Block Decomposition Research Articles

Development of scalable GPU-based direct whole-core depletion calculation methods

GPU (Graphical Processing Unit) acceleration methods are introduced to the depletion solver and the cross section (XS) calculation routines of the direct whole core calculation code, nTRACER, for scalable and superior computing performance. Extensive GPU acceleration besides transport solvers is necessary to resolve the performance degradation in depletion calculations caused by the newly generated nuclides in depleted fuel regions and the poor scalability. A matrix exponential solver employing the Gauss-Seidel-based Chebyshev rational approximate method is implemented to reduce branch divergence and load imbalance in GPU computing in the solution of the Bateman equation. A non-zero element major storage scheme is developed to optimize memory access efficiency on GPUs by exploiting the same sparsity pattern of the depletion matrix in all the depletion regions. XS data are serialized for effective massive parallelization of XS calculation on GPUs. A block decomposition scheme for XS calculation is implemented to reduce the global memory requirement due to the limited memory capacity of commercial GPUs. It is demonstrated that nTRACER can complete a direct whole core depletion calculation for a realistic core in a few hours on a moderate-sized GPU-based heterogeneous computing platform after extensive GPU porting to all the essential routines.

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac–Coulomb operators

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on C^\infty_c(\mathbb{R}^3\setminus\{0\}, \mathbb{C}^4) . In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137

On star-dagger matrices and the core-EP decomposition

The concept of star-dagger matrices was introduced in 1984 by Hartwig and Spindelböck. While they completely characterized the star-dagger matrices by using a block decomposition of the form PQ00 , they also proposed the following open problem: “Can the triangular form PQ0Rbe used to obtain further results on the star-dagger matrices?” In this paper, we have attempted this open problem by using an upper-triangularization of Schur’s type for a square matrix, namely, the core-EP decomposition. Furthermore, similar problems regarding bi-dagger and bi-EP matrices are investigated.

Finite-dimensional representations of map superalgebras

We describe the block decomposition of the categories of finite-dimensional modules for classical map superalgebras and affine superalgebras in terms of spectral characters. En route to showing these block decompositions, we obtain a new description of finite-dimensional irreducible modules for classical map and affine superalgebras, provide formulas for their (super)characters and describe their extension groups. As an application, we establish a relation between the highest weights of a given finite-dimensional irreducible module for a classical map superalgebra with respect to non-conjugate Borel subalgebras.

Right-angled Coxeter groups with totally disconnected Morse boundaries

This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition. We deduce a new proof for the result of Charney-Cordes-Sisto (Complete topological descriptions of certain Morse boundaries, Groups Geom. Dyn. 17(1),157–184 (2023)) that every right-angled Artin group has totally disconnected Morse boundary, and discuss concrete examples of surface amalgams studied by Ben-Zvi (Boundaries of groups with isolated flats are path connected. arXiv:1909.12360, 2019).

Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type Pℓ of a linear form ℓ∈A1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ – which is nilpotent. The first three authors previously determined which partitions of n=dimk⁡A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T[1].We here consider the family GT of graded Artinian quotients A=R/I of R=k[x,y], having arbitrary Hilbert function H(A)=T. The Jordan cell V(EP) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal EP determined by P. These cells give a decomposition of the variety GT into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V(EP), generalizing a result of [1]. In particular, we determine those partitions for which κ(P)=κ(T), the generic number of generators for an ideal defining an algebra A in GT. We also count the number of partitions P of diagonal lengths T having a given κ(P). A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler Ti and partitions Pi, such that V(EP) is the product of the cells V(EPi), and Ti is single-block: GTi is a Grassmannian.

Convergence rates in the functional CLT for [formula omitted]-mixing triangular arrays

We obtain convergence rates (in the Levi–Prokhorove metric) in the functional central limit theorem (CLT) for partial sums Sn=∑j=1nξj,n of triangular arrays {ξ1,n,ξ2,n,…,ξn,n} satisfying some mixing and moment conditions (which are not necessarily uniform in n). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of Sn. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of Sn is “not negligible” in comparison with the (new) number of summands.

Multiscale Fusion Algorithm for Underwater Image Enhancement Based on Color Preservation

Underwater images regularly exhibit color deviation and low contrast attributed to attenuation and scattering associated with wavelength and distance. For solving these two degradation problems, we propose an efficient underwater image enhancement model, which takes a multiscale fusion strategy as a pivot to concurrently achieve color balance and contrast enhancement. Specifically, we target two core degradation problems through two preprocessing steps. On the one hand, the white balance strategy enhances the appearance of the image by suppressing unnecessary colors to solve the color deviation problem. On the other hand, we employ a color-preserving adaptive histogram equalization (CP-AHE) to obtain an image with higher contrast by jointly processing the three color channels. In the fusion phase, a fusion strategy based on the detail-preserving decomposition of the edge-protected structural blocks is used to converge the white balance and CP-AHE inputs. Extensive experiments on common underwater image datasets have shown the advantages of the proposed method over other state-of-the-art underwater image methods in terms of visual and quantitative evaluation.

