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Journal articles on the topic 'Frobenius's theorem'

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Published: 4 June 2021

Last updated: 16 February 2022

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1

Yarzhemsky, V. G. "Induced Representation Method in the Theory of Electron Structure and Superconductivity." Advances in Mathematical Physics 2019 (April 9, 2019): 1–10. http://dx.doi.org/10.1155/2019/4873914.

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It is shown that the application of theorems of induced representations method, namely, Frobenius reciprocity theorem, transitivity of induction theorem, and Mackey theorem on symmetrized squares, makes simplifying standard techniques in the theory of electron structure and constructing Cooper pair wavefunctions on the basis of one-electron solid-state wavefunctions possible. It is proved that the nodal structure of topological superconductors in the case of multidimensional irreducible representations is defined by additional quantum numbers. The technique is extended on projective representations in the case of nonsymmorphic space groups and examples of applications for topological superconductors UPt3 and Sr2RuO4 are considered.

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2

Gadea, Pedro M., and J. Munoz Masqué. "Fibred Frobenius theorem." Proceedings of the Indian Academy of Sciences - Section A 105, no. 1 (February 1995): 31–32. http://dx.doi.org/10.1007/bf02840587.

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3

Kaplan, Gil. "A note on Frobenius–Wielandt groups." Journal of Group Theory 22, no. 4 (July 1, 2019): 637–45. http://dx.doi.org/10.1515/jgth-2018-0140.

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AbstractLet G be a finite group. G is called a Frobenius–Wielandt group if there exists {H<G} such that {U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H. The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space {G/H}, it states in particular that the subgroup {D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G. Let {W=U^{G}}, the normal closure of U in G. Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality {W\cap H=U}. This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with {H<G}: (a) {W=D} and {G=HW}, or (b) {W<D} and {HW<G}. In the latter case, {G/W} is a Frobenius group with a Frobenius complement {HW/W} and Frobenius kernel {D/W}.

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4

Hauser, J., and Zhigang Xu. "An Approximate Frobenius Theorem †." IFAC Proceedings Volumes 26, no. 2 (July 1993): 157–60. http://dx.doi.org/10.1016/s1474-6670(17)49098-7.

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5

Faran, James J. "A synthetic Frobenius theorem." Journal of Pure and Applied Algebra 128, no. 1 (June 1998): 11–32. http://dx.doi.org/10.1016/s0022-4049(97)00034-0.

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6

DANERS, DANIEL, and JOCHEN GLÜCK. "THE ROLE OF DOMINATION AND SMOOTHING CONDITIONS IN THE THEORY OF EVENTUALLY POSITIVE SEMIGROUPS." Bulletin of the Australian Mathematical Society 96, no. 2 (March 29, 2017): 286–98. http://dx.doi.org/10.1017/s0004972717000260.

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We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

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7

FAITH, CARL. "FACTOR RINGS OF PSEUDO-FROBENIUS RINGS." Journal of Algebra and Its Applications 05, no. 06 (December 2006): 847–54. http://dx.doi.org/10.1142/s0219498806001831.

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If R is right pseudo-Frobenius (= PF), and A is an ideal, when is R/A right PF? Our main result, Theorem 3.7, states that this happens iff the ideal A′ of the basic ring B of R corresponding to A has left annihilator F in B generated by a single element on both sides. Moreover, in this case B/A′ ≈ F in mod-B, (see Theorem 3.5), a property that does not extend to R, that is, in general R/A is not isomorphic to the left annihilator of A. (See Example 4.3(2) and Theorem 4.5.) Theorem 4.6 characterizes Frobenius rings among quasi-Frobenius (QF) rings. As an application of the main theorem, in Theorem 3.9 we prove that if A is generated as a right or left ideal by an idempotent e, then e is central (and R/A is then trivially right PF along with R). This generalizes the result of F. W. Anderson for quasi-Frobenius rings. (See Theorem 2.2 for a new proof.). In Proposition 1.6, we prove that a generalization of this result holds for finite products R of full matrix rings over local rings; namely, an ideal A is finitely generated as a right or left ideal iff A is generated by a central idempotent. We also note a theorem going back to Nakayama, Goursaud, and the author that every factor ring of R is right PF iff R is a uniserial ring. (See Theorem 5.1.).

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8

Gong, Xianghong. "A Frobenius–Nirenberg theorem with parameter." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 759 (February 1, 2020): 101–59. http://dx.doi.org/10.1515/crelle-2017-0051.

