My Official PhD thesis is available at http://www.disi.unige.it/dottorato/THESES/2012-01-CaninoD.pdf
They are slides, that I presented on May 7, 2012, while defending my PhD Thesis in Computer Science under the supervision of Professor Leila De Floriani at the DIBRIS Department (Department of Bioengineering, Computer Science, and Systems Engineering) in Genova, Italy.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
The document discusses various topics related to sorting networks and their connections to mathematical structures:
1) It describes primitive sorting networks and their representation as pseudoline arrangements, along with properties of the contact graph and the graph of possible flips between arrangements.
2) It discusses relationships between point sets, minimal sorting networks, and oriented matroids.
3) It covers triangulations, their representation as alternating sorting networks and pseudoline arrangements, and how the brick polytope of these networks gives associahedra.
4) Other topics mentioned include pseudotriangulations, multitriangulations, and how duplicated networks relate to permutahedra.
IRJET- Independent Middle Domination Number in Jump GraphIRJET Journal
This document discusses independent middle domination number (iM(J(G))) in jump graphs. It defines iM(J(G)) as the minimum cardinality of an independent dominating set of the middle graph M(J(G)). The paper obtains several bounds on iM(J(G)) in terms of the vertices, edges, and other parameters of J(G). It also establishes relationships between iM(J(G)) and other domination parameters such as domination number, strong split domination number, and edge domination number. Exact values of iM(J(G)) are determined for some standard jump graphs like paths, cycles, stars, and wheels.
Nelly Litvak – Asymptotic behaviour of ranking algorithms in directed random ...Yandex
There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.
In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
This document summarizes research on edge coloring of complement fuzzy graphs. It defines fuzzy graphs and complement fuzzy graphs. It presents an algorithm to find the complement of any fuzzy graph in O(n2) time. It then defines a coloring function based on α-cut to color the edges of the complement fuzzy graph. The paper provides an example of a fuzzy graph G, calculates its complement Gc, and uses the coloring function to find the chromatic number of Gc in 3 or fewer colors.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
The document discusses various topics related to sorting networks and their connections to mathematical structures:
1) It describes primitive sorting networks and their representation as pseudoline arrangements, along with properties of the contact graph and the graph of possible flips between arrangements.
2) It discusses relationships between point sets, minimal sorting networks, and oriented matroids.
3) It covers triangulations, their representation as alternating sorting networks and pseudoline arrangements, and how the brick polytope of these networks gives associahedra.
4) Other topics mentioned include pseudotriangulations, multitriangulations, and how duplicated networks relate to permutahedra.
IRJET- Independent Middle Domination Number in Jump GraphIRJET Journal
This document discusses independent middle domination number (iM(J(G))) in jump graphs. It defines iM(J(G)) as the minimum cardinality of an independent dominating set of the middle graph M(J(G)). The paper obtains several bounds on iM(J(G)) in terms of the vertices, edges, and other parameters of J(G). It also establishes relationships between iM(J(G)) and other domination parameters such as domination number, strong split domination number, and edge domination number. Exact values of iM(J(G)) are determined for some standard jump graphs like paths, cycles, stars, and wheels.
Nelly Litvak – Asymptotic behaviour of ranking algorithms in directed random ...Yandex
There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.
In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
This document summarizes research on edge coloring of complement fuzzy graphs. It defines fuzzy graphs and complement fuzzy graphs. It presents an algorithm to find the complement of any fuzzy graph in O(n2) time. It then defines a coloring function based on α-cut to color the edges of the complement fuzzy graph. The paper provides an example of a fuzzy graph G, calculates its complement Gc, and uses the coloring function to find the chromatic number of Gc in 3 or fewer colors.
Representing Simplicial Complexes with MangrovesDavid Canino
These slides have been presented at the 22nd International Meshing Roundtable, Orlando, FL, USA. They describe our GPL software, the Mangrove TDS Library: http://mangrovetds.sourceforge.net. It is a C++ tool for the fast prototyping of topological data structures, representing dynamically simplicial and cell complexes.
Higher-order organization of complex networksDavid Gleich
A talk I gave at the Park City Institute of Mathematics about our recent work on using motifs to analyze and cluster networks. This involves a higher-order cheeger inequality in terms of motifs.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Problem Solving by Computer Finite Element MethodPeter Herbert
This document discusses using finite element methods and the cotangent Laplacian to solve partial differential equations numerically. It begins by explaining how to generate simplicial meshes by dividing a region into basic pieces. It then introduces the cotangent Laplacian, which approximates the Laplacian operator, and how it is calculated based on angles in triangles. Finally, it demonstrates applying the cotangent Laplacian to solve sample Dirichlet and Neumann boundary value problems and compares the approximate solutions to exact solutions, showing convergence as the mesh is refined.
This document provides an overview and introduction to graphs as mathematical structures. It defines several types of graphs, including simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs. It also discusses key graph terminology like vertices, edges, adjacency, degrees, and special graph structures like complete graphs, cycles, wheels, hypercubes, bipartite graphs, and subgraphs. The document is a lecture on graph theory from a course on discrete mathematics. It is intended to familiarize students with basic graph concepts and terminology.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
Amirim Project - Threshold Functions in Random Simplicial Complexes - Avichai...Avichai Cohen
This document presents an overview of prior work on threshold functions in random simplicial complexes. It discusses questions about the threshold for the existence of cycles and collapsibility in higher dimensional simplicial complexes. Specifically, it summarizes previous results that established coarse threshold functions of 1/n for the existence of cycles in dimensions greater than 1. It also reviews upper bounds on the critical probability for this threshold. The document outlines prior studies on the collapsability threshold and presents lower bounds on the critical probability. The author's own work is aimed at improving the bounds and determining sharp threshold functions for these properties in random simplicial complexes.
The document describes an algorithm for enumerating 2-level polytopes in fixed dimensions. A 2-level polytope has vertices that are contained in two parallel hyperplanes. The algorithm takes as input a list of (d-1)-dimensional 2-level polytopes and extends each one to d dimensions, computing the closed sets of vertices to obtain new d-dimensional 2-level polytopes. Experimental results show the numbers of 2-level polytopes enumerated for dimensions up to 6. Open questions ask for a more output-sensitive enumeration algorithm and whether the number of d-dimensional 2-level polytopes is exponential in d.
Detecting paraphrases using recursive autoencodersFeynman Liang
Presentation on deep learning applied to natural language processing, presented at University of Cambridge Machine Learning Group's Research and Communication Club 2-11-2015 meeting.
This document defines and provides examples of various graph concepts from discrete mathematics including:
- Complete graphs, cycle graphs, wheel graphs, and hypercubes as special types of simple graphs
- Bipartite graphs and complete bipartite graphs
- Subgraphs and unions of graphs
- Representing graphs using adjacency lists and matrices
- Graph isomorphism
It includes over 20 examples applying these graph concepts and definitions.