Task-based differences in brain state dynamics and their relation to cognitive ability

Transient patterns of interregional connectivity form and dissipate in response to varying cognitive demands. Yet, it is not clear how different cognitive demands influence brain state dynamics, and whether these dynamics relate to general cognitive ability. Here, using functional magnetic resonance imaging (fMRI) data, we characterised shared, recurrent, global brain states in 187 participants across the working memory, emotion, language, and relation tasks from the Human Connectome Project. Brain states were determined using Leading Eigenvector Dynamics Analysis (LEiDA). In addition to the LEiDA-based metrics of brain state lifetimes and probabilities, we also computed information-theoretic measures of Block Decomposition Method of complexity, Lempel-Ziv complexity and transition entropy. Information theoretic metrics are notable in their ability to compute relationships amongst sequences of states over time, compared to lifetime and probability, which capture the behaviour of each state in isolation. We then related task-based brain state metrics to fluid intelligence.We observed that brain states exhibited stable topology across a range of numbers of clusters (K = 2:15). Most metrics of brain state dynamics, including state lifetime, probability, and all information theoretic metrics, reliably differed between tasks. However, relationships between state dynamic metrics and cognitive abilities varied according to the task, the metric, and the value of K, indicating that there are contextual relationships between task-dependant state dynamics and trait cognitive ability.This study provides evidence that the brain reconfigures across time in response to cognitive demands, and that there are contextual, rather than generalisable, relationships amongst task, state dynamics, and cognitive ability.

Three-point energy correlators and the celestial block expansion

We study the three-point energy correlator (EEEC), defined as a matrix element of a product of three energy detectors at different locations on the celestial sphere. Lorentz symmetry implies that the EEEC can be decomposed into special functions called celestial blocks. We compute three-point celestial blocks in an expansion around the collinear limit, where the three detectors approach each other on the celestial sphere. The leading term is a traditional d – 2-dimensional four-point conformal block, and thus the collinear EEEC behaves like a conformally-invariant four-point function in d – 2 dimensions. We obtain the coefficients of the conformal block decomposition for the collinear EEEC at leading nontrivial order in weakly-coupled \U0001d4a9 = 4 SYM and QCD. These data allow us to make certain all-orders predictions for the collinear EEEC in various kinematic limits, including the OPE limit and the double lightcone limit. We also study Ward identities satisfied by the EEEC and compute contact terms in the EEEC in weakly-coupled \U0001d4a9 = 4 SYM. Finally, we study the celestial block expansion of the EEEC in planar \U0001d4a9 = 4 SYM at strong coupling, determining celestial block coefficients to leading and first subleading order at large λ.

Identifying Large Charge operators

The Large Charge sector of Conformal Field Theory (CFT) can generically be described through a semiclassical expansion around a superfluid background. In this work, focussing on U(1) invariant Wilson-Fisher fixed points, we study the spectrum of spinning large charge operators. For sufficiently low spin these correspond to the phonon excitations of the superfluid state. We discuss the organization of these states into conformal multiplets and the form of the corresponding composite operators in the free field theory limit. The latter entails a mapping, built order-by-order in the inverse charge n−1, between the Fock space of vacuum fluctuations and the Fock space of fluctuations around the superfluid state. We discuss the limitations of the semiclassical method, and find that the phonon description breaks down for spins of order n1/2 while the computation of observables is valid up to spins of order n. Finally, we apply the semiclassical method to compute some conformal 3-point and 4-point functions, and analyze the conformal block decomposition of the latter with our knowledge of the operator spectrum.

LSD-induced increase of Ising temperature and algorithmic complexity of brain dynamics

