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Academic literature on the topic 'Entrywise Positivity Preservers'

Author: Grafiati
Published: 6 September 2023

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Journal articles on the topic "Entrywise Positivity Preservers"

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Khare, Apoorva, and Terence Tao. "On the sign patterns of entrywise positivity preservers in fixed dimension." American Journal of Mathematics 143, no. 6 (2021): 1863–929. http://dx.doi.org/10.1353/ajm.2021.0049.

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Khare, Apoorva. "Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities." Transactions of the American Mathematical Society 375, no. 3 (December 22, 2021): 2217–36. http://dx.doi.org/10.1090/tran/8563.

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A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n ⩾ 1 n \geqslant 1 , if the entrywise application of a smooth function f : ( 0 , ∞ ) → R f : (0,\infty ) \to \mathbb {R} preserves the set of n × n n \times n positive semidefinite matrices with positive entries, then f f and its first n − 1 n-1 derivatives are non-negative on ( 0 , ∞ ) (0,\infty ) . In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108]. In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n n nonzero Taylor coefficients at individual points rather than on all of ( 0 , ∞ ) (0,\infty ) . A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series 1 / ( 1 − u j v k ) 1 / (1 - u_j v_k) ), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.
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Belton, Alexander, Dominique Guillot, Apoorva Khare, and Mihai Putinar. "Schur polynomials and matrix positivity preservers." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6408.

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International audience A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
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Dissertations / Theses on the topic "Entrywise Positivity Preservers"

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Vishwakarma, Prateek Kumar. "Positivity preservers forbidden to operate on diagonal blocks." Thesis, 2021. https://etd.iisc.ac.in/handle/2005/5537.

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The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at “opposite ends”, and in both cases the preservers have to be absolutely monotonic. The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
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Books on the topic "Entrywise Positivity Preservers"

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Khare, Apoorva. Matrix Analysis and Entrywise Positivity Preservers. Cambridge University Press, 2021.

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Khare, Apoorva. Matrix Analysis and Entrywise Positivity Preservers. Cambridge University Press, 2022.

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Book chapters on the topic "Entrywise Positivity Preservers"

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"Entrywise Powers Preserving Positivity, Monotonicity, and Superadditivity." In Matrix Analysis and Entrywise Positivity Preservers, 64–71. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.011.

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"Preservers of Loewner Positivity on Kernels." In Matrix Analysis and Entrywise Positivity Preservers, 155–58. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.022.

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"Entrywise Polynomial Preservers and Horn–Loewner-Type Conditions." In Matrix Analysis and Entrywise Positivity Preservers, 213–19. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.029.

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"Entrywise Powers Preserving Positivity in a Fixed Dimension." In Matrix Analysis and Entrywise Positivity Preservers, 40–45. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.008.

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"Entrywise Preservers of Positivity on Matrices with Zero Patterns." In Matrix Analysis and Entrywise Positivity Preservers, 55–63. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.010.

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"Midconvex Implies Continuity, and 2 × 2 Preservers." In Matrix Analysis and Entrywise Positivity Preservers, 46–54. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.009.

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"The Schur Product Theorem and Nonzero Lower Bounds." In Matrix Analysis and Entrywise Positivity Preservers, 16–22. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.005.

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"Stronger Vasudeva and Schoenberg Theorems via Bernstein’s Theorem." In Matrix Analysis and Entrywise Positivity Preservers, 127–34. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.018.

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"Exercises." In Matrix Analysis and Entrywise Positivity Preservers, 264–69. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.035.

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"Exercises." In Matrix Analysis and Entrywise Positivity Preservers, 84–94. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108867122.013.

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