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Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 29 липня 2024

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1

Crâşmăreanu, Mircea. "Particular trace decompositions and applications of trace decomposition to almost projective invariants." Mathematica Bohemica 126, no. 3 (2001): 631–37. http://dx.doi.org/10.21136/mb.2001.134205.

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2

ĐOKOVIĆ, DRAGOMIR Ž., and KAIMING ZHAO. "RATIONAL JORDAN DECOMPOSITION OF BILINEAR FORMS." Communications in Contemporary Mathematics 07, no. 06 (December 2005): 769–86. http://dx.doi.org/10.1142/s0219199705001945.

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Анотація:

This is a continuation of our previous work on Jordan decomposition of bilinear forms over algebraically closed fields of characteristic 0. In this note, we study Jordan decomposition of bilinear forms over any field K0 of characteristic 0. Let V0 be an n-dimensional vector space over K0. Denote by [Formula: see text] the space of bilinear forms f : V0 × V0 → K0. We say that f = g + h, where f, g, [Formula: see text], is a rational Jordan decomposition of f if, after extending the field K0 to an algebraic closure K, we obtain a Jordan decomposition over K. By using the Galois group of K/K0, we prove the existence of rational Jordan decompositions and describe a method for constructing all such decompositions. Several illustrative examples of rational Jordan decompositions of bilinear forms are included. We also show how to classify the unimodular congruence classes of bilinear forms over an algebraically closed field of characteristic different from 2 and over the real field.

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3

Fontanil, Lauro, and Eduardo Mendoza. "Common complexes of decompositions and complex balanced equilibria of chemical reaction networks." MATCH Communications in Mathematical and in Computer Chemistry 87, no. 2 (2021): 329–66. http://dx.doi.org/10.46793/match.87-2.329f.

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A decomposition of a chemical reaction network (CRN) is produced by partitioning its set of reactions. The partition induces networks, called subnetworks, that are "smaller" than the given CRN which, at this point, can be called parent network. A complex is called a common complex if it occurs in at least two subnetworks in a decomposition. A decomposition is said to be incidence independent if the image of the incidence map of the parent network is the direct sum of the images of the subnetworks' incidence maps. It has been recently discovered that the complex balanced equilibria of the parent network and its subnetworks are fundamentally connected in an incidence independent decomposition. In this paper, we utilized the set of common complexes and a developed criterion to investigate decomposition’s incidence independence properties. A framework was also developed to analyze decomposition classes with similar structure and incidence independence properties. We identified decomposition classes that can be characterized by their sets of common complexes and studied their incidence independence. Some of these decomposition classes occur in some biological and chemical models. Finally, a sufficient condition was obtained for the complex balancing of some power law kinetic (PLK) systems with incidence independent and complex balanced decompositions. This condition led to a generalization of the Deficiency Zero Theorem for some PLK systems.

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4

Diestel, Reinhard. "Simplicial minors and decompositions of graphs." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (May 1988): 409–26. http://dx.doi.org/10.1017/s0305004100065026.

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The purpose of this paper is to give natural characterizations of the countable graphs that admit tree-decompositions or simplicial tree-decompositions into primes. Tree-decompositions were recently introduced by Robertson and Seymour in their series of papers on graph minors [7]. Simplicial tree-decompositions were first considered by Halin[6], being the most typical kind of ‘simplicial decomposition’ as introduced by Halin[5] in 1964. The problem of determining which infinite graphs admit a simplicial decomposition into primes has stood unresolved since then; a first solution for simplicial tree-decompositions was given in [2].

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5

AVGUSTINOVICH, S. V., and A. E. FRID. "A UNIQUE DECOMPOSITION THEOREM FOR FACTORIAL LANGUAGES." International Journal of Algebra and Computation 15, no. 01 (February 2005): 149–60. http://dx.doi.org/10.1142/s0218196705002116.

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Анотація:

We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition. Then we prove that for each factorial language, a canonical decomposition exists and is unique.

