Selective synthesis of complex aromatic alkenes. (1,4-Cyclohexadienyl)alkenes by tandem arylation-multistep reduction of .alpha.,.beta.,.gamma.,.delta.-unsaturated ketones

1981 ◽  
Vol 46 (21) ◽  
pp. 4139-4142 ◽  
Author(s):  
Judith S. Ryan Zilenovski ◽  
Stan S. Hall
Keyword(s):  
Selective Synthesis ◽  
Gamma Delta ◽  
Unsaturated Ketones ◽  
Alpha Beta

scholarly journals Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abhishek Mishra ◽  
Vishnu Narayan Mishra ◽  
M. Mursaleen
Keyword(s):  
Double Fourier Series ◽  
Degree Of Approximation ◽  
Trigonometric Approximation ◽  
Lipschitz Class ◽  
Gamma Delta ◽  
Approximation Of Functions ◽  
Matrix Summability ◽  
Alpha Beta ◽  
Generalized Lipschitz Class ◽  
Hausdorff Matrix

AbstractIn this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Nickel-catalyzed conjugate addition of alkynyl groups to .alpha.,.beta.-unsaturated ketones

1980 ◽  
Vol 45 (15) ◽  
pp. 3053-3061 ◽  
Author(s):  
Jeffrey Schwartz ◽  
Denise B. Carr ◽  
Robert T. Hansen ◽  
Fabian M. Dayrit
Keyword(s):  
Conjugate Addition ◽  
Unsaturated Ketones ◽  
Alpha Beta
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