Slides: Total Jensen divergences: Definition, Properties and k-Means++ Cluste...Frank Nielsen
The document defines total Jensen divergences, which are a generalization of total Bregman divergences. Total Jensen divergences incorporate a double-sided conformal factor that makes them invariant to rotations. They reduce to total Bregman divergences when distributions are close. The square root of the total Jensen-Shannon divergence is not a metric. Jensen centroids are not always robust. However, total Jensen k-means++ clustering does not require calculating centroids and provides approximation guarantees.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
A Decomposition-based Approach to Modeling and Understanding Arbitrary ShapesDavid Canino
Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on non- manifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.
1) The document describes using diagonal reference models to analyze the effect of social mobility on long-term limiting illness using data from the UK ONS Longitudinal Survey.
2) Diagonal reference models estimate the effects of stable and mobile social classes on health outcomes. The models were fit separately for men and women using the gnm package in R.
3) The results found higher odds of long-term illness for lower social classes, and that the effect of social mobility was to dilute health inequalities, with larger effects for stable lower classes.
This document provides an overview of multivariate methods for machine learning. It discusses multivariate data with multiple measurements or features. It introduces concepts like covariance, correlation, and estimating these parameters from sample data. It describes techniques for handling missing data. It also covers the multivariate normal distribution and how to model classification problems parametrically when the class distributions are multivariate normal. It discusses estimating class parameters from data and different assumptions that can be made about the covariance matrices.
The tutorial describes existing approaches to model graph databases and different techniques implemented in RDF and Database engines including their main drawbacks when a large volume of interconnected data needs to be traversed.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
These are the slides, presented at the "Smart Tools and Apps in computer Graphics (STAG 2016) conference, held at Genoa (GE), Italy, October 3-4, 2016. My paper is entitled "Manipulating Topological Decompositions of Non-Manifold Shapes", and published in the Eurographics Digital Library.
Representing Simplicial Complexes with MangrovesDavid Canino
These slides have been presented at the 22nd International Meshing Roundtable, Orlando, FL, USA. They describe our GPL software, the Mangrove TDS Library: http://mangrovetds.sourceforge.net. It is a C++ tool for the fast prototyping of topological data structures, representing dynamically simplicial and cell complexes.
Higher-order organization of complex networksDavid Gleich
A talk I gave at the Park City Institute of Mathematics about our recent work on using motifs to analyze and cluster networks. This involves a higher-order cheeger inequality in terms of motifs.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Problem Solving by Computer Finite Element MethodPeter Herbert
This document discusses using finite element methods and the cotangent Laplacian to solve partial differential equations numerically. It begins by explaining how to generate simplicial meshes by dividing a region into basic pieces. It then introduces the cotangent Laplacian, which approximates the Laplacian operator, and how it is calculated based on angles in triangles. Finally, it demonstrates applying the cotangent Laplacian to solve sample Dirichlet and Neumann boundary value problems and compares the approximate solutions to exact solutions, showing convergence as the mesh is refined.
This document provides an overview and introduction to graphs as mathematical structures. It defines several types of graphs, including simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs. It also discusses key graph terminology like vertices, edges, adjacency, degrees, and special graph structures like complete graphs, cycles, wheels, hypercubes, bipartite graphs, and subgraphs. The document is a lecture on graph theory from a course on discrete mathematics. It is intended to familiarize students with basic graph concepts and terminology.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
Amirim Project - Threshold Functions in Random Simplicial Complexes - Avichai...Avichai Cohen
This document presents an overview of prior work on threshold functions in random simplicial complexes. It discusses questions about the threshold for the existence of cycles and collapsibility in higher dimensional simplicial complexes. Specifically, it summarizes previous results that established coarse threshold functions of 1/n for the existence of cycles in dimensions greater than 1. It also reviews upper bounds on the critical probability for this threshold. The document outlines prior studies on the collapsability threshold and presents lower bounds on the critical probability. The author's own work is aimed at improving the bounds and determining sharp threshold functions for these properties in random simplicial complexes.
The document describes an algorithm for enumerating 2-level polytopes in fixed dimensions. A 2-level polytope has vertices that are contained in two parallel hyperplanes. The algorithm takes as input a list of (d-1)-dimensional 2-level polytopes and extends each one to d dimensions, computing the closed sets of vertices to obtain new d-dimensional 2-level polytopes. Experimental results show the numbers of 2-level polytopes enumerated for dimensions up to 6. Open questions ask for a more output-sensitive enumeration algorithm and whether the number of d-dimensional 2-level polytopes is exponential in d.
Detecting paraphrases using recursive autoencodersFeynman Liang
Presentation on deep learning applied to natural language processing, presented at University of Cambridge Machine Learning Group's Research and Communication Club 2-11-2015 meeting.
This document defines and provides examples of various graph concepts from discrete mathematics including:
- Complete graphs, cycle graphs, wheel graphs, and hypercubes as special types of simple graphs
- Bipartite graphs and complete bipartite graphs
- Subgraphs and unions of graphs
- Representing graphs using adjacency lists and matrices
- Graph isomorphism
It includes over 20 examples applying these graph concepts and definitions.
Slides: Total Jensen divergences: Definition, Properties and k-Means++ Cluste...Frank Nielsen
The document defines total Jensen divergences, which are a generalization of total Bregman divergences. Total Jensen divergences incorporate a double-sided conformal factor that makes them invariant to rotations. They reduce to total Bregman divergences when distributions are close. The square root of the total Jensen-Shannon divergence is not a metric. Jensen centroids are not always robust. However, total Jensen k-means++ clustering does not require calculating centroids and provides approximation guarantees.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
A Decomposition-based Approach to Modeling and Understanding Arbitrary ShapesDavid Canino
Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on non- manifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.
1) The document describes using diagonal reference models to analyze the effect of social mobility on long-term limiting illness using data from the UK ONS Longitudinal Survey.
2) Diagonal reference models estimate the effects of stable and mobile social classes on health outcomes. The models were fit separately for men and women using the gnm package in R.
3) The results found higher odds of long-term illness for lower social classes, and that the effect of social mobility was to dilute health inequalities, with larger effects for stable lower classes.
This document provides an overview of multivariate methods for machine learning. It discusses multivariate data with multiple measurements or features. It introduces concepts like covariance, correlation, and estimating these parameters from sample data. It describes techniques for handling missing data. It also covers the multivariate normal distribution and how to model classification problems parametrically when the class distributions are multivariate normal. It discusses estimating class parameters from data and different assumptions that can be made about the covariance matrices.