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AbstractThe Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

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9

Hill, C., and Santiago Simanca. "The super complex Frobenius theorem." Annales Polonici Mathematici 55, no. 1 (1991): 139–55. http://dx.doi.org/10.4064/ap-55-1-139-155.

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10

Sine, Robert. "A nonlinear Perron-Frobenius theorem." Proceedings of the American Mathematical Society 109, no. 2 (February 1, 1990): 331. http://dx.doi.org/10.1090/s0002-9939-1990-0948156-x.

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11

Evstigneev, Igor V., and Sergey A. Pirogov. "Stochastic nonlinear Perron–Frobenius theorem." Positivity 14, no. 1 (January 8, 2009): 43–57. http://dx.doi.org/10.1007/s11117-008-0003-2.

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12

HAFOUTA, YEOR. "Limit theorems for some skew products with mixing base maps." Ergodic Theory and Dynamical Systems 41, no. 1 (August 5, 2019): 241–71. http://dx.doi.org/10.1017/etds.2019.48.

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We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$ together with a $T$-invariant measure whose base map $\unicode[STIX]{x1D703}$ satisfies certain topological and mixing conditions and the maps $T_{\unicode[STIX]{x1D714}}$ on the fibers are certain non-singular distance-expanding maps. Our results hold true when $\unicode[STIX]{x1D703}$ is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by $T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields162 (2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.

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13

GARCÍA ROZAS, J. R., LUIS OYONARTE, and BLAS TORRECILLAS. "ON HOMOLOGICAL FROBENIUS COMPLEXES AND BIMODULES." Glasgow Mathematical Journal 56, no. 3 (August 22, 2014): 629–42. http://dx.doi.org/10.1017/s0017089514000068.

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AbstractWe introduce the concept of homological Frobenius functors as the natural generalization of Frobenius functors in the setting of triangulated categories, and study their structure in the particular case of the derived categories of those of complexes and modules over a unital associative ring. Tilting complexes (modules) are examples of homological Frobenius complexes (modules). Homological Frobenius functors retain some of the nice properties of Frobenius ones as the ascent theorem for Gorenstein categories. It is shown that homological Frobenius ring homomorphisms are always Frobenius.

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14

Kim, Taekyun, and Dae San Kim. "An identity of symmetry for the degenerate Frobenius-Euler Polynomials." Mathematica Slovaca 68, no. 1 (February 23, 2018): 239–43. http://dx.doi.org/10.1515/ms-2017-0096.

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Abstract In this paper, we prove an identity of symmetry for the higher-order degenerate Frobenius-Euler polynomials and derive the recurrence relations and multiplication theorem type result for the degenerate Frobenius-Euler polynomials.

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15

Kedlaya, Kiran S. "Frobenius modules and de Jong’s theorem." Mathematical Research Letters 12, no. 3 (2005): 303–20. http://dx.doi.org/10.4310/mrl.2005.v12.n3.a3.

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16

Chang, K. C., K. Pearson, and T. Zhang. "Perron-Frobenius theorem for nonnegative tensors." Communications in Mathematical Sciences 6, no. 2 (2008): 507–20. http://dx.doi.org/10.4310/cms.2008.v6.n2.a12.

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17

Bidard, Christian, and Guido Erreygers. "Potron and the Perron–Frobenius Theorem." Economic Systems Research 19, no. 4 (December 2007): 439–52. http://dx.doi.org/10.1080/09535310701698563.

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18

Teichmann, Josef. "A Frobenius Theorem on Convenient Manifolds." Monatshefte f�r Mathematik 134, no. 2 (December 1, 2001): 159–67. http://dx.doi.org/10.1007/s006050170005.

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19

Dietzenbacher, Erik. "The non-linear Perron-Frobenius theorem." Journal of Mathematical Economics 23, no. 1 (January 1994): 21–31. http://dx.doi.org/10.1016/0304-4068(94)90033-7.

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20

Wan, Zheyan, and Yilong Wang. "Classification of Spherical Fusion Categories of Frobenius–Schur Exponent 2." Algebra Colloquium 28, no. 01 (January 20, 2021): 39–50. http://dx.doi.org/10.1142/s1005386721000055.

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In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.

