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How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking

Anna Adamaszek, Parinya Chalermsook, Andreas Wiese
unpublished
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axis-parallel rectangles in the plane.  ...  Another important, poorly understood question is whether one can color rectangles with at most O(ω(R)) colors where ω(R) is the size of a maximum clique in the intersection graph of a set of input rectangles  ...  We explain now how to use Lemma 14 in order to prove Theorem 2 and Corollary 3. Rectangle Coloring It is straightforward to see that Lemma 14 implies the coloring algorithm.  ... 
fatcat:6ewrb2ox2nagjpsw5xxezp45zy

An Introduction to Multiparameter Persistence [article]

Magnus Bakke Botnan, Michael Lesnick
2022 arXiv   pre-print
In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future.  ...  However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped  ...  We also thank the authors of [170] for helpful feedback on our discussion of their work in Section 4.4 and for sharing Fig. 4 .3 with us.  ... 
arXiv:2203.14289v1 fatcat:ylrfxfdljrcidondwsiiqytea4

A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data [article]

Min-Chun Wu, Vladimir Itskov
2020 arXiv   pre-print
Topological features of these complexes are then used to infer the intrinsic dimension.  ...  We introduce a construction of a filtration of Dowker complexes, associated to measurements by quasi-convex functions.  ...  Definition 5. 2 . 2 A set of vectors V = {v 1 , ..., v m } ⊆ R d is said to be in general direction if, for every σ ⊆ [m] with |σ| ≤ d, the set of vectors {v i } i∈σ is linearly independent.  ... 
arXiv:2007.03208v1 fatcat:uxg5nqw5fnbu5pr7jxznv3g3nm

The Rules of Variation Expanded, Implications for the Research on Compatible Genomics

Fernando Castro-Chavez
2011 Biosemiotics  
The main focus of this article is to present the practical aspect of the code rules of variation and the search for a second set of genomic rules, including comparison of sequences to understand how to  ...  preserve compatible organisms in danger of extinction and how to generate biodiversity.  ...  The external reality is present in nature independently of how we define, describe or classify it.  ... 
doi:10.1007/s12304-011-9118-0 pmid:21743816 pmcid:PMC3130522 fatcat:fd6thhu5nbhmlit2yold2dts4e

Computational Challenges of Nonlinear Systems [chapter]

Laurette S. Tuckerman
2020 Regularity and Complexity in Dynamical Systems  
In order of increasing complexity, we describe methods for calculating steady states and bifurcation diagrams, linear stability and Floquet analysis, and heteroclinic orbits.  ...  We survey some of the major types of dynamical-systems computations that can be carried out for two or three-dimensional systems of partial differential equations.  ...  This Floquet problem has been solved by an original method [23] : rather than fixing the oscillation amplitude a and calculating the resulting Floquet multipliers, the Floquet multiplier is set to 1 =  ... 
doi:10.1007/978-3-030-44992-6_11 fatcat:l4vqzpmodbhblfev7qssnc53m4

Hierarchical Shape Segmentation and Registration via Topological Features of Laplace-Beltrami Eigenfunctions

Martin Reuter
2009 International Journal of Computer Vision  
This work introduces a method to hierarchically segment articulated shapes into meaningful parts and to register these parts across populations of nearisometric shapes (e.g. head, arms, legs and fingers  ...  The method exploits the isometry invariance of eigenfunctions of the Laplace-Beltrami operator and uses topological features (level sets at important saddles) for the segmentation.  ...  Acknowledgements This work has been funded by a Feodor Lynen fellowship of the Alexander von Humboldt Foundation. The author would like to thank Prof. Nicholas M.  ... 
doi:10.1007/s11263-009-0278-1 fatcat:bpura5zsojhermyh2jgisd2r6y

A Convolutional Persistence Transform [article]

Elchanan Solomon, Paul Bendich
2022 arXiv   pre-print
Other advantages of convolutional persistence are improved stability and robustness to noise, greater flexibility for data-dependent vectorizations, and reduced computational complexity for convolutions  ...  This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in image data.  ...  Left: A complex with four top-dimensional cubes, with function values, indicated using color, extended to vertices and edges via the upper-* rule.  ... 
arXiv:2208.02107v1 fatcat:b5szau3dtfcfzf3kgfr2k7fure

A Novel Edit Distance for Tree-Like Topological Summaries [article]

Matteo Pegoraro
2022 arXiv   pre-print
The metric introduced can be computed with a dynamical integer linear programming approach and we showcase its feasibility and the effectiveness of the whole framework with simulated data sets.  ...  This family of objects is a natural combinatoric generalization of merge trees of real valued functions and hierarchical dendrograms which produces trees enriched with additional information regarding  ...  Acknowledgments This work was carried out as part of my PhD Thesis, under the supervision of Professor Piercesare Secchi.  ... 
arXiv:2108.13108v5 fatcat:gvh7m55dkrbctckwh3xte3njqe

