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Journal articles on the topic 'Fundamental theorem of algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 21 February 2023

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1

Blecher, David P. "On Morita's fundamental theorem for $C^*$-algebras." MATHEMATICA SCANDINAVICA 88, no. 1 (March 1, 2001): 137. http://dx.doi.org/10.7146/math.scand.a-14319.

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We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

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2

Hirschhorn, Michael D. "The Fundamental Theorem of Algebra." College Mathematics Journal 29, no. 4 (September 1998): 276. http://dx.doi.org/10.2307/2687681.

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3

Hirschhorn, Michael D. "The Fundamental Theorem of Algebra." College Mathematics Journal 29, no. 4 (September 1998): 276–77. http://dx.doi.org/10.1080/07468342.1998.11973954.

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4

Perrucci, Daniel, and Marie-Françoise Roy. "Quantitative fundamental theorem of algebra." Quarterly Journal of Mathematics 70, no. 3 (May 15, 2019): 1009–37. http://dx.doi.org/10.1093/qmath/haz008.

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Abstract Using subresultants, we modify a real-algebraic proof due to Eisermann of the fundamental theorem of Algebra (FTA) to obtain the following quantitative information: in order to prove the FTA for polynomials of degree d, the intermediate value theorem (IVT) is required to hold only for real polynomials of degree at most d2. We also explain that the classical proof due to Laplace requires IVT for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

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5

Bowman, Chris, Stephen Doty, and Stuart Martin. "An integral second fundamental theorem of invariant theory for partition algebras." Representation Theory of the American Mathematical Society 26, no. 15 (April 1, 2022): 437–54. http://dx.doi.org/10.1090/ert/593.

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We prove that the kernel of the action of the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.

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6

Bolima, Jethro Elijah, and Katrina Belleza Fuentes. "First and Third Isomorphism Theorems for the Dual B-Algebra." European Journal of Pure and Applied Mathematics 16, no. 1 (January 29, 2023): 577–86. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4675.

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In this paper, some properties of the dual B-homomorphism are provided, along with the natural dual B-homomorphism and the fundamental theorem of dual B-homomorphisms for dual B-algebras. The first and third isomorphism theorems for the dual B algebra are also presented in the paper.

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7

Výborný, Rudolf. "A simple proof of the Fundamental Theorem of Algebra." Mathematica Bohemica 135, no. 1 (2010): 57–61. http://dx.doi.org/10.21136/mb.2010.140682.

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8

Loya, Paul. "Green's Theorem and the Fundamental Theorem of Algebra." American Mathematical Monthly 110, no. 10 (December 2003): 944. http://dx.doi.org/10.2307/3647967.

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9

Loya, Paul. "Green's Theorem and the Fundamental Theorem of Algebra." American Mathematical Monthly 110, no. 10 (December 2003): 944–46. http://dx.doi.org/10.1080/00029890.2003.11920036.

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10

Derksen, Harm. "The Fundamental Theorem of Algebra and Linear Algebra." American Mathematical Monthly 110, no. 7 (August 2003): 620. http://dx.doi.org/10.2307/3647746.

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11

Derksen, Harm. "The Fundamental Theorem of Algebra and Linear Algebra." American Mathematical Monthly 110, no. 7 (August 2003): 620–23. http://dx.doi.org/10.1080/00029890.2003.11920001.

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12

Krantz, Steven. "How Fundamental is the Fundamental Theorem of Algebra?" Mathematics Magazine 93, no. 2 (March 14, 2020): 139–42. http://dx.doi.org/10.1080/0025570x.2020.1704614.

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13

von Sohsten de Medeiros, Airton. "The Fundamental Theorem of Algebra Revisited." American Mathematical Monthly 108, no. 8 (October 2001): 759. http://dx.doi.org/10.2307/2695621.

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14

Vaggione, Diego. "On the Fundamental Theorem of Algebra." Colloquium Mathematicum 73, no. 2 (1997): 193–94. http://dx.doi.org/10.4064/cm-73-2-193-194.

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15

Strang, Gilbert. "The Fundamental Theorem of Linear Algebra." American Mathematical Monthly 100, no. 9 (November 1993): 848. http://dx.doi.org/10.2307/2324660.

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16

Körner, T. W. "On the Fundamental Theorem of Algebra." American Mathematical Monthly 113, no. 4 (April 1, 2006): 347. http://dx.doi.org/10.2307/27641922.