The tutorial describes existing approaches to model graph databases and different techniques implemented in RDF and Database engines including their main drawbacks when a large volume of interconnected data needs to be traversed.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
Similar to Tools for Modeling and Analysis of Non-manifold Shapes (20)
These are the slides, presented at the "Smart Tools and Apps in computer Graphics (STAG 2016) conference, held at Genoa (GE), Italy, October 3-4, 2016. My paper is entitled "Manipulating Topological Decompositions of Non-Manifold Shapes", and published in the Eurographics Digital Library.
A Dimension-Independent and Extensible Framework for Huge Geometric ModelsDavid Canino
Nowadays, gigantic models can be easily produced in many applications and their dimension often exceeds the RAM size in a common workstation. Thus, using an external memory technique is mandatory in this case. In this paper, we define a dimension-independent and extensible framework, called Objects Management in Secondary Memory (OMSM), for managing huge models. The OMSM framework can be easily adapted to the users needs through dynamic plugins, providing many techniques to be integrated in a storing architecture.
An Extensible Framework for Modeling Simplicial ComplexesDavid Canino
The document describes the Mangrove TDS Framework, a tool for rapidly prototyping and designing topological data structures to represent simplicial complexes. The framework is based on graph-based "mangroves" that encode connectivity and can be customized for any modeling needs. It supports implicit representations of ghost simplices to improve query efficiency. The framework has fully implemented six data structures with topological queries.
A Compact Representation for Topological Decompositions of Non-Manifold ShapesDavid Canino
This document presents a compact representation for topological decompositions of non-manifold shapes. It summarizes previous work on representing non-manifold shapes and decomposing them into simpler, almost-manifold components. The key idea of the presented approach is to explicitly expose and combine the combinatorial and structural information of a shape's manifold-connected decomposition using a two-level graph-based representation. This compact representation allows for efficient extraction of topological queries and exposes the global structure and connectivity of meaningful components within complex, non-manifold shapes.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
Tools for Modeling and Analysis of Non-manifold Shapes
1. Tools for Modeling and Analysis of Non-manifold
Shapes
David Canino
Department of Computer Science, Universitá degli Studi di Genova, Italy
PhD. Final Exam
May 7, 2012
David Canino (DISI) May 7, 2012 1 / 1
2. Introduction I
Manifold shapes (Topological Manifold)
Each point has a neighborhood homeomorphic to either
an open ball (internal point), or to a closed half-ball
(boundary point).
Properties
simple structure (topology)
smooth, derivable, . . .
efficient representations
many tools based on manifold shapes.
But they are a subset of all shapes.
Non-manifold Shapes
Shapes which violate manifold conditions.
2 v
1 t 2 v
1 t
df e e
1 t2 t
David Canino (DISI) May 7, 2012 2 / 1
3. Introduction II
Non-manifold Shapes
non-manifold singularities, i.e., points at which the manifold
condition is not satisfied
parts of different dimensions.
Idealization Process
Applied to simpler (manifold) shapes, and produce idealized shapes
Engineering component Idealized Shape FEM simulation
Remove details and simplify shapes for FEM simulations
David Canino (DISI) May 7, 2012 3 / 1
4. Objectives & Contributions
General Objective
Represent simplicial complexes describing non-manifold shapes.
Research Area I - Representation by Topological Data Structures
Two data structures for abstract simplicial complexes in arbitrary dimensions:
I the Incidence Simplicial (IS) data structure
I the Generalized Indexed data structure with Adjacencies (IA).
Mangrove TDS framework
I rapid prototyping of data structures for arbitrary simplicial complexes
Research Area II - Decompositions and Structural Models
Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007
I the Exploded MC-Graph (hyper-graph)
I the Pairwise MC-Graph
I the Compact MC-Graph (hyper-graph)
Mayer-Vietoris (MV) Algorithm for computing Z-homology, which combines:
I the MC-Decomposition (Pairwise MC-Graph)
I the Constructive Homology Theory - Sergeraert and Rubio, 2006
David Canino (DISI) May 7, 2012 4 / 1
5. Simplicial Complexes
Euclidean simplex
Let p a non negative integer, then an Euclidean p-simplex is the linear combination of p + 1
points V = [v0; : : : ; vp] in any Euclidean space En.
Face of a Simplex
Any subset of (k + 1)-vertices in V generates a k-face 0 of , with k p.
Euclidean Simplicial complex
An Euclidean simplicial complex is a set of
simplices in En of dimension at most d, with
0 d n such that:
contains all the faces of each simplex
two simplices in can be either distinct, or
can share a face
C
e
Valid Not Valid
Geometric realizations of abstract simplicial complexes, NOT necessarily embedded in En.
David Canino (DISI) May 7, 2012 5 / 1
6. Some Combinatorial Concepts
Given a p-simplex in a simplicial d-complex , with 0 p d:
Boundary
Collection B() of k-faces of , with 0 k p
Star
Collection St() of simplices with in their boundary
(incident at )
w v
f
t
St(v) = fw; f ; tg, plus their faces
incident at v
Link
Collection Lk() formed by faces of simplices in
St(), which are not incident at
v' w v
f
t
e
f
t
f
e
Lk(v) = fv0; ef ; ftg
Top Simplex
If is not on boundary of other simplices.
w is a top 1-simplex
f is a top 2-simplex
t is a top 3-simplex
David Canino (DISI) May 7, 2012 6 / 1
7. Combinatorial Manifolds
Objective
Provide a combinatorial characterization of topological manifolds.
