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Journal articles on the topic 'Sup-Norm'

Author: Grafiati

Published: 5 April 2025

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1

Grundmeier, Dusty, Lars Simon, and Berit Stensønes. "Sup-norm estimates for $\overline{\partial}$." Pure and Applied Mathematics Quarterly 18, no. 2 (2022): 531–71. http://dx.doi.org/10.4310/pamq.2022.v18.n2.a8.

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2

Temple, Blake. "Sup-norm estimates in Glimm's method." Journal of Differential Equations 83, no. 1 (January 1990): 79–84. http://dx.doi.org/10.1016/0022-0396(90)90069-2.

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3

Young, Robin. "Sup-norm stability for Glimm's scheme." Communications on Pure and Applied Mathematics 46, no. 6 (July 1993): 903–48. http://dx.doi.org/10.1002/cpa.3160460605.

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4

Karafyllis, Iasson, and Miroslav Krstic. "ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 57. http://dx.doi.org/10.1051/cocv/2021053.

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This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.

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5

Brown, Mark. "Weighted sup-norm inequalities and their applications." Communications in Statistics - Theory and Methods 19, no. 11 (January 1990): 4061–81. http://dx.doi.org/10.1080/03610929008830429.

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6

Bertin, Karine, and Vincent Rivoirard. "Maxiset in sup-norm for kernel estimators." TEST 18, no. 3 (April 15, 2008): 475–96. http://dx.doi.org/10.1007/s11749-008-0109-7.

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7

Jorgenson, J., and J. Kramer. "Bounding the sup-norm of automorphic forms." Geometrical and Functional Analysis GAFA 14, no. 6 (December 2004): 1267–77. http://dx.doi.org/10.1007/s00039-004-0491-6.

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8

Blomer, Valentin, and Péter Maga. "The Sup-Norm Problem for PGL(4)." International Mathematics Research Notices 2015, no. 14 (June 19, 2014): 5311–32. http://dx.doi.org/10.1093/imrn/rnu100.

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9

Hinz, Michael. "Sup-norm-closable bilinear forms and Lagrangians." Annali di Matematica Pura ed Applicata (1923 -) 195, no. 4 (June 6, 2015): 1021–54. http://dx.doi.org/10.1007/s10231-015-0503-1.

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10

Nezir, Veysel, and Nizami Mustafa. "c0 can be renormed to have the fixed point property for affine nonexpansive mappings." Filomat 32, no. 16 (2018): 5645–63. http://dx.doi.org/10.2298/fil1816645n.

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P.K. Lin gave the first example of a non-reflexive Banach space (X,||?||) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (l1,||?||1) with the equivalent norm ||?|| given by ||x|| = sup k?N 8k/1+8k ?1,n=k |xn|, for all x = (xn)n?N ? l1. We wonder (c0, ||?||1) analogue of P.K. Lin?s work and we give positive answer if functions are affine nonexpansive. In our work, for x = (?k)k ? c0, we define |||x||| := lim p?? sup ?k?N ?k (?1,j=k |?j|p/2j)1/p where ?k ?k 3, k is strictly increasing with ?k > 2, ?k ? N, then we prove that (c0,|||?|||) has the fixed point property for affine |||?|||-nonexpansive self-mappings. Next, we generalize this result and show that if ?(?) is an equivalent norm to the usual norm on c0 such that lim sup n ?(1/n ?n,m=1 xm + x) = lim sup n ?(1/n ?n,m=1 xm) + ?(x) for every weakly null sequence (xn)n and for all x ? c0, then for every ? > 0, c0 with the norm ||?||? = ?(?)+?|||?||| has the FPP for affine ||?||?-nonexpansive self-mappings.

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11

Moghaddam, Sadaf Fakri, and Alireza Kamel Mirmostafaee. "Numerical Radius of Bounded Operators with ℓ ^p -Norm." Tatra Mountains Mathematical Publications 81, no. 1 (November 1, 2022): 155–64. http://dx.doi.org/10.2478/tmmp-2022-0012.

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Abstract We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓ p -norm, where 1 ≤ p ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓ p -norm, where 1 ≤ p ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓ p -sum of Hilbert spaces where 2 <p < ∞. We also present some applications of our results.

