This paper reports the study of properties of the curve pairs of the Bertrand types using our automated reasoning program based on Wu’s method of mechanical theorem proving for differential geometry. A complete list of results about Bertrand curves in metric and affine spaces is derived mechanically. The list includes most of the known results of various Bertrand curves. We also derive some new...

Perlick’s classification of (3 + 1)-dimensional spherically symmetric and static spacetimes (M, η = − 1 V dt + g) for which the classical Bertrand theorem holds [Perlick V 1992 Class. Quantum Grav. 9 1009] is revisited. For any Bertrand spacetime (M, η) the term V (r) is proven to be either the intrinsic Kepler– Coulomb or the harmonic oscillator potential on its associated Riemannian 3manifold...

In [14] Matsuda and Yorozu.explained that there is no special Bertrand curves in Eⁿ and they new kind of Bertrand curves called (1,3)-type Bertrand curves Euclidean space. In this paper , by using the similar methods given by Matsuda and Yorozu [14], we obtain that bitorsion of the quaternionic curve is not equal to zero in semi-Euclidean space E4^2. Obtain (N,B2) type quaternionic Bertrand cur...

one of the most practical routine tests for convergence of a positive series makes use of the ratio test. if this test fails, we can use rabbe's test. when rabbe's test fails the next sharper criteria which may sometimes be used is the bertrand's test. if this test fails, we can use a generalization of bertrand's test and such tests can be continued in nitely. for simplicity...

This paper provides a simple model of endogenous horizontal product differentiation that has two important implications. First, the model can explain the “empirical Bertrand paradox” – the failure to observe homogeneous product Bertrand oligopoly. If product differentiation is possible at reasonable cost, then Bertrand firms would always invest in product differentiation. Using a quadratic util...

This paper compares Bertrand and Cournot equilibria in a di®erentiated duopoly with R&D (research and development) competition. Cournot competition is shown to induce more R&D e®ort than Bertrand competition. However, the price is lower and output is higher in Bertrand than in Cournot competition. Furthermore, the Bertrand equilibrium is more e±cient than the Cournot equilibrium if either R&D p...

Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822-1900). Specifically, it is a model of price competition between duopoly firms which results in each charging the price that would be charged under perfect competition, known as marginal cost pricing. The model has the following assumptions: There are at least two firms producing ho...

This study consists of four sections. In the first section, objective and background are given. Second section is devoted to basic definitions theorems related 3-dimensional Euclidean spaces. Bertrand-B curve definition given subject was mentioned. third chapter common isoasymptotic surface families were examine, so we obtain necessary sufficient condition for a (with Bishop 2-frame) such that ...