Key Idea
Discrete neighborhood of a simplex is characterized
by St() in a simplicial d-complex
Combinatorial Manifold (p + 1)-simplex
St() is homemorphic to the triangulation of the
(d p)-sphere
Combinatorial Manifold Complex
All simplices are combinatorial manifold
Combinatorial manifold
Combinatorial non-manifold
Problems Restrictions
NOT algorithmically decidable for d 5, Nabutovski, 1996 (not dimension-independent )
David Canino (DISI) May 7, 2012 7 / 1
8. Topological Relations Data Structures
Objective
Connectivity of simplices
Let j the collection of j-simplices
in , and Rk;m k m, then:
e
v 6
v
v
v
v
v
1
2
3
4
5
e
e
e
e
7
8
9
10
f
f
f
f
f
1
2
3
4
5
e
e
e
e
e
1
2
3
4
5
Boundary relations
Rk;m(; 0) if 0 2 B(), with k m
R2;0(f1) = fv; v1; v2g, R2;1(f1) = fe1; e6; e10g
Co-boundary relations
Rk;m(; 0) if 0 2 St(), with k m
R0;1(v) = fe6; : : : ; e10g, R1;2(e10) = ff1; f2g
Adjacency relations
Rk;k (; 0), if and 0 shares a (k 1)-simplex,
with k6= 0
R0;0(; 0), if an edge connects and 0
R0;0(v) = fv1; : : : ; v5g, R2;2(f1) = ff2; f5g
Topological Data Structures
Subset of topological entities (simplices) and topological relations
David Canino (DISI) May 7, 2012 8 / 1
9. Directed Graph Representation (Mangrove) for a
Topological Data Structure
A topological data structure can be represented as a directed graph G = (N;A):
each node n in N describes a simplex
each arc (n; n0 ) describes a topological relation Rk;m(; 0)
Boundary Arc (n; n0 )
If Rk;m(; 0) is a boundary relation
Boundary Graph
Formed by nodes in N + boundary arcs
Co-boundary Arc (n; n0 )
If Rk;m(; 0) is a co-boundary relation
Co-boundary Graph
Formed by nodes in N + co-boundary arcs
Adjacency Arc (n; n0 )
If Rk;k (; 0) is an adjacency-relation
Adjacency Graph
Formed by nodes in N + adjacency arcs
David Canino (DISI) May 7, 2012 9 / 1
10. Data Structures for Simplicial Complexes
There are a lot of representations in the literature, De Floriani and Hui, 2005
Taxonomy (partial)
Dimension-Independent versus Dimension-Specific
Manifold versus Non-Manifold
Incidence-based versus Adjacency-based
Incidence-based (global mangrove)
all simplices
boundary and co-boundary relations
#
Adjacency-based (local mangrove)
vertices and top simplices
adjacency relations
#
Incidence Simplicial (IS) data structure
Dimension-independent variant, restricted
to simplicial complexes, of the
Incidence-Graph (IG), Edelsbrunner,1987
Generalized Indexed data structure with
Adjacencies (IA)
Dimension-independent variant, specific for
non-manifolds, of the IA data structure,
Paoluzzi et al., 1993
David Canino (DISI) May 7, 2012 10 / 1
11. The Incidence Graph (IG) Edelsbrunner, 1987
Abstract simplicial d-complex
Dimension-independent
For each p-simplex :
I boundary relation Rp;p1()
I co-boundary relation Rp;p+1()
Global mangrove (IG-graph)
IG Boundary/Co-boundary Arcs
Correspond to Rp;p1 and Rp;p+1
IG Boundary/Co-boundary Graph
Nodes + IG Boundary/Co-boundary Arcs
v'=5 w v=0
2
1
f
t
e
3 4
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IG
Boundary Graph
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IG
Co-boundary Graph
David Canino (DISI) May 7, 2012 11 / 1
12. Properties of the IG Data Structure
Topological Relations
Can be retrieved in optimal time, i.e., linear in the number of involved simplices
Rp;p1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rp;p1, Rp1;p2, and so on O(1)
Rp;p+1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rk;k+1 and Rk+1;k , for k p O(kRp;q ()k)
R0;0() Combine R0;1 and R1;0 O(kR0;0()k)
Rp;p(), with p6= 0 Combine Rp;p1 and Rp1;p O(kRp;p()k)
Storage Cost
2
Xd
p=1
sp(p + 1)
sp : number of p-simplices
Disadvantages
too verbose
large overhead for manifolds
David Canino (DISI) May 7, 2012 12 / 1
13. The Incidence Simplicial (IS) Data Structure
Key idea: simplify the IG
Boundary relations are constant
No need full co-boundary relations
Abstract simplicial d-complex
Dimension-independent
Encodes all simplices in
For each p-simplex :
I boundary relation Rp;p1()
I partial co-boundary relation Rp
;p+1()
Global Mangrove (IS-Graph)
Partial co-boundary relation Rp
;p+1()
One arbitrary (p + 1)-simplex for each
connected component in Lk().
v'=5 w v=0
2
1
f
t
e
3 4
R 0;1(v) = fw; eg
Important
Rd
1;d Rd1;d
L. De Floriani, A. Hui, D. Panozzo, D. Canino, A Dimension-Independent Data Structure for Simplicial
Complexes, In S. Shontz Ed., Proceedings of the 19th International Meshing Roundtable (IMR 2010), pages
403-420, Springer, 2010 - Chattanooga, Tennessee, USA
David Canino (DISI) May 7, 2012 13 / 1
14. The IS-Graph
IS Boundary Arcs IG Boundary Arcs
Correspond to Rp;p1
IS Boundary Graph IG Boundary Graph
Nodes + IS Boundary Arcs
IS Co-boundary Arcs
Correspond to Rp
;p+1
IS Co-boundary Graph
Nodes + IS Co-boundary Arcs
v'=5 w v=0
2
1
f
t
e
3 4
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IS
Boundary Graph IG Boundary Graph
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IS
Co-boundary Graph
IS-Graph is more compact than IG-Graph
David Canino (DISI) May 7, 2012 14 / 1
15. Storage Cost of the IS Data Structure
Boundary Relations
Xd
p=1
sp(p + 1)
sp : number of p-simplices
+
Partial Co-boundary Relations
Xd
p=1
X
2p
H
Xd
p=1
sp(p + 1)
H : #connected components in Lk()
2D Shapes
Shape IG IS (%)
Armchair 127k 101k 20:5
Cone 14k 11k 21:4
Frame 15k 12k 20
Tower 221k 175k 20:8
21% more compact than IG
3D Shapes
Shape IG IS (%)
Basket 113k 80k 29:2
Flasks 104k 75k 27:9
Sierpinski 917k 688k 24:9
Teapot 219k 163k 25:6
27% more compact than IG
Archive of 62 shapes publicly available at http://ggg.disi.unige.it/nmcollection/
Note
More compact with manifolds =) scalability to manifolds
David Canino (DISI) May 7, 2012 15 / 1
16. Storage Cost of the IS Data Structure (cont’d)
For Manifolds:
Partial co-boundary relation Rp
;p+1 contains:
only one (p + 1)-simplex, if p d
one or two d-simplices, if p = d 1
Remark
Rd
1;d Rd1;d
v v
e
All edges in R0;1(v) versus one edge in R0
;1(v)
Boundary Relations
Xd
p=1
sp(p + 1)
+
Partial Co-boundary
Relations
dX2
p=0
sp + (d + 1)sd
David Canino (DISI) May 7, 2012 16 / 1
17. Topological Relations in the IS Data Structure
