Ding actually first discussed this problem on Hilbert spaces in (6), where he showed that an

isometry between the unit spheres of two Hilbert spaces can be isometrically extended to the whole space.

2003) to

isometry (Keramoti lagoon: present study) to positive allometries (Mediterranean: Mariani et al.

In order to assess if SSD increased or decreased between populations of each species, we analyzed the RMA between-sex allometric slopes of the six measured traits in both species, under the null hypothesis of [beta] = 1 (

isometry).

C(Y) be a surjective linear

isometry. Then the following are equivalent.

and since multiplication by ql q is an

isometry of L2 -because ql q has modulus 1 on T- we obtain

Taking into account that the values "a" and "b" obtained with the exponential equation have a biological interpretation, where "a" is the condition factor and "b" is a relative growth coefficient representing the

isometry that each guapote has in its growth, then "a" and "b" show an inverse relationship (Fig.4).

To determine if the major axis deviated from a vector of

isometry with elements 1/[8.sup.1/2] = 0.354, 1000 bootstrap major axes were obtained by resampling (with replacement) from the 177 families of the ancestral population and, for each, the angle of separation of the axis from

isometry was computed.

My geometry students study transformation and

isometry using these.

Because of the choice of an origin and a frame at the origin, it is clear that if the limit exists it is unique up to a unique

isometry. If we did not fix the origin, the limit metric might not be unique; for example a sequence of paraboloids with the same shape but the origin moving away can converge to the flat plane.

Hence, if a Leibniz shift h is not an

isometry (i.e.

A linear transformation U is called unitary or an

isometry if (U[xi], U[eta]) = ([xi], [eta]) for all [xi], [eta][member of][E.sub.n].

Comparison with

isometry ([H.sub.0] slope = 1 vs [H.sub.1] slope[not equal to]1) were performed using t-test using a significance level of [alpha] = 0.05.

Otherwise, restricted

isometry property (RIP) indicates the sufficient and necessary condition that the determined solution exists [17].

According to the nonlinear adjustment, the length-weight relationship coefficients ([+ or -]SE) were: a = 3.27 [+ or -] 0.50 and b = 1.56 [+ or -] 0.05; this latter constant is the slope of the curve and has been referred as the "

isometry coefficient", as it represents an indication of the symmetry in body form of an organism as it grows (Haddon, 2011).

With the exception of the hyperbolic 3-space, [H.sup.3](-1), simply connected homogeneous Riemannian 3-manifolds with

isometry group of dimension 4 or 6, can be represented by a 2-parameter family E(c, [mu]), where c, [mu] [member of] R.