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Journal articles on the topic 'Splitting'

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Published: 4 June 2021

Last updated: 30 July 2024

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1

Ambos-Spies, Klaus, and Peter A. Fejer. "Degree theoretical splitting properties of recursively enumerable sets." Journal of Symbolic Logic 53, no. 4 (December 1988): 1110–37. http://dx.doi.org/10.1017/s0022481200027961.

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A recursively enumerable splitting of an r.e. set A is a pair of r.e. sets B and C such that A = B ∪ C and B ∩ C = ⊘. Since for such a splitting deg A = deg B ∪ deg C, r.e. splittings proved to be a quite useful notion for investigations into the structure of the r.e. degrees. Important splitting theorems, like Sacks splitting [S1], Robinson splitting [R1] and Lachlan splitting [L3], use r.e. splittings.Since each r.e. splitting of a set induces a splitting of its degree, it is natural to study the relation between the degrees of r.e. splittings and the degree splittings of a set. We say a set A has the strong universal splitting property (SUSP) if each splitting of its degree is represented by an r.e. splitting of itself, i.e., if for deg A = b ∪ c there is an r.e. splitting B, C of A such that deg B = b and deg C = c. The goal of this paper is the study of this splitting property.In the literature some weaker splitting properties have been studied as well as splitting properties which imply failure of the SUSP.

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2

Mishra, Debasisha. "Reverse proper splittings of rectangular matrices." Filomat 29, no. 7 (2015): 1491–99. http://dx.doi.org/10.2298/fil1507491m.

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In this article, we introduce a new splitting for rectangular matrices called reverse proper splitting. We then propose several subclasses of this splitting and also discuss convergence results for these splittings.

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3

Nandi, Ashish, and Jajati Sahoo. "Regularized iterative method for ill-posed linear systems based on matrix splitting." Filomat 35, no. 4 (2021): 1343–58. http://dx.doi.org/10.2298/fil2104343n.

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In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov?s regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The convergence of such a well-posed system is discussed by using different types of matrix splittings. Comparison analysis of both systems are studied by operating certain types of weak splittings. Further, we have extended the double splitting of [Song J. and Song Y, Calcolo 48(3), 245-260, 2011] to double weak splitting of type II for nonsingular symmetric matrices. In addition to that, some more comparison results are presented with the help of such weak double splittings of type I and type II.

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4

Li, Cui-Xia, and Su-Hua Li. "Comparison Theorems of Spectral Radius for Splittings of Matrices." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/573024.

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A class of the iteration method from the double splitting of coefficient matrix for solving the linear system is further investigated. By structuring a new matrix, the iteration matrix of the corresponding double splitting iteration method is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.

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5

Huang, Shaowu, Qing-Wen Wang, Shuxia Wu, and Yaoming Yu. "Extensions of pseudo-Perron-Frobenius splitting related to generalized inverse AT,S(2)." Special Matrices 6, no. 1 (January 1, 2018): 46–55. http://dx.doi.org/10.1515/spma-2018-0005.

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Abstract We in this paper define the outer-Perron-Frobenius splitting, which is an extension of the pseudo- Perron-Frobenius splitting defined in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The findings of this paper generalize some known results in the literatures.

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6

Davey, Brian A., Tomasz Kowalski, and Christopher J. Taylor. "Splittings in varieties of logic." International Journal of Algebra and Computation 31, no. 04 (June 2021): 727–74. http://dx.doi.org/10.1142/s021819672150034x.

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We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

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7

Fichtner, K. H., V. Liebscher, and W. Freudenberg. "Time Evolution and Invariance of Boson Systems Given by Beam Splittings." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 04 (October 1998): 511–31. http://dx.doi.org/10.1142/s0219025798000284.

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Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.

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8

Lee, Bum-Hoon, Shahin Mamedov, and Chanyong Park. "Nucleon mass splitting in the isospin medium." International Journal of Modern Physics A 29, no. 29 (November 20, 2014): 1450170. http://dx.doi.org/10.1142/s0217751x1450170x.

