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is l*t(d) + 5*k(d)?
2*d - 2
Let o(v) = 14*v - 14. Let j(l) = -5*l + 5. Suppose 0 = -3*t - 7 - 11. Give t*o(i) - 17*j(i).
i - 1
Let w(n) = 17*n + 5. Let p(l) = -9*l - 3. Determine 5*p(i) + 3*w(i).
6*i
Let v(h) = -h**2. Let n(j) = 3*j**2. Let d = -2 - -31. Suppose -4*z + 36 = -4*r - 0*z, -r + 5*z - d = 0. Let s be (-2)/(5/(170/r)). Give s*v(p) + 6*n(p).
p**2
Let l(k) = k**3 - k**2 + 5*k + 6. Let i(j) be the first derivative of -j**3/3 + 2*j**2 + 5*j + 5. What is 5*i(z) - 4*l(z)?
-4*z**3 - z**2 + 1
Let v be 1/1*3*-1. Let r(i) = i**2 - 4*i + 1. Let o be r(3). Let p(t) = -4*t + 1. Let m(f) = 5*f - 1. Give o*m(q) + v*p(q).
2*q - 1
Let t(p) = -28 - 5*p - 31 + 54 - 11*p**2. Let x(g) = -17*g**2 - 8*g - 8. What is 8*t(j) - 5*x(j)?
-3*j**2
Let b(l) = l**2 - 4 + 15 + 2*l**2 + l**3 - 3*l**3. Let u(f) be the second derivative of -f**5/20 + f**4/12 + 5*f**2/2 + 2*f. What is 3*b(r) - 7*u(r)?
r**3 + 2*r**2 - 2
Let o(r) = 3*r + 15. Let u(l) = -l - 8. What is -3*o(d) - 7*u(d)?
-2*d + 11
Let k(h) = -3*h**2 - 5*h + 3. Suppose 2*u = -0*u - 10. Let l(x) = -x + 2*x - 2*x**2 + 2 - 4*x + 0*x**2. Give u*l(c) + 3*k(c).
c**2 - 1
Let n(r) = r - 5. Let l be n(-2). Let c(b) = -2*b**2 - 7*b - 3. Let m(i) = -4*i**2 - 13*i - 5. What is l*c(d) + 4*m(d)?
-2*d**2 - 3*d + 1
Let z(h) = 0*h + 3*h + 3 - 3*h**2 + 2*h**2. Let y be 2/(-1) + (10 - 1). Let d(b) = 2*b**2 - 7*b - 7. Give y*z(w) + 3*d(w).
-w**2
Let l(i) = -i + 1. Let n be l(5). Let j(v) = -v + 2. Let o(w) = -1 + 2 + 0*w + 1 - w. What is n*o(q) + 5*j(q)?
-q + 2
Let z(f) = -6*f - 13. Let v(m) = -3*m - 7. Determine -7*v(d) + 4*z(d).
-3*d - 3
Let i(p) = p. Let h(m) = m**3 - m**2 - 1. Let a(x) = 0*x**2 + 0 + 4*x - 2 + 3 + 4*x**2 - 3*x**3. Let v(g) = a(g) + 2*h(g). Give -4*i(c) + v(c).
-c**3 + 2*c**2 - 1
Let d(i) = -5*i**3 + 4*i. Let g = 217 - 226. Let b be -4 - (3 - (-4)/(-2)). Let p = -1 - b. Let v(q) = -11*q**3 + 9*q. Determine g*d(k) + p*v(k).
k**3
Let t(y) = -61*y - 6. Let v(b) = -184*b - 17. What is 17*t(a) - 6*v(a)?
67*a
Let n = -1 + 0. Let j(i) = i - 4. Let l be j(5). Let c(k) = 0*k**2 + 4*k**2 - 5*k**2 + 0 - l. Let s(q) = 2*q**2. Give n*s(d) - c(d).
-d**2 + 1
Let s(u) = u + 1. Let w(l) = 8*l - 7. Determine 6*s(m) - w(m).
-2*m + 13
Let u = -1 - -7. Let w(v) = -2*v**2 + 4*v - 6. Let d(y) = 1. Determine u*d(q) + w(q).
-2*q**2 + 4*q
Let v(x) = x**3 + 3*x**2 + x + 6. Let p be v(-3). Let q(z) = z**2 + 2. Let c(i) = -i**2 - 3. What is p*c(l) + 5*q(l)?
2*l**2 + 1
Let j(k) be the first derivative of -k**2 - 6. Let p(v) = 3*v. Let x(y) = y**3 + 5*y**2 - 7*y - 4. Let f be x(-6). Suppose 0*n + f*n = -8. Give n*j(z) - 3*p(z).
-z
Let g(k) = 5*k**2 + 3*k + 5. Let c(d) = 4*d**2 + 2*d + 4. Let n(i) = -4*i - 8. Let w be n(-4). Let z = 5 - w. Determine z*g(q) + 4*c(q).
q**2 - q + 1
Let j(y) = 8*y**2 - 2*y - 2. Let v = 13 - 14. Let o be v*(3 - (-3 - -7)). Let a(m) = -m**2. Calculate o*j(w) + 6*a(w).
2*w**2 - 2*w - 2
Let b(w) be the third derivative of w**6/24 - w**4/3 + 4*w**3/3 - 2*w**2. Let u(f) = 3*f**3 - 5*f + 5. What is -5*b(o) + 8*u(o)?
-o**3
Let n(c) = -c**3 + c**2 + 1. Let t(f) = 9*f**3 + 4*f**2 + 7. What is 5*n(x) - t(x)?
-14*x**3 + x**2 - 2
Let n(t) = -t + 14 - 30 + 17. Let d(p) = 4*p - 5. Determine -d(b) - 2*n(b).
-2*b + 3
Suppose -3*s = -4*s - 1. Let k(o) = 2*o - 1. Let u(j) = -1. Determine s*k(y) + 2*u(y).
-2*y - 1
Let a(k) = -k**3 + 2*k**2. Let g(m) = -2*m**3 + 3*m**2. Let l be -1*0/(0 - -2). Suppose -6*h + 2*h + 8 = l. What is h*g(z) - 3*a(z)?
-z**3
Let w(f) = -f**3 - 7*f**2 + 2*f - 2. Let r(t) be the third derivative of -t**6/24 - 29*t**5/60 + 3*t**4/8 - 3*t**3/2 - 14*t**2. Determine 2*r(v) - 9*w(v).
-v**3 + 5*v**2
Let n(s) = s - 12. Let w(b) = 3*b - 35. Calculate 17*n(i) - 6*w(i).
-i + 6
Let t(q) = 1 + 2 + 6*q - 5 + 9*q**2 - 4. Let l(o) = 8*o**2 + 5*o - 5. Calculate 6*l(d) - 5*t(d).
3*d**2
Let p = 1 + 0. Let o(n) be the second derivative of n**2/2 + 32*n. Let r(z) = z - 1. Calculate p*r(s) + 2*o(s).
s + 1
Let p(g) = -2*g - 9. Let f(x) = x + 4. Determine 14*f(l) + 6*p(l).
2*l + 2
Suppose 0 = -9*i + 8*i + 1. Let m be (2 - i)*(-6 - 1). Let w(l) = -2*l**2 + 1. Let v(n) = 6*n**2 - 3. Calculate m*w(h) - 2*v(h).
2*h**2 - 1
Suppose 0 = -2*r + 4*d, 10 = d + 4*d. Let y(x) = -r*x**2 + 2 + 14*x - 14*x. Let n(v) = 9*v**2 - 5. Calculate -2*n(c) - 5*y(c).
2*c**2
Let n(k) = -6*k**2 + 2. Let c(i) be the second derivative of -5*i**4/12 + i**2/2 - 6*i. Calculate -5*c(p) + 4*n(p).
p**2 + 3
Let w be (-1)/((-6)/72*-4). Let z(b) = -2*b**3 + b - 3. Let d(m) = -2*m**3 - 2. Calculate w*d(s) + 2*z(s).
2*s**3 + 2*s
Let m(u) = 4*u**3 - 4*u**2 + 6*u. Let o be 105/20*8/6. Let d(w) = 9*w**3 - 9*w**2 + 13*w. Determine o*m(j) - 3*d(j).
j**3 - j**2 + 3*j
Let j = 9 - 8. Let u(x) = -x**3 - x**2 - x + 1. Let i(c) = 2*c**3 + 6*c**2 + 6*c - 6. What is j*i(h) + 6*u(h)?
-4*h**3
Let c(n) = n**2 - n + 2. Let r(f) = -2*f**2 + f - 4. Determine 5*c(m) + 2*r(m).
m**2 - 3*m + 2
Let s(j) = -3*j - 2. Let d be (-258)/(-54) - (-4)/18. Let c(y) = 2*y + 1. Calculate d*s(i) + 8*c(i).
i - 2
Let q(m) = m**2 - 1. Let o(y) be the third derivative of -y**6/120 - y**5/20 - y**4/24 + y**3/2 - y**2 + 4*y. Calculate -o(l) - 3*q(l).
l**3 + l
Let c(t) = t - 1. Let f(y) = -3*y + 3. Suppose -2*a = -4*a + 2. Let k = a + 2. Suppose 2*m - 55 = -k*m. Calculate m*c(j) + 4*f(j).
-j + 1
Let q(v) = -v + 2. Let p(t) be the second derivative of t**3/6 - 3*t**2/2 - 13*t. Determine 5*p(j) + 6*q(j).
-j - 3
Let x be 1 + (-6)/(1 - -1). Let a be 2*(-1)/x*5. Let y(k) = k**3 - 2*k**2 + 2*k. Let v(h) = -2*h**3 + 5*h**2 - 4*h. Calculate a*y(c) + 2*v(c).
c**3 + 2*c
Let n be (28/(-10) - -2)/(24/60). Let u(v) = -v**2. Let o(h) = 7*h**2. What is n*o(z) - 18*u(z)?
4*z**2
Let c(i) = i**2 - i. Suppose -5*j + 2*g = -82, -2*j + 2 = -3*j + 5*g. Let r = -13 + j. Suppose r = -b + 3. Let o(f) = 4*f**2 - 3*f. Determine b*o(v) + 6*c(v).
-2*v**2
Let t(w) = 3*w + 2. Let b be t(0). Let c(d) = -d + 2. Let i(g) = -g - 5*g + 11 + 0. Give b*i(h) - 11*c(h).
-h
Suppose -4 - 2 = -3*c. Let k(t) = -t**2 + t. Let y(r) = 3*r**2 - 2*r. Determine c*k(a) + y(a).
a**2
Let z(x) be the second derivative of x**5/20 + x**4/12 + x**3/3 + 2*x**2 - 7*x. Let q(w) = -w**3 - w**2 - w - 3. What is -4*q(s) - 3*z(s)?
s**3 + s**2 - 2*s
Let d(c) = 3 - 3*c**2 - c**2 + 4*c**3 + 7*c**2 - 3*c. Let f(k) = k**3 + k**2 - k + 1. What is -d(q) + 3*f(q)?
-q**3
Let o(u) = 4*u**2 - 8*u + 1. Let q(a) = -3*a**2 + 7*a - 1. Suppose 2*s + 5*z = -3, 4*s - 27 = 2*s + 5*z. Calculate s*q(v) + 5*o(v).
2*v**2 + 2*v - 1
Let w(x) = 4*x + 14. Let u(y) = -8*y - 27. Calculate -3*u(n) - 5*w(n).
4*n + 11
Let o(j) = -6*j**2 + 25*j - 7. Let y(r) = -3*r**2 + 13*r - 3. Calculate -3*o(i) + 7*y(i).
-3*i**2 + 16*i
Let q(f) = f**2 - f. Let s(k) = 8*k**2 - 7*k - 6. Give 14*q(j) - 2*s(j).
-2*j**2 + 12
Let h(f) = f**2 - 2*f - 2. Let r be (-35)/(-4) + 1/4. Let s(p) = -6*p + 3*p + 0*p + 5*p**2 - r - 6*p. Determine 9*h(z) - 2*s(z).
-z**2
Let b(z) = 7*z + 8. Let c(k) = 6*k + 7. What is 4*b(q) - 5*c(q)?
-2*q - 3
Let a(s) = -1. Let m(h) = 3*h. What is -2*a(f) - m(f)?
-3*f + 2
Let d(c) = -c**2 + 1. Let h(v) be the first derivative of 2*v**3 + v**2 - 4*v + 43. Give -4*d(l) - h(l).
-2*l**2 - 2*l
Let u(w) = 3*w**2 - 2*w + 2. Let g(b) be the second derivative of -b**4/12 - b**3/6 + b**2/2 + 2*b. Determine -2*g(i) + u(i).
5*i**2
Let v(j) = -2545*j**3 - 11*j + 0*j + 2532*j**3. Let l(x) = -6*x**3 - 5*x. Give 9*l(q) - 4*v(q).
-2*q**3 - q
Let l(c) be the second derivative of 5*c**3/6 - 4*c**2 + 2*c. Let b(u) = -9*u + 15. Determine -6*b(w) - 11*l(w).
-w - 2
Let w = -26 - -24. Let r(o) be the third derivative of o**4/12 + 5*o**3/6 - 2*o**2. Let c(d) = 2*d + 4. Calculate w*r(p) + 3*c(p).
2*p + 2
Let z(r) = -4*r**3 + 16*r**2 + 16. Let d(b) = b**3 - b**2 - 1. Give -16*d(n) - z(n).
-12*n**3
Let t(i) = -6*i**2 - 3. Let x be 10 + -9 - (2 - 2). Let w(p) = -p**2 - 1. Determine x*t(q) - 3*w(q).
-3*q**2
Suppose 8 = 3*c - 7. Let v(u) = 2*u - 5. Let z(s) = -7*s - 15. Let t(g) = 20*g + 42. Let d(n) = -6*t(n) - 17*z(n). Determine c*d(h) + 3*v(h).
h
Let m(f) = -13*f. Let j(z) = -1. Calculate 3*j(w) + m(w).
-13*w - 3
Let x(b) = 8*b + 1. Suppose s = -2 - 3. Let z(y) = -7*y - 1. Let o(t) = -t**2 + 5*t. Let m be o(6). Give m*z(r) + s*x(r).
2*r + 1
Let t(p) = 8*p**3 - 9*p**2 - 6*p. Let k(n) = -n**3 + n**2 + n. Determine 6*k(h) + t(
|
{
"pile_set_name": "DM Mathematics"
}
|
Steve Price
Senior Tax Consultant
Steve is a former Inspector of Taxes with over 30 years’ experience of most aspects of HMRC’s work. This included 10 years dealing with enquiries of all types, managing a CT enquiry team and having a significant role in the development of HMRC’s risk profiling. Steve has since been with Markel Tax for almost 10 years achieving excellent results for clients in all types of compliance work. Steve specialises in dispute resolution and is the team’s ADR expert. He also excels in resolving particularly complex and difficult cases. In carrying out this role he provides support to taxpayers and accountants to whatever extent is required, presents his own ADR workshop and writes associated articles for wider publication.
Markel Consultancy Services Limited is an Appointed Representative of Markel International Insurance Company Limited which is authorised by the Prudential Regulation Authority and regulated by the Financial Conduct Authority and the Prudential Regulation Authority. Insurance is underwritten by Markel International Insurance Company Limited. Financial Services Register Number 202570.
|
{
"pile_set_name": "Pile-CC"
}
|
Daniel Boone (book)
Daniel Boone is a book by James Daugherty about the famous pioneer. It won the Newbery Medal for excellence in American children's literature in 1940. It deals with the life, death, and legacy of Daniel Boone.
Work
Daniel Boone, wilderness scout , New York: Viking Press: Junior Literary Guild, 1939.
Category:Newbery Medal-winning works
Category:1939 children's books
Category:Daniel Boone
Category:Children's history books
Category:American biographies
Category:Viking Press books
Category:Works about American history
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Wednesday, December 31, 2014
After
paying our monthly expenses I was able to make my monthly deposit into
savings. I also signed up for the
company 401(k) plan at work since it was open enrollment time.
Since
I've been paying it down every month and all I owe now is for my box spring and
mattress I went ahead and ordered my bed frame, headboard and side table from
eBay. I found a pretty light fixture to
replace the broken one in my bedroom on sale for 50% off at Home Depot
online. I put it on my PayPal account as
well. I worked enough overtime to cover my PayPal payment plus quite a bit
extra and the light fixture.
Christmas
was paid for from my fund that I saved up all throughout the year plus two gift
cards I had. I was pretty happy with how
much I was able to do with so little money this year. I was a little concerned about it but wonderful
sales, coupons and creativity plus a great stash of sewing and craft supplies
help out a lot.
I ended
the month with quite a bit of money left in my checking account. Part of it I used to purchase a set of
quilted organizers to keep my Christmas china in and a set of felt pads to put
between the pieces. I put the rest
toward my PayPal account and made an extra payment.
I received
a Christmas bonus from my employer this month.
I was thrilled because it was totally unexpected. I decided to set it aside to use for a family
vacation next year. We haven’t had one
of those in a very long time.
Looking
back at the year we just had I am amazed at all that I have accomplished
without adding any substantial debt. In
fact I was able to pay down debt, increase savings, fund Christmas, refurnish
an entire house, pay all of our monthly bills, either early or on time, (I
prefer early) and establish an emergency fun.
The best part is I didn’t feel like I was sacrificing that much to
accomplish my financial goals.
With
new goals set for 2015 I will continue my uber thrifty ways and see what more I
can accomplish in the coming year. I'm
thrilled and excited to get started.
Tuesday, December 30, 2014
Well,
I didn’t stay on budget and I have to admit I didn't really try too hard. There were so many great buys and we did A
LOT of entertaining this month. I'm not
unhappy with my final total but I do have a ton of ideas swirling around in my
head of ways to do better in 2015.
My
non food budget came in at $12.98 for a 16# bag of cat food on sale and a bag
of cat litter. That was it. I didn't buy anything else.
This
is what I bought for food this month:
Groceries
$$$
Steak
3.66
Markdown
Steak
3.88
Markdown
Steak
6.86
Markdown
Steak
7.00
Markdown
Ground
Beef
2.56
Markdown
Ground
Beef
2.46
Markdown
Ground
Beef
2.58
Markdown
Ground
Beef
2.42
Markdown
WW
Flour (2)
2.10
Sale
Frozen
vegetables (6)
4.74
Store
Coupon
Soda -
2 liter (3)
2.00
Sale/Coupon
Olives
(3)
2.67
Store
Coupon
Bacon
6.98
Bologna
0.98
Sale
Smoked
sausage
4.68
Sale
Cheese
(6)
10.00
Store
Coupon
Cereal
1.69
Cereal
0.99
Sale/Coupon
Cereal
1.16
Sale/Coupon
Chicken
4.62
Sale
Crackers
(3)
5.00
Store
Coupon
Canned
vegetables (4)
1.00
Sale/Coupon
Stuffing
(2)
0.98
Store
Coupon
Milk -
gal.
2.09
Non-stick
spray (2)
0.98
Store
Coupon
Eggs -
doz. (4)
3.96
Sale
Cheese
(3)
(0.03)
Markdown
Yep,
that's right!
Cereal
(2)
2.26
Sale/Coupon
Soup
(4)
4.36
Sale/Coupon
Weiners
(3)
2.94
Markdown
Coffee
creamer (2)
1.98
Sale
Milk -
gal.
1.88
Store
Coupon
Butter
(4)
10.00
Sale
Albertson's
Special Offer
(5.00)
Store
Coupon
Sour
cream (2)
1.98
Store
Coupon
Cottage
cheese
0.99
Store
Coupon
Sweet
rolls (3)
3.55
Sale/Coupon
Rib eye
steaks
16.72
Sale
Flour -
5#
1.09
Sale/Coupon
Peanut
butter (2)
2.45
Sale/Coupon
Lettuce
0.48
Coupon
Cornmeal
- 5#
3.99
Margarine
(2)
1.84
Bagels
(2)
1.98
Markdown
Shrimp
2.82
Milk -
gal.
2.09
Lettuce
0.98
Mushrooms
2.00
Half
& half
1.69
Grape
tomatoes
2.00
Chicken
breasts - 5#
10.34
Bananas
0.69
Gelatin
0.99
Whipped
topping (3)
2.37
Total
$ 167.47
$167.47 ÷ 31 days ÷3 people = $1.80 per
person per day!
Not
horrible but I know I can do much better next month and I intend to. How did you save money on groceries this
month? Were you able to find some good
sales and stock up on anything?
Monday, December 29, 2014
I
peeled the label off of the empty pumpkin roll scented hand soap dispenser and
used Goo Gone to get the adhesive cleaned off.
I refilled it with soap and put it back on the kitchen sink to use
rather than sending it off to the recycle bin.
I rendered
some bacon fat and refilled my dripping jar that I keep in the refrigerator.
On
Christmas Eve I made a huge salad by clearing out the crisper drawer and using
up a huge variety of veggies I found in there.
There was enough salad to go with our dinner that night and again on Christmas.
On
Christmas morning I collected any reusable gift bags, gift boxes, tissue paper,
bubble wrap, and bows. I put them away
for next year.
I used
up three egg yolks left over after baking macaroons by making a lemon curd for
a cream pie for Christmas dessert. I had
a graham cracker crust in the pantry I had purchased at the dollar store a few
months ago and also some leftover whipped topping in the freezer. I had originally planned to make a pumpkin
pie but the lemon really called to us.
It was delicious by the way.
I made
a pint of Creamy Italian Salad Dressing and another one of 1000 Island.
The day
after Christmas I watched some tutorial videos for free on Youtube so I could
learn a few new crochet stitches. It
inspired me to dig out my knitting bag which is filled with odds and ends of
yarn left from finished projects. As a
result I started my first Christmas gift for 2015.
I
printed six online grocery coupons.
I
shopped the weekly ads online and made up a new grocery list for January. I was excited to see several items from my
"Stock Up" list were in the Albertson's ad.
We watched
Castle on Hulu for free. I also watched The Blacklist after the girls went to bed.
Friday, December 26, 2014
There
are a few things I want to be sure and accomplish while the girls are off
school for the holidays. Some of these
tasks they can do while I'm at work which is a huge help to me.
Wash coats, hats,
scarves & gloves
Wash backpacks
and lunch boxes
Wash bedding and
pillows
Finish raking the
leaves before the next snowstorm
Send thank you
notes for gifts
Organize and put
away the gift wrapping supplies
Organize and put
away the Christmas decorations
Clean the oven
Starting
the new year with everything clean and tidy will help me feel like I am truly
making a fresh start in 2015.
I
ordered a set of quilted china storage cases and some felt pads to place
between each piece. I've always wanted a
set of these and I figured no better time than the present. After pricing out what it would cost to make
my own felt pads it came out to be the same cost as the ones I purchased. I thought these things would be more
expensive than they were but I got everything I needed from Amazon for $36 with
free shipping. Now my Christmas china
will be safely stored away and much more easily accessible.
I
will be making and using homemade greeting cards for all of us to send out our thank
yous. I still have plenty of Christmas
stamps and return address labels to use too.
How
about you? Do you have a list of things
to get done before year end?
Tuesday, December 23, 2014
I
didn't do a lot of decorating this year.
Our lights weren't working so we didn't hang anything up outdoors and
our box of Christmas wreaths is missing so we no longer have those to put up
either. We were hoping they were
"accidentally misplaced" and might reappear on our doorstep one day
but that didn't happen. So we move on.
You've
all seen our Christmas tree and it is very pretty with the strands of still
working lights on it and all of our pretty ornaments on display. A lot of sentimental ones to remind us of
loved ones and times past.
My
mother made the glass block present for me several years ago.
I've
had this Santa quite awhile now.
The
mantle is my favorite. I had a lot of
fun decorating it. Thanks to my next
door neighbor's row of hedges I was able to bring in some fresh greenery. The pink Fenton bowl belonged to my
grandmother and the vintage bulbs came from my grandparents light strings. I just love them. The ornaments in the vase were the very first
ones I ever bought when I moved out on my own.
I
painted the Santa at a ceramics class I took quite some time ago. This year I decided to surround him with my
grandparent's elves. They had those in
their home for as long as I can remember.
The kerosene lamp belonged to my great-grandmother.
And
our table.
My
grandmother's tablecloths have been getting quite a workout since
Thanksgiving. Because of the red chairs
I decided to layer her lace cloth over her red one to break up the sea of red. I like the look. More greenery and a few tea lights for our
tablescape.
|
{
"pile_set_name": "Pile-CC"
}
|
You are here
CBSE schools to have sports period daily for classes 1 to 12
All work and no play makes a child dull. Nowadays students have a very hectic schedule so that they cannot take out time to play games and gently start ignoring sports.
Therefore, it is the responsibility of schools to include physical education in the curriculum so that students can remain fit and active.
When it comes to curriculum CBSE has always made an effort for the overall development of the students. This time the Central Board of Education (CBSE) has come up with a major change in the curriculum of classes 1 to 12 to make the entire teaching and learning process smooth.
Also, students may download NCERT solutionsand refer them while preparing for the CBSE board exam.
To promote life skills and value education, CBSE has decided to combine health and physical education with academics. As per the new directive issued by the CBSE, all students in class 1 to 12 will now have one period of sports per day. Now games and sports will be a compulsory part of academics from class 1 to 12 leading to a well-balanced in all walks of life.
As per the reports, the syllabus will also include children with special needs. As per the CBSE circular, in many games sign language, wheelchairs etc have also been introduced so that the differently-abled students can be incorporated. Along with artificial intelligence and early childhood care, yoga has also been introduced a skill subject from the new academic session.
With the implementation of these new sports guidelines introduced by CBSE is because if a child can do well in certain they will be provided with focused training too. As per the CBSE rules, In case of absence of specialized sports teacher, it’s the class teacher duty to ensure that each student participates in some kind of physical activity.
Now there would be no theory classes and the syllabus will be divided into four strands. The strand one consists of the following activities i.e. athletics, team games, individual games and adventure sports. At least one activity has to take by each student from strand one. Well, students can also change their respective activity throughout the year. The whole class will also have to participate in group activities.
The strand two consists of health and fitness while strand three is for Social Empowerment through Work Education and Action (SEWA) and strand four is for health and activity card (for the record). The strand one will carry 50 marks while the strand two and three will have the least weightage of only 25 marks each out of 100. Besides this, there will be no marks for strand four.
Prior, this year CBSE has also declared to make arts compulsory for all classes 1 o 12. The board has been also instructed every school to compulsorily reserve a minimum of two periods per week, per class, for art education. All the four main streams belong to arts like music, dance, visual arts and theatre should be included in this period.
Latest NCERT & CBSE News
Read the latest news and announcements from NCERT and CBSE below. Important updates relating to your studies which will help you to keep yourself updated with latest happenings in school level education. Keep yourself updated with all latest news and also read articles from teachers which will help you to improve your studies, increase motivation level and promote faster learning
HRD Minister Ramesh Pokhriyal, through a video conference with all the education ministers of the state, discussed the further steps to be taken to smoothen the education process during the Covid-19 lockdown.
The most important topic which was discussed repeatedly on...
It is clear that this pandemic has utterly disrupted an education system that many assert was already losing its relevance. Enhancing professional development digitally is a need in the times of COVID-19, where social distancing and remote interactions is the new...
CBSE has announced the datesheet for the remaining exams of class 10 and class 12 exams. The exams are going to start in July. The class 10 and 12th board exams will be conducted for the remaining 29 papers from July 1 to July 15, 2020, at various centers nominated by...
In enhancing the students studying part, the Government is planning to introduce one standard one channel plan. The lockdown in India has adversely collapsed all the operations including schools, colleges, and workplaces with unanticipated setbacks.
On behalf of the...
CBSE Board has decided to introduce Art-Integrated Project work for classes I to X to promote Art-Integrated Learning in schools to make teaching learning Competency-Based and joyful. As part of this, at least one Art-Integrated Project in each subject shall be taken...
CBSE has issued a public advisory for all students to be careful from unscrupulous persons impersonating themselves as officers/officials of CBSE. These people have been contacting parents and have been telling them that they have access to student marks data for Board...
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{
"pile_set_name": "Pile-CC"
}
|
Alistair and Jonathan Brownlee shared an emotional embrace at the finish line as the brothers took the gold and silver medals in the Olympic Triathlon.
Their younger brother, Edward, said while the pair have a great relationship, they "aren't always like that".
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{
"pile_set_name": "OpenWebText2"
}
|
changelog:
- type: HELM
description: Add Extra ServiceAccount annotations to helm chart for Gloo, Discovery, and Proxy pods.
issueLink: https://github.com/solo-io/gloo/issues/2698
|
{
"pile_set_name": "Github"
}
|
Prognostic significance of arrhythmia inducibility or noninducibility at initial electrophysiologic study in survivors of cardiac arrest.
The value of arrhythmia inducibility or noninducibility at initial electrophysiologic study to predict the likelihood of arrhythmia recurrence was assessed in 150 consecutive survivors of cardiac arrest. Ventricular tachycardia (greater than or equal to 6 beats) or ventricular fibrilation was induced in 113 patients (75%); ventricular arrhythmia could not be induced in 37 patients (25%). During follow-up of a mean of 16 months (range 1 to 72), there were 65 arrhythmia recurrences, 34 of them fatal, in 58 patients. Multivariate regression analysis showed that inducibility at initial study of ventricular tachycardia or ventricular fibrilation was an independent predictor of total arrhythmia recurrence (p less than 0.0001) and fatal arrhythmia recurrence (p = 0.02). At 1 year, 25 +/- 5% of patients with an inducible arrhythmia had a fatal arrhythmia recurrence compared with only 4 +/- 4% of patients without (p = 0.003). The nature of the inducible arrhythmia had no additional predictive value. Inducibility or noninducibility of ventricular arrhythmias at initial electrophysiologic study is a powerful, independent predictor of subsequent arrhythmia recurrence in survivors of cardiac arrest. Patients without inducible arrhythmias have a low frequency of fatal arrhythmia recurrence.
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{
"pile_set_name": "PubMed Abstracts"
}
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Holidays in Paris, France
Book Flight+Hotel
Paris has an incomparable mesmerising quality. The City of Love is probably one of the world’s most visited cities and is known to bring out the romance in the most cynical. Going by foot is the best way to explore the city’s districts, streets, sidewalks and great parks.
Paris has all but exhausted the superlatives that can reasonably be applied to any city. Notre Dame and the Eiffel Tower - at sunrise, at sunset, at night - have been described countless times, as have the Seine and the subtle (and not-so-subtle) differences between the Left and Right Banks. It is busy with thousands of tourists, but the city has managed to keep its real authenticity. Whether you are there for business, fashion or love, you will definitely be swept away.
Radisson Blu Disneyland® Paris ★★★★
Less than 10 minutes from the theme parks and on the grounds of Golf Disneyland®, the Radisson Blu Hotel at Disneyland® Paris combines an ideal location with welcoming features to make any Disney vacation even more memorable. The hotel provides free round-trip shuttle service to the theme parks, located only 7 minutes away, and features 250 contemporary rooms and suites with Free high-speed, wireless Internet. Relax after a day at the parks with 2 on-site restaurants, a spa and fitness center, and 27 meeting rooms with state-of-the-art technology.
Hotel Califormia is just steps away from the famous Champs Elysees, in the heart of a fashionable area offering a wealth of elegant boutiques, cinemas and theatres. The hotel's classic Art Deco architecture is filled with light and opens on a bucolic central patio offering tranquility and relaxation. It is an ideal place to stay whether you are visiting Paris on vacation or for business.
The only five star hotel on the Champs Elysees with exceptional accommodations and intuitive service, ideal for blending work and play. The hotel features stylish event space and is just minutes from some of the city's most elegant boutiques and popular attractions including Christian Dior, Louis Vuitton and Valentino; Arc de Triomphe, Louvre Museum and the Eiffel Tower.
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{
"pile_set_name": "Pile-CC"
}
|
Simultaneous SERS detection of copper and cobalt at ultratrace levels.
We report a SERS-based method for the simultaneous and independent determination of two environmental metallic pollutants, Cu(ii) and Co(ii). This was achieved by exploiting the coordination-sensitive Raman bands of a terpyridine (TPY) derivative for detecting transition metal ions. Changes in the vibrational SERS spectra of dithiocarbamate anchored terpyridine (TPY-DTC) were correlated as a function of each metal ion concentration, with limits of detection comparable to those of several conventional analytical methods. Simultaneous detection of ultratrace levels of Co(ii) in the presence of high Cu(ii) concentration was also demonstrated, supporting the potential of this sensing strategy for monitoring potable water supplies.
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{
"pile_set_name": "PubMed Abstracts"
}
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Search form
Error message
Deprecated function: The each() function is deprecated. This message will be suppressed on further calls in menu_set_active_trail() (line 2405 of /customers/f/2/a/reliasset.com/httpd.www/includes/menu.inc).
Services
The GAP-analysis is an important tool within the field of asset management in relation to supporting decision-making processes on a company's assets. The analysis is used to identify gaps between the current asset performance, and the expected performance.
We have many years of experience with GAP-analyses, and have worked with several prominent Danish companies. By applying quantitative self-assesment methods and qualitative interviews, we are able to asses the information given, and utilize this to support decision-making processes regarding the company's assets.
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{
"pile_set_name": "Pile-CC"
}
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1. Field of the Invention
The present invention relates to a semiconductor LSI design device and designing method used for control devices in which high safety is required.
2. Description of the Related Art
In nuclear power plants, a safety protection system is installed to perform control such as emergency nuclear reactor shutdown, shutoff of various valves for preventing leakage of radioactive materials, and the like on the basis of an input from a radiation measurement device or various other sensors.
In the past, a microcomputer has been used as a control device, but when the same software is used for a redundant system by device multiplexing, there is a possibility of a device multiplexing function being damaged due to a defect of software. Further, when a memory cell of a storage device is irradiated with radiation such as cosmic rays, a phenomenon called a soft error in which charges are lost and data is rewritten is likely to occur, leading to an accidental abnormal operation. In addition, there is a growing demand for tamper resistance such as software rewrite prevention.
For this reason, in a control device of a nuclear power plant or the like in which high safety is required, a processor-less hardwired system is required for the purpose of improving security.
A background art is disclosed in Japanese Patent No. 4371856 (patent document 1).
“An safety instrumentation system of a nuclear reactor constructed using a digital logic mounted on hardware selected from an ASIC and an FPGA, including a digital logic portion which is configured using at least one of a functional unit which is verified in advance at a stage before output logical patterns with respect to all input logical patterns are mounted and a functional module constituted by combining the verified functional units, wherein the functional module is configured only with functional units having the same logical configuration as the verified functional units” is disclosed in patent document 1.
|
{
"pile_set_name": "USPTO Backgrounds"
}
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The discrimination potential of amalgam restorations for identification: part 2.
The standard dental bitewing radiograph is used to detect interproximal caries but it also provides a specific view of the dental restorations that can be duplicated for identification purposes. The antemortem and postmortem bitewing radiographs are often not at the same angle and result in distorted images of the restorations. The aim of this study was to investigate the progressive increase in angulations of a bitewing radiograph of the same restoration and to determine at what angle the image is distorted sufficiently as not to be recognized. Bitewing radiographs were taken of the same two restorations at 5 ̊, 10 ̊, 15 ̊ and 20 ̊ superior, inferior, mesial and distal to the original 0 ̊bitewing radiograph. Twenty examiners were required to determine at what angle the distortion prevented matching of the image with the original bitewing radiograph. The results showed that the image distortion at 15 ̊became suspect but at 20 ̊none of the images could be matched to the original bitewing radiograph.
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{
"pile_set_name": "PubMed Abstracts"
}
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206 F.3d 462 (5th Cir. 2000)
AMERICAN RIVER TRANS. CO., ET AL., Plaintiffs,v.KAVO KALIAKRA SS, ET AL., Defendants,KAVO KALIAKRA SS, her engines, tackle, appurtenances, etc., in rem;UNITED KINGDOM MUTUAL STEAMSHIP ASSURANCE ASSOCIATION (BERMUDA) LTD., in personam, Defendants - Appellees,v.COMPASS CONDO CORP., Appellants.In Re: In the Matter of the Complaint of AROSITA SHIPPING CO., LTD., as owner of the M/V Kavo Kaliakra for exoneration from or limitation of liabilityAROSITA SHIPPING CO. LTD., as owner of the M/V Kavo Kaliakra; GROMAR SHIPPING CO., LTD., as owners of the M/V Kavo Kaliakra; GOURDOMICHALIS MARITIME SA, as owners of the M/V Kavo Kaliakra, Petitioners - Appellees,v.COMPASS CONDO CORP., ET AL., Claimants,COMPASS CONDO CORP., Claimant - Appellant.HORACE NICHOLAS, Plaintiff,v.KAVO KALIAKRA SS, ET AL., Defendants,KAVO KALIAKRA SS, her engines, tackle, appurtenances, etc., in rem; AROSITA SHIPPING CO., LTD., Defendants - Appellees,v.COMPASS CONDO CORP., Appellants.
No. 98-31205
IN THE UNITED STATES COURT OF APPEALS FOR THE FIFTH CIRCUIT
March 8, 2000
Appeal from the United States District Court for the Eastern District of Louisiana
Before POLITZ, JOHN R. GIBSON,* and HIGGINBOTHAM, Circuit Judges.
PATRICK E. HIGGINBOTHAM, Circuit Judge:
1
In this admiralty action, we apply again the principles of Robins Dry Dock & Repair Co. v. Flint.1 Compass Condo Corporationappeals the dismissal of its claims for economic damages arising from the allision of the M/V KAVO KALIAKRA with barges owned by the American River Transportation Company. We AFFIRM the district court's dismissal of Compass's claims for economic damages.
2
* On March 30, 1992, employees of the appellant, Compass Condo Corporation, were engaged as barge washers on a floating barge dock at the Tulane Fleeting Facility. The floating dock was owned by the American River Transportation Company (ARTCO), and Compass's employees were cleaning ARTCO barges. At some point, the M/V KAVO KALIAKRA allided with the ARTCO barges, harming Compass's employees and its equipment. The employees received workers compensation awards under the Longshoreman and Harbor Workers Compensation Act for their personal injuries. Allegedly, as a result of the numerous workers compensation claims, Compass's workers compensation premiums increased.
3
ARTCO filed suit against the owners and operators of the vessel and their insurer. The district court held the defendants liable for the allision, but dismissed Compass's economic damage claims for increased workers compensation premiums. This appeal ensued.
II
4
In Robins Dry Dock the Supreme Court held that a steamship charterer could not recover economic damages when the steamship he chartered was rendered useless to him for a period of days after the defendant negligently broke the propeller.2 The charterer had no property interest in the ship when it was harmed, but instead merely had a contract with the ship's owners.3 The Court noted the general rule that "a tort to the person or property of one man does not make the tort-feasor liable to another merely because the injured person was under a contract with that other unknown to the doer of the wrong."4 In similar cases, this circuit consistently applies the Robins Dry Dock rule to bar recovery for economic damages in negligence that are unconnected to an injury to a property interest.5
5
In this case, Compass's employees were injured by the negligence of the M/V KALIAKRA. As a result of the accident, Compass's employees filed numerous workers compensation claims, which were paid by Compass's insurer. In turn, the M/V KALIAKRA's owners and insurers paid Compass's insurer 100% of the value of the workers compensation claims, which meant that Compass's insurer endured no loss. Compass apparently changed insurance carriers and pays a higher premium. It blames its new higher premiums on the claims filed by Compass's employees after the allision.
6
Assuming that Compass's higher premiums did result in some finite sense from the M/V KALIAKRA's negligence, Compass's claims are barred under our general rule. These economic damages are traceable only to the personal injuries of Compass's employees, but Compass has no property interest in its employees in any relevant sense. Compass did have a property interest in a few thousand dollars worth of equipment which fell overboard during the accident, but Compass's claimed economic damages are unrelated to the loss of that equipment.
7
Compass argues that the rule is old and eroding. This reliance on the age of the rule in resistance to its application is not persuasive. Its age rather attests to itsutility. And we are otherwise unpersuaded of its erosion.
8
First, Compass argues that employers have been allowed to recover from defendants any compensation payments the employer made to its employees after an accident.6 However, such recovery is a form of indemnification, in which the defendant pays the employer the sums paid to the employees by the employer for damage caused by the defendant. The employer's recovery rests on the employee's personal injury.
9
In this case, Compass bore none of the costs of the compensation awards to its employees. Compass's insurer paid those claims and was in turn fully reimbursed by the defendants. Perhaps Compass's insurance rates should not have been raised by its new insurer in a situation in which the predecessor insurer had no loss, but that is a bone Compass must pick with its new insurer and not the defendants. It is precisely the type of remote economic injury rippling at a distant point from the liability event and unanchored by concrete injury to property that we have consistently disallowed.
10
Second, Compass contends that some courts have allowed the recovery of increased insurance premiums which resulted after an insurer was forced to compensate victims of a defendant's negligence, citing Ledex, Inc. v. Healthbath Corp.7 In Ledex, however, the Ohio Supreme Court merely held that a particular state statute, which purported to void all agreements to indemnify employers against payment of compensation to workers, did not bar an employer from seeking to recover increased workers compensation premiums resulting from injuries suffered by its employees at the hands of a third party.8 Ledex did not hold that such damages were compensable, but only that they were not barred by a particular statute.9
11
In sum, we remain unpersuaded of the need to revise the longstanding admiralty rule that economic damages are not recoverable in negligence untethered to an injury to a property interest. As this circuit explained in Akron Corp. v. M/T Cantigny:10 "The rule's purpose is to prevent limitless liability for negligence and the filing of law suits of a highly speculative nature."11 This case is just another example of the type of speculative and potentially unbounded liability the rule aims to preclude.
12
AFFIRMED.
Notes:
*
Circuit Judge of the Eighth Circuit, sitting by designation.
1
275 U.S. 303 (1927).
2
Id. at 307-08.
3
Id. at 308-09.
4
Id. at 309.
5
See State of Louisiana ex rel Guste v. M/V Testbank, 752 F.2d 1019, 1023-24, 1026-27 (5th Cir. 1985) (en banc).
6
See, e.g., Adams v. Texaco, 640 F.2d 618 (5th Cir. 1981).
7
461 N.E.2d 1299 (Ohio 1984).
8
See id. at 1304.
9
See id. at 1303. Compass also cites Tiger Well Service, Inc., 343 So.2d 1158 (La. App. 3d Cir. 1977), for the proposition that increased insurance premiums may be recovered as economic damages. However, in Tiger Well, the plaintiff suffered property damage and was only allowed to recover economic damages to the degree that they flowed from the claim of property damages. See id. at 1158. Thus, the court in Tiger Well did not oppose the rule at issue here.
10
706 F.2d 151 (5th Cir. 1983).
11
Id. at 152.
|
{
"pile_set_name": "FreeLaw"
}
|
Value of tattoo "lotus" in the modern world
Today the youth are very popularget tattoos of flowers. Such images are almost always a symbol of nature, personifying the cycle of its birth, dawn, life and death, rebirth. A living plant gives a push to a person for the realization of his inner world, develops a desire to act in an unconventional way.
The symbol of the lotus in the East
Flowers in cultures of different nations have manyvalues. For example, the tattoo "lotus flower", the meaning of which will be discussed below, came to us from the East. It was the East that filled the lotus with a deep spiritual meaning. Due to the cup-shaped form this flower personifies the feminine. It was the cup that used to be considered the symbol of the continuation of the family, life, birthplace, longevity and prosperity, health and immortality, purity of the soul.
If we consider the value of tattoo "lotus", it can be noted that this flower, namely its structure, personifies the relationship of male and female beginnings.
So, in India, the lotus is correlated with the yoni - female genital organ, which personifies the mother goddess, the beginning of a great sacral force.
The symbol of the lotus in India, Rome and Greece
The meaning of the tattoo "red lotus" came to us exactlyfrom India. The red flower is the emblem of this country and represents a huge creative force. In Egypt, he embodies the intellect and royalty. In ancient Rome and ancient Greece, the lotus was the emblem of the goddess of love, and in Hinduism it represents the whole universe, liberation from sinful life and immortality. In Iran, this plant is associated with cosmic life.
Since the lotus is today a symbol of the East,he expresses chastity, fertility, purity and integrity, spiritual growth, the union of the Moon and the Sun, peace and tranquility, harmony and happiness. The meaning of "lotus" tattoo can vary in some countries, because each culture gives it its own special meaning. Before you figure it out, you need to consider what the lotus is.
What is the lotus?
Lotus is an unusually beautiful flower,sacred in the eyes of the natives, a plant growing on dirty water. It has been growing for many years on submarine grounds, has underwater and above-water leaves. Those of them that float on the water, flat-rounded shape, they sit on powerful straight petioles of various heights, resembling a large funnel. The leaves of the plant are covered with a wax coating, which gives it a mysterious appearance. Drops of water, falling on a sheet, poured, so many remind pearls.
Bearing in itself the symbolism of the East, it servesa symbol of purity. The value of tattoo "lotus" is very interesting. This plant can be found in Buddhist treatises. Since the flower is born in the impure water, in a swamp, it nevertheless appears in the light pure and unblemished. Also, all beings who are born in the "world of samsara", but follow the teachings of the Buddha, eventually get rid of obscurations. In Buddhist art, the lotus image is widely used.
This sacred plant was predetermined by manimportant role. In many works of art, for example, painting, poetry or sculpture, the lotus in modern times has not lost its significance. It continues to be mentioned in verse, it can be found in paintings, sculptures of contemporaries.
Lotus in the modern world
It's no wonder that so many young girls andyoung men choose lotus tattoos. This flower admired and continues to delight a large number of people with its beauty, despite the fact that it grows in marshy water bodies. Anyone who is looking for a designation of tattoo "lotus", will discover a lot of interesting things. He learns that the flower represents the divine birth, tranquility, perfection and harmony. Overcoming the barrier of turbid water, it blooms on the surface of the reservoir, remaining clean and undefiled. So people, according to the religion of the East, if they follow the teachings righteously, eventually become purified, gaining purity of thought and spirit, and give the world the beauty and richness of the soul.
Lotus in the cultures of different countries
The value of tattoo "lotus" today is quite extensive. This is due to the fact that this flower in different cultures and beliefs has a large number of personifications. But they all have something in common: a lotus combines two principles, male and female, consequently, beauty and tranquility, along with stubbornness, the desire to overcome obstacles and boldly go to their goal. Maybe that's why lotus tattoos are chosen as representatives of the weaker sex, and men in the whole world.
Lotus - a huge beautiful flower of gentlePink-white with a transparent core. Anyone who sees him at least once, will fall in love with him forever. The flower emits a sweet aroma that carries man to the land of the rising sun. An ancient legend says: "For a wish to come true, you need to guess it along with the breath of lotus fragrance!" Having become acquainted with this wonderful flower and its history, most young people do not want to part with it, so they want to capture its image on their body.
|
{
"pile_set_name": "Pile-CC"
}
|
Q:
Javascript date time comparison returns false but branch stmt still executed
I am confused with this Javascript behaviour. Check this code.
var NoOfMonthsElapsed = 6; //Should be >= 1 and <= 12
var MsgURL = "about:blank";
var PopupTitle = "ContactInfoUpdate";
var OptionString = "height=165,width=400,menubar=0,toolbar=0,location=1,status=0,resizable=0,status=0,HAlign=center,top=300";
var lastUpdatedDate = crmForm.all.dxb_lastcontactinfoupdatedon.DataValue; //Reads a field with date value = 01 Jan 2010
if (lastUpdatedDate)
{
var month = lastUpdatedDate.getMonth();
var year = lastUpdatedDate.getYear();
var date = lastUpdatedDate.getDate();
month = month + NoOfMonthsElapsed;
year = year + parseInt(month / 11);
month = (month % 11);
var today = new Date();
var showPopupAfterDate = new Date();
showPopupAfterDate.setYear(year);
showPopupAfterDate.setMonth(month);
var alertMsg = "LastUpdatedDate = "+ lastUpdatedDate + "\n"
var alertMsg += "Today = "+ today + "\n"
var alertMsg += "PopupAfterDate = "+ showPopupAfterDate + "\n"
var alertMsg += "Today>showPopupAfterDate = "+ (today>showPopupAfterDate) + "\n"
alert(alertMsg);
if (today>showPopupAfterDate);
{
window.open(MsgURL, PopupTitle, OptionString);
}
}
else
{
window.open(MsgURL, PopupTitle, OptionString);
}
//
// It displays the following output
//
LastUpdatedDate = Wed May 18 20:56:00 UTC+0400 2011
Today = Fri May 18 20:23:49 UTC+0400 2011
PopupAfterDate = Fri Nov 18 20:23:49 UTC+0400 2011
Today>showPopupAfterDate = false
Why today is shown as Fri May 18 2011... though May 18 2011 is Wed
Why PopupAfterDate is shown as Fri Nov 18 2011...
And Even though the dates comparission returns false; The window.open still get executed.
A:
Found the issue:
if (today>showPopupAfterDate) //<-- remove the `;`
{
window.open(MsgURL, PopupTitle, OptionString);
}
Your code is running the if, stopping, then doing the next statement which is the window.open
|
{
"pile_set_name": "StackExchange"
}
|
Many vehicles include a collision avoidance system that comprises a vehicle-mounted sensor to warn a driver of any dangers that may lie ahead on the road. Many vehicles also comprise safety equipment, such as an internal or an external airbag, which is arranged to be deployed if/when the vehicle collides with an object.
|
{
"pile_set_name": "USPTO Backgrounds"
}
|
Aaron Smith - ASIC, CLWM, CID, CIC, CLIA, CGIA
Associations
Certifications
Certified Irrigation Designer
Certified Landscape Water Manager
Certified Irrigation Contractor
Certified Landscape Irrigation Auditor
The Beginning
Since 1982 when I was 10 years old I have been installing irrigation systems. I still remember the first day when all my friends were starting summer vacation and my Dad "Ron Smith" had just started his new irrigation business.