A topic of growing interest in computational neuroscience is the discovery of fundamental principles underlying global dynamics and the self-organization of the brain. In particular, the notion that the brain operates near criticality has gained considerable support, and recent work has shown that the dynamics of different brain states may be modeled by pairwise maximum entropy Ising models at various distances from a phase transition, i.e., from criticality. Here we aim to characterize two brain states (psychedelics-induced and placebo) as captured by functional magnetic resonance imaging (fMRI), with features derived from the Ising spin model formalism (system temperature, critical point, susceptibility) and from algorithmic complexity. We hypothesized, along the lines of the entropic brain hypothesis, that psychedelics drive brain dynamics into a more disordered state at a higher Ising temperature and increased complexity. We analyze resting state blood-oxygen-level-dependent (BOLD) fMRI data collected in an earlier study from fifteen subjects in a control condition (placebo) and during ingestion of lysergic acid diethylamide (LSD). Working with the automated anatomical labeling (AAL) brain parcellation, we first create "archetype" Ising models representative of the entire dataset (global) and of the data in each condition. Remarkably, we find that such archetypes exhibit a strong correlation with an average structural connectome template obtained from dMRI (r = 0.6). We compare the archetypes from the two conditions and find that the Ising connectivity in the LSD condition is lower than in the placebo one, especially in homotopic links (interhemispheric connectivity), reflecting a significant decrease of homotopic functional connectivity in the LSD condition. The global archetype is then personalized for each individual and condition by adjusting the system temperature. The resulting temperatures are all near but above the critical point of the model in the paramagnetic (disordered) phase. The individualized Ising temperatures are higher in the LSD condition than in the placebo condition (p = 9 × 10-5). Next, we estimate the Lempel-Ziv-Welch (LZW) complexity of the binarized BOLD data and the synthetic data generated with the individualized model using the Metropolis algorithm for each participant and condition. The LZW complexity computed from experimental data reveals a weak statistical relationship with condition (p = 0.04 one-tailed Wilcoxon test) and none with Ising temperature (r(13) = 0.13, p = 0.65), presumably because of the limited length of the BOLD time series. Similarly, we explore complexity using the block decomposition method (BDM), a more advanced method for estimating algorithmic complexity. The BDM complexity of the experimental data displays a significant correlation with Ising temperature (r(13) = 0.56, p = 0.03) and a weak but significant correlation with condition (p = 0.04, one-tailed Wilcoxon test). This study suggests that the effects of LSD increase the complexity of brain dynamics by loosening interhemispheric connectivity-especially homotopic links. In agreement with earlier work using the Ising formalism with BOLD data, we find the brain state in the placebo condition is already above the critical point, with LSD resulting in a shift further away from criticality into a more disordered state.

BDRI: block decomposition based on relational interaction for knowledge graph completion

BDRI: block decomposition based on relational interaction for knowledge graph completion

Optical/γ-ray blazar flare correlations: understanding the high-energy emission process using ASAS-SN and Fermi light curves

ABSTRACT Using blazar light curves from the optical All-Sky Automated Survey for Supernovae (ASAS-SN) and the γ-ray Fermi-LAT telescope, we performed the most extensive statistical correlation study between both bands, using a sample of 1180 blazars. This is almost an order of magnitude larger than other recent studies. Blazars represent more than 98 per cent of the AGNs detected by Fermi-LAT and are the brightest γ-ray sources in the extragalactic sky. They are essential for studying the physical properties of astrophysical jets from central black holes. However, their γ-ray flare mechanism is not fully understood. Multiwavelength correlations help constrain the dominant mechanisms of blazar variability. We search for temporal relationships between optical and γ-ray bands. Using a Bayesian Block Decomposition, we detect 1414 optical and 510 γ-ray flares, we find a strong correlation between both bands. Among all the flares, we find 321 correlated flares from 133 blazars, and derive an average rest-frame time delay of only 1.1$_{-8.5}^{+7.1}$ d, with no difference between the flat-spectrum radio quasars, BL Lacertae-like objects or low, intermediate, and high-synchrotron peaked blazar classes. Our time-delay limit rules out the hadronic proton-synchrotron model as the driver for non-orphan flares and suggests a leptonic single-zone model. Limiting our search to well-defined light curves and removing 976 potential but unclear ‘orphan’ flares, we find 191 (13 per cent) and 115 (22 per cent) clear ‘orphan’ optical and γ-ray flares. The presence of ‘orphan’ flares in both bands challenges the standard one-zone blazar flare leptonic model and suggests multizone synchrotron sites or a hadronic model for some blazars.

Spatially localized sparse approximations of deep features for breast mass characterization.

We propose a deep feature-based sparse approximation classification technique for classification of breast masses into benign and malignant categories in film screen mammographs. This is a significant application as breast cancer is a leading cause of death in the modern world and improvements in diagnosis may help to decrease rates of mortality for large populations. While deep learning techniques have produced remarkable results in the field of computer-aided diagnosis of breast cancer, there are several aspects of this field that remain under-studied. In this work, we investigate the applicability of deep-feature-generated dictionaries to sparse approximation-based classification. To this end we construct dictionaries from deep features and compute sparse approximations of Regions Of Interest (ROIs) of breast masses for classification. Furthermore, we propose block and patch decomposition methods to construct overcomplete dictionaries suitable for sparse coding. The effectiveness of our deep feature spatially localized ensemble sparse analysis (DF-SLESA) technique is evaluated on a merged dataset of mass ROIs from the CBIS-DDSM and MIAS datasets. Experimental results indicate that dictionaries of deep features yield more discriminative sparse approximations of mass characteristics than dictionaries of imaging patterns and dictionaries learned by unsupervised machine learning techniques such as K-SVD. Of note is that the proposed block and patch decomposition strategies may help to simplify the sparse coding problem and to find tractable solutions. The proposed technique achieves competitive performances with state-of-the-art techniques for benign/malignant breast mass classification, using 10-fold cross-validation in merged datasets of film screen mammograms.

