A necessary and sufficient condition for the aggregation of linear diophantine equations

Summary The set of the integer values a linear form can take may be called its spectrum. Aggregating two equations means establishing a linear combination of the two left hand side spectra, the coefficients of which are the aggregation weights. If and only if this linear combination does not contain any spectral correlation point other than the right hand side vector, the equation found by aggregation is equivalent to the two given ones.Methods for calculating the spectrum and the spectral correlation set are given.

---

Zusammenfassung Der Wertevorrat einer Diophantischen Linearform möge als deren Spektrum bezeichnet werden. Die Zusammenfassung zweier Diophantischer Gleichungen läuft auf die Bildung einer Linearkombination der Spektren hinaus, die zu den beiden auf der linken Seite stehenden Linearformen gehören. Die Koffizienten der Linearkombination sind gleich den für die Zusammenfassung verwendeten Gewichten. Wenn und nur wenn die erwähnte Linearkombination ausser dem Punkt, dessen Koordinaten die auf der rechten Seite des Gleichungssystems stehenden Zahlen sind, keine anderen Spektral-Korrelationspunkte enthält, ist die durch Zusammenfassung gebildete Gleichung äquivalent dem System der beiden Ausgangsgleichungen.Methoden für die Berechnung von einfachen und korrelierten Spektren werden gegeben.

     

一类四阶奇异边值问题的正解存在的充分必要条件

利用上下解方法和极大值原理给出了一般边界条件下四阶微分方程的奇异迫值问题有C^2[0，1]和C^3[0，1]正解存在的充分必要条件．推广了韦忠礼(1999)的结果。     

A special kind of nonoscillatory second order linear differential equations

In this note the nonoscillatory property ofy +p (x)y = 0, wherep (x) is a periodic pulse function, is considered. Sufficient condition for guaranteeing nonscillation is obtained. We compare the result with obtained by applying Adamov's theorem.     

A note on the existence and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations

This paper is concerned with nonoscillatory solutions of the fourth order quasilinear differential equation

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

A necessary and sufficient condition for input identifiability for linear time-invariant systems

A necessary and sufficient condition for input identifiability for linear autonomous systems is given. The result is based on a finite iterative process and its proof relies on elementary arguments involving matrices, finite dimensional linear spaces, Gronwall’s lemma, and linear differential systems. Our condition is equivalent to the classical condition involving the geometrical concept of controlled invariant [V. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, NJ, 1992, p. 237] and the dimension reduction algorithm that we propose seems to be useful in designing deconvolution methods.     

一类高阶非线性中立型差分方程组非振动解的存在性

研究了一类高阶非线性中立型差分方程组非振动解的存在性.利用Banach空间的压缩映象原理,获得了该方程组存在非振动解的充分条件.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

A necessary and sufficient condition for the global-in-time existence of solutions to stochastic differential and parabolic equations on manifolds

A necessary and sufficient condition of one-sided type for the completeness of a stochastic flow and the corresponding diffusion semigroup on a manifold M is found under the assumption that the space C ₀(M) of functions is invariant. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 69–76, 2007.     

Solution to general fuzzy linear system and its necessary and sufficient condition

This paper investigates general fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matrices of crisp coefficients and x, y fuzzy number vectors. The aim of this paper is twofold. First, by the unique least Euclidean norm solution we solve the systems with non-full rank matrices A, B. Second, we give the new necessary and sufficient condition for a strong fuzzy solution existence. Moreover, some numerical examples are designed.     

A sufficient condition for smoothness of solutions of Navier-Stokes equations

The main theorem in this paper states that if a certan bound is imposed on the associated pressure pertaining to a weak solution of the Navier-Stokes equation then the solution is actually smooth. The proof uses the fact that such a bound implies a bound on the first derivatives of the solution which, in turn, leads to smoothness.     

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u_i)_t=Δu_i+f_i(u₁,…,u_m) with and u_i(x,0)=φ_i(x), i=1,…,m, in a bounded domain Ω⊂Rn. Under suitable assumptions on f_i, he proves that, if φ_i?(1+ε₀)ψ_i in , for some small ε₀>0, then the solutions blow up in a finite time, where ψ_i is a positive solution of Δψ_i+f_i(ψ₁,…,ψ_m)?0, with ψ_i|_∂Di=0 for i=1,…,m. If m=1, the initial value can be negative in a subset of Ω.     

Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

Consider the forced higher-order nonlinear neutral functional differential equation

拟圆的一个充分必要条件

利用拟双曲度量研究了平面拟共形映照中的拟圆，得到了拟圆的一个充分必要条件。