The Lesson
It was not until I was 16 and having spent many summers digging ditches, installing sprinklers and setting valves that I decided that I had enough. I wanted a real job. I soon became a bagger at Albertsons. After my first $80 paycheck for 20 hours work, my Dad smiled at me and gave me my first repair job. I made more in one afternoon than my friends would make in a week. It was a hard-earned lesson but a great one.
Life
The next four years I spent in the United States Air Force as a mechanic learning all about hydraulics, electricity, computers etc. It was during this time that my son Kyle was born.
I left the USAF to go back to what I really liked to do, irrigation. My first adventure was for a national lawn service company as an Irrigation Technician. Soon afterward, I quickly advanced to Irrigation Install Foreman, then Irrigation Manager. I was hungry for more and soon I was back in college to learn how to be an Irrigation Designer.
Professional Career
After college, I was working for Miller Legg, a full-service engineering firm as Head of Irrigation Design. I worked for several years developing a reputation in the irrigation industry as a competent designer. I also met my wife Angie and her daughter Sarah. It was not long before I was recruited by Masuen Consulting, a National Water Resource Consulting firm.
During my time with Masuen Consulting all my life experience with Irrigation, Landscape Management, Water Conservation etc. was tested and honed. My wife gave birth to our daughter Lily and I was born again after accepting Jesus Christ as my Savior. I also earned just about every industry certification possible. This period of my life was an unforgettable, positive experience that has prepared me for my next endeavor
Insight Irrigation LLC
30 years have gone by since I installed my first irrigation system with my Dad and I want to give back to the community where I first started. I created Insight Irrigation LLC as a Christian based business with the highest possible level of service and irrigation experience available.
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{
"pile_set_name": "Pile-CC"
}
|
Domain structure of the large subunit of Escherichia coli carbamoyl phosphate synthetase. Location of the binding site for the allosteric inhibitor UMP in the COOH-terminal domain.
The large subunit of Escherichia coli carbamoyl phosphate synthetase (a polypeptide of 117.7 kDa that consists of two homologous halves) is responsible for carbamoyl phosphate synthesis from NH3 and for the binding of the allosteric activators ornithine and IMP and of the inhibitor UMP. Elastase, trypsin, and chymotrypsin inactivate the enzyme and cleave the large subunit at a site approximately 15 kDa from the COOH terminus (demonstrated by NH2-terminal sequencing). UMP, IMP, and ornithine prevent this cleavage and the inactivation. Upon irradiation with ultraviolet light in the presence of [14C]UMP, the large subunit is labeled selectively and specifically. The labeling is inhibited by ornithine and IMP. Cleavage of the 15-kDa COOH-terminal region by prior treatment of the enzyme with trypsin prevents the labeling on subsequent irradiation with [14C]UMP. The [14C]UMP-labeled large subunit is resistant to proteolytic cleavage, but if it is treated with SDS the resistance is lost, indicating that UMP is cross-linked to its binding site and that the protection is due to conformational factors. In the presence of SDS, the labeled large subunit is cleaved by trypsin or by V8 staphylococcal protease at a site located 15 or 25 kDa, respectively, from the COOH terminus (shown by NH2-terminal sequencing), and only the 15- or 25-kDa fragments are labeled. Similarly, upon cleavage of the aspartyl-prolyl bonds of the [14C]UMP-labeled enzyme with 70% formic acid, labeling was found only in the 18.5-kDa fragment that contains the COOH terminus of the subunit. Thus, UMP binds to the COOH-terminal domain.(ABSTRACT TRUNCATED AT 250 WORDS)
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Kathleen Turner says she had a great time; Butch and Pepe are on stage with the cast. Harry Shearer holds his “Thanks Tom” sign.
Don Pardo announces the next show lineup with Roy “Jaws” Scheider and Billy “Loverboy” Ocean, before signing off as Don “Loverjaws” Pardo.
Bruce Feirstein (Goldeneye, The World is Not Enough and Tomorrow Never Dies), Ron Richards (Not Necessarily The News) and Rosie Shuster are credited with additional sketches.
Final Thoughts:
A bit of a noticeable drop down in quality from the first half of the season; while the cast and writers’ professionalism prevents things from getting unwatchable, the first half of the show has quite a few duller pieces and a recurring sketch that’s starting to lose steam. As a host, Kathleen Turner was game, but she didn’t really bring too much to the sketches (aside from maybe the Joe Franklin Show), and the women in the cast didn’t get much to do tonight. However, there are a few funny unplanned moments this week, one of which livens up an otherwise overlong sketch, and Harry Shearer gets two of the better pieces into what ended up being his final show. As well, even though he only appears in the weaker first half of the show, Billy Crystal demonstrates that he has become the reliable backbone of the cast by this point in the season, doing an impressive six appearances in that span.
SHOW HIGHLIGHTS:
The Joe Franklin Show
MacDouglass-Drummond
Strictly From Blackwell
Green Room
SHOW LOWLIGHTS:
Monologue
Predictions
MVP:
Billy Crystal/Harry Shearer
CAST & GUEST BREAKDOWN
cast
Jim Belushi : absent
: absent Billy Crystal : 6 appearances [Green Room, Do You Know What I Hate? (IV), Nose Hair Trimmer, Fire Dance, The Joe Franklin Show, Boxer]
: 6 appearances [Green Room, Do You Know What I Hate? (IV), Nose Hair Trimmer, Fire Dance, The Joe Franklin Show, Boxer] Mary Gross : 1 appearance [Predictions]; 1 voice-over [Fire Dance]
: 1 appearance [Predictions]; 1 voice-over [Fire Dance] Christopher Guest : 5 appearances [Do You Know What I Hate? (IV), Nose Hair Trimmer, Fire Dance, The Joe Franklin Show, Saturday Night News]
: 5 appearances [Do You Know What I Hate? (IV), Nose Hair Trimmer, Fire Dance, The Joe Franklin Show, Saturday Night News] Rich Hall : 3 appearances [Green Room, Nose Hair Trimmer, Saturday Night News]
: 3 appearances [Green Room, Nose Hair Trimmer, Saturday Night News] Gary Kroeger : 3 appearances [Nose Hair Trimmer, Safeco, Saturday Night News]
: 3 appearances [Nose Hair Trimmer, Safeco, Saturday Night News] Julia Louis-Dreyfus : 1 appearance [Fire Dance]
: 1 appearance [Fire Dance] Harry Shearer : 2 appearances [MacDouglass-Drummond, Strictly From Blackwell]; 1 voice-over [Predictions]
: 2 appearances [MacDouglass-Drummond, Strictly From Blackwell]; 1 voice-over [Predictions] Martin Short : 4 appearances [Fire Dance, The Joe Franklin Show, Scary Lady, Strictly From Blackwell]
: 4 appearances [Fire Dance, The Joe Franklin Show, Scary Lady, Strictly From Blackwell] Pamela Stephenson: 1 appearance [Safeco]
crew and extras
Andy Breckman : 1 appearance [Scary Lady]
: 1 appearance [Scary Lady] Butch : 1 appearance [Green Room]
: 1 appearance [Green Room] Larry David : 1 appearance [Scary Lady]; 1 voice-over [Predictions]
: 1 appearance [Scary Lady]; 1 voice-over [Predictions] Barry Nichols : 1 appearance [Scary Lady]
: 1 appearance [Scary Lady] Pepe : 1 appearance [Green Room]
: 1 appearance [Green Room] Rob Riley : 2 appearances [Green Room, Scary Lady]
: 2 appearances [Green Room, Scary Lady] Rosie Shuster : 1 appearance [Scary Lady]
: 1 appearance [Scary Lady] Dave Wilson: 1 voice-over [Monologue]
guests
Kathleen Turner : 5 appearances [Monologue, Nose Hair Trimmer, The Joe Franklin Show, Scary Lady, Killing Time]
: 5 appearances [Monologue, Nose Hair Trimmer, The Joe Franklin Show, Scary Lady, Killing Time] John Waite: 1 appearance [“Saturday Night”]
REBROADCAST HISTORY:
April 27, 1985
Known alterations:
Safeco and Killing Time removed
Jogger Motel (from 10/31/81) added
Additional screen captures from this episode are available here.
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No author I ever heard of has ever gotten offended by readers wanting ‘moar’, and I’m no exception. The explosion of the “Fire From the Sky” series left me stunned, though in a happy way, and readers clamoring for more is what every author dreams of. Again, I am no exception.
Let me pause here to say “thank you!” to everyone of you, too. It is an honor to be ranked so high on Amazon’s author list, and I owe every bit of that to those of you who have read my books. I am beyond words to describe how exciting it is, or how much it tickles me to hear that you have enjoyed the fruits of my labor.
And it’s perhaps the ultimate compliment for readers to immediately demand more. I simply don’t know what could be better than that.
In the last few weeks I’ve had questions about Parno, Stormcrow and of course “Fire”, and when they could expect the next of whichever series they were interested in. Why this book was first, or second, or wherever it fell in the order.
First of all, I usually work on more than one project at a time, which is why I sometimes have books of different series released so close together. When I am stuck on one I just move over to another and keep going and so forth, back and forth until they’re finished. So I’m always working on something unless I’m taking a break, (which I do about three times a year for no more than one week, mostly to recharge my batteries if you will).
Fire From the Sky and its follow on novels run about 70-100k at the moment. They have clearly drawn lines and the characters are generally a straightforward group at this point which eases the dialogue challenge a little. As the story grows I expect that to change, with complexity rising as it does, and the length of the books as well, in all likelihood.
Stormcrow novels run anywhere from 90-130k words, give or take, and while those novels are complicated and tangled, the main cast of characters, complex though they are, is small and pretty manageable. Dialogue and movement isn’t terribly complicated to put together once I’ve gotten the plot and scenes sketched out.
Parno has become a monster in both scope, size, number of characters and the complexity of those characters. The story has grown from an original three books to now a minimum of five and more likely six or seven. I still have no idea how that happened other than the story wouldn’t do what I told it. That sounds ludicrous I know but. . .it’s amazing what characters you invent decide to do once you start writing. It doesn’t matter how long you’ve had something planned or how well, once you start putting it down, it’s almost guaranteed to change.
And Parno novels at a minimum weigh in at 150k words and every time I end one I know I could have kept going. At some point though I have to start thinking about what the print version will be like and stop.
And this is why some books are faster than others, at least for me. I’m not really familiar with how anyone else does their planning so I may be the odd one out, or I may just be doing the same thing as everyone else in this line of work, I don’t know. My point though is that in the time it takes me to write the next Parno novel, I may very well have been able to write two ‘Fire’ novels and a Stormcrow novel as well. Parno novels are vastly more complicated and have way more moving parts, which slows me down. I like to think it’s worth it and that it makes the Parno series better for it, but it still takes time to put that many wheels into one gear box.
In the time it takes to write the next Stormcrow novel, I still might hammer out two “Fire” novels, again just because the tangled plot of Stormcrow takes longer to work.
So no, I don’t favor one series over the other. I write wherever the inspiration is for that day. Every book has an outline (more like a map, really) so I can pick up where I left off with little to no problem. And I usually write a little bit every day on everything I’m working on at the moment, unless I’m really on a hot streak with one in which case I’ll ride that streak out until I collapse from exhaustion, 😀
So the order of books being finished isn’t a result of my liking one series over another, but simply the fact that each one is so different. They each have their challenges, their advantages, and their drawbacks. Some are easier, some are harder, some are a mix of both.
For those waiting on the next installment of Parno, I’m looking at a Spring release, which I managed to do this year with “Gambit”. One a year in that series is about what I’m going to average I think, and I doubt that would go faster even if it was all I worked on. I would still have times when I couldn’t move the plot along and instead of letting that be idle time, I just work on something else.
For Stormcrow, it may or may not precede the next Parno (tentatively entitled ‘Parno’s Peril’, btw) but I also hope to have it released by Spring, if not sooner.
The next ‘Fire’ novel may well be released around Christmas time.
Remember that all of this is assuming the Lord wills I live so long to get all this done. I do hope that he does of course, but I never take the Lord for granted. He will do as He will.
I hope this helps explain what I know seems to be a complicated system. And it’s just as rough on the proof readers too. Imagine you’re editing the next ‘Stormcrow’ and you suddenly read how Parno is working on the engine of the Celia. Or in ‘Peril’ you read that Clayton has decided to lead a cavalry charge against the Imperial Army. Or Galen working to protect his family farm.
Yeah, that happens. 😀
Let me throw in a special thanks to my wife, as well as Creative Text, my publishers, for all the editing and proofing they do. The thing is, I actually spell really well. What I don’t always do well is type. And I am terrible at catching mistakes. I suffer from a malady that affects many readers in that if I’m already expecting to find something in print, (say because I wrote it, perhaps 😉 ), then I’ll read it whether it’s there or not. That doesn’t happen all the time, but it does happen enough that I miss many errors.
So thank you honey (da wife) and Dan (da publisher) for the great work you do cleaning up my messes.
And thank all of you again for helping me make it to ‘bestseller’ status on Amazon. I will do my utmost to continue measuring up to your expectations, and showing my appreciation for your patronage and your support, as well as your encouragement!
My newest novel, Fire From the Sky, is now available on Amazon. This is the first book in a new Post Apocalyptic series I have been developing for a while now. At this point the series is open ended as I try and come up with as many torments as possible for the protagonists in the series 😀 (insert evil/sinister laugh here)
ANYway, here’s a link where it can be found. If PAW is your thing then I hope it’s worth checking out and you enjoy it.
I have now sent the first volume of my new Post Apoc series “Fire From the Sky” to the publishers. Fire From the Sky is a contemporary story set in the present (well, now it’s the near past, to be honest) and focusing on one family and their friends as they try to survive in a world gone wrong.
UNLIKE Odd Billy Todd and Roland, this book is intended to be the start of a series, so there will be questions that don’t get answers here, probably, and there will be things and people that you may not see again until Book 3 or 4. It is meant to be an ongoing story and at the moment is open ended. I guess as long as I can make the story interesting, I’ll keep writing.
As soon as I know the release date, I’ll post it here and on Facebook.
I have to give a big thanks to my readers as Parno’s Gambit has done well so far. I can’t tell you how exciting it is to know that people are enjoying a story that you put together and wrote basically from scratch. I really appreciate all of you who had taken the time to write and let me know you enjoyed it. As I’ve said before, some nights, when I really wanted to throw my hands up and say “I quit”, that encouragement from you is what kept me working.
Parno’s Gambit, Book III in the Black Sheep of Soulan, will hit the market March 10! Featured here is the dust jacket image for the hardcover edition (YAY) and the blurb from that back cover;
His father murdered by his insane sister. His older brother in exile for attempting a coup against the Crown. His oldest brother very nearly dead at the hands of the same sister and now on the throne only by the grace of a skilled surgeon who wanted to see the man she hoped to marry.
All in all, Parno McLeod’s family troubles have gone from bad to worse of late. And if that wasn’t enough for the young Marshal of the Army, he’s also facing an army that outnumbers his almost three-to-one, fighting on two fronts and stretched thin in numbers, equipment and supplies.
Add to that now an insane sibling controlling an elite regiment of Soulan cavalry and heading south to free her traitorous twin in hopes of putting him on the throne, a flood of refugees from the war torn area of the invasion path of the Norland Imperial Army, spies and saboteurs in their midst, and a sudden startling revelation that despite his successes so far, he is as close to losing the war as he has been since it began.
It’s enough to give anyone a headache.
As Parno struggles to adapt to his new position of importance after so many years of being ignored, he likewise struggles to prevent the enemy from driving any deeper into his Kingdom’s territory and with getting ‘reluctant’ bureaucrats to help him by simply doing their jobs correctly in the King’s name!
Parno has been fortunate, and he is a gifted leader for one so young, but he is young, and inexperienced. He has a plan that will save his people, but he’s about to learn a very important lesson; no plan survives contact with the enemy.
Gambling with his people’s freedom, Parno tries to do too much, too soon, and with too few men. A gambit that he admits is dangerous, and may yet see the capitol of Soulan in flames!
I know it’s been a long road to book Three, but I hope you find it has been worth the wait.
Thank you all so much!
EDITED TO ADD: Just got an update, and Gambit is available for pre-order HERE!
Wow. I had no idea how long I had gone without an entry of any kind. I have been busy as a beaver in a rain forest, and as a result I’ve not posted a single thing for almost a month. Sorry about that.
I’ve been working on my house, which is always fun times, and I’ve been under the weather as well. Throw in a wedding, a revival, Halloween, and the same personal problems that we all have to deal with over the course of our lives, and you get a picture of why it’s been so long between updates.
I’ve finished the short story “Monster of Creasy’s Hollow”, the first chapter of which is posted here as “Terror of Creasy’s Hollow”. It’s still got to make it through the editing process, but as soon as it does, it will go up on Amazon.
Another novel, Roland, has finished the editing stage, and now has to go through the final read and then formatting. Billy had a lot of formatting issues, and I’m trying to keep that from happening with Roland. I’m also trying to work on the formatting problems Billy has, but at 800 pages, it takes a long time.
Meanwhile, I’ve got another novel already complete, with a sequel underway as well, which now must go through the editor. Once it has, it will start it’s way toward being published as well. I’m talking about Parno’s Company of course. I know, I know, I keep saying that, but I’m learning that editing and proof reading take far longer than I thought at first. I do all that I can, but since I’m the writer, I far too often see what I want to see, and miss the mistakes that I’ve made. I’m told that’s a common problem among writers. I know it is for me, as I’ve learned to my chagrin.
In other news, I’ve been prevailed upon, ( is that an actual phrase or did I make that up?) to make the Bonespear Tales into a novel, or at least a short novel, and publish it. I resisted that idea for a long time because I started Bonespear on a whim, and I’ve only been adding to it when I had the time and urge to work on it. It was supposed to just be a fun way to blow off steam. But, it’s proven very popular so far, and I do kinda like old Kalef, sooooo, as soon as I can, it will get the full treatment as well. Possibly before Christmas, but Christmas is coming up waaay faster than I’d planned on.
Today’s election day. Go vote. Let your voice be heard.
I hope all of you had a safe Halloween. Thank you, as always, to those of you who like my work, and let me know it. It means a great deal, I promise you.
Full Disclosure?
I've seen a lot of people who review books and what not posting information about the fact that they receive books from people for free in order to review them. Apparently that's in compliance with some sort of law, or legal action.
I have occasionally posted reviews here on this blog, as anyone who reads it knows. I'm not a professional reviewer, or even really an amateur. I just like to read, and then share what I've read. That's all.
Every book I've ever posted a review for on this blog, I bought. Period. No one paid me or anything else. I just wanted to share my thoughts.
So with no idea whether I needed to state this or not, here ya go.
NC
I am a freelance Photographer born and raised in the Southeast. I have uprooted my life in Macon Georgia for a new life as an unlikely cowgirl in love with a handsome cowboy in Wyoming. I hope you enjoy my photo journal on life, love, and the spirit of Wyoming.
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Q:
WPF - Treeview selected item index
I have a treeview panel. In the panel, there are several child nodes. Some of them are only a header.
The way I create the treeview:
treeviewpaneL.Items.Add(art);
art.Items.Add(prt);
some if statement....
TreeViewItem cldhdr = new TreeViewItem() { Header = "ChildNodes:" };
prt.Items.Add(cldhdr);
TreeViewItem cld = new TreeViewItem() .......
........
.....
cldhdr.Items.Add(cld);
Treeview:
Node1
ChildNodes: (This is header only. It appears if child node exists)
Childnode1
Childnode2
childnode3
Node2
Node3
ChildNodes:
Childnode1
Childnode2
childnode3
Node4
Node5
In my treeview there are also images in front of all nodes. It's a code driven treeview. In the xaml part i have only:
<TreeView x:Name="treeviewpaneL" SelectedItemChanged="treeviewpaneL_SelectedItemChanged" >
</TreeView>
What I want to do is when I click on any of the treeview items, I need to get its index number.
My code is:
private void treeviewpaneL_SelectedItemChanged(object sender, RoutedPropertyChangedEventArgs<object> e)
{
int index = 0;
ItemsControl parent = ItemsControl.ItemsControlFromItemContainer(prt);
foreach (var _item in parent.Items)
{
if (_item == treeviewpaneL.SelectedItem)
{
selectedNodeIndex = index;
MessageBox.Show(selectedNodeIndex.ToString());
break;
}
index++;
}
}
With the code above, I can get the index of Node1,Node2,Node3, Node4 and Node5 as 0,1,2,3,4
What I want is to get the index numbers as:
Node1 = 0
Childnode1 = 1 (Skipping the header)
Childnode2 = 2
Childnode3 = 3
Node2 = 4
....
....
....
What am I missing?
A:
Here is the solution, first of all your "MyTreeViewItem"
public class MyTreeViewItem :TreeViewItem
{
private int _index;
public int Index
{
get { return _index; }
set { _index = value; }
}
public MyTreeViewItem() : base()
{
}
}
and usage;
MyTreeViewItem art = new MyTreeViewItem();
art.Header = "Node1";
art.Index = 1;
MyTreeViewItem prt = new MyTreeViewItem();
prt.Header = "Child1";
prt.Index = 2;
art.Items.Add(prt);
treeviewpaneL.Items.Add(art);
and event;
private void treeviewpaneL_SelectedItemChanged(object sender, RoutedPropertyChangedEventArgs<object> e)
{
MyTreeViewItem selectedItem = e.NewValue as MyTreeViewItem;
if (selectedItem != null)
{
MessageBox.Show("" + selectedItem.Index);
}
}
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---
abstract: 'Consider an arbitrary automorphism of an Enriques surface with its lift to the covering $K3$ surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.'
address:
- 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan'
- 'Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan'
- 'Institute of Mathematics, Jagiellonian University, ul. [Ł]{}ojasiewicza 6, 30-348 Kraków, Poland'
author:
- Yuya Matsumoto
- Hisanori Ohashi
- 'S[Ł]{}awomir Rams'
title: On automorphisms of Enriques surfaces and their entropy
---
[^1]
introduction
============
It is known that the only compact Kähler surfaces that admit automorphisms of positive topological entropy are rational, Enriques, $K3$ surfaces and complex tori (see e.g. [@cantat $\S$ 2.5]). Salem numbers that can be realized as the dynamical degrees of automorphisms of 2-dimensional tori are fully characterized in [@reschke12 Thm 1.1] in terms of values of the minimal polynomials. The same question is solved for rational surfaces in terms of Weyl groups in [@Uehara]. These are exactly the description of the dynamical spectrum $$\Lambda(\mathcal{C})=\{\lambda(f)\in\mathbb{C}\mid \lambda(f)\text{is the dynamical degree of $f\in \mathrm{Aut}(S)$ for some $S\in \mathcal{C}$}\}$$ where the class of surfaces $\mathcal{C}$ is taken to be $2$-tori or rational surfaces. For $K3$ surfaces, the recent preprint [@BF-T] describes the case of degree 22 Salem numbers. Other degrees on $K3$ surfaces and also on Enriques surfaces the description of $\Lambda(\mathcal{C})$ remains open.
The purpose of this note is to give a new property which is satisfied by all automorphisms of Enriques surfaces. As a consequence, we obtain a new constraint on the Salem numbers that appear as the dynamical degrees of automorphisms of Enriques surfaces, namely a property of $\Lambda
(\text{Enriques})$. It should be noted that despite its ergodic interpretation [@cantat $\S$.2.2.2], the problem we consider is purely algebraic, in the sense that the dynamical degree of an automorphism of an Enriques surface $S$ can be detected as the spectral radius of the map it induces on $\mathrm{Num}(S)$ (see $\S$.\[sect-dynamical-degrees\]).
To state the theorem, let $S$ be an Enriques surface and let $\tilde{S}$ be its $K3$-cover. We denote by $\varepsilon$ the covering involution of the double étale cover $\pi: \tilde{S} \rightarrow S$ and put $N$ to denote the orthogonal complement of the $\varepsilon$-invariant sublattice $H^2(\tilde{S},\Z)^{\varepsilon}$ in the lattice $H^2(\tilde{S},\Z)$: $$\label{eq-latN}
N = (H^2(\tilde{S},\Z)^{\varepsilon})^{\perp} \, .$$ Recall that for an arbitrary automorphism $f\in \mathrm{Aut}(S)$, there exists a lift $\tilde{f} \in \mathrm{Aut}(\tilde{S})$. Obviously the lift in question is not unique (given $\tilde{f}$, the automorphism $\tilde{f} \circ \varepsilon$ is also a lift of $f$), but the constraints we prove are valid for any choice of $\tilde{f}$. As is well-known (see e.g. [**[@namikawa]**]{}), the lattice $N$ is stable under the cohomological action $\tilde{f}^*$, hence the restriction $${f_N}:= \tilde f^* \rvert_N$$ is an automorphism (isometry) of $N$. It is easy to see that the order $\ord({f_N})$ is finite, Lemma \[Nfin\]. Here we show a more precise constraint on the order of the map ${f_N}$:
\[th1\] Let $S$ be an Enriques surface and let $f\in \mathrm{Aut}(S)$. Then, the order of ${f_N}$ is an integer which divides at least one of the integers $$120, 90, 84, 72, 56, 48.$$ Equivalently and explicitly, these are one of the 31 integers $$\label{eq-T1}
120,90,84,72,60,56,
48,45,42,40,36,30,28, 24,21,20,18, 16, 15,14,12,10,\dots,1.$$
We use the above theorem to derive the following mod 2 constraint for a Salem number to be the dynamical degree of automorphisms of Enriques surfaces (which refines [@third Lemma 4.1]):
\[mod2\] Let $f$ be an automorphism of an Enriques surface $S$ and let $s_{\lambda}$ be the minimal polynomial of its dynamical degree $\lambda(f)$. Then the modulo $2$ reduction of $s_{\lambda}$ is a product of (some of) the following polynomials $$\begin{aligned}
F_{ 1}(x) &= x + 1, \quad
F_{ 3}(x) = x^2 + x + 1, \quad
F_{ 5}(x) = x^4 + x^3 + x^2 + x + 1, \\
F_{ 7}(x) &= x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, \\
F_{ 9}(x) &= x^6 + x^3 + 1, \\
F_{15}(x) &= x^8 + x^7 + x^5 + x^4 + x^3 + x + 1, \\\end{aligned}$$
Here each $F_m(x) \in \F_2[x]$ is the modulo $2$ reduction of the $m$-th cyclotomic polynomial $\Phi_m(x)\in \mathbb{Z}[x]$. Among these six polynomials, $F_7(x)$ and $F_{15}(x)$ are products of two distinct irreducible factors, each of which is not self-reciprocal, whereas the other four are irreducible.
We do not know whether all the six factors above do appear among factorizations of the minimal polynomial of the map induced by an automorphism of Enriques surfaces, but we are able to give examples where $F_1(x), F_3(x), F_5(x)$ do come up (see Example \[example-existence-3factors\].a). On the other hand, we can check that they all appear from some [*[lattice isometry]{}*]{} of $U\oplus E_8$ by using lattice theory and ATLAS table, for example.
A closely related problem to the description of the dynamical spectrum is to find the minimal nontrivial dynamical degree $1\neq \lambda\in \Lambda(\mathcal{C})$.
This question was answered for complex tori (see [@mcmullen11 Thm 1.3]), rational and $K3$ surfaces by McMullen (see [@mcmullen07; @mcmullen11; @mcmullen16]), whereas the smallest dynamical degree attained by automorphisms of Enriques surfaces is yet to be found ([@third Question 4.5.(3)]). By [@third Remark 4.4], none of the smallest five Salem numbers (including the ones of degree $>10$) can be realized on Enriques surfaces. Although explicit descriptions of automorphism groups of several special families (see [@BP; @MO15]) of Enriques surfaces are known, our present knowledge seems not to be enough to determine the minimal dynamical degree of automorphisms of surfaces in this class. Presently the smallest known dynamical degree of an automorphism of an Enriques surface is the one constructed by Dolgachev [@dolgachev16 Table 2], who found an automorphism of an Enriques surface (of Hesse type) of dynamical degree $\lambda_D=2.08101\ldots$ (see Example \[Dolgachevexample\] for more details). As to this respect, in Example \[MO15family\], we give an additional study on the family of [@MO15] to show that all nontrivial dynamical degrees of automorphisms of Enriques surfaces in [@MO15] are at least $\lambda_D$. The result thus fails to give a new lower bound, but gives a good account for what is going on.
To fill the gap, the constraint given by Thm \[mod2\] works to give the following slightly better theoretical lower bound:
\[th2\] The dynamical degree of an automorphism of an Enriques surface is greater than or equal to the Salem number $\lambda=1.35098\cdots$ given by the polynomial $$x^{10}-x^9-x^6+x^5-x^4-x+1.$$
We use Theorem \[mod2\] together with [@Gross--McMullen], combined with Dolgachev’s example [@dolgachev16 Table 2] to show that the smallest (non-trivial) dynamical degree of an automorphism of an Enriques surface must be one of the 39 Salem numbers which we list in $\S.$\[sect-appendix\] Appendix.
Our approach is inspired by Oguiso’s proof of [@third Thm 1.2]. We consider mod 2 reduction of the cohomological action, and apply results from [@Gross--McMullen] to obtain a detailed picture. It should be noted that our approach does rule out numerous Salem numbers (see Example \[rem-possibilities\]), but Thm \[mod2\] cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface. One possible way of determining the exact minimal value of non-trivial dynamical degrees of automorphisms of Enriques surfaces will be that the remaining $39$ cases can be treated efficiently by some refinement of McMullen’s method [@mcmullen16], but this task exceeds the scope of this paper.
[**Convention:**]{} In this note we work over the field of complex numbers $\C$.
Proof of Theorem \[th1\] {#sect-th1}
========================
We maintain the notation of the introduction: $S$ is assumed to be an Enriques surface and $f\in \mathrm{Aut}(S)$ is an automorphism. As is well-known, the canonical cover $\tilde{S}=\mathrm{Spec}(\mathcal{O}\oplus \mathcal{O}(K_S))$ is a $K3$ surface and the morphism $\pi: \tilde{S} \rightarrow S$ is a double étale cover. We denote by $\varepsilon$ the covering involution of $\pi$. Since the automorphism $f$ preserves the canonical class $K_S\in \mathrm{Pic}(S)$, $f$ lifts to an automorphism $\tilde{f}$ of $\tilde{S}$.
Recall that for an Enriques surface the lattice $\mathrm{Num}(S)$ is the free part of the cohomology group $H^2(S,\Z)$. Let $$M:=H^2(\tilde{S},\Z)^{\varepsilon}$$ be the $\varepsilon$-invariant sublattice of the cohomology lattice $H^2(\tilde{S},\Z)$ and let $N:= M^\perp$ be its orthogonal complement. The direct orthogonal sum $M \oplus N$ is a finite index sublattice of the lattice $H^2(\tilde{S},\Z)$. Moreover, we know by [@namikawa Proposition (2.3)] that $M$ coincides with the pullback of $H^2(S,\Z)$ by $\pi$, hence we have the isomorphisms $$\label{eq-enriques-isomorphisms}
M \simeq \mathrm{Num}(S)(2) \simeq U(2)\oplus E_8(2) \text{ and } N\simeq U\oplus U(2)\oplus E_8(2),$$ where $U$ denotes the unimodular hyperbolic plane and $E_8$ is the unique even unimodular negative-definite lattice of rank 8. Moreover, for a lattice $L$ and $n\in \Q$, $L(n)$ denotes the lattice whose underlying abelian group is the same as $L$ and the bilinear form is multiplied by $n$. Basic facts concerning integral symmetric bilinear forms can be found in [@nikulin-sym].
Since the lift $\tilde{f}$ commutes with the involution $\varepsilon$, the map it induces on the cohomology lattice preserves sublattices $M$ and $N$. We put $${f_M}:= \tilde f^* \rvert_M \mbox{ and } {f_N}:= \tilde f^* \rvert_N \, .$$ Obviously, ${f_N}$ induces an isometry of the quadratic space $N \otimes \R$ of signature $(2,10)$ that preserves the original lattice $N =: N_{\Z} \subset N \otimes \R$ and the Hodge structure of $N$. Since the latter is exactly given by an oriented positive 2-plane in $N \otimes \R$, we have $${f_N}\in O(N_{\Z})\cap (O(2)\times O(10)).$$ Since the right-hand side is a discrete subgroup in a compact group, we obtain the following well-known fact.
\[Nfin\] The map ${f_N}$ is of finite order.
By Lemma \[Nfin\], the characteristic polynomial of ${f_N}$ is a product $$\label{eq-charpolfac}
{p}_N(x)=\det (xI-{f_N}) = \prod_{i=1}^k \Phi_{n_i}(x)$$ of cyclotomic polynomials $\Phi_{n_i}(x)$ for a collection of positive integers $\{n_i :i=1, \ldots, k\}$ with $\sum_{i=1}^k {\varphi}(n_i) = 12$, where ${\varphi}(\cdot)$ stands for the Euler totient function. Obviously, the order of ${f_N}$ is just the least common multiple $$\label{eq-ord-tdfn}
\ord({f_N}) = \operatorname{lcm}\{n_i, i = 1, \ldots,k\}.$$
The integers $m$ for which ${\varphi}(m)\leq 12$ are as follows.
\[table-pphi\] ${\varphi}(m)$ $m$
------------------------------- ---------------------
$12$ $13,21,26,28,36,42$
$10$ $11,22$
$ 8$ $15,16,20,24,30$
$ 6$ $7,9,14,18$
$ 4$ $5,8,10,12$
$ 2$ $3,4,6$
$ 1$ $1,2$
The proof of Thm \[th1\] will be based on the following lemmas.
\[lem1-new\] Let ${p}_N(x)$ be the characteristic polynomial of the map ${f_N}$.
[(a)]{} The reduction $({p}_N(x) \bmod 2)$ is divisible by $(x^2+1)=(x+1)^2$.
[(b)]{} ${p}_N(1) {p}_N(-1)$ is either zero or a square in $\mathbb{Q}^{*}$.
\(a) Let us consider the action of ${f_N}$ on the reduction $N\otimes \mathbb{F}_2 \cong (1/2)N/N$. Obviously, the reduction contains the 10-dimensional ${f_N}$-invariant subspace $$\label{eq-10dim}
N^*/N \subset(1/2)N/N,$$ where $N^*/N$ is the discriminant group of $N$. Thus the reduction $({p}_N(x) \bmod 2)$ is divisible by a degree two polynomial.
In fact, there exists an $11$-dimensional canonical subspace $N^{*,+}$ of the reduction $(1/2)N/N$ containing $N^*/N$. We discuss as follows. The residue group $(1/2)N/N^*$ consists of four residue classes modulo $N^*$ and $N^*$ has the property that for all $y\in N^*$, $(y,y)\in \Z$ (namely $\delta (N)=0$ in Nikulin’s notation.) Hence, the induced quadratic form $(1/2)N/N^*\rightarrow \Q/\Z$ is well-defined. Among the four residue classes, there exists a unique nonzero element whose form value is $1/2$, which corresponds to $N^{*,+}$. (The idea of this proof parallels [@allcock]). Thus we obtain (a).
\(b) Claim follows from [@BF Proposition 5.1].
\[lem-yuya\] The order $\mathrm{ord}(f_N)$ cannot be one of the integers $11, 22, 35, 70$.
We put $p_M(x) = \sum_{i=0}^{10} a_i x^i$ (resp. $p_{N^*/N}(x)$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. of the map induced by ${f_N}$ on the discriminant group $N^*/N$). Since $M \oplus N$ is a finite index sublattice of the unimodular lattice $H^2(\tilde{S}, \Z)$ we have a canonical isomorphism of the discriminant groups $$M^*/M \cong N^*/N \, ,$$ which is $({f_M},{f_N})$-equivariant. Moreover, from $M^*/M = \frac{1}{2}M/M$ and the inclusion we infer $$\label{eq-pnstarn}
p_{N^*/N}(x) = (p_M(x) \bmod 2) \quad \mbox{and} \quad p_{N^*/N}(x) {\mid}(p_N(x) \bmod 2) .$$
Assume $\mathrm{ord}(f_N) \in \{35, 70\}$. Then, by and the table on p. we have $$\label{eq-pnx}
p_N(x) = \Phi_{5 l_1}(x) \Phi_{7 l_2}(x) \Phi_{l_3}(x) \Phi_{l_4}(x) \quad \mbox{ with } l_1, l_2, l_3, l_4 \in \{1,2\} \, .$$ Thus implies that $$(p_M(x) \bmod 2) = p_{N^*/N}(x) = F_5(x) F_7(x) = x^{10} + x^8 + x^6 + x^5 + x^4 + x^2 + 1 ,$$ so the coefficient $a_5$ and the sums $a_0 + a_2 + a_4$, $a_6 + a_8 + a_{10}$ are odd integers. In particular, the polynomial $p_M(x)$ is self-reciprocal (see e.g. [@cantat]), so the product $p_M(1)p_M(-1)$ can be expressed as $$\begin{aligned}
\bigl(\sum_{i:even} a_i\bigr)^2 - \bigl(\sum_{i:odd}a_i\bigr)^2 &=& (2(a_0 + a_2 + a_4))^2 - (2(a_1 + a_3) + a_5)^2 \\
&=& (2 \bmod 4)^2 - (1 \bmod 2)^2 \end{aligned}$$ Thus $p_M(1)p_M(-1) \equiv 3 \bmod 8$, which contradicts [@BF Proposition 5.1].
If $\mathrm{ord}(f_N) \in \{11, 22\}$, then the factorization of $p_N(x)$ is as follows $$\label{eq-pnx11}
p_N(x) = \Phi_{11 l_1}(x) \Phi_{l_2}(x) \Phi_{l_3}(x) \quad \mbox{ with } l_1, l_2, l_3 \in \{1,2\} \, .$$ Thus we have $(p_M(x) \bmod 2) = F_{11}(x) $ and, as in the previous case we obtain $p_M(1)p_M(-1) \equiv 3 \bmod 8$.
The proof of Lemma \[lem-yuya\] shows that the characteristic polynomial $p_M(x) = \sum_{i=0}^{10} a_i x^i$ of ${f_M}$ cannot be a polynomial such that the coefficient $a_5$ and the sums $a_0 + a_2 + a_4 = a_6 + a_8 + a_{10}$ are odd integers, i.e. the modulo 2 reduction $(p_M(x) \bmod 2)$ cannot be one of the polynomials $$F_5(x) F_7(x), F_{11}(x),
F_3(x)^5, F_3(x)^2 F_9(x), F_3(x) F_5(x)^2, F_3(x) F_{15}(x) \, .$$
The next lemma rules out the first two lines of the table on p. . It is also of use in the next section.
\[lem12\] If $\Phi_{m}(x)$ comes up in the factorization , then $${\varphi}(m) < 10 \, .$$
If ${\varphi}(m)=12=\operatorname{rank}N$, then the characteristic polynomial ${p}_N(x)$ equals the cyclotomic polynomial $\Phi_m(x)$. The decomposition of $\Phi_m(x) \bmod 2$ into irreducible factors is given in the table below:
$m$ irreducible decomposition of $\Phi_m \bmod 2$
------- -----------------------------------------------
42,21 $(x^6+x^4+x^2+x+1)(x^6+x^5+x^4+x^2+1)$
36 $(x^6 + x^3 + 1)^2$
28 $(x^3 + x + 1)^2 (x^3 + x^2 + 1)^2$
26,13 $x^{12} + x^{11} + \cdots + x + 1$
Thus ${\varphi}(m) < 12$ by Lemma \[lem1-new\].a.
Suppose that ${\varphi}(m)=10$, namely $m=11,22$. Then $\mathrm{ord}(f_N)$ is a multiple of 11, and taking some power we get a contradiction to Lemma \[lem-yuya\].
After these preparations we can give the proof of Thm \[th1\].
Let $p(x) = \prod_{i=1}^k \Phi_{n_i}(x)$ be a product of cyclotomic polynomials for some $n_1$, $\ldots$ $n_k \in \mathbb{N}$ such that $\sum_{i=1}^k {\varphi}(n_i) = 12$ and ${\varphi}(n_i) < 10$ for $i=1, \ldots, k$. If $p(x)$ is the characteristic polynomial of the map ${f_N}$ induced by an automorphism of an Enriques surface, then it satisfies the conditions (a),(b) of Lemma \[lem1-new\] and the order of ${f_N}$ is given by . An enumeration of all cases (by hand or by a help of a computer) and Lemma \[lem-yuya\] show that $n = \operatorname{lcm}\{n_i, i = 1, \ldots,k\}$ is one of the integers that appear in Theorem \[th1\].
\[rem-finite-ohashi\] To illustrate Theorem \[th1\], we shall give a classification of the case when the order of $f\in \mathrm{Aut}(S)$ is finite. Such automorphisms were classified in [@MO1; @ohashi-birep]. The following table gives the complete classification of the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$.
$\mathrm{ord} (f)$ 1 2 3 4 5 6 8
---------------------- ----- ----- ----- ------- ------ ----- -----
$\mathrm{ord} (f_N)$ 1,2 1,2 3,6 1,2,4 5,10 3,6 4,8
We list here nontrivial examples exhibiting the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$ and leave the proofs to the reader. The pair $(2,1)$ is supplied by No. 18 in [@IO]. When $\mathrm{ord} (f)=3$, $5$ or $6$, $f$ is semi-symplectic by [@MO1 Proposition 4.5] (i.e. $f$ acts trivially on the space $H^0(S, \mathcal{O}(2K_S))$) and we can use the symplectic lift to compute eigenvalues. Examples 1.1 and 1.2 of [@ohashi-birep] give the pairs $(4,4)$ and $(8,8)$. Finally, Example 1.3 of [@ohashi-birep] provides the remaining possibilities for $\mathrm{ord} (f)=4$ or $8$.
As Example \[example-existence-3factors\] shows, the order $\mathrm{ord} (f_N)$ is no longer bounded by $10$ when the order of $f \in \mathrm{Aut}(S)$ is infinite (see ). However, we have very few examples: the question of determining the exact list of possible $\ord({f_N})$ remains open.
Dynamical degrees {#sect-dynamical-degrees}
=================
We maintain the notation of the previous section. For the convenience of the reader we recall the definition of the dynamical degree.
Let $f \in \mathrm{Aut}(S)$. The [**dynamical degree $\lambda(f)$ of**]{} $f$ is defined as the spectral radius of the map $f^*:\mathrm{Num}(S) \rightarrow \mathrm{Num}(S)$. One can show that the map $f^*$ has either none or exactly two eigenvalues away from the unit circle in $\C$. If such two eigenvalues come up, they are real and reciprocal. Thus either $\lambda(f) = 1$ or it is the largest real eigenvalue of the map $f^*$ (for a precise discussion of the above notion and its properties see [@cantat $\S$.2.2.2], [@mcmullen11], [@mcmullen16] and references therein).
After these preparations we are in position to give the proof of Thm \[mod2\] (c.f. [@third proof of Lemma 4.1]).
We put $p_M$ (resp. $p_N$, resp. $p_f$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. ${f_N}$ on $N$, resp. $f^*$ on $\mathrm{Num}(S)$). We assume that $\lambda(f) \neq 1$ and denote the minimal polynomial of $\lambda(f)$ by $s_\lambda$.
By the first isomorphism in , the action of $f^*$ on the discriminant group of the lattice $\mathrm{Num}(S)(2)$ coincides with the action of ${f_M}$ on the discriminant group $M^*/M$. From $M^*/M = \frac{1}{2}M/M$, we obtain the equality of modulo 2 reductions: $$p_M \equiv p_f \bmod 2 .$$ Moreover yields: $$(p_M \bmod 2) {\mid}(p_N \bmod 2) .$$
Let $h$ be an irreducible factor of the reduction $(s_\lambda \bmod 2)$. We have just shown $h$ appears also in the factorization of $(p_N \bmod 2)$. Then, from and Lemma \[lem12\], $h$ divides $(\Phi_m \bmod 2)$ for certain $m$ such that ${\varphi}(m) \leq 8$. Since $(\Phi_{2^e m} \bmod 2)$ is a power of the reduction $(\Phi_m \bmod 2)$, we may assume $m$ to be odd. Hence, by the table on p. , we have $m = 1,3,5,7,9,15$. Thus either $h = F_m$, with $m=1,3,5,9$ or $h$ appears in the factorization of $F_m$, where $m=7,15$. But, for $m=7,15$ we have $F_m = F_{m,1} \cdot F_{m,2}$ where $$\begin{aligned}
F_{7,1} := (x^3 + x + 1),
& \quad F_{7,2} := (x^3 + x^2 + 1), \\
F_{15,1} := (x^4 + x + 1),
& \quad F_{15,2} := (x^4 + x^3 + 1).\end{aligned}$$ Being a Salem polynomial, $s_\lambda$ is self-reciprocal, and so is its modulo $2$ reduction. Since the polynomials $F_{7,1}$ and $F_{7,2}$ are not self-reciprocal, their multiplicities in $(s_\lambda \bmod 2)$ should coincide with those of their reciprocal counterparts. The same holds for $F_{15,1}$ and $F_{15,2}$. This completes the proof.
It is natural to ask which of the six factors given in Thm \[mod2\] do appear in modulo 2 reductions of minimal polynomials of dynamical degrees of automorphisms of Enriques surfaces. We do not know whether the polynomials $F_{7}(x)$, $F_{9}(x)$, $F_{15}(x)$ are realized by automorphisms of Enriques surfaces. To answer the question whether $F_m$ where $m=1,3,5$ come up in $(s_{\lambda} \bmod 2)$, we analyze some automorphisms constructed in [@dolgachev16].
\[example-existence-3factors\] (a) By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ equals[^2] $$\label{eq-f5f3}
p_{f^*}(x) = x^{10} - 6x^9 - 7x^8 - 9x^7 - 6x^6 - 10x^5 - 6x^4 - 9x^3 - 7x^2 - 6x +1 .$$ One can check $p_{f^*}(x) \in \Z[x]$ is irreducible and we have $$(p_{f^*}(x) \bmod 2) = F_5(x) \cdot F_3(x) \cdot F_1^4(x).$$ In particular, combined with the proof of Thm \[mod2\] yields that $$\label{eq-order15}
15 {\mid}\ord({f_N}).$$
\(b) According to [@dolgachev16 Sect. 3.2, Case $m = 4$] there exists an Enriques surface $S$ and $f \in \mathrm{Aut}(S)$ such that $p_{f^*}(x)$ is divisible by the following polynomial $$\label{eq-dolgtocheck}
x^8 - 165x^7 + 223x^6 - 59x^5 - 133x^4 - 59x^3 + 223x^2 - 165x + 1 \, .$$ Since the modulo 2 reduction of the above polynomial factors as the product $(F_3(x) \cdot F_9(x))$, this would imply that $F_9(x)$ can also appear in the reduction of the minimal polynomial $s_{\lambda}$. Unfortunately, there is a misprint in [@dolgachev16 Sect. 3.2, Case $m = 4$]: the term $x^4$ comes with the coefficient $(-144)$ (instead of $(-133)$ as in ). Thus the question whether the factor $F_9(x)$ is possible or not remains open.
The example below shows that even the direct potential refinement of Thm \[mod2\] (i.e. ruling out all/some of the factors $F_7$, $F_9$, $F_{15}$ in the modulo 2 reduction of the characteristic polynomial) cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface.
\[example-strange\] Consider the Salem number $\lambda = 1.64558...$ given by the polynomial $$\label{eq-mini-salem}
s_{\lambda} := x^{10}-x^9-x^8-x^2-x+1 \, .$$ As one can easily check, we have $$(s_{\lambda} \bmod 2) = F_3(x) \cdot F_1^8(x).$$ Thus Thm \[mod2\] does not rule out the above number as the dynamical degree of an automorphism of an Enriques surface. But we can apply [@Gross--McMullen Theorem 6.1]. Suppose that the Salem number given by is the dynamical degree of an automorphism of an Enriques surface $S$. The lattice $\mbox{Num}(S)$ is of rank 10, so $s_{\lambda}$ is the full characteristic polynomial of the induced automorphism on $\mbox{Num}(S)$. Also $\mathrm{Num}(S)$ is even unimodular, so by [@Gross--McMullen Theorem 6.1] both $|s_{\lambda}(\pm 1)|$ must be squares, which is not the case. This contradiction shows that cannot be the minimal polynomial of the dynamical degree of an automorphism of an Enriques surface.
Presently, the smallest known non-trivial dynamical degree is the one constructed by Dolgachev:
\[Dolgachevexample\] By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ has the Salem polynomial $$\label{eq-mp-lambdaD}
x^4-x^3-2x^2-x+1$$ as a factor. The largest real root of the above polynomial is $$\lambda_D := 2.08101... \, .$$ This gives the smallest known value of non-trivial dynamical degree of an automorphism of an Enriques surface.
\[MO15family\] As an another example, let us consider the Enriques surface $S$ in the family of [@MO15]. In that paper, it is proved that $\mathrm{Aut}(S)\simeq C_2^{*4}\rtimes \mathfrak{S}_4$. We here prove that every element in this group has dynamical degree at least $\lambda_D$.
In fact, [@MO15] exhibits 10 smooth rational curves $E_i,\ E_{ij}\ (i,j=1,\dots,4,\ i\neq j)$ on $S$ which form a rational basis of $\mathrm{Num}(S)$. We denote by $L$ the sublattice generated by them. The symmetric group $\mathfrak{S}_4$ acts on the indices of generators, while the generators $s_i$ of the four cyclic groups $C_2$ act by reflections in some divisors $G_i\in L$ of self-intersection $-2$. In particular, $\mathrm{Aut}(S)$ preserves $L$. We use the relations $(G_i, E_{kl})=0$ (for all $i,k,l$) and $(G_i, l)\in 2\mathbb{Z}$ for all $l\in L$.
