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Journal articles on the topic 'Sylvester's theorem'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Chen, Tungyang. "A New Look at Sylvester's Theorem in Matrix Theory." Journal of Mechanics 14, no. 4 (1998): 209–15. http://dx.doi.org/10.1017/s172771910000023x.

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ABSTRACTBy diagonalizing a matrix via a similarity transformation, we provide a new and direct proof of Sylvester's theorem in matrix theory. Several known theorems are reconstructed. In some places we offer new connections which are unnoticed in the literature before.

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2

Wan, Zhe-Xian. "A Generalization of Witt's Theorem and Sylvester's Law of Nullity." Algebra Colloquium 15, no. 02 (2008): 181–84. http://dx.doi.org/10.1142/s1005386708000175.

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3

Kraft, Jürgen. "Singularity of Monomial Curves in A3 and Gorenstein Monomial Curves in A4." Canadian Journal of Mathematics 37, no. 5 (1985): 872–92. http://dx.doi.org/10.4153/cjm-1985-047-8.

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Let 2 ≦ s ∊ N and {n1, …, ns) ⊆ N*. In 1884, J. Sylvester [13] published the following well-known result on the singularity degree S of the monomial curve whose corresponding semigroup is S: = 〈n1, …, ns): If s = 2, thenLet K: = –Z\S andfor all 1 ≦ i ≦ s. We introduce the invariantof S involving a correction term to the Milnor number 2δ [4] of S. As a modified version and extension of Sylvester's result to all monomial space curves, we prove the following theorem: If s = 3, thenWe prove similar formulas for s = 4 if S is symmetric.

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4

Burm, Jacqueline, and Paul Fishback. "Period-3 Orbits via Sylvester's Theorem and Resultants." Mathematics Magazine 74, no. 1 (2001): 47. http://dx.doi.org/10.2307/2691153.

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5

Burm, Jacqueline, and Paul Fishback. "Period-3 Orbits via Sylvester's Theorem and Resultants." Mathematics Magazine 74, no. 1 (2001): 47–51. http://dx.doi.org/10.1080/0025570x.2001.11953032.

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6

Perov, A. I., and I. D. Kostrub. "The Vandermonde matrix in the commutative case." Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ 517, no. 1 (2024): 33–37. http://dx.doi.org/10.31857/s2686954324030057.

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In complex Banach algebra, under the condition of separateness and spectral separateness, the conditions for the reversibility of the Vandermonde matrix are formulated and proved. The necessary and sufficient signs of reversibility of the Vandermonde matrix are given. Analogs of Sylvester's theorem are formulated.

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7

Laishram, Shanta, Sudhir Singh Ngairangbam, and Ranjit Singh Maibam. "Yet another generalization of Sylvester's theorem and its application." Publicationes Mathematicae Debrecen 95, no. 1-2 (2019): 1–17. http://dx.doi.org/10.5486/pmd.2019.8217.

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8

Tuan, Tran Thanh, and Tran Ngoc Trung. "The dispersion of Rayleigh waves in orthotropic layered half-space using matrix method." Vietnam Journal of Mechanics 38, no. 1 (2016): 27–38. http://dx.doi.org/10.15625/0866-7136/38/1/6191.

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In this paper, the secular equation of Rayleigh surface waves propagating in an orthotropic layered half-space is derived by the matrix method. All the layers and the half-space are assumed to have identical principle axes. The explicit form of the matrizant for each layer is obtained by the Sylvester's theorem. The derived secular equation takes only real values and depends only on the dimensionless variables and dimensionless material parameters. Hence, it is convenient in numerical calculation.

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9

Dennis, M. R. "Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere." Journal of Physics A: Mathematical and General 37, no. 40 (2004): 9487–500. http://dx.doi.org/10.1088/0305-4470/37/40/011.

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10

Akritas, Alkiviadis, Gennadi Malaschonok, and Panagiotis Vigklas. "Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences." Serdica Journal of Computing 8, no. 1 (2015): 29–46. http://dx.doi.org/10.55630/sjc.2014.8.29-46.

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In 1971 using pseudo-divisions - that is, by working in Z[x] - Brown and Traub computed Euclid’s polynomial remainder sequences (prs’s) and (proper) subresultant prs’s using sylvester1, the most widely known form of Sylvester’s matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little known form of Sylvester’s matrix of 1853 to initially compute Sturm sequences in Z[x] without pseudodivisions - that is, by working in Q[x]. We then extend our work in Q[x] and, despite

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11

Cai, Jianping, Xiaofeng Wu, and Shuhui Chen. "Chaos Synchronization Criteria and Costs of Sinusoidally Coupled Horizontal Platform Systems." Mathematical Problems in Engineering 2007 (2007): 1–10. http://dx.doi.org/10.1155/2007/86852.

