We consider the positive solution of the perturbed simple pendulum problem u′′(r) + N − 1 r u′(r)− g(u(t)) + λ sinu(r) = 0, with 0 < r < R, u′(0) = u(R) = 0. To understand well the shape of the solution uλ when λ 1, we establish the leading and second terms of ‖uλ‖q (1 ≤ q < ∞) with the estimate of third term as λ → ∞. We also obtain the asymptotic formula for uλ(R) as λ→∞.

Theorem 1.1. Suppose u is a Leray-Hopf solution to the Navier-Stokes equation (1.1) with initial data u0 ∈ L(R) and blows up as t → T . Then (1) (T − t) 14‖∇xu(t)‖L2(R3) → 0, as t → T ; (2) (T − t) 1 2‖u(t)‖L∞(R3) → 0, as t → T. Here u : (x, t) ∈ R × (0, T ) → R is called a weak solution of (1.1) if it is a Leray-Hopf solution. Precisely, it satisfies (1) u ∈ L(0, T ;L(R)) ∩ L(0, T ;H(R)), (2) ...

Einstein’s equations are solved for spherically symmetric universes composed of dust with tangential pressure provided by angular momentum, L(R), which differs from shell to shell. The metric is given in terms of the shell label, R, and the proper time, τ , experienced by the dust particles. The general solution contains four arbitrary functions of R M(R), L(R), E(R) and τ0(R). The solution is ...

Consider applying the Conjugate Residual (CR) method, which is a Krylov subspace type iterative solver, to systems of linear equations Ax = b or least squares problems min x∈Rn ‖b −Ax‖2, where A is singular and nonsymmetric. We will show that when R(A)⊥ = kerA, the CR method can be decomposed into the R(A) and kerA components, and the necessary and sufficient condition for the CR method to conv...

in this paper, we investigate the existence of positive solutions for the ellipticequation $delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $omega$ of $r^{n}$, $ngeq2$, with navier boundary conditions. we show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

In this manuscript, we introduce  a new class of fuzzy problems, namely ``fuzzy inclusion linear systems" and   propose a fuzzy solution set for it. Then, we present a theoretical discussion about the relationship between  the fuzzy solution set of a  fuzzy inclusion linear system and the algebraic solution of a fuzzy linear system. New necessary and sufficient conditions are derived for obtain...

Let Fq be a finite field with q elements with prime power q and let r > 1 be an integer with q ≡ 1 (mod r). In this paper, we present a refinement of the Cipolla-Lehmer type algorithm given by H. C. Williams, and subsequently improved by K. S. Williams and K. Hardy. For a given r-th power residue c ∈ Fq where r is an odd prime, the algorithm of H. C. Williams determines a solution of X = c in O...

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given matrices X ∈ Rn×p, B ∈ Rp×p and A0 ∈ Rr×r , find a matrix A ∈ Rn×n such that ‖XTAX − B‖ = min, s. t. A([1, r]) = A0, where A([1, r]) is the r×r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n× n matrix à with Ã([1, r])...

In this paper, a sufficient and necessary condition for the matrix equations B AX = and , D XC = where , , n m n m B A × × ∈ ∈ R R , p n C × ∈R and , p n D × ∈ R to have a common symmetric positive semi-definite solution X is established, and if it exists, a representation of the solution set X S is given. An optimal approximation between a given matrix n n X × ∈ R ~ and the affine subspace X S...

The computation oj quasistalionary nonhnear two-dimensional magnetic jield leadslo thejollowingproblem There is given a bounded domain Q and an open nonempty set R <= Q We are looking jor the magnetic vectorpotentialu(xu x2, t) which satisjies 1) a certain nonhnear parabolic équation and_an initial condition in R , 2) a nonhnear elhptic équation in S ~ O. — R which is the stationary case of the...