Now from $(G_i, E_{kl})=0$, the six-dimensional subspace $V$ generated by $E_{ij}$ is stable under $\mathrm{Aut}(S)$. Since it is negative-definite, we see that any Salem number on $S$ has degree at most four, the dimension of the complement to $V$. Thus for any element in $\mathrm{Aut}(S)$, the characteristic polynomial $F$ decomposes into degree four (on $L_{\mathbb{C}}/V_{\mathbb{C}}$) and degree six (on $V_{\mathbb{C}}$). Moreover, since $G_i$ intersects evenly with all $l\in L$, it acts trivially on $L/2L$. Hence the degree four part of $(F \bmod 2)$ decomposes either into linear factors or has only one non-linear factor $x^2+x+1$, which arises when the residue class has order 3 in $\mathfrak{S}_4$. Looking through the list in the Appendix, we get the assertion. (This is something unfortunate, although.)
We use Theorem \[mod2\] on low-degree Salem numbers that do not exceed $\lambda_D$.
\[rem-possibilities\] One can check that there are exactly 133 Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$. (The list of small Salem numbers can be found in [@mossinghoff].) Oguiso’s criterion [@third Lemma 4.1] shows that 28 among them cannot be dynamical degrees of automorphisms of Enriques surfaces. To go further, combining Theorem \[mod2\] and [@Gross--McMullen] we can prove that 65 more cannot be realized by automorphisms of Enriques surfaces. For the convenience of readers we list the 133 Salem numbers in question, their minimal polynomials and their modulo 2 reductions in $\S.$\[sect-appendix\] Appendix. Those numbers excluded by modulo 2 reductions are marked “impossible”. Non-existence that results from [@Gross--McMullen Theorem 6.1] (see Example \[example-strange\]) is marked “impossible (\*)”. (In some cases both apply.)
In conclusion, [*there remain $39$ Salem numbers as candidates for the minimal dynamical degree of automorphisms of Enriques surfaces.*]{}
On the arXiv[^3] we attach two lists, in plain text format, of the coefficients of the minimal polynomials of (1) these $39$ candidates, and (2) all $133$ Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$.
Finally we can give the proof of Corollary \[th2\].
There are only finitely many Salem numbers less than $\lambda = 1.35098...$ and of degree $\leq 10$ (there are none of degree $\leq 6$, one of degree $8$, and six of degree $10$). By a straightforward calculation each of those numbers violates the condition of Thm \[mod2\].
For explicit factorization see the first seven entries of the table in $\S.$\[sect-appendix\] Appendix.
[**Acknowledgement:**]{} This project was completed during the workshop “New Trends in Arithmetic and Geometry of Algebraic Surfaces” in BIRS (Banff, Canada). The authors would like to thank the organizers and the staff of BIRS for very good working conditions. We thank the referee for inspiring comments and calling our attention to the paper [@BF].
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Appendix: List of low-degree Salem numbers up to $\lambda_D$ {#sect-appendix}
============================================================
Below we list all Salem numbers of degree $\leq 10$ up to the dynamical degree $\lambda_D$ of Dolgachev’s example and show the minimal polynomial $s_\lambda$ and its mod 2 factorization. Some of them are impossible by Thm \[mod2\] and marked “impossible”. Others marked “impossible (\*)” are those excluded by [@Gross--McMullen Theorem 6.1].
As noted in Remark \[rem-possibilities\], the list of the coefficients of the minimal polynomials are available in plain text format on the arXiv.
---- ----- ------------ ------------------------------------------------------------------ -------------------
\# deg value minimal polynomial $s_\lambda$
factorization of $s_\lambda \bmod 2$ conclusion
10 1.17628... $x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
10 1.21639... $x^{10}-x^6-x^5-x^4+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 1.23039... $x^{10}-x^7-x^5-x^3+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
10 1.26123... $x^{10}-x^8-x^5-x^2+1$
\* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
8 1.28063... $x^8-x^5-x^4-x^3+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
10 1.29348... $x^{10}-x^8-x^7+x^5-x^3-x^2+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
10 1.33731... $x^{10}-x^9-x^5-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
1 10 1.35098... $x^{10}-x^9-x^6+x^5-x^4-x+1$
\* $(x^2 + x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$
8 1.35999... $x^8-x^7+x^6-2x^5+x^4-2x^3+x^2-x+1$
\* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible
10 1.38363... $x^{10}-x^9-x^7+x^6-x^5+x^4-x^3-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
2 6 1.40126... $x^6-x^4-x^3-x^2+1$
\* $(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)$
3 8 1.42500... $x^8-x^7-x^5+x^4-x^3-x+1$
\* $(x^4 + x + 1) (x^4 + x^3 + 1)$
10 1.43100... $x^{10}-x^9-x^8+x^7-x^5+x^3-x^2-x+1$
\* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
10 1.44842... $x^{10}-2x^9+2x^8-2x^7+x^6-x^5+x^4-2x^3+2x^2-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
4 8 1.45798... $x^8-x^6-x^5-x^3-x^2+1$
\* $(x + 1)^2 (x^6 + x^3 + 1)$
10 1.47235... $x^{10}-x^9-x^6-x^4-x+1$
\* $(x + 1)^6 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*)
10 1.47960... $x^{10}-2x^8-2x^7+x^6+3x^5+x^4-2x^3-2x^2+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
5 6 1.50613... $x^6-x^5-x^3-x+1$
\* $(x^2 + x + 1)^3$
10 1.51386... $x^{10}-x^7-2x^6-x^5-2x^4-x^3+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
8 1.52306... $x^8-x^7-x^6+x^4-x^2-x+1$
\* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible
6 10 1.53292... $x^{10}-x^9-x^8+x^5-x^2-x+1$
\* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$
8 1.54719... $x^8-2x^7+2x^6-3x^5+3x^4-3x^3+2x^2-2x+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
7 6 1.55603... $x^6-x^5-x^4+x^3-x^2-x+1$
\* $(x^3 + x + 1) (x^3 + x^2 + 1)$
8 6 1.58234... $x^6-x^4-2x^3-x^2+1$
\* $(x + 1)^6$
10 1.59070... $x^{10}-2x^9+x^8-2x^6+3x^5-2x^4+x^2-2x+1$
\* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
10 1.59700... $x^{10}-x^8-x^7-x^6-x^5-x^4-x^3-x^2+1$
\* $(x^2 + x + 1)^2 (x^6 + x^3 + 1)$ impossible (\*)
10 1.59866... $x^{10}-2x^9+x^8-x^7+2x^6-3x^5+2x^4-x^3+x^2-2x+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
9 8 1.60544... $x^8-2x^7+x^6-x^4+x^2-2x+1$
\* $(x^4 + x^3 + x^2 + x + 1)^2$
10 1.62501... $x^{10}-x^9-x^8-x^7+x^6+x^5+x^4-x^3-x^2-x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
10 1.62754... $x^{10}-2x^9+2x^7-x^6-x^5-x^4+2x^3-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 6 1.63557... $x^6-2x^5+2x^4-3x^3+2x^2-2x+1$
\* $x^6 + x^3 + 1$
8 1.64003... $x^8-2x^6-x^5+x^4-x^3-2x^2+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
10 1.64558... $x^{10}-x^9-x^8-x^2-x+1$
\* $(x + 1)^8 (x^2 + x + 1)$ impossible (\*)
10 1.65740... $x^{10}+x^9-2x^7-4x^6-5x^5-4x^4-2x^3+x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
11 8 1.66104... $x^8-2x^7+x^6-x^5+x^4-x^3+x^2-2x+1$
\* $(x^2 + x + 1) (x^3 + x + 1) (x^3 + x^2 + 1)$
10 1.66929... $x^{10}-x^9-x^7-2x^5-x^3-x+1$
\* $(x + 1)^2 (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
10 1.67310... $x^{10}-2x^9+x^8-x^7+x^5-x^3+x^2-2x+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
12 8 1.68491... $x^8-x^7-x^6-x^2-x+1$
\* $(x + 1)^2 (x^2 + x + 1)^3$
10 1.69017... $x^{10}-x^9-2x^8+x^7+x^6-x^5+x^4+x^3-2x^2-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
13 8 1.69350... $x^8-x^7-x^5-x^4-x^3-x+1$
\* $(x^4 + x + 1) (x^4 + x^3 + 1)$
10 1.71336... $x^{10}-2x^9+x^8-2x^6+2x^5-2x^4+x^2-2x+1$
\* $(x + 1)^{10}$ impossible (\*)
14 4 1.72208... $x^4-x^3-x^2-x+1$
\* $x^4 + x^3 + x^2 + x + 1$
10 1.73694... $x^{10}-x^9-x^7-x^6-x^5-x^4-x^3-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
10 1.74492... $x^{10}-2x^9+2x^8-3x^7+2x^6-3x^5+2x^4-3x^3+2x^2-2x+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
10 1.74601... $x^{10}-x^9-x^8-x^7+2x^5-x^3-x^2-x+1$
\* $(x + 1)^4 (x^3 + x + 1) (x^3 + x^2 + 1)$ impossible (\*)
10 1.75173... $x^{10}-2x^9+x^8-x^7+x^6-2x^5+x^4-x^3+x^2-2x+1$
\* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
10 1.75309... $x^{10}-x^8-x^7-2x^6-3x^5-2x^4-x^3-x^2+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
10 1.76015... $x^{10}-2x^9+x^8-2x^7+2x^6-x^5+2x^4-2x^3+x^2-2x+1$
\* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
10 1.76400... $x^{10}-2x^9+x^7-x^5+x^3-2x+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
10 1.76690... $x^{10}-2x^8-2x^7+x^5-2x^3-2x^2+1$
\* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*)
10 1.77056... $x^{10}-3x^9+4x^8-5x^7+5x^6-5x^5+5x^4-5x^3+4x^2-3x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
15 6 1.78164... $x^6-x^5-x^4-x^2-x+1$
\* $(x + 1)^4 (x^2 + x + 1)$
10 1.78840... $x^{10}-x^9-2x^7-x^5-2x^3-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
8 1.79607... $x^8-x^7-x^6-x^4-x^2-x+1$
\* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible
10 1.79978... $x^{10}-3x^8-3x^7+2x^6+5x^5+2x^4-3x^3-3x^2+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
16 8 1.80017... $x^8-3x^7+4x^6-5x^5+5x^4-5x^3+4x^2-3x+1$
\* $(x^4 + x + 1) (x^4 + x^3 + 1)$
17 10 1.80501... $x^{10}-2x^9+x^7-x^6+x^5-x^4+x^3-2x+1$
\* $(x^2 + x + 1)^3 (x^4 + x^3 + x^2 + x + 1)$
18 8 1.80978... $x^8-x^7-2x^5-2x^3-x+1$
\* $(x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$
8 1.81161... $x^8-2x^7+x^5-x^4+x^3-2x+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
10 1.82383... $x^{10}-x^8-2x^7-2x^6-2x^5-2x^4-2x^3-x^2+1$
\* $(x + 1)^{10}$ impossible (\*)
19 10 1.82514... $x^{10}-x^9-2x^8+x^6+x^5+x^4-2x^2-x+1$
\* $(x^2 + x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$
20 6 1.83107... $x^6-2x^5+x^3-2x+1$
\* $x^6 + x^3 + 1$
21 8 1.83488... $x^8-x^6-2x^5-3x^4-2x^3-x^2+1$
\* $(x^4 + x^3 + x^2 + x + 1)^2$
10 1.84835... $x^{10}-x^9-x^8-x^6-x^5-x^4-x^2-x+1$
\* $(x^2 + x + 1)^5$ impossible (\*)
8 1.84959... $x^8+x^7-x^6-4x^5-5x^4-4x^3-x^2+x+1$
\* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible
10 1.85312... $x^{10}-2x^9+x^8-2x^7+2x^6-2x^5+2x^4-2x^3+x^2-2x+1$
\* $(x + 1)^{10}$ impossible (\*)
10 1.85712... $x^{10}-2x^9+x^8-x^7-x^5-x^3+x^2-2x+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
10 1.86264... $x^{10}-3x^9+3x^8-x^7-3x^6+5x^5-3x^4-x^3+3x^2-3x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
22 8 1.86406... $x^8-x^7-2x^6+2x^4-2x^2-x+1$
\* $(x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$
10 1.86876... $x^{10}-x^9-2x^7-3x^5-2x^3-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
10 1.87573... $x^{10}-2x^9-x^8+3x^7-3x^5+3x^3-x^2-2x+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
23 4 1.88320... $x^4-2x^3+x^2-2x+1$
\* $(x^2 + x + 1)^2$
10 1.88996... $x^{10}-x^9-x^8-x^6-2x^5-x^4-x^2-x+1$
\* $(x + 1)^2 (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*)
10 1.89360... $x^{10}-x^9-2x^8+x^6+x^4-2x^2-x+1$
\* $(x + 1)^6 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*)
10 1.89663... $x^{10}-2x^9+x^7-x^6-x^4+x^3-2x+1$
\* $(x + 1)^4 (x^6 + x^3 + 1)$ impossible (\*)
10 1.89910... $x^{10}-2x^9+x^6-x^5+x^4-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 1.90562... $x^{10}-x^8-2x^7-3x^6-3x^5-3x^4-2x^3-x^2+1$
\* $(x^3 + x + 1) (x^3 + x^2 + 1) (x^4 + x^3 + x^2 + x + 1)$ impossible (\*)
10 1.90830... $x^{10}-x^9-x^8-x^7-x^5-x^3-x^2-x+1$
\* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
10 1.91112... $x^{10}-2x^8-2x^7-x^6-x^5-x^4-2x^3-2x^2+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 1.91445... $x^{10}-x^9-x^7-3x^6-x^5-3x^4-x^3-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
24 8 1.91649... $x^8-x^7-x^6-x^5-x^3-x^2-x+1$
\* $(x + 1)^4 (x^4 + x^3 + x^2 + x + 1)$
8 1.92062... $x^8-3x^7+3x^6-2x^5+x^4-2x^3+3x^2-3x+1$
\* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible
10 1.92606... $x^{10}-x^9-x^8-x^7-x^6+x^5-x^4-x^3-x^2-x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
25 8 1.92678... $x^8-2x^6-2x^5-x^4-2x^3-2x^2+1$
\* $(x^2 + x + 1)^4$
10 1.92990... $x^{10}-x^9-x^8-2x^7+x^6+x^4-2x^3-x^2-x+1$
\* $(x + 1)^2 (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*)
10 1.93231... $x^{10}-2x^9+2x^8-4x^7+3x^6-5x^5+3x^4-4x^3+2x^2-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 1.93295... $x^{10}-3x^9+3x^8-2x^7+x^5-2x^3+3x^2-3x+1$
\* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$ impossible (\*)
10 1.93637... $x^{10}-2x^9+x^5-2x+1$
\* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*)
10 1.94005... $x^{10}-2x^9+x^8-x^7-x^6-x^4-x^3+x^2-2x+1$
\* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
26 6 1.94685... $x^6-x^5-x^4-x^3-x^2-x+1$
\* $(x^3 + x + 1) (x^3 + x^2 + 1)$
10 1.94998... $x^{10}-x^9-2x^8-x^7+x^6+3x^5+x^4-x^3-2x^2-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
8 1.95530... $x^8-2x^7-x^5+3x^4-x^3-2x+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
27 6 1.96355... $x^6-2x^5-x^4+3x^3-x^2-2x+1$
\* $(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)$
10 1.97209... $x^{10}-2x^9-x^6+3x^5-x^4-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
28 6 1.97481... $x^6-2x^5+x^4-2x^3+x^2-2x+1$
\* $(x + 1)^6$
29 6 1.98779... $x^6-2x^4-3x^3-2x^2+1$
\* $x^6 + x^3 + 1$
30 8 1.99400... $x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$
\* $(x^4 + x^3 + x^2 + x + 1)^2$
10 1.99703... $x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
31 10 1.99852... $x^{10}-2x^9+x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$
\* $(x + 1)^2 (x^2 + x + 1)^4$
10 2.00145... $x^{10}-x^9-2x^8-x^7+x^6+2x^5+x^4-x^3-2x^2-x+1$
\* $(x + 1)^4 (x^2 + x + 1)^3$ impossible (\*)
10 2.00289... $x^{10}-3x^9+3x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
10 2.00573... $x^{10}-2x^9-2x+1$
\* $(x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^2$ impossible (\*)
10 2.00624... $x^{10}-x^8-3x^7-3x^6-4x^5-3x^4-3x^3-x^2+1$
\* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
10 2.00947... $x^{10}-2x^9+x^8-x^7-2x^6+x^5-2x^4-x^3+x^2-2x+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
32 8 2.01128... $x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$
\* $(x^2 + x + 1) (x^6 + x^3 + 1)$
10 2.01335... $x^{10}-x^9-2x^7-2x^6-3x^5-2x^4-2x^3-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
10 2.01488... $x^{10}-2x^8-2x^7-2x^6-3x^5-2x^4-2x^3-2x^2+1$
\* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*)
10 2.01600... $x^{10}-3x^9+2x^8+x^7-3x^6+3x^5-3x^4+x^3+2x^2-3x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
10 2.01671... $x^{10}-2x^9-x^8+2x^7+x^6-3x^5+x^4+2x^3-x^2-2x+1$
\* $(x^3 + x + 1) (x^3 + x^2 + 1) (x^4 + x^3 + x^2 + x + 1)$ impossible (\*)
33 8 2.02202... $x^8-2x^7-2x+1$
\* $(x + 1)^8$
10 2.02344... $x^{10}-2x^9-x^7+2x^6-x^5+2x^4-x^3-2x+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
10 2.02739... $x^{10}-x^9-x^8-x^7-x^6-2x^5-x^4-x^3-x^2-x+1$
\* $(x + 1)^2 (x^2 + x + 1)^2 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*)
34 8 2.03064... $x^8-x^7-3x^5-x^4-3x^3-x+1$
\* $(x^4 + x + 1) (x^4 + x^3 + 1)$
10 2.03298... $x^{10}-2x^9+x^6-3x^5+x^4-2x+1$
\* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible
10 2.03579... $x^{10}-2x^9-x^6+2x^5-x^4-2x+1$
\* $(x + 1)^6 (x^2 + x + 1)^2$ impossible (\*)
10 2.03890... $x^{10}-x^9-2x^8-2x^7+2x^6+3x^5+2x^4-2x^3-2x^2-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
35 6 2.04249... $x^6-3x^5+3x^4-3x^3+3x^2-3x+1$
\* $(x^3 + x + 1) (x^3 + x^2 + 1)$
10 2.04414... $x^{10}-x^9-2x^8-x^7+x^6+x^5+x^4-x^3-2x^2-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
10 2.04776... $x^{10}-3x^9+3x^8-3x^7+2x^6-x^5+2x^4-3x^3+3x^2-3x+1$
\* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible
10 2.04817... $x^{10}-x^9-2x^8-x^5-2x^2-x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
36 8 2.04952... $x^8-x^7-x^6-x^5-2x^4-x^3-x^2-x+1$
\* $(x + 1)^4 (x^4 + x^3 + x^2 + x + 1)$
10 2.05286... $x^{10}-x^9-x^8-2x^7-x^5-2x^3-x^2-x+1$
\* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$ impossible (\*)
10 2.05353... $x^{10}-2x^9-x^8+2x^7-x^5+2x^3-x^2-2x+1$
\* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
10 2.05523... $x^{10}-x^9-x^8-x^7-x^6-3x^5-x^4-x^3-x^2-x+1$
\* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*)
37 10 2.05631... $x^{10}-2x^9-x^7+x^6+x^5+x^4-x^3-2x+1$
\* $(x^2 + x + 1)^3 (x^4 + x^3 + x^2 + x + 1)$
10 2.05819... $x^{10}-x^9-3x^7-x^6-3x^5-x^4-3x^3-x+1$
\* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible
38 8 2.06017... $x^8-3x^7+2x^6+x^5-3x^4+x^3+2x^2-3x+1$
\* $(x^4 + x + 1) (x^4 + x^3 + 1)$
10 2.06226... $x^{10}-2x^8-3x^7-2x^6-x^5-2x^4-3x^3-2x^2+1$
\* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible
10 2.06420... $x^{10}-2x^9+x^8-2x^7-x^5-2x^3+x^2-2x+1$
\* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible
10 2.06557... $x^{10}-x^8-3x^7-4x^6-5x^5-4x^4-3x^3-x^2+1$
\* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible
8 2.06972... $x^8-2x^7+x^5-3x^4+x^3-2x+1$
\* $x^8 + x^5 + x^4 + x^3 + 1$ impossible
10 2.07416... $x^{10}-3x^9+4x^8-6x^7+6x^6-7x^5+6x^4-6x^3+4x^2-3x+1$
\* $x^{10} + x^9 + x^5 + x + 1$ impossible
39 4 2.08101... $x^4-x^3-2x^2-x+1$
\* $(x + 1)^2 (x^2 + x + 1)$ $\exists$ example
---- ----- ------------ ------------------------------------------------------------------ -------------------
[^1]: Y. M. is supported by JSPS KAKENHI Grant Numbers 15H05738 and 16K17560. H. O. is supported by JSPS KAKENHI 15K17521. S. R. is partially supported by National Science Centre, Poland, grant 2014/15/B/ST1/02197.
[^2]: There is a misprint in [@dolgachev16 Sect. 4.5, Table 2]: the terms $x^4$, $x^6$ appear with coefficient $6$ in $p_{f^*}(x)$. In we give the correct formula, but the misprint is irrelevant for us because we consider modulo 2 reduction.
[^3]: `https://arxiv.org/abs/1707.02563`
|
{
"pile_set_name": "ArXiv"
}
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The American Health Care Act, which narrowly won passage in the House on Thursday, could transform the nation’s health insurance system and create a new slate of winners and losers.
While the Senate will probably demand changes, this bill, if it becomes law in its current form, will repeal and replace large portions of the Affordable Care Act (Obamacare). It will change the rules and subsidies for people who buy their own insurance coverage, and make major cuts to the Medicaid program, which funds care for the poor and disabled.
Any sizable change in our complex health care system leaves some people and businesses better or worse off. For some, insurance will become more affordable — or their taxes will be lower. Others will lose out on financial support or health care coverage. You can see how you might be affected in our summary of winners and losers.
Winners
High-income earners: The bill eliminates two taxes on individuals earning more than $200,000 or couples earning more than $250,000: a 0.9 percent increase on the Medicare payroll tax, and a 3.8 percent tax on investment income. It also allows people to save more money in tax-excluded health savings accounts, a change most useful to people with enough money to have savings.
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Q:
Return value from Oracle function using Mybatis
I have looked at a number of StackOverflow questions on this issue but can't find one that makes any sense. This one comes closest but doesn't show how to get the return value from the function.
Here's my mapper call:
public Long callMyFunction(@Param("recordId") Long recordId, @Param("otherId") Long otherId, @Param("date") Date date, @Param("comments") String comments);
Here's the mapper XML:
<select id="callMyFunction" parameterType="map" statementType="CALLABLE" resultType="java.lang.Long">
{ #{resultId,javaType=java.lang.Long,jdbcType=NUMERIC,mode=OUT} = call MYSCHEMA.MYPACKAGE.my_function(
#{recordId,jdbcType=NUMERIC,mode=IN},
#{otherId,jdbcType=NUMERIC,mode=IN},
#{date,jdbcType=DATE,mode=IN},
#{comments,jdbcType=VARCHAR,mode=IN})}
</select>
The call works, but the return value (resultId) is always null.
Can anybody spot the problem?
A:
If you want to return directly the result, then the SQL call must be:
SELECT MYSCHEMA.MYPACKAGE.my_function(...) FROM DUAL
If you want to keep with calling the function in the procedure call style, that means the result is an OUT parameter (you env declared it OUT).
The minimum change would consist in adding a parameter to the mapper method signature:
public Long callMyFunction(@Param("recordId") Long recordId, @Param("otherId") Long otherId, @Param("date") Date date, @Param("comments") String comments, @Param("resultIdContainer") Map<String, Long> resultIdContainer);
In the XML: forget the resultType, this is for selects. And the call:
{ #{resultIdContainer.resultId, javaType=java.lang.Long,jdbcType=NUMERIC,mode=OUT} = call ...
Not that I use here a map to contain the resutlId: an indirection is required: the function will write the parameter 'result' value somewhere you can read later (after your mapper call), you can also use a class with a resultId property.
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{
"pile_set_name": "StackExchange"
}
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BOAVISTA
1
0
0
THE PARADISE LIES HERE
If Cabo Verde postcards were once crowned by the paradisiacal beaches of the island of Sal, in recent years, the island of Boa Vista has also been affirmed as one of the favorite destinations of tourists who seek to enjoy the charms that the “Sun & Sea” have to offer
to those who visit Cabo Verde.
Discovered on May 3rd, 1480, the island of Boa Vista holds a history rich in culture and traditions, such as weaving, ceramics, music and literature, aside from the beautiful white sand beaches, which extend for over 55 Kilometers and offer a diversity of nautical sports, from diving, to discover the treasures of sunken and stranded ships, to windsurfing.
To talk about Boa Vista is to talk about the ruins of the Curral Velho and the Viana Desert to be exploited. An inhospitable and even lunar scenery. There are those who call it the extension of the Sahara, in the middle of the Atlantic Ocean, where white sand dunes stretching for 5 kilometers, which make Boa Vista offer a landscape diversity beyond its paradisiacal beaches.
WHERE TO STAY
Boa Vista has a wide range of accommodation, both inside and outside the city of Sal Rei. Hotels, condominiums, aparthotels and all inclusive resorts. Offers for all tastes and financial availabilities, but the ideal is to book reservations before arriving at the island.
WHERE TO EAT
There are several restaurants and cafes / bars in the city, some themed, where you can enjoy a breakfast, or a good meal, mainly based on fish and shellfish. There are restaurants next to the sea area, ideals for a hot summer night. Offers do not lack, you just have to choose.
WHERE TO BUY
There are at least two local markets, where you can find varied handicrafts, from Cabo Verde to West African Coast parts, in the center of the city. Aside from the markets, there are several souvenir shops in Sal Rei, including at the airport.
ITINERARY OF BEACHES
Similarly to the island of Maio and Sal, much of the beauty and enchantment of the island of Boa Vista is due to its beautiful beaches. Some really close to the city like Cruz Beach, or Estoril beach, others more deserted and virgin, like Santa Monica, and others even with stranded boats like the Cape of Santa Maria.
Swimming, diving, paddling or windsurfing. The really hard part is choosing. Therefore, it is best to venture, preferably with a guide, on a trip around the main beaches of the island. Do not forget the hat and sunscreen. The sun of Boa Vista is strong and shines throughout the day. Watch out for unguarded waves and beaches. At Santa Monica they impose respect.
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{
"pile_set_name": "Pile-CC"
}
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# RUN: not llvm-mc -arch=hexagon -filetype=asm %s 2> %t; FileCheck %s < %t
#
# Check that changes to a read-only register is caught.
{ c9:8 = r1:0 }
# CHECK: error: Cannot write to read-only register
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{
"pile_set_name": "Github"
}
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"Just like the railway budget, there are no specifics in this budget there are no roadmaps," he said in a reaction to the budget.
The union budget for the year 2014-15 presented by Finance Minister Arun Jaitley on Thursday was greeted with mixed response.
Most of the political leaders said the budget was ‘in the right direction’ and would ‘pay long-term dividends’, however the opposition parties slammed the budget saying it had nothing for the common man.
The Union Budget for 2014-15 presented by Finance Minister Arun Jaitley in the Parliament on Thursday left income tax rates unchanged but provided sops to small and marginal assessees by raising the threshold exemption limit from Rs 2 lakh to Rs 2.5 lakh and investments under 80C by Rs 50,000 to Rs 1.5 lakh while promising not to bring tax changes with retrospective effect.
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{
"pile_set_name": "Pile-CC"
}
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/* Copyright 2016 The TensorFlow Authors. All Rights Reserved.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
==============================================================================*/
syntax = "proto3";
package tensorflow;
option cc_enable_arenas = true;
option java_outer_classname = "ClusterProtos";
option java_multiple_files = true;
option java_package = "org.tensorflow.distruntime";
// This file contains protos to be used when defining a TensorFlow
// cluster.
//
// EXAMPLES
// --------
//
// 1. A single-process cluster, containing "/job:local/task:0".
//
// Cluster:
// job { name: 'local' tasks { key: 0 value: 'localhost:2222' } }
//
// Server:
// cluster { $CLUSTER } job_name: 'local' task_index: 0
//
// 2. A two-process cluster, containing "/job:local/task:{0,1}".
//
// Cluster:
// job { name: 'local' tasks { key: 0 value: 'localhost:2222' }
// tasks { key: 1 value: 'localhost:2223' } }
//
// Servers:
// cluster { $CLUSTER } job_name: 'local' task_index: 0
// cluster { $CLUSTER } job_name: 'local' task_index: 1
//
// 3. A two-job cluster, containing "/job:worker/task:{0,1,2}" and
// "/job:ps/task:{0,1}".
//
// Cluster:
// job { name: 'worker' tasks { key: 0 value: 'worker1:2222' }
// tasks { key: 1 value: 'worker2:2222' }
// tasks { key: 2 value: 'worker3:2222' } }
// job { name: 'ps' tasks { key: 0 value: 'ps0:2222' }
// tasks { key: 1 value: 'ps1:2222' } }
//
// Servers:
// cluster { $CLUSTER } job_name: 'worker' task_index: 0
// cluster { $CLUSTER } job_name: 'worker' task_index: 1
// cluster { $CLUSTER } job_name: 'worker' task_index: 2
// cluster { $CLUSTER } job_name: 'ps' task_index: 0
// cluster { $CLUSTER } job_name: 'ps' task_index: 1
// Defines a single job in a TensorFlow cluster.
message JobDef {
// The name of this job.
string name = 1;
// Mapping from task ID to "hostname:port" string.
//
// If the `name` field contains "worker", and the `tasks` map contains a
// mapping from 7 to "example.org:2222", then the device prefix
// "/job:worker/task:7" will be assigned to "example.org:2222".
map<int32, string> tasks = 2;
}
// Defines a TensorFlow cluster as a set of jobs.
message ClusterDef {
// The jobs that comprise the cluster.
repeated JobDef job = 1;
}
|
{
"pile_set_name": "Github"
}
|
For months, Hillary Clinton and Bernie Sanders seemed to run for president in parallel universes. Clinton focused her criticism on Republicans and rarely targeted Sanders. She did not even say his name in public until long after the first debate. They raced like lap swimmers, each in their own lane.
But with snow settling on New Hampshire and a loss likely in the Granite State on Tuesday, Clinton and her allies have turned their fire on the Vermont senator. In the last weeks, Clinton, her husband Bill, daughter Chelsea, allied super PACs, surrogates and members of Congress have by turns accused Sanders of hypocrisy, inconsistency and cutting corners. The Clintons, well-accustomed to decades of intense scrutiny by Republicans, now are giving Sanders a more thorough appraisal than he has ever had in his career.
Frustrated Clinton aides believe that Sanders is a politician like any other. Sanders has flip-flopped on gun control, Wall Street donations and immigration reform, they say, changing his views when it is politically expedient.
It is a view of Sanders that has become increasingly visible on the campaign trail. “The purity bubble is about to burst,” David Brock, the architect behind the pro-Clinton super PAC Correct the Record, told TIME.
Clinton world’s focus on Sanders in recent weeks has been sundry and repeated. Her allies foresee a fine-toothed scrutiny of Sanders’ record, saying that he has benefitted in his career so far by escaping much scrutiny.
“I think there’s a lot that has not yet been said and written about Senator Sanders and what he’s actually done,” New Hampshire Democratic Sen. Jeanne Shaheen, who is campaigning for Clinton, told TIME. “This is someone who has been in Washington almost 30 years. You don’t stay in Washington that long unless you’re a politician and you understand politics.”
The Clinton campaign’s more heated tone reflects a growing realization among campaign aides that the primary against Sanders will be a long and grueling fight. The 74-year-old insurgent Vermont senator is poised to win in New Hampshire on Tuesday, and barely lost to Clinton in Iowa. He has said he will continue campaigning all the way to the Democratic convention in July, and will likely force Clinton to expend money and energy for months against him. With an open spigot of cash from his robust online fundraising operation, Sanders has little reason to end his campaign early.
Clinton has begun to forcefully strike back at Sanders for questioning her Wall Street donations. At an event in a high-ceilinged building in Manchester on Monday, Clinton suggested Sanders was hypocritical for attacking her over her banking donors. Clinton pointed to a recent report that Sanders had indirectly received finance industry donations for his 2006 campaign. “Senator Sanders took about $200,000 from Wall Street firms. Not directly, but through the Democratic Senate Campaign Committee,” Clinton said. (Some of the DSCC cash for Sanders’ race originally came from Wall Street institutions.)
In their latest debate on Thursday, Clinton called Sanders’ criticism over her Wall Street donations a “very artful smear,” suggesting Sanders was insulting her integrity. “If you’ve got something to say, say it, directly,” Clinton said.
Former President Bill Clinton has been perhaps the most adamant critic of Sanders in recent days. At a campaign event on Sunday, he questioned Sanders’ honesty for misleadingly suggesting in his campaign ads that he received endorsements from newspapers.
“Bernie took what they said was good about him and put it in his own endorsements,” the former president said on referring to an advertisement that suggested Sanders had an endorsement from the Valley News. (It has not.)
And in a rebuke to Sanders for railing against the establishment, Bill Clinton said on Monday in Manchester that Sanders was being divisive and avoiding a substantive policy debate. “We can’t be in a place where we demonize everybody is against us,” he said. “Where everybody who’s on the other side is part of some mythical establishment. … We have to do this together, and producing results together.”
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In New Hampshire last month, Chelsea Clinton said Sanders wanted to “dismantle Obamacare, dismantle the [Children’s Health Insurance Program], dismantle Medicare, and dismantle private insurance,” referring to his plan for universal health care coverage, which has been criticized as thin and unworkable. Chelsea Clinton said she worried “that would strip millions and millions and millions of people off their health insurance.”
Some Clinton supporters welcome a more combative candidate, saying they want a closer look at Sanders’ record.
“There’s a lot of stuff that comes out about Hillary. She’s been scrutinized, scrutinized, scrutinized. I don’t see any of that about Bernie coming out—and there are things,” said Barbara Marzelli, who runs a gardening business in New Hampshire. “It’s like everybody’s throwing spaghetti at the wall and seeing what sticks. They haven’t started to fling spaghetti at Bernie.”
Hillary Clinton has also questioned whether Sanders is a true progressive, pointing to a mixed record on immigration and gun control. She has said Sanders following the gun lobby’s lead on a 2005 gun control bill, and aides point to Sanders’ vote against immigration reform in 2007. “The only person on [the debate] stage who voted against Ted Kennedy’s comprehensive immigration bill was Senator Sanders,” Clinton campaign pollster Joel Benenson said Friday at a breakfast with reporters in Manchester, N.H. “I don’t see how he gets to go out and decide who progressive is when he voted against that.”
Brock, who runs Correct the Record, a super PAC that coordinates with Clinton’s campaign, outlined the main lines of attack against Sanders in an interview, pointing to what Brock called a lack of direct foreign policy experience, sketchily drawn policy proposals—and “hypocrisy.” Sanders’ hypocrisy “goes to to the question of his fundraising at the DSCC and the financial sector,” Brock said. “It goes to pledges that he’s not going to run a negative campaign and breaking the pledges.”
Sanders’ campaign responded on Monday to the criticism. “It is very disturbing that, as the Clinton campaign struggles through Iowa and New Hampshire, they have become increasingly negative and dishonest,” campaign manager Jeff Weaver said.
Meanwhile, Clinton and her aides have kept up more familiar forms of criticism. On the stump, Clinton points out that Sanders’ policy proposals would be extraordinarily difficult to pass and says Sanders would be unlikely to deliver, as a candidate and as a president. “We don’t want to over-promise,” Clinton said in a high school gym in southern New Hampshire on Monday night. “The last thing we want is promises that can’t be met.”
They have also pointed to Sanders’ lack of foreign policy experience. “I think you really saw on that stage someone who can… be our commander-in-chief, protect our national security,” campaign manager Robby Mook said last week. “I think you saw Senator Sanders fail that test.”
Whatever happens in New Hampshire on Tuesday, it is clear that some voters have been waiting for Clinton to open up on Sanders.
“Nobody is immaculate except Mary, the mother of God,” said Barbara Waldmann, a retired schoolteacher from New York. “No one is immaculate.”
-With additional reporting by Jay Newton-Small
Contact us at letters@time.com.
|
{
"pile_set_name": "OpenWebText2"
}
|
Thursday, July 29, 2010
Lisa M here - Nicole asked me to share some photos and thoughts about beloved friends and supporters left behind in Orlando. All of the following comments and captions from Orlando were written by Nicole. I've tried to organize them as best as I can.
Orlando - One week and a million years away.
Debbie surprised me with a visit shortly after July 4th to deliver Wrenn's beautiful sign. Not only did she make one for Tanner, she made several inspirational magnets for the nurses. It's a wonderful way to remember Wrenn and our family. I can't wait to bring Wrenn home and decorate her nursery, which will include her sign made from Las Vegas, artwork by Tanner and all the well wishes while in the NICU.
On the same day that Debbie delivered her beautiful signs, Nicole Neufeld stopped by and posed for a picture with Wrenn. It was great to finally meet. We both felt like we had been friends for many years and not just weeks!
The Amazing People at Winnie Palmer Hospital
Nurse Kristi and Respiratory Specialist Vanessa with Wrenn who is 13 weeks here.
Respiratory Specialists: Also known as my therapists and cheerleaders. They were great and really comforted us as we waited for the news of Wrenn's lung condition-- Ann, Micki, Julie and Perri
Ann's sister knitted a beautiful baby blanket for Wrenn and gave it to us the morning we left for St. Louis. Can't thank you enough for thinking of us-- God bless!
Two of Wrenn's night nurses: Aunt Carol and Theresa. I slept soundly knowing she was in your hands each night but the circles under my eyes never went away! Thanks again for the CD and onesies but for especially rocking my girl to sleep when I really needed to get some rest.
Being put on the vent pretty much killed Wrenn's therapy sessions for sucking on the binky. But I'm sure that all of Meredith's sessions left a lasting impression.
Patti, on the right, is on the transport team as a respiratory specialist. She rocked Wrenn many, many times when I was unable to do so myself. She also spent several moments holding my hand on really rough days. I miss our talks, Patti. HUGS!
A snapshot of Tanner, Chaplain Gail, Vanessa, Ann and Uncle Collin.
Gail baptized my husband as well as Wrenn in the first two weeks in the NICU. Gail's purpose in our lives was introduced by a higher power when she gave us Nicole Neufeld's phone number; the six degree type of intervention will forever give me goose bumps on how we all happened to meet. One day it will be a great story.
Back to St. Louis
We now return to St. Louis.... where Tanner is adjusting well, playing with the play kitchen at sibling day care:
We never forget the reason for this blog... the one thing that has brought us all together. Here she is: Wrenn asleep.
Wednesday, July 28, 2010
St. LouisChildren's Hospital has done a wonderful job in making us feel at home. There are several retreats throughout the hospital that allow us to get away without actually leaving the hospital. On the same floor as the NICU there is a Ronald McDonald Room; we can do laundry, watch TV, play games with Tanner or visit with other families over a cup of tea. In many ways we already feel settled even though our future residence is still unknown (we are on the list for the Ronald McDonald house but the list is long). On the 8th floor there is an outside area to get some sun and take in the view of the city. Since most of our time is spent indoors, this is a place I visit often to help catch my breath.
The flight was bumpy for Wrenn but the doctors at the NICU did wonders getting her stable those first 48 hours. I was unsure about how she'd do on the trip and say what could have been a real goodbye back in Orlando. I think it will take a lot more than that to knock down my little firecracker. She is still fighting hard and is making us proud at how well she has transitioned to her new home.
As our roller coaster ride continues, I am thankful that we are surrounded by such wonderful nurses and doctors here in St. Louis. They have already learned that Wrenn is feisty, which I am told is a good thing if you are in need of a transplant. I'm sure the nurses will be experts on Wrenn's personality very soon! :-)
Wrenn was switched back to the ventilator this morning and is stable. She is still on minimal sedation but continues to open her eyes and memorize new faces and voices. She listens to her CD's, some familiar (like the one Aunt Carol gave her back in Orlando) and some new (a CD from Jaimie; a nurse we had the first few days Wrenn arrived). :-)
If possible, I hope to hold Wrenn in the next few days. I am cautious and don't want to do anything that will upset her. I rely on the nurses to advise me on this when the time is right.
Tanner is enjoying his time in the sibling daycare room and visiting his sister's new private room. One of our new friends, Liz and Dan, based here in St. Louis, brought Tanner some books and toys to our hotel. I cannot thank them enough for taking the time to do this. You guys are awesome. :) HUGS
Wrenn received your cards and letters this week. As usual, the prayers and kind words always come at my darkest moments. Please keep praying for her lungs to heal, answers to come our way, and of course, the perfect set of lungs, should that be her path.
Jason and I feel the love and support from everyone involved and we thank you for carrying us through this difficult road. I will update as you as time permits. :-) I want to again thank Lisa M. and Nicole N. for everything they have done for our family and this blog. ((((HUGS))))
Sunday, July 25, 2010
The first few days in St. Louis have been very busy for the Parris family. Nicole has kept me up to date via text messages when she gets a chance. I will share some of them, as well as some photos she sent me via MMS.
July 25 3:17 AMWrenn had a rough flight as do most surfactant babies. Their lungs are heavily affected by changes in altitude, which make it difficult for the transport team to keep their CO2 levels low.Once Wrenn arrived in St. Louis, the medical team had to stabilize her. Her hemoglobin level was low and her CO2 level was the highest it's ever been. She was placed on an oscillator, which is a large machine, more powerful than a ventilator but gentler on the lungs.
The medical team has done a lot of blood work on Wrenn, but first they put in a Broviac line for easy withdrawals. She did great during and after that surgery. To help raise her hemoglobin level and lower her CO2 level, Wrenn was given a blood transfusion. She will receive more transfusions during the transplant process.
The NICU team has been very methodical in how they've treated Wrenn. Their goal is to take a critically ill baby and bring her to a stable condition - keep her comfortable, stable, and make sure all her other organs are working well. This includes her heart, which now beats at a much slower rate (90-140).
The goal of the NICU team is to wean Wrenn from the oscillator back to the ventilator, and if possible, back to the Camula flow like she had in Florida before she needed the ventilator. However, the odds of doing this prior to transplant are not good because during the trip she once again developed Persistent Pulmonary Hypertension of the Newborn (PPHN). This often happens during a traumatic experience. I am told this is due to her lung problem as is her enlarged heart. Post transplant, both conditions will hopefully disappear.
Liz, a wonderful woman, and her husband Dan, met Wrenn today and also delivered some toys to Tanner to play with that their two boys enjoyed when they were little. I met Liz through my friend Pam who has stood by my side throughout this entire ordeal. Liz's father is a transplant surgeon in Houston,, so she knows about what we are facing. I want to thank her for taking the time out of her schedule to visit us at the hospital, and for thinking of Tanner who is also struggling to adjust to his new environment. (Lisa's note - just take one look at his face. It speaks a thousand words.)
July 25 9:12 AMI'd like to say a word of thanks to Nicole Neufeld, who had a Fay's Friends gift bag waiting for us in Wrenn's room. I'll take a photo so you can post it later. I also owe you material about Debbie's beautiful signs and Nicole's visit to Orlando. I haven't even cracked open my laptop yet but I will in the next few days.
July 25 9:53 AMEverything is so methodical here. It takes the emotion out of it somehow. All business with one common goal. It's crazy.
July 25 11:50 AMNews flash. Just met a dad two doors down whose daughter is being tested for a surfactant deficiency. I'll give you more information as I hear. It's a freaking epidemic.
July 25 10:02 PMThe photographer took pictures of Wrenn today. They do this for parents when kids enter and leave NICU. I'll load those tomorrow. Wrenn had a better day today. She is moving around her. 90% oxygen. Not as swollen and doing well with feedings.
I hope Nicole doesn't mind my posting her texts. But I think it's important for people to realize what a typical day is like for her. Finally, I'd like to share some photos of Wrenn's present world with you. Welcome to her room.
And now, please observe a moment of thoughtful silence, prayer, and/or communion with the universe to appeal for help for what rests peacefully, for the moment, inside.
Rest well, sweet Wrenn. We love you so much.
Friday, July 23, 2010
The past week has been an emotional roller coaster for the Parris family. I'm sorry that I haven't posted more regular updates. I feel as if I'm always waiting for good news, and by the time there is more news, good or bad, the old news seems really old. But I hound Nicole and Jason for news constantly, so that I can pass it along, and I do want to share the details.
Here is the sequence of recent events:
Preceding weeks: Wrenn's breathing began to deteriorate slowly. At one point she'd been on 7 liters and about 78% oxygen, but her need for oxygen began to increase to the point where she was back up to 8 liters at 99 percent.
Saturday July 17: Wrenn's Golf Tournament was held at North Shore Golf Club in Orlando, FL. It was a resounding success and some pictures are displayed at the end of this post. Over $10,500 was raised to help pay for Wrenn-related expenses.
Sunday, July 18: Wrenn's breathing deteriorated to the point where she could no longer breathe without the help of a ventilator. On this sad and difficult day, Wrenn was sedated and strapped down as the breathing tube was inserted down her throat. No longer can her parents hold her or see her smile. On the plus side, her color is much better and finally her poor little lungs can rest and let the ventilator do the work for her. Here is a picture of Wrenn, very different from the ones you've seen posted on this blog up until this point.
Tuesday, July 20: The St. Louis Children's Hospital Lung Transplant team approved Wrenn for evaluation. Wrenn's Lear jet was scheduled to leave Thursday morning. Her parents quickly threw their necessary belongings together and planned to leave as soon as they saw her in the morning to say goodbye. Here are some photos of the card the medical team at Winnie Palmer Hospital lovingly made for Wrenn. We are pretty sure just about every hospital employee signed it!
Thursday, July 22: Nicole, Jason and Tanner left in the morning for the 16+ hour car ride from Orlando to St. Louis. Wrenn's jet took off in the early afternoon. The Parris family stopped around midnight at a hotel. Here is Tanner and Buzz watching Toy Story in the car. (Tanner, by the way, was an excellent traveler!)
Friday, July 23 (Today): Jason, Nicole and Tanner arrived in St. Louis at about 1pm. They are currently meeting back up with Wrenn and, I'm sure, getting to know doctors and medical staff at the new hospital, as I wait anxiously for news.
Please stay tuned via the Facebook page (link to the right). I will try to post regular blogs every couple of days with details, but breaking news is easy to put up quickly on Facebook, anytime, from wherever I may be.
"I met my husband 17 years ago. We have played together in many a charity golf tournament. Never in a million years did we think we would be on the receiving end of one." -- Nicole Parris
It is easy during trying times like these to feel nothing but despair. But I don't. So many people are helping this incredible baby - it's enough to restore anyone's faith in the kindness of their fellow human beings! At Wrenn's Golf Tournament on July 17, nearly 30 teams of golfers came out to play in support of Wrenn. 22 local business sponsored holes on the course, and over $7000 worth of goods and services were donated for raffle prizes. Jason and Nicole both gave emotional speeches of thanks to the golfers at the luncheon/raffle after the tournament. Here are just a few snapshots of this wonderful, inspirational day.
The players enjoy lunch, donated by Nature's Table, following the tournament. In the background, you can see some of the many prizes that would soon be awarded:
Jason played in the tournament with his many, many friends. Here he is with some of them (Bob and Carey, plus Tanner):
Another team of golfers (Dean, Greg, Joe and Dave):
The Red Bull team, a big donation giver for the day:
And yet another team. I don't know who that second guy on the right is, but he sure is cute! (Just kidding. It's my husband Brian with his teammates Mark, Mike and Rob.)
Marshall English customized a "Wrenn" painting of a hole at North Shore during the raffle drawing. Please see his website (http://marshallenglish.com) to purchase one of these gorgeous works of art! Remember that a portion of all proceeds provides assistance to Wrenn's family.
Jason and Manny, who I was told "gives Jason and the other guys my money every week!"
My kids and I working the raffle ticket sales as final team scores are posted in the background:
There are many more pictures from this wonderful day, and I will sprinkle others in the posts in upcoming weeks. For now, I would like to send out a HUGE THANK YOU once again on behalf of the Parris family to Lee Siske who organized the tournament and to everyone else who helped in any way big or small. A charity golf tournament is a great way to raise funds but even by usual standards, this tournament was a huge success, and that is no surprise to me. Wrenn is a very special baby and she and her parents are well loved in our community. While it saddens us that the tournament took place on the last day Wrenn breathed without a ventilator, it was a very special day and the proceeds will help her parents make it through the next few months.
That being said, the road ahead is still long for the Parris family, and even this great fundraiser won't support them throughout the entire journey. Please pass on word about Wrenn to everyone you know so that they can help in whatever way they can. Thanks for reading and for keeping Wrenn in your thoughts and prayers.
"I would like to tell you a little about Wrenn. She is a very special girl. She has never known any life other than the one she has lived in the hospital, but she smiles at us all day, every day." -- Nicole Parris
Friday, July 16, 2010
No, not really….but it sure feels like it some days. I wanted to share with you a photo of my refrigerator freezer. I pump every three to four hours so that Wrenn can benefit from my breast milk but as you can see, my freezer has had enough! I’m always excited when the milk tech requests 30 to 40 bottles. I can only imagine what the Winnie Palmer NICU freezer looks like. Haha!
Thursday, July 15, 2010
It is very difficult trying to make decisions for Wrenn when her condition is so unknown. Her degree of severity, I am told, is the worst because she has a double mutation. It really makes sense for her to be seen by the experts so we agreed to have her flown to St. Louis. We will know the date within the next few weeks. The doctors there can perform their own tests and will hopefully put her on the list for a transplant, which is what she needs to be able to function in this world as a normal baby. As with any transplant, there are always risks of complications, infections, rejections and death while waiting. The unknown is so scary and I battle the ‘what if’s’ every single moment of the day. I question every decision I make and already wonder if flying her to St. Louis is the right choice. Wrenn’s doctor told me that I’ll live a life filled with guilt if I try and question every road I take should the outcome not be favorable. I wish I could just flip a coin but as one good friend recently said, “I know you won’t!”
