Counting and locating the solutions of polynomial systems of maximum likelihood equations, I

Counting and locating the solutions of polynomial systems of maximum likelihood equations, I

Let μ be a p-dimensional vector, and let Σ1 and Σ2 be p × p positive definite covariance matrices. On being given random samples of sizes N1 and N2 from independent multivariate normal populations Np(μ,Σ1) and Np(μ,Σ2), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters μ, Σ1, and Σ2. It is well-known that the likelihood equations...

Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations, Ii: the Behrens-fisher Problem

Let μ be a p-dimensional vector, and let Σ1 and Σ2 be p × p positive definite covariance matrices. On being given random samples of sizes N1 and N2 from independent multivariate normal populations Np(μ,Σ1) and Np(μ,Σ2), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters μ, Σ1, and Σ2. We shall prove that for N1, N2 > p there are, ...

existence and approximate $l^{p}$ and continuous solution of nonlinear integral equations of the hammerstein and volterra types

بسیاری از پدیده ها در جهان ما اساساً غیرخطی هستند، و توسط معادلات غیرخطی ‎‏بیان شد‎‎‏ه اند. از آنجا که ظهور کامپیوترهای رقمی با عملکرد بالا، حل مسایل خطی را آسان تر می کند. با این حال، به طور کلی به دست آوردن جوابهای دقیق از مسایل غیرخطی دشوار است. روش عددی، به طور کلی محاسبه پیچیده مسایل غیرخطی را اداره می کند. با این حال، دادن نقاط به یک منحنی و به دست آوردن منحنی کامل که اغلب پرهزینه و ...

Counting Stable Solutions of Sparse Polynomial Systems

Coupled systems of equations with entire and polynomial functions

We consider the coupled system$F(x,y)=G(x,y)=0$,where$$F(x, y)=bs 0 {m_1}   A_k(y)x^{m_1-k}mbox{ and } G(x, y)=bs 0 {m_2} B_k(y)x^{m_2-k}$$with entire functions $A_k(y), B_k(y)$.We    derive a priory estimates  for the sums of the rootsof the considered system andfor the counting function of  roots.