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Journal articles on the topic 'Matrice partitions'

Author: Grafiati

Published: 22 June 2023

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1

Carayol, Cécile. "La Ligne rouge de Hans Zimmer. Matrice d’un « nouvel Hollywood » électro-minimaliste et contemplatif." Revue musicale OICRM 5, no. 2 (November 30, 2018): 79–102. http://dx.doi.org/10.7202/1054148ar.

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À travers une étude comparative de plusieurs films au contexte narratif contemplatif comme La Ligne Rouge (Terrence Malick, 1998), partition-matrice qui a marqué une nette évolution dans l’esthétique zimmerienne, Hannibal (Ridley Scott, 2001), Da Vinci Code (de Ron Howard, 2006) « synthèse la plus raffinée des influences du minimalisme » (Berthomieu 2013, p. 698), jusqu’à des partitions que Hans Zimmer a composées pour Christopher Nolan comme Inception (2010) et Interstellar (2014), cet article montre de quelle manière Zimmer parvient pleinement à imposer un nouveau courant musical à Hollywood en intégrant une écriture épurée imprégnée notamment par le minimalisme d’Arvo Pärt à des boucles élaborées par des synthétiseurs ou des sons électroniques : si les hommages ciblés à des oeuvres d’Arvo Pärt sont propices à souligner le tourment intérieur ou le recueillement sombre, Zimmer reprend également des principes plus généraux de cette forme de minimalisme – souvent une oscillation immuable et répétée à l’infini autour d’un accord parfait mineur – presque systématiquement mêlés à cette énergie créative de timbres hybrides, afin de créer une autre temporalité apportant une forme d’inéluctable à l’image tout en maintenant empathie et synchronisme discret comme soutiens à l’action (La Ligne rouge, Batman Begins, Da Vinci Code, Inception). La quinte – seule, en ostinato ou répétée sur un motif – quintessence du tintinnabuli zimmerien (au-delà de l’accord parfait pärtien), souligne l’instant suspendu (La Ligne rouge, Hannibal, Interstellar), tandis qu’une forme de radicalisation de ce minimalisme qui va parfois jusqu’à la négation de toute mélodie, remplacée par une note unique, devenue texture abstraite, ou par un cluster diatonique en blend mode (Da Vinci Code, Interstellar), évoque le désespoir, la mort, ou le néant. Loin d’être un « monde » qui « se réduit alors au vide d’un présent sans rêve » (Berthomieu 2004, p. 75), l’écriture électro-minimaliste et contemplative de Zimmer, marquée par une cohérence narrative forte, est connectée au programme esthétique des films auxquels elle se destine.

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Tomescu, Mihaela Aurelia, Lorentz Jäntschi, and Doina Iulia Rotaru. "Figures of Graph Partitioning by Counting, Sequence and Layer Matrices." Mathematics 9, no. 12 (June 18, 2021): 1419. http://dx.doi.org/10.3390/math9121419.

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A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures.

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Lambkin, Christine L. "Partitioned Bremer support localises significant conflict in bee flies (Diptera : Bombyliidae : Anthracinae)." Invertebrate Systematics 18, no. 4 (2004): 351. http://dx.doi.org/10.1071/is04004.

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Partitioned Bremer support examination of combined cladistic analyses indicates nodes at which the support from the partitions differs, and also identifies the location of character disagreement generated by the combination of data matrices. Significant character incongruence was found between mtDNA sequence data and adult morphological characters from three tribes of bee flies (Diptera : Bombyliidae : Anthracinae : Villini, Anthracini, Exoprosopini). Partitioned Bremer support quantitatively reveals the location of significant conflict between characters from the different partitions. Examination of several partitioned support measures show which characters contributed to the disagreement at that node. A very high value of increased support at another node occurred upon combination of the data partitions and was also examined using these support measures.

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4

Zhang, Yi, Xinwang Liu, Jiyuan Liu, Sisi Dai, Changwang Zhang, Kai Xu, and En Zhu. "Fusion Multiple Kernel K-means." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (June 28, 2022): 9109–17. http://dx.doi.org/10.1609/aaai.v36i8.20896.

