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^2 + \big((p+q)-2(k_2+k_3)\big)\|z\|^2\right\}\right] \\ &{\leqslant}\exp\left[ 2\left\{\tilde\delta\|p-q\|^2 + \big((p+q)-2(k_2+k_3)\big)\|z\|^2\right\}\right] \\ &= \exp\left[ 2\left\{\tilde\delta\|p-q\|^2 + (p+q-2(k_2+k_3))\|z\|^2\right\}\right] {\leqslant}\exp\left[ 2\left\{\tilde\delta\|p-q\|^2 + (p+q)\|z\|^2\right\}\right]\end{aligned}$$By simply scaling, we have the desired. $rmk:goldfar$ If M is a Riemannian manifold, then d(p,q) is the Riemannian distance between p,q\in M. Also, if M is a Euclidean manifold, then d(p,q) is the euclidean distance between p,q\in {\mathds{R}}^n. It is worth pointing out that in either of the settings, the convexity of d(p,q) implies its scale-invariance. Let p,q\in M be such that d(p,q) {\leqslant}R. From we know that for all z\in\mathfrak{T}_{p,q},$$\exp\left[-\lambda\cdot d(p,q)\right] {\leqslant}\tilde\delta\|z\|^2 {\leqslant}\tilde\delta \big(d(p,q)\big)^2 Now, for the case of Riemannian manifolds, we may use the bound that $d(p,q)^2 {\leqslant}R^2$. The case for Euclidean manifolds is

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Like me on Facebook! Monday, March 8, 2010 Valentines Day Special Hey everyone, I am finally getting around to showing off the top that I made for my Valentine’s Day gift to my husband! If you know anything about me, you know I am a sucker for an oversize shirt. I even wore an oversize blouse during a blizzard for a week and then had it repaired. It was so cold, I didn’t get a picture before I replaced it. This top is knit from the « Treasure From the Sea » pattern by Colette. As a marine biologist, this was a fun project. I chose a yarn that had a great sheen and a blue color that would look good on my husband. I made sure to keep all measurements in mind when I knit the top and I love the fit. I got several compliments on it while wearing it to work yesterday. I will be wearing this top in a very special outing with my husband on Monday, March 22! I hope you guys have a great week! I’m looking forward to getting on with my knitting, planning my scrapbooking schedule, and learning how to use my camera! My Profile What I Blog About I am a working mother who has plenty of interests and plenty of time to kill. I enjoy knitting, scrapbooking, genealogy, books, movies, theater, and learning French. I will be blogging about these things and my personal life and sharing my opinions on things going on in the world. I am a proud Marine mom and wife who finds joy in simple pleasures. Gifts to the Sea Fairfax, Va. — An international science competition has yielded a big prize: $1.8 million. And it’s not money, but a gift to the world’s oceans from a retired bottle company. The Ocean Conservancy’s BLUE Ocean Project is organizing a “Donation to the Sea” contest that will end with an exhibit and auction at the National Aquarium in Baltimore. Anyone who gives the organization a monetary gift will receive half of what’s donated, and the other half will go to ocean conservation. “We’re going to be giving out more than$1.8 million to organizations that are working on ocean conservation,” said Devin Hughes, the project’s director