Volume 59 , issue 4 ( 2007 ) | back |

Generalized (co)homology and Morse complex | 153$-$160 |

Correlation analysis: Exact permutation paradigm | 161$-$170 |

**Abstract**

For a general class of problems, the permutation of observations is the only possible method of truly constructing exact tests of significance. The exact sampling distribution of a test statistic for an experiment compiled by the permutation approach requires no reference to a population distribution and therefore no requirement that it should conform to a mathematically definable frequency distribution. Algorithms for the exact permutation distribution of correlation coefficients is presented and implemented. As an illustrative example, critical values for Pearson's product moment and Spearman's rank correlation coefficients are produced for Charles Darwin's data on the heights of cross and self fertilized plants.

**Keywords:** Permutation test, Monte Carlo test, p-value, algorithm, paired observations, correlation.

**MSC:** 62E15

Forcing signed domination numbers in graphs | 171$-$179 |

**Abstract**

We initiate the study of forcing signed domination in graphs. A function $f:V(G)\longrightarrow \{-1,+1\}$ is called {\it signed dominating function} if for each $v\in V(G)$, ${\sssize\sum}_{u\in N[v]}f(u)\geq 1$. For a signed dominating function $f$ of $G$, the {\it weight} $f$ is $w(f)={\sssize\sum}_{v\in V}f(v)$. The {\it signed domination number} $\gamma_s(G)$ is the minimum weight of a signed dominating function on $G$. A signed dominating function of weight $\gamma_s(G)$ is called a $\gamma_s(G)$-{\it function}. A $\gamma_s(G)$-function $f$ can also be represented by a set of ordered pairs $S_f=\{(v, f(v)): v\in V\}$. A subset $T$ of $S_f$ is called a {\it forcing subset\/} of $S_f$ if $S_f$ is the unique extension of $T$ to a $\gamma_s(G)$-function. The {\it forcing signed domination number} of $S_f$, $f(S_f,{\gamma_s})$, is defined by $f(S_f,{\gamma_s})=\min\{|T|: \mbox{$T$ is a forcing subset of\/ } S_f\}$ and the {\it forcing signed domination number} of $G$, $f(G,{\gamma_s})$, is defined by $f(G,{\gamma_s})=\min\{f(S_f,{\gamma_s}): S_f \;\;\tx{is a}\; \gamma_s(G)\mbox{-function}\}$. For every graph $G$, $f(G,\gamma_s)\geq 0$. In this paper we show that for integer $a,b$ with $a$ positive, there exists a simple connected graph $G$ such that $f(G,\gamma_s)=a$ and $\gamma_s(G)=b$. The forcing signed domination number of several classes of graph, including paths, cycles, Dutch-windmills, wheels, ladders and prisms are determined.

**Keywords:** Forcing signed domination number, signed domination number.

**MSC:** 05C15

Dense sets, nowhere dense sets and an ideal in generalized closure spaces | 181$-$188 |

**Abstract**

In this paper, concepts of various forms of dense sets and nowhere dense sets in generalized closure spaces have been introduced. The interrelationship among the various notions have been studied in detail. Also, the existence of an ideal in generalized closure spaces has been settled.

**Keywords:** Generalized closure spaces, isotonic spaces, dense sets, nowhere dense sets, ideals.

**MSC:** 54A05

Some curvature conditions of the type $2\times 4$ on the submanifolds satisfying Chen's equality | 189$-$196 |

**Abstract**

Submanifolds of the Euclidean spaces satisfying equality in the basic Chen's inequality have, as is known, many interesting properties. In this paper, we discuss on such submanifolds the curvature conditions of the form $E_2\cdot F_4=0$, where $E_2$ is the Ricci or the Einstein curvature operator, $F_4$ is any of the standard curvature operators $R, Z, P, K, C$, and $E_2$ acts on $F_4$ as a derivation.

**Keywords:** Submanifolds, curvature conditions, basic equality.

**MSC:** 53B25, 53C40

A note on boundary values for the Poisson transform | 197$-$204 |

**Abstract**

We determine boundary values in distributional and pointwise sense for the Poisson transform of a certain class of weighted distributions.

**Keywords:** $S^{\prime}$-convolution; weighted distributions; Poisson kernel.

**MSC:** 46F20, 46F05, 46F12

Notes on doubly warped and doubly twisted product CR-submanifolds of Kaehler manifolds | 205$-$210 |

**Abstract**

B. Y. Chen studied warped product CR-submanifolds [7] and twisted product CR-submanifolds [6] in Kaehler manifolds. In this paper, we have checked the existence of other product CR-submanifolds such as doubly warped and doubly twisted products in Kaehler manifolds.

**Keywords:** Doubly warped product; doubly twisted product; CR-submanifold; Kaehler manifold.

**MSC:** 53C40, 53C42, 53C15

Gluing and Piunikhin-Salamon-Schwarz isomorphism for Lagrangian Floer homology | 211$-$228 |

**Abstract**

We prove Floer gluing theorem in the case of objects of mixed type, that incorporate both Morse gradient trajectories and holomorphic discs with Lagrangian boundary conditions.

**Keywords:** Lagrangian submanifolds; Floer homology; Morse theory; gluing.

**MSC:** 53D40, 57R58, 53D12