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Multiple kernel clustering aims to seek an appropriate combination of base kernels to mine inherent non-linear information for optimal clustering. Late fusion algorithms generate base partitions independently and integrate them in the following clustering procedure, improving the overall efficiency. However, the separate base partition generation leads to inadequate negotiation with the clustering procedure and a great loss of beneficial information in corresponding kernel matrices, which negatively affects the clustering performance. To address this issue, we propose a novel algorithm, termed as Fusion Multiple Kernel k-means (FMKKM), which unifies base partition learning and late fusion clustering into one single objective function, and adopts early fusion technique to capture more sufficient information in kernel matrices. Specifically, the early fusion helps base partitions keep more beneficial kernel details, and the base partitions learning further guides the generation of consensus partition in the late fusion stage, while the late fusion provides positive feedback on two former procedures. The close collaboration of three procedures results in a promising performance improvement. Subsequently, an alternate optimization method with promising convergence is developed to solve the resultant optimization problem. Comprehensive experimental results demonstrate that our proposed algorithm achieves state-of-the-art performance on multiple public datasets, validating its effectiveness. The code of this work is publicly available at https://github.com/ethan-yizhang/Fusion-Multiple-Kernel-K-means.

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Adm, Mohammad, Shaun Fallat, Karen Meagher, Shahla Nasserasr, Sarah Plosker, and Boting Yang. "Achievable multiplicity partitions in the inverse eigenvalue problem of a graph." Special Matrices 7, no. 1 (January 1, 2019): 276–90. http://dx.doi.org/10.1515/spma-2019-0022.

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Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

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Liu, Jiyuan, Xinwang Liu, Siwei Wang, Sihang Zhou, and Yuexiang Yang. "Hierarchical Multiple Kernel Clustering." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 10 (May 18, 2021): 8671–79. http://dx.doi.org/10.1609/aaai.v35i10.17051.

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Current multiple kernel clustering algorithms compute a partition with the consensus kernel or graph learned from the pre-specified ones, while the emerging late fusion methods firstly construct multiple partitions from each kernel separately, and then obtain a consensus one with them. However, both of them directly distill the clustering information from kernels or graphs to partition matrices, where the sudden dimension drop would result in loss of advantageous details for clustering. In this paper, we provide a brief insight of the aforementioned issue and propose a hierarchical approach to perform clustering while preserving advantageous details maximumly. Specifically, we gradually group samples into fewer clusters, together with generating a sequence of intermediary matrices of descending sizes. The consensus partition with is simultaneously learned and conversely guides the construction of intermediary matrices. Nevertheless, this cyclic process is modeled into an unified objective and an alternative algorithm is designed to solve it. In addition, the proposed method is validated and compared with other representative multiple kernel clustering algorithms on benchmark datasets, demonstrating state-of-the-art performance by a large margin.

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Zoghlami, Mohamed Ali, Minyar Sassi Hidri, and Rahma Ben Ayed. "Consensus-Driven Cluster Analysis: Top-Down and Bottom-Up Based Split-and-Merge Classifiers." International Journal on Artificial Intelligence Tools 26, no. 04 (August 2017): 1750018. http://dx.doi.org/10.1142/s021821301750018x.

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Consensus clustering is used in data analysis to generate stable results out of a set of partitions delivered by stochastic methods. Typically, the goal is searching for the socalled median (or consensus) partition, i.e. the partition that is most similar, on average, to all the input partitions. In this paper we address the problem of combining multiple fuzzy clusterings without access to the underlying features of the data while basing on inter-clusters similarity. We are concerned of top-down and bottom-up based consensus-driven fuzzy clustering while splitting and merging worst clusters. The objective is to reconcile a structure, developed for patterns in some dataset with the structural findings already available for other related ones. The proposed classifiers consider dispersion and dissimilarity between the partitions as well as the corresponding fuzzy proximity matrices. Several illustrative numerical examples, using both synthetic data and those coming from available machine learning repositories, are also included. The experimental component of the study shows the efficiency of the proposed classifiers in terms of quality and runtime.