Friday, July 2, 2010
Wrenn's doctor finally had a chance to sit down and talk to St. Louis about our situation. And here is what they said.
They are waiting for Wrenn to declare herself. Dr. Sweet has seen babies get worse over time and some get better. They can't give her a new lung until she is in NEED of one. What they can't tell us is how long we might have to wait to find out how her particular mutation progresses.
What they did say was that some babies who were worse off than Wrenn got better. Those babies were vented and came off the ventilator and could be weaned down on the oxygen and liters. This information amazed me because if you are born with limited surfactant protein, I can't imagine growing it or getting it later on in life. The doctor had a hard time explaining this to me but in a nutshell, as she grows she uses less of the surfactant protein and more of her lung capacity. So asking you all to pray that Wrenn's lungs heal or better distribute surfactant protein is NO joke. There is a possibility that she won't need a lung transplant ever, IF she can get weaned off the oxygen and liters. Right now she is still at the highest levels but stable, which is great!
For the time being, we are not going to send her to St. Louis to be evaluated because she would still have to be shipped back to either Orlando or VA. It's risky and could hurt her in the long run. She is NOT on the list for a transplant until she declares needing one.
We have chosen to keep her in Orlando for now. Within the next two months she might be transferred across the street to Arnold Palmer Children's hospital where she will continue to grow and receive therapy to keep her developmentally up to date.
The hard part is that she is still in the hospital on HIGH levels of oxygen and on 8 liters. The only way she could come home is to be trached and vented and would require 24 hour care in our home by a nurse. NOT an option for us right now. She needs to get down on the liters. (We need to pray for her to go from 8 to 1 or zero!) Time will tell if she is the baby that will improve or decline.
Please continue to pray for her condition to improve. I really picture her leaving the hospital in her car seat with her lungs intact. I would love for all the nurses and doctors to witness a miracle and I totally believe that with this disease, anything is possible!
Today she looked right at me for several seconds and then smiled, twice... it was the best day ever to see that.. I took it as a sign that all will be okay... it's been a while since I've been able to say that. For now, I am going to just believe.
Wrenn's Baby Breath Fund
Click here to DONATE through the National Transplant Assistance Fund. Any help you can give will go directly towards Wrenn's medical expenses. Thank you!
Wrenn's Story
Wrenn was born on April 14, 2010. She was a little early, but not alarmingly so, and at 18 inches long and just under six pounds she was pink and pudgy and looked to be the picture of newborn health.
Unfortunately, Wrenn quickly developed breathing difficulties and needed full-time oxygen or her little lungs would collapse. Doctors suspected that Wrenn might have a rare genetic disorder called Surfactant Protein Deficiency.
Shortly after her birth a series of tests were run, and on her one month birthday her parents found out that Wrenn does, indeed, have the incredibly rare genetic disorder known as ABCA-3 Surfactant Protein Deficiency.
Wrenn needed new lungs and in September 2010, two months after her family relocated to St. Louis, she received a double lung transplant.
This site has been created as a space for Wrenn's family to keep friends and family updated on Wrenn's progress. She's a fighter, so we expect lots of updates!
Wrenn's parents have good health insurance, but they're still going to have huge medical bills to deal with, so we hope that anyone who can will donate to Wrenn's Baby Breath Fund using the NTAF link above.
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If you are looking for a way to show off your love for your country and do it in a stylish way, then patriotic hoodies might be the perfect thing for you. These are not the same old hoodies that can seem a bit frumpy, there are newer cut hoodies that are made with a […]
Home workouts are all the rage these days and for good reason! With gyms closed due to the Coronavirus pandemic, the only places you can work out these days are in your home or outdoors. For those of us who love the machines at the gym, this can be a big struggle. However, the good […]
If you have been looking for a truly fun workout that utilizes the whole body and gets you the kinds of results you’ve been dreaming about, look no further than boxing for exercise. What’s so great about boxing is that it is a full body workout that really tightens your muscles, and gives you a […]
If you are interested in getting in shape and appearing your best but don’t like the idea of going to the gym to exercise, then perhaps looking into a home workout plan might be a good option for you. While seemingly simple, having a good home workout plan that you stick to can be a […]
Everyone is transitioning towards organic products these days. People are finally beginning to prioritize their health and body by really restricting what goes into it! This is definitely a good thing overall, but it can be difficult to find quality organic products that are effective and affordable. This is especially true when it comes to […]
It might seem like the logical option, especially if you’ve been drinking. If you refuse the breathalyzer test, then the police cannot know what your blood-alcohol level is, right? Not only is that incorrect, but you’re setting yourself up for more trouble than just a DUI. Understanding Implied Consent In America, driving is a privilege […]
For a lot of people, there just doesn’t seem to be enough time in a day. Some people are balancing school, work, a family, social life, down time, and various hobbies or skills they’re developing, and it quickly ends up eating away at sleep and well being. When handling a busy lifestyle, it’s important to […]
Have you ever heard of driving a lemon? It refers to a vehicle, often a new one, that ends up having various defects and issues. These can affect its value, utility, and safety. So many people find their new ride is tainted that there are even lemon law attorneys to help them file a claim […]
NFL Hall of Famer Coach and co-founder of The National Child ID Program, Mike Singletary is in the news making headlines again! This time it’s for his role in the fight against the Coronavirus pandemic. “It is time to re-evaluate our priorities,” said the Hall of Famer also known as “the heart of the defense” […]
COVID-19 is thought to spread mainly through close contact from person-to-person. And some people without symptoms may be able to spread the virus. We are still learning about how the virus spreads and the severity of the illness it causes. For now, we know that the virus can spread: Between people who are in close […]
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{
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Immunomodulation of lambs following treatment with a proteasome preparation from infective larvae of Trichostrongylus colubriformis.
Proteinases are known to be capable of prolonging the survival of endoparasites in a host. We were therefore interested in knowing whether immunization of lambs against a proteasome (multisubunit proteinases) preparation obtained from Trichostrongylus colubriformis infective third-stage larvae (L3) would have any effect on the immune response to a single challenge infection with the same organism. A total of 21 penned lambs aged 8 months were divided into 3 equal groups. Group 1 was immunized on three occasions with increasing amounts of a proteasome-enriched fraction obtained from infective L3. Group 2 was given a similar amount of protein from the initial supernatant of homogenized larvae. Group 3 (controls) received adjuvant plus saline solution only. All groups were challenged with 60,000 infective T. colubriformis larvae at 28 days after the last immunization. Significant protection was obtained only when the initial supernatant extract was used to immunize lambs. The proteasome preparation seemed to have immunosuppressive effects through the stimulation of nonspecific IgE production. Significantly lower levels of specific IgE were observed in lambs immunized with the proteasome-enriched fraction, and levels of specific IgG antibodies were increased. We suggest that proteasome fractions of T. colubriformis may serve as useful preparations for the study of mechanisms of IgE production in parasitized sheep.
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{
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'use strict';
/**
* @ngdoc directive
* @name webAppApp.directive:FileEditor
* @description
* # FileEditor
*/
angular.module('webAppApp')
.directive('fileEditor', function () {
return {
templateUrl: 'views/directives/fileEditor.html',
scope: {
ngModel: '=',
ngDisabled: '=',
filesLocationUrl: '=',
filesUploadUrl: '='
},
restrict: 'E',
link: function postLink(scope, element, attrs) {
},
controller: ['$scope', 'api', 'FileUploader', function ($scope, api, FileUploader) {
var uploader = new FileUploader({
url: api.getStorageEndpoint(),
removeAfterUpload: true,
autoUpload: true
});
$scope.uploader = uploader;
$scope.showProgress = false;
$scope.configuration = '';
uploader.onBeforeUploadItem = function () {
$scope.showProgress = true;
};
uploader.onAfterAddingFile = function () {
uploader.queue[uploader.queue.length - 1].headers.Authorization = "Bearer " + api.getToken();
};
uploader.filters.push({
name: 'customFilter',
fn: function (item /*{File|FileLikeObject}*/, options) {
return this.queue.length < 10;
}
});
uploader.onErrorItem = function () {
$scope.uploadedFileUrl = '';
};
uploader.onCompleteItem = function (fileItem, response) {
debugger;
$scope.ngModel = response.url;
$scope.showProgress = false;
};
function isUrl(str) {
return !!str && str.indexOf('http') > -1;
}
function reload() {
api.getConfiguration().then(function (response) {
$scope.configuration = response.url;
});
}
reload();
}]
};
});
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{
"pile_set_name": "Github"
}
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What new hardware and software is likely in the pipeline for the bestselling $35 credit card-sized board?
Image: Matt Richardson mrichardson23@gmail.com
This year, the $35 Raspberry Pi leaped forward a generation, with the release of the third iteration of the credit card-sized computer.
But while 2017 will likely be a somewhat quieter year, there are still upgrades in the pipeline and new hardware to look forward to.
Here are the a few predictions about what you should, and shouldn't, expect concerning the best-selling board next year.
Launch of the Compute Module 3
Expect to see the Raspberry Pi powering far more appliances in 2017, following the release of the Compute Module 3 (CM3).
Due to be launched "very early next year", the CM3 will pack the same quad-core Broadcom BCM2837 processor and 1GB memory used on the Pi 3 onto a slimmer and smaller board.
SEE: Raspberry Pi: The smart person's guide
The compact design of the Compute Module, which comes with 4GB eMMC Flash storage, makes it better suited to being built into electronic products.
The CM3 marks a significant leap forward in processing power, since the previous Compute Module was based on the first-generation, single-core Raspberry Pi, which is up to ten times slower than the third-generation board.
When released, it will also be the first Compute Module to run Windows 10 IoT Core, a cut-down version of Windows 10 designed to support Internet of Things appliances.
Following the launch of the Compute Module, the electronics giant NEC has pledged to build a custom version of the CM3 into 40-, 48-, and 55-inch screens.
Raspberry Pi 3 Model A release
Originally expected to release last year, the Raspberry Pi 3 Model A now looks likely to be launched in the coming year.
Like the Pi 1 Model A, the board will be a version of the Pi 3 that has no Ethernet and only one USB port, but that sells for a cheaper price. Unlike the Pi 1 Model A, the lack of an Ethernet port and single USB will be compensated for by the inclusion of Wi-Fi and Bluetooth connectivity in the Pi 3 Model A.
Speaking in February, the Raspberry Pi co-creator Eben Upton said he expected the Pi 3 Model A to launch in the middle of this year, so a 2017 release date seems likely, however the Raspberry Pi Foundation didn't respond when asked for an update on when the Pi 3 Model A was likely to launch.
New operating systems
The Raspberry Pi is already home to a plethora of operating systems, but 2017 promises to see even more OSes ported to the $35 board.
Making this possible will be ongoing work to unlock more of the visual processing power of the board using the open-source VC4 graphics driver, which is continuing to be refined by Eric Anholt.
Speaking earlier this year, Upton said that this open-source driver, combined with the faster, 64-bit processor on the Pi 3, would make it easier to port over new operating systems.
New operating systems continue to be brought to the Pi, most recently the preview release of Android Things, a bare bones version of Google's OS focused on IoT devices rather than mobile.
Don't expect the Raspberry Pi 4
Upton has all but ruled out the release of the next generation of Raspberry Pi in 2017.
While the third-generation board was only released one year after the second, the leap to the fourth generation will likely be more tricky.
The rapid release of the Pi 3 is something of a one-off, he said earlier this year, made possible by the availability of the board's more powerful chipset and a fall in the cost of producing the Pi, which enabled Wi-Fi and Bluetooth to be added without increasing the price.
"We're kind of at the end of that particular roadmap. I would expect a longer pause, a couple of years at least, before any kind of major bump to the platform," he said at the time.
Don't expect more memory
Upton has also cautioned against expecting the Pi's memory to be increased beyond 1GB in the near future.
In an interview this autumn, Upton said that in the short term, it was unlikely to be possible to squeeze more memory onto to board without increasing the price and size of the Pi.
While he suggested that 2GB "might be feasible in a few years' time", he said that in the meantime the Foundation would focus on optimising software to run on the 1GB of memory on the Raspberry Pi 3.
Open Source Weekly Newsletter You don't want to miss our tips, tutorials, and commentary on the Linux OS and open source applications. Delivered Tuesdays Sign up today
Read more about the Raspberry Pi
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{
"pile_set_name": "OpenWebText2"
}
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1. Field of the Invention
The present invention relates to a liquid crystal display (LCD) device, and more particularly, to an LCD device and a method of fabricating the LCD device.
2. Description of the Related Art
Presently, LCD devices are being developed as the next generation of display devices because of their light weight, thin profile, and low power consumption. In general, an LCD device is a non-emissive display device that displays images using a refractive index difference utilizing optical anisotropy properties of a liquid crystal material that is interposed between an array (TFT) substrate and a color filter substrate. Among the various type of LCD devices commonly used, active matrix LCD (AM-LCD) devices have been developed because of their high resolution and superiority in displaying moving images. The AM-LCD device includes a thin film transistor (TFT) in each pixel region as a switching device, a pixel electrode in each pixel region, and a second electrode used for a common electrode.
FIG. 1 is a perspective view of an LCD device according to the related art. In FIG. 1, first and second substrates 10 and 30 are arranged to face each other with a liquid crystal material 50 interposed therebetween. On an inner surface of the first substrate 10, a color filter layer 12 and a common electrode 16, which functions as an electrode for applying an electric field to the liquid crystal layer 50, are subsequently formed. The color filter layer 12 includes a color filter for passing only a specific wavelength of light, and a black matrix (not shown) that is disposed at the boundary of the color filter and shields light from a region in which alignment of the liquid crystal material is uncontrollable. On an inner surface of the second substrate 30, a plurality of gate lines 32 and a plurality of data lines 34 are formed in a matrix array defining individual pixel regions P. A TFT T, which functions as a switching device, is disposed in each pixel region P where a gate line 32 and a data line 34 cross. A pixel electrode 46 is connected to the TFT T. Although not shown, the thin film transistor T includes a gate electrode to where a gate voltage is applied, source and drain electrodes for applying a data voltage to the pixel electrode.
First and second polarizing plates 52 and 54, which transmit only light parallel to a polarizing axis, are disposed on outer surfaces of the first and second substrates 10 and 30, respectively. An additional light source, such as a backlight, is disposed under the polarizing plate 54. Although not shown, first and second alignment layers contacting the liquid crystal layer 50 are formed on inner surfaces of the first and second substrates. The surfaces of the first and second alignment layers are aligned along predetermined rubbing directions, respectively.
FIGS. 2A and 2B are schematic views showing a liquid crystal display device according to the related art. FIG. 2A is a plan view and FIG. 2B is a cross-sectional view taken along a line II—II of FIG. 2A.
As shown in FIG. 2A, a plurality of pixel regions P are defined on a substrate 60. A black matrix 64 is formed on the substrate 60 and has a plurality of open portions 62. Each of the open portions 62 corresponds to each of the pixel regions P.
A color filter layer 66 is formed over the substrate 60 having the black matrix 64 and includes red, green and blue sub-color filters 66a, 66b and 66c. The red, green and blue sub-color filters 66a, 66b and 66c are located in the open portions 62. In addition, a common electrode 68 and an alignment layer 70 are sequentially formed over the black matrix 64 and the color filter layer 66 on the substrate 60.
The black matrix 64 is made of one of chromium (Cr)-based materials and resin materials. Typically, the black matrix material is made of resin materials. A black matrix formed by coating a resin material is formed thicker than a black matrix formed by depositing a Cr-based material. Therefore, a large step difference is created by a portion of the color filter layer 66 that overlaps edges of the black matrix 64, as shown in FIG. 2B. When a rubbing direction rl of the substrate 60 is from a right upper portion to a left lower portion in 45 degrees as shown in FIG. 2A (similarly, the rubbing direction is from right to left in the cross sectional view of FIG. 2B), a rubbing defect occurs at a first sloped side SR, as shown in FIG. 2B, that causes light leakage. This light leakage severely degrades a black picture state.
To improve an aperture ratio by reducing an align margin, a structure in which both the thin film transistors and color filters are formed on the same substrate has been suggested. In such a structure, the black matrix has a thicker thickness than the black matrix shown in FIG. 2B in order to effectively block light due to an increased step difference caused by forming a the thin film transistor. Thus, a rubbing defect caused by an even larger step difference becomes more severe. Hereinafter, color filter layer on thin film transistor (COT) type LCD device in which the color filter device is formed on the substrate having the thin film transistor will be explained.
FIGS. 3A and 3B are schematic views showing a COT type LCD device according to the related art. FIG. 3A is a plan view and FIG. 3B is a cross sectional view taken along a line III—III of FIG. 3A.
As shown in FIGS. 3A and 3B, a gate line 72 is formed on a substrate 70 along a first direction, a data line 80 is formed along a second direction orthogonal to the first direction. A thin film transistor T is located adjacent to a crossing of the gate line 72 and the data line 80.
A color filter layer 86 that includes red, green and blue sub-color filters 86a, 86b and 86c is formed over the thin film transistor T, the gate line 72 and the data line 80 on the substrate 70. Each of the red, green and blue sub-color filters 86a, 86b and 86c is located in a pixel region P defined by the gate line 72 and the data line 80. Specifically, each of the red, green and blue sub-color filters 86a, 86b and 86c overlaps portions of the gate and data lines 72 and 80. A black matrix 88 is formed over the thin film transistor T as an island pattern. An insulating layer (not shown) is formed over the entire surface of the color filter layer 86 and the black matrix 88. A pixel electrode 92 is formed on the insulating layer and is connected to the thin film transistor T.
As shown in FIG. 3B, the black matrix 88 is thickly formed to minimize parasitic capacitance in the COT structure. Therefore, when the rubbing direction rl is from a right upper portion to a left lower portion at a 45 degree angle from one corner of the pixel electrode, as shown in FIG. 3A, and from right to left, as shown in FIG. 3B, light leakage caused by a rubbing defect in region RD can occur.
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"pile_set_name": "USPTO Backgrounds"
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Q:
VBA 2010 Add Query Table: Application-defined or Object-Defined Error
If it were functioning properly, the following code would grab the name of a table from a textbox, concatenate it with 'describe ' and then put the results in a query table. As it stands, the code fails with 'Run Time Error 1004: Application-Defined or Object-Defined error'
Private Sub cmdNew_Click()
Dim TableName As String
Dim NewSheet As Excel.Worksheet
Dim ConnString As String
Dim SQLStatement As String
ConnString = ConnString = "DSN=REMOVED;UID=REMOVED;;DBQ= REMOVED;DBA=W;APA=T;EXC=F;FEN=T;QTO=T;FRC=10;FDL=10;LOB=T;RST=T;BTD=F;BNF=F;BAM=IfAllSuccessful;NUM=NLS;DPM=F;MTS=T;MDI=F;CSR=F;FWC=F;FBS=64000;TLO=O;MLD=0;ODA=F;"
TableName = ActiveSheet.txtTableName.Text
SQLStatement = "desc " & TableName
Set NewSheet = Sheets.Add
NewSheet.Name = TableName
Set qry = NewSheet.QueryTables.Add(ConnString, NewSheet.Range("A1"), SQLStatement)
qry.Refresh
End Sub
A:
Assuming you are querying an Oracle DB, the DESC keyword is for sorting a column in descending order, try using the keyword DESCRIBE in full.
If that doesn't work then run your query against the DB server outside of Excel, and see if it gives you a more helpful message?
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"pile_set_name": "StackExchange"
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Background {#Sec1}
==========
Dumbo, the famous Disney cartoon character, is an elephant with oversized ears that enable it to fly. Some real-life animals with abnormal external ears are also named "dumbo" such as the *dumbo* mouse and *dumbo* rat \[[@CR1], [@CR2]\]. Almost all mammals have outer ears (pinna) of variable sizes. The main function of the pinna is to collect sound waves and direct them into the ear. In some species, pinna also serve functions such as dissipating heat and signaling mood. In humans, different congenital pinna malformations are observed. Microtia (OMIM 600674) is an external ear developmental malformation characterized by a small, abnormally shaped pinna \[[@CR3]\]. It ranges from mild structural abnormalities to complete absence of the ear, affects one or both ears, and occurs as isolated or syndromic birth defects. The concha-type microtia is considered a mild form of microtia, with remnant ear lobule, concha, acoustic meatus, tragus, and incisura intertragica \[[@CR4]\]. Because of the variety of severity and forms, it is hard to estimate microtia prevalence---reported data varies from 0.83 to 17.4 in 10,000 live births worldwide \[[@CR3]\]. The causes of microtia among most patients are unknown, although some risk factors have been reported, such as gestational exposure to teratogens, maternal diabetes, and higher maternal parity \[[@CR3]\].
Genetic studies have made great progress in understanding ear development and function by identifying underlying genetic defects of certain diseases, especially in hearing loss, but less is known about genetic control of external ear morphogenesis. To date, only *HOXA2* mutations have been reported as responsible for isolated bilateral microtia with or without hearing loss in humans \[[@CR5]--[@CR7]\]. Single-gene defects and chromosomal aberrations have also been reported in different microtia-associated syndromes \[[@CR3], [@CR8]\]. Nevertheless, efforts in finding coding region mutations in genes responsible for microtia-associated syndromes have failed for isolated microtia. Among these, the H6 family homeobox 1 transcription factor gene (*HMX1*) in 4p16.1 deserves special attention. It plays an important role downstream of embryonic patterning genes in lateral facial mesenchyme differentiation \[[@CR1]\]. In human, recessive loss of function mutations in *HMX1* have been associated with oculoauricular syndrome (OAS, OMIM 142992) characterized by malformation of the external ear and eyes \[[@CR9], [@CR10]\]. A linkage locus of 10-Mb encompassing 4p16 has been reported in a five-generation Chinese family with isolated bilateral microtia \[[@CR11]\].
In the human genome, 98% of sequences are non-coding but harbor many regulatory elements that direct the precise spatial and temporal expression of coding genes. Comparing genomic sequences from diverse vertebrate species has revealed numerous highly conserved non-coding regions near developmental regulatory genes, particularly transcription factors and these regions are considered to have potential regulatory functions \[[@CR12]\]. These functional regions are collectively referred to conserved non-coding elements (CNEs). The general features of CNEs were noticed, including their non-random distribution in line with key developmental regulatory target genes across genomes, the distinguished sequence features with AT-rich and runs of identical nucleotides, the overlapping with transcription factor binding sites and known function as developmental enhancers in many cases \[[@CR12]\].Human diseases and phenotypic changes have been associated with alterations in CNEs \[[@CR13]--[@CR18]\]. One of the well-characterized example is the *SHH* ZRS enhancer, in which point mutations and copy number variations could result in limb malformation in both human and other species \[[@CR19]--[@CR21]\]. In wild populations of animals, a CNE proximal to the Hmx1 was also noticed and proved to be associated with external ear development \[[@CR22]\].Structural variants (SVs) such as deletions, duplications, insertions and inversions can disrupt or rearrange functional genomic elements \[[@CR23], [@CR24]\]. The genetic etiology of many diseases such as limb malformation and autism has been proven to relate to rare inherited SVs in coding gene cis-regulatory elements \[[@CR25]--[@CR27]\]. For ear development, evidence in mice implicated an evolutionarily conserved enhancer region (ECR) downstream of *Hmx1* as an important regulatory element driving ear development. Hoxa2, Meis and Pbx can act cooperatively on a 32 bp core sequence within the ECR to regulate *Hmx1* expression \[[@CR28]\]. Mutations in *Hmx1* coding region and SVs involving the *Hmx1*-ECR region have been found in animals with dysmorphic external ears, including 'dumbo' or 'misplaced ears' in mice, 'dumbo' in rats, 'crop ear' in highland cattle, and 'short ear' in Altay sheep (Table [1](#Tab1){ref-type="table"}) \[[@CR1], [@CR2], [@CR10], [@CR29], [@CR30]\]. In human genome, \~ 600 bp conserved sequence homologous to mouse *Hmx1*-ECR was also observed. However, whether genetic changes affecting this region are associated with human ear malformations is unknown.Table 1Genomic changes and dysmorphic outer ear phenotypes across speciesPhenotype descriptionPhenotype/disease entrySpeciesGenomic changesInheritanceReferencesEnlarged ear pinnae with a distinctive ventrolateral shift, microphthalmic anomaliesDumbo (*dmbo*)MouseNonsense mutation in *Hmx1* exon1RecessiveMunroe et al. \[[@CR1]\]Laterally-protruding ears and microphthalmic anomaliesMisplaced ears (*mpe*)Mouse8 bp deletion in *Hmx1* exon2RecessiveMunroe et al. \[[@CR1]\]Congenital malformations of the pinna and modest reduction in ocular sizeDumbo (dmbo)Rat5777 bp deletion encompassing Hmx1-ECRRecessiveQuina et al. \[[@CR2]\]Moderately to severely truncated earCrop earHighland cattle76 bp Hmx1-ECR duplicationDominantKoch et al. \[[@CR29]\]Shorter and thicker earShort earAltay sheep76 bp Hmx1-ECR duplicationDominantHe et al. \[[@CR30]\]Ophthalmic anomalies and external ear abnormalitiesOculoauricular syndrome (OAS)Human26 bp deletion in HMX1 coding regionRecessiveSchorder et al. \[[@CR10]\]Bilateral external ear malformation/cup earConcha type microtiaHumanDuplications involving HMX1-ECRDominantThis study
In the present study, we show that duplications involving *HMX1*-ECR are associated with human isolated bilateral concha-type microtia. A \~ 600 bp human ECR sequence may function as a tissue-specific enhancer regulating *HMX1* expression and response to *HOXA2* in the lateral facial mesenchyme that contributes to outer ear development.
Materials and methods {#Sec2}
=====================
Subjects {#Sec3}
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Five Han Chinese families with isolated bilateral microtia were included in the present study (Fig. [1](#Fig1){ref-type="fig"}). All patients were clinically evaluated, and digital photographs were taken to document ear phenotypes in affected individuals. Family 1 (F1) consisted of 25 individuals including 10 affected individuals with microtia in four generations. Family 2 (F2) and Family 3 (F3) each had six affected individuals in four generations. Family 4 (F4) and Family 5 (F5) are nuclear families with an affected child and affected mother. The ear malformations are consistent within the five families (Fig. [1](#Fig1){ref-type="fig"}b--q). We also recruited 53 patients with oculoauricular syndrome, six patients with severe bilateral isolated microtia and two patients with bilateral syndromic microtia. Blood samples from all available family members were collected following informed consent. The study was reviewed and approved by the institutional review board of the Chinese Academy of Medical Sciences.Fig. 1Five families with isolated bilateral Concha-Type Microtia. **a** Pedigree of five families (F1-F5). Individuals with available blood samples are indicated with an asterisk. **b**--**q** Identical pinna phenotypes in five families. All patients have identical bilateral concha-type microtia phenotype, and representative individuals from each family are shown. IV-4 (**b**), III-8 (**c**), IV-6 (**d**), IV-5 (**e**) in F1; III-4 (**f**, **g**), III-6 (**h**), IV-7 (**i**) in F2; IV-1 (**j**, **k**), III-1 (**l**), IV-2 (**m**) in F3; II-1 in F4 (**n**, **o**), II-1 in F5 (**p**, **q**)
Genotyping, whole genome linkage and haplotype analysis {#Sec4}
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Affymetrix Genome-Wide Human SNP array 5.0 was used to perform whole genome linkage analysis in the four-generation F1 family. Genomic DNA samples from 10 individuals were genotyped following the manufacturer's instructions. Genotype calling and quality control were performed with the Affymetrix Genotyping Console 2.1 package. Parametric multipoint linkage analysis was performed using MERLIN v.1.1.2 under the assumptions of autosomal-dominant inheritance with 99% penetrance, a disease allele frequency of 0.1%, and equal SNP allele frequency (50%). Genotyping and data analysis were accomplished at the CapitalBio Corporation (Beijing, China). Selected polymorphic micro-satellite markers within candidate disease loci were genotyped. Polymorphic micro-satellite markers and amplification primers are summarized in Additional file [1](#MOESM1){ref-type="media"}: Table S1.
Whole genome sequencing {#Sec5}
-----------------------
Ten individuals from F1 (III2, III3, III4, III5, III8, III9, IV1, IV4, IV6, IV7) underwent whole genome sequencing (WGS) using the NEBNext Ultra II DNA Library Prep kit for Illumina (New England Biolabs, Ipswich, MA, USA) and a HiSeq X Ten sequencer (Illumina, San Diego, CA, USA). Reads were aligned to the GRCh37/hg19 human reference sequence using the Burrows-Wheeler Aligner (BWA, v.0.7.8-r455) and variant calling was performed with SAMtools (v.1.0) and annotated using ANNOVAR (v.2015Dec14). Picard (v.1.111) was used to merge BAM files of the same sample and filter out duplicate reads marked. SNP/Indel, CNV, and SV variants were called and classified by SAMtools (v.1.0), Control-FREEC (v.V7.0), and CREST (v.V0.0.1), respectively. WGS was performed and bioinformatic analysis accomplished at the Novogene Corporation (Beijing, China).
Microarray analysis {#Sec6}
-------------------
Genomic copy number changes at locus 4p16.1 in F2-5 were further tested by the Laboratory of Clinical Genetics of Peking Union Medical College Hospital using a High-Resolution Array CGH analysis (SurePrint G3 Human 1x1M; Agilent Technologies, Santa Clara, CA, USA). One patient from each family was selected to undergo microarray analysis. The experiment and data analysis were performed according to the manufacturer's instructions. In brief, patient and control DNA were labeled and combined to hybridize to the 60mer oligonucleotide-based microarray. The resulting fluorescent signals were automatically scanned by the Agilent SureScan Microarray Scanner. Agilent CytoGenomics software was then used to extract and translate the signal into log ratios for further analysis of copy number changes.
Real-time quantitative PCR (qPCR) and Gap-PCR {#Sec7}
---------------------------------------------
We performed qPCR to confirm the *HMX1*-ECR duplication and determine the extent of duplications in different families. qPCR primer sequences and amplicon positions are given in Additional file [2](#MOESM2){ref-type="media"}: Table S2. qPCR assays were performed using SYBR premix Ex Taq (TaKara Bio., Dalian, China), and reactions were run in a Rotor-gene 6000 real-time rotary analyzer (Qiagen, Hilden, Germany) as previously reported \[[@CR19]\]. Data were analyzed by Rotor Gene Q series software (Qiagen, Hilden, Germany). The relative copy number (RCN) of the target sequence was determined by the comparative ΔΔCt method where ΔCt = (mean Ct~Target~) − (mean Ct~Reference~) and ΔΔCt = ΔCt~patient~ − ΔCt~control~. An RCN of \~ 1.5 indicated a heterozygous duplication. For F1 and F2, Gap-PCR was designed according to the extent of duplication implicated by qPCR assays. q20 forward and q3 reverse primers were used for Gap-PCR in F1, while q35 forward and q8 reverse primers were used for Gap-PCR in F2 (Additional file [2](#MOESM2){ref-type="media"}: Table S2). Breakpoint junctions were detected by direct Sanger sequencing of Gap-PCR products.
Dual-luciferase activity assay {#Sec8}
------------------------------
hECR and mECR fragments were PCR amplified from genomic DNA and inserted into the pGL4.23 firefly luciferase vector (Promega, Madison, WI, USA) using either a restriction digest strategy or the In-Fusion cloning kit (TaKaRa Bio, Beijing, China). The human HOXA2 cDNA sequence was inserted into the multiple cloning site of the pcDNA3.1(+) vector (Invitrogen, Carlsbad, CA, USA) using *HindIII* and *BamHI*. All plasmids were sequenced to confirm correct fragment insertion. Primers for plasmid construction are summarized in Additional file [3](#MOESM3){ref-type="media"}: Table S3. COS-1 cells were plated into 24-well plates 1 day before transfection and grown until 70--90% confluent. For each well, 500 ng luciferase reporter vector was transfected into the cells using Lipofectamin™ 3000 Reagent (Invitrogen, Carlsbad, CA, USA) with or without the pcDNA3.1 expression vector, with 25 ng of the pRL-TK Renilla luciferase vector used as an internal control to normalize transfection efficiency. 24 h post-transfection, cells were harvested and lysed with 100 μl passive lysis buffer (Promega, Madison, WI, USA). The firefly and renilla luciferase activities for each 20 μl cell lysate were measured by the Microplate Luminometer Centro LB 960 (Berthold, Germany). Relative luciferase activity was calculated by the ratio of firefly luciferase activities/renilla luciferase activities as fold change compared to pGL4.23. Assays were conducted as indicated in the dual luciferase reporter assay system manual (Promega, Madison, WI, USA). Normalized luciferase activity fold change (mean ± SD) of three experiments with six duplicates each is reported.
Results {#Sec9}
=======
Mapping of a susceptibility locus on 4p16.1 {#Sec10}
-------------------------------------------
Genome-wide linkage analysis in F1 suggested three candidate loci: a 20 Mb interval on 4p16.1, a 25 Mb interval on 4q, and a 2 Mb interval on 5q. Genotyping of selected polymorphic microsatellite markers within candidate regions of 4q and 5q showed no co-segregation status in F1. However, one polymorphic microsatellite marker (CHLC.GATA151E03) on 4p16.1 co-segregated with phenotype in F1. Fine mapping using Affymetrix SNP 5.0 microarray probe-sets refined the critical region to 1.9 Mb between rs4696668 to rs16891285 (chr4:8061832--9954880, hg19) with a HLOD score of 1.8. The interval includes 13 protein-coding genes including *HMX1*, yet we found no potential coding region mutations in these genes by sanger sequencing.
Identification of the *HMX1*-ECR duplication in five families with isolated bilateral concha-type microtia {#Sec11}
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We further performed WGS in 10 members of F1. Consistent with the previous sanger sequencing result, no potential mutations were identified in the coding region. However, WGS implicated a \~ 95 Kb duplication in the critical interval in six patients, but not in two unaffected members or two unrelated members in the family (Fig. [2](#Fig2){ref-type="fig"}a). The duplication encompasses a partial intragenic region between *CPZ* and *HMX1*, and involves the \~ 600 bp evolutionarily conserved region downstream of *HMX1* (*HMX1*-ECR). qPCR assays designed within a 600 bp critical region confirmed the duplication and detected full-segregation status in F1 (Fig. [2](#Fig2){ref-type="fig"}c). This finding prompted us to detect copy number changes in other families with isolated bilateral concha-type microtia. There are limited probes within the identified duplicated region designed in commercial array CGH systems, decreasing our accuracy and efficiency in CNV detection. Nevertheless, SurePrint G3 Human 1x1M microarray implicated increased copy number in a 46.2 Kb intergenic region (chr4: 8677567--8723767, hg19) between *CPZ* and *HMX1* in four additional families with the identical phenotype same as F1 (Fig. [2](#Fig2){ref-type="fig"}b). Duplications were confirmed by qPCR assay in the *HMX1*-ECR region in all four additional families (Fig. [2](#Fig2){ref-type="fig"}c).Fig. 2Detected duplications involving the long range *HMX1* Enhancer in five families. **a** Whole genome sequencing indicated duplications in F1. Red bar shows the duplicated region. **b** Duplications detected by array-CGH in F2-F5. Blue arrows show where the probes detected 3 copies. **c** qPCR assays in the *HMX1*-ECR region confirm the duplication and co-segregation status with phenotype in five families
Determination of the duplication extent and critical region {#Sec12}
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To determine the size of the duplications in different families, multiple qPCR assays were designed to cover a region of 253 Kb (chr4:8617326--8871246, hg19) encompassing *CPZ*, *HMX1*, and their intergenic region (Additional file [2](#MOESM2){ref-type="media"}: Table S2). Extent and overlapping regions of duplications in five families were detected (Fig. [3](#Fig3){ref-type="fig"}). We performed qPCR assays on one affected individual per family to determine the extent of duplication in each family and identified duplications of 94.6 Kb, 147 Kb, 185--213 Kb, 49.8--55.9 Kb, and 67.4--104 Kb in F1, F2, F3, F4 and F5, respectively (Fig. [3](#Fig3){ref-type="fig"}a). We detected the precise duplicated segment and breakpoints by gap-PCR and sanger sequencing in F1 (chr4:8638135--8732725, hg19) and F2 (chr4: 8677560--8,824,629, hg19) (Fig. [3](#Fig3){ref-type="fig"}b). In F3, F4 and F5, multiple qPCR assays detected the boundary regions harboring the breakpoints (Fig. [3](#Fig3){ref-type="fig"}c). All identified duplications contained a 21.8 Kb overlapping region (chr4:8,684,896--8,706,719, hg19) harboring the *HMX1*-ECR.Fig. 3Extent and overlapping regions of duplications in five families. A. Schematic diagram showing detected duplications, 600 bp core hECR sequences, and CNVs in the database of genomic variants. The blue line highlights the ECR region. B. Chromatogram of the breakpoint junctions in F1 and F2. C. A series of qPCR assays detected the extent of duplications in F3, F4 and F5
A 600 bp sequence within the duplicated region shows enhancer activity increased by HOXA2 {#Sec13}
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A 594 bp *Hmx1*-ECR region has been demonstrated to be a specific enhancer determining endogenous Hmx1 lateral facial expression patterns in mouse \[[@CR28]\]. Thus, a 600 bp human sequence (hECR) in the identified duplicated region homologous to the 594 bp mouse sequence (mECR) was tested for enhancer function by dual luciferase assay (Fig. [4](#Fig4){ref-type="fig"}). As a result, constructs containing hECR showed increased luciferase activity compared to the empty group (replicate = 3, *p *\< 0.0001), suggesting hECR enhancer activity (Fig. [4](#Fig4){ref-type="fig"}a). However, the induced luciferase activity was significantly lower in the hECR than in the mECR group, which is regulated by the Hox-Pbx-Meis complex (Fig. [4](#Fig4){ref-type="fig"}a). *HOXA2* mutations were reported in patients with isolated bilateral microtia without hearing loss. In the luciferase assay, co-transfection with human *HOXA2* expression vectors led to an 8.14-fold increase in enhancer activation, indicating that the hECR is responsive to *HOXA2* (Fig. [4](#Fig4){ref-type="fig"}b).Fig. 4Human ECR within the duplicated region shows an enhancer activity increased by HOXA2. **a** hECR showed increased luciferase activity. **b** hECR is responsive to *HOXA2. \*\*\*\*p *\< *0.0001*
Detection of *HMX1*-ECR CNVs in patients with other types of microtia {#Sec14}
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To determine whether *HMX1*-ECR CNVs associate with other ear malformations, we performed qPCR assays in the 600 bp *HMX1*-ECR in 53 patients with unilateral lobule-type microtia, six patients with isolated bilateral lobule-type microtia, one patient with bilateral concha-type microtia with preauricular sinus, and one patient with bilateral concha-type microtia with atrial septal defect. No duplications or deletions were detected in these microtia cases. In health population, some duplications involving the *HMX1*-ECR region were documented in the database of genomic variants (DGV) \[[@CR31]\], but they were relatively large in size and involved other nearby genes (at least *CPZ*) at the same time. There were also some duplications only involving the *CPZ* and *HMX1* intergenic region, but they did not contain the *HMX1*-ECR. While deletions only involving the intragenic region between *CPZ* and *HMX1*, and containing *HMX1*-ECR were also documented in the DGV database (Fig. [3](#Fig3){ref-type="fig"}a).
Discussion {#Sec15}
==========
Microtia is phenotypically and etiologically heterogeneous. Little is known about the genetic background underlying microtia. Among candidate loci for microtia, Chromosome 4p16 deserves special attention. A partial deletion from the short arm of chromosome 4 (4p deletion) results in Wolf-Hirschhorn syndrome (WHS, OMIM\#194190) featuring a distinct craniofacial phenotype and intellectual disability \[[@CR32]\]. WHS patients with pure and translocated forms of monosomy 4p16.1 → pter (M4p16.1) have different types of external ear malformation such as poorly rolled descending helix edge, short ear lobes, or deep or long concha \[[@CR33]\]. By studying 72 oculoauriculovertebral spectrum (OAVS) patients with highly heterogeneous phenotypes involving ears, eyes, face, neck and other organs, Bragagnolo et al. observed recurrent chromosomal imbalances predominantly in chromosome 4 in four patients \[[@CR34]\]. Balikova et al. reported on a large family with autosomal-dominantly inherited microtia, eye coloboma, and imperforation of the nasolacrimal duct, and found the phenotype linked to a cytogenetically visible alteration at 4p16 consisting of five copies of a copy-number-variable region \[[@CR35]\]. Li et al. reported a 10 Mb susceptibility locus for isolated bilateral microtia on 4p15.32--4p16.2 in a 5-generation Chinese family \[[@CR11]\].
*HMX1* harbored in 4p16.1, also known as *NKX5*-*3,* is an important transcription factor in craniofacial structure development, especially in eye and ear. Expression of *Hmx1* was observed in the external ear, lens, and retina of mice as early as E13.5. In humans, *HMX1* expression was observed in the optic vesicle in the 5--6-week embryonic period and in the developing pinna and auricular mesenchymatous cells at the 20-week fetus period \[[@CR10]\]. The different expression patterns of *HMX1* in ear and eye development suggest that there may be different regulatory elements determining strict spatial--temporal expression. Meanwhile, homozygous mutation in the human *HMX1* gene leads to abrogation of gene function causing oculoauricular syndrome (OAS, OMIM \#612109) affecting both the eye and external ear \[[@CR9], [@CR10], [@CR36]\]. Thus, isolated microtia and syndromic microtia without eye affects are unlikely to be caused by mutations in the *HMX1* coding region. Accordingly, we found no potential *HMX1* coding region mutations in 120 OAVS patients by whole exome sequencing (unpublished data).
Conserved non-coding elements (CNEs) are sequences outside of protein coding regions highly conserved across diverse vertebrate species \[[@CR37]\]. They may act as cis-regulatory modules (CRMs) that interact with nearby genes to determine tissue-specific gene expression, and they are enriched near transcription factor genes expressed during embryogenesis, suggesting a possible role in regulating the expression of essential developmental genes \[[@CR38], [@CR39]\]. CNEs are required for normal development, and mutations in CNEs have been established as causal for human diseases and subtle phenotypic changes that likely lead to decreased fitness over evolutionary time \[[@CR12]\]. Dickel et al. created knock out mice with individual or pairwise deletion of four CNEs near *ARX*, the essential neuronal transcription factor \[[@CR40]\]. These knockout mice showed substantial alterations of neuron populations and structural brain defects that potentially detrimental in the wild, although they were viable and fertile in laboratory conditions. *Rosin* et al. showed that *Hmx1* has such a CNE that functions as a strong and highly dynamic lateral facial enhancer \[[@CR28]\]. The CNE is a \~ 600 bp evolutionarily conserved region (ECR) with a 32 bp core sequence containing consensus binding sites for Hoxa2, Pbx, and Meis, and it has tissue-specific enhancer function in the craniofacial mesenchyme which contributes to the pinna. Genomic structural variations disrupting the ECR enhancer role associate with loss of Hmx1 expression specifically in the first and second branchial arch (BA1 and BA2) mesenchyme, leading to dysmorphic outer ears across species (Table [1](#Tab1){ref-type="table"}). Genomic findings in human patients with isolated bilateral concha-type microtia reinforce the enhancer role of *HMX1*-ECR in conserved pinna developmental processes. We also noticed that hECR has weaker enhancer activity compared to mECR via luciferase assay. However, it remains unclear whether the difference in the relative size of the pinna between human and mice is related to the level of enhancer activity.
The core sequence of hECR is highly homologous to mECR including the consensus binding sites of HOXA2, PBX and MEIS \[[@CR22], [@CR28]\]. In dual luciferase assays, co-transfection of HOXA2 and hECR resulted in increased expression level, suggesting that the hECR may also be regulated by the HOX-PBX-MEIS complex. HOX, PBX and MEIS are all homeobox proteins involved in transcriptional regulation by forming heterodimers and are essential contributors to developmental programs. Genes encoding this homeoprotein complex associate with congenital anomalies with craniofacial phenotypes. *HOXA2* is the only reported gene responsible for isolated microtia to date. Patients with homozygous mutations in HOXA2 display more severe microtia than hECR duplicated carriers, presenting middle ear deformities and hearing loss \[[@CR7]\]. *PBX1* mutations lead to congenital kidney and urinary tract anomalies with or without hearing loss, abnormal ears, or developmental delay \[[@CR41]\]. MEIS2 mutations associate with cleft palate, cardiac defects, and mental retardation \[[@CR42], [@CR43]\]. These findings suggest that *HOXA2, PBX1,* and *MEIS2* act early in patterning of the branchial arch region and transactivate *HMX1* by binding to hECR. Therefore, we speculate that any genetic changes affecting hECR regulation by the HOX-PBX-MEIS complex may lead to developmental defects involving ears and eyes.
Regulatory elements and their target gene clusters often exist in the same local chromatin interaction regions, called topologically associated domains (TADs), to ensure that the regulatory elements are specific to their target genes rather than other nearby genes \[[@CR44]\]. Boundaries between TADs are required and provide an insulator function to prohibit interference between opposing activities of neighboring domains \[[@CR44], [@CR45]\]. We used the 3D Genome Browser (<http://promoter.bx.psu.edu/hi-c/>) to visualize the chromatin interaction surrounding *HMX1*. According to Hi-C profile data from human embryonic stem cells, *HMX1* and *CPZ* are in two different TADs, while *hECR* sequences appear in the same TAD with *HMX1* but not with *CPZ* (Additional file [4](#MOESM4){ref-type="media"}: Figure S1). The detected duplications in isolated bilateral concha-type microtia patients are in the same TAD with *HMX1,* and do not interrupt the TAD boundary. They contain hECR but not the HMX1 gene. Therefore, these duplications may result in overexpression of *HMX1* by increasing the number of local enhancers but not the coding gene. Meanwhile, the copy number variation (nsv1014219) in the DGV database detected in normal population involves the hECR, the CPZ gene, and the boundary between two TADs. Therefore, due to the insulator effect of TAD boundaries, the increased hECR could not interact with *HMX1*, thus it probably does not change gene expression level.
Notably, the size of the hECR region (600 bp) is small and its copy number changes could be missed by chromosomal microarray analysis (CMA). Duplications detected in the present study range from \~ 50 to \~ 200 Kb. Although they were implicated in the Agilent SurePrint G3 Human CGH 1X1M microarray analysis, they could not be automatically detected in standard analysis process due to limited probes designed within the region. The genomic findings in these patients indicate the importance of checking the *HMX1*-ECR copy number status and highlight the necessity for custom designed microarrays with higher probe density covering this region.
Conclusions {#Sec16}
===========
In this study, we found various genomic duplications involving the *HMX1*-ECR long range enhancer in five families with isolated bilateral concha-type Microtia. The *HMX1*-ECR duplications were specifically associated with isolated bilateral concha-type microtia but not with other ear malformations or syndromic microtia. We add to evidence in humans that copy number variations in *HMX1*-ECR, a conserved non-coding elements (CNEs), associates with ear malformations, as in other species. We provide additional evidence that the dosage sensitive effects of *HMX1* may result in different types of ear malformations. Unveiling genetic causes of isolated microtia provides an entry point into understanding the regulatory network for common lateral facial birth defects and complex syndromes involving external ear malformations. Meanwhile, the results could be used for genetic counseling and screening for isolated bilateral concha-type microtia.
Supplementary information
=========================
{#Sec17}
**Additional file 1: Table S1.** Polymorphic micro-satellite markers and primers. **Additional file 2: Table S2.** qPCR primers in 4p16.1. **Additional file 3: Table S3.** Primers for plasmid construction. **Additional file 4: Figure S1.** Visualization of chromatin interaction surrounding *HMX1*.
CNVs
: Copy number variations
ECR
: Evolutionarily conserved enhancer region
HMX1
: The H6 family homeobox 1 transcription factor gene
SNP
: Single nucleotide polymorphism
aCGH
: Array comparative genomic hybridization
qPCR
: Quantitative real-time polymerase chain reaction
OAS
: Oculoauricular syndrome
SVs
: Structural variants
WGS
: Whole genome sequencing
hECR
: Human sequence of evolutionarily conserved enhancer region
mECR
: Mouse sequence of evolutionarily conserved enhancer region
DGV
: Database of genomic variants
WHS
: Wolf--Hirschhorn syndrome
OAVS
: Oculoauriculovertebral spectrum
CNEs
: Conserved non-coding elements
CRMs
: Cis-regulatory modules
BA1 and BA2
: First and second branchial arch
TADs
: Topologically associated domains
CMA
: Chromosomal microarray analysis
**Publisher\'s Note**
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Nuo Si and Xiaolu Meng contributed equally to this work
Supplementary information
=========================
**Supplementary information** accompanies this paper at 10.1186/s12967-020-02409-6.
We thank all participants in the study.
NS, BP and conceived and designed the study. XL, CL, MY, YZ, CW, PG, LZ, LL, ZL, ZZ, ZC, BP and HJ carried out the study of the clinical part. NS, XM, ZL, ZQ, LW and BP performed the genetic analysis. NS, XM and BP wrote the manuscript. NS and BP revised the manuscript. XZ, HJ contributed to supervision. XZ, HJ, BP and ZL contributed to funding acquisition. All authors read and approved the final manuscript.
This study was financially supported by the National Key Research and Development Program of China (2016YFC0905100), the CAMS Innovation Fund for Medical Sciences (CIFMS) (2016-I2M-1-002), the National Natural Science Foundation of China (81571863, 81871574), the Key Laboratory of craniofacial congenital malformation of Chinese Academy of Medical Sciences (2018PT31051).
All data generated or analyzed during this study were included in this published article and its additional files.
The study was approved by institutional review board of Chinese Academy of Medical Sciences, and all participants signed written informed consent.
Not applicable.
The authors declare that they have no competing interests.
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Abnormal secretory response to parasympathomimetic and sympathomimetic stimulations from the submaxillary gland of rats treated with reserpine.
