| 41. |
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Discrete Geometry and Mathematical Morphology : First International Joint Conference, DGMM 2021, Uppsala, Sweden, May 24–27, 2021, Proceedings
- 2021
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Samlingsverk (redaktörskap) (refereegranskat)abstract
- This book constitutes the proceedings of the First IAPR International Conference on Discrete Geometry and Mathematical Morphology, DGMM 2021, which was held during May 24-27, 2021, in Uppsala, Sweden.The conference was created by joining the International Conference on Discrete Geometry for computer Imagery, DGCI, with the International Symposium on Mathematical Morphology, ISMM.The 36 papers included in this volume were carefully reviewed and selected from 59 submissions. They were organized in topical sections as follows: applications in image processing, computer vision, and pattern recognition; discrete and combinatorial topology; discrete geometry - models, transforms, visualization; discrete tomography and inverse problems; hierarchical and graph-based models, analysis and segmentation; learning-based approaches to mathematical morphology; multivariate and PDE-based mathematical morphology, morphological filtering.The book also contains 3 invited keynote papers.
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| 42. |
- Gäfvert, Oliver, 1991-
(författare)
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Topological and geometrical methods in data analysis
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns two related data analysis pipelines, using topological and geometrical methods respectively, to extract relevant information. The first pipeline, referred to as the topological data analysis (TDA) pipeline, constructs a filtered simplicial complex on a given data set in order to describe its shape. The shape is described using a persistence module, which characterizes the topological features of the filtration, and the final step of the pipeline extracts algebraic invariants from this object. The second pipeline, referred to as the geometric data analysis (GDA) pipeline, associates an algebraic variety to a given data set and aims to describe the structure of this variety. Its structure is described using homology, an invariant which for most algebraic varieties can only be computed numerically using sampling methods.In Paper A we consider invariants on multi-parameter persistence modules. We explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi-parameter persistence modules with respect to the interleaving distance and so called simple noise systems. For one parameter, we recover the standard barcode information. For more than one parameter we prove that the constructed invariants are in general NP-hard to calculate. A consequence is that computing the feature counting function, proposed by Scolamiero et. al. (2016), is NP-hard.In Paper B we introduce an efficient algorithm to compute a minimal presentation of a multi-parameter persistent homology module, given a chain complex of free modules as input. Our approach extends previous work on this problem in the 2-parameter case, and draws on ideas underlying the F4 and F5 algorithms for Gröbner basis computation. In the r-parameter case, our algorithm computes a presentation for the homology of C ->F A ->G B, with modules of rank l,n,m respectively, in O(r2nr+1 + nrm + nr-1m2 + rn2 l) arithmetic operations. We implement this approach in our new software Muphasa, written in C++. In preliminary computational experiments on synthetic TDA examples, we compare our approach to a version of a classical approach based on Schreyer's algorithm, and find that ours is substantially faster and more memory efficient. In the course of developing our algorithm for computing presentations, we also introduce algorithms for the closely related problems of computing Gröbner bases for the image and kernel of the morphism G. This algorithm runs in time O(nrm + nr-1m2) and memory O(n2 + mn + nr + K), where K is the size of the output.Paper C analyzes the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in RN. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDD) of a certain variety called the hypothesis variety as the number of points in the configuration increases. Finally, we establish a connection to complexity of architectures of polynomial neural networks. For the problem of fitting an (N-1)-sphere to a configuration of m points in RN, we give a closed formula for the algebraic complexity of the hypothesis variety as m grows for the case of N=1. For the case N>1 we conjecture a generalization of this formula supported by numerical experiments.In Paper D we present an efficient algorithm to produce a provably dense sample of a smooth compact variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et. al. (2019).
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| 43. |
- Rowlett, Julie, 1978
(författare)
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Donut choirs and Schiemann's symphony
- 2021
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Ingår i: Oberwolfach Reports. - 1660-8941 .- 1660-8933. ; 27, s. 33-36
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Abstract: Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral non-isometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is: what is the lowest dimension in which such isospectral non-isometric pairs exist? This question can be equivalently formulated in analytic, geometric, and number theoretic language. We will explore all three formulations and describe Schiemann's solution to this question, as well as some new results and open problems. This talk is based on joint work with Erik Nilsson and Felix Rydell (PhD students at KTH, Stockholm).
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| 44. |
- Gerken, Jan, 1991, et al.
(författare)
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Geometric deep learning and equivariant neural networks
- 2023
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Ingår i: Artificial Intelligence Review. - : Springer Nature. - 1573-7462 .- 0269-2821. ; 56:12, s. 14605-14662
-
Tidskriftsartikel (refereegranskat)abstract
- We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning.
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| 45. |
- Aiyappan, S., et al.
