pp 515-524
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 $𝜁(1/2+it)$ is bounded by $(\log t)^{\frac{1}{2}+o(1)}$, we show that for any given 𝛼 > 0 the interval $(X, X+\sqrt{X}(\log X)^{1/2+o(1)}]$ contains an integer having no prime factor exceeding $X^𝛼$ for all 𝑋 sufficiently large.
pp 525-534
Weakly Distributive Modules. Applications to Supplement Submodules
Engin Büyükaşik Yilmaz M Demirci
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.
pp 535-561
Orthogonal Symmetries and Clifford Algebras
Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.
pp 563-582
Embedding Relations Connected with Strong Approximation of Fourier Series
We consider the embedding relation between the class $W^q H^𝜔_𝛽$, 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.
pp 583-591
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.
pp 593-609
Gromov Hyperbolicity in Cartesian Product Graphs
Junior Michel José M Rodríguez José M Sigarreta María Villeta
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 𝛿(𝑋) the sharp hyperbolicity constant of 𝑋, i.e. $𝛿(X)=\inf\{𝛿≥ 0:X\, \text{is}𝛿-\text{hyperbolic}\}$. In this paper we characterize the product graphs 𝐺_{1} × 𝐺_{2} which are hyperbolic, in terms of 𝐺_{1} and 𝐺_{2}: the product graph 𝐺_{1} × 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} × 𝐺_{2},𝛿(𝐺_{1}),𝛿(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.
pp 611-618
An almost sure Invariance Principle for Trimmed Sums of Random Vectors
Let $\{X_n;n≥ 1\}$ 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 $\{|X_k|;k≤ n\}$. Define $S_n=\sum_{k≤ n}X_k$ 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.
pp 619-629
A very Simple Proof of Pascal's Hexagon Theorem and some Applications
Nedeljko Stefanović Miloš Milošević
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.
pp 631-637 Subject Index
pp 639-641 Author Index
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Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.