Rats treated with 0.5 mg/kg of reserpine per day for 7 days were anesthetized and submaxillary saliva was collected and analyzed for Na+, K+, Ca++ and protein concentrations. Salivary secretion was elicited by i.p. injections of carbamylcholine (50-100 mug/kg), phenylephrine (5 mg/kg) and isoproterenol (10 mg/rat). Saliva was also collected from untreated controls. Submaxillary glands were excised from both groups of animals at the termination of the secretory response, homogenized and analyzed. Glands from other animals were removed in the resting state and similarly processed. Pretreatment with reserpine resulted in decreased volumes of salvia and in elevated salivary concentrations of Ca++ and protein. Saliva from the reserpine-treated animals secreted in response to carbamylcholine had higher concentrations of Na+ and K+ than control saliva, particularly at the low rates of flow. Saliva secreted after stimulation with the two sympathomimetic secretagogues had lower concentrations of these two ions. Resting glands from the treated animals showed significant elevations in protein and Ca++ content and a significant decrease in K+ content. At the end of the secretory response to the three secretagogues, glands from treated animals showed a significantly higher Na+ content and a significantly lower K+ content than control glands. It is concluded that pretreatment with reserpine alters the secretory response of the rat submaxillary gland to both parasympathomimetic and sympathomimetic stimulation. This alteration results from a toxic lesion caused by reserpine in the salivary cells, which involves changes in their permeability to ions and in their energy resources. These in turn, result in an abnormal stimulus-secretion coupling mechanism. The possibility that the toxic lesion is related to alterations in Ca++ homeostasis is discussed.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Julie Hecht
Julie Hecht is a contemporary American fiction writer specializing in interlacing short stories.
Personal life
Hecht has purposely revealed very little about her personal life. According to her publisher's website, she lives in the winter on the east end of Long Island, New York, and spends summer and fall in Massachusetts. In an interview with Publisher's Weekly, Hecht said that the good reaction she got from her fellow schoolchildren gave her the idea to keep writing. "It's nice to look at a group of people and see them all smiling and laughing," she said. Hecht is somewhat reclusive about publicity, rarely giving interviews and avoiding the internet. She prefers to write by hand, sitting on a couch, and faxing her work back and forth to a typist for editing.
Awards
O. Henry Award for "I Want You, I Need You, I Love You" (1979, third prize)
Bibliography
Do the Windows Open? (1997)
Was This Man a Genius?: Talks with Andy Kaufman (2001)
The Unprofessionals: A Novel (2003)
Happy Trails to You: Stories (2008)
References
External links
Hecht's work with The New Yorker
Hecht's work with Harper's Magazine
Interview with Gigantic
Category:American short story writers
Category:Living people
Category:Guggenheim Fellows
Category:Year of birth missing (living people)
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|
New insights into the origin, structure and role of CD52: a major component of the mammalian sperm glycocalyx.
The sperm glycocalyx represents the primary interface between the male gamete and its environment, and gamete interaction inevitably involves interaction with this structure. Thus, it has potential significance as a target for antibodies that inhibit sperm function. Still, little is known about the components and biological role of the sperm glycocalyx. Despite the apparent complexity of the sperm membrane, surface carbohydrate labelling experiments show a high selectivity suggesting that carbohydrate side chains of CD52, an unusually short, bipolar glycopeptide of epididymal origin, form major components of the sperm glycocalyx in all mammalian species investigated. Acquisition of the highly sialylated, lipid-anchored CD52 antigen is one of the few well-defined modifications that occur to the sperm membrane during epididymal passage. It would explain changes in lectin-binding patterns and also the remarkable surface charge differences occurring during epididymal transit, most probably attributable to its terminal sialic acid residues. CD52 seems to be immunodominant on human spermatozoa, and antibodies directed against it can agglutinate and completely immobilize human sperm in the presence of complement. Expression of the same peptide backbone in lymphocytes had largely discounted its consideration as a candidate for contraceptive development. However, the recent proof of male-specific modifications indicates the feasibility of this approach.
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5/7/17
Throughout the years I've been blogging, I've been keeping a record of the books my kids have gotten for their birthdays. This weekend my little one turned 14, and here's what he's got. I myself like to see what books other people get, and I think it's an interesting illustration of why picking "books for an x-year-old kid" is so futile an endeavor unless you really know the kid. CatStronauts: Mission Moon, and Race to Mars, by Drew Brockington. (these are graphic novels that are cute as all get out. I gave him a Catstronauts teeshirt a few years ago, and he wore it to pieces. These just came out, but I have known for months that he would be getting them).
The Northern Crusades, by Eric Christiansen (he wants to learn more about these, and very annoyingly I got rid of my own book about the Northern Crusades five or so years ago, which just goes to show that you should never get rid of any books ever. On the other hand, my book was probably out of date, and of course one should never offer inferior history to one's children.....Quite possible I will end up reading it and then telling him about it; my oral version will probably be more interesting. But in large part I bought it for him as a conscious act to show respect for his intellectual curiosity and capability).
Special bonus and thanks to Clarion books (thanks!), Ghosts of Greenglass House, by Kate Milford (yay!). He wants to wait till Christmastime so I can read it out loud to him and his brother like I did for Greenglass House. Which would indeed be sweet seasonal warmth. But I will read it to myself long before then....
Nice to hear that kids are still getting books for birthday presents! These look like interesting books. I like to hear he wants to learn about the Crusades. Such an interesting chapter in history. Thanks for the post.
Happy belated birthing-day to you! So many good books here! I'm so very excited to hear that another Greenglass House book is coming out! but I've put Catstronauts on hold in the meantime, and added the Manual of Aeronautics to my son's wishlist. He doesn't know about it, but would want it desperately.
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Fallout 4 Creation Kit and mod support are now available
The Fallout 4 Creation Kit and built-in mod support through Bethesda.net are now live as part of the 1.5 beta update.
“For Fallout 4, our goal was to make Mods easier and more accessible than ever before—for both the creators and the players. By building an all-new system with Bethesda.net we’ve made a huge leap forward in achieving that,” Bethesda said. “You can now browse and search for the latest and greatest Mods, choose your favorites, post feedback, and install them—all within the game. Simply select Mods from the main menu, and start browsing.”
For those who aspire to making those mods, the Creation Kit is available as a free download through the Bethesda.net launcher. “It allows you to touch almost anything you can imagine, from how game systems work to the creatures, the dialog, and level design,” Lead Level Designer Joel Burgess explained. “It's something that's been important to us as a studio, to make sure we support our community in this way.”
The built-in mod support is available through the 1.5 update, which is still in beta, so you'll need to be opted in if you want to leap into it right now. To do that, just right-click Fallout 4 in your Steam library, select Settings, then Betas, and then “beta” from the drop-down menu. It'll take a bit for the game to update but when it does, it'll appear as “Fallout 4 [Beta]” in your library.
Bethesda asked users to provide feedback in the Fallout 4 Creation Kit forum, so it can "update and evolve Mods and the Creation Kit based on your experiences." More information, including a range of tutorials on how to actually use the thing, can be found in the official Creation Kit Wiki.
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Q:
How do you compare using .NET types in an NHibernate ICriteria query for an ICompositeUserType?
I have an answered StackOverflow question about how to combine to legacy CHAR database date and time fields into one .NET DateTime property in my POCO
here (thanks much Berryl!). Now i am trying to get a custom ICritera query to work against that very DateTime property to no avail. here's my query:
ICriteria criteria =
Session.CreateCriteria<InputFileLog>()
.Add(Expression.Gt(MembersOf<InputFileLog>.GetName(x => x.FileCreationDateTime), DateTime.Now.AddDays(-14)))
.AddOrder(Order.Desc(Projections.Id()))
.CreateCriteria(typeof(InputFile).Name)
.Add(Expression.Eq(MembersOf<InputFile>.GetName(x => x.Id), inputFileName));
IList<InputFileLog> list = criteria.List<InputFileLog>();
And here's the query it's generating:
SELECT this_.input_file_token as input1_9_2_,
this_.file_creation_date as file2_9_2_,
this_.file_creation_time as file3_9_2_,
this_.approval_ind as approval4_9_2_,
this_.file_id as file5_9_2_,
this_.process_name as process6_9_2_,
this_.process_status as process7_9_2_,
this_.input_file_name as input8_9_2_,
gonogo3_.input_file_token as input1_6_0_,
gonogo3_.go_nogo_ind as go2_6_0_,
inputfile1_.input_file_name as input1_3_1_,
inputfile1_.src_code as src2_3_1_,
inputfile1_.process_cat_code as process3_3_1_
FROM input_file_log this_
left outer join go_nogo gonogo3_ on this_.input_file_token=gonogo3_.input_file_token
inner join input_file inputfile1_ on this_.input_file_name=inputfile1_.input_file_name
WHERE this_.file_creation_date > :p0 and
this_.file_creation_time > :p1 and
inputfile1_.input_file_name = :p2
ORDER BY this_.input_file_token desc;
:p0 = '20100401',
:p1 = '15:15:27',
:p2 = 'LMCONV_JR'
The query is exactly what i would expect, actually, except it doesn't actually give me what i want (all the rows in the last 2 weeks) because in the DB it's doing a greater than comparison using CHARs instead of DATEs. I have no idea how to get the query to convert the CHAR values into a DATE in the query without doing a CreateSQLQuery(), which I would like to avoid. Anyone know how to do this?
UPDATE:
I've been looking into trying to use Projections.SqlFunction() or formulas to accomplish this, but to no avail so far. Here's the code i have using SqlFunction(), but i get an NHibernate.QueryException : property does not map to a single column: FileCreationDateTime error:
DateTime twoWeeksAgo = DateTime.Now.AddDays(-14);
ICriteria criteria =
Session.CreateCriteria<InputFileLog>()
.Add(Restrictions.Gt(Projections.SqlFunction("to_date", NHibernateUtil.DateTime, Projections.Property(MembersOf<InputFileLog>.GetName(x => x.FileCreationDateTime))), twoWeeksAgo))
//.Add(Expression.Gt(MembersOf<InputFileLog>.GetName(x => x.FileCreationDateTime), DateTime.Now.AddDays(-14)))
.AddOrder(Order.Desc(Projections.Id()))
.CreateCriteria(typeof(InputFile).Name)
.Add(Expression.Eq(MembersOf<InputFile>.GetName(x => x.Id), inputFileName));
I'm sure i'm doing something wrong here and it doesn't like it still anyway because FileCreationDateTime uses a custom ICompositeUserType which splits the .NET DateTime property into two Oracle SQL CHAR columns (see this StackOverflow question for details).
A:
I finally figure this out! here's the code (for some reason StackOverflow is making some of the methods names in the this first code snippet the syntax color of a type):
IList<InputFileLog> list = null;
DateTime twoWeeksAgo = DateTime.Now.AddDays(-14);
IProjection datePropProj =
DefaultStringFileCreationDateTimeType.GetFileCreationDateToDateSQLProjection();
IProjection timePropProj =
DefaultStringFileCreationDateTimeType.GetFileCreationTimeToDateSQLProjection();
IProjection dateConstProj =
DefaultStringFileCreationDateTimeType.GetFileCreationDateToDateSQLFunction(twoWeeksAgo);
IProjection timeConstProj =
DefaultStringFileCreationDateTimeType.GetFileCreationTimeToDateSQLFunction(twoWeeksAgo);
ICriteria criteria =
Session.CreateCriteria<InputFileLog>()
.Add(Restrictions.Or(Restrictions.GtProperty(datePropProj, dateConstProj),
Restrictions.And(Restrictions.EqProperty(datePropProj, dateConstProj),
Restrictions.GeProperty(timePropProj, timeConstProj))))
.AddOrder(Order.Desc(Projections.Id()))
.CreateCriteria(typeof(InputFile).Name)
.Add(Expression.Eq(MembersOf<InputFile>.GetName(x => x.Id), inputFileName));
list = criteria.List<InputFileLog>();
And here's the methods i used to create the SQLProjections and SQLFunctions. i put them in my ICompositeUserType (DefaultStringFileCreationDateTime) that i used for the custom type mapping on the FileCreationDateTime property.
public class DefaultStringFileCreationDateTime : ICompositeUserType
{
.
.
.
public const string DotNetDateFormat = "yyyyMMdd";
public const string DotNetTimeFormat = "HH:mm:ss";
public const string DbDateFormat = "YYYYMMDD";
public const string DbTimeFormat = "HH24:MI:SS";
private const string _nullDateRepresentationInDb = "00000000";
public struct DatabaseFieldNames
{
/// <summary>
/// File creation date column name.
/// </summary>
public const string FileCreationDate = "file_creation_date";
/// <summary>
/// File creation time column name.
/// </summary>
public const string FileCreationTime = "file_creation_time";
}
public static IProjection GetFileCreationDateToDateSQLProjection()
{
return ProjectionUtil.GetToDateSQLProjection(DatabaseFieldNames.FileCreationDate, DbDateFormat, NHibernateUtil.DateTime);
}
public static IProjection GetFileCreationTimeToDateSQLProjection()
{
return ProjectionUtil.GetToDateSQLProjection(DatabaseFieldNames.FileCreationTime, DbTimeFormat, NHibernateUtil.DateTime);
}
public static IProjection GetFileCreationDateToDateSQLFunction(DateTime dt)
{
return ProjectionUtil.GetToDateSQLFunction(dt, DotNetDateFormat, DbDateFormat);
}
public static IProjection GetFileCreationTimeToDateSQLFunction(DateTime dt)
{
return ProjectionUtil.GetToDateSQLFunction(dt, DotNetTimeFormat, DbTimeFormat);
}
}
I was already using the consts DatabaseFieldNames struct for the PropertyNames member implementation, so I was able to reuse these hard-coded column names for the Projections i needed as well.
Here's the Projection utility class where the generic to_date methods live:
public class ProjectionUtil
{
public static IProjection GetToDateSQLProjection(
string columnName, string dbToDateFormat, IType returnType)
{
return Projections.SqlProjection(
string.Format("to_date({0}, '{1}') as {0}", columnName, dbToDateFormat),
new string[] { columnName },
new IType[] { returnType });
}
public static IProjection GetToDateSQLFunction(
DateTime dt, string dotNetFormatString, string dbFormatString)
{
return Projections.SqlFunction(
"to_date",
NHibernateUtil.DateTime,
Projections.Constant(dt.ToString(dotNetFormatString)),
Projections.Constant(dbFormatString));
}
}
Finally, here's the Oracle SQL that NHibernate generates:
SELECT
this_.input_file_token as input1_9_2_,
this_.file_creation_date as file2_9_2_,
this_.file_creation_time as file3_9_2_,
this_.approval_ind as approval4_9_2_,
this_.file_id as file5_9_2_,
this_.process_name as process6_9_2_,
this_.process_status as process7_9_2_,
this_.input_file_name as input8_9_2_,
gonogo3_.input_file_token as input1_6_0_,
gonogo3_.go_nogo_ind as go2_6_0_,
inputfile1_.input_file_name as input1_3_1_,
inputfile1_.src_code as src2_3_1_,
inputfile1_.process_cat_code as process3_3_1_
FROM
input_file_log this_
left outer join go_nogo gonogo3_ on this_.input_file_token=gonogo3_.input_file_token
inner join input_file inputfile1_ on this_.input_file_name=inputfile1_.input_file_name
WHERE
(
to_date(file_creation_date, 'YYYYMMDD') > to_date(:p0, :p1) or
(
to_date(file_creation_date, 'YYYYMMDD') = to_date(:p2, :p3) and
to_date(file_creation_time, 'HH24:MI:SS') >= to_date(:p4, :p5)
)
) and
inputfile1_.input_file_name = :p6
ORDER BY this_.input_file_token desc;
:p0 = '20100415',
:p1 = 'YYYYMMDD',
:p2 = '20100415',
:p3 = 'YYYYMMDD',
:p4 = '18:48:48',
:p5 = 'HH24:MI:SS',
:p6 = 'LMCONV_JR'
can't believe i got this one! i thought i was going to have to resort to an ISQLQuery for sure!
|
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|
INTRODUCTION
============
Testosterone replacement therapy (TRT) for the treatment of testosterone deficiency syndrome (TDS) has been widely used in the South Korea for decades.[@B1] The aim of TRT is to establish a physiologically normal concentration of serum testosterone to correct for androgen deficiency, relieve its symptoms, and prevent long-term sequelae. The target organs of testosterone exist throughout the whole body and include the skin, muscle, liver, kidney, brain, bone marrow, and male reproductive organs.[@B2] However, in terms of the prostate, TRT has possible side effects including development of prostate cancer and worsening symptoms of benign prostatic hyperplasia (BPH), because prostate growth is dependent on the presence of androgens, and androgens play an important role in the development of BPH. Nevertheless, there is no direct correlation between serum testosterone levels in men and the risk of developing prostate cancer.[@B3],[@B4] Furthermore, there are no compelling data suggesting that TRT contributes to worsening of lower urinary tract symptoms (LUTS) or promotion of urinary retention.[@B5] Instead, an increasing body of literature with short-term follow up has recently demonstrated positive effects on LUTS and uroflow parameters.[@B6]-[@B9] We therefore evaluated the correlation between TRT and LUTS for patients who had been treated with TRT for over a year, in patients with TDS concomitant with moderate LUTS. To the best of our knowledge, this is the first data published on this issue particularly in a Korean population.
MATERIALS AND METHODS
=====================
1. Institutional strategy for TRT
---------------------------------
Between January 2006 and January 2011, 383 patients underwent TRT using intramuscular injection of 1,000 mg of testosterone undecanoate (Nebido™, Bayer Pharma AG, Berlin, Germany) in our institution. The institutional policy for initiation of TRT was the complaint of erectile dysfunction as the main symptom of TDS with a serum testosterone level less than 3.5 ng/ml. For all, a complete medical history regarding concomitant medical disease and medication from other departments or institutions was carefully taken and physical examination was performed. To estimate the efficacy of TRT, serum testosterone levels were routinely measured during the first visit, then 3, 6, and 12 months later; the time of blood sampling was 7 AM to 11 AM. The amount of total and free testosterone were measured separately, both by radioimmunoassay. Also, the changes in symptoms induced by TDS were assessed by a self-administered aging male symptoms (AMS) questionnaire, which was performed again routinely a year after TRT. The exclusion criteria for TRT in our institution were a prior history of treated hypogonadism, history of prostate cancer, active systemic disease, human immunodeficiency virus, psychosis, or history of sleep apnea. In case of initial prostate specific antigen (PSA) over 4.0 ng/ml, TRT was selectively performed for the pathologically confirmed BPH cases.
2. The main assessment of prostatic status
------------------------------------------
To investigate the baseline status of the prostate, patients underwent transrectal ultrasonography to determine the prostate volume, serum PSA, uroflowmetery parameters including maximal flow rate, voiding volume, and post-voiding residual urine. The International Prostate Symptom Score (IPSS) questionnaire was performed before TRT. In the IPSS questionnaire, the storage symptom score was defined as the sum of items 2, 4 and 7 and the voiding symptom score was defined as the sum of items 1, 3, 5, and 6. The serum PSA, uroflowmetry, and IPSS were routinely re-evaluated a year after TRT, except for a case of a complaint of aggravation of LUTS that required clinical assessment for the status of the prostate. According to the decision of the treating physician, BPH medications including alpha blockers or phytotherapy were additionally administered after initiation of TRT. However, 5-alpha reductase inhibitor was not used during TRT. At the beginning of this series, we did not have any regular restriction on the usage of LUTS medication during TRT, considering the potential development or aggravation of prostatic disease.
3. Patient selection and statistical analysis
---------------------------------------------
Among all potential subjects, 246 patients who had completed a minimum of a year of follow up, with a normal digital rectal examination and PSA level of less than 4 ng/ml, and had undergone a prostate evaluation after a year of TRT were recruited for this study by retrospective chart review. In addition, another group of 17 patients who had moderate LUTS, which was defined as an initial IPSS over 8 points but below 19 with a maximal flow rate of at least 10 ml/s, but who did not take any BPH medication or use of phosphodiesterase type 5 inhibitors daily during testosterone replacement were selected from the 246 cases of TRT. Then the outcomes were assessed separately for the two groups.
The primary end point of this study was changes in the IPSS and uroflowmetry parameters including maximal flow rate (Qmax), voiding volume, and post voiding residual urine (PVR) during a year. Secondary efficacy and safety was assessed according to changes in serum testosterone, AMS score, Body Mass Index (BMI), and serum PSA.
For statistical analysis of the whole patient, to evaluate the goodness-of-fit, the Kolmogorov-Smirnov test was performed first, then Wilcoxon\'s signed rank test or a paired t-test was performed to compare several parameters before and after TRT. In the 17 selected cases with LUTS without BPH medication, the changes in each parameter were compared by Wilcoxon\'s signed rank test, assuming p\<0.05 to be significant. To compare the no-BPH medication patients and their counterparts, the Mann-Whitney U test was applied. All statistical analyses were performed using SPSS™ version 17.0 (SPSS Inc., Chicago, IL, USA).
RESULTS
=======
1. Outcomes from 246 patients with a year follow up
---------------------------------------------------
The median age with interquartile range (IQR) of the 246 patients with TRT was 58.5 (52\~64.2) years old and the mean follow up period for TRT was 14.7 (13\~19.3) months. Before TRT, the median (IQR) total and free testosterone levels were 2.51 (1.97\~3.06) ng/ml and 8.12 (6.46\~10.49) ng/ml, respectively. The median baseline prostate volume was 24.2 (19.7\~29.7) ml.
After a year of TRT, the AMS score and serum testosterone level including total and free testosterone were significantly increased ([Table 1](#T1){ref-type="table"}). The BMI was also significantly decreased compared to baseline (p\<0.001). In contrast, the IPSS and uroflowmetry parameters did not show any significant change from the baseline status. Interestingly, the serum PSA level remained unchanged after a year of TRT (p=0.078).
2. Outcomes from the 17 patients with moderate LUTS without BPH medication
--------------------------------------------------------------------------
The median (IQR) age of the 17 patients was 53 (52\~63) years old, and median follow up period during TRT was 15.1 (14\~16) months. In all of the patients, the total and free testosterone increased after TRT, and this change was statistically significant ([Table 2](#T2){ref-type="table"}). However, the change in AMS score and BMI was not significant in this patient group (p=0.154 and 1.000, respectively).
Regarding the prostate status, when comparing with the counterpart group (n=229), these selected 17 patients with moderate LUTS without BPH medication had similar baseline prostate characteristics in all variables including prostate volume, IPSS, Qmax, voiding volume, and serum PSA except PVR, but increased PVR in the no-BPH medication group (21 ml vs. 10 ml, [Table 3](#T3){ref-type="table"}). All of the selected patients had a minimal Qmax over 10 (10.84\~45) ml/s.
The total IPSS had decreased significantly in the no-BPH medication group (p=0.028). One patient with a baseline prostate volume of 27.2 ml had reported an increased IPSS score from 10 to 12, and one patient with a baseline prostate volume of 31 ml had an unchanged IPSS score of 11. However, the maximal flow rate from both patients was maintained above 20 ml/s during TRT. Other than these two cases, the IPSS score was diminished in the rest of the patients, both in storage symptoms (items 2, 4, and 7) and voiding symptoms (items 1, 3, 5, and 6; [Table 2](#T2){ref-type="table"}). However, uroflowmetery variables including Qmax and PVR were maintained after a year of TRT. The serum PSA level also remained unchanged (p=0.144). No patients experienced urinary retention, BPH-related surgery, or admission for urinary tract infection during follow up.
DISCUSSION
==========
Although contraindications for testosterone supplementation in aging men are controversial due to the lack of large-scale studies, the greatest concerns for contemporary TRT are potential side effects on the prostate. It is well-known that the presence of androgen is required for the development of BPH, and antiandrogen agents can decrease prostate volume in patients with BPH,[@B10] suggesting that TRT could result in worsening of LUTSs. However, recent research on the relationship between TRT and the progression of LUTS revealed outcomes opposite these traditional concerns. Tenover[@B6] performed a short-term clinical study using 100 mg of testosterone enanthate every 3 weeks for 3 months and reported that short-term TRT did not significantly increase prostate volume or PVR. From a double-blind controlled study design, Holmäng et al[@B7] indicated that TRT contributed to an increase of 12% in volume of the prostate gland after 8 months of treatment, but did not affect urodynamic data or IPSS. Similarly, in a double-blind placebo-controlled study, Marks et al[@B8] demonstrated that 6 months of testosterone administration normalized serum androgen levels in aging men with hypogonadism but had little effect on prostatic tissue androgen levels or on their urinary symptoms and urodynamic data. Recently, Shigehara et al[@B9] reported the outcome from a year-long randomized control study on 42 patients with TDS concomitant with LUTS, demonstrating improvement not only in IPSS but also in Qmax only by TRT.
In this study, we found that TRT for hypogonadal men with moderate LUTS could contribute to improvement of subjective LUTS. IPSS sub-scores also showed improvements both in storage (2, 4, 7) and voiding (1, 3, 5, 6) symptoms. On the other hand, there was no statistical difference in terms of uroflowmetery parameters including Qmax, the relatively higher Qmax in our selected group may have affected the outcomes. Like the outcomes reported by other researchers, the serum PSA level remained unchanged, and it is noteworthy that no clinical progression of BPH including urinary retention, surgery for BPH, or admission for urinary tract infection occurred during more than a year of follow up.
Possible explanations for the improvement of LUTS and bladder function by testosterone therapy have been suggested. Indeed, there are several similarities in the etiologies of erectile dysfunction and LUTS, such as metabolic syndrome, autonomic nervous activity, nitric oxide (NO) activity, and rho-kinase activity.[@B11] Testosterone has been shown to have an effect on rho-kinase activity, and rho-kinase activity in the urinary tract may at least in part depend on testosterone.[@B12] Increased rho-kinase activity coincides with the development of LUTS in aging men with BPH, and inhibition of rho-kinase in the rat model is thought to decrease prostatic smooth muscle cell proliferation and to decrease adrenergic contractions.[@B13] NO is also one of the mediators of dilatation of the urethra and bladder neck.[@B11] Chamness et al[@B14] reported that testosterone had an effect on NO synthase in the urinary tract and suggested the possibility of improving urinary symptoms by increasing NO production.
By an uncontrolled prospective study, Karazindiyanoğlu and Cayan[@B15] indicated that TRT significantly increased maximal bladder capacity and compliance, and decreased detrusor pressure at maximal flow, based on the results of pressure-flow analysis.[@B15] Celayir[@B16] observed the changes in rabbit bladder tissue after bilateral orchiectomy and demonstrated that testosterone injection significantly increased bladder capacity and compliance. By observing bladders of castrated rats receiving androgen therapy for 28 days, Madeiro et al[@B17] had reported improvement in terms of the number of vessels, epithelial thickness, and quantity of muscular fibers compared to untreated controls. Cayan et al[@B18] reported that, in rats, testosterone treatment resulted in a significantly higher bladder smooth muscle/collagen ratio than controls. These findings suggest that TRT can affect bladder muscle contractility and compliance.
This study had inherent limitations mainly due to its retrospective study design, including absence of urodynamic studies and follow up prostate volume after a year. Also, the small number of the patients with moderate LUTS without BPH medication lessened the power of the statistical analysis. The relatively highly maintained Qmax in the 17 selected men may bias the outcome toward a more favorable direction. In addition, the relatively high follow up loss rate of this study among 383 TRT men (29.5%) may also have biased the outcome toward positive results. Finally, due to the absence of a placebo group, we cannot draw conclusions about the subjective improvements, particularly in the Qmax without TRT, and the constitutive impact on LUTS. This is the reason that a larger prospective study design is still required on this issue, considering the lifelong impact of contemporary TRT.
CONCLUSIONS
===========
Over a year of TRT for selected patients with moderate LUTS and no BPH medication improved both storage and voiding symptoms, without causing clinical progression of BPH or increasing PSA. Thus, TRT can be a safe strategy even in TDS patients with moderate LUTS but maintaining a relatively high uroflow rate.
######
Comparison of parameters at baseline and after TRT in 246 patients
Values are presented as mean (range).
TRT: testosterone replacement therapy, AMS: aging male symptom, BMI: body mass index, IPSS: International Prostate Symptom Score, QoL: quality of life, Qmax: maximal flow rate, PVR: post voiding residual urine, PSA: prostate specific antigen.
######
Comparison of parameters at baseline and after TRT in 17 selected patients with moderate LUTS without BPH medication
Values are presented as mean (range).
TRT: testosterone replacement therapy, LUTS: lower urinary tract symptom, BPH: benign prostate hyperplasia, AMS: aging male symptom, BMI: body mass index, IPSS: International Prostate Symptom Score, Qmax: maximal flow rate, PVR: post voiding residual urine, PSA: prostate specific antigen.
######
Comparison of baseline characteristics in patients with or without BPH-related medication
Values are presented as mean (range).
BPH: benign prostate hyperplasia, BMI: body mass index, AMS: aging male symptom, IPSS: International Prostate Symptom Score, Qmax: maximal flow rate, PVR: post voiding residual urine, PSA: prostate specific antigen.
|
{
"pile_set_name": "PubMed Central"
}
|
HLA-B*40:356, identified by next-generation sequence based typing in a Chinese tuberculosis patient.
HLA-B*40:356 differs from B*40:02:01 by only one nucleotide transition, C>A 1040 in exon 6.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Elastic stable intramedullary nailing versus Kirschner wire pinning: outcome of severely displaced proximal humeral fractures in juvenile patients.
Significantly displaced juvenile proximal humeral fractures (Neer-Horowitz type 3 and 4) usually require reduction and fixation. The most commonly used fixation methods are Kirschner wire (K-wire) pinning or retrograde elastic stable intramedullary nailing (ESIN). However, results comparing the long-term outcome of both methods are absent in the literature. The aim of this study was to provide an outcome comparison of both techniques. Included were 40 patients treated between 1998 and 2008 and who had complete records concerning operation time, duration of hospital stay, and time until implant removal. The assessment of clinical (Disabilities of Arm, Shoulder and Hand [DASH] and Constant-Murley scores) and radiologic long-term outcome was possible in 31 patients (78%). Preoperative, postoperative and follow-up radiographs of these patients were evaluated for angular deformity, reduction, and remodeling. The mean follow-up of the 31 patients (16 ESIN; 15 K-wire) was 5.8 ± 3.6 (standard deviation) years. The operative time of the primary fixation procedure was shorter in the ESIN group (P < .001), but the hospital stay and the time until implant removal were significantly longer. No significant difference was seen between the groups at follow-up for the mean DASH (ESIN, 1.44; K-wire, 1.66) or Constant-Murley (ESIN, 89.5; K-wire, 92) scores. The neck-shaft angle was significantly improved by reduction in both groups (P < .001) and remained unchanged at follow-up. ESIN and K-wire pinning have a favorable and comparable functional outcome and therefore seem to be adequate methods for treating Neer-Horowitz type 3 and 4 proximal humeral fractures in juvenile patients. The initially achieved improvement of the neck-shaft angle can be maintained at long-term follow-up.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Q:
#define negative numbers?
Simple question, but I cannot seem to find the answer. Is it okay to use #define to define a negative number, as in:
#define kGravity -9.8
XCode is changing the 9.8 to the color of my numbers I have set (purple), but the - is being shown as the same color as the define statement (orange).
Is this legal? Will it compile?
A:
Did you try it? It should work fine.
However, I would encourage you to not use pre-processor macros, but instead use real constants.
static const float kGravity = -9.8f;
Preprocessor directives are a bit frowned upon, in general. Here's some more info on the subject: #define vs const in Objective-C
A:
It is absolutely legal to define negative constants with #define. What you discovered is most likely a bug in Xcode's code coloring, which will probably be fixed in one of the future revisions.
|
{
"pile_set_name": "StackExchange"
}
|
Another solid month of job gains expected for February
U.S. employers are expected to follow the best three-month burst of hiring in 17 years with another solid round of job gains in February.
Still, the result is unlikely to match the furious pace of November through January, when 1 million positions were added. Economic growth probably isn't fast enough to sustain such a large gain. And harsh winter weather might have discouraged some hiring.
Advertisement
Economists have forecast a job gain of 240,000 and a drop in unemployment to a near-normal 5.6 percent from 5.7 percent, according to the data firm FactSet. That would be evidence of a job market that continues to outshine others around the world.
The February jobs report will be released at 5:30 a.m. Pacific time.
"People are pretty optimistic about the U.S. economy, and they're hiring," said Frank Friedman, interim CEO of Deloitte, the consulting firm that counts 80 percent of the Fortune 500 as clients.
A bright outlook among employers has translated into a robust average of 267,000 jobs added monthly over the past 12 months. That means there are 3.2 million more Americans earning paychecks now than at the start of 2014. That additional income, along with sharply lower gas prices, has left more Americans able to spend.
The steady hiring may also finally be forcing wages up. Average hourly earnings rose 0.5 percent in January, the most in six years. Economists did caution against reading too much into one month's figure. Most expect a more modest average wage gain in February.
Friday's jobs report will come less than two weeks before the next policy meeting of the Federal Reserve, which is considering when to raise interest rates from record lows. Tim Hopper, chief economist at TIAA-CREF, suggested that the strengthening job market and tentative signs of pay increases give the Fed room to move toward raising short-term rates.
Most analysts expect the Fed to pave the way for higher rates by adjusting the statement it issues after its March meeting, to be followed by the first hike in June or September.
It may turn out that some temporary factors held back job growth in February. Snow and ice storms in the Midwest and parts of the Southeast closed some businesses and possibly delayed hiring. Boston and other parts of the Northeast have been hit by enormous snowfalls.
Investment bank UBS estimates that such factors lowered February's job gain by 25,000. Construction companies, auto dealers, and retailers are the sectors most likely to have been affected by winter storms and unseasonably cold weather.
Several industries may also be hiring less than in recent months or even cutting back. Oil and gas drilling companies have cut jobs in response to the 60 percent drop in oil prices since summer. Applications for unemployment aid have risen in such oil-heavy states as Texas, Oklahoma and North Dakota.
In January, retailers reported a sizable job gain, which most economists don't think is sustainable.
Manufacturers may also have pulled back in the face of weaker growth overseas. A survey of manufacturing firms shows that export orders have shrunk for two months. The U.S. dollar has also soared in value compared with the euro and Japan's yen, thereby squeezing profits for American multinationals that operate overseas.
Yet the U.S. job market and economy, for all their obstacles, are still outdoing those of other major nations. Though Europe and Japan are showing signs of growing more than last year, their economies remain feeble. The euro currency union's unemployment rate has started to fall, but at 11.2 percent it remains nearly twice the U.S. level.
The U.S. economy expanded at a breakneck annual pace of 4.8 percent in last year's spring and summer, only to slow to a tepid 2.2 percent rate in the final three months of 2014. Many economists estimate that growth is picking up slightly in the current quarter to an annual rate of 2.5 percent to nearly 3 percent.
Advertisement
Economists remain bullish about hiring despite the slowdown in growth. The fourth quarter's slowdown occurred largely because companies reduced their stockpiles of goods, which translated into lower factory output.
But companies focus more on consumer demand in making hiring decisions, and demand was strong in the October-December quarter. Americans stepped up their spending by the most in four years.
And though consumers are saving much of the cash they have from cheaper gas, spending in January still rose at a decent pace after adjusting for lower prices.
Mark Zandi, chief economist at Moody's Analytics, expects the economy to grow 3 percent this year, which would be first time it's reached that level in a decade. That's fast enough to support hiring of about 250,000 a month, he said.
|
{
"pile_set_name": "Pile-CC"
}
|
1. Field of the Invention
The present invention relates generally to inserting advertisements in web pages, or more generally any medium in which ad performance can be measured and quantified. In particular, the present invention is directed towards determining an optimal allocation of advertisements in response to specified objectives and constraints.
2. Description of the Related Art
Web site advertising has traditionally included a variety of models. In one model, known as an auction model, advertisers bid for space on a web site. In the auction model, an advertising space, also known in the art as an impression, is given to the highest bidder (or bidders). In another model, called a guarantee model, the site guarantees a certain number of impressions to an advertiser in exchange for a fixed fee. The guarantee model also supports additional arrangements—for example, a site can guarantee that an impression will receive a minimum number of clicks, or will result in a minimum number of transactions.
The number of impressions available on a site is known as the site's inventory. Inventory is traditionally hard to precisely measure because of difficulties in estimating site traffic, since the amount of traffic on a site can be highly variable, and hence the number of impressions available in the future is unknown. In addition, available inventory is hard to track because of the existence of guarantee-based contracts. Consider the following example, which illustrates the difficulty of tracking inventory. Assume that a site has five keywords in its inventory of slots to which impressions can be allocated. Assume also that the site charges an advertiser a first amount for each impression and a second amount for each click on that impression. Assume the following amounts for four clients:
Client A pays $0.25 for clicks on keywords 1, 2 and 3, and $0.05 for each displayed impression. Client 1 has been guaranteed 10,000 impressions.
Client B pays $0.35 for clicks on keywords 2, 3 and 4, and $0.10 for each displayed impression. Client 2 has been guaranteed 15,000 impressions.
Client C pays $0.85 for clicks on keywords 2 and 4, and $0.05 for each displayed impression. Client 3 has been guaranteed 5,000 impressions.
Client D pays $0.05 for clicks on keywords 1 and 3, and $0.02 for each impression. Client 4 has been guaranteed 15,000 impressions.
Now assume that a potential client, client E, is willing to pay $0.20 for clicks on keywords 1 and 5, and is willing to pay $0.02 for each impression. Client E demands a guarantee of 11,000 impressions. The site has estimated its inventory of keyword 5 at 6,000 impressions, and inventory of keyword 1 as 4,000 impressions.
To determine whether accepting the contract with client E is possible, the site operator must determine whether an additional 1,000 impressions can be delivered for keywords 1 and 5 by shifting the existing allocation of keywords and not violating any guarantees made to the other advertisers. As the number of advertisers, keywords and guarantees grows, this problem becomes increasingly complex. Accommodating even a single new advertiser may necessitate the rescheduling of hundreds or even thousands of other advertisers (and their associated creatives) due to a ripple effect caused by moving one advertiser's allocation, which then causes another advertiser's allocation to be moved, etc.
There is therefore a need in the art for an efficient way to allocate advertising impressions across a web site while maximizing revenue for the site and honoring guarantees made to advertisers.
|
{
"pile_set_name": "USPTO Backgrounds"
}
|
648 F.2d 598
UNITED STATES of America, Plaintiff-Appellee,v.Ali Asghar TAHERI, Defendant-Appellant.
No. 79-1711.
United States Court of Appeals,Ninth Circuit.
Argued and Submitted Jan. 8, 1981.Decided June 1, 1981.
Kenneth McMullan, San Diego, Cal., for defendant-appellant.
Hector E. Salitrero, Asst. U. S. Atty., argued, M. James Lorenz, U. S. Atty., Hector E. Salitrero, Asst. U. S. Atty., on the brief, San Diego, Cal., for plaintiff-appellee.
Appeal from the United States District Court for the Southern District of California.
Before TRASK, SNEED and SCHROEDER, Circuit Judges.
SCHROEDER, Circuit Judge:
1
This is an appeal from a conviction on three counts of charges related to drug possession.1 The case was tried to the Court on stipulated facts. Two issues are presented for review. The first is the admissibility of evidence of heroin seized pursuant to a warrant, but originally discovered in an initial search which had been conducted without a warrant and without probable cause. The second issue is whether, assuming the heroin was inadmissible, the defendant's post-arrest consent to other searches, which yielded opium, was sufficient to purge the primary taint. We reverse all of the convictions.
2
On May 30, 1979, an informant of unknown reliability supplied a DEA agent with the description of a person who was allegedly selling heroin from a certain motel. The agent verified with the motel clerk that a man who matched the informant's description, appellant Ali Taheri, had been staying at the motel but had hurriedly checked out. A few days later, the motel clerk advised the agent that Taheri had returned to the motel and had stated that a package was to arrive for him in the near future. That afternoon the clerk called again and reported that a package had arrived.
3
An agent went to the motel and examined the package, which had suffered during handling in the mail and was held together by a rubberband. While the agent was handling the box, the top opened and the agent observed numerous folded paper bindles inside. He then removed the rubberband and took out one of the bindles, causing some brown powder to fall out. The agent took the powder to the DEA office for testing, and it was determined to be heroin. The government does not seriously dispute the illegality of that search of the package and seizure of the sample.
4
At that point, the agent reported the situation to an assistant U.S. Attorney. On the advice of the assistant U.S. Attorney, agents took a Customs detector dog to the motel. The dog "alerted" on the package, indicating the presence of drugs. The fact of the dog's alert, without mention of the earlier search of the package, was provided to a U.S. magistrate, who issued a search warrant. The agents then, armed with the warrant, returned to the motel, seized 58 of the bindles and substituted bindles of powder for the heroin. One heroin bindle was also left in the package.
5
The following evening, after Taheri returned to the motel and claimed the package, he was arrested with the contents of the package in his possession. After his arrest, he was advised of his rights and gave written consent to searches of his vehicle and his room in another hotel. Opium was found in both places.
ADMISSIBILITY OF THE HEROIN
6
The first issue to be decided on appeal is whether the heroin should have been suppressed because it was discovered by means of an unlawful search of the package followed by the equally unlawful seizure and testing of one of the bindles. The government argues that the subsequent "alert" by the dog, which formed the basis for the search warrant and which preceded the seizure of the remaining bindles, was an "independent source" which purged the taint of the initial unlawful activity. Wong Sun v. United States, 371 U.S. 471, 83 S.Ct. 407, 9 L.Ed.2d 441 (1963). Thus the government's position is that the taint of an unlawful discovery of evidence is purged by a "rediscovery" by means acceptable for obtaining a warrant.2
7
The test as laid down by this Court, however, is whether "anything seized illegally, or any leads gained from that illegal activity, tend significantly to direct the investigation toward the specific evidence sought to be suppressed." United States v. Cales, 493 F.2d 1215, 1216 (9th Cir. 1974). In this case, the illegally gained knowledge that the substance in the package was heroin formed the impetus for the use of the detector dog. The dog's alert therefore cannot be considered an independent source which removed the taint of the original illegal search. The initial search was part of the same criminal investigation leading to this prosecution and revealed the same evidence which was the subject of the motion to suppress. See the discussion in United States v. Bacall, 443 F.2d 1050 (9th Cir.), cert. denied, 404 U.S. 1004, 92 S.Ct. 565, 30 L.Ed.2d 557 (1971), in which a later investigation uncovered separate evidence of a different crime.
8
The government's position cannot be reconciled with the policy behind the exclusionary rule: the effective deterrence of unlawful searches and seizures. Tehan v. United States ex rel. Shott, 382 U.S. 406, 413, 86 S.Ct. 459, 463, 15 L.Ed.2d 453 (1966); Linkletter v. Walker, 381 U.S. 618, 636, 85 S.Ct. 1731, 1741, 14 L.Ed.2d 601 (1965). The subsequent use of the dog and the securing of the warrant amounted to no more than a post hoc justification for using information that had already been illegally obtained. To permit evidence to be admitted under these circumstances would encourage police officers to ignore the dictates of the fourth amendment in conducting initial investigations. As this Court has recently stated:
9
Mechanical application of the traditional Wong Sun "independent source" analysis where a search warrant is subsequently commissioned albeit supported by an affidavit that relies upon independent evidence, would allow police officers to treat the warrant requirement as merely an ex post facto formality. See United States v. Griffin, 502 F.2d 959 (6th Cir. 1974), cert. denied, 419 U.S. 1050, 95 S.Ct. 626, 42 L.Ed.2d 645 (1974) (per curiam)
10
United States v. Allard, 634 F.2d 1182, 1185 n.3 (9th Cir. 1980).
11
This is not a case challenging the sufficiency of a warrant on the ground that the supporting affidavit, in addition to containing legal evidence sufficient to establish probable cause, also referred to illegal evidence. See, e. g., United States v. DiMuro, 540 F.2d 503, 515 (1st Cir. 1976), cert. denied sub nom. Hurley v. United States, 429 U.S. 1038, 97 S.Ct. 733, 50 L.Ed.2d 749 (1977); Howell v. Cupp, 427 F.2d 36, 38 (9th Cir. 1970). Nor is this a case in which the police had legally obtained probable cause before engaging in an illegal, confirmatory search. Krauss v. Superior Court of San Joaquin County, 5 Cal.3d 418, 422-23, 487 P.2d 1023, 1027, 96 Cal.Rptr. 455, 458 (1971).3 In this case the agents had no probable cause before the illegal search, and the efforts to obtain a warrant were merely attempts to validate information unlawfully gained. The evidence of the heroin was not admissible.
SEIZURE OF THE OPIUM
12
The second issue is whether the appellant's post-arrest consent to search is sufficient to avoid exclusion of the opium. The only basis for the appellant's arrest was the illegally seized evidence. As a result, the arrest is also illegal since it was not supported by any independent factual basis providing probable cause at the time it was made. See United States v. Marchand, 564 F.2d 983, 991 (2d Cir. 1977), cert. denied, 434 U.S. 1015, 98 S.Ct. 732, 54 L.Ed.2d 760 (1978); W. LaFave, 3 Search and Seizure § 11.4(e) (1978). Shortly after the appellant was illegally arrested, confronted with the illegally seized evidence, and advised of his Miranda rights, he executed a written consent to search both his car and hotel room where the opium was discovered.
13
The government seeks to justify admission of the opium evidence because of the appellant's consent. Even assuming that the consent was voluntary, however, see Dunaway v. New York, 442 U.S. 200, 216, 99 S.Ct. 2248, 2259, 60 L.Ed.2d 824 (1979), the evidence found as a result of that consent must nonetheless be suppressed if the unconstitutional conduct was not sufficiently attenuated from the subsequent seizure to avoid exclusion of the evidence. Brown v. Illinois, 422 U.S. 590, 602, 95 S.Ct. 2254, 2261, 45 L.Ed.2d 416 (1975); Dunaway v. New York, 442 U.S. at 216, 99 S.Ct. at 2258; United States v. Perez-Esparza, 609 F.2d 1284, 1288 (9th Cir. 1979). See also United States v. Perez-Castro, 606 F.2d 251 (9th Cir. 1979) (inculpatory statements on morning following illegal arrest not sufficiently attenuated). In this case there was no sufficient attenuation. The government, which bears the burden of showing admissibility in these circumstances, United States v. Perez-Esparza, 609 F.2d at 1289, points to no intervening events or lapse of time which would show Taheri's consent was "sufficiently an act of free will to purge the primary taint of the unlawful invasion." Id., quoting Wong Sun v. United States, 371 U.S. 471, 486, 83 S.Ct. 407, 416-17, 9 L.Ed.2d 441 (1963). See also United States v. Jones, 608 F.2d 386, 392 (9th Cir. 1979). The opium was also inadmissible.
14
The convictions are reversed and the mandate shall issue forthwith.
1
The defendant was found guilty on one count of conspiracy to possess a controlled substance in violation of 21 U.S.C. § 846 and two counts of possession of a controlled substance with intent to distribute in violation of 21 U.S.C. § 841(a)(1)
2
We assume without deciding that use of the dog under these circumstances did not require a warrant. See United States v. Solis, 536 F.2d 880 (9th Cir. 1976)
3
The California Supreme Court has recently re-evaluated the problems inherent in the Krauss position that there is no exploitation of a police illegality within the meaning of Wong Sun when a police officer possesses lawfully acquired information sufficient to support the issuance of a search warrant, but conducts an illegal search before actually seeking the warrant. In People v. Cook, 22 Cal.3d 67, 98-99, 583 P.2d 130, 148-49, 148 Cal.Rptr. 605, 623-24 (1978), the court, noting that the ensuing illegal search would increase the risk of invading innocent privacy and was therefore constitutionally unreasonable, stated:
After Krauss, a police officer need not rely solely on lawfully obtained probable cause; he can instead achieve "certain cause" by conducting an unlawful confirmatory search, thus saving himself the time and trouble of obtaining and executing a warrant if he does not find the evidence. He can safely engage in this conduct because Krauss teaches him that if the evidence does turn up in the course of the illegal search, he will still be allowed to seize it later in a second "search" under color of a warrant. The latter prospect thus gives him strong incentive to proceed with the warrantless entry. Yet every time he fails to find the suspected evidence, he has also invaded the privacy of a citizen innocent of any wrongdoing. The second "search" is therefore constitutionally unreasonable because it significantly contributes to increasing the risk of such invasions of privacy.
Krauss was therefore overruled insofar as it held to the contrary.
|
{
"pile_set_name": "FreeLaw"
}
|
More
Cats Prefer Interacting With People Over Food, Study Finds
CBS Local – Cats do care about their humans. Really. According to a new study, cats care about people even more than food.
The study took both domestic and sheltered cats and gave them different choices of stimuli. The cats could choose between food, toys, scent or humans. In a surprising twist to most, the cats chose that human interaction over food. Though food came in second.
“Increasingly cat cognition research is providing evidence of their complex socio-cognitive and problem solving abilities,” the study read, which was led by Kristyn R. Vitale Shreve. “Nonetheless, it is still common belief that cats are not especially sociable or trainable.”
The study defended cats, saying that the public perception is incorrect due to a ‘lack of knowledge.’
“This disconnect may be due, in part, to a lack of knowledge of what stimuli cats prefer, and thus may be most motivated to work for,” the study read. “The current study investigated domestic cat preferences at the individual and population level using a free operant preference assessment.”
