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Patrick Clarke

Publications and source records attributed to Patrick Clarke.

10 recordsLinked to original sources
arXiv2018

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

math.AG↗
arXiv2016

Birationality and Landau-Ginzburg models

We introduce a new technique for approaching birationality questions that arise in the mirror symmetry of complete intersections in toric varieties. As an application we answer affirmatively and conclusively the question of Batyrev-Nill (2008) about the birationality of Calabi-Yau families associated to multiple mirror nef-partitions. This completes the progress in this direction made by Li's breakthrough (2016). In the process, we obtain results in the theory of Borisov's nef-partitions (1993) and provide new insight on the geometric content of the multiple mirror phenomenon.

math.AG↗
arXiv2015

Curved A-infinity-categories: adjunction and homotopy

We develop a theory of curved A-infinity-categories around equivalences of their module categories. This allows for a uniform treatment of curved and uncurved A-infinity-categories which generalizes the classical theory of uncurved A-infinity algebras. Furthermore, the theory is sufficiently general to treat both Fukaya categories and categories of matrix factorizations, as well as to provide a context in which unitification and categorification of pre-categories can be carried out. Our theory is built around two functors: the adjoint algebra functor U_e and the functor Q_*. The bulk of the paper is dedicated to proving crucial adjunction and homotopy theorems about these functors. In addition, we explore the non-vanishing of the module categories and give a precise statement and proof the result known as "Positselski-Kontsevich vanishing".

math.AG↗
arXiv2015

Dual fans and mirror symmetry

We show that the mirror constructions of Greene-Plesser, Berglund-Hubsch, Batryev-Borsov, Givental and Hori-Vafa can be expressed in terms of what we call dual fans. To do this, we associate to a pair of dual fans a pair of toric Landau-Ginzburg models, and we describe a process by which each of the mirror constructions listed also produces a pair of toric Landau-Ginzburg models. Replacing mirror pairs by toric Landau-Ginzburg models is reversible, and our main result is the dual fan models and the mirror pairs models coincide.

math.AG↗
arXiv2014

Quasi-Splines and their moduli

We study what we call quasi-spline sheaves over locally Noetherian schemes. This is done with the intention of considering splines from the point of view of moduli theory. In other words, we study the way in which certain objects that arise in the theory of splines can be made to depend on parameters. In addition to quasi-spline sheaves, we treat ideal difference-conditions, and individual quasi- splines. Under certain hypotheses each of these types of objects admits a fine moduli scheme. The moduli of quasi-spline sheaves is proper, and there is a natural compactification of the moduli of ideal difference-conditions. We include some speculation on the uses of these moduli in the theory of splines and topology, and an appendix with a treatment of the Billera-Rose homogenization in scheme theoretic language.

math.AG↗
arXiv2013

A proof of the birationality of certain BHK-mirrors

We generalize and give an elementary proof of Kelly's refinement of Shoemaker's result on the birationality of certain BHK-mirrors. Our approach uses a construction that is equivalent to the Krawitz generalization of the duality of Berglund-Hubsch.

math.AG↗
arXiv2013

Bidirectional MMSE Algorithms for Interference Mitigation in CDMA Systems over Fast Fading Channels

This paper presents adaptive bidirectional minimum mean-square error (MMSE) parameter estimation algorithms for fast-fading channels. The time correlation between successive channel gains is exploited to improve the estimation and tracking capabilities of adaptive algorithms and provide robustness against time-varying channels. Bidirectional normalized least mean-square (NLMS) and conjugate gradient (CG) algorithms are devised along with adaptive mixing parameters that adjust to the time-varying channel correlation properties. An analysis of the proposed algorithms is provided along with a discussion of their performance advantages. Simulations for an application to interference suppression in DS-CDMA systems show the advantages of the proposed algorithms.

cs.IT↗
arXiv2013

Resource Allocation and Interference Mitigation Techniques for Cooperative Multi-Antenna and Spread Spectrum Wireless Networks

This chapter presents joint interference suppression and power allocation algorithms for DS-CDMA and MIMO networks with multiple hops and amplify-and-forward and decode-and-forward (DF) protocols. A scheme for joint allocation of power levels across the relays and linear interference suppression is proposed. We also consider another strategy for joint interference suppression and relay selection that maximizes the diversity available in the system. Simulations show that the proposed cross-layer optimization algorithms obtain significant gains in capacity and performance over existing schemes.

cs.IT↗
arXiv2008

Duality for toric Landau-Ginzburg models

We introduce a duality construction for toric Landau-Ginzburg models, applicable to complete intersections in toric varieties via the sigma model / Landau-Ginzburg model correspondence. This construction is shown to reconstruct those of Batyrev-Borisov, Berglund-H"ubsch, Givental, and Hori-Vafa. It can be done in more general situations, and provides partial resolutions when the above constructions give a singular mirror. An extended example is given: the Landau-Ginzburg models dual to elliptic curves in (P^1)^2 .

math.AG↗