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6

Breiding, Paul, and Nick Vannieuwenhoven. "On the average condition number of tensor rank decompositions." IMA Journal of Numerical Analysis 40, no. 3 (June 20, 2019): 1908–36. http://dx.doi.org/10.1093/imanum/drz026.

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Abstract We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random rank-$r$ decomposition, given by factor matrices with independent and identically distributed standard normal entries, is infinite. This entails that it is expected and probable that such a rank-$r$ decomposition is sensitive to perturbations of the tensor. Moreover, it provides concrete further evidence that tensor decomposition can be a challenging problem, also from the numerical point of view. On the other hand, we provide strong theoretical and empirical evidence that tensors of size $n_1~\times ~n_2~\times ~n_3$ with all $n_1,n_2,n_3 \geqslant 3$ have a finite average condition number. This suggests that there exists a gap in the expected sensitivity of tensors between those of format $n_1\times n_2 \times 2$ and other order-3 tensors. To establish these results we show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions with an infinite geometric condition number is bounded from below by the inverse of this condition number. That is, we prove one inequality towards a so-called condition number theorem for the tensor rank decomposition.

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7

Broer, Abraham. "Decomposition Varieties in Semisimple Lie Algebras." Canadian Journal of Mathematics 50, no. 5 (October 1, 1998): 929–71. http://dx.doi.org/10.4153/cjm-1998-048-6.

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AbstractThe notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

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8

FOULIS, DAVID J., SYLVIA PULMANNOVÁ, and ELENA VINCEKOVÁ. "TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA." Journal of the Australian Mathematical Society 89, no. 3 (December 2010): 335–58. http://dx.doi.org/10.1017/s1446788711001042.

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Анотація:

AbstractEffect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

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9

Theriault, Stephen D. "Homotopy Decompositions Involving the Loops of Coassociative Co-H Spaces." Canadian Journal of Mathematics 55, no. 1 (February 1, 2003): 181–203. http://dx.doi.org/10.4153/cjm-2003-008-5.

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AbstractJames gave an integral homotopy decomposition of ∑Ω∑X, Hilton-Milnor one for Ω(∑X ∨ ∑Y), and Cohen-Wu gave p-local decompositions of Ω∑X if X is a suspension. All are natural. Using idempotents and telescopes we show that the James andHilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative co-H spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative co-H space.

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10

GUTIERREZ, MAURICIO, and ADAM PIGGOTT. "RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS." Bulletin of the Australian Mathematical Society 77, no. 2 (April 2008): 187–96. http://dx.doi.org/10.1017/s0004972708000105.

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AbstractWe show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

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11

Borup, Lasse, and Morten Nielsen. "Frame Decomposition of Decomposition Spaces." Journal of Fourier Analysis and Applications 13, no. 1 (February 2007): 39–70. http://dx.doi.org/10.1007/s00041-006-6024-y.

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12

Hauenstein, Jonathan D., Luke Oeding, Giorgio Ottaviani, and Andrew J. Sommese. "Homotopy techniques for tensor decomposition and perfect identifiability." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 753 (August 1, 2019): 1–22. http://dx.doi.org/10.1515/crelle-2016-0067.

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Анотація:

AbstractLetTbe a general complex tensor of format{(n_{1},\dots,n_{d})}. When the fraction{\prod_{i}n_{i}/[1+\sum_{i}(n_{i}-1)]}is an integer, and a natural inequality (called balancedness) is satisfied, it is expected thatThas finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions ofT, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats{(3,4,5)}and{(2,2,2,3)}which have a unique decomposition as the sum of six and four decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the onlyidentifiablecases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.

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13

Goh, Say Song, S. L. Lee, Zuowei Shen, and W. S. Tang. "Construction of Schauder decomposition on banach spaces of periodic functions." Proceedings of the Edinburgh Mathematical Society 41, no. 1 (February 1998): 61–91. http://dx.doi.org/10.1017/s001309150001943x.