This is a big development for cat lovers, who can show this to all their snobby dog loving friends who deride their cats for a so-called lack of personality.
|
{
"pile_set_name": "Pile-CC"
}
|
package zfs
import (
"bytes"
"context"
"fmt"
"io/ioutil"
"os"
"os/exec"
"time"
log "github.com/sirupsen/logrus"
)
func doSimpleZFSCommand(cmd *exec.Cmd, description string) error {
errBuffer := bytes.Buffer{}
cmd.Stderr = &errBuffer
err := cmd.Run()
if err != nil {
readBytes, readErr := ioutil.ReadAll(&errBuffer)
if readErr != nil {
return fmt.Errorf("error reading error: %v", readErr)
}
return fmt.Errorf("error running ZFS command to %s: %v / %v", description, err, string(readBytes))
}
return nil
}
// zfsCommandWithRetries runs a given command, it will retry as long as ctx is not cancelled.
func zfsCommandWithRetries(ctx context.Context, description, name string, arg ...string) error {
var err error
for {
select {
case <-ctx.Done():
return fmt.Errorf("deadline exceeded, last error: %s", err)
default:
cmd := exec.Command(name, arg...)
bts, err := cmd.CombinedOutput()
if err != nil {
log.WithFields(log.Fields{
"error": err,
"description": description,
"output": string(bts),
}).Warn("[zfsCommandWithRetries] failed to read error buffer after command failed")
time.Sleep(500 * time.Millisecond)
continue
}
// success!
return nil
}
}
}
// TODO: why not use a logger and using a file with it
// then tune the level? This is used aalllll over the codebase.
func LogZFSCommand(filesystemId, command string) {
// Disabled by default; we need to change the code and recompile to enable this.
if false {
f, err := os.OpenFile(os.Getenv("POOL_LOGFILE"), os.O_APPEND|os.O_WRONLY|os.O_CREATE, 0644)
if err != nil {
panic(err)
}
defer f.Close()
fmt.Fprintf(f, "%s # %s\n", command, filesystemId)
}
}
|
{
"pile_set_name": "Github"
}
|
Feminist media critic Anita Sarkeesian is facing a barrage of criticism since her unprovoked outburst at popular YouTuber Carl “Sargon of Akkad” Benjamin at Vidcon 2017 last week.
The feminist berated Benjamin before an audience, calling him a “garbage human” for criticizing her work on YouTube. Since then, Sarkeesian has been claiming victimhood — describing Benjamin’s presence at her panel as an act of intimidation in a blog post, and in an interview on Polygon where she called for the creation of a blacklist for those who “harass” her.
Despite Sarkeesian’s claims, game developers are now accusing her of promoting harassment. One prominent creator spoke out against her for publishing a piece of fiction about murdering him. The feminist critic often rails against video game players, claiming they direct abuse and misogynist harassment at women in the gaming industry.
With enough “fuck you” money to speak out against the feminist outrage mob on social media without fear of reprisal, billionaire Minecraft developer Markus Persson called out the social justice ideologue and her supporters.
“I thought you were ok after our brief talk at my house,” tweeted Persson. “You seemed sane and level headed. Unfortunately, you have turned more evil.”
“She’s an empty vessel, a wrench, a tool to be used. She’s a sinister greedy fuck,” he added.
Not one to mince words, Persson said that Anita’s behavior was akin to that of a “cunt.” He also took aim at gaming website Polygon for supporting her false narrative that demonized Carl Benjamin and his friends as “misogynist harassers.” Persson called the website “toxic filth” that is “actively harming any healing that could ever happen.”
Adding to the discussion was Randy Pitchford, the CEO and owner of Gearbox Software, the game studio best known for “Borderlands.” Despite being an outspoken progressive liberal who often voices his support for feminist causes, Pitchford told Persson that he was subject to Sarkeesian’s bullying.
“She posted a fiction story about murdering me,” wrote Pitchford. “My child read it – that’s how I found out about it. In person later, she did not apologize.”
Sarkeesian has not responded to these claims, and the murder fantasy remains live on the official Tumblr account for Feminist Frequency.
|
{
"pile_set_name": "OpenWebText2"
}
|
Q:
C libcurl force "Content-Type"
I have the follwing C code with libcurl to upload a file to my webserver, almost ok the only problem I need the upload to be "Content-Type: application/vnd.ms-excel" but it gets "Content-Type: application/octet-stream", tried with headers but now luck. In PHP with curl is easy, I just do curl_setopt($ch, CURLOPT_POSTFIELDS, array("importfile" => "@".$file_path . ";type=application/vnd.ms-excel" and works ok. Any help? Ideas?
#include <stdio.h>
#include <string.h>
#include <curl/curl.h>
int main(int argc, char *argv[])
{
CURL *curl;
CURLcode res;
struct curl_httppost *formpost=NULL;
struct curl_httppost *lastptr=NULL;
struct curl_slist *headerlist=NULL;
static const char buf[] = "Expect:";
curl_global_init(CURL_GLOBAL_ALL);
curl_formadd(&formpost, &lastptr, CURLFORM_COPYNAME, "importfile", CURLFORM_FILE, "document.csv", CURLFORM_END);
curl_formadd(&formpost, &lastptr, CURLFORM_COPYNAME, "action", CURLFORM_COPYCONTENTS, "upload", CURLFORM_END);
curl = curl_easy_init();
headerlist = curl_slist_append(headerlist, buf);
if(curl) {
curl_easy_setopt(curl, CURLOPT_URL, "http://www.myhost.com/upload/upload.php");
if ( (argc == 2) && (!strcmp(argv[1], "noexpectheader")) )
curl_easy_setopt(curl, CURLOPT_HTTPHEADER, headerlist);
curl_easy_setopt(curl, CURLOPT_HTTPPOST, formpost);
res = curl_easy_perform(curl);
if(res != CURLE_OK)
fprintf(stderr, "curl_easy_perform() failed: %s\n",
curl_easy_strerror(res));
curl_easy_cleanup(curl);
curl_formfree(formpost);
curl_slist_free_all (headerlist);
}
return 0;
}
Tried to add in the HEADER as "Content-Type: application/vnd.ms-excel" but is not right, I get:
Content-Type: application/vnd.ms-excel
in the headers but I need to have:
------WebbbbFormBoundaryXaI1UTNwqAyKWvLT
Content-Disposition: form-data; name="importfile"; filename="document.csv"
Content-Type: application/vnd.ms-excel
instead I get:
------WebbbbFormBoundaryXaI1UTNwqAyKWvLT
Content-Disposition: form-data; name="importfile"; filename="document.csv"
Content-Type: application/octet-stream
Thanks for help guys, answer is:
curl_formadd(&formpost, &lastptr, CURLFORM_COPYNAME, "importfile", CURLFORM_FILE, "document.csv", CURLFORM_CONTENTTYPE, "application/vnd.ms-excel", CURLFORM_END);
A:
You're looking for curl_formadd's option CURLFORM_CONTENTTYPE.
|
{
"pile_set_name": "StackExchange"
}
|
Atypical Kawasaki Disease in a 4 Years Old Child with Mumps.
Kawasaki disease is an acute febrile condition seen in children. However, it is also well recognized that some patients do not fulfill the classic diagnostic criteria for the diagnosis of Kawasaki disease. The incomplete form of Kawasaki disease is termed as 'Incomplete KD' or 'Atypical KD'. This is a case of 4 years old child with fever and mumps. He had bilateral cervical adenitis. Patient failed to respond to IV antibiotics fulfilled the criteria of incomplete Kawasaki disease. The child was managed with high dose aspirin until the child was afebrile for 48 hours. Kawasaki disease is a common vasculitis in children. Atypical cases might be missed if there is concomitant viral illness. Hence the identification and management of Kawasaki disease is paramount to decrease the mortality related to the cardiac disease. Keywords: bilateral cervical adenitis; fever and mumps; failed to respond IV antibiotics; incomplete kawasaki disease.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Petition for Writ of Mandamus Dismissed and Memorandum Opinion filed
January 29, 2008
Petition for
Writ of Mandamus Dismissed and
Memorandum Opinion filed January 29, 2008.
In The
Fourteenth Court of
Appeals
____________
NO. 14-07-01025-CV
____________
IN RE JOHN GRAY, Relator
ORIGINAL PROCEEDING
WRIT OF MANDAMUS
M E M O R A N D U M O P I N I O N
On November 30, 2007, relator John Gray filed a petition
for writ of mandamus. See Tex. Gov=t Code Ann. '22.221 (Vernon
2004); see also Tex. R. App. Proc. 52. In his petition, relator asks
this court to direct respondent David Patronella, Justice of the Peace of
Harris County, to (1) return to him certain mail (from relator to the clerk of
the 230th District Court of Harris County) that was erroneously received by a
member of respondent=s staff; or (2) forward such mail to the
clerk of the 230th District Court of Harris County at the proper address.
Texas
Government Code Section 22.221 authorizes this court to issue writs of mandamus
(1) against a judge of a district or county court in the court of appeals=s district or (2) where necessary to
enforce this court=s jurisdiction. Tex. Gov=t Code Ann. '22.221 (Vernon 2004). Relator has
not claimed or shown that the relief he requests is necessary to enforce this
court=s jurisdiction, and this court has no
independent authority to issue a writ of mandamus against a justice of the
peace. See, e.g., Easton v. Franks, 842 S.W.2d 772 (Tex. App.CHouston [1st Dist.] 1992, orig.
proceeding); Simpson v. Morgan, 779 S.W.2d 509 (Tex. App.CBeaumont 1989, orig. proceeding). This proceeding
is, therefore, dismissed for lack of jurisdiction.
PER CURIAM
Petition Dismissed and Memorandum
Opinion filed January 29, 2008.
Panel consists of Justices Yates,
Guzman, and Brown.
Do Not Publish B Tex. R. App. Proc. 47.2(b).
|
{
"pile_set_name": "FreeLaw"
}
|
load("@tf_runtime//tools:mlir_to_bef.bzl", "glob_tfrt_lit_tests")
licenses(["notice"])
# GPU tests
glob_tfrt_lit_tests(
data = [":test_utilities"],
default_tags = ["requires-gpu-nvidia"],
tags_override = {
"tf_biasadd.benchmark.mlir": ["manual"],
"tf_conv2d.benchmark.mlir": ["manual"],
"tf_fusedbatchnorm.benchmark.mlir": ["manual"],
"tf_reduction.benchmark.mlir": ["manual"],
"tf_relu.benchmark.mlir": ["manual"],
"multi_gpu.mlir": [
"manual",
"notap",
],
},
tfrt_translate = "@tf_runtime//backends/gpu:tfrt_gpu_translate",
)
# Bundle together all of the test utilities that are used by tests.
filegroup(
name = "test_utilities",
testonly = True,
srcs = [
"@llvm-project//llvm:FileCheck",
"@tf_runtime//tools:bef_executor",
"@tf_runtime//tools:bef_name",
],
)
|
{
"pile_set_name": "Github"
}
|
El Mourouj
El Mourouj (Arabic: المروج) is a town and commune in the southern suburbs of Tunis in the Ben Arous Governorate, Tunisia. It became a commune in 1991. It has 104 538 inhabitants as of 2014, making it the most populous commune in the Ben Arous Governorate.
See also
List of cities in Tunisia
References
Category:Populated places in Ben Arous Governorate
Category:Communes of Tunisia
Category:Tunisia geography articles needing translation from French Wikipedia
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Preventing infectious disease with passive immunization.
Antibodies can prevent infectious diseases by providing passive immune protection. Here we review successful clinical trials of passive immunization and consider some of the unique qualities monoclonal antibodies are now beginning to offer for developing methods for passive immunization against a wide range of infectious diseases.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Steve Munro (disambiguation)
Steve Munro may refer to:
Steve Munro, a Canadian transit advocate
Steve Munro (rugby union), see West of Scotland F.C.
|
{
"pile_set_name": "Wikipedia (en)"
}
|
351 F.3d 360
UNITED STATES of America, Appellee,v.Thomas J. BERNARD, Appellant.
No. 03-1352.
United States Court of Appeals, Eighth Circuit.
Submitted: October 21, 2003.
Filed: December 8, 2003.
Michael J. Tasset, argued, Oakland, NE, for appellant.
Steven A. Russell, argued, Asst. U.S. Atty., Lincoln, NE, for appellee.
Before RILEY, BEAM, and SMITH, Circuit Judges.
RILEY, Circuit Judge.
1
Thomas Bernard (Bernard) appeals the district court's1 dismissal of his 28 U.S.C § 2255 habeas motion challenging a restitution order in excess of $27,000,000. The district court concluded the relief sought by Bernard was beyond the scope of the statute. We agree and affirm.
2
In July 2000, Bernard pled guilty to two counts of bank fraud. The district court held an evidentiary hearing and determined the amount of loss exceeded $20,000,000. The court sentenced Bernard to 54 months imprisonment and ordered restitution in the amount of $27,534,980.03. Bernard did not file a direct appeal.
3
In December 2001, Bernard filed a 28 U.S.C. § 2255 motion challenging his restitution order on the basis the district court failed to consider evidence of Bernard's ability to pay restitution, as required by 18 U.S.C. §§ 3663(a)(1) and 3664(a). The district court granted Bernard an evidentiary hearing, and the government filed a motion to dismiss for lack of jurisdiction. Although the district court did not adopt the government's argument that the court lacked subject matter jurisdiction, the district court dismissed Bernard's habeas motion, ruling that 28 U.S.C. § 2255 "cannot be utilized by a federal prisoner who challenges only the restitution portion of his sentence." The district court concluded that, upon Bernard's release from prison, 18 U.S.C. § 3664(k) would provide an appropriate remedy.2
4
The issue of whether 28 U.S.C. § 2255 affords relief to a prisoner challenging the restitution portion of his sentence is one of first impression in this circuit. We believe the plain and unambiguous language of the statute—"[a] prisoner in custody ... claiming the right to be released"—precludes a restitution challenge. We join a majority of circuits in holding that a federal prisoner cannot challenge the restitution portion of his sentence using 28 U.S.C. § 2255, because this statute affords relief only to prisoners claiming a right to be released from custody. See Kaminski v. United States, 339 F.3d 84, 87 (2d Cir.2003); United States v. Kramer, 195 F.3d 1129, 1130 (9th Cir. 1999); United States v. Hatten, 167 F.3d 884, 887 (5th Cir.1999); Blaik v. United States, 161 F.3d 1341, 1342 (11th Cir.1998); Barnickel v. United States, 113 F.3d 704, 706 (7th Cir.1997); Smullen v. United States, 94 F.3d 20, 25 (1st Cir.1996); see also Obado v. New Jersey, 328 F.3d 716, 717-18 (3d Cir.2003); cf. United States v. Watroba, 56 F.3d 28, 29 (6th Cir.1995) (concluding habeas movant was precluded from challenging the imposition of a fine and supervised release in a 28 U.S.C. § 2255 motion); but see Weinberger v. United States, 268 F.3d 346, 351 n. 1 (6th Cir.2001) (finding an ineffective assistance of counsel claim regarding restitution is cognizable under 28 U.S.C. § 2255). Because the relief Bernard requests does not qualify as a "right to be released," as dictated by 28 U.S.C. § 2255, we affirm the dismissal of his habeas motion.
Notes:
1
Honorable Warren K. Urbom, United States District Judge for the District of Nebraska
2
Although not necessary to our holding, we believe the district court's conclusion, that 18 U.S.C. § 3664(k) is an appropriate future remedy, is correct
|
{
"pile_set_name": "FreeLaw"
}
|
Q:
To find out if a div has a scrollbar
If a div is set as overflow:auto, how can I find out (using plain JS) if a scrollbar (Vertical / Horizontal) has been rendered?
A:
Check if clientHeight is smaller than scrollHeight.
|
{
"pile_set_name": "StackExchange"
}
|
Millard Marine
Event Listing
We know you love the water
...so we really think you should get in and see the team at Millard Marine in Bunbury! Based in Bunbury, one of the most beautiful and prosperous locations in Western Australia, Millard Marine has been in operation for nearly 40 years selling new boats, used boats and helping you with all your boating requirements.
Millard’s have it all covered, selling everything from new and pre loved boats, outboard motors to finance and insurance, as well as a massive range of accessories to get the most out of your adventures on the water!
So, what are you waiting for? Come and visit the team at Millard Marine!
Comments / Ratings
Dealing with them for approx 10yrs. They are impeccable and excellent people to deal with
Robin
Sales rep Donna Brown was excellent in presentation & knew what she was talking about. I would have no problems in recommending her to family & friends. I will be definitely be back to see her in the future!!
|
{
"pile_set_name": "Pile-CC"
}
|
Post navigation
[Audio] Kate Boy’s ‘In Your Eyes’
Early last month we featured Northern Lights, the début single from Sweedish ElectroPop band Kate Boy. We still don’t know that much about them, but since then they’ve generated quite a buzz. This new trio is set to release the track early next year on IAMSOUND Records and we can now have a listen to Northern Lights’ B-side. The pounding In Your Eyes.
In Your Eyes presents a huge sound, with a sonic pallet not dissimilar to mid-80s Depeche Mode, with razor sharp digital synths and an almost Industrial feel. The rousing vocal slips between chilly ScandiPop and euphoric exuberance with ease which the rhythm heavy music is compellingly relentless whist holing on to a tundral edge. There have, of course, been many comparisons to The Knife (something seemingly unavoidable for Scandinavian ElectroPop acts), but Kate Boy display more of an IndiePop sensibility in their tunes, and we can’t wait to hear more.
|
{
"pile_set_name": "Pile-CC"
}
|
Enumamula Agriculture Market
Enumamula Agriculture Market is an agriculture market located in Enumamula, Warangal, Telangana. It is second-biggest grain market in Asia. It is spread over 117 acres.
History
The market is run by the Agriculture Market Committee of Marketing Department, Government of Telangana. The total income, as of 2017, is around 20 crores.
The Market
The market is very dynamic for selling. It has 450 commission agents (adithidar), 300 traders, 800 administrative staff, and thousands of laborers. It is divided into different yards based on the product like Mirchi Yard, Cotton Yard etc.
The market serves as a big market for red chillies for Warangal and neighboring regions of Nalgonda, Khammam, Adilabad and Karimnagar. Traders from Guntur and Maharashtra purchase chillies in the market.
e-NAM project
The market was selected one among 40 markets in Telangana by Government of India for National Agricultiral Marketing Project (NAM project) in 2016. Using of electronic weighing scales was one of the reasons for its selection. This project helps with its new electronic trading platform called e-NAM, any trader from participating markets can buy produce at the market and also reduce middlemen.
References
Category:Economy of Telangana
Category:Warangal (urban) district
Category:Agriculture in Telangana
Category:Agricultural marketing in India
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Central Florida cheap eats
CaptionAmazonas Latin Grill
Where: Where: 8273 S. John Young Parkway, Orlando. Contact: 407-903-9535 The lowdown: The menu at Amazonas Latin Grill in Orlando covers the islands and parts of South America. But make no mistake: Venezuela rules this kitchen. Lots of food for a fair price, and the staff is eager to help you explore the menu. That's my kind of foodie adventure. Try the lechon and the asado negro. The lechon was a rich Cuban-style shredded pork plate. The asado negro was a large slab of beef eye round, blanketed in a delicious thick brown gravy. Most entrees come with two sides, but you can pile them on at $1.29 a serving.
Where: Where: 8273 S. John Young Parkway, Orlando. Contact: 407-903-9535 The lowdown: The menu at Amazonas Latin Grill in Orlando covers the islands and parts of South America. But make no mistake: Venezuela rules this kitchen. Lots of food for a fair price, and the staff is eager to help you explore the menu. That's my kind of foodie adventure. Try the lechon and the asado negro. The lechon was a rich Cuban-style shredded pork plate. The asado negro was a large slab of beef eye round, blanketed in a delicious thick brown gravy. Most entrees come with two sides, but you can pile them on at $1.29 a serving.
Dining out on a budget can be a challenge, but there are many opportunities throughout Central Florida. Wing houses and chain sub shops and fast foods are a given, but we've assembled a few eateries that you may have overlooked. Ready to take a budget-conscious culinary journey? Simply turn the page.
|
{
"pile_set_name": "Pile-CC"
}
|
<?php
/*
* Copyright 2014 Google Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License"); you may not
* use this file except in compliance with the License. You may obtain a copy of
* the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
* License for the specific language governing permissions and limitations under
* the License.
*/
class Google_Service_Appengine_OperationMetadataV1Beta extends Google_Collection
{
protected $collection_key = 'warning';
protected $createVersionMetadataType = 'Google_Service_Appengine_CreateVersionMetadataV1Beta';
protected $createVersionMetadataDataType = '';
public $endTime;
public $ephemeralMessage;
public $insertTime;
public $method;
public $target;
public $user;
public $warning;
/**
* @param Google_Service_Appengine_CreateVersionMetadataV1Beta
*/
public function setCreateVersionMetadata(Google_Service_Appengine_CreateVersionMetadataV1Beta $createVersionMetadata)
{
$this->createVersionMetadata = $createVersionMetadata;
}
/**
* @return Google_Service_Appengine_CreateVersionMetadataV1Beta
*/
public function getCreateVersionMetadata()
{
return $this->createVersionMetadata;
}
public function setEndTime($endTime)
{
$this->endTime = $endTime;
}
public function getEndTime()
{
return $this->endTime;
}
public function setEphemeralMessage($ephemeralMessage)
{
$this->ephemeralMessage = $ephemeralMessage;
}
public function getEphemeralMessage()
{
return $this->ephemeralMessage;
}
public function setInsertTime($insertTime)
{
$this->insertTime = $insertTime;
}
public function getInsertTime()
{
return $this->insertTime;
}
public function setMethod($method)
{
$this->method = $method;
}
public function getMethod()
{
return $this->method;
}
public function setTarget($target)
{
$this->target = $target;
}
public function getTarget()
{
return $this->target;
}
public function setUser($user)
{
$this->user = $user;
}
public function getUser()
{
return $this->user;
}
public function setWarning($warning)
{
$this->warning = $warning;
}
public function getWarning()
{
return $this->warning;
}
}
|
{
"pile_set_name": "Github"
}
|
Q:
Initialize 64 by 64 numpy of (0,0) tuples in python?
Is it possible to create a numpy of any arbitrary data structure, for example tuples? If yes, how do I initialize it without writing it out? (Obviously, I don't want to write out 64 by 64 array)
A:
Another way:
value = np.empty((), dtype=object)
value[()] = (0, 0)
a = np.full((64, 64), value, dtype=object)
Some trickery is required here to ensure that numpy does not try to iterate the tuple, hence the initial wrapping in an object array
|
{
"pile_set_name": "StackExchange"
}
|
Q:
Configuration и xml
Помогите, не могу разобраться. Я начал изучать Spring (IoC Concept), разобрал как можно конфигурировать бины через xml и отдельно через аннотации, но не пойму возможно ли создать бин в java-классе с аннотацией @Configuration и заполнить его поля, например, через сеттеры() в xml конфигурации?
A:
Пример конфигурирования самого бина непосредственно при его инициализации в контексте, и да - нет, Вы не сможете через сеттеры с помощью хмл конфигурировать бин
@Bean
@ServiceActivator(inputChannel = "tcpOutChannel")
public TcpSendingMessageHandler tcpOut(AbstractServerConnectionFactory connectionFactory) {
TcpSendingMessageHandler gateway = new TcpSendingMessageHandler();
gateway.setConnectionFactory(connectionFactory);
return gateway;
}
|
{
"pile_set_name": "StackExchange"
}
|
In several thought experiments using the File API I've wanted to create a
Blob for data that I haven't materialized. It seems that a way to create a
blob backed by an arbitrary data source would be useful. In particular, I
would like to see a blob constructor that takes a URL and size as well as
one that takes a callback.
A URL constructed blob could use a byte range request when a FileReader
requests a slice of the blob. i.e the internal implementation could be
reasonably efficient.
A callback backed blob would be a bit more complicated. Maybe something
like the interface below, though you'd probably need another level of
indirection in order to deal with concurrency.
interface BlobDataProvider : EventTarget {
void getDataSlice(long long start, long long end);
void abort();
readonly attribute any result;
readonly attribute unsigned short readyState;
attribute [TreatNonCallableAsNull] Function? onloadstart;
attribute [TreatNonCallableAsNull] Function? onprogress;
attribute [TreatNonCallableAsNull] Function? onload;
attribute [TreatNonCallableAsNull] Function? onabort;
attribute [TreatNonCallableAsNull] Function? onerror;
attribute [TreatNonCallableAsNull] Function? onloadend;
}
Alternatively, exposing the (internal) interface FileReader uses to
interact with a Blob would remove the magic from a Blob and let a user
synthesize their own blob.
--
Steve
|
{
"pile_set_name": "Pile-CC"
}
|
"(electrical buzzing)" "(buzzing)" "(woman gasping)" "(gasping)" "(machine starts)" "Hello?" "." "Anyone here?" "." "Hello." "(roaring static)" "(electronic hissing)" "(screams)" "(electrical buzz)" "Numbers... where's the goddamn numbers?" "." "There's something..." "Oh shit." "I mean, they're my numbers..." "Damn!" "Don't I at least get a shot with my numbers, you stupid fucks?" ".!" "I want a chance!" "I want a chance like everyone else." "(woman screaming)" "Oh my God." "Oh, you bastards." "Our Father Which art in heaven, hallowed be Thy name." "Thy kingdom come, Thy will be done, on earth as it is in heaven." "Give us this day our daily bread, and forgive us our debts as we forgive our debtors." "Lead us not into temptation" "Lead us not into temptation, but deliver us from evil." "For Thine... is the kingdom... and the power... and the glory, forever and ever." "Amen." "(electronic hissing)" "Hello?" "." "Are you all right?" "." "Can you hear me?" "." "Are you hurt?" "." "(man growling) (woman screaming)" "Don't move." "Don't make a sound." "I said, "Don't move."" "Are you alone?" "." "Blink once for yes, twice for no." "Is there anyone else in the other room?" "." "All right." "All right." "I'm going to release you." "But I can take you down again, as quick as the first time." "So don't scream, understand?" "." "Once for yes, twice for no." "Are you alone?" "." "All right." "All right." "What the hell do you think you're doing?" "." "How many rooms have you been in?" "." "What is this place?" "." "Man:" "I don't know." "You tell me, how many rooms you've been in?" "." "I don't know, five or six." "What about you?" "." "Uh..." "this is my third." "You sure you didn't see anyone else?" "." "No." "Have you?" "." "No." "So who are you?" "." "Who are you?" "." "(loud static) What the fuck are those noises?" "." "Relax." "This is just for show." "Pretend that you're scared." "Should be easy." "All right." "Get down here now, or she's dead." "Some hero." "I have that effect on men." "Glad to see you haven't lost your sense of humor." "(electronic hissing)" " Is this really necessary?" "." " Just hang on." "Woman:" "Isn't that the same" "Man:" "Yes." "(electronic hissing)" "Man:" "Hey!" "Come back!" "(electronic hissing)" "Wait." "Man:" "Jesus Christ." "Hey!" "Wait!" "Where the fuck do you think you're going?" "." "This guy sure moves quickly." "He's just gone." "Maybe something made him go, maybe you should come back." "This place is awful." "Girl:" "Please don't hurt me." "Woman:" "Don't be frightened." "Don't worry." "There's nobody else here." "Just me." "I'm Kate." "What's your name?" "." "Sasha." "That's a beautiful name." "Don't worry." "Everything's going to be all right, okay?" "." "Do you understand me?" "." "I'm just blind, not retarded." "(electronic hissing)" "(struggling)" "Finally." "I was beginning to wonder if I was the only one in here." "I kept hoping I would find some other people." "I've been wondering around these rooms for hours." "Stay away from me!" "No one's going to hurt you." "Well, certainly not." "No." "No." "I come in peace, young lady." "I'm Jerry." "This is Sasha." "She's blind." "And she's very scared." "What a bummer, to be blind in this place." "Do you know what this place is?" "." "No." "No." "No." "I don't suppose, either of you two could" "let me know what we're doing in here?" "." "No." "Oh..." "Jerry Whitehall." "Kate Filmore." "Kate:" "What are you doing?" "." "I'm marking the rooms." "This is the fourth room you've been in?" "." "Yeah." "I thought you said you've been wandering these rooms for hours." "Yeah." "That's the weird thing isn't it?" "." "Each one of these rooms, has six of these doors, or portals." "But no matter how many portals I go through," "I always wound up in the same three rooms." "Until now." "It's as if the rooms were moving around or something." "Yeah." "But I don't feel any motion, do you?" "." "No." "It's just so strange." "I ran into this guy-- (rumbling)" "Oh no." "It's getting closer!" " What is?" "." " I don't know." "Something's coming after us, and I don't think it likes us." "I don't hear any people." "I think it's just motors." "He wants us dead." "We have to get out of here right now." "Any suggestion on which way to go next?" "." "Your guess is as good as mine." "I don't know." "Please, we have to move!" "Uh... here." "Okay." "Sasha, we're going to have to climb a bit of a ladder." "Hold on to me." " Move quickly." " Get me out of here!" "Man:" "Help!" " Help me!" " Jerry:" "Oh God!" "There's a guy hanging." "I can't hold him much longer!" " Jerry:" "Okay." "I'm coming." " Kate:" "Oh my God." "Sasha:" "What's happening?" "." "Man:" "Hurry!" "Jerry:" "Hang on." "Hang on." "Hang on." "Man:" "Grab his..." "Jerry:" "I'm coming." "I'm coming." "Here..." "Good girl." "Good girl." "Sasha, you're going to be okay." "Just stay here." "You're going to be okay." "(electronic hissing)" "There you are." "You've found some friends, huh?" "." "Help me get him down." "Fantastic." " Man:" "Give me some space." " Kate:" "I'm here." " Man:" "I can grab his legs." " Jerry:" "I'm loosening the belt." "Man:" "All right." "All right." "I got him." "I'm losing it." "Ready, you got it?" "." " Man:" "Yeah." " Hurry." "There it goes." "I know you want a cigarette, but not yet." " Jerry:" "Slowly." "Easy." " Kate:" "I've got his legs." "Kate:" "Lean him up against this wall." "Okay." "Watch his head." "That's it." "Just make sure he's upright." "He's alive." "He's breathing." "Man:" "Good." "He's military." "Colonel Thomas H. Maguire, Department of Defense," "Jesus Christ." "He's Pentagon, hi-tech liaison." "You... how the hell would you know?" "." "A buddy of mine's doing 10- 20 upstate for cracking the mainframe on the Pentagon two years ago." "I helped him write the code." "Maybe that's why I'm in here." "You know for sure we're in prison?" "." " I'm assuming" " You know what they say." "(laughing)" "Thanks." "That's helpful." "Jerry Whitehall." "Thanks." "Simon Grady." "That's Kate." "Sasha." "I'm sorry." "I didn't get your name." "Max Reisler." " Thanks." " Kate:" "Sasha..." "Sasha..." "Jerry, could you..." "Jerry:" "Sure, of course." "Simon:" "Come on, sit him up." "Jerry:" "Let's get you sitting up straight here." "Sasha, you okay?" "." "This man's been beaten badly." "He's been tortured." "Look at his hand." " Cigarette burns." " Jesus." "Sasha:" "They're coming for us next." "Don't you worry." "I won't let anyone hurt you." "I hope it's that easy, Kate." "There's nothing in here." "What do you think?" "." "I figure he knows something he didn't want anyone to know." "That's usually why people are tortured." "Max:" "How do you know, Mr. Cigarette Burns?" "." "What are you, an expert?" "." "As a matter of fact, you weedy little shit" "(electronic hissing)" "Sasha:" "They're coming!" "They're coming!" "They're coming." "They're coming." "(Simon yelps) Oh, hello." "Do you know where the showers are?" "." "I seem to be lost, again." "I always get lost in this gym." "I don't know why my daughter-in-law insists on bringing me here." "It's all the way across town." "Jerry and Simon:" "Whoa." "Simon:" "Watch your step." "Watch your step." "Old lady:" "Thank you, dear." "Oh my goodness." "I'm afraid I can't help you with the showers." "I'm Jerry." "You're not little Jerry Reisback, are you?" "." "No." "Thank goodness." "He's my paperboy, and I owe him $10." "Right." "Well, this is Simon." "Kate and Sasha." "Hello." "And uh..." "Max." "Max." " Old Lady:" "Max." " Jerry:" "And you are?" "." "I'm Mrs. Paley." "How do you do Mrs. Paley?" "." "You don't happen to know why you're here, do you?" "." "Oh dear." "I was never very good at philosophy." "(Sasha giggling)" "Do any of us know what's going on here?" "." "Sorry." "I've been trying to get a handle on the configuration of these rooms." "Jerry:" "All I can say is" "Simon:" "They just don't make any sense." "Jerry:" "That's right." "They sure don't." "Max:" "It's as if the rooms are moving around very quickly." "Jerry:" "There's got to be some kind of logic to it." "These rooms just seem to repeat." "You go in one direction, the room just loops back on itself." "(electronic hissing)" "It's getting closer." "Sasha, do you know what it is?" "." "Not really." "But I can hear it." "All the time, even when you don't." "And it sounds it feels wrong." "Mrs. Paley:" "Maybe we're in hell." "All right, let's get some real answers." "His answer was suicide." "Not exactly a comforting thought." "Soldier, up on your feet." "Careful, he's hurt." "Come on, pumpkin, up you come." "Stop it." "He's in serious condition." "Stop it." "Listen to me, sweetheart" "I'm not your sweetheart, asshole." "All right." "This man's been tortured." "You want to be next when they come for us?" "." "Who exactly are they?" "." "Well, I'm going to find out." "Come on." "Stop it." "Stop it." "(moaning) Kate:" "Are you trying to kill us?" "." "It's not going to help us if he dies, you know?" "." "Jerry:" "You've got no information whatsoever." " It's here." " Jerry:" "Jesus... what the fuck?" ".!" "Simon:" "Come on." "He's conscious." "Kate:" "Sit him up." "Sit him up." "Simon:" "Come on, soldier." "Talk to me, soldier." "Talk to me." "What's going on here?" "." "Colonel:" "Are you speaking to me?" "." "Yes." "Jesus Christ." "Yes." "Just tell me, how do we get out of here?" "." "I don't know." "You figure out the code, you get out." "The first one had rules..." "The first one?" "." "What are you talking about?" "." "Leave him alone." "He's delirious." "He knows something, all right?" "." "I'll carry him myself, just" "Colonel:" "The first one had numbers..." "Simon:" "What do you mean, "the first one"?" "." " Kate:" "Let him go." " Max:" "I'm out of here." "(chattering)" "(Mrs. Paley whimpers)" "Kate:" "Oh my God." "Are you okay?" "." "Did you hit your head?" "." "Yeah." "I slipped." "It's a wall." "It wiggled." " Kate:" "You sure you're okay" " Max:" "Look." "There it goes again." "Jerry:" "That--that--that" "That's-- that's-- I don't know." "No!" "Max:" "Holy fuck." "What is that?" "." "Simon:" "I don't know." "But I don't think we should stick around to find out." " Come on, let's go." " Jerry:" "Come on." "Kate:" "Come on, Sasha, I'm here." "I'm right behind you, honey." "Jerry:" "Hurry, let's go!" "Easy, easy..." "Simon:" "For Christ's sake." "Kate:" "What the" "Jerry:" "Come on." "There you go." " Kate:" "Grab right there." " Sasha:" "Where are you?" "." "Simon:" "Come on, Colonel." "It's time for us to go." "I'm not done with you yet." "Colonel:" "I'm not going." "What did you do that for?" "." "I'm not going anywhere soon." "Well, fuck you." "Come on, Kate." "Let's get out of here." "Come on, where's the key?" "." "Do you have the key?" "." "Colonel:" "You mean this?" "." "Kate:" "Yeah." "No." "What did you do that for?" "." "You don't honestly think you can escape?" "." "We have to try." "Oh my God." "This is gonna hurt." "(Simon yelling)" "Kate, get the hell up here!" "No." "I'm not leaving without him." "For God's sake, he's a dead man." "Move." "Colonel:" "God." "Simon:" "There's no time left." "Let's go." "(Kate yelling)" "(Colonel yelling)" "Oh my God." "There's got to be a way." "Get the-- get the hell out of there." " Oh my God." " Simon:" "Leave him, he's dead." "(Colonel screaming)" "Oh my God." "(Colonel screaming continues)" "Kate:" "Let me go." "(Colonel screaming)" "Kate:" "Let me go." "It's still coming." "Come on." "So what the hell happened back there?" "." "I don't know." "Mrs. Paley:" "Hello." "And where did you come from?" "." "What's your name?" "." "I'm" " I'm Kate." "Don't you remember me?" "." "I'm Mrs. Paley." "The Colonel was our only link to this place." "Maybe not." "I designed the door panels in here." "The touch sensors." "What?" "." "I was freelancing for a subcontractor, I..." "You didn't think this was worth mentioning before?" "." "I signed a confidentiality" "Given our current situation, I'd say it's null and void." " Jesus Christ!" " The legal" "What the hell is this place?" "." "I don't know." "You don't think the guy that makes the toilets in the space shuttle, gets to see the plans for the rest of it?" "." "You must have had some idea what they were building." "It was experimental." "It was a prototype." "For what?" "." " Jerry:" "I'm not sure." " Simon:" "For what?" ".!" "Jerry:" "Leading-edge stuff." "There were rumors..." "What kind of rumors, Jerry?" "." "What rumors?" "." "Quantum teleportation." "Pardon?" "." "They were just rumors." "You mean, like..." ""Beam me up, Scotty"?" "." "Kate:" "Okay." "Now this is getting ridiculous." "What do you know, Jerry?" "." "Nothing." " Simon:" "You're lying." " Jerry:" "Why would I lie?" "." "Because both you and the Colonel work for them." "Them?" "." "Who the hell are "them" anyway?" "." "What?" "." "The government?" "." "The mafia?" "." "Aliens?" "." "Mrs. Paley:" "Oh my goodness." "It's a tesseract." "Jerry:" "Christ, she's losing it." "Isn't it beautiful?" "." "Kate:" "Isn't what beautiful, Mrs. Paley?" "." "Holy shit." "If you look at it from just the right angle..." "What did you call it again, Mrs. Paley?" "." "It's a tesseract, sweetheart." "Tesseract?" "." "Tesseract." "It's-- It's a tesseract." "How do you know it's a tesser... act, or whatever you call it?" "." "Is the second act beginning already?" "." "We should be getting back to our seats." "Excuse me." "Kate:" "Yeah." "Is she for real?" "." "(Jerry laughs)" "A tesseract, Holy Christ, of course." "A tesseract." "(laughing)" "Jerry, you okay?" "." "Yeah, yeah." "I can't believe I didn't see it before." "It's been staring at us in the face the whole time." "(laughing)" "Maybe you could share with the rest of the class." "Oh." "Of course." "A tesseract... it's another name for a hypercube." "A what?" "." "A hypercube-- a four-dimensional cube." "Four dimensions?" "." "All the elements are there." "Rooms repeating." "Rooms folding in on themselves." "Teleportation." "It could all very well add up." "Look." "Here." "See?" "." "Let's call one dimension, length, and represent that with a simple line." "Now, two dimensions are length and width, which can be represented by a single square." "Now if we extend that square, one more dimension... we get..." "a cube, which has three dimensions." "Length, width, and depth." "Come on, we've all passed the eighth grade." "Hush." "Jerry:" "Here's the really funky part." "(nervous laughter)" "If you take this cube, and extend it one more dimension, we get..." "Kate:" "A tesseract." "Max:" "I thought time was considered to be the fourth dimension." "Jerry:" "Sure." "That's one idea." "But what if you have a fourth spatial dimension?" "." " Max:" "There's no such thing." " Simon:" "Why don't you shut up?" "." "Okay." "Okay." "Let's just say that we are in this hyper" " Jerry:" "Cube." " Kate:" "Cube." "Whatever." "Does this diagram show us how to get out?" "." "Well, uh... no." "A hypercube isn't supposed to be real." "It's just a theoretical construct." "Oh." "Well that makes me feel better." "Is there a theory on how we might get out of this... theoretical construct?" "." "I don't know." "Mrs. Paley:" "Don't worry, dear." "It's just a matter of time." "That's so much clearer." "Thank you." "Max:" "Great." "Anyway... here's my theory for what it's worth:" "we are all... unwilling participants..." "in a game show, and are being videotaped as we speak." "Although I don't condone forcible abduction of innocent people, we can take comfort in the fact that once the show airs, we'll be able to parlay this into lucrative sponsorship deals." "Kate:" "Okay, Max." "And what's the object of this game show?" "." "Hmm..." "I don't know." "To get out alive?" "." "Simon:" "I think we should make a map." "I don't think a map is really going to help us." "Look at that." "That wasn't there a minute ago." "Did you write that?" "." "No." "I left my wings at home." "I just marked this room as 10 before it." "Somebody's been here before us." "Thanks." "A really large someone." "Why go through the trouble of writing it on the ceiling?" "." "60,659 rooms?" "." "Christ." "This place must be huge." "Mrs Paley:" "Oh yes, yes." "In a hypercube, there could be 60 million rooms." "She could be right." "There's a comforting thought." "(electronic hissing)" "What the hell are you doing?" "." "Yowza." "The ratings on this show just doubled." "Simon:" "What do you see?" "." "Max:" "A girl." "Simon:" "Oh Jesus." "Max:" "I'm going in." " I think I'm going to puke." " Simon:" "You all right?" "." "Okay." "I just... whoa..." "All:" "Hey." "Hey." "Hey." " Simon:" "Get back here." " Kate:" "What's happening?" "." " Jerry:" "What's going on?" "." " Max:" "Help!" "Simon:" "Try to give me your hand." " Max:" "Help me!" " Simon:" "You okay?" "." "Kate:" "What the hell is going on in there?" "." "Simon:" "Max, don't be an idiot." " Jerry:" "Come on back." " Max:" "No, it's okay." " The gravity shifted." " Max:" "She needs help," " she may be hurt." " Wow." "Are we going in there or what?" "." "Maybe she knows something." "What about our..." "less agile members?" "." "I have an idea." "(electronic hissing)" "Hello." "Kate:" "Whoa." "I'm all right." "Max:" "Miss?" "." "Hey, Miss." "Hey." "Wake up." "She's on her way." "I guess it's time for plan B." "Give her mouth-to-mouth and see if it works." "Think I should?" "." "Oh, very funny." "(woman muttering)" "Hi." "What's wrong?" "." "What the hell?" "." "Where am I?" "." "That's the million dollar question." "(laughing)" "Does she have to keep cackling like that?" "." "Come on, let's get this over with." "All right." "This is going to be some kind of..." "Yeah, Mrs. Paley." "Just keep laughing there." "This is more fun than I've had since my thirteenth birthday." "Glad somebody is having fun." "This reminds me of that rope swing we used to have at home." "For God's sake, Mrs. Paley, this isn't a game!" "Come on." "Come on down." "Simon, you guys, keep her steady." "All right, all right." "Come on." "Okay." "And you're down." "That's good." "Simon:" "It's okay." "She's down." " Mrs. Paley:" "We've landed." " Kate:" "Yeah." "Simon:" "It's your turn, you'll have to climb around the sides." "Mrs. Paley:" "Well, this is a lot like the other ones, isn't it?" "." "Simon:" "Christ." "This is going to be fun." "Hello there." "I'm Mrs. Paley." "Who the hell are you people?" ".!" "Well..." "I'm Kate, and this is Mrs. Paley." "And I am Julia." "And this is..." "Hey." "Is this your jacket?" "." "No." "Well then how did I..." "I must have had more to drink last night than I thought." "There's nothing in here." "How the hell did I get here?" "." "Have we been kidnapped?" "." "That's a really good question." "Does anyone remember how we got here?" "." "Last thing I remember was going to sleep." "Where?" "." "In my bed." "Where do you live?" "." "Lincoln, Nebraska." "Kate, I am a healthy, married, male white engineer, who enjoys reading horror novels and eating chocolate ice cream, as well as climbing around psycho-jungle-gyms." "Okay." "I remember driving home from Maine State Hospital." "It was late, and I just wanted to get home," "I've been working long hours." "I'm a psychotherapist." "Simon." "New Haven." "I was out for a drink." "What do you do?" "." "A consultant." "Management consultant." "Max:" "Yeah, right." "And I'm Santa Claus." "Okay, Max." "How about you?" "." "I live in Palo Alto." "I design computer games." "The last thing I remember, I fell asleep on my keyboard." "Sasha." "And where do you live?" "." "New Mexico." "I was doing my homework in the kitchen, like I always do." "Nothing ever happens to me." "The last thing I remember... is taking the dog for a walk." "Julia?" "." "I was at an after party for a premiere in Santa Monica." "Are you an actress?" "." "No." "No, I'm an attorney." "Let's just say if I was kidnapped it was for the ransom." "So we've got Silicon Valley, Connecticut, Nebraska," "Hollywood, Maine." "Mrs. Paley, where exactly do you live?" "." "In that high-rise at the corner of Riverside and 94th." "Could you give me a lift home, dear?" "." "I seem to have lost my daughter-in-law." "Ignorance is bliss." "This doesn't make any sense." "Simon:" "Yeah, well..." "I think they probably flew us here." "Drugged us and flew us out here in private or military jets." "That's what I think." " Max:" "Where is here?" "." " Simon:" "I don't know." "Your guess is as good as mine." "Jerry:" "They could have built an hypercube structure anywhere." "And if this thing really does fold space, we could be literately anywhere." "What the hell" "What?" "." "Here is that number again." "6-0-6-5-9" "This can't be the same room." "Jesus!" "It's not, because now they're in a string of other numbers." "There just can't be that many rooms." "Maybe they are not room numbers." "Maybe somebody else figured something out." "We are definitely not alone." "Check this out." "What is it?" "." "What's happening?" "." "Julia, is that your watch?" "." "No." "No, it's mine." "It" "Well, this is bizarre." "Max:" "What, somebody else uses a watch just like yours?" "." "Somebody is playing games with us." "Shit-- look-- this is the watch that my wife gave me on my 40th." "See?" "." "This is the watch we've just found." "Maybe it's your game that we are playing, Jerry." "You designed the doors, your watch." "What's next?" "." " No, I" " For the love of God, Mrs. Paley" "Mrs. Paley, please don't." "Oh, my goodness, I think he's still alive." " Mrs. Paley, please." " Simon:" "Get her down." "Easy, careful." "Max:" "Is somebody in there?" "." " Simon:" "What do you see?" "." " Kate:" "Jerry?" "." "Oh, God." "There's a guy down here." "Oh!" " The gravity's shifting." " Simon:" "What's happening?" "." "You okay?" "." "Are you all right?" "." "Kate:" "Simon, what's going on?" "." "My God." "Hey, hey, fellow." "Oh, Jesus." " Be careful." " Ugh!" " Is he alive?" "." " I don't think so." " Let's roll him over." " Jerry:" "Yeah, yeah..." "Jerry:" "Oh God, is he ripe." "What-- what is that?" "." "Oh!" "It looks pretty dead to me." "Mrs. Paley:" "I hope he's still alive." "He's been here for a while, Mrs. Paley." "Mrs. Paley:" "Oh, poor man." "Who do you think wrote these numbers?" "." "He must have written them himself, It's all upside down." "It's the same writing as those numbers we keep seeing." "Oh, no!" "It's poor Dr. Rosenzweig." "You know him?" "." "He'll never get his Nobel Prize now." " Who the hell is he?" "." " Max:" "Holy Christ..." "Phil Rosenzweig." "Last year he was the leading theoretical physicist in his field." " Which was?" "." " Jerry:" "Quantum chaos." "Oh God, Phil Rosenzweig." "I read his book" "(gasps) (all scream)" "(exhales)" " Simon:" "Oh Jesus." " Kate:" "Oh my God." "Simon:" "Is he--?" "." "Kate, is he--?" "." " Yeah, he's gone." " You are positive?" "." "Uh-huh." "Sasha:" "Somebody--?" "." "I'm here, Sasha." "Got you." "Does anyone understand these numbers?" "." "What was he writing?" "." "Beats me." "May be he was trying to calculate his way out of here." "Looks like he ran out of time." "How did old Mrs. Paley know him?" "." " I don't know." " Hey, look." "More numbers." "Jerry:" "Same handwriting." "Max:" "The question is, did he solve it before he died?" "." "(slams, yells)" "God." "She's okay." "Are you okay?" "." "He must have finished it here." "Look, there it is again-- 6-0-6-5-9." "Max:" "How does that get us out of here?" "." " It's got to mean something." " Yeah, but what?" "." "I don't know, but we better remember it." "You know, I wonder if is he's got a..." "Max:" "What are you doing?" "." "That's disgusting." "Found a pen." "Hope you have better luck than he did." "Mrs. Paley:" "Oh no, this can't be happening." "Oh dear." "Oh dear, just wait." "Oh, I'm such an idiot." "Oh, no." "Wait a minute." "She's gonna drive me insane." "Mrs. Paley:" "Oh, it was right here." " What is it, Mrs. Paley?" "." " Mrs. Paley:" "Izon, where are you?" "." "I'm such an idiot." " Did she say Izon?" "." " Mrs. Paley:" "Izon, where are you?" "." "You come back here this instant." "Who are you looking for Mrs. Paley?" "." "Have you seen my dog?" "." "He's a darling little Shi-Tzu." "Darn it." "(gasps)" "Her dog's name is "Izon"?" "." "That's weird." "I thought Izon was a" "Simon and Jerry:" "A weapons manufacturer." "Yes." "(Mrs. Paley sniffles)" "And you guys would know that because..." "I read the papers." " Julia:" "Mrs. Paley..." " Mrs. Paley:" "Poor puppy..." "Mrs. Paley, are you sure you didn't leave the dog at home?" "." "No." "He was right here." "I'm such an idiot." "Simon:" "This breaks my heart." "Mrs. Paley..." "Mrs. Paley, what do you do for living?" "." "Oh, nothing." "I'm retired." "I shouldn't have let him off the leash." "Before you retired, what did you do?" "." "Nothing very exciting." "I was a theoretical mathematician." "Where did you work?" "." "Oh, my head hurts." "At Skippy Research Affiliates-- what do they called it?" "." "A think-tank in Washington state." "She worked for Skippy Research Affiliates and her dog's name was Izon?" "." " Mrs. Paley?" "." " Hmm?" "." "Is it possible that you worked for Izon Research Affiliates and your dog's name is Skippy?" "." "How did you know where I worked?" "." "Oh, God." "She worked for a weapons manufacturer?" "." "She's a good friend of our Nobel wanna-be." "I don't think we should trust Mrs. Funnyfarm." "I'm not crazy, and I'm not hard of hearing either." "I told you no one would believe me." "Simon:" "Mrs. Paley, what sort of research did you do at Izon?" "." "General, I will not be party to this insanity!" "I'm not a general." "I don't care what Alex Trusk says." "It's impossible." "And what's more, it's inhuman." " Who's Alex?" "." " Max:" "Did she say Alex Trusk?" "." "Oh Christ." "We're dead." "Jerry:" "This is really getting ridiculous." "Who's Alex Trusk?" "." "Alex Trusk?" "." "Hacker extraordinare?" "." "He's a legend." "This is exactly the kind of twisted maze he'd create." "Alex Trusk doesn't even exist." "Some things should never be created." "They exist for theoretical purposes only." "It would never last." "Can't you understand that?" "." "Okay, you're a shrink." "Can you tell me what the hell's