• Volume 120, Issue 5

November 2010,   pages  515-641

• Integers without Large Prime Factors in Short Intervals: Conditional Results

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of $\zeta(1/2+it)$ is bounded by $(\log t)^{\frac{1}{2}+o(1)}$, we show that for any given $\alpha &gt; 0$ the interval $(X, X+\sqrt{X}(\log X)^{1/2+o(1)}]$ contains an integer having no prime factor exceeding $X^\alpha$ for all 𝑋 sufficiently large.

• Weakly Distributive Modules. Applications to Supplement Submodules

In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules. We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that 𝜋-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive $\oplus$-supplemented module is quasi-discrete.

• Orthogonal Symmetries and Clifford Algebras

Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.

• Embedding Relations Connected with Strong Approximation of Fourier Series

We consider the embedding relation between the class $W^q H^\omega_\beta$, including only odd functions and a set of functions defined via the strong means of Fourier series of odd continuous functions. We establish an improvement of a recent theorem of Le and Zhou [Math. Inequal. Appl. 11(4)(2008) 749--756] which is a generalization of Tikhonov’s results [Anal. Math. 31(2005) 183--194]. We also extend the Leindler theorem [Anal. Math. 31(2005) 175--182] concerning sequences of Fourier coefficients.

• Uniqueness of Singular Solution of Semilinear Elliptic Equation

In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established.

• Gromov Hyperbolicity in Cartesian Product Graphs

If 𝑋 is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2], [x_2x_3]$ and $[x_3x_1]$ in 𝑋. The space 𝑋 is 𝛿-hyperbolic (in the Gromov sense) if any side of 𝑇 is contained in a 𝛿-neighborhood of the union of the two other sides, for every geodesic triangle 𝑇 in 𝑋. If 𝑋 is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of 𝑋, i.e. $\delta(X)=\inf\{\delta\geq 0:X\, \text{is}\delta-\text{hyperbolic}\}$. In this paper we characterize the product graphs $G_1\times G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the product graph $G_1\times G_2$ is hyperbolic if and only if $G_1$ is hyperbolic and $G_2$ is bounded or $G_2$ is hyperbolic and $G_1$ is bounded. We also prove some sharp relations between the hyperbolicity constant of $G_1\times G_2,\delta(G_1),\delta(G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.

• An almost sure Invariance Principle for Trimmed Sums of Random Vectors

Let $\{X_n;n\geq 1\}$ be a sequence of independent and identically distributed random vectors in $\mathfrak{R}^p$ with Euclidean norm $|\cdot p|$, and let $X^{(r)}_n=X_m$ if $|X_m|$ is the 𝑟-th maximum of $\{|X_k|;k\leq n\}$. Define $S_n=\sum_{k\leq n}X_k$ and $^{(r)}S_n=S_n-(X^{(1)}_n+\cdots+X^{(r)}_n)$. In this paper a generalized strong invariance principle for the trimmed sums $^{(r)}S_n$ is derived.

• A very Simple Proof of Pascal's Hexagon Theorem and some Applications

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.

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• # Proceedings – Mathematical Sciences

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