This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The is based on the persistence diagram (PD), mathematical tool for capturing shapes multiscale data. input to PDs given by an atomic configuration and output expressed as 2D histograms. Then, specific distributions such curves islands in identify meaningful shape characteristics configuration. Although can be applied wide variety disordered systems, it here silica glass,...

The characterization of the medium-range (MRO) order in amorphous materials and its relation to short-range is discussed. A new topological approach extract a hierarchical structure presented, which robust against small perturbations allows us distinguish it from periodic or random configurations. This method called persistence diagram (PD) introduces scales many-body atomic structures facilitate size shape characterization. We first illustrate representation perfect crystalline PDs. Then,...

In this work, we propose a new invariant for 2D persistence modules called the compressed multiplicity and show that it generalizes notions of dimension vector rank invariant. addition, module M, an "interval-decomposable replacement" δ⁎(M) (in split Grothendieck group category modules), which is expressed by pair interval-decomposable modules, is, its positive negative parts. We M if only equal to in group. Furthermore, even not necessarily interval-decomposable, preserves M. provide...

In the persistent homology of filtrations, indecomposable decompositions provide persistence diagrams. However, in almost all cases multidimensional persistence, classification modules is known to be a wild problem. One direction consider subclass interval-decomposable modules, which are direct sums interval representations. We introduce definition pre-interval representations, more natural algebraic definition, and study relationships between pre-interval, interval, thin show that over...

Single-cell RNA sequencing (scRNA-seq) can determine gene expression in numerous individual cells simultaneously, promoting progress the biomedical sciences. However, scRNA-seq data are high-dimensional with substantial technical noise, including dropouts. During analysis of data, such noise engenders a statistical problem known as curse dimensionality (COD). Based on statistics, we herein formulate reduction method, RECODE (resolution dimensionality), for random sampling noise. We show that...

Where do firms innovate? Mapping their locations and directions in technological space is challenging due to its high dimensionality. We propose a new method characterize firms' inventive activities via topological data analysis (TDA) that represents high-dimensional shape graph. Applying this 333 major patents 1976–2005 reveals hitherto undocumented industry dynamics: some remain undifferentiated; others develop unique portfolios. Firms with trajectories, which we define measure...

Abstract A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules using their restrictions onto lines, giving family 1D modules. We give constructive proof that any module with finite support can be found as restriction some indecomposable support. As consequences our construction, we are able to exhibit whose has holes well an containing all line restrictions. Finally, also show finite-rectangle-decomposable n D $$(n+1)$$ <mml:math...

While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence proven more difficult to apply. One reason is the serious drawback of no longer having concise and complete descriptor analogous diagrams former. We propose simple algebraic construction illustrate existence infinite families indecomposable modules over regular grids sufficient size. On top providing constructive proof representation type, we also provide...

A paper by Mischaikow and Nanda [14] uses filtered acyclic matchingsto form a Morse filtration for complex. The is smallerin size, yet has persistent homology equivalent to that of the original. We give anextension matchings case zigzag complexes prove theMorse complex similarly obtained isomorphic thatof present an algorithm compute agiven some numerical examples. Since complexis smaller in calculations its tend complete faster thanthose original complex.DOI :...

Where do firms innovate? Mapping their locations and directions in technological space is challenging due to its high dimensionality. We propose a new method characterize firms’ inventive activities via topological data analysis (TDA) that represents high-dimensional shape graph. Applying this 333 major patents 1976–2005 reveals substantial heterogeneity: some remain undifferentiated; others develop unique portfolios. Firms with trajectories, which we define measure graph-theoretically as...

While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence proven more difficult to apply. One reason is the serious drawback of no longer having concise and complete descriptor analogous diagrams former. We propose simple algebraic construction illustrate existence infinite families indecomposable modules over regular grids sufficient size. On top providing constructive proof representation type, we also provide...

In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across filtration. this work, we extend setting, propose use bipath homology, which can be used to study persistence pair filtrations connected at their ends, compare two filtrations. interval-decomposability is guaranteed, provide an algorithm for computing diagrams discuss interpretation diagrams.

In recent years, the use of data-driven methods has provided insights into underlying patterns and principles behind culinary recipes. this exploratory work, we introduce topological data analysis, especially persistent homology, in order to study space particular, homology analysis provides a set recipes surrounding multiscale "holes" existing We then propose method generate novel ingredient combinations using combinatorial optimization on information. made biscuits combinations, which were...

Abstract In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across filtration. this work, we extend setting, propose use bipath , which can be used to study persistence pair filtrations connected at their ends, compare two filtrations. interval-decomposability is guaranteed, provide an algorithm for computing diagrams discuss interpretation diagrams.

In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes notions of dimension vector rank invariant. addition, module $M$, an "interval-decomposable replacement" $\delta^{\ast}(M)$ (in split Grothendieck group category modules), which is expressed by pair interval-decomposable modules, is, its positive negative parts. We $M$ if only equal to in group. Furthermore, even not necessarily interval-decomposable,...

We study persistence modules defined on commutative ladders. This class of frequently appears in topological data analysis, and the theory algorithm proposed this paper can be applied to these practical problems. A new algebraic framework deals with as representations associative algebras Auslander-Reiten is develop theoretical algorithmic foundations. In particular, we prove that ladders length less than 5 are representation-finite explicitly show their quivers. Furthermore, a...