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21

Tateishi, Hiroshi. "Perron-Frobenius theorem for multi-valued mappings." Kodai Mathematical Journal 15, no. 2 (1992): 155–64. http://dx.doi.org/10.2996/kmj/1138039594.

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22

Khurana, Dinesh, and Anjana Khurana. "A Theorem of Frobenius and Its Applications." Mathematics Magazine 78, no. 3 (June 1, 2005): 220. http://dx.doi.org/10.2307/30044160.

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23

Meng, Wei, and Jiangtao Shi. "On an inverse problem to Frobenius’ theorem." Archiv der Mathematik 96, no. 2 (January 14, 2011): 109–14. http://dx.doi.org/10.1007/s00013-010-0211-4.

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24

DHARMADASA, H. KUMUDINI, and WILLIAM MORAN. "A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM." Bulletin of the Australian Mathematical Society 100, no. 2 (February 18, 2019): 317–22. http://dx.doi.org/10.1017/s0004972719000042.

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Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

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25

Kloeden, P. E., and A. M. Rubinov. "A generalization of the Perron–Frobenius theorem." Nonlinear Analysis: Theory, Methods & Applications 41, no. 1-2 (July 2000): 97–115. http://dx.doi.org/10.1016/s0362-546x(98)00267-3.

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26

Khurana, Dinesh, and Anjana Khurana. "A Theorem of Frobenius and Its Applications." Mathematics Magazine 78, no. 3 (June 2005): 220–25. http://dx.doi.org/10.1080/0025570x.2005.11953330.

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27

Han, Chong-Kyu. "GENERALIZATION OF THE FROBENIUS THEOREM ON INVOLUTIVITY." Journal of the Korean Mathematical Society 46, no. 5 (September 1, 2009): 1087–103. http://dx.doi.org/10.4134/jkms.2009.46.5.1087.

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28

Gautier, Antoine, Francesco Tudisco, and Matthias Hein. "The Perron--Frobenius Theorem for Multihomogeneous Mappings." SIAM Journal on Matrix Analysis and Applications 40, no. 3 (January 2019): 1179–205. http://dx.doi.org/10.1137/18m1165037.

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29

Lundell, Albert T. "A short proof of the Frobenius theorem." Proceedings of the American Mathematical Society 116, no. 4 (April 1, 1992): 1131. http://dx.doi.org/10.1090/s0002-9939-1992-1145422-4.

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30

Bolsinov, A. V., and K. M. Zuev. "A formal Frobenius theorem and argument shift." Mathematical Notes 86, no. 1-2 (August 2009): 10–18. http://dx.doi.org/10.1134/s0001434609070025.

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31

Sury, B. "Frobenius and his Density theorem for primes." Resonance 8, no. 12 (December 2003): 33–41. http://dx.doi.org/10.1007/bf02839049.

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32

Cerveau, D., and A. Lins Neto. "Frobenius theorem for foliations on singular varieties." Bulletin of the Brazilian Mathematical Society, New Series 39, no. 3 (September 2008): 447–69. http://dx.doi.org/10.1007/s00574-008-0016-2.

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33

Adachi, Toshiaki, and Toshikazu Sunada. "Twisted Perron-Frobenius theorem and L-functions." Journal of Functional Analysis 71, no. 1 (March 1987): 1–46. http://dx.doi.org/10.1016/0022-1236(87)90014-0.

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34

Bekes, Robert A., and Peter J. Hilton. "Induction and restriction as adjoint functors on representations of locally compact groups." International Journal of Mathematics and Mathematical Sciences 16, no. 1 (1993): 61–66. http://dx.doi.org/10.1155/s0161171293000067.

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35

Chang, K. C. "Nonlinear extensions of the Perron–Frobenius theorem and the Krein–Rutman theorem." Journal of Fixed Point Theory and Applications 15, no. 2 (June 2014): 433–57. http://dx.doi.org/10.1007/s11784-014-0191-2.

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36

Gugnin, D. V. "Polynomially dependent homomorphisms. Uniqueness theorem for Frobeniusn-homomorphisms." Russian Mathematical Surveys 62, no. 5 (October 31, 2007): 993–95. http://dx.doi.org/10.1070/rm2007v062n05abeh004460.

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37

Beasley, Leroy B., Alexander E. Guterman, Sang-Gu Lee, and Seok-Zun Song. "Frobenius and Dieudonné theorems over semirings." Linear and Multilinear Algebra 55, no. 1 (January 2007): 19–34. http://dx.doi.org/10.1080/03081080500361256.

