Updated: July 5th, 2023 15:40

2021 journal article

LOG-CONCAVE POLYNOMIALS, I: ENTROPY AND A DETERMINISTIC APPROXIMATION ALGORITHM FOR COUNTING BASES OF MATROIDS

*DUKE MATHEMATICAL JOURNAL*, *170*(16), 3459–3504.

Source: Web Of Science

Added: November 29, 2021

2021 article

Log-Concave Polynomials IV: Approximate Exchange, Tight Mixing Times, and Near-Optimal Sampling of Forests

*STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING*, pp. 408–420.

author keywords: Counting and Sampling; Near-Linear Time Algorithm; Random Walk; Exchange Property; Discrete Optimization

TL;DR:
It is shown that the down-up random walk, started from an arbitrary point in the support, mixes in time O(klogk), and tight mixing time bounds are proved for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials.
(via Semantic Scholar)

UN Sustainable Development Goal Categories

11. Sustainable Cities and Communities
(Web of Science)

15. Life on Land
(OpenAlex)

Source: Web Of Science

Added: July 18, 2022

2021 article

Log-Concave Polynomials in Theory and Applications (Tutorial)

*STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING*, pp. 12–12.

author keywords: Log-Concave Polynomials; Matroids; Approximate Counting; Approximate Sampling; High Dimensional Expanders

TL;DR:
This tutorial will introduce the theory and applications of log-concave polynomials, survey some of the recent developments in this area, and survey the random cluster model in certain regimes.
(via Semantic Scholar)

Source: Web Of Science

Added: July 18, 2022

2021 journal article

Positively hyperbolic varieties, tropicalization, and positroids

*ADVANCES IN MATHEMATICS*, *383*.

Source: Web Of Science

Added: May 24, 2021

2019 journal article

Computing complex and real tropical curves using monodromy

*JOURNAL OF PURE AND APPLIED ALGEBRA*, *223*(12), 5232–5250.

Source: Web Of Science

Added: August 5, 2019

2019 article

Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid

*PROCEEDINGS OF THE 51ST ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING (STOC '19)*, pp. 1–12.

author keywords: Approximating Counting; Approximate Sampling; High-Dimensional Expanders; Geometry of Polynomials

TL;DR:
It is shown that a high dimensional walk on a weighted simplicial complex mixes rapidly if for every link of the complex, the corresponding localized random walk on the 1-skeleton is a strong spectral expander, and an FPRAS is designed to count the number of bases of any matroid given by an independent set oracle.
(via Semantic Scholar)

UN Sustainable Development Goal Categories

11. Sustainable Cities and Communities
(Web of Science)

15. Life on Land
(OpenAlex)

Source: Web Of Science

Added: April 27, 2020

2019 journal article

Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

*INTERNATIONAL MATHEMATICS RESEARCH NOTICES*, *2019*(1), 33–54.

Source: Web Of Science

Added: February 4, 2019

2018 article

Log-Concave Polynomials, Entropy, and a Deterministic Approximation Algorithm for Counting Bases of Matroids

*2018 IEEE 59TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS)*, pp. 35–46.

author keywords: matroid; deterministic counting; entropy; log-concave polynomial

Source: Web Of Science

Added: January 21, 2019

2015 conference paper

A small frame and a certificate of its injectivity

*2015 International Conference on Sampling Theory and Applications (SAMPTA)*, 197–200.

Source: NC State University Libraries

Added: August 6, 2018

2015 journal article

An algebraic characterization of injectivity in phase retrieval

*Applied and Computational Harmonic Analysis*, *38*(2), 346–356.

author keywords: Phase retrieval; Algebraic geometry

TL;DR:
It is shown that any vector is uniquely determined from 4 M − 4 generic measurements, and the set of frames defining non-injective measurements with the projection of a real variety is identified and bound its dimension.
(via Semantic Scholar)

Source: Crossref

Added: February 24, 2020

2015 journal article

Computing Hermitian determinantal representations of hyperbolic curves

*INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION*, *25*(8), 1327–1336.

author keywords: Hyperbolic polynomials; determinantal representations; interlacing; Hermitian matrices of linear forms

TL;DR:
An algorithm is presented that reduces a large part of the problem to linear algebra and its numerical implementation is discussed, which is computationally intensive but effective.
(via Semantic Scholar)

UN Sustainable Development Goal Categories

11. Sustainable Cities and Communities
(Web of Science)

Source: Web Of Science

Added: August 6, 2018

2015 journal article

Hyperbolic polynomials, interlacers, and sums of squares

*Mathematical Programming*, *153*(1), 223–245.

Source: Crossref

Added: December 11, 2020

2015 journal article

Quartic spectrahedra

*MATHEMATICAL PROGRAMMING*, *151*(2), 585–612.

Source: Web Of Science

Added: August 6, 2018

2013 journal article

Determinantal representations of hyperbolic plane curves: An elementary approach

*Journal of Symbolic Computation*, *57*, 48–60.

author keywords: Hyperbolic polynomials; Determinantal representations; Interlacing; Hermitian matrices of linear forms

TL;DR:
By allowing for Hermitian matrices instead, this work shows that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extends a classical construction of determinantal representations of Dixon from 1902.
(via Semantic Scholar)

Source: Crossref

Added: August 28, 2020

2013 journal article

The entropic discriminant

*Advances in Mathematics*, *244*, 678–707.

UN Sustainable Development Goal Categories

10. Reduced Inequalities
(OpenAlex)

Source: Crossref

Added: August 28, 2020

2012 journal article

Real radical initial ideals

*Journal of Algebra*, *352*(1), 392–407.

Source: Crossref

Added: December 11, 2020

2012 journal article

The Central Curve in Linear Programming

*Foundations of Computational Mathematics*, *12*(4), 509–540.

Source: Crossref

Added: August 28, 2020

2011 journal article

Edges of the Barvinok–Novik Orbitope

*Discrete & Computational Geometry*, *46*(3), 479–487.

author keywords: Moment curve; Toeplitz operator; Orbitope; Convex hull of a curve

TL;DR:
Results of Smilansky prove tightness for k=2 and prove the conjecture that for all k the Barvinok–Novik orbitope is tight.
(via Semantic Scholar)

Source: Crossref

Added: January 5, 2021

2011 journal article

Quartic curves and their bitangents

*Journal of Symbolic Computation*, *46*(6), 712–733.

author keywords: Plane curves; Bitangents; Determinantal representations; Sums of squares; Semidefinite programming; Gale duality

TL;DR:
Interwoven is an exposition of much of the 19th century theory of plane quartics, which expresses Vinnikov quartics as spectrahedra and positive Quartics as Gram matrices and explores the geometry of Gram spectahedra.
(via Semantic Scholar)

Source: Crossref

Added: August 28, 2020

2009 journal article

Lower bounds for optimal alignments of binary sequences

*Discrete Applied Mathematics*, *157*(15), 3341–3346.

author keywords: Sequence alignment; Parametric analysis; Computational biology

TL;DR:
The maximum number of distinct optimal alignment summaries over all pairs of length n sequences is @Q(n^2^/^3), thereby disproving the ''n conjecture''.
(via Semantic Scholar)

Source: Crossref

Added: January 5, 2021

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