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Using the AdS/CFT correspondence, we investigate a nucleon mass splitting and pion–nucleon coupling in the isospin medium. We find that there exists a nucleon mass splitting which is exactly given by the half of the meson mass splitting because nucleon has the half isospin charge of the charged mesons. In addition, we also investigate the pion–nucleon and four pion interactions which requires the modification of the known Abelian-type unitary gauge fixing because of the non-Abelian structure of the isospin medium. After finding the non-Abelian unitary gauge fixing, we find that in the isospin medium the couplings only for π0 π0 π+ π- and π0 π0π0 π0 of four pion interactions shift, while there is no pion–nucleon coupling splitting in spite of the nucleon's and meson's mass splittings.

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9

Weiden, Peter. "Splitting and Pill Splitting." Psychiatric Services 58, no. 2 (February 2007): 163. http://dx.doi.org/10.1176/ps.2007.58.2.163.

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10

SAITO, TOSHIO. "MERIDIONALLY INCOMPRESSIBLE SURFACES AND THE DISTANCE OF (1,1)-SPLITTINGS." Journal of Knot Theory and Its Ramifications 12, no. 07 (November 2003): 1009–21. http://dx.doi.org/10.1142/s0218216503002925.

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In this paper, we show that for any torus giving a (1,1)-splitting of (M,K), every closed orientable surface of genus g>0 which is meridionally incompressible can be isotoped to intersect the the splitting torus in a circle. As an application, we estimate the distance, which is defined in [16], of (1,1)-splittings of (1,1)-knots whose exteriors contain closed orientable meridionally incompressible surfaces.

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11

Bolognini, Davide, and Ulderico Fugacci. "Betti splitting from a topological point of view." Journal of Algebra and Its Applications 19, no. 06 (June 27, 2019): 2050116. http://dx.doi.org/10.1142/s0219498820501169.

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A Betti splitting [Formula: see text] of a monomial ideal [Formula: see text] ensures the recovery of the graded Betti numbers of [Formula: see text] starting from those of [Formula: see text] and [Formula: see text]. In this paper, we introduce an analogous notion for simplicial complexes, using Alexander duality, proving that it is equivalent to a recursive splitting condition on links of some vertices. We provide results ensuring the existence of a Betti splitting for a simplicial complex [Formula: see text], relating it to topological properties of [Formula: see text]. Among other things, we prove that orientability for a manifold without boundary is equivalent to the admission of a Betti splitting induced by the removal of a single facet. Taking advantage of our topological approach, we provide the first example of a monomial ideal which admits Betti splittings in all characteristics but with characteristic-dependent resolution. Moreover, we introduce new numerical descriptors for simplicial complexes and topological spaces, useful to deal with questions concerning the existence of Betti splitting.

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12

Halsband, Aurélie. "Embryo-Splitting und reproduktives Klonen." Zeitschrift für Praktische Philosophie 9, no. 1 (August 19, 2022): 15–40. http://dx.doi.org/10.22613/zfpp/9.1.1.

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Mithilfe der Verfahren des Embryo-Splittings werden unter Laborbedingungen aus einem Embryo mehrere, genetisch identische Embryonen gewonnen. Gegenwärtig wird in Fachbeiträgen debattiert, die in der Nutztierzucht etablierten Verfahren auf die humane Reproduktionsmedizin auszuweiten. Eine solche Anwendung wird derzeit flächendeckend als reproduktives Klonen verstanden und ist in allen Staaten per Gesetz oder Richtlinie untersagt. Bei der Prüfung ausgewählter Einwände gegen die prinzipielle Zulässigkeit einer Anwendung des Embryo-Splittings als assistierter Reproduktionstechnologie zeigt sich, dass sich die Einwände gegen das reproduktive Klonen nicht auf den Fall des Splittings übertragen lassen. Insbesondere liegt dem Splitting nicht die ‚narzisstische‘ Replikation des Genoms einer bereits lebenden Person zugrunde. Die Gleichsetzung mit anderen Formen des reproduktiven Klonens verdeckt darüber hinaus eigene regulative Fragen zum Umgang mit gesplitteten Embryonen, die hinsichtlich einer potenziellen klinischen Anwendung adressiert werden müssen. Schließlich wird deutlich, dass die Maßgabe, Embryonen zu schützen, bisher einseitig als ausreichender Grund für eine restriktive Regulierung des Splittings herangezogen wurde. Diese Argumentation erweist sich in einer pluralistischen Gesellschaft jedoch als unzureichend.