happening?" "." "I... umm..." "I'd say that seeing the dead man triggered some sort of emotional response." "Flashbacks, post-traumatic stress" "So she is connected to this, yeah?" "." "Mrs. Paley:" "Skippy..." "Ooh, I think I hear him." "Skippy?" "." "Mrs. Paley, Mrs. Paley, don't" "Skippy?" "." "Mrs. Paley." "(electronic hissing)" "(hissing continues)" "Oh my." "Help me, please." "(gasps)" "(stabbing) Uh, uh." "No, don't trust the old cunt." "She lied about everything." "(screaming)" "(locking, yelling)" "(yelling continues)" "What a" " Jesus." "(yelling continues)" "Sasha:" "What just happened?" "." "It took my head, man." "I lost my fucking head!" "Paley opened... the portal... (all talking)" "Wait, wait, I have an idea." "I think we're all tired." "We need to calm down, everybody." "Hold it." "Wait a minute!" "(whistles)" "I have an idea." "Let me guess, you designed the floor too." "You better have a really good fucking explanation, Jerry." "I know what just happened there was a little... shocking, but it actually makes total sense if we're-- in a really multi-dimensional quantum environment." "Julia:" "English, please." "One fundamental idea of a quantum universe, is that parallel realities can actually exist simultaneously." "How do you know that, Jerry?" "." "All you designed were the door panels!" "I read it in Rosenzweig book, it was a big part of this theory." "What if whoever designed this stinking thing somehow managed to create a place where parallel realities can crossover like that?" "." "So you are saying we just saw Simon and Mrs. Paley" " in a parallel universe?" "." " Yes, yes!" "Stop it!" "A universe where things turn out a little differently." " Fuck." " Simon:" "Uh, please." "Think about it, think about it." "A few minutes before," " when you found the watch," " Yeah." "and we realized that somebody else had been marking numbers in the rooms, right?" "." "That's when I thought maybe we should leave markers behind, in case we doubled back and got side-tracked." "Things like a piece of clothing or jewelry." "I think what a pity it would be to leave my watch." "But I would do it if I had to." "You're saying that in an alternate reality we'd already gotten ourselves into such trouble that you decided that is okay to leave your watch?" "." " Yes!" " For crying out loud!" "What a load of crap!" "There's got to be a logical explanation." "What would you think that is?" "." "This is just an optical illusion or something?" "." "Yeah, sure." "Why not?" "." "Julia:" "Finally, yeah." "A sane idea." "Kate:" "Oh, that's a sane idea." "Jerry:" "All right." "If Max is right, why don't you go and open that door?" "." " Simon:" "Oh, come on." " Jerry:" "Have a look." "Max:" "Excuse me?" "." "If you're so sure it's an optical illusion, open the door and have a look." "You might want to stand to the side, in case that thing..." "Fuck off, Jerry!" "Simon:" "Okay, hold it!" "All right?" "." "Don't open it." "Say Jerry's right," "I think all of this is a hoax, okay?" "." "I think Jerry's either full of shit or part of this experiment." " Jerry:" "Simon, I never..." " Simon:" "Shut the fuck up!" "I'm agreeing with you, Max." "I think we're all pumped so full of LSD and I think we're hidden in some CIA hospital in Area 51, or whatever." "But let's just say, on the off- chance that Jerry is actually right." "Then what happens if whatever the fuck it was in there that killed the guy, killed me, what happens if that fucking thing gets in here?" ".!" "What happens then?" ".!" "You're all crazy." "(knife snicks)" "Max:" "Hey, man, what are you doing?" "." "Just don't fucking open the door." "What are you so scared of anyway?" "." "I just saw my fucking head taken off by something or other." "How's that for starters?" "." "Jerry:" "Where did you get that knife?" "." "It's mine." "I collect knives." "Max:" "He had it when I first saw him." " Kate:" "He sure did." " Jerry:" "That's convenient." "He's the only guy with a knife here, if anyone was a part of the experiment... maybe you." "I know!" "I know!" "I've got it!" "I've got it!" "(laughing)" "Why don't we all just get some ice cream and everyone will feel a lot better about everything?" "." "(evil laughing)" "You're funny, Mrs. Paley." "You really are a funny old lady." "You're also cute and senile." "You leave her alone, she doesn't understand." "All right, she's already admitted working for Izon, one of the most powerful weapons manufacturers in the world." "She recognized the figures on the wall, and she knew the dead guy." "No." "Maybe my alter ego wasn't all that wrong." "Maybe we shouldn't trust dear old Mrs. Paley." "Sasha:" "Maybe you're the one we shouldn't trust." "Simon:" "I beg your pardon?" "." "Maybe you're the one we shouldn't trust." "Oh." "I'm going to ignore that, little girl, because you're a cripple." "I think we should keep moving, huh?" "." "After you." "(electronic hissing)" "Sasha:" "Kate, I'm really thirsty." "Me too, sweetheart." "Me too." "Mrs. Paley:" "I'm exhausted." "Are we there yet?" "." "Jerry:" "Come on, Mrs. Paley." "Just hold on a little longer." "Kate:" "Who's this Alex Trusk person?" "." "Max:" "Are you kidding?" "." "He programmed the virus that crashed the Tokyo stock exchange two years ago." "He's the one that broke into the Air Traffic Control Grid, and crashed those two Air Force jets in Nevada to protest military spending." "Jerry:" "You forgot the one about Alex Trusk being the first genetically engineered superhuman, bred in a test tube, who now lives and walks amongst us." "Kate:" "That sounds a bit far-fetched." "Jerry:" "Don't worry." "Alex Trusk is a conspiracy theorist's wet dream." "He doesn't exist." "Max:" "Fine." "Just keep fooling yourself." "Simon:" "Would you people just shut up and keep moving?" "." "Julia:" "Do you really believe this stuff about parallel universes?" "." "Kate:" "If you asked me yesterday, I'd say no." "But this place changes your perception about what's possible." "Look, let's just keep this between us for now." "Yeah." "Okay." "Here's the deal." "I'm a private investigator." "I'm on a case, missing persons." "Given our current situation, this counts as irony." "You're right." "I think they stuck me in here because of her" "Becky Young." "They've emptied my pockets except for my knife and this." "Maybe it's a message or a warning." "Here's the kicker." "Guess who she worked for?" "." "Izon?" "." "Sasha:" "Kate." "Kate." "Kate, you have to wake up, there's" "Kate:" "What is it?" "." "What's wrong?" "." "There's something here." "Do you see anything?" "." "You're probably just dreaming or some" "Whoa." "There is-- there is something." "Simon:" "What the hell is that?" "." "Kate." "Kate, what do you see?" "." "It's a square." "It's just floating." "Max:" "What is it?" "." "Mrs. Paley:" "It's beautiful." "Simon:" "You recognize this too, Mrs. Paley?" "." "Mrs. Paley:" "Well, not exactly" " Sasha:" "What just happened?" "." " Now there's two." "Jerry:" "It's multiplying." " Mrs. Paley:" "Oh my God" " Julia:" "Maybe it's the way out." "Everyone:" "Oh." "Wow." "Max:" "If we're drunk, I wanna know what this stuff is." "Kate:" "Maybe that's what a four-dimensional object looks like." "Simon:" "If that's the way out, how do we use it?" "." "I hate it." "It sounds wrong." "It's stunning." "The math of it." "It's a perfect quadrangular oscillation." "It's moving." "Mrs. Paley, no!" "(screaming)" "Ahh." "Simon:" "Get the fuck out of the way!" "Jerry, you're hurt." "No." "I'm okay." "Max:" "Look out." "Stay low." "Whoa!" "Go on." "Kate:" "Come on, Mrs. Paley." "It's time to go." "Over here." "It looks safe." " Don't stare at it." "Go." " Max:" "It's coming back." "Come on, Max." "Max:" "Go, Julia." "Go." " I want to get it." " No." "We've got to go." "Oh my God." "Go." "Go up the stairs." "Kate:" "Come on, Mrs. Paley." "Keep going" "(Sasha screaming)" "Kate!" "Sasha." "(screaming)" "Jerry." " Oh Jesus!" " No." " Sasha." " No." "Just don't do it." " Just get in here." " What the fuck are you doing?" "." "Sasha:" "Kate!" "She's gone." "You want to end up like Jerry?" "." " Sasha." " Kate!" "Sasha:" "Kate, I'm here." "Kate." "You're a dead woman." "Oh... oh Jesus." "Sasha." "Kate, help." " Kate, help!" " I'm coming to get you." "Help me!" "Drop to the ground, Sasha." "Now." "Help me." "Okay, Sasha." "Now... uh..." "I can see you, Sasha." "Flatten yourself against the wall, on the ground." " Crawl towards me." " No." "I can't." "I'm scared." "You've gotta move, or I can't come and get you." "Move towards me." "No." "Stop, Sasha!" "Don't, Sasha." "Get down." "Damn it, Kate, what do you want me to do?" "." "Just keep moving until you hit that corner, and then stay put." "Kate:" "Okay?" "." "Sasha?" "." "Okay?" "." "I'm almost there." "Just crawl on." "Keep your side to the ground." "You can do it." "You've made it." "I'm right behind you." "There you are." "Keep your knees in." "Sasha:" "Kate, thank God." "There you are." "You're okay." "You're okay..." "What just happened?" "." "I don't know." "I don't know, Sasha." "Sweet Jesus." "It's shrinking so it can get us." "Please let it be quick." "Okay." "We're going to make a run for it." " I can't." " Yes you can." "Just hold onto me." "We're going to go to our right." "No." "We're going to go to the left." "All right?" "." " Okay?" "." " Okay." "On the count of three..." "just hold on to me." "One, two, three..." "I've got you." "Oh my God!" "Sasha, no." "It responds to our movements." "Don't move." "Stay still." "Don't" "Oh my God." "Is it gone?" "." "Yeah." "It's gone." "Thank you." "We did it together." "Let's get out of here before it comes back." "My glasses." "You're glasses are gone." "Just go up the ladder." " Kate." " Yeah?" "." "What?" "." " I can smell it." " What?" "." "Did someone die in here?" "." "Yeah." " Yeah." "Jerry died." " Oh no." "He was so nice." "We've got to go." "We've got to go, honey." "Just get up there." "One foot after the other." " Where are the others?" "." " I think we're on our own." "You're doing good." "I see a pattern." "Anyone connected to this-- this hypercube... eventually dies a very violent, and painful death." "(Mrs. Paley whimpering)" "I would hate for that to happen to you, Mrs. Paley." "I would." "So drop the act and tell us how we get out of here." "Huh?" "." "Julia:" "Simon." "Simon, that's enough." " Shut up." " Hey, Simon..." " Shut up." " Okay." "Okay, Mrs. Paley." "You ready to talk?" "." "'Cause I'm all ears." "I really don't need this operation, Doctor." "I'm in perfect health." "Just ask my daughter-in-law over there." "She'll explain." "This joke's really getting stale." "I'm going to give you one more chance, Mrs. Paley." "You tell me what I need to know, or I'm going to have to kill you." "Do you understand?" "." "I'm going to kill you." "It's your choice." "Julia:" "Get away from her!" "You have a choice to make." "Right now." "Do you want to get out of here or not?" "." " Yes, but" " There are no buts!" "Hey guys, maybe" "All right." "(crackling)" "Oh shit." "It's here." "It's gonna take my head off." "Let's go." "What about Mrs. Paley?" "." "Goddamn it." " Simon:" "I'm not gonna die here." " Julia:" "You can't leave her." " Simon:" "Why should I help her?" "." " Julia:" "She'll die." " Max:" "Forget about her." " I'm gonna regret this." "Max:" "Julia!" "(screaming)" " Simon:" "I'm trying to get this off." " Max:" "Come on." "Hurry." "Simon:" "Shit!" "Would someone give me a hand?" "." "Would somebody help me pick her up?" "." "Shit!" "Let go of me." "Shit!" "I'm not staying here." "I'm out of here." "Let go of me." " Simon:" "Jesus Christ." " Julia:" "Oh my God." "Oh God." "(muffled screams)" "Simon:" "Where do you think you're going?" "." "Max:" "Hurry up." "Get back here." "What the hell?" "." "6-0-6-5-9." "It could mean anything." "Kate:" "Right." "Maybe it's a coordinate of some kind but in four dimensions." "Some..." "I don't know." "Sasha:" "For all we know, it could be somebody's birthday, or a zip code, or just a serial number." "I wish I was just..." "smarter." "Why do you think you're in here?" "." "I don't know." "Maybe I was just in the wrong place at the wrong time." "Damn it." "There's got to be some sort of meaning to this." "6-0-6-5-9." "6-0-6-5-9." "Maybe they just like to watch us squirm." "Max:" "Alex Trusk is the only one with the resources and brains to create something like this." "Julia:" "Okay, Max." "Say you're right." "Say it is Alex Trusk who's responsible for all of this." "Does it help us get out?" "." "No, we're still stuck here, with no clue." "And if this is someone's game, it isn't a fair one." "Max, do you think" "We've got to keep moving." "Come on." "Max?" "." "(voice in slow motion)" "Max, what's wrong with you?" "." "(voice in fast forward)" "(voice in slow motion)" "Holy shit!" "Variable time-speed rooms." " Pretty wild." " I guess you could say that." "(voice in slow motion continues)" "(electronic hissing)" "(groaning, grunting)" "Holy jumping Jesus!" "Get your hands off me." " Hello, Jerry." " Who the hell are you?" "." " You don't recognize me?" "." " Where am I?" "." " How do you know my name?" "." " You don't know me?" "." "No." "Where the hell am I?" "." "You don't--?" "." "Wait, wait." "You don't remember me?" "." " No." " What about the senile old broad," " or the little blind kid?" "." " What?" "." "Or that thing that chopped you up like little pieces of deli fucking meat?" "." "I'm a friend-- I mean you no harm." "Let me go, and I'll be on my way." "No come on, you" "You mean me no harm?" "." "You're fucking hilarious, Jerry." "How long have you been in here?" "." "I don't know." "I just woke up." "You just woke up?" "." "Holy shit!" "Holy shit." "Maybe you were right before, huh?" "." "You or the-- whatever" "This place does crossover parallel alternate realities, huh?" "." "Have you been drinking?" "." "I really don't give a shit anymore." "We are talking about alternate realities." "Can I ask you a question?" "." "Do you mind?" "." "Yeah, okay, sure-- just don't hurt me, huh?" "." "Are you hungry?" "." "Are you... hungry?" "." "Yeah, I guess..." "my stomach's" "'Cause I am hungry." "I'm fucking starving." "(electronic hissing)" "(electronic turndown)" "(electronic hissing)" "Oh my God." "Kate, what is it?" "." "Kate?" "." "What do you see?" "." "(locking)" "Kate, what's going on?" "." "I don't know." "I mean" "I think in some other reality, things didn't turn out so well." "Again with the Alex Trusk?" "." "Why are you so afraid of this Alex Trusk?" "." "Because he's ruthless." "He's a hi-tech genius whose morals make Muammar Qaddafi" "look like Mother Teresa." "Why are you panicking?" "." "Because he's throwing anybody connected to this thing in here to fucking die!" "(echoes)" "Max, you're not connected." "Can you keep a secret?" "." "Sure." "I love secrets." "I designed a computer game called "Relativity,"" "where contestants linked to each over the Internet do battle in a 3-D environment, using different time signatures." "What are you saying?" "." "Are you saying that you designed all this?" "." "No." "Just these variable time speeds rooms." "Just the concept." "It was for a game." "Now here we are." "Wow." "This is really some game, Max." "Do you have other little secrets you want to tell me?" "." "People are dead, okay?" "." "I'm not responsible for this." "It was just a game." "I can't even sell the stupid thing." "It's all involved in a lawsuit." "What kind of lawsuit?" "." "I don't know, some" "Right, you're a lawyer." "A company called Cyber Thrill stole it from me." "Cyber Thrill?" "." "You've heard of them?" "." "Yeah." "Are you into computer games?" "." "Settle." "Excuse me?" "." "I strongly recommend that you settle." "You'll never win." " How do you know that?" "." " Because... you're not up against Cyber Thrill." "They're just a subsidiary." "Who owns them?" "." "Izon." "How do you know that?" "." "Because I... represent them." "(ticking)" "(distant woman's voice)" "(electronic hissing)" "I must be hearing things." "(distant voices)" "(distant voices continue)" "(electronic hissing)" "Jerry:" "Finally." "I was wondering if I was the only one in here." "Do you need a hand there?" "." "You are upside-down." "Do you have any idea what's going on here?" "." "Be careful." "Watch your step." " Crazy." " (locking)" "Simon:" "You are not the only person here, Jerry." "There's lots of people just like you." "Jerry:" "How do you know my name?" "." "Simon:" "We've met before, Jerry." "I get this mouth-watering feeling when I talk to you." " What do you mean?" "." " Come here, look at this." "Jerry:" "Hey, hey." "Agh!" " Agh, no!" " Take this!" "And this!" "(Jerry screaming)" "Why don't you just go?" "." "I'm blind, Kate." "I'm a burden." "You can move faster without me." "Sasha, please, I know how you're feeling." "No you don't." "You have no idea how I'm feeling." " I know this seems hopeless" " It is hopeless." "No, it's not." "I'm gonna figure this out." "(giggling)" ""Figure it out."" "Trust me, precious." "If I haven't figured it out, you sure as shit aren't going to." "What did you just say?" "." "I'm sorry, Kate." "I didn't mean to" "No, no." "Wait a second." "Back that up." "Why do you think that you should be able to figure this out?" "." "I wasn't kidnapped." "When I found out they were putting people in here," "I tried to blow the whistle on them." "So they came after me." "I escaped into the one place they wouldn't dare follow me-- in here." "Poetic justice, don't you think?" "." "Who are you?" "." "Max was right." "Jerry was wrong." "I exist." "Oh my God." "Sasha." "Of course." "Sasha is the nickname for Alexandra." "Together:" "Alex Trusk." "Pleased to make your acquaintance." "Damn it." "This is the worst nightmare I've ever had." "I just I wish I'd wake up." "You really think this is all a dream?" "." "(sighs)" "Definitely." "In the real world I'd never kiss you." "Oh, really?" "." "Yeah, you're not my type." "You're not my type, either." "(ticking)" "(electronic hissing)" "Aa-ah!" "Ya-aa!" "Becky Young." "Yeah." "Rebecca Young." "Yeah." "Who... are you?" "." "Who am I?" "." "I'm Simon Grady." "Your parents hired me to find you." " Really?" "." " Yeah." "Thank God." "Yeah." "I'm happy to see you too, Becky." "I really am." "I've been wandering these rooms for hours." "(stabbing)" "Ahhgh!" "(distant voices)" "You gotta love these parallel universes." "You built this thing?" "." "I gave them the key to build it-- better than they wanted." "I gave them a real hypercube." "How do we get out?" "." "Kate, this isn't a game." "There's no happy ending." "(sighs)" "This place is out of control." "It's not stable." "Does this number mean nothing?" "." "6-0-6-5-9." "It's got to mean something." "It's everywhere." "I'm sorry, Kate." "It's over." "No, it is not over." "Not yet." "(electronic hissing)" "I refuse to die here." "(electronic hissing)" "What the hell?" "." "(crackling)" "Oh my God." "We've got to move." "Come on." "We've got to get out of here." " Leave me." " No, you're coming with me." " (roaring)" " Oh my God." "We got to get out of here." "Just go up as fast as you can." "Faster, faster." "Keep going." "Okay." "Good girl." "What the hell?" "." "Oh my God." "Okay, we've got to get out of here." "This is insane." "It's Jerry's diagram." "It's all the numbers." "They're all just suddenly in here." "It's Jerry's markings-- that dead physicist's equations" "and that damn colonel's corpse, just hanging there as if we never even rescued him?" "." "Everything keeps appearing over and over again." "Sasha-- please, you have to have some idea what this means." "All the realities are starting to collapse into one space." "And what happens to us when they-- "it" collapses?" "." "The whole thing implodes." "It's only a matter of time." "I'm not gonna just stand here while it happens." "(electronic hissing)" "Kate:" "I'm not leaving him behind." " Oh my God." " Man:" "Kate!" " Kate:" "Let me go." " Get the hell up here!" "Kate:" "Oh my God." "Sasha:" "Kate." "We don't have much time." "(gasps)" "No." "I refuse to die here." "I wish I had your spirit." "Hello, gorgeous." "Miss me?" "." "Huh?" "." "Come on in." "The water's fine." "Agh!" "(Sasha screaming)" "Hello, Kate." "Let her go." "Who are you?" "." "Yeah, it's me-- good old Simon." "Do you remember this?" "." "Do you?" "." "I've waited a long time for pay back." "That was just..." "seconds ago." "Don't be so stupid, Kate." "Time works differently in this place." "Hey, just-- okay." "Just let her go." "She's just a kid." "She'll be a dead little kid, unless you come here to papa." "Okay." "Just let her go-- we can figure something out, okay?" "." "You take her place." "What do you want?" "." "I'm hungry." " Simon?" "." " Huh?" "." "Sasha:" "There's no point." "We're all dead anyway." "Oh, oh." "Okay." "No!" "Oh, come on Kate." "The fun's just beginning." "Oh my God." "What the hell?" "." " Yah!" " Agh!" "I'm really gonna miss you." "(panting)" "Jesus" "Of course, 6-0-6-5-9." "It's an expiration date." "He figured out when it's gonna implode." "Kate:" "Leave me." "Man:" "Hello, Kate." "Welcome back." "So, you figured it out." "Yes, sir." "No time to spare." "The device-- any luck?" "." "We'll take that to Darcy and see if anything recorded on it." "(gunshot)" "(phone ringing)" "Sir?" "." "Yes, sir." "Phase two is terminated." "I see." "Yes, sir." "Right away, sir."
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GADTs are obviously currently a hot topic in functional programming research. Most of the papers focus only on the GADT mechanism (how type checking works etc.). The only example that one usually sees is the "typed evaluator".
I am not an expert on this topic, and I'd like to know more about how they would actually be useful in practical programming.
For example, I wonder how a parser would look like if it is impossible to construct "wrong" ASTs. Would type checking then effectively take place during parsing? How would a type error in the parsed program be detected and thrown?
What other interesting applications exist? In general, how do GADTs change the programming model?
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Cathetopteron amoena
Cathetopteron amoena is a species of beetle in the family Cerambycidae, and the only species in the genus Cathetopteron. It was described by Hamilton in 1896.
References
Category:Hemilophini
Category:Beetles described in 1896
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Women and men from majority Muslim areas of Russia, including Chechnya, have traveled to Syria and Iraq to fight alongside the Islamic State there, and dozens have begun to return as the group has lost most of its territory.
The restive region of Chechnya was the site of two separatist wars in the 1990s. Following those conflicts, Mr. Kadyrov, with the backing of the Russian government, has cracked down on dissent. Human rights groups have accused his forces of abuses including torture, kidnapping and murder.
Grozny once had a substantial ethnic Russian, Christian population but most of them fled during the wars. The church that was attacked Saturday is in the center of the city and was at the heart of some of the battles of the 1990s.
Mr. Kadyrov condemned Saturday’s assault and said he immediately went to the church and was with the police as they responded. He vowed to crush anyone who attempted attacks in his region.
“I once again very seriously declare that you can try to commit any actions aimed at undermining the security of the residents of Grozny and other settlements,” he said. “But anyone who makes the first step along this path will be immediately destroyed.” He made the statement via the Telegram messaging app, where he regularly posts information after being kicked off both Instagram and Facebook after he was added to a United States sanctions list.
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Transductional efficacy and safety of an intraperitoneally delivered adenovirus encoding an anti-erbB-2 intracellular single-chain antibody for ovarian cancer gene therapy.
We have previously shown that adenoviral-mediated delivery of an anti-erbB-2 intracellular single-chain antibody (sFv) causes specific cytotoxicy in erbB-2-overexpressing ovarian carcinoma cells. Furthermore, intraperitoneal delivery of the anti-erbB-2 sFv enhances survival and reduces tumor burden in a xenograft model of human ovarian carcinoma in SCID mice. These findings have led to an RAC-approved Phase I clinical trial for patients with ovarian cancer. In this report, we show that expression of the anti-erbB-2 sFv could be readily detected in target tumor cells by in situ hybridization methodology. PCR analysis of DNA extracted from various murine tissues demonstrated that the anti-erbB-2 sFv remained localized to the peritoneum. Delivery of the sFv to the non-erbB-2-overexpressing REN mesothelial and Hep G2 hepatocellular carcinoma cell lines was not deleterious to either one, affirming the tumor specificity of this gene therapy strategy. In addition, histopathological analysis of various tissues showed that adenoviral-mediated delivery of the anti-erbB-2 sFv to immunocompetent mice with either primary exposure or previous vector challenge at different doses produced no abnormal changes when compared to untreated animals. These findings suggest that adenoviral-mediated delivery of the anti-erbB-2 sFv in a human context can be effectively assayed, is potentially free of vector-associated toxicity, and retains biologic utility based on tumor specificity.
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"pile_set_name": "PubMed Abstracts"
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Gnaphosa fallax
Gnaphosa fallax is a ground spider species found in Hungary.
See also
List of Gnaphosidae species
References
External links
Category:Gnaphosidae
Category:Ground spiders of Europe
Category:Spiders described in 1879
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"pile_set_name": "Wikipedia (en)"
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Inhibition of osteoblast function in vitro by aminobisphosphonates.
Bisphosphonates are analogues of pyrophosphate, a key physicochemical inhibitor of mineralisation. We examined the direct actions of bisphosphonates on the function of cultured osteoblasts derived from rat calvariae. Treatment with zoledronate, the most potent bisphosphonate studied, reduced osteoblast number at concentrations > or = 100 nM and was strongly toxic at 10 microM, causing a threefold decrease in osteoblast viability after 2 days and a 90% decrease in cell numbers after 14 days. In control osteoblast cultures on plastic, abundant formation of 'trabecular' mineralised bone matrix nodules began after 10 days. Continuous exposure to zoledronate inhibited bone mineralisation at concentrations as low as 10 nM. Pamidronate and clodronate exerted similar effects but at higher doses > or = 1 and > or = 10 microM, respectively). Short-term or intermittent exposure of osteoblasts to zoledronate and pamidronate (1-10 microM) was sufficient to inhibit bone mineralisation by > or = 85%. Zoledronate but not pamidronate or clodronate also strongly inhibited osteoblast alkaline phosphatase activity at concentrations > or = 100 nM and soluble collagen production at concentrations > or = 1 microM. We additionally studied the effects of zoledronate on osteoblasts cultured on dentine, a bone-like mineralised substrate, observing similar inhibitory effects, although at concentrations 10-100-fold higher; this shift presumably reflected adsorption of zoledronate to dentine mineral. Thus, zoledronate blocked bone formation in two ways: first, a relatively non-toxic, selective inhibition of mineralisation at concentrations in the low nanomolar range and second, a cytotoxic inhibition of osteoblast growth and function at concentrations > or = 1 microM. Although no data are available on the bisphosphonate concentrations that osteoblasts could be exposed to in vivo, our results are consistent with earlier observations that bisphosphonates may inhibit bone formation.
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"pile_set_name": "PubMed Abstracts"
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Friday, February 25, 2011
Mortal Kombat Banned in Australia
The upcoming Mortal Kombat game is brutal as hell. Meaning, and this shouldn't come as too great a shock to most of you, it's been "banned" in Australia.
I say "banned" because, like all other recent cases of a commercial title being barred from release Down Under, it's actually been "refused classification" (as Australia lacks an adults-only rating, and this was deemed too violent for an MA15+ badge), which isn't technically the same as a ban.
Though as the game is now unable to be sold at retail in Australia, it's pretty much the same thing. Semantics!
The bad news for Aussies is that they'll now have to import the game, which is also technically illegal. The good news is, with the Australian dollar at an all-time high, it's a good 20-30% cheaper to buy from overseas anyway.
It's expected/hoped that, at a meeting of State Attorneys-General later this year, an adults-only rating for video games in Australia will be introduced. Which will of course come too late for Mortal Kombat and its publisher, Warner Bros.Mortal Kombat Banned In Australia [Kotaku AU]
-------------------------------------------------------
Just passing this one along. Interested in the title myself, thank GOD Canada isn't stuck in the 60's like the Aussies. :S
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How diagnosis-related group 559 will change the US Medicare cost reimbursement ratio for stroke centers.
Thrombolysis for acute ischemic stroke saves societal costs, but hospitals that practice acute stroke care appear to shoulder the burden of the cost, which exceeds reimbursement. With creation of the diagnosis-related group (DRG) 559, the US Centers for Medicare and Medicaid Services pays hospitals approximately US $6000 more per case when thrombolysis is administered. We sought to determine the total cost of, and reimbursement for, acute stroke treatment with thrombolysis at a single stroke center and the economic impact of DRG 559. Between September 2001 and December 2004, we collected data on all patients with acute stroke who received thrombolysis. We identified all hospital costs and reimbursement per patient. Financial results were expressed as a cost-reimbursement ratio: average total cost to average total reimbursement per patient. We then reanalyzed data using the projected Medicare hospital reimbursement with DRG 559. Sixty-seven patients with stroke (mean age, 72 years) were treated (mean length of stay, 4.4 days; mean stroke severity, National Institutes of Health Stroke Scale score of 15; and symptomatic intracranial hemorrhage rate, 7%). The cost-reimbursement ratio was 1.41 (95% CI=0.98 to 2.28) before DRG 559 and estimated to be 0.82 (95% CI=0.66 to 0.97) after DRG 559. Our hospital costs have traditionally exceeded Medicare reimbursement for the acute care of thrombolyzed patients with ischemic stroke, but with DRG 559, a new economically favorable cost-reimbursement ratio for hospitals will be established.
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{
"pile_set_name": "PubMed Abstracts"
}
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The Messenger: A Soul Mate Story
It was a beautiful sunny day in Seattle, which is so unusual for the 4th of July; a day for fireworks and
celebration honoring our country's independence. That day, little did I know that I was soon to receive a
very special message from the most unusual messenger.
It was early in the afternoon and I was seriously busy refinishing some very old windows out on my back porch.
All of the doors and windows throughout my home where open, letting in the light and fresh air.
My life was certainly going through some changes. Only a month before, I had been struggling with a relationship
that I knew needed to change form. A few days earlier I had said my goodbyes. I had always heard that when
one door closes another door opens, but I didn't know if I was fully prepared to step into the new door that
was presenting itself so quickly. You see, I was finally realizing that the dearest male friend that I had
ever known, was the soul mate I had been looking for my entire life. What a shock to my senses this was!
I had always thought of myself as an intuitive person, but I didn't see this one coming. It was like I had
a veil pulled over my eyes.
Needless to say, I was filled with bliss from this new awareness. But also scared and a bit unsure at the
same time. They say that the chance of someone single in their mid forties finding true love, is like finding
an old vintage "oat penny" in a jar full of pocket change--highly unlikely. (What do 'they' know
:-)
I knew in my heart that my friend Kirk was a soul mate, but my head was swimming in some type of fear that
I couldn't really fully understand. So, I did what I do when I need to process so many emotions. I worked
on something tangible in the physical world that needed to be fixed.
As I was refinishing the old window frames, I began to notice layer after layer of different colored paint.
It seemed like I was going through the generations of a family tree as each layer was exposed. Each of the
colors of paint was sharing a story about different cycles in my life. As each layer exposed itself, I had
some sort of revelation. I can't really tell you every thought that went through my mind, but it seemed so
very revealing at the time.
There came a point when I was so full of thought that I knew I needed to take a break. So, I decided to walk
into my magical dining room where the beautiful stained glass windows surrounded my favorite plants. It was
at that very moment when something unusual caught my eye. I have always had an affinity for butterflies and
love to watch them outside, but this was the first time I had ever seen one indoors.
"Wow!", I thought to myself. "This is the most unusual looking butterfly." I have seen
Monarch butterflies up close, but never one like this before. So I slowly walked up to take a better look.
I was in awe. This butterfly was so very large, black, white, and orange and incredibly beautiful. He seemed
quite calm really; much calmer than I was for sure.
I wondered, "Now, what does a person do with a butterfly in their home?" I can tell you what I thought.
I was concerned that he was going to die if he stayed inside any longer. It is unnatural for a butterfly
not to be free. So, I wondered what to do.
Just then I heard thoughts in my head telling me to fill myself with love and then to place my hand beside
this radiant butterfly. So I did just that. As I was feeling this incredible sense of love oozing from my
being, I watched him so intently while pressing my hand close to him against the window pane. In my
amazement he slowly and steadily crept over to my hand and I watched him carefully step right into the center
of my palm. I was motionless and in shock. Why would a butterfly as unusually beautiful as this one come
inside my home and end up in my palm?
Just then I heard another thought fill my senses, and this time it seemed to be coming from the Butterfly!
"Trust dear one. Trust your life and trust your heart. Here you are with the door open, and the light
is shining the way. I am here to show you that you now have your independence. Do with it as you wish. In
going through the door before you, your life will have more freedom than you have ever known before."
This was so weird. I had this incredible peace fill every inch of my being. I just knew this butterfly was
sent to me as a messenger, to tell me everything was okay; that my love for my wonderful friend and soul
mate Kirk, was perfect, so not to fear.
I slowly walked the twenty-five or so steps through my dining room and out onto my front porch. The butterfly
contently lay in the palm of my hand, not about to leave until he knew I received my message. At the point
of mutual acknowledgement I raised my hands to the sky and watched as the butterfly so elegantly ascended
from my palm up into the sky.
I stood there for a while and watched him circle the yard, as if he was smiling back at me, saying his goodbyes.
I thought to myself, "What a magical messenger indeed." From that moment on, I have never turned
back.
This soul mate story was published in," Romancing the Soul, True Soul Mate Stories from around the World
and Beyond." Find over 60 stories of soul mate connections and an explaination of various types of soul mates.
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---
abstract: 'Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four dimensional Lorentzian manifold. The conditions for existence of symmetry operators for the different equations are seen to be related. Computer algebra tools have been developed and used to systematically reduce the equations to a form which allows geometrical interpretation.'
address: |
$^1$ Albert Einstein Institute, Am Mühlenberg 1, D-14476 Potsdam, Germany\
$^2$ The University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh, EH9 3JZ, UK
author:
- 'Lars Andersson$^1$, Thomas Bäckdahl$^2$, and Pieter Blue$^2$'
title: Second order symmetry operators
---
Introduction
============
The discovery by Carter [@carter:1968PhRv..174.1559C] of a fourth constant of the motion for the geodesic equations in the Kerr black hole spacetime, allowing the geodesic equations to be integrated, together with the subsequent discovery by Teukolsky, Chandrasekhar and others of the separability of the spin-$s$ equations for all half-integer spins up to $s=2$ (which corresponds to the case of linearized Einstein equations) in the Kerr geometry, provides an essential tool for the analysis of fields in the Kerr geometry. The geometric fact behind the existence of Carter’s constant is, as shown by Walker and Penrose [@PenroseWalker:1970], the existence of a Killing tensor. A Killing tensor is a symmetric tensor $K_{ab} = K_{(ab)}$, satisfying the equation $\nabla_{(a} K_{bc)} = 0$. This condition implies that the quantity $K = K_{ab} \dot \gamma^a \dot \gamma^b$ is constant along affinely parametrized geodesics. In particular, viewed as a function on phase space, $K$ Poisson commutes with the Hamiltonian generating the geodesic flow, $H = \dot \gamma^a \dot \gamma_a$.
Carter further showed that in a Ricci flat spacetime with a Killing tensor $K_{ab}$, the operator ${\mathcal K} = \nabla_a K^{ab} \nabla_b$, which may be viewed as the “quantization” of $K$, commutes with the d’Alembertian $\mathcal H = \nabla^a \nabla_a$, which in turn is the “quantization” of $H$, cf. [@carter:1977PhRvD..16.3395C]. In particular, the operator $\mathcal K$ is a symmetry operator for the wave equation $\mathcal H \phi = 0$, in the sense that it maps solutions to solutions. The properties of separability, and existence of symmetry operators, for partial differential equations are closely related [@miller:MR0460751]. In fact, specializing to the Kerr geometry, the symmetry operator found by Carter may be viewed as the spin-0 case of the symmetry operators for the higher spin fields as manifested in the Teukolsky system, see eg. [@kalnins:miller:williams:1989JMP....30.2360K; @kalnins:miller:williams:1989JMP....30.2925K].
In this paper we give necessary and sufficient conditions for the existence of second order symmetry operators, for massless test fields of spin 0, 1/2, 1, on a globally hyperbolic Lorentzian spacetime of dimension 4. (As explained in Section \[sec:indepspinors\], the global hyperbolicity condition can be relaxed.) In each case, the conditions are the existence of a conformal Killing tensor or Killing spinor, and certain auxiliary conditions relating the Weyl curvature and the Killing tensor or spinor. We are particularly interested in symmetry operators for the spin-1 or Maxwell equation. In this case, we give a single auxiliary condition, which is substantially more transparent than the collection previously given in [@KalMcLWil92a]. For the massless spin-1/2 or Dirac-Weyl equation, our result on second order symmetry operators represents a simplification of the conditions given by McLenaghan, Smith and Walker [@mclenaghan:smith:walker:2000RSPSA.456.2629G] for the existence of symmetry operators of order two. The conditions we find for spins 1/2 and 1 are closely related to the condition found recently for the spin-0 case for the conformal wave equation by Michel, Radoux and [Š]{}ilhan [@michel:radoux:silhan:2013arXiv1308.1046M], cf. Theorem \[thm:intro:spin0\] below.
A major motivation for the work in this paper is provided by the application by two of the authors [@andersson:blue:2009arXiv0908.2265A] of the Carter symmetry operator for the wave equation in the Kerr spacetime, to prove an integrated energy estimate and boundedness for solutions of the wave equation. The method used is a generalization of the vector fields method [@klainerman:MR784477] to allow not only Killing vector symmetries but symmetry operators of higher order. In order to apply such methods to fields with non-zero spin, such as the Maxwell field, it is desirable to have a clear understanding of the conditions for the existence of symmetry operators and their structure. This serves as one of the main motivations for the results presented in this paper, which give simple necessary and sufficient conditions for the existence of symmetry operators for the Maxwell equations in a 4-dimensional Lorentzian spacetime.
The energies constructed from higher order symmetry operators correspond to conserved currents which are not generated by contracting the stress energy tensor with a conformal Killing vector. Such conserved currents are known to exist eg. for the Maxwell equation, as well as fields with higher spin on Minkowski space, see [@anco:pohjanpelto:2003RSPSA.459.1215A] and references therein. In a subsequent paper [@andersson:backdahl:blue:currents] we shall present a detailed study of conserved currents up to second order for the Maxwell field.
We will assume that all objects are smooth, we work in signature $(+,-,-,-)$, and we use the 2-spinor formalism, following the conventions and notation of [@PenRin84; @PenRin86]. For a translation to the Dirac 4-spinor notation, we refer to [@PenRin84 Page 221]. Recall that $\Lambda/24$ is the scalar curvature, $\Phi_{ABA'B'}$ the Ricci spinor, and $\Psi_{ABCD}$ the Weyl spinor. Even though several results are independent of the existence of a spin structure, we will for simplicity assume that the spacetime is spin. The 2-spinor formalism allows one to efficiently decompose spinor expressions into irreducible parts. All irreducible parts of a spinor are totally symmetric spinors formed by taking traces of the spinor and symmetrizing all free indices. Making use of these facts, any spinor expression can be decomposed in terms of symmetric spinors and spin metrics. This procedure is described in detail in Section 3.3 in [@PenRin84] and in particular by Proposition 3.3.54.
This decomposition has been implemented in the package *SymManipulator* [@Bae11a] by the second author. *SymManipulator* is part of the *xAct* tensor algebra package [@xAct] for *Mathematica*. The package *SymManipulator* includes many canonicalization and simplification steps to make the resulting expressions compact enough and the calculations rapid enough so that fairly large problems can be handled. A Mathematica 9 notebook file containing the main calculations for this paper is available as supplementary data at <http://hdl.handle.net/10283/541>.
We shall in this paper consider only massless spin-$s$ test fields. For the spin-$0$ case the field equation is the conformal wave equation $$\label{eq:confwave1}
(\nabla^a \nabla_a + 4 \Lambda) \phi = 0,$$ for a scalar field $\phi$, while for non-zero spin the field is a symmetric spinor $\phi_{A \cdots F}$ of valence $(2s,0)$ satisfying the equation $$\label{eq:massless}
\nabla^A{}_{A'} \phi_{A \cdots F} = 0 .$$ In this paper we shall restrict our considerations to spins $0,1/2,1$. For $s \geq 3/2$, equation implies algebraic consistency conditions, which strongly restrict the space of solutions in the presence of non-vanishing Weyl curvature. Note however that there are consistent equations for fields of higher spin, see [@PenRin84 §5.8] for discussion.
Recall that a Killing spinor of valence $(k,l)$ is a symmetric spinor $L_{A_1\cdots A_k}{}^{A'_1 \cdots A'_l}$, $$\label{eq:KSgeneral}
\nabla_{(A_1}{}^{(A'_1} L_{A_2\cdots A_{k+1})}{}^{A'_2 \cdots A'_{l+1})} = 0.$$ A valence $(1,1)$ Killing spinor is simply a conformal Killing vector, while a valence $(2,0)$ Killing spinor is equivalent to a conformal Killing-Yano 2-form. On the other hand, a Killing spinor of valence $(2,2)$ is simply a traceless symmetric conformal Killing tensor. It is important to note that , and are conformally invariant if $\phi$ and $\phi_{A \cdots F}$ are given conformal weight $-1$, and $L^{A_1\cdots A_kA'_1 \cdots A'_l}$ is given conformal weight $0$. See [@PenRin84 sections 5.7 and 6.7] for details.
Recall that a symmetry operator for a system ${\mathcal H}\varphi=0$, is a linear partial differential operator ${\mathcal K}$ such that ${\mathcal H}{\mathcal K}\varphi=0$ for all $\varphi$ such that ${\mathcal H}\varphi=0$. We say that two operators ${\mathcal K}_1$ and ${\mathcal K}_2$ are equivalent if ${\mathcal K}_1-{\mathcal K}_2={\mathcal F}{\mathcal H}$ for some differential operator ${\mathcal F}$. We are interested only in *non-trivial* symmetry operators, i.e. operators which are not equivalent to the trivial operator $0$. For simplicity, we will only consider equivalence classes of symmetry operators.
To state our main results, we need two auxiliary conditions.
\[def:auxcond\] Let $L_{AB}{}^{A'B'}$ be a Killing spinor of valence $(2,2)$.
1. \[point:A0\] $L_{AB}{}^{A'B'}$ satisfies auxiliary condition \[point:A0\] if there is a function $Q$ such that $$\begin{aligned}
\nabla_{AA'}Q={}&\tfrac{1}{3} \Psi_{ABCD} \nabla^{(B|B'|}L^{CD)}{}_{A'B'}
+ \tfrac{1}{3} \bar\Psi_{A'B'C'D'} \nabla^{B(B'}L_{AB}{}^{C'D')} \nonumber\\&
+ L^{BC}{}_{A'}{}^{B'} \nabla_{(A}{}^{C'}\Phi_{BC)B'C'}
+ L_{A}{}^{BB'C'} \nabla^{C}{}_{(A'}\Phi_{|BC|B'C')} .
\label{eq:A0}\end{aligned}$$
2. \[point:A1\] $L_{AB}{}^{A'B'}$ satisfies auxiliary condition \[point:A1\] if there is a vector field $P_A{}^{A'}$ such that $$\begin{aligned}
\nabla_{(A}{}^{(A'}P_{B)}{}^{B')}= {}&L{}^{CDA'B'} \Psi_{ABCD}
- L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} .
\label{eq:A1}\end{aligned}$$
Under conformal transformations such that $L^{ABA'B'}$, $P^{AA'}$ and $Q$ are given conformal weight $0$, the equations and are conformally invariant.
We start by recalling the result of Michel et al. for the spin-$0$ conformal wave equation, which we state here in the case of a Lorentzian spacetime of dimension 4.
\[thm:intro:spin0\] Consider the conformal wave equation $$\label{eq:confwave}
(\nabla^a \nabla_a + 4 \Lambda ) \phi = 0$$ in a 4-dimensional Lorentzian spacetime. There is a non-trivial second order symmetry operator for if and only if there is a non-zero Killing spinor of valence $(2,2)$ satisfying condition \[point:A0\] of Definition \[def:auxcond\].
Previous work on the conformal wave equation was done by [@kamran:mclenaghan:1985LMaPh...9...65K], see also Kress [@kress:thesis], see also [@kalnins:miller:1983JMP....24.1047K]. Symmetry operators of general order for the Laplace-Beltrami operator in the conformally flat case have been analyzed by Eastwood [@eastwood:MR2180410].
Next we consider fields with spins 1/2 and 1. The massless spin-1/2 equations are
\[eq:spin1/2eqs\] $$\begin{aligned}
\nabla^A{}_{A'} \phi_A &= 0, \label{eq:spin1/2left} \\
\intertext{and its complex conjugate form}
\nabla_A{}^{A'} \chi_{A'} &= 0, \label{eq:spin1/2right}\end{aligned}$$
which we shall refer to as the left and right Dirac-Weyl equations [^1]. Analogously with the terminology used by Kalnins et al. [@KalMcLWil92a] for the spin-1 case, we call a symmetry operator $\phi_A \mapsto \lambda_A$, which takes a solution of the left equation to a solution of the left equation a symmetry operator of the *first kind*, while an operator $\phi_A \mapsto \chi_{A'}$ which takes a solution of the left equation to a solution of the right equation a symmetry operator of the *second kind*.
If one considers symmetry operators in the Dirac 4-spinor notation, a 4-spinor would correspond to a pair of 2-spinors $(\phi_A, \varphi_{A'})$. Therefore a symmetry operator $(\phi_A, \varphi_{A'})\mapsto (\lambda_A, \chi_{A'})$ for a 4-spinor is formed by a combination of symmetry operators of first $\phi_A\mapsto \lambda_{A}$, and second $\phi_A\mapsto \chi_{A'}$ kind, together with complex conjugate versions of first $\varphi_{A'}\mapsto \chi_{A'}$, and second $\varphi_{A'}\mapsto \lambda_A$ kind symmetry operators.
\[thm:intro:spin1/2\] Consider the Dirac-Weyl equations in a Lorentzian spacetime of dimension 4.
1. There is a non-trivial second order symmetry operator of the first kind for the Dirac-Weyl equation if and only if there is a non-zero Killing spinor of valence $(2,2)$ satisfying auxiliary conditions \[point:A0\] and \[point:A1\] of definition \[def:auxcond\].
2. There is a non-trivial second order symmetry operator of the second kind for the Dirac-Weyl equation if and only if there is a non-zero Killing spinor $L_{ABC}{}^{A'}$ of valence $(3,1)$, such that the auxiliary condition $$\begin{aligned}
0={}&\tfrac{3}{4} \Psi_{ABCD} \nabla^{FA'}L^{CD}{}_{FA'}
+ \tfrac{5}{6} \Psi_{B}{}^{CDF} \nabla_{(A}{}^{A'}L_{CDF)A'}\nonumber\\&
+ \tfrac{5}{6} \Psi_{A}{}^{CDF} \nabla_{(B}{}^{A'}L_{CDF)A'}
- \tfrac{3}{5} L_{B}{}^{CDA'} \nabla_{(A}{}^{B'}\Phi_{CD)A'B'}\nonumber\\&
- \tfrac{3}{5} L_{A}{}^{CDA'} \nabla_{(B}{}^{B'}\Phi_{CD)A'B'}
+ \tfrac{4}{3} L^{CDFA'} \nabla_{(A|A'|}\Psi_{BCDF)}.
\label{eq:A1/2*} \end{aligned}$$ is satisfied.
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1. [Under conformal transformations such that $\widehat L^{ABCA'}=L^{ABCA'}$, the equation is conformally invariant. ]{}
2. [We remark that the auxiliary condition \[point:A0\], appears both in Theorem \[thm:intro:spin1/2\], and for the conformal wave equation in Theorem \[thm:intro:spin0\].]{}
In previous work, Benn and Kress [@benn:kress:2004CQGra..21..427B] showed that a first order symmetry operator of the second kind for the Dirac equation exists exactly when there is a valence $(2,0)$ Killing spinor. See also Carter and McLenaghan [@carter:mclenaghan:1979PhRvD..19.1093C], and Durand, Lina, and Vinet [@durand:lina:vinet:1988PhRvD..38.3837D] for earlier work. The conditions for the existence of a second order symmetry operator for the Dirac-Weyl equations in a general spacetime were considered in [@mclenaghan:smith:walker:2000RSPSA.456.2629G], see also [@fels:kamran:1990RSPSA.428..229F]. The conditions derived here represent a simplification of the conditions found in [@mclenaghan:smith:walker:2000RSPSA.456.2629G]. Further, we mention that symmetry operators of general order for the Dirac operator on Minkowski space have been analyzed by Michel [@michel:2014arXiv1208.4052].