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Анотація:

This paper deals with Schauder decompositions of Banach spaces X2π of 2π-periodic functions by projection operators Pk onto the subspaces Vk, k = 0,1,…, which form a multiresolution of X2π,. The results unify the study of wavelet decompositions by orthogonal projections in the Hilbert space on one hand and by interpolatory projections in the Banach space C2π on the other. The approach, using “orthogonal splines”, is constructive and leads to the construction of a Schauder decomposition of X2π and a biorthogonal system for X2π, and its dual X2π. Decomposition and reconstruction algorithms are derived from the construction.

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14

Kröger, Hannes, and Jörg Hartmann. "Extending the Kitagawa–Oaxaca–Blinder decomposition approach to panel data." Stata Journal: Promoting communications on statistics and Stata 21, no. 2 (June 2021): 360–410. http://dx.doi.org/10.1177/1536867x211025800.

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The Kitagawa–Oaxaca–Blinder decomposition approach has been widely used to attribute group-level differences in an outcome to differences in endowment, coefficients, and their interactions. The method has been implemented for Stata in the popular oaxaca command for cross-sectional analyses. In recent decades, however, research questions have been more often focused on the decomposition of group-based differences in change over time, for example, diverging income trajectories, as well as decomposition of change in differences between groups, for example, change in the gender pay gap over time. We review five existing methods for the decomposition of changes in group means and contribute an extension that takes an interventionist perspective suitable for applications with a clear before–after comparison. These decompositions of levels and changes over time can be implemented using the xtoaxaca command, which works as a postestimation command for different regression commands in Stata. It is built to maximize flexibility in modeling and implements all decomposition techniques presented in this article.

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15

Gibbons, Courtney R., Robert Huben, and Branden Stone. "Recursive strategy for decomposing Betti tables of complete intersections." International Journal of Algebra and Computation 29, no. 07 (October 23, 2019): 1165–91. http://dx.doi.org/10.1142/s0218196719500450.

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Анотація:

In the spirit of Boij–Söderberg theory, we introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to investigate stability and compatibility of the Boij–Söderberg decompositions of related diagrams; indeed, when the biggest generating degree is sufficiently large, the alternative algorithm produces the Boij–Söderberg decomposition. We also provide a detailed analysis of the Boij–Söderberg decomposition for Betti diagrams of codimension four complete intersections where the largest generating degree satisfies the size condition.

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16

BELLENOT, STEVEN F. "SKIPPED BLOCKING AND OTHER DECOMPOSITIONS IN BANACH SPACES." International Journal of Mathematics 17, no. 02 (February 2006): 129–41. http://dx.doi.org/10.1142/s0129167x06003400.

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Анотація:

Necessary and sufficient conditions are given for when a sequence of finite-dimensional subspaces (Xn) can be blocked to be a skipped blocking decompositon (SBD). These are very similar to known results about blocking of biorthogonal sequences. A separable space X has PCP, if and only if, every norming decomposition (Xn) can be blocked to be a boundedly complete SBD. Every boundedly complete SBD is a JT-decomposition.

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17

Voigtlaender, Felix. "Structured, compactly supported Banach frame decompositions of decomposition spaces." Dissertationes Mathematicae 575 (2022): 1–179. http://dx.doi.org/10.4064/dm804-5-2021.

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18

Hernandez, Bryan, Deza Amistas, Ralph John De la Cruz, Lauro Fontanil, Aurelio de los Reyes, and Eduardo Mendoza. "Independent, Incidence Independent and Weakly Reversible Decompositions of Chemical Reaction Networks." MATCH Communications in Mathematical and in Computer Chemistry 87, no. 2 (2021): 367–96. http://dx.doi.org/10.46793/match.87-2.367h.

