• Volume 124, Issue 3

August 2014,   pages  281-469

• Kirchoff Index of Graphs and some Graph Operations

Let 𝑇 be a rooted tree, 𝐺 a connected graph, $x,y\in V(G)$ be fixed and $G_i$’s be $|V(T)|$ disjoint copies of 𝐺 with $x_i$ and $y_i$ denoting the corresponding copies of 𝑥 and 𝑦 in $G_i$, respectively. We define the 𝑇-repetition of 𝐺 to be the graph obtained by joining $y_i$ to $x_j$ for each $i\in V(T)$ and each child 𝑗 of 𝑖. In this paper, we compute the Kirchhoff index of the 𝑇-repetition of 𝐺 in terms of parameters of 𝑇 and 𝐺. Also we study how $Kf(G)$ behaves under some graph operations such as joining vertices or subdividing edges.

• Diagrams for Certain Quotients of $PSL(2, \mathbb{Z}[i])$

Actions of the Picard group $PSL(2,\mathbb{Z}[i])$ on $PL(F_p)$, where $p\equiv 1(\mathrm{mod} 4)$, are investigated through diagrams. Each diagram is composed of fragments of three types. A technique is developed to count the number of fragments which frequently occur in the diagrams for the action of the Picard group on $PL(F_p)$. The conditions of existence of fixed points of the transformations are evolved. It is further proved that the action of the Picard group on $PL(F_p)$ is transitive. A code in Mathematica is developed to perform the calculation.

• Positive Integer Solutions of the Diophantine Equation $x^2 - L_n xy + (-1)^n y^2 = \pm 5^r$

In this paper, we consider the equation $x^2-L_n xy+(-1)^n y^2=\pm 5^r$ and determine the values of 𝑛 for which the equation has positive integer solutions 𝑥 and 𝑦. Moreover, we give all positive integer solutions of the equation $x^2-L_n xy+(-1)^n y^2=\pm 5^r$ when the equation has positive integer solutions.

• Schematic Harder–Narasimhan Stratification for Families of Principal Bundles and 𝛬-modules

Let 𝐺 be a reductive algebraic group over a field 𝑘 of characteristic zero, let $X\to S$ be a smooth projective family of curves over 𝑘, and let 𝐸 be a principal 𝐺 bundle on 𝑋. The main result of this note is that for each Harder–Narasimhan type 𝜏 there exists a locally closed subscheme $S^\tau (E)$ of 𝑆 which satisfies the following universal property. If $f:T\to S$ is any base-change, then 𝑓 factors via $S^\tau (E)$ if and only if the pullback family $f^∗E$ admits a relative canonical reduction of Harder–Narasimhan type 𝜏. As a consequence, all principal bundles of a fixed Harder–Narasimhan type form an Artin stack. We also show the existence of a schematic Harder–Narasimhan stratification for flat families of pure sheaves of 𝛬-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of $\mathcal{O}$-modules proved earlier by Nitsure. This again has the implication that 𝛬-modules of a fixed Harder–Narasimhan type form an Artin stack.

• Some Hermite–Hadamard Type Inequalities for Geometrically Quasi-Convex Functions

In the paper, we introduce a new concept ‘geometrically quasi-convex function’ and establish some Hermite–Hadamard type inequalities for functions whose derivatives are of geometric quasi-convexity.

• Interpolation for a subclass of $H^\infty$

We introduce and characterize two types of interpolating sequences in the unit disc $\mathbb{D}$ of the complex plane for the class of all functions being the product of two analytic functions in $\mathbb{D}$, one bounded and another regular up to the boundary of $\mathbb{D}$, concretely in the Lipschitz class, and at least one of them vanishing at some point of $\overline{\mathbb{D}}$.

• Time-Periodic Solution of a 2D Fourth-Order Nonlinear Parabolic Equation

By using the Galerkin method, we study the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a fourth-order nonlinear parabolic equation in 2D case.

• Stability of a Simple Levi–Civitá Functional Equation on Non-Unital Commutative Semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation $f(x+y)=f(x)h(y)+f(y)$ and its pexiderization $f(x+y)=g(x) h(y)+k(y)$ on non-unital commutative semigroups by investigating the functional inequalities $|f(x+y)-f (x)h(y)-f(y)|\leq \epsilon$ and $|f(x+y)-g(x)h(y)-k(y)|\leq \epsilon$, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.

• On the Stability of the $L^p$-Norm of the Riemannian Curvature Tensor

We consider the Riemannian functional $\mathcal{R}_p(g)=\int_M|R(g)|^p dv_g$ defined on the space of Riemannian metrics with unit volume on a closed smooth manifold 𝑀 where $R(g)$ and $dv_g$ denote the corresponding Riemannian curvature tensor and volume form and $p\in (0,\infty)$. First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for $\mathcal{R}_p$ for certain values of 𝑝. Then we conclude that they are strict local minimizers for $\mathcal{R}_p$ for those values of 𝑝. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for $\mathcal{R}_p$ for certain values of 𝑝.

• Growth of Fundamental Group for Finsler Manifolds with Integral Ricci Curvature Bound

In this paper, an upper bound on the growth of fundamental group for a class of Finsler manifolds with integral Ricci curvature bound is given. This generalizes the corresponding results with pointwise Ricci curvature in literature.

• A New Proof of the Theorem: Harmonic Manifolds with Minimal Horospheres are Flat

In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem. We also confirm that if a harmonic manifold 𝑀 has same volume density function as $\mathbb{R}^n$, then 𝑀 is flat.

• Geometry of the Cotangent Bundle with Sasakian Metrics and its Applications

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.

• Periodic Diffeomorphisms on Homotopy $E(4)$ Surfaces

• Some Limit Theorems for Negatively Associated Random Variables

Let $\{X_n,n\geq 1\}$ be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the $L^p$-convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.

• On Quadratic Variation of Martingales

We give a construction of an explicit mapping

$$\Psi: D([0,\infty),\mathbb{R})\to D([0,\infty),\mathbb{R}),$$

where $D([0,\infty), \mathbb{R})$ denotes the class of real valued r.c.l.l. functions on $[0,\infty)$ such that for a locally square integrable martingale $(M_t)$ with r.c.l.l. paths,

$$\Psi(M.(\omega))=A.(\omega)$$

gives the quadratic variation process (written usually as $[M,M]_t$) of $(M_t)$. We also show that this process $(A_t)$ is the unique increasing process $(B_t)$ such that $M_t^2-B_t$ is a local martingale, $B_0=0$ and

$$\mathbb{P}((\Delta B)_t=[(\Delta M)_t]^2, 0 &lt; \infty)=1.$$

Apart from elementary properties of martingales, the only result used is the Doob’s maximal inequality. This result can be the starting point of the development of the stochastic integral with respect to r.c.l.l. martingales.

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