• Volume 122, Issue 1

February 2012,   pages  1-152

• On a Paper of S S Pillai

In 1935, Erdös proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

• Generalizations of some Zero Sum Theorems

Given an abelian group 𝐺 of order 𝑛, and a finite non-empty subset 𝐴 of integers, the Davenport constant of 𝐺 with weight 𝐴, denoted by $D_A(G)$, is defined to be the least positive integer 𝑡 such that, for every sequence $(x_1,\ldots,x_t)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{j_1},\ldots,x_{j_l})$ and $a_i\in A$ such that $\sum^l_{i=1}a_ix_{j_i}=0$. Similarly, for an abelian group 𝐺 of order $n,E_A(G)$ is defined to be the least positive integer 𝑡 such that every sequence over 𝐺 of length 𝑡 contains a subsequence $(x_{j_1},\ldots,x_{j_n})$ such that $\sum^n_{i=1}a_ix_{j_i}=0$, for some $a_i\in A$. When 𝐺 is of order 𝑛, one considers 𝐴 to be a non-empty subset of $\{1,\ldots,n-1\}$. If 𝐺 is the cyclic group $\mathbb{Z}/n\mathbb{Z}$, we denote $E_A(G)$ and $D_A(G)$ by $E_A(n)$ and $D_A(n)$ respectively.

In this note, we extend some results of Adhikari et al(Integers 8(2008) Article A52) and determine bounds for $D_{R_n}(n)$ and $E_{R_n}(n)$, where $R_n=\{x^2:x\in(\mathbb{Z}/n\mathbb{Z})^∗\}$. We follow some lines of argument from Adhikari et al(Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z.58(1953) 459–484;66(1956) 88–110;61(1955) 429–434).

• A New Class of Lattice Paths and Partitions with 𝑛 Copies of 𝑛

Agarwal and Bressoud (Pacific J. Math. 136(2)(1989) 209–228) defined a class of weighted lattice paths and interpreted several 𝑞-series combinatorially. Using the same class of lattice paths, Agarwal (Utilitas Math. 53(1998) 71–80; ARS Combinatoria 76(2005) 151–160) provided combinatorial interpretations for several more 𝑞-series. In this paper, a new class of weighted lattice paths, which we call associated lattice paths is introduced. It is shown that these new lattice paths can also be used for giving combinatorial meaning to certain 𝑞-series. However, the main advantage of our associated lattice paths is that they provide a graphical representation for partitions with 𝑛+𝑡 copies of 𝑛 introduced and studied by Agarwal (Partitions with 𝑛 copies of 𝑛, Lecture Notes in Math., No. 1234 (Berlin/New York: Springer-Verlag) (1985) 1–4) and Agarwal and Andrews (J. Combin. Theory A45(1)(1987) 40–49).

• Optimal Combinations Bounds of Root-Square and Arithmetic Means for Toader Mean

We find the greatest values 𝛼1 and 𝛼2, and the least values 𝛽1 and 𝛽2, such that the double inequalities $𝛼_1S(a,b)+(1-𝛼_1)A(a,b) < T(a,b) < 𝛽_1S(a,b)+(1-𝛽_1)A(a,b)$ and $S^{𝛼_2}(a,b)A^{1-𝛼_2}(a,b) < T(a,b) < S^{𝛽_2}(a,b)A^{1-𝛽_2}(a,b)$ hold for all 𝑎,𝑏>0 with 𝑎 ≠ 𝑏. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, $S(a,b)=[(a^2+b^2)/2]^{1/2},A(a,b)=(a+b)/2$, and $T(a,b)=\frac{2}{𝜋}\int^{𝜋/2}_{0}\sqrt{a^2\cos^2𝜃+b^2\sin^2𝜃}d𝜃$ denote the root-square, arithmetic, and Toader means of two positive numbers 𝑎 and 𝑏, respectively.