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Анотація:

Chemical reaction networks (CRNs) are directed graphs with reactant or product complexes as vertices, and reactions as arcs. A CRN is weakly reversible if each of its connected components is strongly connected. Weakly reversible networks can be considered as the most important class of reaction networks. Now, the stoichiometric subspace of a network is the linear span of the reaction vectors (i.e., difference between the product and the reactant complexes). A decomposition of a CRN is independent (incidence independent) if the direct sum of the stoichiometric subspaces (incidence maps) of the subnetworks equals the stoichiometric subspace (incidence map) of the whole network. Decompositions can be used to study relationships between positive steady states of the whole system (induced from partitioning the reaction set of the underlying network) and those of its subsystems. In this work, we revisit our novel method of finding independent decomposition, and use it to expand applicability on (vector) components of steady states. We also explore CRNs with embedded deficiency zero independent subnetworks. In addition, we establish a method for finding incidence independent decomposition of a CRN. We determine all the forms of independent and incidence independent decompositions of a network, and provide the number of such decompositions. Lastly, for weakly reversible networks, we determine that incidence independence is a sufficient condition for weak reversibility of a decomposition, and we identify subclasses of weakly reversible networks where any independent decomposition is weakly reversible.

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19

CHODA, MARIE. "VON NEUMANN ENTROPY AND RELATIVE POSITION BETWEEN SUBALGEBRAS." International Journal of Mathematics 24, no. 08 (July 2013): 1350066. http://dx.doi.org/10.1142/s0129167x13500663.

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Анотація:

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

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20

Demyanovich, Yuri, and Le Thi Nhu Bich. "Discrete and Continuous Wavelet Expansions." WSEAS TRANSACTIONS ON MATHEMATICS 21 (February 23, 2022): 58–67. http://dx.doi.org/10.37394/23206.2022.21.9.

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Анотація:

This paper proposes a new approach to the construction of wavelet decomposition, which is suitable for processing a wide range of information flows. The proposed approach is based on abstract functions with values in linear topological spaces. It is defined by embedded spaces and their projections. The proposed approach allows for adaptive ways of decomposition for the initial flow depending on the speed changes of the last one. The initial information flows can be real number flows, flows of complex and p-adic numbers, as well as flows of (finite or infinite) vectors, matrices, etc. The result is illustrated with examples of spline-wavelet decompositions of discrete flows, and also with the example of the decomposition of a continuous flow.

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21

Šiljak, D. D., and A. I. Zečević. "A nested decomposition algorithm for parallel computations of very large sparse systems." Mathematical Problems in Engineering 1, no. 1 (1995): 41–57. http://dx.doi.org/10.1155/s1024123x95000068.

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Анотація:

In this paper we present a generalization of the balanced border block diagonal (BBD) decomposition algorithm, which was developed for the parallel computation of sparse systems of linear equations. The efficiency of the new procedure is substantially higher, and it extends the applicability of the BBD decomposition to extremely large problems. Examples of the decomposition are provided for matrices as large as250,000×250,000, and its performance is compared to other sparse decompositions. Applications to the parallel solution of sparse systems are discussed for a variety of engineering problems.

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22

Rutter, John W. "The group of homotopy self-equivalence classes using an homology decomposition." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (March 1988): 305–15. http://dx.doi.org/10.1017/s0305004100064872.

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Анотація:

Exact sequences which can be used for calculating the group (X) of homotopy classes of homotopy self-equivalences of a space X are known in the case of cellular decompositions (some references are given in § 1 of [8]), and in the case of Postnikov decompositions (for example 3·1 of [5]). Here we obtain exact sequences which can be used to calculate (X) using a simply-connected homology decomposition.

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23

Fekete, Dorottya, Sandra Palau, Juan Carlos Pardo, and Jose Luis Pérez. "Backbone Decomposition of Multitype Superprocesses." Journal of Theoretical Probability 34, no. 3 (February 27, 2021): 1149–78. http://dx.doi.org/10.1007/s10959-021-01076-7.

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AbstractIn this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

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24

Birkar, Caucher, and Zhengyu Hu. "Polarized pairs, log minimal models, and Zariski decompositions." Nagoya Mathematical Journal 215 (September 2014): 203–24. http://dx.doi.org/10.1017/s0027763000010953.

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Анотація:

AbstractWe continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (X,B) be a projective log canonical pair. We will show that (X,B) has a log minimal model if eitherKX+Bbirationally has a Nakayama–Zariski decomposition with nef positive part, or ifKX+Bis big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (X,B+P), where (X,B) is a usual projective pair and wherePis nef, and we study the birational geometry of such pairs.