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12

Manucci, Mattia, Jose Vicente Aguado, and Domenico Borzacchiello. "Sparse Data-Driven Quadrature Rules via ℓ^p-Quasi-Norm Minimization." Computational Methods in Applied Mathematics 22, no. 2 (February 15, 2022): 389–411. http://dx.doi.org/10.1515/cmam-2021-0131.

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Abstract This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p {\ell^{p}} -quasi-norm minimization. Compared to ℓ 1 {\ell^{1}} -norm minimization, the choice of 0 < p < 1 {0<p<1} provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the ℓ 1 {\ell^{1}} -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both ℓ 1 {\ell^{1}} -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.

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13

Bang, Sungwan, and Myoungshic Jhun. "Adaptive sup-norm regularized simultaneous multiple quantiles regression." Statistics 48, no. 1 (August 30, 2012): 17–33. http://dx.doi.org/10.1080/02331888.2012.719512.

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14

Boyer, E., L. Lister, and B. Shader. "Sphere-of-influence graphs using the sup-norm." Mathematical and Computer Modelling 32, no. 10 (November 2000): 1071–82. http://dx.doi.org/10.1016/s0895-7177(00)00191-6.

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15

Gaı¨ffas, Stéphane. "Sharp estimation in sup norm with random design." Statistics & Probability Letters 77, no. 8 (April 2007): 782–94. http://dx.doi.org/10.1016/j.spl.2006.11.017.

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16

Konakov, Valentin, and Vladimir Panov. "Sup-norm convergence rates for Lévy density estimation." Extremes 19, no. 3 (March 11, 2016): 371–403. http://dx.doi.org/10.1007/s10687-016-0246-4.

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17

El-Nouty, Charles. "On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors." Journal of Applied Mathematics and Stochastic Analysis 2008 (March 17, 2008): 1–16. http://dx.doi.org/10.1155/2008/160303.

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We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

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18

Li, Jiaojiao, and Meixia Dou. "New Model for L<sup>2</sup> Norm Flow." Journal of Applied Mathematics and Physics 03, no. 07 (2015): 741–45. http://dx.doi.org/10.4236/jamp.2015.37089.

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19

Chen, Qihong. "Minimax control of an elliptic variational bilateral problem." ANZIAM Journal 44, no. 4 (April 2003): 539–59. http://dx.doi.org/10.1017/s1446181100012918.

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AbstractThis paper deals with a minimax control problem for semilinear elliptic variational inequalities associated with bilateral constraints. The control domain is not necessarily convex. The cost functional, which is to be minimised, is the sup norm of some function of the state and the control. The major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation (variational inequality) and the nonsmooth cost functional (the sup norm). In this paper, the existence conditions and the Pontryagin-type necessary conditions for optimal controls are established.

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20

Ashyralyev, Allaberen, Yasar Sozen, and Pavel E. Sobolevskii. "A Note on the Parabolic Differential and Difference Equations." Abstract and Applied Analysis 2007 (2007): 1–16. http://dx.doi.org/10.1155/2007/61659.

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The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(E)=C(ℝ,E)with norm‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder spaceCα(E)=Cα(ℝ,E)with norm‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα),0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.

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21

Galicer, Daniel, Martín Mansilla, and Santiago Muro. "The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials." Mathematische Nachrichten 293, no. 2 (December 11, 2019): 263–83. http://dx.doi.org/10.1002/mana.201800404.

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22

Blomer, Valentin. "EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE." Journal of the Institute of Mathematics of Jussieu 19, no. 2 (April 2, 2018): 581–96. http://dx.doi.org/10.1017/s1474748018000142.

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Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

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23

Harutyunyan, Anahit, and Wolfgang Lusky. "Toeplitz operators on weighted spaces of holomorphic functions." MATHEMATICA SCANDINAVICA 103, no. 1 (September 1, 2008): 40. http://dx.doi.org/10.7146/math.scand.a-15067.

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We define a notion of Toeplitz operator on certain spaces of holomorphic functions on the unit disk and on the complex plane which are endowed with a weighted sup-norm. We establish boundedness and compactness conditions, give norm estimates and characterize the essential spectrum of these operators for many symbols.