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17

Leung, T. W. "On the ‘Fundamental theorem of Algebra’." International Journal of Mathematical Education in Science and Technology 26, no. 1 (January 1995): 83–88. http://dx.doi.org/10.1080/0020739950260110.

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18

Körner, T. W. "On the Fundamental Theorem of Algebra." American Mathematical Monthly 113, no. 4 (April 2006): 347–48. http://dx.doi.org/10.1080/00029890.2006.11920315.

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19

Armentano, Diego, and Michael Shub. "Smale’s Fundamental Theorem of Algebra Reconsidered." Foundations of Computational Mathematics 14, no. 1 (May 8, 2013): 85–114. http://dx.doi.org/10.1007/s10208-013-9155-y.

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20

Strang, Gilbert. "The Fundamental Theorem of Linear Algebra." American Mathematical Monthly 100, no. 9 (November 1993): 848–55. http://dx.doi.org/10.1080/00029890.1993.11990500.

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21

von Sohsten de Medeiros, Airton. "The Fundamental Theorem of Algebra Revisited." American Mathematical Monthly 108, no. 8 (October 2001): 759–60. http://dx.doi.org/10.1080/00029890.2001.11919809.

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22

Shipman, Joseph. "Improving the Fundamental Theorem of Algebra." Mathematical Intelligencer 29, no. 4 (September 2007): 9–14. http://dx.doi.org/10.1007/bf02986170.

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23

Akritas, A. G., G. I. Malaschonok, and P. S. Vigklas. "The SVD-Fundamental Theorem of Linear Algebra." Nonlinear Analysis: Modelling and Control 11, no. 2 (May 18, 2006): 123–36. http://dx.doi.org/10.15388/na.2006.11.2.14753.

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Given an m × n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and AT; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N(A) is the orthogonal complement of R(AT). These four subspaces tell the whole story of the Linear System Ax = y. So, for example, the absence of N(AT) indicates that a solution always exists, whereas the absence of N(A) indicates that this solution is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In “Classical” Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl’s excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the “SVD-Fundamental Theorem” of Linear Algebra, and apply it in solving systems of linear equations.

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24

Zhang, Wenchao, Roman Yavich, Alexei Belov-Kanel, Farrokh Razavinia, Andrey Elishev, and Jietai Yu. "Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras." Mathematics 10, no. 22 (November 11, 2022): 4214. http://dx.doi.org/10.3390/math10224214.

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This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice Wn-algebras associated with sln, which is by far the simplest known approach concerning constructing such algebras until now.

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25

BASU, SOHAM. "STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING." Bulletin of the Australian Mathematical Society 104, no. 2 (January 18, 2021): 249–55. http://dx.doi.org/10.1017/s0004972720001434.

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AbstractWithout resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

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26

Sen, Anindya. "Fundamental Theorem of Algebra -- Yet Another Proof." American Mathematical Monthly 107, no. 9 (November 2000): 842. http://dx.doi.org/10.2307/2695742.

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27

Gamboa, Ruben, and John Cowles. "The Fundamental Theorem of Algebra in ACL2." Electronic Proceedings in Theoretical Computer Science 280 (October 29, 2018): 98–110. http://dx.doi.org/10.4204/eptcs.280.8.

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28

Dunham, William. "Euler and the Fundamental Theorem of Algebra." College Mathematics Journal 22, no. 4 (September 1991): 282. http://dx.doi.org/10.2307/2686228.

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29

Dunham, William. "Euler and the Fundamental Theorem of Algebra." College Mathematics Journal 22, no. 4 (September 1991): 282–93. http://dx.doi.org/10.1080/07468342.1991.11973397.

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30

Sen, Anindya. "Fundamental Theorem of Algebra—Yet Another Proof." American Mathematical Monthly 107, no. 9 (November 2000): 842–43. http://dx.doi.org/10.1080/00029890.2000.12005281.

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31

Kalantari, Bahman, and Bruce Torrence. "The Fundamental Theorem of Algebra for Artists." Math Horizons 20, no. 4 (April 2013): 26–29. http://dx.doi.org/10.4169/mathhorizons.20.4.26.

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32

Ansari-Piri, E. "A class of factorable topological algebras." Proceedings of the Edinburgh Mathematical Society 33, no. 1 (February 1990): 53–59. http://dx.doi.org/10.1017/s001309150002887x.

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The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

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33

ALIABADI, MOHSEN. "A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA." Bulletin of the Australian Mathematical Society 97, no. 3 (March 28, 2018): 382–85. http://dx.doi.org/10.1017/s0004972718000035.

