial problem of computing the first$$\ell $$homotopy groups of a finite simplicial complex of size bounded by$$(sd)^{k^{O(\ell )}}$$. 

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4. Gardner formula for Ising perceptron models at small densities

   Bolthausen, E.; Nakajima, S; Sun, N; Xu, C (January 2022, Conference on Learning Theory)

   We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth–Mezard 1989, see also Gardner–Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero. 

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5. Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

   https://doi.org/10.1142/S1793042120400126  

   Folsom, Amanda (March 2021, International Journal of Number Theory)

   null (Ed.)

   In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity. 

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