Let $ A,B \in \mathbb N$ with $A\gt 1, B\gt 1$ and $\gcd(A,B)=1$, $k\geq 2$ be an integer coprime with $AB$, and let $\lambda\in\{1,2,4\}$ be such that if $\lambda=4$, then $A\ne 4$ and $B\ne 4$; and if $k$ is even, then $\lambda=4$. In this paper, we shall describe all solutions of the equation $$ AX^2+BY^2=\lambda k^Z,\,\, X,Y,Z\in \mathbb Z,\,\gcd(X,Y)=1,\,\,Z\gt 0 $$ with $X|^*A$ or $Y|^*B$, where the symbol $X|^*A$ means that every prime divisor of $X$ divides $A$. Then, using this result, we give some more general results on the number of solutions of the equation $la^x+mb^y=\lambda c^z$, $x\gt 1$, $y\gt 1$, $z\gt 1$. In addition, using Cao's result on Pell equation, we obtain some im- provement of Terai's results on the equations $a^x+2=c^z, a^x+4=c^z$ and $a^x+2^y=c^z$.

In this paper, we define rectifying curves in Minkowski space-time and characterize null, pseudo null and partially null rectifying curves in terms of their curvatures. Also, we give some explicit equations of null, pseudo null and partially null rectifying curves in $E^4_1$.

We establish local and global existence results for a critical case of Dirac-Klein-Gordon equations in one space dimension, employing a null form estimate, a bilinear estimate and a fixed point argument.

In 1995, S. Mercourakis introduced the concept of uniformly weakly convergent sequences and characterized such sequences as those with the property that any of its subsequences is Ces`aro-summable. In this paper, we present a Tauberian theorem for such kind of convergence. As a consequence, we prove that the uniformly pointwise convergence and the uniform convergence of a sequence of complex-valued functions coincide under a suitable Tauberian condition. This result affirmatively answers a question raised by S. Mercourakis concerning the Fourier series of a continuous function on the circle group T. In this paper, a result of Banach type is also established for uniformly weakly convergent sequences. Our result generalizes the work of Mercourakis.

Petty's conjectured projection inequality is a famous open problem in convex bodies theory. In this paper, it is shown that a $L_p$-version of the Petty's conjectured projection inequality. As its applications, we give a reverse of the Blaschke-Santaló inequality and consider the monotony of volumes for convex body and its $L_p$-Petty projection body, respectively. Otherwise, we also give the reverses of the $L_p$-Petty projection inequality.

In this paper a solution of the following difference equation$\ $was investigated \begin{equation*} x_{n+1}=\frac{x_{n-(5k+9)}}{1+x_{n-4}x_{n-9}...x_{n-(5k+4)}},\ n=0,1,2,... \end{equation*} where $x_{-(5k+9)},x_{-(5k+8)},...,x_{-1},x_{0}\in (0,\infty ).$

The purpose of this paper is to introduce and to discuss the concept of orthogonality in the locally convex spaces, and obtaining some results on orthogonality in locally convex spaces similar to orthogonality of normed spaces. We shall obtain some characterizations of the best approximation and the best coapproximation in locally convex spaces.

Since the investigation of impulsive delay differential equations are inchoate, the literature on delay epidemic models with pulse vaccination is not extensive. In this paper, we propose a delay SIR epidemic model with impulsive effect and analyze the global dynamics behaviors of the model. Using the discrete dynamical system determined by the stroboscopic map, we show that there exists an ’infection-free’ periodic solution in the model. Further, we prove that the ’infection-free’ periodic solution is globally attractive when the period of impulsive effect is less than some critical value. The condition for the permanence of population for epidemic model with pulse vaccination is given, which means the epidemic disease to spread around. We conclude that time delay and pulse bring great effects on the dynamics of the model. It illustrated theoretically that we should look carefully after the infectious patients to shorten the convalescence period (or infections period). Lastly, the biologic discussion and strategy for the elimination of infectious diseases are also given. In this paper, the main feature is that we introduce time delay together with pulse into epidemic model, and we investigate their effects on the dynamics of model.

Let $f_i, i=1,2,\ldots,m $ be transcendental meromorphic functions of order less than $\frac{1}{2}$ with at most finitely many poles and at least one of them has positive lower order. Let $g = f_m \circ f_{m-1}\circ \cdots \circ f_1.$ Then either $g$ has no unbounded Fatou components or at least one unbounded Fatou component $g$ is multiply connected.

We give the sufficient and necessary conditions of Bochner-Orlicz sequence spaces equipped with Orlicz norm that have the $\lambda$ property and uniform $\lambda$ property, respectively. The results show that the $\lambda$ property can not be lifted from $X$ to $l_{_{M}}(X)$.