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24

Ba, Demba Bocar. "Estimate by the L² Norm of a Parameter Poisson Intensity Discontinuous." Research Journal of Mathematics and Statistics 6, no. 1 (February 25, 2014): 1–5. http://dx.doi.org/10.19026/rjms.6.5805.

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25

Templier, Nicolas. "Hybrid sup-norm bounds for Hecke–Maass cusp forms." Journal of the European Mathematical Society 17, no. 8 (2015): 2069–82. http://dx.doi.org/10.4171/jems/550.

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26

Erkkilä, Paula, and Jari Taskinen. "Sup-norm estimates for Bergman-projections on regulated domains." MATHEMATICA SCANDINAVICA 102, no. 1 (March 1, 2008): 111. http://dx.doi.org/10.7146/math.scand.a-15054.

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We give sufficient and necessary conditions for the boundedness of generalized Bergman projections on the space $L^\infty_v(\Omega)$. The conditions depend on the geometry of the simply connected domain $\Omega \subset \mathsf C$.

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27

Fornaess, John Erik. "Sup-Norm Estimates for $\overline \partial$ in $C^2$." Annals of Mathematics 123, no. 2 (March 1986): 335. http://dx.doi.org/10.2307/1971275.

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28

Cuong, Le. "Interior estimates in sup-norm for generalized potential vectors." Complex Variables and Elliptic Equations 58, no. 6 (June 2013): 813–20. http://dx.doi.org/10.1080/17476933.2011.622043.

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29

Yukich, J. E., M. B. Stinchcombe, and H. White. "Sup-norm approximation bounds for networks through probabilistic methods." IEEE Transactions on Information Theory 41, no. 4 (July 1995): 1021–27. http://dx.doi.org/10.1109/18.391247.

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30

LEMMENS, BAS, and MICHAEL SCHEUTZOW. "On the dynamics of sup-norm non-expansive maps." Ergodic Theory and Dynamical Systems 25, no. 3 (June 2005): 861–71. http://dx.doi.org/10.1017/s0143385704000665.

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31

Huang, Bingrong. "Sup-Norm and Nodal Domains of Dihedral Maass Forms." Communications in Mathematical Physics 371, no. 3 (January 30, 2019): 1261–82. http://dx.doi.org/10.1007/s00220-019-03335-5.

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32

Steiner, Raphael S. "Sup-norm of Hecke–Laplace eigenforms on $$S^3$$." Mathematische Annalen 377, no. 1-2 (May 7, 2020): 543–53. http://dx.doi.org/10.1007/s00208-020-01996-5.

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33

Pomper, Markus. "Types over c(k) spaces." Journal of the Australian Mathematical Society 77, no. 1 (August 2004): 17–28. http://dx.doi.org/10.1017/s1446788700010120.

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AbstractLet K be a compact Hausdorff space and C(K) the Banach space of all real-valued continuous functions on K, with the sup norm. Types over C(K) (in the sense of Krivine and Maurey) are represented here by pairs (l, u) of bounded real-valued functions on K, where l is lower semicontinuous and u is upper semicontinuous, l ≤ u and l(x) = u(x) for every isolated point x of K. For each pair the corresponding type is defined by the equation τ(g) = max{║l + g║∞, ║u + g║∞} for all g ∈ C(K), where ║·║∞ is the sup norm on bounded functions. The correspondence between types and pairs (l, u) is bijective.

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34

Aryasomayajula, Anilatmaja, and Baskar Balasubramanyam. "Estimates of automorphic cusp forms over quaternion algebras." International Journal of Number Theory 14, no. 04 (May 2018): 1143–70. http://dx.doi.org/10.1142/s1793042118500719.

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In this paper, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [A. Aryasomayajula, Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight, Arch. Math. 106(2) (2016) 165–173; J. S. Friedman, J. Jorgenson and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, in Arbeitstagung Bonn 2013, Progress in Mathematics, Vol. 319 (Birkhäuser/Springer, Cham, 2016), pp. 127–154] to higher dimensions.

