In the present paper sufficient conditions of the existence of integral manifolds for impulsive differential equations are obtained. The investigations are carried on by means of piecewise continuous functions which are analogues of Lyapunov’s functions.

For a kind of singular non-continuous operators, we prove that unbounded continua of the solution set exist. As applications, we give global structure of the solution set to some impulsive singular integrodifferential boundary value problems.

In the present paper sufficient conditions for the existence of affinity integral manifolds of linear singularly perturbed systems of impulsive differential equations are obtained.

Let $R$ be a ring and $d$ a derivation of $R$. We consider the following three conditions: (a) every quasi-prime $d$-ideal of $R$ is prime, (b) any weak associated prime of every $d$-ideal of $R$ is a $d$-ideal and (c) every $d$-prime $d$-ideal of $R$ is prime. In this paper we show that if $R$ is a Laskerian ring, then the two conditions (a) and (b) are equivalent. Furthermore we show that if $R$ is a strongly Laskerian ring, then any $d$-prime $d$-ideal of $R$ is quasi-prime, and then the three conditions (a), (b) and (c) are equivalent.

We consider families of complex cubic fields introduced by Ishida. Using the Voronoi continued fraction expansion, we find all the reduced principal ideals.

The expected relative entropy (or the expected divergence) between finite probability distribution $Q$ on $\{1,2,\dots ,\ell \}$ and its empirical one obtained from the sample of size $n$ drawn from $Q$ is computed and is found to be given asymptotically by $(\ell –1)(\log e)/2n$ which is independent of $Q$. A method to compute the entropy of the binomial distribution more accurately than before is also given.

In this article, we give a new idea to prove analyticity of solutions to analytic nonlinear elliptic equations.

Let $(M_i,g_i)$ be a certain almost Hermitian $2n$-manifold $M_i$ with a Hermitian metric $g_i$ for $i=1,2$, which is more general than an almost $L$ manifold (a Käshlerian manifold is known to be a special almost $L$ manifold). Let $Spe{c}^p(M_i,g_i)$ denote the spectrum of the real Laplacian on $p$-forms on $M_i$. The purpose of this paper is to show that for some special values of $p$ and $n$, if $Spe{c}^p(M_1,g_1)=Spe{c}^p(M_2,g_2)$, then $(M_1,g_1)$ is of constant holomorphic sectional curvature $H_1$ if and only if $(M_2,g_2)$ is of constant holomorphic sectional curvature $H_2$, and $H_2=H_1$. The corresponding results on almost $L$ manifolds were obtained by C. C. Hsiung and C. X. Wu (The spectral geometry of almost $L$ manifolds, Bull. Inst. Math. Acad. Sinica, 23 (1995), 229–241).