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25

Al‐Jawahri, Zeineb, and Morten Nielsen. "On homogeneous decomposition spaces and associated decompositions of distribution spaces." Mathematische Nachrichten 292, no. 12 (October 22, 2019): 2496–521. http://dx.doi.org/10.1002/mana.201700474.

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26

Mohsenipour, Maryam, and Ghadir Sadeghi. "Atomic decompositions of martingale Hardy-Lorentz spaces and interpolation." Filomat 31, no. 19 (2017): 5921–29. http://dx.doi.org/10.2298/fil1719921m.

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Анотація:

In this paper, we establish atomic decompositions for the martingale Hardy-Lorentz spaces. As an application, with the help of atomic decomposition, some interpolation theorems with a function parameter for these spaces are proved.

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27

NOTBOHM, D. "HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS." Journal of the London Mathematical Society 64, no. 2 (October 2001): 472–88. http://dx.doi.org/10.1112/s0024610701002459.

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Анотація:

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

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28

Jakubík, Ján. "Direct product decomposition of $MV$-algebras." Czechoslovak Mathematical Journal 44, no. 4 (1994): 725–39. http://dx.doi.org/10.21136/cmj.1994.128490.

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29

Flippo, Olaf E., and Alexander H. G. Rinnooy Kan. "Decomposition in general mathematical programming." Mathematical Programming 60, no. 1-3 (June 1993): 361–82. http://dx.doi.org/10.1007/bf01580620.

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30

Paúl, Pedro J., Carmen Sáez, and Juan M. Virués. "Locally Convex Spaces with Toeplitz Decompositions." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 19–40. http://dx.doi.org/10.1017/s1446788700001555.

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Анотація:

AbstractA Toeplitz decomposition of a locally convez space E into subspaces (Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the locally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex properties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective tensor products and spaces with Cesàro bases.

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31

Bagchi, Susmit. "Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings." Symmetry 12, no. 3 (March 12, 2020): 450. http://dx.doi.org/10.3390/sym12030450.

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Анотація:

In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

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32

Casselman, W. "Introduction to the Schwartz Space of T\G." Canadian Journal of Mathematics 41, no. 2 (April 1, 1989): 285–320. http://dx.doi.org/10.4153/cjm-1989-015-6.

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Анотація:

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

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33

Prabawa, Harsa Wara, Rizky Rosjanuardi, and Elah Nurlaelah. "Problem Decomposition Skills, Mathematical Maturity, and Their Relation to Mathematics Problem-Solving in A Computer Science Learning Class." Jurnal Kependidikan: Jurnal Hasil Penelitian dan Kajian Kepustakaan di Bidang Pendidikan, Pengajaran dan Pembelajaran 9, no. 3 (September 16, 2023): 946. http://dx.doi.org/10.33394/jk.v9i3.8258.

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Анотація:

This study investigates how students represent ideas when decomposing mathematical problems and how their mathematical maturity influences the problem-solving process. The method used in this research is explorative research. The subject of this research was six Computers Science Education Department students at the Indonesian Education University. The instrument used task-based interviews. Data analysis used the concept of Miles and Huberman, including data reduction, presentation, and drawing conclusions. The research found that problem decomposition skills, mathematical maturity, and their relation to solving mathematical problems in computer science learning classes influenced one another. Decomposition skills were influenced by how basic math skills are taught, so they can affect students' maturity in solving math problems.

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34

ANTONIOU, IOANNIS, COSTAS KARANIKAS, and STANISLAV SHKARIN. "DECOMPOSITIONS OF SPACES OF MEASURES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 01 (March 2008): 119–26. http://dx.doi.org/10.1142/s0219025708003014.

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Анотація:

Let 𝔐 be the Banach space of σ-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of 𝔐 is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon–Nikodým theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen–Wintner purity theorem for our decompositions.

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35

Zabolotnii, Ya, and I. Denega. "About one problem on extremal decomposition." Issues of Analysis 28, no. 3 (November 2021): 141–50. http://dx.doi.org/10.15393/j3.art.2021.10410.