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35

Blomer, Valentin, Gergely Harcos, Péter Maga, and Djordje Milićević. "The sup-norm problem for GL(2) over number fields." Journal of the European Mathematical Society 22, no. 1 (August 30, 2019): 1–53. http://dx.doi.org/10.4171/jems/916.

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36

Saha, Abhishek. "Hybrid sup-norm bounds for Maass newforms of powerful level." Algebra & Number Theory 11, no. 5 (July 12, 2017): 1009–45. http://dx.doi.org/10.2140/ant.2017.11.1009.

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37

Orasch, Markus, and William Pouliot. "Tabulating weighted sup-norm functionals used in change-point analysis." Journal of Statistical Computation and Simulation 74, no. 4 (April 2004): 249–76. http://dx.doi.org/10.1080/0094965031000148703.

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38

Assing, Edgar. "On sup-norm bounds part II: GL(2) Eisenstein series." Forum Mathematicum 31, no. 4 (July 1, 2019): 971–1006. http://dx.doi.org/10.1515/forum-2018-0014.

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Abstract In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for {\mathrm{GL}_{2}} over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalize those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field {\operatorname{\mathbb{Q}}} had been considered.

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39

Blomer, Valentin, Gergely Harcos, and Péter Maga. "On the global sup-norm of GL(3) cusp forms." Israel Journal of Mathematics 229, no. 1 (January 2019): 357–79. http://dx.doi.org/10.1007/s11856-018-1805-y.

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40

Mhaskar, H. N., and E. B. Saff. "Where does the sup norm of a weighted polynomial live?" Constructive Approximation 1, no. 1 (December 1985): 71–91. http://dx.doi.org/10.1007/bf01890023.

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41

Lemmens, Bas, Michael Scheutzow, and Colin Sparrow. "Transitive actions of finite abelian groups of sup-norm isometries." European Journal of Combinatorics 28, no. 4 (May 2007): 1163–79. http://dx.doi.org/10.1016/j.ejc.2006.02.003.

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42

Nazarova, Larisa. "Climate dynamics of the Beloe Sea catchment area." Environment. Technology. Resources. Proceedings of the International Scientific and Practical Conference 2 (June 17, 2015): 232. http://dx.doi.org/10.17770/etr2015vol2.238.

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The climate of the Beloe Sea catchment area (total size 717.7 km<sup>2</sup>, 714 of which belongs to Russia) is described. The territory is characterized by several geographic zones, thus substantial diversity of climatic conditions is observed. Climate variability in the region was studied using data from the longest available instrumental observations at weather stations (WS) and gauge sites of the Russian Federal Hydrometeorology and Environmental Monitoring Agency located in the study area, covering the period from the beginning of observations at the stations until 2012-2013. The data obtained were statistically processed with due regard to the research tasks. Modern observational data are analyzed to distinguish the changes in the climatic regime of the main parameters, i.e. air temperature, precipitation, sunshine length, etc. Since 1989, the stable increase of mean annual air temperature (1-2<sup>о</sup>C) over a climatic norm is observed. The most intensive warming is typical for the winter months. The analysis of changes in precipitation over the study area demonstrates the stable increase of annual sums, deviation of which from the climatic norm in the first decade of XXI century is about 50-100 mm

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43

Caraballo, Tomás, Marta Herrera-Cobos, and Pedro Marín-Rubio. "Robustness of time-dependent attractors in H¹-norm for nonlocal problems." Discrete & Continuous Dynamical Systems - B 23, no. 3 (2018): 1011–36. http://dx.doi.org/10.3934/dcdsb.2018140.

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44

Li, Xiang, and Xingsong Zhang. "The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group." Journal of Function Spaces 2021 (August 23, 2021): 1–9. http://dx.doi.org/10.1155/2021/7612482.

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In this article, we define a kind of truncated maximal function on the Heisenberg space by M γ c f x = sup 0 < r < γ 1 / m B x , r ∫ B x , r f y d y . The equivalence of operator norm between the Hardy-Littlewood maximal function and the truncated maximal function on the Heisenberg group is obtained. More specifically, when 1 < p < ∞ , the L p norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When p = 1 , we get the equivalence of weak norm L 1 ⟶ L 1 , ∞ and M ̇ 1 , λ ⟶ W ̇ M 1 , λ . Those results are generalization of previous work on Euclid spaces.