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13

BARTOLINI, LORETTA. "INCOMPRESSIBLE ONE-SIDED SURFACES IN EVEN FILLINGS OF FIGURE 8 KNOT SPACE." Journal of Knot Theory and Its Ramifications 21, no. 07 (April 7, 2012): 1250070. http://dx.doi.org/10.1142/s0218216512500708.

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In the closed, non-Haken, hyperbolic class of examples generated by (2p, q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio [Formula: see text] and determined to be unique in all cases. When applied to one-sided Heegaard splitting, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splitting.

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14

Speth, Raymond L., William H. Green, Shev MacNamara, and Gilbert Strang. "Balanced Splitting and Rebalanced Splitting." SIAM Journal on Numerical Analysis 51, no. 6 (January 2013): 3084–105. http://dx.doi.org/10.1137/120878641.

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15

Sutton, Gregory P. "Splitting hairs about splitting muscles." American Journal of Obstetrics and Gynecology 165, no. 4 (October 1991): 1575. http://dx.doi.org/10.1016/s0002-9378(12)90802-9.

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16

Sutton, Gregory P. "Splitting hairs about splitting muscles." American Journal of Obstetrics and Gynecology 165, no. 5 (November 1991): 1575. http://dx.doi.org/10.1016/0002-9378(91)90408-j.

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17

Du, Kun. "Unstabilized dual Heegaard splittings of 3-manifolds." Journal of Knot Theory and Its Ramifications 25, no. 06 (May 2016): 1650032. http://dx.doi.org/10.1142/s0218216516500322.

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Let [Formula: see text] be a [Formula: see text]-manifold, [Formula: see text] be an essential planar surface which cuts [Formula: see text] into two 3-manifolds [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] [Formula: see text] is a Heegaard splitting of [Formula: see text], [Formula: see text] is the dual Heegaard splitting of [Formula: see text] and [Formula: see text] along [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, we give a condition of unstabilized dual Heegaard splittings of [Formula: see text]-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of [Formula: see text]-manifolds for any Hempel’s distance.

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18

LEE, JUNG HOON. "Parity criterion for unstabilized Heegaard splittings." Mathematical Proceedings of the Cambridge Philosophical Society 149, no. 1 (March 16, 2010): 115–25. http://dx.doi.org/10.1017/s0305004110000095.

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AbstractWe give a parity condition of a Heegaard diagram implying that it is unstabilized. As applications, we show that Heegaard splittings of 2-fold branched coverings of n-component, n-bridge links in S3 are unstabilized, and we also construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.

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19

Soskova, Mariya I., and S. Barry Cooper. "How enumeration reductibility yields extended Harrington non-splitting." Journal of Symbolic Logic 73, no. 2 (June 2008): 634–55. http://dx.doi.org/10.2178/jsl/1208359064.

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§1. Introduction. Sacks [16] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [11] proved the existence of a c.e. a < 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [10] showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′.Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson [15] showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov [1] showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable). Cooper [4] and Yates (see Miller [13]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [5], and Slaman and Steel [17] (see also Downey [9]), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails.We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.

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20

Lai, Junjiang, and Zhencheng Fan. "Stability for discrete time waveform relaxation methods based on Euler schemes." AIMS Mathematics 8, no. 10 (2023): 23713–33. http://dx.doi.org/10.3934/math.20231206.