Results on monochromatic vertex disconnection of graphs

<abstract><p>Let $ G $ be a vertex-colored graph. A vertex cut $ S $ of $ G $ is called a <italic>monochromatic vertex cut</italic> if the vertices of $ S $ are colored with the same color. A graph $ G $ is <italic>monochromatically vertex-disconnected</italic> if any two nonadjacent vertices of $ G $ have a monochromatic vertex cut separating them. The <italic>monochromatic vertex disconnection number</italic> of $ G $, denoted by $ mvd(G) $, is the maximum number of colors that are used to make $ G $ monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: $ mvd(G) $, connectivity and block decomposition. We determine the value of $ mvd(G) $ for some well-known graphs, and then characterize $ G $ when $ n-5\leq mvd(G)\leq n $ and all blocks of $ G $ are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph $ G $ with $ mvd(G) = k $ for any $ k $. Finally, we study the Erdős-Gallai-type results for $ mvd(G) $, and completely solve them.</p></abstract>

Block decomposition for Herz spaces associated with ball Banach function spaces

Block decomposition for Herz spaces associated with ball Banach function spaces

The free field representation for the GL(1∣1) WZW model revisited

The GL(1∣1) WZW model in the free field realization that uses the bc system is revisited. By bosonizing the bc system we describe the Neveu-Schwarz and Ramond sector modules and in terms of the subspaces of a given fermion number l. We show that there are two sectors of mutually local operators, each consists of all Neveu-Schwarz operators and of Ramond operators with either integer or half-integer spins. Conformal blocks and structure constants are found for operators that correspond the highest weight vectors of the spaces . The crossing and braiding matrices are considered and the hexagon and pentagon equations are shown to be satisfied for typical modules. The degenerate case of conformal blocks with atypical (logarithmic) modules as intermediate states is considered. The known conformal block decomposition of correlation functions in the degenerate case is shown to be related to the degeneration splitting in the crossing and braiding relations. The scalar product in atypical modules is discussed. The decomposition of unity in the full correlation functions in the degenerate case in terms of this scalar product is explained.

Twin SVM for conditional probability estimation in binary and multiclass classification

In this paper, we estimate the conditional probability function by presenting a new twin SVM model (CPTWSVM) in binary and multiclass classification problems. The motivation of CPTWSVM is to implement the empirical risk minimization on training data, which is hard to realize in traditional twin SVMs. In each subproblem of CPTWSVM, it measures the empirical risk and outputs the corresponding probability estimate of each class, which eliminates the problems of inconsistent measurement in twin SVMs. Though an additional discriminant objective function is introduced, the optimization problem size of each subproblem is smaller than conditional probability SVM, and is solved by block decomposition algorithm efficiently. In addition, we extend CPTWSVM to multiclass classification by estimating the conditional probability of each class, and maintaining the above properties. Numerical experiments on benchmark and real application datasets demonstrate that CPTWSVM outputs the estimate of probability and the data projection well, resulting in better generalization ability than some leading TWSVMs communities, in terms of binary and multiclass classification.

Combining Algorithmic Information Dynamics Concepts and Machine Learning for Electroencephalography Analysis: What Can We Get?

Electroencephalography (EEG) as an example of electrophysiological monitoring methods has a rather long history of successful application for the diagnosis and treatment of diseases, and this success would not have been possible without effective methods of mathematical, and more recently, computer analysis. Most of these methods are based on statistics. Among the methods of EEG analysis, there is a group of methods that use different versions of Shannon’s entropy estimation as a “main component” and that do not differ significantly from traditional statistical approaches. Despite the external similarity, another approach is to use the Kolmogorov–Chaitin definition of complexity and the concepts of algorithmic information dynamics. The algorithmic dynamics toolbox includes techniques (e.g., block decomposition method) that appear to be applicable to EEG analysis. The current paper is an attempt to use the block decomposition method along with the recent addition to the management of EEG data provided by machine learning, with the ultimate goal of making this data more useful to researchers and medical practitioners.