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Dalfó, Cristina, and Miquel Àngel Fiol. "A general method to obtain the spectrum and local spectra of a graph from its regular partitions." Electronic Journal of Linear Algebra 36, no. 36 (July 12, 2020): 446–60. http://dx.doi.org/10.13001/ela.2020.5225.

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It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the $C$-local multiplicities of any class of vertices $C$ is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.

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9

Shen, Shuhui, and Xiaojun Zhang. "Constructions of Goethals–Seidel Sequences by Using k-Partition." Mathematics 11, no. 2 (January 6, 2023): 294. http://dx.doi.org/10.3390/math11020294.

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In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences.

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Benatia, Akrem, Weixing Ji, Yizhuo Wang, and Feng Shi. "Sparse matrix partitioning for optimizing SpMV on CPU-GPU heterogeneous platforms." International Journal of High Performance Computing Applications 34, no. 1 (November 14, 2019): 66–80. http://dx.doi.org/10.1177/1094342019886628.

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Sparse matrix–vector multiplication (SpMV) kernel dominates the computing cost in numerous applications. Most of the existing studies dedicated to improving this kernel have been targeting just one type of processing units, mainly multicore CPUs or graphics processing units (GPUs), and have not explored the potential of the recent, rapidly emerging, CPU-GPU heterogeneous platforms. To take full advantage of these heterogeneous systems, the input sparse matrix has to be partitioned on different available processing units. The partitioning problem is more challenging with the existence of many sparse formats whose performances depend both on the sparsity of the input matrix and the used hardware. Thus, the best performance does not only depend on how to partition the input sparse matrix but also on which sparse format to use for each partition. To address this challenge, we propose in this article a new CPU-GPU heterogeneous method for computing the SpMV kernel that combines between different sparse formats to achieve better performance and better utilization of CPU-GPU heterogeneous platforms. The proposed solution horizontally partitions the input matrix into multiple block-rows and predicts their best sparse formats using machine learning-based performance models. A mapping algorithm is then used to assign the block-rows to the CPU and GPU(s) available in the system. Our experimental results using real-world large unstructured sparse matrices on two different machines show a noticeable performance improvement.

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11

Walker, Grant. "Horizontal partitions and Kleshchev's algorithm." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (July 1996): 55–60. http://dx.doi.org/10.1017/s0305004100074661.

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We shall work with rational Mn(K)-modules, where K is an infinite field of prime characteristic p > 0, and Mn(K) is the full matrix semigroup of n × n matrices over K. Recall that equivalence classes of simple rational Mn(K)-modules are parametrized by the set of all partitions λ = (λ1, λ2, …, λl) of length l = l(λ) ≤ n, and that the socle L(λ) of the Schur module (or dual Weyl module) H0(λ) is a simple Mn(K)-module whose highest weight corresponds to the partition λ.

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12

Li, Yimeng, Marcello Ruta, and Matthew A. Wills. "Craniodental and Postcranial Characters of Non-Avian Dinosauria Often Imply Different Trees." Systematic Biology 69, no. 4 (November 26, 2019): 638–59. http://dx.doi.org/10.1093/sysbio/syz077.