Genes, Evolution and Intelligence

Thomas J. Bouchard
2014 Behavior Genetics  
A review of a large body of evidence demonstrates that there is a g factor in a wide range of species and that, in the species studied, it relates to brain size and is heritable.  ...  In spite of numerous studies with considerable statistical power, we know of very few genes that influence g and the effects are very small. Nevertheless, g appears to be highly polygenic.  ...  Trichromatic color vision has evolved independently in monkeys but not in other mammals (Jacobs and Rowe 2004) .  ... 
doi:10.1007/s10519-014-9646-x pmid:24604063 fatcat:7ywlstxkqbg7hoc7zxtpk5cgxm

Distributing Persistent Homology via Spectral Sequences [article]

Álvaro Torras Casas
2020 arXiv   pre-print
We present bases of persistence modules, and give motivation as for the advantages of using them.  ...  Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. Finally, we review PerMaViss, a method implementing these ideas.  ...  ACKNOWLEDGEMENTS I would like to thank my supervisor Dr. Ulrich Pennig who suggested this topic and has been very helpful and supportive in the development of these ideas.  ... 
arXiv:1907.05228v2 fatcat:em6gkpef7rdpne56tqbhqqykrq

Orthogonal layers of parallelism in large-scale eigenvalue computations [article]

Andreas Alvermann, Georg Hager, Holger Fehske
2022 arXiv   pre-print
An analysis in terms of the communication metric predicts that scalability can be restored if, and only if, one implements the two orthogonal layers of parallelism via different distributed vector layouts  ...  The performance model quantifies to which extent scalability and parallel efficiency are lost due to communication overhead.  ...  Instead of collecting further benchmark data or tables for the communication metric 𝜒 we like to emphasize that both can be generated easily with the tools in the FD and ScaMaC repositories.  ... 
arXiv:2209.01974v1 fatcat:5a57jg4wbzd3lfyo773naanh6y

An Algorithmic Approach to Collective Behavior

Bernard Chazelle
2014 Journal of statistical physics  
This article examines the merits and challenges of an algorithmic approach to the emergence of collective order.  ...  While powerful tools from dynamics and statistical mechanics have been brought to bear, sometimes with great success, an algorithmic perspective has been lacking.  ...  Acknowledgments I wish to thank the anonymous referees for many useful comments and suggestions. This work was supported in part by NSF Grants CCF-0832797, CCF-0963825, and CCF-1016250.  ... 
doi:10.1007/s10955-014-1140-6 fatcat:gaysmo7xffaabnfbns4xatb4aa

Controlled release of multiphoton quantum states from a microwave cavity memory

Wolfgang Pfaff, Christopher J. Axline, Luke D. Burkhart, Uri Vool, Philip Reinhold, Luigi Frunzio, Liang Jiang, Michel H. Devoret, Robert J. Schoelkopf
2017 Nature Physics  
Our system can serve as the backbone of a microwave quantum network, paving the way towards error-correctable distribution of quantum information and the transfer of highly non-classical states to hybrid  ...  Importantly, the release process can be controlled precisely on fast time scales, allowing us to generate entanglement between the cavity and the traveling mode by partial conversion.  ...  A powerful way to tame complexity when scaling up a quantum system is to construct it as a network.  ... 
doi:10.1038/nphys4143 fatcat:i3yhdy53rjcxdewbomi6l6qsri

M5-brane Sources, Holography, and Argyres-Douglas Theories [article]

Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni
2021 arXiv   pre-print
Our solutions are obtained in 7d gauged supergravity, where they take the form of a warped product of AdS_5 and a "half-spindle."  ...  Our solutions feature a realization of supersymmetry that is distinct from the usual topological twist, as well as an interesting Stückelberg mechanism involving the gauge field associated to a generator  ...  Acknowledgments We are grateful to Nikolay Bobev, Simone Giacomelli, Yifan Wang for interesting conversations and correspondence. The work of IB is supported in part by NSF grant PHY-1820784.  ... 
arXiv:2106.01322v1 fatcat:me54zku2zjbgbat6dlpk4kxfxq

Zero-temperature 2D Ising model and anisotropic curve-shortening flow [article]

H. Lacoin, F. Simenhaus, F. L. Toninelli
2013 arXiv   pre-print
To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.  ...  Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result.  ...  Acknowledgments The authors would like to thank P. Cardaliaguet for enlightening discussions and valuable comments on various analytical aspects of the present work.  ... 
arXiv:1112.3160v2 fatcat:s53txnxjw5cpjlbi3vl7dchazq
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