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《Indagationes Mathematicae》2022,33(4):801-815
We consider the irreducibility of polynomial where is a negative integer. We observe that the constant term of vanishes if and only if . Therefore we assume that where is a non-negative integer. Let and more general polynomial, let where with are integers such that . Schur was the first to prove the irreducibility of for . It has been proved that is irreducible for . In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either is irreducible or is linear factor times irreducible polynomial. This is a consequence of the estimate whenever has a factor of degree and . This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey. 相似文献
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Padavattan Govindaswamy Patrick J. Carroll Yurij A. Mozharivskyj Mohan Rao Kollipara 《Journal of Chemical Sciences》2006,118(4):319-326
The reaction of [{(η5-C5Me5)M(μ-Cl)Cl}2] {where M = Rh (1), Ir (2)} with functionalized phosphine viz., diphenyl-2-pyridylphosphine (PPh2Py) in dichloromethane solvent yield neutral ϰ1-P-coordinated rhodium and iridium complexes [(η5-C5Me5)RhCl2(κ1-P-PPh2Py)]3 and [(η5-C5Me5) IrCl2(ϰ1-P-PPh2Py)]4. Reaction of complexes 1 and 2 with the ligand PPh2Py in methanol under reflux give bis-substituted complexes such as [(η5-C5Me5)RhCl(ϰ1-P-PPh2Py)2]+
5 and [(η5-C5Me5)IrCl(ϰ1-P-PPh2Py)2]+
6, whereas stirring in methanol at room temperature gives P-, N-chelating complexes of the type [(η5-C5Me5)RhCl(ϰ2-P-N-PPh2Py)]+ 7 and [(η5-C5Me5)IrCl(ϰ2-P-N-PPh2Py)]+
8. Neutral ϰ1-P-coordinated complexes [(η5-C5Me5)RhCl2(ϰ1-P-PPh2Py)]3 and [(η5-C5Me5)IrCl2(ϰ1-P-PPh2Py)]4 easily undergo conversion to the cationic P-, N-chelating complexes [(η5-C5Me5)RhCl(ϰ2-P-N-PPh2Py)]+
7 and [(η5-C5Me5) IrCl(ϰ2-P, N-PPh2Py)]+
8 on stirring in methanol at room temperature. These complexes are characterized by FT-IR and FT-NMR spectroscopy as well as
analytical methods. The molecular structures of the representative complexes [(η5-C5Me5)RhCl2(ϰ1-P-PPh2Py)]3, [(η5-C5Me5)IrCl2(ϰ1-P-PPh2Py)]4 and hexafluorophosphate salt of complex [(η5-C5Me5)IrCl(ϰ2-P-PPh2Py)2]+
6 are established by single-crystal X-ray diffraction methods 相似文献
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M. Georgelin A. Pocheau 《The European Physical Journal B - Condensed Matter and Complex Systems》1998,4(2):169-174
The polycrystalline perovskitelike manganese oxides La1-xAxMnO (A = Na, and K, ) have been fabricated by sol-gel technique. For all the compositions explored in this work, the average manganese oxidation
state is practically constant, at for A = Na, and for A = K, respectively. A close relationship is confirmed to hold between the Curie temperature (Tc) and the bond distance of Mn-O. Results of magnetic measurements show that these materials can be utilized as suitable candidates
for magnetic refrigerants with wide applied temperature span, for their significant entropy change and the easily tuned Curie
temperature.
Received: 12 September 1997 / Revised: 18 December 1997 / Accepted: 21 January 1998 相似文献
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In this paper, the authors studied certain properties of the estimate of Liang and Krishnaiah (1985, J. Multivariate Anal. 16, 162–172) for multivariate binary density. An alternative shrinkage estimate is also obtained. The above results are generalized to general orthonormal systems. 相似文献
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Yoel Shkolnisky Mark Tygert Vladimir Rokhlin 《Applied and Computational Harmonic Analysis》2006,21(3):413-420
Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae. 相似文献
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G. Dattoli 《Journal of Mathematical Analysis and Applications》2002,269(2):716-725
Bilateral generating functions are those involving products of different types of polynomials. We show that operational methods offer a powerful tool to derive these families of generating functions. We study cases relevant to products of Hermite polynomials with Laguerre, Legendre and other polynomials. We also propose further extensions of the method which we develop here. 相似文献