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How do you make the cast-on row for a sock as loose as the ribbing? I've been ripping out the cast-on row after the sock is finished (a blind cast-on), but that doesn't leave a neat edge.
For two positive continuous functions $f$ and $g$ with support $\bb{R}^+$, let $g$ be strictly increasing. For an arbitrary $f$, can I find a $g$ such that $f(g^{-1}(x)) \left| \frac{d}{dx} g^{-1}(x) \right|$ is strictly increasing when $f$ is strictly increasing and increasing-then-decreasing when $f$ is increasing-then-decreasing? If not, can I show that no such $g$ exists?
How can I convince Angry Toddler to wear panties to daycare if the Dora the Explorer panties aren't enough to make her want to?
7 comments:
3. Have you tried Jasmine/other Disney princess underpants?
1. I cast on and then space them out loosely. Use a lot of play in the yarn while you do the first row.
We've tried Dora panties, Carebear panties, and Hello Kitty panties. She likes panties with rainbows and butterflies on them. For the past few days, though, she just wants diapers. Maybe the pressure is getting to her.
Part of the sock problem might be the way I learned to cast on in the round. I make a crochet chain and join it, then cast on via sc through the loops of the chain (hard to explain without the yarn in my hand). I wonder if going to a circular needle would solve my issues.
I just had an idea. What if I cast on double the number of stitches required and then drop every other one on the first round?
I use a cable cast-on. Make the first anchor loop, and knit through it. Slip the new knit stitch from the right-hand needle to the left, next to the base loop. For subsequent stitches, instead of going through the previous stitch, go through the spaces between the two previous stitches. This makes a nice-looking, slightly stretchy cast-on row. (The knit-on cast-on is very similar, but you continue knitting through the previous stitches, instead of between them).
If you need even more looseness in the cast-on row, use two regular needles or one bigger needle to hold the stitches while you create them, then for your first row knit onto the regular-size needle for the pattern.
Here's a good reference for these two: http://www.wonderful-things.com/newknit1.htm
AP, I think that idea might work. It'll also give a nice nubby pattern up at the top.
my word down below is CNRYARUN. "Seeing our yarn". Pretty good!
2) g(x) = x always satisfies the requirement. (Assuming that g^{-1} is the inverse function rather that 1/g(x))
In general you will need additional requirements -- ie. if g is bounded above you might have difficulty meeting both conditions on f.
We can easily construct one nontrivial example. Denote S_i as x such that f(x) is increasing and S_d as x such that f(x) is decreasing. Then all continuous functions g(x) which are 1:1 on S_i and S_d satisfy your requirements (assuming I understand what you are really asking.)
Best of luck!
You've surely figured this out by now--3 years later--but if you use double points you have a few options: do your cast on holding 2 together or even just using 1 that is 2-3 sizes larger than you need for your sock, that will fix the problem.
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