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36

Bank, Randolph E., and Christian Wagner. "Multilevel ILU decomposition." Numerische Mathematik 82, no. 4 (June 1, 1999): 543–76. http://dx.doi.org/10.1007/s002110050430.

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37

Buzdin, A., and G. Wittum. "Two-frequency decomposition." Numerische Mathematik 97, no. 2 (April 1, 2004): 269–95. http://dx.doi.org/10.1007/s00211-003-0459-8.

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38

Islam, Md Shariful, Mir Shariful Islam, Md Zavid Iqbal Bangalee, AFM Khodadad Khan, and Amal Halder. "Approximate Solution of Systems of Volterra Integral Equations of Second Kind by Adomian Decomposition Method." Dhaka University Journal of Science 63, no. 1 (January 28, 2015): 15–18. http://dx.doi.org/10.3329/dujs.v63i1.21761.

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Анотація:

Real life problems that arise in different branches of science and social science, in the form of differential and integral equations are non-linear in nature. However, methods developed in Mathematics, usually, are suitable for the linear system. In this article, we talk on approximating solution of system of Volterra integral equations of second kind in an analytic way using Adomian decomposition method in Mathematica. DOI: http://dx.doi.org/10.3329/dujs.v63i1.21761 Dhaka Univ. J. Sci. 63(1): 15-18, 2015 (January)

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39

Antoniou, I., and S. A. Shkarin. "Decay spectrum and decay subspace of normal operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (December 2001): 1245–55. http://dx.doi.org/10.1017/s0308210500001372.

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Анотація:

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

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40

SKALSKI, ADAM, and JOACHIM ZACHARIAS. "WOLD DECOMPOSITION FOR REPRESENTATIONS OF PRODUCT SYSTEMS OF C*-CORRESPONDENCES." International Journal of Mathematics 19, no. 04 (April 2008): 455–79. http://dx.doi.org/10.1142/s0129167x08004765.

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Анотація:

Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C*-correspondences over [Formula: see text], generalizing the classical result for a doubly commuting pair of isometries due to Słociński. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are provided. Possibilities of extending isometric representations to fully coisometric ones are discussed.

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41

Ellefsen, Kai Olav, Joost Huizinga, and Jim Torresen. "Guiding Neuroevolution with Structural Objectives." Evolutionary Computation 28, no. 1 (March 2020): 115–40. http://dx.doi.org/10.1162/evco_a_00250.

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Анотація:

The structure and performance of neural networks are intimately connected, and by use of evolutionary algorithms, neural network structures optimally adapted to a given task can be explored. Guiding such neuroevolution with additional objectives related to network structure has been shown to improve performance in some cases, especially when modular neural networks are beneficial. However, apart from objectives aiming to make networks more modular, such structural objectives have not been widely explored. We propose two new structural objectives and test their ability to guide evolving neural networks on two problems which can benefit from decomposition into subtasks. The first structural objective guides evolution to align neural networks with a user-recommended decomposition pattern. Intuitively, this should be a powerful guiding target for problems where human users can easily identify a structure. The second structural objective guides evolution towards a population with a high diversity in decomposition patterns. This results in exploration of many different ways to decompose a problem, allowing evolution to find good decompositions faster. Tests on our target problems reveal that both methods perform well on a problem with a very clear and decomposable structure. However, on a problem where the optimal decomposition is less obvious, the structural diversity objective is found to outcompete other structural objectives—and this technique can even increase performance on problems without any decomposable structure at all.

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42

Rios-Avila, Fernando. "Recentered influence functions (RIFs) in Stata: RIF regression and RIF decomposition." Stata Journal: Promoting communications on statistics and Stata 20, no. 1 (March 2020): 51–94. http://dx.doi.org/10.1177/1536867x20909690.

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Анотація:

Recentered influence functions (RIFs) are statistical tools popularized by Firpo, Fortin, and Lemieux (2009 , Econometrica 77: 953–973) for analyzing unconditional partial effects on quantiles in a regression analysis framework (unconditional quantile regressions). The flexibility and simplicity of these tools have opened the possibility to extend the analysis to other distributional statistics using linear regressions or decomposition approaches. In this article, I introduce one function and two commands to facilitate the use of RIFs in the analysis of outcome distributions: rifvar() is an egen extension used to create RIFs for a large set of distributional statistics, rifhdreg facilitates the estimation of RIF regressions enabling the use of high-dimensional fixed effects, and oaxaca_rif implements Oaxaca–Blinder decomposition analysis (RIF decompositions).