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Abstract Despite the increasing importance of molecular sequence data, morphology still makes an important contribution to resolving the phylogeny of many groups, and is the only source of data for most fossils. Most systematists sample morphological characters as broadly as possible on the principle of total evidence. However, it is not uncommon for sampling to be focused on particular aspects of anatomy, either because characters therein are believed to be more informative, or because preservation biases restrict what is available. Empirically, the optimal trees from partitions of morphological data sets often represent significantly different hypotheses of relationships. Previous work on hard-part versus soft-part characters across animal phyla revealed significant differences in about a half of sampled studies. Similarly, studies of the craniodental versus postcranial characters of vertebrates revealed significantly different trees in about one-third of cases, with the highest rates observed in non-avian dinosaurs. We test whether this is a generality here with a much larger sample of 81 published data matrices across all major dinosaur groups. Using the incongruence length difference test and two variants of the incongruence relationship difference test, we found significant incongruence in about 50% of cases. Incongruence is not uniformly distributed across major dinosaur clades, being highest (63%) in Theropoda and lowest (25%) in Thyreophora. As in previous studies, our partition tests show some sensitivity to matrix dimensions and the amount and distribution of missing entries. Levels of homoplasy and retained synapomorphy are similar between partitions, such that incongruence must partly reflect differences in patterns of homoplasy between partitions, which may itself be a function of modularity and mosaic evolution. Finally, we implement new tests to determine which partition yields trees most similar to those from the entire matrix. Despite no bias across dinosaurs overall, there are striking differences between major groups. The craniodental characters of Ornithischia and the postcranial characters of Saurischia yield trees most similar to the “total evidence” trees derived from the entire matrix. Trees from these same character partitions also tend to be most stratigraphically congruent: a mutual consilience suggesting that those partitions yield more accurate trees. [Dinosauria; homoplasy; partition homogeneity.]

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13

MARTIN, PAUL. "TEMPERLEY-LIEB ALGEBRAS FOR NON-PLANAR STATISTICAL MECHANICS — THE PARTITION ALGEBRA CONSTRUCTION." Journal of Knot Theory and Its Ramifications 03, no. 01 (March 1994): 51–82. http://dx.doi.org/10.1142/s0218216594000071.

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We give the definition of the Partition Algebra Pn(Q). This is a new generalisation of the Temperley–Lieb algebra for Q-state n-site Potts models, underpinning their transfer matrix formulation on arbitrary transverse lattices. In Pn(Q) subalgebras appropriate for building the transfer matrices for all transverse lattice shapes (e.g. cubic) occur. For [Formula: see text] the Partition algebra manifests either a semi-simple generic structure or is one of a discrete set of exceptional cases. We determine the Q-generic and Q-independent structure and representation theory. In all cases (except Q = 0) simple modules are indexed by the integers j ≤ n and by the partitions λ ˫ j. Physically they may be associated, at least for sufficiently small j, to 2j 'spin' correlation functions. We exhibit a subalgebra isomorphic to the Brauer algebra.

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14

Bożejko, Marek, and Wojciech Bożejko. "Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 03 (September 2015): 1550020. http://dx.doi.org/10.1142/s0219025715500204.

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The main purpose of this paper is an explicit construction of generalized Gaussian process with function tb(V) = bH(V), where H(V) = n - h(V), h(V) is the number of singletons in a pair-partition V ∈ 𝒫2(2n). This gives another proof of Theorem of A. Buchholtz15 that tb is positive definite function on the set of all pair-partitions. Here there are some new combinatorial formulas presented. Connections with free additive convolutions probability measure on ℝ are also done. There are new positive definite functions on permutations presented. What is more, it is proven that the function H is norm (on the group S(∞) = ⋃S(n)).

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15

Hughes, D. R., and N. M. Singhi. "Partitions in Matrices and Graphs." European Journal of Combinatorics 12, no. 3 (May 1991): 223–35. http://dx.doi.org/10.1016/s0195-6698(13)80088-1.

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Davenport, Dennis, Neil Hindman, Imre Leader, and Dona Strauss. "Multiply partition regular matrices." Discrete Mathematics 322 (May 2014): 61–68. http://dx.doi.org/10.1016/j.disc.2014.01.003.

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17

Claesson, Anders, Mark Dukes, and Martina Kubitzke. "Partition and composition matrices." Journal of Combinatorial Theory, Series A 118, no. 5 (July 2011): 1624–37. http://dx.doi.org/10.1016/j.jcta.2011.02.001.

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18

Deuber, Walter A., Neil Hindman, Imre Leader, and Hanno Lefmann. "Infinite partition regular matrices." Combinatorica 15, no. 3 (September 1995): 333–55. http://dx.doi.org/10.1007/bf01299740.