Rp;p1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rp;p1, Rp1;p2, and so on O(1)
Rd1;d () Directly encoded O(1)
Rp;q (), p q Recursively combine Rk
;k+1 and Rk+1;k , for k p O(kSt()k)
R0;0() Combine R0;1 and R1;0 O(kSt()k)
Rp;p(), with p6= 0 Combine Rp;p1 and Rp1;p O(kSt()k)
IS star-graph of a p-simplex
Subgraph G of the IS-graph:
nodes representing simplices in St()
IS boundary arcs restricted to St()
IS co-boundary arcs restricted to St()
Co-boundary relation Rp;q()
breadth-first traversal of G
examine top simplices in St()
and their faces
linear in kSt()k
Co-boundary relations are optimal only for simplicial 2- and 3-complexes in E3
Experiments show that they are about less than 10% slower than in the IG
David Canino (DISI) May 7, 2012 17 / 1
18. The Generalized Indexed Data Structure with
Adjacencies (IA)
Key Idea
More compact encoding for a simplicial
d-complex
The Indexed data structure with
Adjacencies (IA), Paoluzzi et al., 1993
vertices, plus d-simplices
boundary relation Rd;0 for
d-simplices
adjacency relation Rd;d for
d-simplices
only for manifolds
The IA data structure
Abstract simplicial d-complex
Dimension-independent
Encodes vertices and top simplices
Adjacency-based
Probably, the most compact representation
for non-manifolds (with respect to the state of
the art)
Non-manifold variant of the Extended IA
(EIA) data structure, De Floriani, et al. 2003
For manifolds, it reduces to the EIA data
structure (scalable)
Local Mangrove (IA-Graph)
D. Canino, L. De Floriani, K. Weiss, IA*: An Adjacency-Based Representation for Non-Manifold Simplicial
Shapes in Arbitrary Dimensions, Computer Graphics, 35(3):747-753, Elsevier Press, Shape Modeling
International 2011 (SMI 2011), Poster
David Canino (DISI) May 7, 2012 18 / 1
19. The IA data structure - Definition
Represents an abstract simplicial d-complex
Boundary relation Rp
;0()
Vertices of a top p-simplex ,
for 1 p d
R 1;0(w) = f1; 2g,
R2
;0(f1) = f1; 3; 4g
Adjacency relation Rp
;p()
Top p-simplices sharing a
(p 1)-simplex with a top
p-simplex , with 2 p d
R 2;2(f1) = ff2; f3; f4g,
R23
2(f5) = ff6g,
;R3(t1) = ft2g
; For Manifolds
IA reduces to EIA
5
3
6
3 2
7 4
1
2
8
9
10
f
11 13
12
14
w
f f
f
t
t
f
f
1
1
4
5
6
2
e
v
only one d-simplex in
Rd for each vertex
0
;Rd1;d : empty
at most one d-simplex
in Rd;d
p-cluster
Maximal collection of adjacent top
p-simplices
2-clusters: ff1; f2; f3; f4g; ff5; f6g
3-cluster: ft1; t2g
Partial co-boundary relation R0
;p(v)
Arbitrary top p-simplex for each
p-cluster in St(v), with 2 p d
R 0;1(v) = fwg,
R 0;2(v) = ff2; f5g, R0
;3(v) = ft1g
Partial co-boundary relation Rp
1;p()
Top p-simplices incident at a
(p 1)-face of a top p-simplex,
with 2 p d (more than two)
R 1;2(e) = ff1; f2; f3; f4g
David Canino (DISI) May 7, 2012 19 / 1
20. The IA data structure - Non-Manifold Adjacency
Key Idea
A (p 1)-face of a top p-simplex is non-manifold if it is shared by more than two top
p-simplices.
5
3
6
3 2
7 4
1
2
8
9
10
f
11 13
12
14
w
f f
f
t
t
f
f
1
1
4
5
6
2
e
v
Manifold Adjacency - At most two top p-simplices in St()
Encode only the other top p-simplex adjacent to along
;2(f5) = ff6g, R2
R2
;2(f6) = ff5g
Non-Manifold Adjacency (Otherwise)
Encode Rp
;p() along as Rp
1;p()
;2(fi ) = R 1;2(e) = ff1; f2; f3; f4g, with i = 1; : : : ; 4
R2
Consequences
Compact encoding of Rp
;p, Rp
1;p stored only once
Partial characterization of non-manifold
(p 1)-simplices
David Canino (DISI) May 7, 2012 20 / 1
21. The IA-Graph
Formed by nodes representing vertices, top simplices, and (some) non-manifold simplices, plus:
IA Boundary Arcs (IA Boundary Graph)
Correspond to Rp
;0 (vertices and top simplices)
IA Co-boundary Arcs (IA Co-boundary Graph)
Correspond to R 0;p (vertices and top simplices)
IA Adjacency Arcs (IA Adjacency Graph)
Correspond to Rp
;p and Rp
1;p
1,11,12,14
1,3,7
1,3
1,12,13,14
1,8,9 1,9,10 1,3,4 1,3,5 1,3,6
IA
Adjacency Graph
IA
Boundary Graph
IA
Co-boundary Graph
David Canino (DISI) May 7, 2012 21 / 1
22. Storage Cost of the IA Data Structure
TS data structure, De Floriani et al., 2003
Variant of the IA data structure
Simplicial 2-complexes in R3
NMIA data structure, De Floriani and Hui, 2003
Variant of the IA data structure
Simplicial 3-complexes in R3
2D Shapes
Shape IS TS IA
Armchair 101k 69:3k 69:1k
Cone 11k 7:8k 7:8k
Frame 12k 8:1k 8:1k
Tower 175k 124k 122k
IS is 1:28 times more expensive than IA
About 5% more compact than TS
3D Shapes
Shape IS NMIA IA
Basket 80k 33k 33k
Flasks 75k 29:6k 29:4k
Sierpinski 688k 197k 197k
Teapot 163k 85k 84:6k
IS is 2:4 times more expensive than IA
Abot 5% more compact than NMIA
Results
the most compact for non-manifolds
small overhead for manifolds (EIA)
Exception: Laced Ring, Gurung et al., 2011
3 times more compact (compression
scheme)
2D manifolds, no editing
David Canino (DISI) May 7, 2012 22 / 1
23. Topological Relations in the IA* Data Structure
Given a simplicial d-complex , a simplex not directly encoded is represented by its vertices :
Rp
;0() Directly encoded O(1)
Rp;q (), p q Generate faces of O(1)
R0;k (v) (top) Expand R 0;k (v) by Rk;k O(#top k-simplices in St(v))
R0;p(v) (any) Select p-simplices in St(v) from top simplices in St(v) O(#top simplices in St(v))
Rp;q (), p q Select q-simplices in St() from top simplices in St(v) O(#top simplices in St(v))
p
Rd;d () Directly encoded O(1)
R0;0(v) Combine R0;1 and R1;0 O(#top simplices in St(v))
Rp;p() Extract Rp and combine Rp;p+1 and Rp+1;p O(#top simplices in St(v))
;: v is a vertex in Rp;0()
Co-boundary relations are optimal only for simplicial 2- and 3-complexes in E3
Basic Operation (optimal)
Retrieving top k-simplices in St(v):
Breadth-first visit of each
k-cluster in R 0;k (v)
Transitive closure of Rk
;k
Linear in #top k-simplices in St(v)
Experimental Comparisons for Co-boundary
vertex-based: 30% faster than IS
edge-based: 10% slower than IS
face-based: 15% slower than IS
David Canino (DISI) May 7, 2012 23 / 1
24. The Mangrove Topological Data Structure (TDS)
Framework
The Mangrove TDS Framewok
Rapid prototyping of topological data structures for simplicial complexes
Satisfies completely design choices of Sieger and Botsch, 2011 for generic frameworks
(probably the first in the literature, independently designed and implemented):