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<abstract><p>Stability properties of discrete time waveform relaxation (DWR) methods based on Euler schemes are analyzed by applying them to two dissipative systems. Some sufficient conditions for stability of the considered methods are obtained; at the same time two examples of instability are given. To investigate the influence of the splitting functions and underlying numerical methods on stability of DWR methods, DWR methods based on different splittings and different numerical schemes are considered. The obtained results show that the stabilities of waveform relaxation methods based on an implicit Euler scheme are better than those based on explicit Euler scheme, and that the stabilities of waveform relaxation methods based on the classical splittings such as Gauss-Jacobi and Gauss-Seidel splittings are worse than those based on the eigenvalue splitting presented in this paper. Finally, numerical examples that confirm the theoretical results are presented.</p></abstract>

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21

Kracht, Marcus. "Splittings and the finite model property." Journal of Symbolic Logic 58, no. 1 (March 1993): 139–57. http://dx.doi.org/10.2307/2275330.

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AbstractAn old conjecture of modal logics states that every splitting of the major systems K4, S4, G and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ/f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive.

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22

Ley, R. "Splitting." Academic Psychiatry 34, no. 1 (January 1, 2010): 70. http://dx.doi.org/10.1176/appi.ap.34.1.70.

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23

Fuad, Tracy. "Splitting." Pleiades: Literature in Context 40, no. 1 (2020): 121–22. http://dx.doi.org/10.1353/plc.2020.0024.

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24

Isaacs, David, and Anne Preisz. "Splitting." Journal of Paediatrics and Child Health 55, no. 10 (October 2019): 1159–60. http://dx.doi.org/10.1111/jpc.14532.

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25

Layton, Lynne. "Splitting." Psychoanalytic Dialogues 34, no. 2 (March 3, 2024): 149–50. http://dx.doi.org/10.1080/10481885.2024.2325912.

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26

Yeston, Jake. "Virtues of splitting up water-splitting." Science 356, no. 6336 (April 27, 2017): 393.7–394. http://dx.doi.org/10.1126/science.356.6336.393-g.

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27

Carr, Emily R. "Retinoschisis: splitting hairs on retinal splitting." Clinical and Experimental Optometry 103, no. 5 (October 29, 2019): 583–89. http://dx.doi.org/10.1111/cxo.12977.

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28

GOURMELON, NIKOLAZ. "Adapted metrics for dominated splittings." Ergodic Theory and Dynamical Systems 27, no. 6 (December 2007): 1839–49. http://dx.doi.org/10.1017/s0143385707000272.

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AbstractA Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

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29

Ching, Terence H. W., Lena Jelinek, Marit Hauschildt, and Monnica T. Williams. "Association Splitting for Obsessive-Compulsive Disorder: A Systematic Review." Current Psychiatry Research and Reviews 15, no. 4 (January 15, 2020): 248–60. http://dx.doi.org/10.2174/2352096512666190912143311.

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Background: Association splitting is a cognitive technique that targets obsessions in obsessive-compulsive disorder (OCD) by weakening biased semantic associations among OCDrelevant concepts. Objective: In this systematic review, we examine studies on the efficacy of association splitting for reducing OCD symptoms. Methods: Following PRISMA guidelines, six studies were included, with diversity in sample characteristics, mode of administration (i.e., self-help vs therapist-assisted), language of administration, comparator groups, etc. Results: Results indicated that association splitting, as a self-help intervention, was efficacious in reducing overall OCD symptom severity, specific OCD symptoms (i.e., sexual obsessions), subclinical unwanted intrusions, and thought suppression, with small-to-large effect sizes (e.g., across relevant studies, ds = .28-1.07). Findings were less clear when association splitting was administered on a therapist-assisted basis as an add-on to standard cognitive-behavior therapy (CBT). Nonetheless, across studies, the majority of participants reported high acceptability, ease of comprehension, and adherence to daily association splitting practice. Conclusion: Although association splitting is an efficacious and acceptable self-help intervention for OCD symptoms, future studies should include appropriate comparison groups, conduct longitudinal assessments, examine efficacy for different symptom dimensions, and assess changes in semantic networks as proof of mechanistic change. There should also be greater representation of marginalized groups in future studies to assess association splitting’s utility in circumventing barriers to face-to-face CBT. Ethical considerations are also discussed.