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19

Reinhart, Georg Martin. "Determinants of Partition Matrices." Journal of Number Theory 56, no. 2 (February 1996): 283–97. http://dx.doi.org/10.1006/jnth.1996.0018.

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20

Brinkworth, Andrew R., Robert Sansom, and Matthew A. Wills. "Phylogenetic incongruence and homoplasy in the appendages and bodies of arthropods: why broad character sampling is best." Zoological Journal of the Linnean Society 187, no. 1 (May 9, 2019): 100–116. http://dx.doi.org/10.1093/zoolinnean/zlz024.

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Abstract Notwithstanding the rapidly increasing sampling density of molecular sequence data, morphological characters still make an important contribution to our understanding of the evolutionary relationships of arthropod groups. In many clades, characters relating to the number and morphological specialization of appendages are ascribed particular phylogenetic significance and may be preferentially sampled. However, previous studies have shown that partitions of morphological character matrices often imply significantly different phylogenies. Here, we ask whether a similar incongruence is observed in the appendage and non-appendage characters of arthropods. We apply tree length (incongruence length difference, ILD) and tree distance (incongruence relationship difference, IRD) tests to these partitions in an empirical sample of 53 published neontological datasets for arthropods. We find significant incongruence about one time in five: more often than expected, but markedly less often than in previous partition studies. We also find similar levels of homoplasy in limb and non-limb characters, both in terms of internal consistency and consistency relative to molecular trees. Taken together, these findings imply that sampled limb and non-limb characters are of similar phylogenetic utility and quality, and that a total evidence approach to their analysis is preferable.

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Maze, Gérard. "Partitions modulo n and circulant matrices." Discrete Mathematics 287, no. 1-3 (October 2004): 77–84. http://dx.doi.org/10.1016/j.disc.2004.04.015.

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SHARMA, RAHUL, SUBHAJIT NANDY, and S. P. BHATTACHARYYA. "ON SOLVING ENERGY-DEPENDENT PARTITIONED REAL SYMMETRIC MATRIX EIGENVALUE PROBLEM BY A PARALLEL GENETIC ALGORITHM." Journal of Theoretical and Computational Chemistry 07, no. 06 (December 2008): 1103–20. http://dx.doi.org/10.1142/s0219633608004428.

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An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of a serial as well as parallel genetic algorithm (GA). The proposed method is tested on two matrices (up to 2000 × 2000) with an increasing number of processors in a master–slave architecture. A comparison is made with the Jacobi–Davidson method in serial mode as implemented in the JDQZ-package. Different partition sizes are used. Traditionally used Löwdin's method is also tested in both serial and parallel modes. The advantages and disadvantages of the parallel GA-based method in solving the partitioned eigenvalue problem are analyzed.

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Almestady, Mohammed S., and Alun O. Morris. "Fischer Matrices for Projective Representations of Generalized Symmetric Groups." Algebra Colloquium 16, no. 03 (September 2009): 449–62. http://dx.doi.org/10.1142/s1005386709000431.

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The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

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Hindman, Neil, and Imre Leader. "Image Partition Regularity of Matrices." Combinatorics, Probability and Computing 2, no. 4 (December 1993): 437–63. http://dx.doi.org/10.1017/s0963548300000821.

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Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

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Ghorbel, M., and M. Ben Farah. "Dirichlet partition on symmetric matrices." Indian Journal of Pure and Applied Mathematics 46, no. 1 (February 2015): 73–83. http://dx.doi.org/10.1007/s13226-015-0109-8.

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Sang, Hai Feng, Qing Chun Li, and Xin Min Li. "The New Criterion for Determining Nonsigularity of Matrices." Applied Mechanics and Materials 380-384 (August 2013): 1262–65. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1262.

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We discuss the problem of nonsingularity of complex partitioned matrices and give the new criterion for determining nonsigularity of complex partitioned matrices and irreducible partitioned matrices by the additive approach. And we obtain the new condition for a matrix to be a positive stable matrix.

