We investigate connections between pairs of Riemannian metrics whose sum is a (tensor) product of a covector field with itself. As a special result is constructed one-to-one mapping between the classes of Euclidean and Lorentzian metrics. The existence of Lorentzian metrics on a differentiable manifold is discussed. We point the possibility that any physical theory based on Lorentzian metric(s)...

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and onl...

recently, rahimi et al. [comp. appl. math. 2013, in press] dened the conceptof quadrupled xed point in k-metric spaces and proved several quadrupled xed pointtheorems for solid cones on k-metric spaces. in this paper some quadrupled xed point resultsfor t-contraction on k-metric spaces without normality condition are proved. obtainedresults extend and generalize well-known comparable result...

1. Vector Spaces 1 2. Norms 4 2.1. Unit Vectors 6 2.2. Convexity 9 3. Inner Product Spaces 9 3.1. Induced Norms and the CBS Inequality 12 3.2. Orthogonal Sets of Vectors 15 3.3. Best Approximations and Least Squares Problems 17 4. Linear Operators 20 4.1. Operator Norms 22 5. Metric Spaces and Analysis 23 5.1. Metric Spaces 23 5.2. Calculus and Analysis 25 6. Numerical Analysis 26 6.1. Big Oh N...

In this paper, we study warped product manifolds admitting $\tau$-quasi Ricci-harmonic(RH) metrics. We prove that the metric of fibre is harmonic Einstein when RH metric. also provide some conditions for $M$ to be a manifold. Finally, necessary and sufficient $g$ by using differential equation system.

The theory of Finsler metric was introduced by Paul Finsler, in 1918. author defines this using the Minkowski norm instead inner product. Therefore, geometry is a more general and includes Riemannian metric. In present work, metric, we investigate position vector rectifying, normal osculating curves Finslerian 3-space $\mathbb{F}^{3}$. We obtain characterizations these Furthermore, show that re...

An iterated function system consists of a complete metric space (X,d) and finite family contractions f1,⋯,fn:X→X. A generalized comprises defined on the Cartesian product Xm with values in X. In this paper, we want to investigate systems more general setting b-metric spaces. We prove that such admits unique attractor and, under some further restrictions b-metric, it depends continuously paramet...

One critical aim of product family design is to offer distinct variants that attract a variety of market segments while maximizing the number of common parts to reduce manufacturing cost. Several indices have been developed for measuring the degree of commonality in existing product lines to compare product families or assess improvement of a redesign. In the product family optimization literat...

Defining product metrics requires a rigorous and disciplined approach, because useful metrics depend, to a very large extent, on one's goals and assumptions about the studied software process. Unlike in more mature scientific fields, it appears difficult to devise a "universal" set of metrics in software engineering, that can be used across application environments. We propose an approach for t...