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The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

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34

Mahmoudi, Mohammad Gholamzadeh. "A proof of the fundamental theorem of algebra using Hurwitz’s theorem." Elemente der Mathematik 76, no. 1 (January 19, 2021): 38–39. http://dx.doi.org/10.4171/em/427.

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35

Velleman, Daniel J. "Another Proof of the Fundamental Theorem of Algebra." Mathematics Magazine 70, no. 3 (June 1, 1997): 216. http://dx.doi.org/10.2307/2691268.

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36

de Sousa, JosÃÂ© Carlos, and Oliveira Santos. "Another Proof of the Fundamental Theorem of Algebra." American Mathematical Monthly 112, no. 1 (January 1, 2005): 76. http://dx.doi.org/10.2307/30037388.

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37

Farzadfard, Hojjat. "The Fundamental Theorem of Algebra Via Newton’s Identities." American Mathematical Monthly 128, no. 10 (October 19, 2021): 942–45. http://dx.doi.org/10.1080/00029890.2021.1977887.

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38

Suzuki, Jeff. "Lagrange's Proof of the Fundamental Theorem of Algebra." American Mathematical Monthly 113, no. 8 (October 1, 2006): 705. http://dx.doi.org/10.2307/27642032.

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39

Cipra, Barry. "A Bicentennial for the Fundamental Theorem of Algebra." Math Horizons 7, no. 2 (November 1, 1999): 5–7. http://dx.doi.org/10.1080/10724117.1999.12088460.

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40

Velleman, Daniel J. "Another Proof of the Fundamental Theorem of Algebra." Mathematics Magazine 70, no. 3 (June 1997): 216–17. http://dx.doi.org/10.1080/0025570x.1997.11996539.

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41

de Sousa Oliveira Santos, José Carlos. "Another Proof of the Fundamental Theorem of Algebra." American Mathematical Monthly 112, no. 1 (January 2005): 76–78. http://dx.doi.org/10.1080/00029890.2005.11920171.

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42

Suzuki, Jeff. "Lagrange's Proof of the Fundamental Theorem of Algebra." American Mathematical Monthly 113, no. 8 (October 2006): 705–14. http://dx.doi.org/10.1080/00029890.2006.11920355.

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43

Chakraborty, Bikash. "Fundamental Theorem of Algebra – A Nevanlinna Theoretic Proof." Resonance 24, no. 2 (February 2019): 239–44. http://dx.doi.org/10.1007/s12045-019-0773-9.

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44

Velleman, Daniel J. "The Fundamental Theorem of Algebra: A Visual Approach." Mathematical Intelligencer 37, no. 4 (November 5, 2015): 12–21. http://dx.doi.org/10.1007/s00283-015-9572-7.

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45

Armentano, Diego, and Michael Shub. "Erratum to: Smale’s Fundamental Theorem of Algebra Reconsidered." Foundations of Computational Mathematics 14, no. 4 (June 14, 2014): 861. http://dx.doi.org/10.1007/s10208-014-9211-2.

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46

Machado, Armando. "Complex eigenvalues before the Fundamental Theorem of Algebra." Linear and Multilinear Algebra 57, no. 6 (September 2009): 561–66. http://dx.doi.org/10.1080/03081080801984103.

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47

de Jong, Theo. "Lagrange Multipliers and the Fundamental Theorem of Algebra." American Mathematical Monthly 116, no. 9 (November 1, 2009): 828–30. http://dx.doi.org/10.4169/000298909x474882.

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48

Baltus, Christopher. "D'Alembert's proof of the fundamental theorem of algebra." Historia Mathematica 31, no. 4 (November 2004): 414–28. http://dx.doi.org/10.1016/j.hm.2003.12.001.

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49

Magyar, Zoltán, and Zoltán Sebestyén. "On the Definition of C*-Algebras II." Canadian Journal of Mathematics 37, no. 4 (August 1, 1985): 664–81. http://dx.doi.org/10.4153/cjm-1985-035-7.

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The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

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50

Nawata, Norio. "Fundamental Group of Simple C*-algebras with Unique Trace III." Canadian Journal of Mathematics 64, no. 3 (June 1, 2012): 573–87. http://dx.doi.org/10.4153/cjm-2011-052-0.

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Abstract We introduce the fundamental group ℱ(A) of a simple σ-inital C*-algebra A with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple C*-algebras with Unique Trace I and II by Nawata andWatatani. Our definition in this paper makes sense for stably projectionless C*-algebras. We show that there exist separable stably projectionless C*-algebras such that their fundamental groups are equal to ℝ×+ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

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