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Some algebraic sufficient criteria for synchronizing two horizontal platform systems coupled by sinusoidal state error feedback control are derived by the Lyapunov stability theorem for linear time-varying system and Sylvester's criterion. The state variables are restricted in a subregion in order to obtain easily verified criteria. The validity of these algebraic criteria is illustrated with some numerical examples. A new concept, synchronization cost, is introduced based on a measure of the magnitude of the feedback control. The minimal synchronization cost as well as optimal coupling streng

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12

Mccall, Martin W. "On vacuum negative refraction, the effective medium and Sylvester's Inertia Theorem: the resolution of a paradox." Journal of Modern Optics 55, no. 2 (2008): 333–40. http://dx.doi.org/10.1080/09500340701446987.

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13

McCall, Martin W. "On vacuum negative refraction, the effective medium and Sylvester's Inertia Theorem: the resolution of a paradox." Journal of Modern Optics 55, no. 6 (2008): 1023. http://dx.doi.org/10.1080/09500340802048146.

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14

Shiue, Peter J. S., Anthony G. Shannon, Shen C. Huang, Michael R. Schwob, and Rama Venkat. "Algorithms for representing positive odd integers as the sum of arithmetic progressions." Notes on Number Theory and Discrete Mathematics 30, no. 4 (2024): 665–80. http://dx.doi.org/10.7546/nntdm.2024.30.4.665-680.

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This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al.

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15

Stetsyuk, Petro, Olha Khomiak, and Oleksander Davydov. "Using the Ellipsoid Method for Sylvester's Problem and its Generalization." Cybernetics and Computer Technologies, no. 1 (March 29, 2024): 27–46. http://dx.doi.org/10.34229/2707-451x.24.1.3.

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Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The general

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16

Chen, Xiaomin. "The Sylvester-Chvatal Theorem." Discrete & Computational Geometry 35, no. 2 (2005): 193–99. http://dx.doi.org/10.1007/s00454-005-1216-9.

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17

Yakoubsohn, Jean-Claude. "On Newton's rule and Sylvester's theorems." Journal of Pure and Applied Algebra 65, no. 3 (1990): 293–309. http://dx.doi.org/10.1016/0022-4049(90)90108-t.

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18

Pretorius, L. M., and K. J. Swanepoel. "A generalised Sylvester-Gallai Theorem." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 26, no. 1 (2007): 8–13. http://dx.doi.org/10.4102/satnt.v26i1.118.

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We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:Let S be a ﬁnite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia) the colour different from k and is collinear with A and B. Then all the points in S are collinear.This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.

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19

Braichev, G. G. "The Sylvester problem and uniqueness sets in classes of entire functions." Contemporary Mathematics. Fundamental Directions 70, no. 1 (2024): 25–37. http://dx.doi.org/10.22363/2413-3639-2024-70-1-25-37.

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In this paper, we study the problem of finding, by a chosen sequence of complex numbers tending to infinity, the widest possible class of entire functions in a given scale for which this sequence is a uniqueness set. Within the framework of this general problem, we establish uniqueness theorems in various classes of entire functions, distinguished by restrictions on the type and indicator under a refined order. In particular, we complement the previously proven uniqueness theorem, using the concept of the Sylvester circle of the indicator diagram of an entire function of exponential type. We d

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20

LIN, MINGHUA, and HARALD K. WIMMER. "THE GENERALIZED SYLVESTER MATRIX EQUATION, RANK MINIMIZATION AND ROTH’S EQUIVALENCE THEOREM." Bulletin of the Australian Mathematical Society 84, no. 3 (2011): 441–43. http://dx.doi.org/10.1017/s0004972711002334.

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21

Özmen, Nejla. "Bilateral and Bilinear generating functions for the Modified Generalized Sylvester polynomials." Facta Universitatis, Series: Mathematics and Informatics 33, no. 2 (2018): 279. http://dx.doi.org/10.22190/fumi1802279o.

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The present study deals with some new properties for the Modified generalized Sylvester polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties and also some special cases for these polynomials. In addition, we derive a theorem giving certain families of bilateral generating functions for the modified generalized Sylvester polynomials and the generalized Lauricella functions. Finally, we get several interesting results of this theorem.

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22

Alladi, Krishnaswami. "A multi-dimensional extension of Sylvester’s identity." International Journal of Number Theory 13, no. 10 (2017): 2487–504. http://dx.doi.org/10.1142/s179304211750138x.

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We obtain by combinatorial means, a multi-dimensional extension of Sylvester’s famous identity of 1882 that generalized Euler’s Pentagonal Numbers Theorem. We also provide a purely [Formula: see text]-hypergeometric proof.

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23

Munagi, Augustine O., and Francisco Javier de de Vega. "An Extension of Sylvester’s Theorem on Arithmetic Progressions." Symmetry 15, no. 6 (2023): 1276. http://dx.doi.org/10.3390/sym15061276.