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Reiter, Clifford A. "Random Markov matrices and partitions of integers." ACM SIGAPL APL Quote Quad 22, no. 3 (March 1992): 7–9. http://dx.doi.org/10.1145/142263.142265.

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Borodin, Alexei, Vadim Gorin, and Eugene Strahov. "Product Matrix Processes as Limits of Random Plane Partitions." International Mathematics Research Notices 2020, no. 20 (January 16, 2019): 6713–68. http://dx.doi.org/10.1093/imrn/rny297.

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AbstractWe consider a random process with discrete time formed by squared singular values of products of truncations of Haar-distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of squared singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.

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Hong, Kyungpyo, Ho Lee, Hwa Jeong Lee, and Seungsang Oh. "Small knot mosaics and partition matrices." Journal of Physics A: Mathematical and Theoretical 47, no. 43 (October 9, 2014): 435201. http://dx.doi.org/10.1088/1751-8113/47/43/435201.

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Szulc, Tomasz, Ljiljana Cvetković, and Maja Nedović. "Scaling technique for Partition-Nekrasov matrices." Applied Mathematics and Computation 271 (November 2015): 201–8. http://dx.doi.org/10.1016/j.amc.2015.08.136.

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Chen, Shizhao, Jianbin Fang, Chuanfu Xu, and Zheng Wang. "Adaptive Hybrid Storage Format for Sparse Matrix–Vector Multiplication on Multi-Core SIMD CPUs." Applied Sciences 12, no. 19 (September 29, 2022): 9812. http://dx.doi.org/10.3390/app12199812.

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Optimizing sparse matrix–vector multiplication (SpMV) is challenging due to the non-uniform distribution of the non-zero elements of the sparse matrix. The best-performing SpMV format changes depending on the input matrix and the underlying architecture, and there is no “one-size-fit-for-all” format. A hybrid scheme combining multiple SpMV storage formats allows one to choose an appropriate format to use for the target matrix and hardware. However, existing hybrid approaches are inadequate for utilizing the SIMD cores of modern multi-core CPUs with SIMDs, and it remains unclear how to best mix different SpMV formats for a given matrix. This paper presents a new hybrid storage format for sparse matrices, specifically targeting multi-core CPUs with SIMDs. Our approach partitions the target sparse matrix into two segmentations based on the regularities of the memory access pattern, where each segmentation is stored in a format suitable for its memory access patterns. Unlike prior hybrid storage schemes that rely on the user to determine the data partition among storage formats, we employ machine learning to build a predictive model to automatically determine the partition threshold on a per matrix basis. Our predictive model is first trained off line, and the trained model can be applied to any new, unseen sparse matrix. We apply our approach to 956 matrices and evaluate its performance on three distinct multi-core CPU platforms: a 72-core Intel Knights Landing (KNL) CPU, a 128-core AMD EPYC CPU, and a 64-core Phytium ARMv8 CPU. Experimental results show that our hybrid scheme, combined with the predictive model, outperforms the best-performing alternative by 2.9%, 17.5% and 16% on average on KNL, AMD, and Phytium, respectively.

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Trenkler, Götz, Bernhard Schipp, Heinz Neudecker, and Liu Shuangzhe. "Generalized Inverses of Partitioned Matrices." Econometric Theory 9, no. 3 (June 1993): 530–33. http://dx.doi.org/10.1017/s0266466600007945.

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Phillips, Peter C. B. "Generalized Inverses of Partitioned Matrices." Econometric Theory 8, no. 3 (September 1992): 426–27. http://dx.doi.org/10.1017/s0266466600013062.

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Jeong, In-Jee, and Sasha Sodin. "A limit theorem for stochastically decaying partitions at the edge." Random Matrices: Theory and Applications 05, no. 04 (October 2016): 1650016. http://dx.doi.org/10.1142/s2010326316500167.

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We provide a combinational proof of the fact that the sequence of rows of certain stochastically decaying partitions, after appropriate scaling in time and length, converges to the Airy line ensemble. Our argument is based on a modification of the moment method, and it highlights the similarity between random partitions and random matrices.

