Journal of applied and computational topology最新文献

英文中文

Large deviation principle for persistence diagrams of random cubical filtrations 随机立方体过滤持久图的大偏差原理

Journal of applied and computational topology

Pub Date : 2024-01-25 DOI: 10.1007/s41468-023-00161-6

Shu Kanazawa, Yasuaki Hiraoka, Jun Miyanaga, Kenkichi Tsunoda

引用次数: 0

Topological and metric properties of spaces of generalized persistence diagrams 广义持久图空间的拓扑和度量特性

Journal of applied and computational topology

Pub Date : 2024-01-23 DOI: 10.1007/s41468-023-00157-2

Peter Bubenik, Iryna Hartsock

引用次数: 0

Morse theoretic signal compression and reconstruction on chain complexes. 莫尔斯理论信号压缩与链复合物重建

Journal of applied and computational topology

Pub Date : 2024-01-01 Epub Date: 2024-09-30 DOI: 10.1007/s41468-024-00191-8

Stefania Ebli, Celia Hacker, Kelly Maggs

At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matchings, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.

在拓扑数据分析（TDA）和机器学习的交叉点上，细胞信号处理领域近年来发展迅速。在这种情况下，复数单元上的每个信号都要利用组合拉普拉斯和由此产生的霍奇分解进行处理。与此同时，离散莫尔斯理论已被广泛应用，在保持复数全局拓扑特性的同时，通过减小复数的大小来加快计算速度。在本文中，我们提供了一种利用代数离散莫尔斯理论工具在链复数上进行信号压缩和重建的方法。其主要目标是通过变形回缩来缩小和重建基于链复数及其单元上的信号集，同时尽可能保留复数和信号的全局拓扑结构。我们首先证明，基于实度的有限维链复数的任何变形回缩都等价于莫尔斯匹配。然后，我们将研究信号在特定类型的莫尔斯匹配下是如何变化的，并证明其重构误差在霍奇分解的特定成分上是微不足道的。此外，我们还提供了一种以最小重构误差计算莫尔斯匹配的算法。

{"title":"Morse theoretic signal compression and reconstruction on chain complexes.","authors":"Stefania Ebli, Celia Hacker, Kelly Maggs","doi":"10.1007/s41468-024-00191-8","DOIUrl":"https://doi.org/10.1007/s41468-024-00191-8","url":null,"abstract":"At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matchings, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 8","pages":"2285-2326"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11541343/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142634048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric characterization of the persistence of 1D maps. 一维地图持久性的几何表征

Journal of applied and computational topology

Pub Date : 2024-01-01 Epub Date: 2023-06-17 DOI: 10.1007/s41468-023-00126-9

Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, Morteza Saghafian

We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.

本文用几何方法刻画了在持久同调中配对的一维映射的临界点，从而得到了关于持久图对称性定理和这种映射的变异定理的初等证明。特别是，我们确定分支点和网络端点作为不对称的唯一来源，并将持续同源的循环基与稳定婚姻问题的一个版本联系起来。我们的分析为维护排序列表集合及其持久性图提供了快速算法的基础。

引用次数: 0

Metric geometry of spaces of persistence diagrams. 持久图空间的度量几何。

Journal of applied and computational topology

Pub Date : 2024-01-01 Epub Date: 2024-09-03 DOI: 10.1007/s41468-024-00189-2

Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $D_{p}$ , $1 \leq p \leq \infty$ , that assign, to each metric pair (X, A), a pointed metric space $D_{p} (X, A)$ . Moreover, we show that $D_{\infty}$ is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that $D_{p}$ preserves several useful metric properties, such as completeness and separability, for $p \in [1, \infty)$ , and geodesicity and non-negative curvature in the sense of Alexandrov, for $p = 2$ . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on $D_{p} (X, A)$ , $1 \leq p \leq \infty$ , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $D_{p} (R^{2 n}, Δ_{n})$ , $1 \leq n$ and $1 \leq p < \infty$ , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.

持久图是拓扑数据分析中的核心对象。在本文中，我们将研究持久图空间的局部和全局几何特性。为此，我们构建了一系列函数 D p , 1 ≤ p ≤ ∞ , 为每个度量对 (X, A) 分配一个尖度量空间 D p ( X , A ) 。此外，我们证明 D ∞ 在度量对的格罗莫夫-豪斯多夫收敛性方面是连续的，并证明 D p 保留了几个有用的度量特性，如对于 p∈ [ 1 , ∞ ) 的完备性和可分性，以及大地性和非大地性。的情况下，D p 保留了几个有用的度量特性，如完整性和可分性；在 p = 2 的情况下，D p 保留了大地性和亚历山德罗夫意义上的非负曲率。对于后一种情况，我们描述了空图处方向空间的度量。我们还证明了在 D p ( X , A ) 上的博尔概率度量的弗雷谢特均值集，1 ≤ p ≤ ∞，具有有限第二矩和紧凑支持，是非空的。作为几何框架的一个应用，我们证明了欧氏持久图空间 D p ( R 2 n , Δ n ) , 1 ≤ n 且 1 ≤ p ∞ 具有无限覆盖维、豪斯多夫维、渐近维、阿苏阿德维和阿苏阿德-纳加塔维。