(författare)
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HOMOGENIZATION OF A NON-PERIODIC OSCILLATING BOUNDARY VIA PERIODIC UNFOLDING
- 2022
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Ingår i: DIFFERENTIAL EQUATIONS & APPLICATIONS. - : Element d.o.o.. - 1847-120X .- 1848-9605. ; 14:1, s. 31-47
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Tidskriftsartikel (refereegranskat)abstract
- This paper deals with the homogenization of an elliptic model problem in a two-dimensional domain with non-periodic oscillating boundary by the method of periodic unfolding. For the non-periodic oscillations, a modulated unfolding is used. The L-2 convergence of the solutions and their fluxes are shown, under natural hypotheses on the domain.
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| 46. |
- Andersson, Johan, et al.
(författare)
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On the universality of the Epstein zeta function
- 2020
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Ingår i: Commentarii Mathematici Helvetici. - 0010-2571 .- 1420-8946. ; 95:1, s. 183-209
-
Tidskriftsartikel (refereegranskat)abstract
- We study universality properties of the Epstein zeta function E-n(L,s) for lattices L of large dimension n and suitable regions of complex numbers s. Our main result is that, as n -> infinity, E-n(L,s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses a novel combination of an approximation result for Dirichlet polynomials, a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as n -> infinity, E-n(L-1,s) - E-n(L-2,s) is universal in the full half-plane to the right of the critical line as E-n(L,s) varies over all pairs of n-dimensional lattices. Finally, we prove a more classical universality result for E-n(L,s) in the s-variable valid for almost all lattices L of dimension n. As part of the proof we obtain a strong bound of E-n(L,s) on the critical line that is subconvex for n >= 5 and almost all n-dimensional lattices L.
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| 47. |
- Atashipour, Rasoul, 1983, et al.
(författare)
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On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation
- 2023
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Ingår i: European Journal of Mechanics, A/Solids. - : Elsevier BV. - 0997-7538. ; 97
-
Tidskriftsartikel (refereegranskat)abstract
- A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.
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| 48. |
- Galizia, Roberto, et al.
(författare)
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Regions of reduced dynamics in dynamic networks
- 2021
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 31:6
-
Tidskriftsartikel (refereegranskat)abstract
- We consider complex networks where the dynamics of each interacting agent is given by a nonlinear vector field and the connections between the agents are defined according to the topology of undirected simple graphs. The aim of the work is to explore whether the asymptotic dynamic behavior of the entire network can be fully determined from the knowledge of the dynamic properties of the underlying constituent agents. While the complexity that arises by connecting many nonlinear systems hinders us to analytically determine general solutions, we show that there are conditions under which the dynamical properties of the constituent agents are equivalent to the dynamical properties of the entire network. This feature, which depends on the nature and structure of both the agents and connections, leads us to define the concept of regions of reduced dynamics, which are subsets of the parameter space where the asymptotic solutions of a network behave equivalently to the limit sets of the constituent agents. On one hand, we discuss the existence of regions of reduced dynamics, which can be proven in the case of diffusive networks of identical agents with all-to-all topologies and conjectured for other topologies. On the other hand, using three examples, we show how to locate regions of reduced dynamics in parameter space. In simple cases, this can be done analytically through bifurcation analysis and in other cases we exploit numerical continuation methods.
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| 49. |
- Hietala, Linnea, 1988
(författare)
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A combinatorial description of certain polynomials related to the XYZ spin chain
- 2020
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Ingår i: Symmetry, Integrability and Geometry - Methods and Applications. - 1815-0659. ; 16, s. 1-26
-
Tidskriftsartikel (refereegranskat)abstract
- We study the connection between the three-color model and the polynomials q_n(z) of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for q_n(z) in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that q_n(z) has positive integer coefficients. We prove the weaker statement that q_n(z + 1) and (z + 1)^(n(n+1))q_n(1/(z + 1)) have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.
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| 50. |
- Jansson, Erik, 1996
(författare)
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Geometric Discretization in Shape analysis
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Discretizations in shape analysis is the main theme of this licentiate thesis, which comprises two papers. The first paper considers the problem of finding a parameterized time-dependent vector field that warps an initial set of points to a target set of points. The parametrization introduces a restriction on the number of available vector fields. It is shown that this changes the geometric setting of the matching problem and equations of motion in this new setting are derived. Computational algorithms are provided, together with numerical examples that emphasize the practical importance of regularization. Further, the modified problem is shown to have connections with residual neural networks, meaning that it is possible to study neural networks in terms of shape analysis. The second paper concerns a class of spherical partial differential equations, commonly found in mathematical physics, that describe the evolution of a time-dependent vector field. The flow of the vector field generates a diffeomorphism, for which a discretization method based on quantization theory is derived. The discretization method is geometric in the sense that it preserves the underlying Lie--Poisson structure of the original equations. Numerical examples are provided and potential use cases of the discretization method are discussed, ranging from compressible flows to shape matching.
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