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38

Rampazzo, Franco. "Frobenius-type theorems for Lipschitz distributions." Journal of Differential Equations 243, no. 2 (December 2007): 270–300. http://dx.doi.org/10.1016/j.jde.2007.05.040.

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39

Kurt, Burak. "A note on the Apostol type q-Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems." Filomat 30, no. 1 (2016): 65–72. http://dx.doi.org/10.2298/fil1601065k.

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The main subject of this study is to define and investigate for the Apostol type Frobenius-Euler polynomials. We give some identities for these polynomials. We generalize the Srivastava-Pint?r addition theorems between the Bernoulli polynomials and Apostol type Frobenius-Euler polynomials

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40

Knapp, Wolfgang, and Peter Schmid. "Frobenius groups of low rank." Archiv der Mathematik 117, no. 2 (April 20, 2021): 121–27. http://dx.doi.org/10.1007/s00013-021-01611-2.

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AbstractLet G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

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41

FAITH, CARL, and DINH VAN HUYNH. "WHEN SELF-INJECTIVE RINGS ARE QF: A REPORT ON A PROBLEM." Journal of Algebra and Its Applications 01, no. 01 (March 2002): 75–105. http://dx.doi.org/10.1142/s0219498802000070.

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Theorems of Osofsky and Kato imply that a right and left self-injective one-sided perfect ring is quasi-Frobenius (= QF). The corresponding question for one-sided self-injective one or two-sided perfect rings remains open, even assuming that the ring is semiprimary. The latter version of the problem is known as Faith's Conjecture (FC). We survey results on QF rings, especially those obtained in connection with FC. We also review various results that provide partial answers to another problem of Faith: Is a right FGF ring necessarily QF? On this topic, we provide a new result, namely that if all factor rings of R are right FGF, then R is QF (Theorem 6.1). In Sec. 7 we review results concerning the question of when a D-ring is QF. Sections 8 and 9 are devoted respectively to IF rings, and to Σ-injective rings and Σ-CS rings.

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42

GREFERATH, MARCUS, ALEXANDR NECHAEV, and ROBERT WISBAUER. "FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES." Journal of Algebra and Its Applications 03, no. 03 (September 2004): 247–72. http://dx.doi.org/10.1142/s0219498804000873.

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The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

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43

Melnick, Karin. "A Frobenius theorem for Cartan geometries, with applications." L’Enseignement Mathématique 57, no. 1 (2011): 57–89. http://dx.doi.org/10.4171/lem/57-1-3.

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44

Ding, Jiu, and Temple H. Fay. "The Perron-Frobenius Theorem and Limits in Geometry." American Mathematical Monthly 112, no. 2 (February 1, 2005): 171. http://dx.doi.org/10.2307/30037416.

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45

Chikhladze, Dimitri. "The Tannaka Representation Theorem for Separable Frobenius Functors." Algebras and Representation Theory 15, no. 6 (May 26, 2011): 1205–13. http://dx.doi.org/10.1007/s10468-011-9285-3.

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46

Montanari, Annamaria, and Daniele Morbidelli. "A Frobenius-type theorem for singular Lipschitz distributions." Journal of Mathematical Analysis and Applications 399, no. 2 (March 2013): 692–700. http://dx.doi.org/10.1016/j.jmaa.2012.10.040.

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47

Corrádi, K., and E. Horváth. "Steps towards an elementary proof of frobenius' theorem." Communications in Algebra 24, no. 7 (January 1996): 2285–92. http://dx.doi.org/10.1080/00927879608825700.

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48

Gaubert, Stéphane, and Jeremy Gunawardena. "The Perron-Frobenius theorem for homogeneous, monotone functions." Transactions of the American Mathematical Society 356, no. 12 (March 23, 2004): 4931–50. http://dx.doi.org/10.1090/s0002-9947-04-03470-1.

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49

Hill, C. Denson, and Michael Taylor. "The complex Frobenius theorem for rough involutive structures." Transactions of the American Mathematical Society 359, no. 1 (August 16, 2006): 293–322. http://dx.doi.org/10.1090/s0002-9947-06-04067-0.

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50

Bapat, R. B. "A max version of the Perron-Frobenius theorem." Linear Algebra and its Applications 275-276 (May 1998): 3–18. http://dx.doi.org/10.1016/s0024-3795(97)10057-x.

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