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Liu, Pei Qiang, Da Ming Zhu, Qing Song Xie, and Jin Jie Xiao. "An Exact Algorithm for 3-Biclique Vertex Partition." Applied Mechanics and Materials 40-41 (November 2010): 189–94. http://dx.doi.org/10.4028/www.scientific.net/amm.40-41.189.

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The paper aims to study the problem of biclustering for gene expression data, which arises in the program of characterizing DNA clone libraries, especially in the oligonucleotide fingerprinting of ribosomal RNA genes method. Gene expression data are arranged in data matrices. The goal of biclustering is to find a submatrix, i.e., subset of rows and a subset of columns. If each element of a matrix is 0 or 1, biclustering is closely related to finding bicliques in a bipartite. The k-BVP (short for k biclique vertex partition problem) is to decide whether the vertices of a bipartite can be partitioned into k groups, and each group induce a biclique. 2-BVP can be solved in polynomial time, but it is an open problem whether or not k-BVP is in P for all k3. On the one hand, present an O(2|V|-3) algorithm to decide whether or not a bipartite graph contains a 3 biclique vertex partition. On the other hand, give an algorithm to produce simulation data. The testing results show that the algorithm can find a 3-BVP of a bipartite if there exist a 3-BVP in the bipartite.

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Avgustinovich, S. V., A. C. Lobstein, and F. I. Solov'eva. "Intersection matrices for partitions by binary perfect codes." IEEE Transactions on Information Theory 47, no. 4 (May 2001): 1621–24. http://dx.doi.org/10.1109/18.923749.

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Cao, Lei, and Zhi Chen. "Partitions of the polytope of doubly substochastic matrices." Linear Algebra and its Applications 563 (February 2019): 98–122. http://dx.doi.org/10.1016/j.laa.2018.10.024.

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P., Suma, and Sunil Jacob J. "Integer partitions revisited using multisets and binary matrices." Malaya Journal of Matematik 8, no. 1 (2020): 282–87. http://dx.doi.org/10.26637/mjm0801/0047.

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Pinelis, Iosif. "Nonnegative sum-symmetric matrices and optimal-score partitions." Positivity 24, no. 3 (July 13, 2019): 533–38. http://dx.doi.org/10.1007/s11117-019-00692-2.

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Xia, Ming-yuan, and Tianbing Xia. "A family of C-partitions and T-matrices." Journal of Combinatorial Designs 7, no. 4 (1999): 269–81. http://dx.doi.org/10.1002/(sici)1520-6610(1999)7:4<269::aid-jcd4>3.0.co;2-l.

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Liu, Peiqiang, Daming Zhu, Jinjie Xiao, Qingsong Xie, and Yanyan Mao. "Partition of a Binary Matrix intok(k ≥ 3) Exclusive Row and Column Submatrices Is Difficult." Mathematical Problems in Engineering 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/934630.

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A biclustering problem consists of objects and an attribute vector for each object. Biclustering aims at finding a bicluster—a subset of objects that exhibit similar behavior across a subset of attributes, or vice versa. Biclustering in matrices with binary entries (“0”/“1”) can be simplified into the problem of finding submatrices with entries of “1.” In this paper, we consider a variant of the biclustering problem: thek-submatrix partition of binary matrices problem. The input of the problem contains ann×mmatrix with entries (“0”/“1”) and a constant positive integerk. Thek-submatrix partition of binary matrices problem is to find exactlyksubmatrices with entries of “1” such that theseksubmatrices are pairwise row and column exclusive and each row (column) in the matrix occurs in exactly one of theksubmatrices. We discuss the complexity of thek-submatrix partition of binary matrices problem and show that the problem is NP-hard for anyk≥3by reduction from a biclustering problem in bipartite graphs.

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Faber, Carel, and Rahul Pandharipande. "Hodge integrals, partition matrices, and the λgconjecture." Annals of Mathematics 157, no. 1 (January 1, 2003): 97–124. http://dx.doi.org/10.4007/annals.2003.157.97.