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Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts f

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24

Barak, B., Z. Dvir, A. Wigderson, and A. Yehudayoff. "Fractional Sylvester-Gallai theorems." Proceedings of the National Academy of Sciences 110, no. 48 (2012): 19213–19. http://dx.doi.org/10.1073/pnas.1203737109.

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25

Wiseman, James A., and Paul R. Wilson. "A sylvester theorem for conic sections." Discrete & Computational Geometry 3, no. 4 (1988): 295–305. http://dx.doi.org/10.1007/bf02187914.

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26

Chv�tal, Vasek. "Sylvester?Gallai Theorem and Metric Betweenness." Discrete and Computational Geometry 31, no. 2 (2004): 175–95. http://dx.doi.org/10.1007/s00454-003-0795-6.

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27

ROWELL, MICHAEL. "A NEW EXPLORATION OF THE LEBESGUE IDENTITY." International Journal of Number Theory 06, no. 04 (2010): 785–98. http://dx.doi.org/10.1142/s1793042110003204.

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We introduce a new combinatorial proof of the Lebesgue identity which allows us to find a new finite form of the identity. Using this new finite form we are able to make new observations about special cases of the Lebesgue identity, namely the "little" Göllnitz theorems and Sylvester's identity.

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28

CH. Harisha. "Periodic Solution for Pair of Matrix Sylvester Volterra Integro-Dynamic System on Time Scales." Advances in Nonlinear Variational Inequalities 28, no. 4s (2025): 105–15. https://doi.org/10.52783/anvi.v28.3228.

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In this paper, we study the periodic solution for a pair of matrix Sylvester Volterra integro-dynamic (VID) systems on time scales. We use the vectorization operator to transform the matrix Sylvester VID into the Kronecker product VID system on time scales. This approach allows us to demonstrate the conditions under which periodic solutions exist for these complex systems. By leveraging the Banach fixed point theorem, we can effectively address the intricacies involved in the periodicity of solutions within the context of time scales. We can establish new results regarding the existence of per

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29

Tran, Quang Hung. "109.30 A generalisation of Sylvester’s theorem with an application." Mathematical Gazette 109, no. 575 (2025): 347–51. https://doi.org/10.1017/mag.2025.10096.

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30

Villiers, Michael De. "Generalising a problem of Sylvester." Mathematical Gazette 96, no. 535 (2012): 78–81. http://dx.doi.org/10.1017/s0025557200003995.

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The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. Much to his surprise, however, the author recently came across the following result and theorem by Sylvester (1814-1897) in [1] that involves an interesting application of forces that relate to the Euler line (segment). This result is also mentioned in [2] without proof or reference to Sylvester.

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31

Lin, X. B. "Another Brief Proof of the Sylvester Theorem." American Mathematical Monthly 95, no. 10 (1988): 932. http://dx.doi.org/10.2307/2322387.

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32

Cohen, Alex, and Frank de Zeeuw. "A Sylvester–Gallai theorem for cubic curves." European Journal of Combinatorics 103 (June 2022): 103509. http://dx.doi.org/10.1016/j.ejc.2022.103509.

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33

Lin, X. B. "Another Brief Proof of the Sylvester Theorem." American Mathematical Monthly 95, no. 10 (1988): 932–33. http://dx.doi.org/10.1080/00029890.1988.11972119.

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34

Czapliński, Adam, Marcin Dumnicki, Łucja Farnik, et al. "On the Sylvester–Gallai theorem for conics." Rendiconti del Seminario Matematico della Università di Padova 136 (2016): 191–203. http://dx.doi.org/10.4171/rsmup/136-13.

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35

Pretorius, Lou M., and Konrad J. Swanepoel. "The Sylvester–Gallai theorem, colourings and algebra." Discrete Mathematics 309, no. 2 (2009): 385–99. http://dx.doi.org/10.1016/j.disc.2007.12.027.

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36

Akritas, Alkiviadis, Gennadi Malaschonok, and Panagiotis Vigklas. "On a Theorem by Van Vleck Regarding Sturm Sequences." Serdica Journal of Computing 7, no. 4 (2014): 389–422. http://dx.doi.org/10.55630/sjc.2013.7.389-422.

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In 1900 E. B. Van Vleck proposed a very efficient method tocompute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizingone of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots takeplace. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” inVan Vleck’s theorem, rectified it but did not extend his method, so that italso works in the cases of: (a) complete Sturm sequences with pivot, and (b)incomplete Sturm sequences.Despite its importance, the Pell-Gordon Theorem for polynomials in

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37

Bezdek, A., F. Fodor, and I. Talata. "Sylvester-type theorems for unit circles." Discrete Mathematics 241, no. 1-3 (2001): 97–101. http://dx.doi.org/10.1016/s0012-365x(01)00146-7.

