Jachymski [ Proc. Amer. Math. Soc., 136 (2008), 1359-1373] gave modified version of a Banach fixed point theorem on a metric space endowed with a graph. In the present paper, (G, Φ)-graphic contractions have been de ned by using a comparison function and studied the existence of fixed points. Also, Hardy-Rogers G-contraction have been introduced and some fixed point theorems have been proved. S...

We establish the Bonnet-Myers theorem, Laplacian comparison and Bishop-Gromov volume theorem for weighted Finsler manifolds as well spacetimes, of Ricci curvature bounded below by using weight function. These theorems are formulated with $\epsilon$-range introduced in our previous paper, that provides a natural viewpoint interpolating conditions different effective dimensions. Some results new ...

The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq ...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...

In this paper we introduce some stability criteria for impulsive fuzzy system of differential equations with finite delay in states. Firstly, a new comparison principle for fuzzy differential system compared to crisp ordinary differential equation, based on a notion of upper quasi-monotone nondecreasing, in N dimentional state space is presented. Furthermore, in order to analyze the stability o...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...

In the setting of generic $\beta$-ensembles, we use loop equation hierarchy to prove a local law with optimal error up constant, valid on any scale including microscopic. This has following consequences. (i) The rigidity ordered particles is order $(\log N)/N$ in bulk spectrum. (ii) Fluctuations satisfy central limit theorem covariance corresponding logarithmically correlated field; particular ...

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.

The problem of relating volume to degree for maps between Riemannian manifolds is a fundamental one. Gromov’s Volume Comparison Theorem [Gr] gives such a relation for maps into negatively curved manifolds. In this paper we extend Gromov’s theorem to locally symmetric manifolds of nonpositive curvature. We derive this as a consequence of the following result, which we believe to be of independen...

We introduce a new type of symmetrization in starlike domains in Riemannian manifolds that maintains the Ricci curvature in the radial direction. We prove that this symmetrization is volume increasing. We get, as its direct consequence, a generalization of Bishop’s volume comparison theorem. Moreover, this generalization shows that this kind volume comparison theorem is qualitative in nature, i...