{"title":"Metric geometry of spaces of persistence diagrams.","authors":"Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis","doi":"10.1007/s41468-024-00189-2","DOIUrl":"https://doi.org/10.1007/s41468-024-00189-2","url":null,"abstract":"Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors <math><msub><mi>D</mi> <mi>p</mi></msub> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , that assign, to each metric pair (X, A), a pointed metric space <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> . Moreover, we show that <math><msub><mi>D</mi> <mi>∞</mi></msub> </math> is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that <math><msub><mi>D</mi> <mi>p</mi></msub> </math> preserves several useful metric properties, such as completeness and separability, for <math><mrow><mi>p</mi> <mo>∈</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo></mrow> </math> , and geodesicity and non-negative curvature in the sense of Alexandrov, for <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <msup><mrow><mi>R</mi></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>,</mo> <msub><mi>Δ</mi> <mi>n</mi></msub> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>n</mi></mrow> </math> and <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 8","pages":"2197-2246"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11541355/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142634046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Defining logical obstruction with fixpoints in epistemic logic 用不动点定义认知逻辑中的逻辑障碍

Journal of applied and computational topology

Pub Date : 2023-11-14 DOI: 10.1007/s41468-023-00151-8

Susumu Nishimura

引用次数: 0

Generalization of graph network inferences in higher-order graphical models 高阶图模型中图网络推理的泛化

Journal of applied and computational topology

Pub Date : 2023-11-13 DOI: 10.1007/s41468-023-00147-4

Yicheng Fei, Xaq Pitkow

Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.

概率图形模型为描述复杂的统计结构提供了一个强大的工具，在科学和工程领域有许多实际应用，从控制机械臂到理解神经元计算。这些图形模型面临的一个主要挑战是，对于一般图形来说，像边缘化这样的推断是难以处理的。这些推断通常由分布式消息传递算法(如Belief Propagation)来近似，该算法在带有循环的图上并不总是表现良好，对于复杂的连续概率分布也不总是容易指定。这种困难经常出现在包括难以处理的高阶交互的表达图形模型中。在本文中，我们定义了循环因子图神经网络(RF-GNN)来实现对涉及多变量相互作用的图模型的快速近似推理。在多个图形模型族上的实验结果表明，该方法对不同大小的图具有分布外泛化能力，并表明该方法在该领域优于信念传播(BP)。此外，我们在真实世界的低密度奇偶校验数据集上测试了RF-GNN作为基准，以及其他基线模型，包括BP变体和其他GNN方法。总体而言，我们发现RF-GNNs在高噪声水平下优于其他方法。

{"title":"Generalization of graph network inferences in higher-order graphical models","authors":"Yicheng Fei, Xaq Pitkow","doi":"10.1007/s41468-023-00147-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00147-4","url":null,"abstract":"Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"61 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The topology of randomized symmetry-breaking distributed computing 随机对称破缺分布式计算的拓扑结构

Journal of applied and computational topology

Pub Date : 2023-11-13 DOI: 10.1007/s41468-023-00150-9

Pierre Fraigniaud, Ran Gelles, Zvi Lotker

Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.

在过去的20年里，通过代数拓扑来研究分布式计算已经成为许多重大突破的来源，特别是在设计下界或不可能结果方面。尽管有数百个结果考虑确定性算法，但没有一个适用于随机算法。本文旨在从代数拓扑的角度研究随机同步分布式计算。我们通过研究同步无故障匿名系统中广泛的(无输入)对称破坏任务，例如领导者选举，来实现这一目标。我们设计了一个拓扑框架，它允许分析这些任务并确定它们的可解决性。关键的技术观察是，与确定性算法不同，确定性算法的可解性意味着描述协议的拓扑复合体可以全局映射到输出协议中，在我们的框架中，可解性是“局部”确定的，即，对于协议复合体的每个单纯形单独确定，而不需要任何全局一致性。作为一个有趣的应用，我们导出了解决共享内存和消息传递模型中领导者选举的充要条件，其中提供给节点的随机性之间可能存在相关性。我们发现领导人选举的可解性与具有相关随机性的政党数量有关，根据具体的通信模型，可以直接或通过其最大公约数。

{"title":"The topology of randomized symmetry-breaking distributed computing","authors":"Pierre Fraigniaud, Ran Gelles, Zvi Lotker","doi":"10.1007/s41468-023-00150-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00150-9","url":null,"abstract":"Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Simplicial branching random walks 简单分支随机游走

Journal of applied and computational topology

Pub Date : 2023-11-10 DOI: 10.1007/s41468-023-00148-3

Ron Rosenthal

引用次数: 0

Rigorous computation in dynamics based on topological methods for multivector fields 基于拓扑方法的多向量场动力学严格计算

Journal of applied and computational topology

Pub Date : 2023-11-04 DOI: 10.1007/s41468-023-00149-2

Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela

Abstract Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.

受Mrozek等人(common Nonlinear Sci numerical Simul 108:106 - 226,2022)的理论结果的启发，我们提出了一种平面ODE的横向元胞分解算法。然后，我们使用相关的组合多向量场算法检测周期轨道的Conley指数的孤立不变量集的存在性，并允许一个组合poincarcarr截面。这种结构与Mrozek等人(2022)的理论结果相结合，产生了一种自动化计算机辅助证明ODE周期解存在性的方法。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Journal of applied and computational topology

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