• Bounds on Gromov Hyperbolicity Constant in Graphs

If 𝑋 is a geodesic metric space and 𝑥1,𝑥2,𝑥3 $\in$ 𝑋, a geodesic triangle 𝑇={𝑥1,𝑥2,𝑥3} is the union of the three geodesics [𝑥1,𝑥2], [𝑥2,𝑥3] and [𝑥3𝑥1] 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 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$: 𝑋 is 𝛿-hyperbolic}. In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs. • The Cartan Matrix of a Centralizer Algebra The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension. • A Note on TI-Subgroups of Finite Groups A subgroup 𝐻 of a finite group 𝐺 is called a TI-subgroup if$H\cap H^x=1$or 𝐻 for any$x\in G$. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified. • The Cohomology of Orbit Spaces of Certain Free Circle Group Actions Suppose that$G=\mathbb{S}^1$acts freely on a finitistic space 𝑋 whose (mod 𝑝) cohomology ring is isomorphic to that of a lens space$L^{2m-1}(p;q_1,\ldots,q_m)$or$\mathbb{S}^1×\mathbb{C}P^{m-1}$. The mod 𝑝 index of the action is defined to be the largest integer 𝑛 such that$𝛼^n≠ 0$, where$𝛼\in H^2(X/G;\mathbb{Z}_p)$is the nonzero characteristic class of the$\mathbb{S}^1$-bundle$\mathbb{S}^1\hookrightarrow X→ X/G$. We show that the mod 𝑝 index of a free action of 𝐺 on$\mathbb{S}^1×\mathbb{C}P^{m-1}$is 𝑝-1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free 𝐺-action on$\mathbb{S}^1×\mathbb{C}P^{m-1}$. It is note worthy that the mod 𝑝 index for free 𝐺-actions on the cohomology lens space is not defined. • Precise Asymptotics for Complete Moment Convergence in Hilbert Spaces Let$\{X, X_n;n≥ 1\}$be a sequence of i.i.d. random variables taking values in a real separable Hilbert space$(H,\|\cdot p\|)$with covariance operator$\sum$. Set$S_n=\sum^n_{i=1}X_i,n≥ 1$. We prove that for 1 < 𝑝 < 2 and$r>1+p/2$, \begin{multline*}\lim\limits_{𝜀\searrow 0}𝜀^{(2r-p-2)/(2-p)}\sum\limits^∞_{n=1}n^{r/p-2-1/p}E\{\|S_n\|-𝜎𝜀 n^{1/p}\}+\\ =𝜎^{-(2r-2-p)/(2-p)}\frac{p(2-p)}{(r-p)(2r-p-2)}E\|Y\|^{2(r-p)/(2-p)},\end{multline*} where 𝑌 is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator 𝛴 , and 𝜎2 is the largest eigenvalue of 𝛴 . • An 𝑛-Dimensional Pseudo-Differential Operator Involving the Hankel Transformation An 𝑛-dimensional pseudo-differential operator (p.d.o.) involving the 𝑛-dimensional Hankel transformation is defined. The symbol class$H^m$is introduced. It is shown that p.d.o.'s associated with symbols belonging to this class are continuous linear mappings of the 𝑛-dimensional Zemanian space$H_𝜇(I^n)$into itself. An integral representation for the p.d.o. is obtained. Using the Hankel convolution, it is shown that the p.d.o. satisfies a certain 𝐿1-norm inequality. • Annihilating Power Values of Co-Commutators with Generalized Derivations Let 𝑅 be a prime ring with its Utumi ring of quotient 𝑈,𝐻 and 𝐺 be two generalized derivations of 𝑅 and 𝐿 a noncentral Lie ideal of 𝑅. Suppose that there exists$0≠ a\in R$such that$a(H(u)u-uG(u))^n=0$for all$u\in L$, where 𝑛 ≥ 1 is a fixed integer. Then there exist$b',c' \in U$such that$H(x)=b'x+xc',G(x)=c'x$for all$x\in R$with$ab'=0$, unless 𝑅 satisfies 𝑠4, the standard identity in four variables. • Remarks on Hausdorff Measure and Stability for the 𝑝-Obstacle Problem (1 < 𝑝 < 2) In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation $$\mathrm{div}(\nabla u|^{p-2}\nabla u)=f\cdot p𝜒\{u>0\},\quad 1 < p < 2,$$ where 𝑓 is a positive, Lipschitz function. We prove that the free boundary has finite$(N-1)\$-Hausdorff measure and stability property, which completes previous works by Caffarelli (J. Fourier Anal. Appl. 4(4--5) (1998) 383--402) for 𝑝=2, and Lee and Shahgholian (J. Differ. Equ. 195 (2003) 14--24) for 2 < 𝑝 < ∞.

• Existence of Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems

In this paper, we establish the existence of at least one and two positive solutions for the system of higher order boundary value problems by using the Krasnosel'skii fixed point theorem.

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