Some recent developments on complex multivariate distributions

In this paper, the author gives a review of the literature on complex multivariate distributions. Some new results on these distributions are also given. Finally, the author discusses the applications of the complex multivariate distributions in the area of the inference on multiple time series.     

On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.     

Hybrid numerical methods for periodic initial value problems involving second-order differential equations

Computer simulation of problems in celestial mechanics often leads to the numerical solution of the system of second-order initial value problems with periodic solutions. When conventional methods are applied to obtain the solution, the time increment must be limited to a value of the order of the reciprocal of the frequency of the periodic solution.In this paper hybrid methods of orders four and six which are P-stable are developed. Further, the adaptive hybrid methods of polynomial order four and trigonometric order one have also been discussed. The numerical results for the undamped Duffing equation with a forced harmonic function are listed.     

Inverse linear multistep methods for the numerical solution of initial value problems of second order differential equations

In this paper inverse linear multistep methods for the numerical solution of second order differential equations are presented. Local accuracy and stability of the methods are defined and discussed. The methods are applicable to a class of special second order initial value problems, not explicitly involving the first derivative. The methods are not convergent, but yield good numerical results if applied to problems they are designed for. Numerical results are presented for both the linear and nonlinear initial value problems.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

Bis coumarinyl bis triazolothiadiazinyl ethane derivatives: Synthesis,antiviral activity evaluation,and molecular docking studies

A series of novel 3,3′-(3,3′-(dihydroxy/hydroxyethane-1,2-diyl)bis(7H-[1,2,4]triazolo[3,4-b][1,3,4]thiadiazine-6,3-diyl))bis(2H-chromen-2-ones) were prepared by the condensation of thiocarbohydrazide with tartaric acid or malic acid followed by various 3-(2-bromoacetyl)-2H-chromen-2-ones in two steps with good yields. All the synthesized compounds were characterized by analytical and spectral (IR, ¹H NMR, ¹³C NMR, and mass) data. These synthesized bis(triazolothiadiazinyl coumarin) compounds were evaluated for broad spectrum of antiviral activity. Among all the tested compounds, compound 5f exhibited antiviral activity against H1N1 virus. The molecular docking studies of these compounds against H1N1 neuraminidase enzyme were performed. The binding affinity and binding values were compared with standard drugs.     

Synthesis of Novel 3‐(3‐(5‐Methylisoxazol‐3‐yl)‐7H‐[1,2,4]Triazolo [3,4‐b][1,3,4]Thiadiazin‐6‐yl)‐2H‐Chromen‐2‐Ones

A novel series of coumarin substituted triazolo‐thiadiazine derivatives were designed and synthesized by using 5‐methyl isoxazole‐3‐carboxylic acid ( 1 ), thiocarbohydrazide ( 2 ), and various substituted 3‐(2‐bromo acetyl) coumarins ( 4a , 4b , 4c , 4e , 4d , 4f , 4g , 4h , 4i , 4j ). Fusion of 5‐methyl isoxazole‐3‐carboxylic acid with thiocarbohydrazide resulted in the formation of the intermediate 4‐amino‐5‐(5‐methylisoxazol‐3‐yl)‐4H‐1,2,4‐triazole‐3‐thiol ( 3 ). This intermediate on further reaction with substituted 3‐(2‐bromo acetyl) coumarins under simple reaction conditions formed the title products 3‐(3‐(5‐methylisoxazol‐3‐yl)‐7H‐[1,2,4]triazolo[3,4‐b][1,3,4]thiadiazin‐6‐yl‐2H‐chromen‐2‐ones ( 5a , 5b , 5c , 5d , 5e , 5f , 5g , 5h , 5i , 5j ) in good to excellent yields. All the synthesized compounds were well characterized by physical, analytical, and spectroscopic techniques.     

On the limit of the largest eigenvalue of the large dimensional sample covariance matrix

Summary In this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist.Research sponsored by the Air Force Office of Scientific Research under Contract F49620-C-0008. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereonThe work of this author was done when he was working at the Center for Multivariate Analysis, University of Pittsburgh.