In the present paper we introduce Poisson type integrals for orthogonal expansions. We first give some direct computations for the moments and compute the rates of convergence by means of the modulus of continuity and the Lipschitz functionals; and also we prove that our results are stronger and more general than the results obtained by Toczek and Wachnicki [J. Approx. Theory 116 (2002), 113-125]. We obtain a statistical approximation theorem by using the concept of T−statistical convergence which is a (non-matrix) summability transformation. Furthermore, we give a general Voronovskaya type theorem for these operators. Finally, introducing a higher order generalization of Poisson integrals we discuss their approximation properties.

It is known that regular vector-fields generate convergent discrete and continuous descent methods for minimizing functions. In this note we present necessary and sufficient conditions for regularity of descent vectorfields associated with convex and Lipschitz functions defined on general Banach spaces.

In this paper, a class of geometric inequalities for the volumes of two $n-$simplexes and their $k-$subsimplexes are established. The results are generalizations to several dimensions of the well-known Neuberg-Pedoe inequality of two triangles.

Let $D$ denote a principle ideal domain with identity element $1$. Fix three elements $a, b, d$ in $D$ with ${\rm gcd}(a, b)=d$. We show there exist two elements $x, y$ in $D$ with either ${\rm gcd}(a, y)=1$ or ${\rm gcd}(b, x)=1$ such that $ax+by=d$. Moreover we show there exist $x, y$ in $D$ such that ${\rm gcd}(a, y)=1,$ ${\rm gcd}(b, x)=1$ and $ax+by=d$ if and only if for each prime divisor $p$ of $d$ with a complete set of residues modulo $p$ containing exactly $2$ elements, the power of $p$ appearing in the factorization of $a$ is different to that of $b$. We apply our results to the study of double-loop networks..

In this paper, the inequality of Hardy-Littlewood-Sobolev type is established for the generalized Riesz potentials generated by the generalized shift operator with the functions in Sobolev spaces $W_{p,v}^{m}(\mathbf{R}_{n}^{+}).$

For any differential system $\vec{V}$ of class $C^1$ on an $n$-dimensional compact, smooth, and boundaryless riemannian manifold $M$, we consider the Liao frame skew-product flow on the reduced orthonormal frame bundle ${\mathcal C}^{\sharp}(M,\vec{V})$ naturally induced by $\vec{V}$ and, using some technical ideas due to S.~Liao, we prove a `reduced reordering' theorem and a `reduced spectrum' theorem. As consequences, we also provide a reordering lemma for the natural skew-product flow $({\mathcal F}(k),\{\mathcal V_t\})$ on the flag bundles ${\mathcal F}(k)$ of the tangent bundle ${TM}$, and give two characteristic spectra for parallelepiped. In addition, we obtain the uniformity of some non-uniformly expanding (resp.~contracting) sets.

The concept of Apollonius circle and Apollonius axes of an allowable triangle in an isotropic plane will be introduced. Some statements about relationships between introduced concepts and some other previously studied geometric concepts about triangle will be investigated in an isotropic plane and some analogies with the Euclidean case will be also considered.

Let $R$ be a ring with unity. For a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$, the system $R[x;\alpha,\delta]$ forms an abelian nearring under addition and substitution operations. In this paper we extend the study of annihilator conditions on nearring of polynomials to skew nearring ($R[x;\alpha,\delta]$, $+$, $\circ$), when $R$ is an $\alpha$-rigid ring. Also, we give a characterization of $\alpha$-rigid rings. An example to show that ``$\alpha$-rigid condition on $R$" is not superfluous is given.

A Lagrangian submanifold in a Kaehler manifold is said to be Hamiltonian-stationary (or simply $H$-stationary) if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In an earlier paper \cite{cg}, $H$-stationary Lagrangian submanifolds of constant curvature in the complex projective 3-space $CP^3$ with positive relative nullity are classified. In this paper we completely classify $H$-stationary Lagrangian submanifolds of constant curvature in the complex hyperbolic 3-space $CH^3$ with positive relative nullity. As an immediate by-product, several explicit new families of $H$-stationary Lagrangian submanifolds in $CH^3$ are obtained.

Let $CC(X)$ be the collection of all non-empty compact, convex subsets of a complex Banach space $X$ endowed with the usual Hausdorff metric $h$. We shall define a natural weak topology ${\cal T}_w$ on $CC(X)$ and investigate properties of ${\cal T}_w$-convergent sequences. Our main result is a theorem which states that if $A_n$, $A\in CC(X)$ and $A_n$ is ${\cal T}_w$-convergent to $A$, then there exists a sequence $\{B_n\}$ (each $B_n$ is a finite convex combination of $A_k$'s) such that $B_n$ converges to $A$ with respact to the Hausdorff metric $h$.

The sets of vertical vectors in the contingent cone, the intermediate tangent cone and the Clarke tangent cone to the graph of a multifunction between normed spaces at a given point are estimated or computed by exact formulae under some suitable assumptions. The obtained results sharpen and complement the results of Dien and Yen [Acta Math., Vietnam. Vol. 10(1) (1985), 144-147] where the set of vertical vectors in the Clarke tangent cone was considered.