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Weinstein, Madeleine. "Real symmetric matrices with partitioned eigenvalues." Linear Algebra and its Applications 633 (January 2022): 281–89. http://dx.doi.org/10.1016/j.laa.2021.10.016.

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Bekker, Paul A. "The positive semidefiniteness of partitioned matrices." Linear Algebra and its Applications 111 (December 1988): 261–78. http://dx.doi.org/10.1016/0024-3795(88)90064-x.

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Bollt, Erik, Paweł Góra, Andrzej Ostruszka, and Karol Życzkowski. "Basis Markov Partitions and Transition Matrices for Stochastic Systems." SIAM Journal on Applied Dynamical Systems 7, no. 2 (January 2008): 341–60. http://dx.doi.org/10.1137/070686111.

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Jucys, A. A. "The bijection between plane partitions and nonnegative integer matrices." Lithuanian Mathematical Journal 35, no. 2 (April 1995): 163–67. http://dx.doi.org/10.1007/bf02341495.

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Liang, Zhi-he, and Shi-xin Liang. "On d-row (Column) Antimagic Matrices and Subset Partitions." Acta Mathematicae Applicatae Sinica, English Series 37, no. 1 (January 2021): 192–200. http://dx.doi.org/10.1007/s10255-021-0990-3.

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Nguyen, Van Vu, Jianchun Li, Emre Erkmen, Mehrisadat Makki Alamdari, and Ulrike Dackermann. "FRF Sensitivity-Based Damage Identification Using Linkage Modeling for Limited Sensor Arrays." International Journal of Structural Stability and Dynamics 18, no. 08 (August 2018): 1840002. http://dx.doi.org/10.1142/s0219455418400023.

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This paper presents a novel method to localize and quantify damage in a jack arch structure by introducing a linkage modeling technique to overcome issues caused by having limited sensors. The main strategy in the proposed Frequency Response Function (FRF)-based sensitivity model updating approach is to divide the specimen into partitions. The Young’s modulus of each partition is then updated to detect stiffness reduction caused by damage. System Equivalent Reduction Expansion Process (SEREP) is used to reduce the full finite element (FE) model to a linkage model. The number of measured degrees of freedom (DOFs) is then expanded to the linkage model using the mass and stiffness matrices of the linkage model for the synthesis of interpolated FRFs. The FRF sensitivities are then formulated using the linkage model along with the interpolated FRFs to iteratively calculate the values of the updating parameters until convergence is achieved. The methodology and theory behind this procedure are discussed and verified using a numerical and experimental study. The successful implementation of this method has the potential to detect the location and severity of damage where sensor placement is limited.

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DING, KEQUAN. "ROOK PLACEMENTS AND CLASSIFICATION OF PARTITION VARIETIES B\ Mλ." Communications in Contemporary Mathematics 03, no. 04 (November 2001): 495–500. http://dx.doi.org/10.1142/s0219199701000469.

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Let M be the set of m by n complex matrices of rank m. Let λ = (λ1,…,λm) be a partition with λi ≥ λi + 1 for 1 ≤ i ≤ m - 1 and λ1 = n. A Ferrers board Fλ is a right justified subarray in a m by n matrix with the length of the ith row being λi, and define Mλ={a ∈ M|ai,j = 0 if (i,j) ∉ Fλ}. Let B be the Borel subgroup of the general linear group GLm (C) consisting of upper triangular matrices. Define B\Mλ={Ba|a∈ Mλ}. The quotient space B\Mλ is a projective variety called a partition variety associated to λ. In this note, we classify partition varieties B\Mλ according to their homology and cohomology groups (up to isomorphisms).

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Adm, Mohammad, Shaun Fallat, Karen Meagher, Shahla Nasserasr, Sarah Plosker, and Boting Yang. "Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]." Special Matrices 8, no. 1 (November 22, 2020): 235–41. http://dx.doi.org/10.1515/spma-2020-0117.

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AbstractWe correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.

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