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38

Jakimczuk, Rafael. "Restricted partitions." International Journal of Mathematics and Mathematical Sciences 2004, no. 36 (2004): 1893–96. http://dx.doi.org/10.1155/s0161171204306502.

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We prove a known partitions theorem by Bell in an elementary and constructive way. Our proof yields a simple recursive method to compute the corresponding Sylvester polynomials for the partition. The previous known methods to obtain these polynomials are in general not elementary.

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39

Nilov, F. K., and A. A. Polyanskii. "A Sylvester–Gallai Type Theorem for Abelian Groups." Mathematical Notes 110, no. 1-2 (2021): 110–17. http://dx.doi.org/10.1134/s0001434621070117.

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40

Pambuccian, Victor. "A Reverse Analysis of the Sylvester-Gallai Theorem." Notre Dame Journal of Formal Logic 50, no. 3 (2009): 245–60. http://dx.doi.org/10.1215/00294527-2009-010.

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41

Kupitz, Yaakov S. "On a generalization of the Gallai-Sylvester theorem." Discrete & Computational Geometry 7, no. 1 (1992): 87–103. http://dx.doi.org/10.1007/bf02187827.

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42

Pinchasi. "Gallai—Sylvester Theorem for Pairwise Intersecting Unit Circles." Discrete & Computational Geometry 28, no. 4 (2002): 607–24. http://dx.doi.org/10.1007/s00454-002-2892-3.

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43

Mandelkern, M. "A constructive version of the Sylvester–Gallai theorem." Acta Mathematica Hungarica 150, no. 1 (2016): 121–30. http://dx.doi.org/10.1007/s10474-016-0624-z.

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44

Kharazishvili, Alexander. "On Some Mutual Positions of Hyperplanes in a Finite-Dimensional Affine Space." Georgian Mathematical Journal 13, no. 1 (2006): 101–8. http://dx.doi.org/10.1515/gmj.2006.101.

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Abstract Several combinatorial questions and facts connected with certain types of mutual positions of finitely many hyperplanes in a finite-dimensional affine space are considered. An application of one of such facts to a multi-dimensional version of the well-known Sylvester theorem is presented.

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45

López-Aguayo, Daniel, and Florian Luca. "Sylvester’s Theorem and the Non-Integrality of a Certain Binomial Sum." Fibonacci Quarterly 54, no. 1 (2016): 44–48. http://dx.doi.org/10.1080/00150517.2016.12427837.

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46

Ho, Chungwu, Tian-Xiao He, and Peter J. S. Shiue. "Representations of positive integers as sums of arithmetic progressions, II." Notes on Number Theory and Discrete Mathematics 29, no. 2 (2023): 260–75. http://dx.doi.org/10.7546/nntdm.2023.29.2.260-275.

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As mentioned in the first part of this paper, our paper was motivated by two classical papers on the representations of integers as sums of arithmetic progressions. One of them is a paper by Sir Charles Wheatstone and the other is a paper by James Joseph Sylvester. Part I of the paper, though contained some extensions of Wheatstone’s work, was primarily devoted to extensions of Sylvester’s Theorem. In this part of the paper, we will pay more attention on the problems initiated by of Wheatstone on the representations of powers of integers as sums of arithmetic progressions and the relationships

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47

Saradha, N., T. N. Shorey, and R. Tijdeman. "Some extensions and refinements of a theorem of Sylvester." Acta Arithmetica 102, no. 2 (2002): 167–81. http://dx.doi.org/10.4064/aa102-2-5.

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48

Elkies, Noam, Lou M. Pretorius, and Konrad J. Swanepoel. "Sylvester–Gallai Theorems for Complex Numbers and Quaternions." Discrete & Computational Geometry 35, no. 3 (2006): 361–73. http://dx.doi.org/10.1007/s00454-005-1226-7.

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49

Nathanson, Melvyn B. "106.03 Real-rooted polynomials and a generalised Hermite-Sylvester theorem." Mathematical Gazette 106, no. 565 (2022): 120–24. http://dx.doi.org/10.1017/mag.2022.18.

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50

Yuan, Xiaodan. "ON THE DIOPHANTINE EQUATION." JP Journal of Algebra, Number Theory and Applications 63, no. 5 (2024): 459–80. http://dx.doi.org/10.17654/0972555524028.

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Based on the Fibonacci-Sylvester algorithm, we introduce a new elementary method in terms of the Fibonacci-Sylvester algorithm for studying the Erdos-Straus conjecture (ESC), that is, the case of all positive integer solutions of the Diophantine equation . Here, only elementary methods are used to provide the general solution expressions for its all positive integer solutions. Using this new method, we provide a new proof of the Mordell theorem, which states that has a expression as the sum of three unit fractions for every natural number , except possibly for the numbers of the form with . In

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