I flexibility - representation of topological data structures (mangroves)
I efficiency - plugins-oriented architecture
I easy-to-use - common interface programming)
Any data structure is supported, without restrictions, including for non-manifolds
Implicit representations of simplices not encoded in a local mangrove (ghost simplices)
The Mangrove TDS Library
Written in C++ (meta-programming techniques)
Common programming interface of the Mangrove TDS framework
We have submitted an article to an international conference, currently under review
Mangrove TDS Library will be released as GPL software at
http://sourceforge.net/projects/mangrovetds/
David Canino (DISI) May 7, 2012 24 / 1
25. The Mangrove TDS Framework - Basic Concepts
Key Idea
A topological data structure is a mangrove
Primitives customized for a p-simplex :
BOUNDARY - boundary B()
STAR - star St()
ADJACENCY - adjacency relation
Rp;p()
LINK - link Lk()
IS_MANIFOLD - checks if is manifold
(when possible)
In this context
Mangrove dynamic plugin in the system
Current Implementations (but extensible)
IG, IS, IA data structures
TS data structure
I adjacency-based
I simplicial 2-complexes in E3
I De Floriani et al., 2003
NMIA data structure
I adjacency-based
I simplicial 3-complexes in E3
I De Floriani and Hui, 2003
SIG data structure
I incidence-based
I dimension-independent
I De Floriani et al., 2004
up to now, only for simplicial complexes
extensible also for cell complexes
Current frameworks partially support non-manifolds through a predefined representation
David Canino (DISI) May 7, 2012 25 / 1
26. The Mangrove TDS Framework - Ghost Simplices
Ghost p-simplex
Not directly encoded in a local mangrove
Explicit Representation
Set of vertices V = fv0; : : : ; vpg
too knownledge
no efficiency for any queries
Implicit Representation
A p-simplex can be either:
a top p-simplex , or
a p-face of a top t-simplex 0, p t
GhostSimplexPointer reference
(t; i; p; pi)
i is the identifier of 0
pi is the identifier of as p-face of 0
0 pi
t + 1
pi + 1
Advantages
less knowledge is required
fixed-length representation
does not depend on an enumerations
of faces
Disadvantage?
a not unique representation
a GhostSimplexPointer reference for
each top simplex in St()
David Canino (DISI) May 7, 2012 26 / 1
27. Explicit Representation of Ghost Simplices
Key Idea
A ghost p-simplex is described by (p + 1)
positions of vertices in V0 Rt;0(0) as
= [k0; : : : ; kp]
Enumeration Rule (Simplicial Homology)
The i-th (p 1)-face i of is defined as
i = [k0; : : : ; ki1; ki+1; : : : ; kp]
Consequence
Partial order relation , such that
i , i.e., i is a face of
Two Hasse diagrams for all t 1
Storage cost of Hasse diagrams
2
Xd
t=1
Xt
p=0
t + 1
p + 1
0,1,2,3
1,2,3 0,2,3 0,1,3 0,1,2
2,3 1,3 1,2 0,3 0,2 0,1
0 1 2 3
(3; 0; 2; 2): vertices in positions [0; 1; 3] in R3;0(t0)
0,1,2,3
0,1,2 0,3,4 1,3,5 2,4,5
2,3 1,3 1,2 0,3 0,2 0,1
0 1 2 3
Same lattice in terms of immediate subfaces
(3; 0; 2; 2) formed by edges [1; 3; 5]
Explicit representation of in O(1)
David Canino (DISI) May 7, 2012 27 / 1
28. Experimental Results on Our Mangroves
We have compared the efficiency of queries on our six mangroves within the Mangrove TDS
Framework
Our results
there is not any data structure optimal for all tasks (advantages vs disadvantages)
in any case, most of queries tend to be more efficient on the IA data structure:
I BOUNDARY is 30% more efficient than IS for a top simplex
I STAR is 35% more efficient than IS for vertices
I LINK is 3X more efficient than IS
Conversely, STAR is within 10% slower than IS for ghost simplices
These improvements are due to the GhostSimplexPointer references, which improve the
expressive power of a local mangrove
David Canino (DISI) May 7, 2012 28 / 1
29. Decomposition Approach
Key Idea
Complex topology of a non-manifold shape offers valuable information for:
decomposing a shape into relevant components with a simpler topology
expose the structure of a shape (connections among components)
Topological data structure Structural model (shape
decomposition)
Semantic model (future
work)
Structural Model for Non-Manifolds
Components joint together at
non-manifold singularities
shape annotation and retrieval
identification of form features
computation of Z-homology
David Canino (DISI) May 7, 2012 29 / 1
30. Manifold-Connected (MC) Decomposition Hui and De Floriani, 2007
Given a simplicial d-complex and k d:
Manifold (k 1)-path (MC-Adjacency)
Sequence of k-simplices in , where each of
simplices is adjacent through a manifold
(k 1)-simplex, bounding at most two
k-simplices
Manifold-Connected (MC) k-Complex
Formed by all k-simplices in connected by a
manifold (k 1)-path
MC-Decomposition
Collection of MC k-Complexes in
MC k-Complexes are the equivalence classes versus MC-Adjacency, and become unique if
restricted to top k-simplices in
David Canino (DISI) May 7, 2012 30 / 1
31. Manifold-Connected (MC) Decomposition (cont’d)
MC-Decomposition
Decomposition of a simplicial complex into its MC-Complexes (MC-components)
Unique, decidable, and dimension-independent (also for high dimensions)
Discrete counterpart of Whitney stratification (1965);
MC-Components
decidable superclass of manifolds
contains some singularities
connected through singularities
It can be represented by a two-level graph-based data structure
David Canino (DISI) May 7, 2012 31 / 1
32. Representing the MC-Decomposition
Two-level Graph-based Data Structure
the lower level describes a non-manifold shape by any mangrove (topological model)
the upper level describes the connectivity of MC-components through a graph-based data
structure (structural model)
MC-graph G = (N;A)
each node in N one MC-component (direct references to top simplices in );
each arc a = (n1; n2; : : : ; nk ) in A intersection of MC-components described by
n1; n2; : : : ; nk (common singularities, as direct references to simplices in )
Relating MC-Components and singularities
the number of MC-Components partially characterizes a
singularity
needs IS_MANIFOLD (no dimension-independent)
efficiency depends on the properties of mangrove