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30

Zhou, Naying, Hongxing Zhang, Wenfang Liu, and Xin Wu. "A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes." Astrophysical Journal 927, no. 2 (March 1, 2022): 160. http://dx.doi.org/10.3847/1538-4357/ac497f.

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Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

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31

Ei, Shin-ichiro, Yasumasa Nishiura, and Kei-ichi Ueda. "2n-splitting or edge-splitting? — A manner of splitting in dissipative systems —." Japan Journal of Industrial and Applied Mathematics 18, no. 2 (June 2001): 181–205. http://dx.doi.org/10.1007/bf03168570.

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32

Rybkina, Anna A., Alevtina A. Gogina, Artem V. Tarasov, Ye Xin, Vladimir Yu Voroshnin, Dmitrii A. Pudikov, Ilya I. Klimovskikh, et al. "Origin of Giant Rashba Effect in Graphene on Pt/SiC." Symmetry 15, no. 11 (November 12, 2023): 2052. http://dx.doi.org/10.3390/sym15112052.

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Intercalation of noble metals can produce giant Rashba-type spin–orbit splittings in graphene. The spin–orbit splitting of more than 100 meV has yet to be achieved in graphene on metal or semiconductor substrates. Here, we report the p-type graphene obtained by Pt intercalation of zero-layer graphene on SiC substrate. The spin splitting of ∼200 meV was observed at a wide range of binding energies. Comparing the results of theoretical studies of different models with the experimental ones measured by spin-ARPES, XPS and STM methods, we concluded that inducing giant spin–orbit splitting requires not only a relatively close distance between graphene and Pt layer but also the presence of graphene corrugation caused by a non-flat Pt layer. This makes it possible to find a compromise between strong hybridization and increased spin–orbit interaction. In our case, the Pt submonolayer possesses nanometer-scale lateral ordering under graphene.

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33

Bernášek, Karel, Marián Grocký, Martin Burian, and Jan Lang. "Stretched Gelatin Phantom for Detection of Residual Dipolar Couplings in MR Spectra and Data Analysis of Carnosine." Journal of Spectroscopy 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/4596542.

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Peak splitting due to the residual dipolar coupling (RDC) represents a potentially applicable spectral parameter for diagnostic purposes. Several of the skeletal muscle metabolites were previously reported to display the RDC splitting inin vivoMR spectra. We constructed anin vitromodel consisting of mechanically stretched gelatin cylinder soaked with the muscle metabolite carnosine. We describe the preparation procedure of an upscaled 50 mL stretched gelatin sample with carnosine that can be used as a phantom for setting-up and testing of spectroscopic measurements of RDC in a MR scanner. We also report on analysis of the RDC splittings in1H and13C high resolution MR spectra of carnosine.

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34

Amiranashvili, Shalva, and Raimondas Čiegis. "STABILITY OF THE HIGHER-ORDER SPLITTING METHODS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH AN ARBITRARY DISPERSION OPERATOR." Mathematical Modelling and Analysis 29, no. 3 (June 27, 2024): 560–74. http://dx.doi.org/10.3846/mma.2024.20905.

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The numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step.

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35

Li, Zhe, Zhesheng MA, Ming Xiong, Nicheng Shi, Danian Ye, and Alexandr Khomyakov. "Quadrupole splitting distributions in triclinic astrophyllite." European Journal of Mineralogy 15, no. 4 (July 28, 2003): 707–10. http://dx.doi.org/10.1127/0935-1221/2003/0015-0707.

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36

Biswas, Indranil. "Connections on a Parabolic Principal Bundle, II." Canadian Mathematical Bulletin 52, no. 2 (June 1, 2009): 175–85. http://dx.doi.org/10.4153/cmb-2009-020-2.

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AbstractIn Connections on a parabolic principal bundle over a curve, I we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

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Schlegel, M., O. Knoth, M. Arnold, and R. Wolke. "Implementation of splitting methods for air pollution modeling." Geoscientific Model Development Discussions 4, no. 4 (November 15, 2011): 2937–72. http://dx.doi.org/10.5194/gmdd-4-2937-2011.

