Sign-Changing Solutions and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with a Reaction Term Nonzero at Zero

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, in particular, the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero, and get new multiple solutions and sign-changing solutions theorems. Moreover, the sign-changing solutions change sign exactly once under some conditions.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

Error of One-leg Methods for Singular Perturbation Problems with Delays

This paper is concerned with the error behaviour of one-leg methods applied to some classes of one-parameter multiple stiff singularly perturbed problems with delays. We derive the global error estimates of A-stable one-leg methods with linear interpolation procedure.     

Matrix Product-Form Solution for an LCFS-PR Single-Server Queue with Multiple Arrival Streams Governed by a Markov Chain

This paper considers a stationary single-server queue with multiple arrival streams governed by a Markov chain, where customers are served on an LCFS preemptive-resume basis. Service times of customers from each arrival stream are generally distributed and service time distributions for different arrival streams may be different. Under these assumptions, it is shown that the stationary joint distribution of queue strings representing from which arrival stream each customer in the system arrived and remaining service times of respective customers in the system has a matrix product-form solution, where matrices constituting the solution are given in terms of the infinitesimal generator of a certain Markov chain. Compared with the previous works, the result in this paper is more general in the sense that general service time distributions are allowed, and it has the advantage of computational efficiency. Note also that the result is a natural extension of the classical result for the LCFS-PR M/G/1 queue. Further, utilizing the matrix product-form solution, we derive a new expression of the vector Laplace–Stieltjes transform of the stationary distribution of unfinished work in the work-conserving single-server queue with multiple arrival streams governed by a Markov chain, which is given by the sum of matrix-geometric series.     

Canonical correlation and reduction of multiple time series

It is shown that a degenerate rankd-variate stationary time series can be reduced to a full rank time series of lower dimension via an orthogonal transformationT provided that , the canonical correlation between past and future of the time series is strictly less than one. Procedures for estimation of rank of the multiple time series,T and testing =1 are outlined, the latter is related to testing the unit root hypothesis in ARMA models.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan