Regular Decomposition Research Articles

Nodal auxiliary space preconditioners for mixed virtual element methods

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.

Hilbert complexes with mixed boundary conditions part 3: Biharmonic complexes

We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings that follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved.

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension

An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension

A network community detection method with integration of data from multiple layers and node attributes

AbstractMultilayer networks are in the focus of the current complex network study. In such networks, multiple types of links may exist as well as many attributes for nodes. To fully use multilayer—and other types of complex networks in applications, the merging of various data with topological information renders a powerful analysis. First, we suggest a simple way of representing network data in a data matrix where rows correspond to the nodes and columns correspond to the data items. The number of columns is allowed to be arbitrary, so that the data matrix can be easily expanded by adding columns. The data matrix can be chosen according to targets of the analysis and may vary a lot from case to case. Next, we partition the rows of the data matrix into communities using a method which allows maximal compression of the data matrix. For compressing a data matrix, we suggest to extend so-called regular decomposition method for non-square matrices. We illustrate our method for several types of data matrices, in particular, distance matrices, and matrices obtained by augmenting a distance matrix by a column of node degrees, or by concatenating several distance matrices corresponding to layers of a multilayer network. We illustrate our method with synthetic power-law graphs and two real networks: an Internet autonomous systems graph and a world airline graph. We compare the outputs of different community recovery methods on these graphs and discuss how incorporating node degrees as a separate column to the data matrix leads our method to identify community structures well-aligned with tiered hierarchical structures commonly encountered in complex scale-free networks.

Traces for Hilbert complexes

We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable “regular decompositions” with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout.

The elasticity complex: compact embeddings and regular decompositions

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.

Spatial transcriptomics dimensionality reduction using wavelet bases

Background: Spatially resolved transcriptomics (ST) measures gene expression along with the spatial coordinates of the measurements. The analysis of ST data involves significant computation complexity. In this work, we propose a gene expression dimensionality reduction algorithm that retains spatial structure. Methods: We combine the wavelet transformation with matrix factorization to select spatially-varying genes. We extract a low-dimensional representation of these genes. We adopt an Empirical Bayes perspective, imposing regularization through the prior distribution of factor genes. Additionally, we visualize the extracted representations, providing an overview of global spatial patterns. We illustrate the performance of our methods through spatial structure recovery and gene expression reconstruction using a simulation and real data analysis. Results: In real data experiments, our method identifies spatial structure of gene factors and outperforms regular decomposition regarding reconstruction error. We find a connection between the fluctuation of gene patterns and wavelet estimates, and this allows us to provide smoother visualizations. We develop the package and share the workflow generating reproducible quantitative results and gene visualization. The package is available at https://github.com/OliverXUZY/waveST. Conclusions: We have proposed a pipeline for dimensionality reduction that respects spatial structure. Both simulations and real data experiments demonstrate that wavelet and shrinkage techniques show positive results in spatially resolved transcriptomics data. We highlight the idea of combining image processing techniques and statistical methods for application in a spatial genomics context

Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from Pauly and Schomburg (2021, 2022) and recent results on the elasticity Hilbert complex with empty or full boundary conditions from Pauly and Zulehner (2020, 2022).

Implementation of ICT (Information, Communication, and Technology) Based Fiqih Learning in Class XII of Madrasah Alyah Ar-Raudhatul Hasanah Medan

In order for the presentation of the material provided by the fiqh teachers to be more effective and efficient, it is necessary to apply learning that can motivate and stimulate students to be enthusiastic in learning. One of them is by utilizing ICT (information, communication, and technology) or ICT-based learning media (information, communication, and technology) or what we often call ICT (technology, information, and communication). This study aims to explain how the implementation of ICT-based learning media (information, communication, and technology) in the learning process of fiqh education in madrasah aliyah. This research method is qualitative research. The nature of this research is descriptive, namely regular decomposition analysis, so the whole concept of providing understanding and explanation to get a description of the results. The data analysis that the writer uses in this research is an analysis of the tracking process and systematic arrangement, field notes that the authors have obtained from interviews, observations and other materials so that the authors can report the results of the research. The results of this author's qualitative research are: (1) The implementation of ICT-based fiqh learning (information, communication, and technology) in class XII Madrasah Aliyah Pesantren Ar-Raudhatul Hasanah Medan City, (2) The implementation of ICT media (information, communication, and technology) focusing on Fiqh lessons in Class XII Madrasah Aliyah Pesantren Ar-Raudhatul Hasanah Medan City, (3) There are supporting and inhibiting factors in the implementation of fiqh learning by using ICT (information, communication, and technology) utilizing infocus media in class XII Madrasah Aliyah Pesantren Ar-Raudhatul Hasanah Medan City.

Evaluation Analysis of Educational Programs

This study aims to understand more deeply about the concept of evaluating educational programs by reviewing the article that has been made before. The method used in this study is qualitative with the type of literature review research or literature review. Literature research or literature review is research that examines or critically reviews the knowledge, ideas, or findings contained in the academic literature. The data source is searched through google scholar using the keyword 'educational program'. The nature of this study is descriptive analysis, which is the regular decomposition of the data that has been obtained. The results of this study, namely the evaluation of educational programs, will have a positive impact on the development of educational programs. Evaluation of educational programs is very important to provide recommendations as a consideration in determining decisions on programs that have been implemented. Evaluation of educational programs can determine the right decision according to the results of the evaluation. The evaluation will also measure the achievement of each program that has been implemented in accordance with the desired objectives.