D. Canino, L. De Floriani, A Decomposition-based Approach to Modeling and Understanding Arbitrary Shapes,
9th Eurographics Italian Chapter Conference, Eurographics Association, 2011
David Canino (DISI) May 7, 2012 32 / 1
33. Graph-based Data Structures
1 MC-component of dimension 1: C4
3 MC-components of dimension 2: C1, C2, C3
Pairwise MC-Graph
An arc intersection of two MC-Components, formed by
a subset of singularities
(partial)
Exploded MC-Graph (Hyper-graph)
A hyper-arc a singularity , and connects all
MC-components sharing
Compact MC-Graph (Hyper-graph)
An hyper-arc corresponds to a maximal set of singularities
common to several MC-components
David Canino (DISI) May 7, 2012 33 / 1
34. Experimental Results
We have combined our MC-Graphs with six mangroves in our library (18 different versions)
2D shapes (Storage cost)
Shape C P E
Armchair 10:7k 10:8k 11:2k
Cone 1:2k 1:2k 1:2k
Frame 2:2k 2:7k 2:3k
Tower 20:9k 86:8k 28:6k
3D shapes (Storage cost)
Shape C P E
Basket 4k 4k 4k
Flasks 4k 4:1k 4:4k
Sierpinski 180k 180k 180k
Teapot 25:6k 103:5k 26:2k
The Compact MC-Graph provides the most compact representation
2D shapes (Running Times in ms)
Shape IA IS IG
Armchair 4k 10:8k 11:2k
Cone 4k 7k 15k
Frame 212 283 5:3k
Tower 8:1k 8:5k 440k
3D shapes (Running Times in ms)
Shape IA IS IG
Basket 4k 8k 17k
Flasks 2:4k 6:7k 383k
Sierpinski 2:9k 7:6k 537k
Teapot 6:4k 22:3k 1M
The IA data structure is the most suitable for retrieving MC-Components
David Canino (DISI) May 7, 2012 34 / 1
35. Experimental Results (cont’d)
2D shapes (Storage cost)
Shape C C+IA IG
Armchair 10:7k 69:1k 127k
Cone 1:2k 9k 14k
Frame 2:2k 10:3k 15k
Tower 20:9k 142:9k 221k
3D shapes (Storage cost)
Shape C C+IA IG
Basket 4k 69:1k 127k
Cone 4k 33:4k 104k
Frame 180k 377k 917k
Tower 25:6k 110:2k 219k
The structural model Compact MC-Graph + IA data structure is:
about 63% of IG for 2D shapes (37% more compact than IG)
about 39% of IG for 3D shapes (61% more compact than IG)
David Canino (DISI) May 7, 2012 35 / 1
36. Iterative Computation of Z-homology
Objective
Computing Z-homology of a non-manifold shape
Mayer-Vietoris (MV) Algoritm
modular, iterative, and dimension-independent
the MC-Decomposition - Pairwise MC-Graph
the Constructive Homology Theory - Sergeraert
and Rubio, 2006
Basic idea
Combine:
homology of its MC-components
homology of the intersection of MC-components
45
nodes, 79 arcs ! (Z; Z27; Z5)
Joint Project with INRIA Rhone Alpes,
Grenoble, France
D. Boltcheva, D. Canino, S. Merino, J.-C. Léon, L. De Floriani, F. Hétroy, An Iterative Algorithm for Homology
Computation on Simplicial Shapes, Computer-Aided Design, 43(11):1457-1467, Elsevier Press, SIAM
Conference on Geometric and Physical Modeling (GD/SPM 2011)
David Canino (DISI) May 7, 2012 36 / 1
37. Classical Approach
Associate an algebraic object, namely a chain-complex (;D), to a simplicial complex from
which we extract the Z-homology (Betti numbers, generators, torsion coefficients)
sequence of chain-groups p
Group of p-chains linear combinations of
oriented k-simplices
(;D) : 0 0
: : :
dp1
p1
dp
: : :d
0
0
Smith Normal Form (SNF), Munkres,1999
Incidence Matrix Ip is reduced through
Gaussian eliminations to its Smith Normal
Form (SNF) Np:
Np =
0
p1
0 0 0
: 0 0 Id
B@
p
0 : : :
p
l
p1
m 0 0 0
1
CA
where:
is a diagonal matrix, with i 2 Z
Ip = Pp1NpPp (basis change)
D sequence of boundary operators dp
Describes the oriented boundary Bo(
p
i ) of a
p-simplex
p
i in terms of its immediate
p1
j by the incidence matrix Ip
subfaces
Incidence Matrix Ip of order p
Ip
j;i =
8
:
p1
j62 Bo(
0 if
p
i )
p1
j 2 Bo(
1 if +
p
i )
p1
j 2 Bo(
1 if
p
i ):
Problems of this approach
The Z-homology is retrieved from Np
not constructive
not feasible for large shapes
the SNF is cubic
David Canino (DISI) May 7, 2012 37 / 1
38. The General Idea of the MV Algorithm
Input: a simplicial d-complex discretizing a non-manifold shape
Output: the Z-homology (Betti numbers, generators, torsion-coefficients)
First step
compute SNF reductions of
all MC-components
Generic step
Given components A and B
two components such that
A B6= ;, we compute
(A B) from A, B, and
(A B)
Sergeraert and Rubio, 2006
In the Pairwise MC-Graph:
store N in the node describing N
collapse the arc connecting A and B
Last step
retrieve the Z-homology from the last node
David Canino (DISI) May 7, 2012 38 / 1
39. Critical Properties of the MC-Decomposition
Shape s MS(%) MG(%)
Armchair 32k 38:4 0:07
Balance 24k 31:4 0:004
Bi-Twist 9k 45:5 0:6
Carter 24k 45 0:12
Chandelier 55k 11:8 0:05
Frame 4k 8 0:8
Twist 7k 65:5 0:9
s: total number of simplices
MS: maximum size of a MC-Component
MG: maximum size of the intersection of two
MC-components
Property #1
Guarantees a small size of the intersection
between two MC-Components
Property #2
Produces subcomplexes smaller than the
input shape
#
Good properties for the MV algorithm
Suitable for computations
Consequence #1
Small MC-components reduce time
complexity of the SNF reductions
Consequence #2
Small intersections make the cone
reductions possible (while merging the
MC-components)
David Canino (DISI) May 7, 2012 39 / 1
40. Experimental Results
We have exploited:
the SNF algorithm provided by Moka Modeller, G. Damiand, LIRIS, Lyon, France (no
optimizations), http://moka-modeler.sourceforge.net
our Pairwise MC-Graph + IS data structure
Shape SNFs(MB) SNFt (ms) MVs(%) MVt(%) Result
Armchair 0:6 60 88 320 (Z; 0; Z5)
Bi-Twist 80 1:2 107 73 380 (Z; Z4; Z3)
Carter 567 7:7 107 79 450 (Z; Z27; Z5)
Twist 50 2:2 106 55 160 (Z; Z2; Z2)
SNFs : storage cost of the SNF algorithm (MB)
SNFt : running time of the SNF algorithm (ms)
MVs : reduction in storage cost of the MV algorithm (% wrt SNFs )
MVt : reduction in running time of the MV algorithm (% wrt SNFt )
The MV algorithm is an effective tool for computing the Z-homology
#
Reductions in storage cost and running times wrt the SNF algorithm
David Canino (DISI) May 7, 2012 40 / 1
41. Experimental Results (cont’d)
MC-Decomposition + (Z; Z2; Z2) for the Twist
shape
MC-Decomposition + (Z; Z4; Z3) for the Twist
shape
David Canino (DISI) May 7, 2012 41 / 1
42. Conclusions and Future Works
What we have done
Several tools for simplicial complexes describing non-manifold shapes.