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43

Bauer, S., M. C. Crabb, and M. Spreafico. "The classifying space of the gauge group of an SO(3)-bundle over S2." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 4 (August 2001): 767–83. http://dx.doi.org/10.1017/s0308210500001098.

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Анотація:

Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension. The stable decomposition is related to a description of the integral cohomology ring.

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44

Plesken, Wilhelm. "Gauss-Bruhat decomposition as an example of Thomas decomposition." Archiv der Mathematik 92, no. 2 (February 2009): 111–18. http://dx.doi.org/10.1007/s00013-008-2786-6.

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45

Jiang, Zhi-jie. "A result about the atomic decomposition of Bloch-type space in the polydisk." AIMS Mathematics 8, no. 5 (2023): 10822–34. http://dx.doi.org/10.3934/math.2023549.

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Анотація:

<abstract><p>The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball.</p></abstract>

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46

Pavešić, Petar. "Reducibility of self-homotopy equivalences." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 137, no. 2 (2007): 389–413. http://dx.doi.org/10.1017/s0308210505000843.

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Анотація:

We describe a new general method for the computation of the group Aut(X) of self-homotopy equivalences of a space. It is based on the decomposition of Aut(X) induced by a factorization of X into a product of simpler spaces. Normally, such decompositions require assumptions (‘induced equivalence property’, ‘diagonalizability’), which are strongly restrictive and difficult to check. We derive computable homological criteria for an analogous assumption, called reducibility, and then show that these criteria are satisfied when the so-called atomic decomposition of the space is used. This essentially reduces the computation of Aut(X) to the computation of the group of self-equivalences of its atomic factors, and the computation of certain homotopy sets between those factors.

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47

Yokota, Yoshiyuki. "On quantumSU(2) invariants and generalized bridge numbers of knots." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (May 1995): 545–57. http://dx.doi.org/10.1017/s0305004100073369.

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Анотація:

Consider a knotKin a closed, oriented 3-manifoldM. A genusgHeegaard decomposition ofMis said to be agenus g bridge decompositionofKif it also decomposesKinto trivial arcs in each handlebody. Thegenus g bridge numberofK, denoted bybg(K), is defined as the minimal number of trivial arcs in a handlebody among all the genusgbridge decompositions ofK[2, 13].

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48

Selick, P., S. Theriault, and J. Wu. "Functorial Decompositions of Looped Coassociative Co-H Spaces." Canadian Journal of Mathematics 58, no. 4 (August 1, 2006): 877–96. http://dx.doi.org/10.4153/cjm-2006-036-3.

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AbstractSelick and Wu gave a functorial decomposition of Ω∑X for path-connected, p-local CW-complexes X which obtained the smallest nontrivial functorial retract Amin(X) of Ω∑X. This paper uses methods developed by the second author in order to extend such functorial decompositions to the loops on coassociative co-H spaces.

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49

Durukan-Odabaşı, Şafak, та Uğur Odabaşı. "The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle". Symmetry 15, № 10 (18 жовтня 2023): 1930. http://dx.doi.org/10.3390/sym15101930.

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Анотація:

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations.

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50

SATO, FUMIHIRO, and KAZUNARI SUGIYAMA. "MULTIPLICITY ONE PROPERTY AND THE DECOMPOSITION OF b-FUNCTIONS." International Journal of Mathematics 17, no. 02 (February 2006): 195–229. http://dx.doi.org/10.1142/s0129167x0600345x.

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Анотація:

Recently, extensive calculations have been made on b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions which seem to be correlated to the decomposition of representations. In the present paper, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Furthermore, we give some criteria for the multiplicity one property. Our method can be applied equally to non-regular prehomogeneous vector spaces.

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