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45

Huy, Vu Nhat, and Ngoc Huy Nguyen. "Landau-Kolmogorov and Gagliardo-Nirenberg Inequalities for Differential Operators in Lorentz Spaces." Journal of the Indian Mathematical Society 89, no. 3-4 (August 23, 2022): 317. http://dx.doi.org/10.18311/jims/2022/25986.

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<p>In this paper, we establish some Landau-Kolmogorov inequalities and Gagliardo-Nirenberg inequalities for di?erential operators generated by polynomials. We illustrate the relation between ||P(D)f||<sub>N?</sub> and ||f||N?, ||D<sup>m</sup>(P(D)f)||<sub>N?</sub> as follows</p><p>||P(D)f||<sub>N?</sub> K<sub>1</sub>(E)||f||<sub>N?</sub> + K<sub>2</sub>(E)||D<sup>m</sup>(P(D)f)||<sub>N?</sub></p><p>for all E > 0, where ||.||N? is the norm in Lorentz spaces N?(R). The corresponding inequalities in L<sup>p</sup>(R<sup>n</sup>) is also obtained.</p>

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46

Trigub, Roald. "On the best approximation of non-integer constants by polynomials with integer coefficients." Ukrainian Mathematical Bulletin 20, no. 2 (June 27, 2023): 283–307. http://dx.doi.org/10.37069/1810-3200-2023-20-2-8.

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The exact decrease rate of the best approximations of non-integer numbers by polynomials with integer coefficients of growing degrees is found on a disk in the complex plane, a cube in $\mathbb{R}^{d}$, and a ball in $\mathbb{R}% ^{d}$. The $\sup $-norm is used in the first two cases, and the norm in $% L_{p}$, $1\leq p<\infty $, is applied in the third one. Detailed comments are given (two remarks at the end of the paper).

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47

PINSKY, ROSS G. "REGULARITY PROPERTIES OF THE DONSKER–VARADHAN RATE FUNCTIONAL FOR NON-REVERSIBLE DIFFUSIONS AND RANDOM EVOLUTIONS." Stochastics and Dynamics 07, no. 02 (June 2007): 123–40. http://dx.doi.org/10.1142/s0219493707001998.

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The Donsker–Varadhan rate functional I(μ) for Markov processes has a simple form in the case that the generator of the process is self-adjoint, or equivalently, that the Markov process is reversible. In particular, in the case of a reversible diffusion generated by a second-order elliptic operator L on a compact manifold, this allows one to give a simple necessary and sufficient criterion on the measure μ in order that I(μ) < ∞, and it also shows that I(μ) is continuous as a function of the coefficients of L in the sup-norm topology. In this paper, we first show that the same criterion for finiteness holds for non-reversible diffusions, and then show that I(μ) is locally Lipschitz continuous as a function of the coefficients of the generator L in the sup-norm topology. We then prove similar results in the setting of random evolutions.

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48

Shi, Yang, and Xuehua Yang. "Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation." Electronic Research Archive 32, no. 3 (2024): 1471–97. http://dx.doi.org/10.3934/era.2024068.

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<abstract><p>This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Numerical results verify the theoretical conclusions.</p></abstract>

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49

Mir, Abdullah. "Some inequalities for self-inversive rational functions with prescribed poles." Publications de l'Institut Math?matique (Belgrade) 107, no. 121 (2020): 109–16. http://dx.doi.org/10.2298/pim2021109m.

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We establish some inequalities for self-inversive rational functions with prescribed poles in the sup-norm on the unit circle in the complex plane. Generalizations of polynomial inequalities of Malik and O?Hara and Rodriguez are obtained for such rational functions

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50

Berthet, Ph, and Z. Shi. "Small ball estimates for Brownian motion under a weighted sup-norm." Studia Scientiarum Mathematicarum Hungarica 36, no. 1-2 (June 1, 2000): 275–89. http://dx.doi.org/10.1556/sscmath.36.2000.1-2.17.

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