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The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra
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To every finite metric space X, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial pX({nx : x ∈ X}). This is…
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This work considers the refined problem of characterizing the set of entrywise powers preserving positivity for matrices with a zero pattern encoded by G, and studies how the geometry of $G$ influences the set $\mathcal{H}_G$.
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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…
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We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace…
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For every finite simple connected graph $G = (V,E)$, we introduce an invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained by dividing the determinant of the distance matrix…
Positivity of Hadamard powers of tridiagonal matrices
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Let Mn(R) denotes the set of all n× n matrices over R. A matrix A ∈ Mn(R) is called nonnegative if all its entries are nonnegative. In this paper, every matrix belongs to Mn(R), and is nonnegative…
Positivity of Hadamard powers of a few band matrices
- MathematicsThe Electronic Journal of Linear Algebra
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Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some…
Post-composition transforms of totally positive kernels
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Author(s): Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai | Abstract: The composition operators preserving total non-negativity and total positivity for various classes of…
A Panorama of Positivity. I: Dimension Free
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This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks)…
Totally positive kernels, Polya frequency functions, and their transforms
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The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on…
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