Research Area I - Representation by Topological Data Structures
Two data structures for abstract simplicial complexes in arbitrary dimensions:
I the Incidence Simplicial (IS) data structure
I the Generalized Indexed data structure with Adjacencies (IA).
Mangrove TDS framework
I rapid prototyping of data structures for arbitrary simplicial complexes
Research Area II - Decompositions and Structural Models
Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007
I the Exploded MC-Graph (hyper-graph)
I the Pairwise MC-Graph
I the Compact MC-Graph (hyper-graph)
Mayer-Vietoris (MV) Algorithm for computing Z-homology, which combines:
I the MC-Decomposition (Pairwise MC-Graph)
I the Constructive Homology Theory - Sergeraert and Rubio, 2006
David Canino (DISI) May 7, 2012 42 / 1
43. Conclusions and Future Works (Cont’d)
Several improvements about the different topics
Topological data structures
Extend the IS and IA:
towards cell complexes, like quad and hexahedral shapes (IS)
reconstructions of shapes from point data in high dimension, Rips complexes (IA)
editing operations (multi-resolution models for non-manifolds)
Mangrove TDS Library
release as GPL
new mangroves and new implementations of topological data structures
extension towards cell complexes
MC-Decomposition
semantic models over the MC-Decomposition
identification of 2-cycles (components bounding a void) in the shape
David Canino (DISI) May 7, 2012 43 / 1
44. Conclusions and Future Works (Cont’d)
MV Algorithm
Improve the efficiency of the MV Algorithm:
shape of generators
use optimized versions of the SNF algorithm
transform MC-components into almost manifolds, and exploit more efficient methods for
manifolds
David Canino (DISI) May 7, 2012 44 / 1
45. My Papers
1 D. Canino, L. De Floriani, A Decomposition-based Approach to Modeling and Understanding Arbitrary
Shapes, 9th Eurographics Italian Chapter Conference, Eurographics Association, 2011
2 D. Boltcheva, D. Canino, S. Merino, J.-C. Léon, L. De Floriani, F. Hétroy, An Iterative Algorithm for
Homology Computation on Simplicial Shapes, Computer-Aided Design, 43(11):1457-1467, Elsevier
Press, SIAM Conference on Geometric and Physical Modeling (GD/SPM 2011)
3 D. Canino, L. De Floriani, K. Weiss, IA*: An Adjacency-Based Representation for Non-Manifold Simplicial
Shapes in Arbitrary Dimensions, Computer Graphics, 35(3):747-753, Elsevier Press, Shape Modeling
International 2011 (SMI 2011), Poster
4 D. Canino, A Dimension-Independent and Extensible Framework for Huge Geometric Models, 8th
Eurographics Italian Chapter Conference, Eurographics Association, 2010, Poster
5 L. De Floriani, A. Hui, D. Panozzo, D. Canino, A Dimension-Independent Data Structure for Simplicial
Complexes, In S. Shontz Ed., Proceedings of the 19th International Meshing Roundtable, pages
403-420, Springer, 2010
6 D. Canino, An Extensible Framework for Huge Geometric Models, Technical Report DISI-TR-09-08, 2009
David Canino (DISI) May 7, 2012 45 / 1
46. Thank for your attention and patience. Any questions?
David Canino (DISI) May 7, 2012 46 / 1
47. Interesting Papers
L. De Floriani, D. Greenfieldboyce, and A. Hui, A Data Structure for Non-manifold Simplicial d-complexes,
In Proceedings of the 2nd Eurographics Symposium on Geometry Processing (SGP ’04), pages 83-92,
ACM Press, 2004
L. De Floriani and A. Hui, A Scalable Data Structure for Three-dimensional Non-manifold Objects, In
Proceedings of the 1st Eurographics Symposium on Geometry Processing (SGP ’03), pages 72-82, ACM
Press, 2003
L. De Floriani and A. Hui, Data Structures for Simplicial Complexes: an Analysis and a Comparison, In
Proceedings of the 3rd Eurographics Symposium on Geometry Processing (SGP ’05), pages 119-128,
ACM Press, 2005
L. De Floriani, P. Magillo, E. Puppo, and D. Sobrero, A Multi-resolution Topological Representation for
Non-manifold Meshes, Computer-Aided Design, 36(2):141-159, 2003
H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, 1987
A. Hui and L. De Floriani, A Two-level Topological Decomposition for Non-Manifold Simplicial Shapes, In
Proceedings of the ACM Symposium on Solid and Physical Modeling, pages 355-360, ACM Press, 2007
J. Munkres, Algebraic Topology, Prentice Hall, 1999
A. Nabutovsky, Geometry of the Space of Triangulations of a Compact Manifold, Communications in
Mathematical Physics, 181:303-330, 1996.
A. Paoluzzi, F. Bernardini, C. Cattani, and V. Ferrucci, Dimension-Independent Modeling with Simplicial
Complexes, ACM Transactions on Graphics, 12(1):56-102, 1993
D. Sieger and M. Botsch, Design, Implementation, and Evaluation of the Surface_Mesh Data Structure.
In S. Shontz, editor, Proceedings of the 20th International Meshing Roundtable, pages 533â˘A
S¸ 550.
Springer, 2011.
F. Sergeraert and J. Rubio, Constructive Homological Algebra and Applications, 2006,
http://www-fourier.ujf-grenoble.fr/sergerar/Papers/
David Canino (DISI) May 7, 2012 47 / 1