By Ken Milburn; ebrary, Inc
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The theorems of Berkeley mathematician Marina Ratner have guided key advances within the realizing of dynamical structures. Unipotent flows are well-behaved dynamical platforms, and Ratner has proven that the closure of each orbit for this type of circulate is of an easy algebraic or geometric shape. In Ratner's Theorems on Unipotent Flows, Dave Witte Morris offers either an straightforward advent to those theorems and an account of the evidence of Ratner's degree category theorem.
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Additional info for Digital abstract and macro photography : Includes index
Then I duplicated the original layer, flipped it upside down, and then moved and rotated it until it filled the hole I just made. Next, I refined the cutout edge by using a softened eraser and used the History brush to paint back in areas that I'd overpainted. I wanted to "paint" the reflective surfaces with portions of other images that reminded me of the legendary motorcycle lifestyle. One by one, I opened images of another motorcycle, an octopus sculpture from Burning Man, the bare back of a tattooed lady, and a portrait of a costumed lady.
I also wanted to darken the grass a bit more to emphasize the life in the leaves in contrast to the cold, hard stone in the sand. Bringing lighting and contrast into the image was a matter of delicately changing the brightness of specific areas of color along the brightness curve in the Curves command dialog. In Photoshop, you accomplish this by dragging an eyedropper cursor to a specific area of brightness and then clicking while the Cmd/Ctrl key is pressed. This places a point along the curve that can be raised or lowered to brighten or darken that particular color.
Once the image was ideally positioned, I flattened the image and used the Clone Stamp to fix any disconnected lines. The Healing Brush and Patch tool were also used to blend textures and edges when the Clone Stamp lacked the right subtlety. You could further interpret this image using just about any technique described in this book. What I thought would work best, however, would be to simply re-interpret color and line. The image at left was treated with two different processes on overlaid duplicates of the original layer.