Hilbert complexes with mixed boundary conditions part 1: de Rham complex

We show that the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well.

Multiple scale method for a wave packet propagating in a nonlinear planar optical waveguide with gain and loss

The article deals with the propagation of a wave packet in a planar waveguide with optical nonlinearity of the third order. On one side of the waveguide there is an active medium and on the other side a conventional dielectric with losses is applied. Gain and loss are described by the imaginary part of the linear dielectric susceptibility of the corresponding signs for all three layers. As a result of a regular decomposition procedure using the multiple-scale method, a reduced (2 + 1) model for the propagation of a quasimonochromatic wave of a single mode is mathematically correctly derived. The envelope of the wave packet is described by the modification of the Ginzburg–Landau equation. Nontrivial expressions for the model parameters are established through the initial physical characteristics of the problem.

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

In this work, an adaptive edge element method is developed for anH(curl)-elliptic constrained optimal control problem. We use the lowest-order Nédélec’s edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition forH(curl)-functions and the standard bubble function techniques, we derive thea posteriorierror estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonalL2-projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy-norm andL2-norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

New constructions of strongly regular Cayley graphs on abelian non p-groups

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9iv4 for any odd positive integer v>1 and any i=0,1. His result covers all orders of abelian non p-groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups Gi, i=1,2,…,s, of order qi4 such that 2m|qi+1, there exists a decomposition of the complete graph on the abelian group G=G1×G2×⋯×Gs by strongly regular Cayley graphs with negative Latin square type parameters (u2,c(u+1),−u+c2+3c,c2+c), where u=q12q22⋯qs2 and c=(u−1)/m. Such strongly regular decompositions were previously known only when m=2 or G is a p-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.

On strongly regular designs admitting fusion to strongly regular decomposition

AbstractA strongly regular decomposition of a strongly regular graph is a partition of the vertex set into two parts on which the induced subgraphs are strongly regular, or cliques or cocliques. Strongly regular designs (srd's) as defined by D. G. Higman are coherent configurations of rank 10 with two fibers in which the homogeneous configuration on each fiber is a strongly regular graph. Haemers and Higman proved the equivalence between strongly regular decompositions, excluding special cases, and srd's with certain parameter conditions. Here we obtain this result by examining the srd's that admit a fusion to a strongly regular graph on the full vertex set. We derive equivalent conditions to Theorem 2.8 of Haemers and Higman by elementary methods. Incorporating recent works of Hanaki and Klin and Reichard, a table of feasible parameter sets for this class of srd's is presented along with a discussion of known constructions. In two cases, nonexistence is observed due to nonexistence of the strongly regular graph obtained through fusion. One of these is also ruled out by Hobart's generalized Krein conditions, applied to srd's. As strongly regular decompositions of the complete graph have sparked interest with recent papers we observe that in our situation this occurs only when the constituent graphs are also complete and the design is trivial.

Complexes from Complexes

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

Convergence of the Hiptmair--Xu Preconditioner for Maxwell's Equations with Jump Coefficients (i): Extensions of the Regular Decomposition

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509...

Selection of representative slices for generation expansion planning using regular decomposition

In power and energy system planning tools, the temporal detail is often reduced by selecting representative slices out of longer time series. Various methods exist for the selection task, but they may prove slow or otherwise unfavourable in practical applications. Here, a generalized clustering algorithm, referred to as regular decomposition, is presented and applied to a power system planning study covering countries in the Northern Europe. The algorithm is compared with other selection methods, and the comparison is repeated with various number of representative slices and in three carbon price scenarios in order to provide more robust results. When selecting four weeks or more, regular decomposition is shown to perform relatively well compared to the other selection methods in terms of the total costs resulting from the power system model runs. When applied to inter-annual time series, regular decomposition is demonstrated to scale well. Although random sampling shows the most stable performance overall, the results indicate the need to test several methods for each system. Moreover, the results highlight the need to include net load peaks in the selected slices and to carefully estimate their position in the time series. A two-stage method for including net load peaks is presented.

Chordal graphs in triangular decomposition in top-down style

In this paper, we first prove that when the associated graph of a polynomial set is chordal, a particular triangular set computed by a general algorithm in top-down style for computing the triangular decomposition of this polynomial set has an associated graph as a subgraph of this chordal graph. Then for Wang's method and a subresultant-based algorithm for triangular decomposition in top-down style and for a subresultant-based algorithm for regular decomposition in top-down style, we prove that all the polynomial sets appearing in the process of triangular decomposition with any of these algorithms have associated graphs as subgraphs of this chordal graph. These theoretical results can be viewed as non-trivial polynomial generalization of existing ones for sparse Gaussian elimination, inspired by which we further propose an algorithm for sparse triangular decomposition in top-down style by making use of the chordal structure of the polynomial set. The effectiveness of the proposed algorithm for triangular decomposition, when the polynomial set is chordal and sparse with respect to the variables, is demonstrated by preliminary experimental results.