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Abstract. Explicit time integration methods are characterized by a small numerical effort per time step. In the application to multiscale problems in atmospheric modeling, this benefit is often more than compensated by stability problems and step size restrictions resulting from stiff chemical reaction terms and from a locally varying Courant-Friedrichs-Lewy (CFL) condition for the advection terms. Splitting methods may be applied to efficiently combine implicit and explicit methods (IMEX splitting). Complementarily multirate time integration schemes allow for a local adaptation of the time step size to the grid size. In combination these approaches lead to schemes which are efficient in terms of evaluations of the right hand side. Special challenges arise when these methods are to be implemented. For an efficient implementation it is crucial to locate and exploit redundancies. Furthermore the more complex program flow may lead to computational overhead which in the worst case more than compensates the theoretical gain in efficiency. We present a general splitting approach which allows both for IMEX splittings and for local time step adaptation. The main focus is on an efficient implementation of this approach for parallel computation on computer clusters.

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Marriott, Jennifer L., and Roger L. Nation. "Splitting tablets." Australian Prescriber 25, no. 6 (December 1, 2002): 133–35. http://dx.doi.org/10.18773/austprescr.2002.131.

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Lerner, Jeff, Jim Siderov, JL Marriott, RL Nation, and G. Sivagnanam. "Splitting tablets." Australian Prescriber 26, no. 2 (April 1, 2003): 27–29. http://dx.doi.org/10.18773/austprescr.2003.019.

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Crossland, Richard. "Splitting Up." Modern Believing 49, no. 2 (April 2008): 21–25. http://dx.doi.org/10.3828/mb.49.2.21.

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Higuchi, Russell. "Splitting Hairs?" Science News 133, no. 23 (June 4, 1988): 355. http://dx.doi.org/10.2307/3972243.

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Schwarzweller, Christoph. "Splitting Fields." Formalized Mathematics 29, no. 3 (September 1, 2021): 129–39. http://dx.doi.org/10.2478/forma-2021-0013.

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Summary. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial p ∈ F [X] as the smallest field extension of F, in which p splits into linear factors. From this follows, that for a splitting field E of p we have E = F (A) where A is the set of p’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to F -isomorphims i.e. isomorphisms i with i|F = Id F . We prove that two splitting fields of p ∈ F [X] are F -isomorphic using the well-known technique [4], [3] of extending isomorphisms from F 1 → F 2 to F 1(a) → F 2(b) for a and b being algebraic over F 1 and F 2, respectively.

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Durie, Robin. "Splitting Time." Philosophy Today 44, no. 2 (2000): 152–68. http://dx.doi.org/10.5840/philtoday200044244.

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Marshall, K. F. "Fee-splitting." British Dental Journal 201, no. 7 (October 2006): 417–19. http://dx.doi.org/10.1038/sj.bdj.4814130.

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Rusk, Nicole. "Splitting Cas9." Nature Methods 12, no. 5 (April 29, 2015): 384–85. http://dx.doi.org/10.1038/nmeth.3387.

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Sterns, Pat. "SPLITTING CARES." Nursing 15, no. 8 (August 1985): 6. http://dx.doi.org/10.1097/00152193-198508000-00005.

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Picard, Richard R., and Kenneth N. Berk. "Data Splitting." American Statistician 44, no. 2 (May 1990): 140. http://dx.doi.org/10.2307/2684155.

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Viner, R. "Splitting hairs." Archives of Disease in Childhood 86, no. 1 (January 1, 2002): 8–10. http://dx.doi.org/10.1136/adc.86.1.8.

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Anderson, Michael T. "splitting theorem." Duke Mathematical Journal 68, no. 1 (October 1992): 67–82. http://dx.doi.org/10.1215/s0012-7094-92-06803-7.

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Morrison, Tony, and Jacquie Brown. "Splitting Siblings." Adoption & Fostering 10, no. 4 (December 1986): 47–51. http://dx.doi.org/10.1177/030857598601000412.

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