For the spin-1 case, we similarly have the left and right Maxwell equations
\[eq:spin1eqs\] $$\begin{aligned}
\nabla^B{}_{A'} \phi_{AB} &= 0, \label{eq:spin1left} \\
\nabla_A{}^{B'} \chi_{A'B'} &= 0 \label{eq:spin1right}\end{aligned}$$
The left-handed and right-handed spinors $\phi_{AB}$, $\chi_{A'B'}$ represent an anti-self-dual and a self-dual 2-form, respectively. Each equation in is thus equivalent to a real Maxwell equation, cf. [@PenRin84 §3.4]. Analogously to the spin-1/2 case, we consider second order symmetry operators of the first and second kind.
\[thm:intro:spin1\] Consider the Maxwell equations in a Lorentzian spacetime of dimension 4.
1. \[point:first:maxwell\] There is a non-trivial second order symmetry operator of the first kind for the Maxwell equation if and only if there is a non-zero Killing spinor of valence $(2,2)$ such that the auxiliary condition \[point:A1\] of definition \[def:auxcond\] is satisfied.
2. \[point:second:maxwell\] There is a non-trivial second order symmetry operator of the second kind for the Maxwell equation if and only if there is a non-zero Killing spinor $L_{ABCD}$ of valence $(4,0)$.
Note that no auxiliary condition is needed in point *(\[point:second:maxwell\])* of Theorem \[thm:intro:spin1\]. The conditions for the existence of second order symmetry operators for the Maxwell equations have been given in previous work by Kalnins, McLenaghan and Williams [@KalMcLWil92a], see also [@kalnins:mclenaghan:williams:1992grra.conf..129K], following earlier work by Kalnins, Miller and Williams [@kalnins:miller:williams:1989JMP....30.2360K], see also [@kress:thesis]. In [@KalMcLWil92a], the conditions for a second order symmetry operator of the second kind were analyzed completely, and agree with the condition given in point *(\[point:second:maxwell\])* of Theorem \[thm:intro:spin1\]. However, the conditions for a second order symmetry operator of the first kind stated there consist of a set of five equations, of a not particularly transparent nature. The result given here in point *(\[point:first:maxwell\])* of Theorem \[thm:intro:spin1\] provides a substantial simplification and clarification of this previous result.
The necessary and sufficient conditions given in theorems \[thm:intro:spin0\], \[thm:intro:spin1/2\], \[thm:intro:spin1\] involve the existence of a Killing spinor and auxiliary conditions. The following result gives examples of Killing spinors for which the auxiliary conditions \[point:A0\], \[point:A1\] and equation are satisfied.
\[prop:factorize\] Let $\xi^{AA'}$ and $\zeta^{AA'}$ be (not necessarily distinct) conformal Killing vectors and let $\kappa_{AB}$ be a Killing spinor of valence $(2,0)$.
1. The symmetric spinor $\xi_{(A}{}^{(A'} \zeta_{B)}{}^{B')}$ is a Killing spinor of valence $(2,2)$, which admits solutions to the auxiliary conditions \[point:A0\] and \[point:A1\].
2. The symmetric spinor $\kappa_{AB}\bar{\kappa}_{A'B'}$ is also a Killing spinor of valence $(2,2)$, which admits solutions to the auxiliary conditions \[point:A0\] and \[point:A1\].
3. The spinor $\kappa_{(AB}\xi_{C)}{}^{C'}$ is a Killing spinor of valence $(3,1)$, which satisfies auxiliary equation .
4. The spinor $\kappa_{(AB}\kappa_{CD)}$ is a Killing spinor of valence $(4,0)$. \[point:factoredvalence4KS\]
The point *(\[point:factoredvalence4KS\])* is immediately clear. The other parts will be proven in Section \[sec:factorizations\].
We now consider the following condition $$\begin{aligned}
0={}&\Psi_{(ABC}{}^{F}\Phi_{D)FA'B'}. \label{eq:intro:AlignmentPsiPhi}\end{aligned}$$ relating the Ricci curvature $\Phi_{ABA'B'}$ and the Weyl curvature $\Psi_{ABCD}$. A spacetime where holds will be said to satisfy the aligned matter condition. In particular this holds in Vacuum and in the Kerr-Newman class of spacetimes. Under the aligned matter condition we can show that the converse of Proposition \[prop:factorize\] part *(\[point:factoredvalence4KS\])* is true. The following theorem will be proved in Section \[sec:factorValence4\].
\[thm:Valence4Factorization\] If the aligned matter condition is satisfied, $\Psi_{ABCD}\neq 0$ and $L_{ABCD}$ is a valence $(4,0)$ Killing spinor, then there is a valence $(2,0)$ Killing spinor $\kappa_{AB}$ such that $$\begin{aligned}
L_{ABCD}={}&\kappa_{(AB}\kappa_{CD)}.\end{aligned}$$
If $\Psi_{ABCD}=0$, the valence $(4,0)$ Killing spinor will still factor but in terms of valence $(1,0)$ Killing spinors, which then can be combined into valence $(2,0)$ Killing spinors. However, the two factors might be distinct.
A calculation shows that if holds, $\kappa_{AB}$ is a valence $(2,0)$ Killing spinor, then $\xi^{AA'} = \nabla^{BA'} \kappa^{A}{}_{B}$ is a Killing vector field. Taking this fact into account, we have the following corollary to the results stated above. It tells that generically one can generate a wide variety of symmetry operators from just a single valence $(2,0)$ Killing spinor.
\[cor:sufficient\] Consider the massless test fields of spins 0, 1/2 and 1 in a Lorentzian spacetime of dimension 4. Assume that there is Killing spinor $\kappa_{AB}$ (not identically zero) of valence $(2,0)$. Then there are non-trivial second order symmetry operators for the massless spin-$s$ field equations for spins 0 and 1, as well as a non-trivial second order symmetry operator of the first kind for the massless spin-$1/2$ field.
If, in addition, the aligned matter condition holds, and $\xi_{AA'} = \nabla^{B}{}_{A'} \kappa_{AB}$ is not identically zero, then there is also a non-trivial second order symmetry operators of the second kind for the massless spin-$1/2$ field.
We end this introduction by giving a simple form for symmetry operators for the Maxwell equation, generated from a Killing spinor of valence $(2,0)$.
\[Thm:SymopMaxwellSimple\] Let $\kappa_{AB}$ be a Killing spinor of valence $(2,0)$ and let $$\begin{aligned}
\Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$ Define the potentials
\[eq:intro:ABdef\] $$\begin{aligned}
A_{AA'}={}&\bar{\kappa}_{A'}{}^{B'} \nabla_{BB'}\Theta_{A}{}^{B}
- \tfrac{1}{3} \Theta_{A}{}^{B} \nabla_{BB'}\bar{\kappa}_{A'}{}^{B'},\\
B_{AA'}={}&\kappa_{A}{}^{B} \nabla_{CA'}\Theta_{B}{}^{C}
+ \tfrac{1}{3} \Theta_{A}{}^{B} \nabla_{CA'}\kappa_{B}{}^{C} . \end{aligned}$$
Assume that $\phi_{AB}$ is a solution to the Maxwell equation in a Lorentzian spacetime of dimension 4. Let $A_{AA'}, B_{AA'}$ be given by . Then
$$\begin{aligned}
\chi_{AB}={}&\nabla_{(B}{}^{A'}A_{A)A'},\\
\omega_{A'B'}={}&\nabla^{B}{}_{(A'}B_{|B|B')}\end{aligned}$$
are solutions to the left and right Maxwell equations, respectively.
The proof can be found in sections \[sec:symopfirstmaxwellfact\] and \[sec:symopsecondmaxwellfact\]. The general form of the symmetry operators for spins 0, 1/2 and 1 is discussed in detail below.
The symmetry operators of the Maxwell equation can in general be written in potential form. See Theorem \[Thm::SymOpFirstKind\] and Theorem \[Thm::SymOpSecondKind\].
The method used in this paper can also be used to show that the symmetry operators $R$-commute with the Dirac and Maxwell equations. Recall that an operator $S$ is said to $R$-commute with a linear PDE $L\phi=0$ if there is an operator $R$ such that $LS=RL$. Even providing a formula for the relevant $R$ operators would require additional notation, so we have omitted this result from this paper.
Overview of this paper {#overview-of-this-paper .unnumbered}
----------------------
In Section \[sec:prel\] we define the fundamental operators $\sDiv, \sCurl, \sCurlDagger, \sTwist$ obtained by projecting the covariant derivative of a symmetric spinor on its irreducible parts. These operators are analogues of the Stein-Weiss operators discussed in Riemannian geometry and play a central role in our analysis. We give the commutation properties of these operators, derive the integrability conditions for Killing spinors, and end the section by discussing some aspects of the methods used in the analysis. Section \[sec:spin0\] gives the analysis of symmetry operators for the conformal wave equation. The results here are given for completeness, and agree with those in [@michel:radoux:silhan:2013arXiv1308.1046M] for the case of a Lorentzian spacetime of dimesion 4. The symmetry operators for the Dirac-Weyl equation are discussed in Section \[sec:spin1/2\] and our results for the Maxwell case are given in Section \[sec:spin1\]. Special conditions under which the auxiliary conditions can be solved is discussed in Section \[sec:factorizations\]. Finally, Section \[sec:symopfactored\] contains simplified expressions for the symmetry operators for some of the cases discussed in Section \[sec:factorizations\].
Preliminaries {#sec:prel}
=============
Fundamental operators
---------------------
Let $S_{k,l}$ denote the vector bundle of symmetric spinors with $k$ unprimed indices and $l$ primed indices. We will call these spinors symmetric valence $(k,l)$ spinors. Furthermore, let $\mathcal{S}_{k,l}$ denote the space of smooth ($C^\infty$) sections of $S_{k,l}$.
For any $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$, we define the operators $\sDiv_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k-1,l-1}$, $\sCurl_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k+1,l-1}$, $\sCurlDagger_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k-1,l+1}$ and $\sTwist_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k+1,l+1}$ as
$$\begin{aligned}
(\sDiv_{k,l}\varphi)_{A_1\dots A_{k-1}}{}^{A_1'\dots A_{l-1}'}\equiv{}&
\nabla^{BB'}\varphi_{A_1\dots A_{k-1}B}{}^{A_1'\dots A_{l-1}'}{}_{B'},\\
(\sCurl_{k,l}\varphi)_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l-1}'}\equiv{}&
\nabla_{(A_1}{}^{B'}\varphi_{A_2\dots A_{k+1})}{}^{A_1'\dots A_{l-1}'}{}_{B'},\\
(\sCurlDagger_{k,l}\varphi)_{A_1\dots A_{k-1}}{}^{A_1'\dots A_{l+1}'}\equiv{}&
\nabla^{B(A_1'}\varphi_{A_1\dots A_{k-1}B}{}^{A_2'\dots A_{l+1}')},\\
(\sTwist_{k,l}\varphi)_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l+1}'}\equiv{}&
\nabla_{(A_1}{}^{(A_1'}\varphi_{A_2\dots A_{k+1})}{}^{A_2'\dots A_{l+1}')}.\end{aligned}$$
1. [ These operators are all conformally covariant, but the conformal weight differs between the operators. See [@PenRin86 Section 6.7] for details.]{}
2. [The left Dirac-Weyl and Maxwell equations can be written as $(\sCurlDagger_{1,0}\phi)_{A'}=0$ and $(\sCurlDagger_{2,0}\phi)_{AA'}=0$ respectively. Similarly the right equations can be written in terms of the $\sCurl$ operator. ]{}
The operator $\sDiv_{k,l}$ only makes sense when $k\geq 1$ and $l\geq 1$. Likewise $\sCurl_{k,l}$ is defined only if $l \geq 1$ and $\sCurlDagger_{k,l}$ only if $k\geq 1$. To make a clean presentation, we will use formulae where invalid operators appear for some choices of $k$ and $l$. However, the operators will always be multiplied with a factor that vanishes for these invalid choices of $k$ and $l$. From the definition it is clear that the complex conjugates of $(\sDiv_{k,l}\varphi)$, $(\sCurl_{k,l}\varphi)$, $(\sCurlDagger_{k,l}\varphi)$ and $(\sTwist_{k,l}\varphi)$ are $(\sDiv_{l,k}\bar\varphi)$, $(\sCurlDagger_{l,k}\bar\varphi)$, $(\sCurl_{l,k}\bar\varphi)$ and $(\sTwist_{l,k}\bar\varphi)$ respectively, with the appropriate indices.
The main motivation for the introduction of these operators is the irreducible decomposition of the covariant derivative of a symmetric spinor field.
For any $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$, we have the irreducible decomposition $$\begin{aligned}
\nabla_{A_1}{}^{A_1'}\varphi{}_{A_2\dots A_{k+1}}{}^{A_2'\dots A_{l+1}'}={}&
(\sTwist_{k,l}\varphi){}_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l+1}'}\nonumber\\
&-\tfrac{l}{l+1}\bar\epsilon^{A_1'(A_2'}(\sCurl_{k,l}\varphi){}_{A_1\dots A_{k+1}}{}^{A_3'\dots A_{l+1}')}\nonumber\\
&-\tfrac{k}{k+1}\epsilon_{A_1(A_2}(\sCurlDagger_{k,l}\varphi){}_{A_3\dots A_{k+1})}{}^{A_1'\dots A_{l+1}'}\nonumber\\
&+\tfrac{kl}{(k+1)(l+1)}\epsilon_{A_1(A_2}\bar\epsilon^{A_1'(A_2'}(\sDiv_{k,l}\varphi){}_{A_3\dots A_{k+1})}{}^{A_3'\dots A_{l+1}')}.\label{eq:IrrDecGeneralDer}\end{aligned}$$
It follows from in [@PenRin84 Proposition 3.3.54] that the irreducible decomposition must have this form. The coefficients are then found by contracting indices and partially expanding the symmetries.
With this notation, the Bianchi system takes the form
$$\begin{aligned}
(\sDiv_{2,2} \Phi)_{AA'}={}&-3 (\sTwist_{0,0} \Lambda)_{AA'},\\
(\sCurlDagger_{4,0} \Psi)_{ABCA'}={}&(\sCurl_{2,2} \Phi)_{ABCA'}.\end{aligned}$$
In the rest of the paper we will use these equations every time the left hand sides appear in the calculations.
With the definitions above, a Killing spinor of valence $(k,l)$ is an element $L_{A\cdots F}{}^{ A'\dots F'} \in \ker \sTwist_{k,l}$, a conformal Killing vector is a Killing spinor of valence $(1,1)$, and a trace-less conformal Killing tensor is a Killing spinor of valence $(2,2)$. We further introduce the following operators, acting on a valence $(2,2)$ Killing spinor.
For $L_{AB}{}^{ A'B'} \in \ker \sTwist_{2,2}$, define
$$\begin{aligned}
(\ObstrZero L)_A{}^{A'} \equiv {}&\tfrac{1}{3} \Psi_{ABCD} (\sCurl_{2,2} L)^{BCDA'}
+ L^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'}\nonumber\\
& + \tfrac{1}{3} \bar\Psi^{A'}{}_{B'C'D'} (\sCurlDagger_{2,2} L)_{A}{}^{B'C'D'}
+ L_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}. \\
(\ObstrOne L)_{AB}{}^{A'B'} \equiv {}&L{}^{CDA'B'} \Psi_{ABCD}
- L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} \end{aligned}$$
The operators $\ObstrZero$ and $\ObstrOne$ are the right hand sides of and in conditions \[point:A0\] and \[point:A1\] respectively. They will play an important role in the conditions for the existence of symmetry operators.
Given a conformal Killing vector $\xi^{AA'}$, we follow [@MR2056970 Equations (2) and (15)], see also [@anco:pohjanpelto:2003RSPSA.459.1215A], and define a conformally weighted Lie derivative acting on a symmetric valance $(2s,0)$ spinor field as follows
For $\xi^{AA'} \in \ker \sTwist_{1,1}$, and $\varphi_{A_1\dots A_{2s}}\in \mathcal{S}_{2s,0}$, we define $$\begin{aligned}
\hat{\mathcal{L}}_{\xi}\varphi_{A_1\dots A_{2s}}\equiv{}&\xi^{BB'} \nabla_{BB'}\varphi_{A_1\dots A_{2s}}+s \varphi_{B(A_2\dots A_{2s}} \nabla_{A_1)B'}\xi^{BB'}
+ \tfrac{1-s}{4} \varphi_{A_1\dots A_{2s}} \nabla^{CC'}\xi_{CC'}.\end{aligned}$$
This operator turns out to be important when we describe first order symmetry operators. See Section \[sec:symopsecondDiracfactored\] for further discussion.
Commutator relations
--------------------
Let $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$ and define the standard commutators $$\square_{AB}\equiv \nabla_{(A|A'|}\nabla_{B)}{}^{A'} \qquad \text{ and }\qquad \square_{A'B'}\equiv \nabla_{A(A'}\nabla^{A}{}_{B')}.$$ Acting on spinors, these commutators can always be written in terms of curvature spinors as described in [@PenRin84 Section 4.9].
\[lemma:commutators\] The operators $\sDiv$, $\sCurl$, $\sCurlDagger$ and $\sTwist$ satisfies the following commutator relations
$$\begin{aligned}
(\sDiv_{k+1,l-1} &\sCurl_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}\nonumber\\*
={}&\tfrac{k}{k+1}(\sCurl_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}
-\square_{B'C'}\varphi{}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'B'C'},
\quad k\geq 0, l\geq 2, \label{eq:DivCurl}\\
(\sDiv_{k-1,l+1} &\sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k-2}}{}^{A_1'\dots A_{l}'}\nonumber\\*
={}&\tfrac{l}{l+1}(\sCurlDagger_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k-2}}{}^{A_1'\dots A_{l}'}
-\square_{BC}\varphi{}_{A_1\dots A_{k-2}}{}^{BCA_1'\dots A_{l}'},
\quad k\geq 2, l\geq 0,\label{eq:DivCurlDagger}\\
(\sCurl_{k+1,l+1}&\sTwist_{k,l} \varphi){}_{A_1\dots A_{k+2}}{}^{A_1'\dots A_{l}'}\nonumber\\*
={}&\tfrac{l}{l+1}(\sTwist_{k+1,l-1} \sCurl_{k,l} \varphi){}_{A_1\dots A_{k+2}}{}^{A_1'\dots A_{l}'}
-\square_{(A_1A_2}\varphi{}_{A_3\dots A_{k+2})}{}^{A_1'\dots A_{l}'},
\quad k\geq 0, l\geq 0,\label{eq:CurlTwist}\\
(\sCurlDagger_{k+1,l+1}&\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}\nonumber\\*
={}&
\tfrac{k}{k+1}(\sTwist_{k-1,l+1} \sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}
-\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l+2}')},
\quad k\geq 0, l\geq 0,\label{eq:CurlDaggerTwist}\\
(\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
={}&
-(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
+\tfrac{l(l+2)}{(l+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
&-\tfrac{l+2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
-\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'},
\quad k\geq 1, l\geq 0,\label{eq:DivTwistCurlCurlDagger}\\
(\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
={}&
-(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurlDagger_{k+1,l-1}\sCurl_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
+\tfrac{k(k+2)}{(k+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
&-\tfrac{k}{k+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
-\tfrac{k+2}{k+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'},
\quad k\geq 0, l\geq 1,\label{eq:DivTwistCurlDaggerCurl}\\
(\sCurl_{k-1,l+1}&\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
={}&
(\sCurlDagger_{k+1,l-1}\sCurl_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
+(\tfrac{1}{k+1}-\tfrac{1}{l+1})(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\*
&-\square_{(A_1}{}^{B}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
+\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'},
\quad k\geq 1, l\geq 1.\label{eq:CurlCurlDagger}\end{aligned}$$
We first observe that and are related by complex conjugation. Likewise and as well as and are also related by complex conjugation. Furthermore, is given by the difference between and . It is therefore enough to prove , and . We consider each in turn.
- [We first prove . We partially expand the symmetry, identify the commutator in one term, and commute derivatives in the other. $$\begin{aligned}
(\sDiv_{k+1,l-1}& \sCurl_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}\\
={}&\nabla^{BB'}\nabla_{(A_1}{}^{C'}\varphi_{A_2\dots A_k B)}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\
={}&\tfrac{1}{k+1}\nabla^{B(B'}\nabla_{B}{}^{C')}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}
+\tfrac{k}{k+1}\nabla^{B(B'}\nabla_{(A_1}{}^{C')}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\
={}&-\tfrac{1}{k+1}\square^{B'C'}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}
+\tfrac{k}{k+1}\epsilon^B{}_{(A_1}\square^{B'C'}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\
&+\tfrac{k}{k+1}\nabla_{(A_1}{}^{C'}\nabla^{BB'}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\
={}&
\tfrac{k}{k+1}(\sCurl_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}
-\square_{B'C'}\varphi{}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'B'C'}.\end{aligned}$$ ]{}
- [ To prove , we first partially expand the symmetrization over the unprimed indices in the irreducible decomposition and symmetrizing over the primed indices. This gives $$\begin{aligned}
\nabla_{A_1}{}^{(A_2'}\varphi_{A_2\dots A_k B}{}^{A_3'\dots A_{l+2}')}={}&
(\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}B}{}^{A_2'\dots A_{l+2}'}
-\tfrac{1}{k+1}\epsilon_{A_1B}(\sCurlDagger_{k,l}\varphi){}_{A_2\dots A_{k}}{}^{A_2'\dots A_{l+2}'}\nonumber\\
&-\tfrac{k-1}{k+1}\epsilon_{A_1(A_2}(\sCurlDagger_{k,l}\varphi){}_{A_3\dots A_{k})B}{}^{A_2'\dots A_{l+2}'}.\label{eq:Irrdecvarphihelp}\end{aligned}$$ Using the definitions of $\sTwist$ and $\sCurlDagger$, commuting derivatives and using , we have $$\begin{aligned}
(\sTwist_{k-1,l+1} \sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}
={}&
\nabla_{(A_1}{}^{(A_1'}\nabla^{|B|A_2'}\varphi_{A_2\dots A_{k})B}{}^{A_3'\dots A_{l+2}')} \nonumber\\
={}&
\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')}
+\nabla^{B(A_1'}\nabla_{(A_1}{}^{A_2'}\varphi_{A_2\dots A_{k})B}{}^{A_3'\dots A_{l+2}')}\nonumber\\
={}&
\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')}
+\nabla^{B(A_1'}(\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}B}{}^{A_2'\dots A_{l+2}')}\nonumber\\
&-\tfrac{1}{k+1}\epsilon_{(A_1|B|}\nabla^{B(A_1'}(\sCurlDagger_{k,l}\varphi){}_{A_2\dots A_{k})}{}^{A_2'\dots A_{l+2}')}\nonumber\\
={}&\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')}
+(\sCurlDagger_{k+1,l+1}\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}\nonumber\\
&+\tfrac{1}{k+1}(\sTwist_{k-1,l+1}\sCurlDagger_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}.\end{aligned}$$ Isolating the $\sCurlDagger\sTwist$-term gives . ]{}
- [ Finally to prove , we assume $k\geq 1$ and observe $$\begin{aligned}
(\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\
={}&-\nabla^B{}_{B'}\nabla_{(B}{}^{(B'}\varphi_{A_1\dots A_k)}{}^{A_1'\dots A_l')}\nonumber\\
={}&
-\tfrac{1}{k+1}\nabla^B{}_{B'}\nabla_{B}{}^{(B'}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_l')}
-\tfrac{k}{k+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')}\nonumber\\
={}&
\tfrac{1}{k+1}(\sDiv_{k+1,l+1}\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
-\tfrac{k}{(k+1)^2}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\
&-\tfrac{k}{k+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')},\end{aligned}$$ where we in the last step used the irreducible decomposition on the first term. We can solve for the $\sDiv\sTwist$-term from which it follows that $$\begin{aligned}
(\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\
={}&
-\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
-\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')}\nonumber\\
={}&
-\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
-\tfrac{1}{l+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}\nonumber\\
&-\tfrac{l}{l+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(A_1'}\varphi_{A_2\dots A_k)B}{}^{A_2'\dots A_l')B'}
\nonumber\\
={}&
-\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
-\tfrac{1}{l+1}\nabla_{(A_1}{}^{B'}\nabla^{B}{}_{|B'|}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}\nonumber\\
&
-\tfrac{2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
-\tfrac{l}{l+1}\nabla_{(A_1}{}^{(A_1'}\nabla^{|B|}{}_{|B'|}\varphi_{A_2\dots A_k)B}{}^{A_2'\dots A_l')B'}\nonumber\\
&-\tfrac{l}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
-\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}
\nonumber\\
={}&
-(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}
+\tfrac{l(l+2)}{(l+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\
&-\tfrac{l+2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}
-\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}.\end{aligned}$$ ]{}
The operators $\sDiv$, $\sCurl$, $\sCurlDagger$ and $\sTwist$ together with the irreducible decomposition and the relations in Lemma \[lemma:commutators\] have all been implemented in the *SymManipulator* package version 0.9.0 [@Bae11a].
Integrability conditions for Killing spinors {#sec:integrabilitycond}
--------------------------------------------
Here we demonstrate a procedure for obtaining an integrability condition for a Killing spinor of arbitrary valence. Let $\kappa_{A_1\dots A_k}{}^{A'_1\dots A'_l}\in \ker \sTwist_{k,l}$. By applying the $\sCurl$ operator $l+1$ times to the Killing spinor equation, and repeatedly commute derivatives with we get $$\begin{aligned}
0={}& (\underbrace{\sCurl_{k+l+1,1}\sCurl_{k+l,2} \cdots \sCurl_{k+2,l}\sCurl_{k+1,l+1}}_{l+1}\sTwist_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}\nonumber\\
={}&\frac{l}{l+1}(\sCurl_{k+l+1,1}\sCurl_{k+l,2} \cdots \sCurl_{k+2,l}\sTwist_{k+1,l-1}\sCurl_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}+\text{curvature terms}\nonumber\\
={}&\frac{1}{l+1}(\sCurl_{k+l+1,1}\sTwist_{k+l,0}\sCurl_{k+l-1,1} \cdots \sCurl_{k+1,l-1}\sCurl_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}+\text{curvature terms}\nonumber\\
={}&\text{curvature terms}.\end{aligned}$$ Here, the curvature terms have $l-m$ derivatives of $\kappa$ and $m$ derivatives of the curvature spinors, where $0\leq m\leq l$. The main idea behind this is the observation that the commutator acting on a spinor field without primed indices only gives curvature terms. In the same way we can use to get $$\begin{aligned}
0={}& (\underbrace{\sCurlDagger_{1,k+l+1}\sCurlDagger_{2,k+l} \cdots \sCurlDagger_{k,l+2}\sCurlDagger_{k+1,l+1}}_{k+1}\sTwist_{k,l}\kappa)_{A'_{1}\dots A'_{k+l+2}}\nonumber\\
={}&\text{curvature terms}.\end{aligned}$$
Splitting equations into independent parts {#sec:indepspinors}
------------------------------------------
In our derivation of necessary conditions for the existence of symmetry operators, it is crucial that, at each fixed point in spacetime, we can freely choose the values of the Dirac-Weyl and the Maxwell field and of the symmetric components of any given order of their derivatives. The remaining components of the derivatives to a given order, which involve at least one pair of antisymmetrized indices, can be solved for using the field equations or curvature conditions. See sections \[sec:ReductionDirac\] and \[sec:ReductionMaxwell\] for detailed expressions. In the literature, the condition that the symmetric components can be freely and independently specified but that no other parts can be is referred to as the exactness of the set of fields [@PenRin84 Section 5.10]. The symmetric components of the derivatives are exactly those that can be expressed in terms of the operator $\sTwist$. One can show that, in a globally hyperbolic spacetime, the Dirac-Weyl and Maxwell fields each form exact sets. However, it is not necessary for the spacetime to be globally hyperbolic for this condition to hold. If the spacetime is such that the fields fail to form an exact set, then our methods still give sufficient conditions for the existence of symmetry operators, but they may no longer be necessary.
The freedom to choose the symmetric components is used in this paper to show that equations of the type $L^{ABA'}(\sTwist_{1,0}\phi)_{ABA'}+M^{A}\phi_{A}=0$ with $(\sCurl_{1,0}^\dagger\phi)_{A'}=0$ forces $L^{(AB)A'}=0$ and $M^A=0$ because $(\sTwist_{1,0}\phi)_{ABA'}$ and $\phi_A$ can be freely and independently specified at a single point. Similar arguments involving derivatives of up to third order are also used.
In several places we will have equations of the form $$\begin{aligned}
\label{eq:ProductExample}
0=S^{ABC}{}_{A'}(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C},\end{aligned}$$ where $T_{A}$ and $(\sTwist_{1,0}\phi)_{ABA'}$ are free and independent. In particular all linear combinations of the form $(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C}$ will then span the space of spinors $W_{ABC}{}^{A'}=W_{(AB)C}{}^{A'}$. As the equation is linear we therefore get $$\begin{aligned}
0=S^{ABC}{}_{A'}W_{ABC}{}^{A'},\end{aligned}$$ for all $W_{ABC}{}^{A'}=W_{(AB)C}{}^{A'}$. We can then make an irreducible decomposition $$\begin{aligned}
W_{ABC}{}^{A'}={}&W_{(ABC)}{}^{A'}
- \tfrac{2}{3} W_{(A}{}^{D}{}_{|D|}{}^{A'}\epsilon_{B)C},\end{aligned}$$ which gives $$\begin{aligned}
0={}&(- \tfrac{1}{3} S_{B}{}^{C}{}_{CA'}
- \tfrac{1}{3} S^{C}{}_{BCA'}) W^{BA}{}_{A}{}^{A'}
- S_{ABCA'} W^{(ABC)A'}.\end{aligned}$$ As $W_{ABC}{}^{A'}$ is free, its irreducible components $W_{(ABC)}{}^{A'}$ and $W_{A}{}^{D}{}_{D}{}^{A'}$ are free and independent. We can therefore conclude that
$$\begin{aligned}
0={}&S_{B}{}^{C}{}_{CA'}+ S^{C}{}_{BCA'},\\
0={}& S_{(ABC)A'}.\end{aligned}$$
Observe that we only get the symmetric part in the last equation due to the symmetry of $W_{(ABC)}{}^{A'}$.
Instead of introducing a new spinor $W_{ABC}{}^{A'}$ we will in the rest of the paper work directly with the irreducible decomposition of $(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C}$ and get $$\begin{aligned}
0={}& (- \tfrac{1}{3} S_{A}{}^{C}{}_{CA'}
- \tfrac{1}{3} S^{C}{}_{ACA'}) T_{B} (\sTwist_{1,0} \phi)^{ABA'}
- S_{ABCA'} T^{(A}(\sTwist_{1,0} \phi)^{BC)A'}.\end{aligned}$$ The formal computations will be the same, and by the argument above, the symmetrized coefficients for the irreducible parts $T_{B} (\sTwist_{1,0} \phi)^{ABA'}$ and $T^{(A}(\sTwist_{1,0} \phi)^{BC)A'}$ will individually have to vanish.
The conformal wave equation {#sec:spin0}
===========================
For completeness we give here a detailed description of the symmetry operators for the conformal wave equation.
The equation $$\begin{aligned}
(\square+4 \Lambda) \phi={}&0,\end{aligned}$$ has a symmetry operator $\phi \rightarrow \chi$ , with order less or equal to two, if and only if there are spinors $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that
$$\begin{aligned}
(\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\
(\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&0,\\
(\sTwist_{0,0} Q)_{A}{}^{A'}={}&\tfrac{2}{5}(\ObstrZero L)_A{}^{A'}.\label{eq:auxcondwave}\end{aligned}$$
The symmetry operator then takes the form $$\begin{aligned}
\chi={}&- \tfrac{3}{5} L^{ABA'B'} \Phi_{ABA'B'} \phi
+ Q \phi
+ \tfrac{1}{4} \phi (\sDiv_{1,1} P)
+ \tfrac{1}{15} \phi (\sDiv_{1,1} \sDiv_{2,2} L)
+ P^{AA'} (\sTwist_{0,0} \phi)_{AA'}\nonumber\\
& + \tfrac{2}{3} (\sDiv_{2,2} L)^{AA'} (\sTwist_{0,0} \phi)_{AA'}
+ L^{ABA'B'} (\sTwist_{1,1} \sTwist_{0,0} \phi)_{ABA'B'}.\label{eq:wavesymop1}\end{aligned}$$
The existence of $Q$ satisfying is exactly the auxiliary condition \[point:A0\]. The proof can also be carried out using the same technique as in the rest of the paper.
The Dirac-Weyl equation {#sec:spin1/2}
=======================
The following theorems imply Theorem \[thm:intro:spin1/2\].
\[Thm::SymOpFirstKindDirac\] There exists a symmetry operator of the first kind for the Dirac-Weyl equation $\phi_{A}\rightarrow \chi_{A}$, with order less or equal to two, if and only if there are spinor fields $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that
$$\begin{aligned}
(\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\
(\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{1}{3} (\ObstrOne L)_{AB}{}^{A'B'},\label{eq:ObstrDiracSecond1}\\
(\sTwist_{0,0} Q)_{A}{}^{A'}={}&\tfrac{3}{10} (\ObstrZero L)_{A}{}^{A'}.\label{eq:ObstrDiracSecond2}\end{aligned}$$
The symmetry operator then takes the form $$\begin{aligned}
\chi_{A}={}&- \tfrac{8}{15} L^{BCA'B'} \Phi_{BCA'B'} \phi_{A}
+ Q \phi_{A}
+ \tfrac{1}{2} \phi^{B} (\sCurl_{1,1} P)_{AB}
+ \tfrac{2}{9} \phi^{B} (\sCurl_{1,1} \sDiv_{2,2} L)_{AB}
+ \tfrac{3}{8} \phi_{A} (\sDiv_{1,1} P)\nonumber\\
& + \tfrac{2}{15} \phi_{A} (\sDiv_{1,1} \sDiv_{2,2} L)
+ P^{BA'} (\sTwist_{1,0} \phi)_{ABA'}
+ \tfrac{8}{9} (\sDiv_{2,2} L)^{BA'} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\
& + \tfrac{2}{3} (\sCurl_{2,2} L)_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}
+ L^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}.\label{eq:diracsymop1}\end{aligned}$$
1. [ Observe that is the auxiliary condition \[point:A1\] for existence of a symmetry operator of the first kind for Maxwell equation, and is the auxiliary condition \[point:A0\] for existence of a symmetry operator for the conformal wave equation. ]{}
2. [ With $L_{ABA'B'}=0$ the first order operator takes the form $$\begin{aligned}
\chi_{A}={}&
\hat{\mathcal{L}}_{P}\phi_{A}+ Q \phi_{A}.\end{aligned}$$ ]{}
\[Thm::SymOpSecondKindDirac\] There exists a symmetry operator of the second kind for the Dirac-Weyl equation $\phi_{A}\rightarrow \omega_{A'}$, with order less or equal to two, if and only if there are spinor fields $L_{ABC}{}^{A'}=L_{(ABC)}{}^{A'}$ and $P_{AB}=P_{(AB)}$ such that
$$\begin{aligned}
(\sTwist_{3,1} L)_{ABCD}{}^{A'B'}={}&0,\label{eq:TwistL31}\\
(\sTwist_{2,0} P)_{ABC}{}^{A'}={}&0,\label{eq:TwistPDirac2}\\
0={}&- \tfrac{9}{8} \Psi_{ABCD} (\sDiv_{3,1} L)^{CD}
+ \tfrac{9}{5} L_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\
&
- \tfrac{5}{2} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} L)_{B)CDF}
- 2 L^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}. \label{eq:DiracSecondKindObstr}\end{aligned}$$
The operator takes the form $$\begin{aligned}
\omega_{A'}={}&- \tfrac{1}{2} L_{BCDB'} \Phi^{CD}{}_{A'}{}^{B'} \phi^{B}
+ \tfrac{2}{3} \phi^{B} (\sCurlDagger_{2,0} P)_{BA'}
+ \tfrac{1}{4} \phi^{B} (\sCurlDagger_{2,0} \sDiv_{3,1} L)_{BA'}
+ P^{BC} (\sTwist_{1,0} \phi)_{BCA'}\nonumber\\
& + \tfrac{3}{4} (\sDiv_{3,1} L)^{BC} (\sTwist_{1,0} \phi)_{BCA'}
+ \tfrac{3}{4} (\sCurlDagger_{3,1} L)_{BCA'B'} (\sTwist_{1,0} \phi)^{BCB'}
+ L^{BCDB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'}.\end{aligned}$$
The scheme for deriving integrability conditions in Section \[sec:integrabilitycond\] can be used to show that $$\begin{aligned}
0={}&- \tfrac{2}{5} L_{(ABC}{}^{A'}(\sCurl_{2,2} \Phi)_{DFH)A'}
+ 3 L_{(AB}{}^{LA'}(\sTwist_{4,0} \Psi)_{CDFH)LA'}
+ 5 \Psi_{(ABC}{}^{L}(\sCurl_{3,1} L)_{DFH)L}\nonumber\\
& + \tfrac{3}{4} \Psi_{(ABCD}(\sDiv_{3,1} L)_{FH)},\end{aligned}$$ follows from . Despite the superficial similarity of this equation to the condition , we conjecture that does not follow from .
Reduction of derivatives of the field {#sec:ReductionDirac}
-------------------------------------
In our notation, the Dirac-Weyl equation $\nabla^{A}{}_{A'}\phi_{A}=0$, takes the form $(\sCurlDagger_{1,0} \phi)_{A'}=0$. We see that the only remaining irreducible part of $\nabla_{A}{}^{A'}\phi_{B}$ is $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$. By commuting derivatives we see that all higher order derivatives of $\phi_{A}$ can be reduced to totally symmetrized derivatives and lower order terms consisting of curvature times lower order symmetrized derivatives.
Together with the Dirac-Weyl equation, the commutators , , give
$$\begin{aligned}
(\sDiv_{2,1} \sTwist_{1,0} \phi)_{A}={}&-6 \Lambda \phi_{A},\\
(\sCurl_{2,1} \sTwist_{1,0} \phi)_{ABC}={}&- \Psi_{ABCD} \phi^{D},\\
(\sCurlDagger_{2,1} \sTwist_{1,0} \phi)_{AA'B'}={}&- \Phi_{ABA'B'} \phi^{B}.\end{aligned}$$
The higher order derivatives can be computed by using the commutators , , together with the equations above and the Bianchi system to get
$$\begin{aligned}
(\sDiv_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{AB}{}^{A'}={}&\tfrac{5}{6} \phi^{C} (\sCurl_{2,2} \Phi)_{ABC}{}^{A'}
+ \tfrac{10}{3} \Phi_{(A}{}^{CA'B'}(\sTwist_{1,0} \phi)_{B)CB'}
- \tfrac{16}{3} \phi_{(A}(\sTwist_{0,0} \Lambda)_{B)}{}^{A'}\nonumber\\
& - 12 \Lambda (\sTwist_{1,0} \phi)_{AB}{}^{A'}
+ \tfrac{3}{2} \Psi_{ABCD} (\sTwist_{1,0} \phi)^{CDA'},\\
(\sCurl_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCD}{}^{A'}={}&\Phi_{(AB}{}^{A'B'}(\sTwist_{1,0} \phi)_{CD)B'}
+ \tfrac{5}{2} \Psi_{(ABC}{}^{F}(\sTwist_{1,0} \phi)_{D)F}{}^{A'}\nonumber\\
& - \tfrac{1}{10} \phi_{(A}(\sCurl_{2,2} \Phi)_{BCD)}{}^{A'}
- \tfrac{1}{2} \phi^{F} (\sTwist_{4,0} \Psi)_{ABCDF}{}^{A'},\\
(\sCurlDagger_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{AB}{}^{A'B'C'}={}&\tfrac{8}{3} \Phi^{C}{}_{(A}{}^{(A'B'}(\sTwist_{1,0} \phi)_{B)C}{}^{C')}
- \tfrac{2}{9} \phi_{(A}(\sCurlDagger_{2,2} \Phi)_{B)}{}^{A'B'C'}\nonumber\\
& - \tfrac{2}{3} \phi^{C} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'B'C'}
- \bar\Psi^{A'B'C'}{}_{D'} (\sTwist_{1,0} \phi)_{AB}{}^{D'}.\end{aligned}$$
Using irreducible decompositions and the equations above, one can in a systematic way reduce any third order derivative of $\phi_A$ in terms of $\phi_A$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$ and $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$.
First kind of symmetry operator for the Dirac-Weyl equation
-----------------------------------------------------------
The general second order differential operator, mapping a Dirac-Weyl field $\phi_{A}$ to $\mathcal{S}_{1,0}$ is equivalent to $\phi_{A}\rightarrow \chi_{A}$, where $$\begin{aligned}
\chi_{A}={}&N_{A}{}^{B} \phi_{B}
+ M_{A}{}^{BCA'} (\sTwist_{1,0} \phi)_{BCA'}
+ L_{A}{}^{BCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'},\label{eq:chiDiracDef}\\
\intertext{and}
L_{ABCD}{}^{A'B'}={}&L_{A(BCD)}{}^{(A'B')},\qquad
M_{ABC}{}^{A'}={}M_{A(BC)}{}^{A'}.\label{eq:SymLMDirac1}\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed in Section \[sec:ReductionDirac\]. The symmetries comes from the symmetries of $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$. To be able to make a systematic treatment of the dependence of different components of the coefficients, we will use the irreducible decompositions
$$\begin{aligned}
L_{ABCD}{}^{A'B'}={}&\underset{4,2}{L}{}_{ABCD}{}^{A'B'}
+ \tfrac{3}{4} \underset{2,2}{L}{}_{(BC}{}^{A'B'}\epsilon_{D)A},\\
M_{ABC}{}^{A'}={}&\underset{3,1}{M}{}_{ABC}{}^{A'}
+ \tfrac{2}{3} \underset{1,1}{M}{}_{(B}{}^{A'}\epsilon_{C)A},\\
N_{AB}={}&\underset{2,0}{N}{}_{AB}
- \tfrac{1}{2} \underset{0,0}{N}{} \epsilon_{AB}.\end{aligned}$$
where $$\begin{aligned}
\underset{2,2}{L}{}_{AB}{}^{A'B'}\equiv{}&L^{C}{}_{ABC}{}^{A'B'},&
\underset{1,1}{M}{}_{A}{}^{A'}\equiv{}&M^{B}{}_{AB}{}^{A'},&
\underset{0,0}{N}{}\equiv{}&N^{A}{}_{A},\\
\underset{4,2}{L}{}_{ABCD}{}^{A'B'}\equiv{}&L_{(ABCD)}{}^{A'B'},&
\underset{3,1}{M}{}_{ABC}{}^{A'}\equiv{}&M_{(ABC)}{}^{A'},&
\underset{2,0}{N}{}_{AB}\equiv{}&N_{(AB)}.\end{aligned}$$ We use the convention that a spinor with underscripts $\underset{k,l}{T}{}$ is a totally symmetric valence $(k,l)$ spinor. Using these spinors, we can rewrite as $$\begin{aligned}
\chi_{A}={}&- \tfrac{1}{2} \underset{0,0}{N}{} \phi_{A}
- \underset{2,0}{N}{}_{AB} \phi^{B}
- \tfrac{2}{3} \underset{1,1}{M}{}^{BA'} (\sTwist_{1,0} \phi)_{ABA'}
- \underset{3,1}{M}{}_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\
& - \tfrac{3}{4} \underset{2,2}{L}{}^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}
- \underset{4,2}{L}{}_{ABCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDA'B'}.\end{aligned}$$
The condition for the operator $\phi_{A}\rightarrow \chi_{A}$ to be a symmetry operator is $$(\sCurlDagger_{1,0} \chi)_{A'}=0.\label{eq:chiDirac}$$ The definition of the $\sCurlDagger$ operator, the Leibniz rule for the covariant derivative, and the irreducible decomposition allows us to write this equation in terms of the fundamental operators acting on the coefficients and the field. Furthermore, using the results from the previous subsection, we see that this equation can be reduced to a linear combination of the spinors $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $\phi_A$. For a general Dirac-Weyl field and an arbitrary point on the manifold, there are no relations between these spinors. Hence, they are independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the field to the $\sTwist$ operator, we can therefore study the different order derivatives in separately. We begin with the highest order, and work our way down to order zero.
### Third order part
The third order derivative term of is $$\begin{aligned}
\label{eq:thirdorderpartdirac}
0={}&- \underset{4,2}{L}{}^{ABCDB'C'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ We will now use the argument from Section \[sec:indepspinors\] to derive equations for the coefficients in a systematic way. To get rid of the free index in equation we multiply with an arbitrary spinor field $T^{A'}$ to get $$\begin{aligned}
0={}&- \underset{4,2}{L}{}^{ABCDB'C'}T^{A'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ From the argument in Section \[sec:indepspinors\] and the observation that $T^{A'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}$ is irreducible we conclude that $$\begin{aligned}
\underset{4,2}{L}{}_{ABCD}{}^{A'B'}={}&0.\end{aligned}$$
### Second order part
The second order derivative terms of can now be reduced to $$\begin{aligned}
0={}&- \underset{3,1}{M}{}^{ABCB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}
+ \tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})^{ABCB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}\nonumber\\
& + \tfrac{3}{4} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{ABCB'C'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A'}$, but here $(\sTwist_{2,1} \sTwist_{1,0} \phi)^{ABCB'C'}T^{A'}$ is not irreducible. Therefore, we decompose it into irreducible parts and get $$\begin{aligned}
0={}&\tfrac{3}{4} T^{(A'}(\sTwist_{2,1} \sTwist_{1,0} \phi)^{|ABC|B'C')} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'}
\nonumber\\
& + \bigl(\tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})^{ABCB'}- \underset{3,1}{M}{}^{ABCB'} \bigr)
T^{A'}(\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}.\end{aligned}$$ The argument in Section \[sec:indepspinors\] tells that the coefficients of the different irreducible parts have to vanish individually which gives
$$\begin{aligned}
(\sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'B'C'}={}&0,\label{eq:TwistL22Dirac1}\\
\underset{3,1}{M}{}_{ABC}{}^{A'}={}&\tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}.\end{aligned}$$
### First order part
The first order derivative terms of are $$\begin{aligned}
0={}&- \underset{2,0}{N}{}^{AB} (\sTwist_{1,0} \phi)_{ABA'}
+ \tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})^{AB} (\sTwist_{1,0} \phi)_{ABA'}
- \tfrac{1}{2} (\sDiv_{3,1} \underset{3,1}{M}{})^{AB} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\
& - \tfrac{2}{3} \underset{2,2}{L}{}_{A}{}^{CB'C'} \Phi_{BCB'C'} (\sTwist_{1,0} \phi)^{AB}{}_{A'}
- 6 \Lambda \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\
& + \tfrac{4}{3} \underset{2,2}{L}{}_{A}{}^{C}{}_{B'}{}^{C'} \Phi_{BCA'C'} (\sTwist_{1,0} \phi)^{ABB'}
+ \tfrac{5}{3} \underset{2,2}{L}{}_{A}{}^{C}{}_{A'}{}^{C'} \Phi_{BCB'C'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\
& + \tfrac{3}{4} \underset{2,2}{L}{}^{CD}{}_{A'B'} \Psi_{ABCD} (\sTwist_{1,0} \phi)^{ABB'}
+ \tfrac{3}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi_{A'B'C'D'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\
& - (\sCurlDagger_{3,1} \underset{3,1}{M}{})_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'}
+ \tfrac{2}{3} (\sTwist_{1,1} \underset{1,1}{M}{})_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A'}$ and decompose $(\sTwist_{1,0} \phi)^{ABB'}T^{A'}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives
$$\begin{aligned}
0={}&- \underset{2,0}{N}{}_{AB}
+ \tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})_{AB}
- \tfrac{1}{4} (\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}
- \tfrac{1}{2} \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{B)CA'B'},\\
0={}&-6 \Lambda \underset{2,2}{L}{}_{AB}{}^{A'B'}
+ \tfrac{3}{4} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD}
+ \tfrac{3}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}
- \tfrac{1}{2} (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}\nonumber\\
& + 3 \underset{2,2}{L}{}_{(A}{}^{C(A'|C'|}\Phi_{B)C}{}^{B')}{}_{C'}
+ \tfrac{2}{3} (\sTwist_{1,1} \underset{1,1}{M}{})_{AB}{}^{A'B'}.\end{aligned}$$
Using the commutators and together with , this reduces to
$$\begin{aligned}
\underset{2,0}{N}{}_{AB}={}&\tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})_{AB}
- \tfrac{1}{6} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB},\\
(\sTwist_{1,1} \underset{1,1}{M}{})_{AB}{}^{A'B'}={}&- \tfrac{3}{8} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD}
+ \tfrac{3}{8} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}
+ (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}.\label{eq:TwistM11Dirac1a}\end{aligned}$$
Isolating the $\sTwist$ terms in leads us to make the ansatz $$\begin{aligned}
\underset{1,1}{M}{}_{A}{}^{A'}={}&- \tfrac{3}{2} P_{A}{}^{A'}
+ (\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'},\label{eq:M11AnsatzDirac1}\end{aligned}$$ where $P_{A}{}^{A'}$ is undetermined. With this ansatz, the first order equations reduce to
$$\begin{aligned}
(\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&\tfrac{1}{4} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD}
- \tfrac{1}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\
={}&\tfrac{1}{4} (\ObstrOne \underset{2,2}{L}{})_{AB}{}^{A'B'},\label{eq:TwistP11Dirac1b}\\
\underset{2,0}{N}{}_{AB}={}&- \tfrac{1}{2} (\sCurl_{1,1} P)_{AB}
+ \tfrac{1}{6} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}\label{eq:N20Dirac1b}.\end{aligned}$$
### Zeroth order part {#sec:zerothorderDirac1}
Using the equations above, the zeroth order derivative terms of are $$\begin{aligned}
0={}&\phi^{A} (-2 \Lambda \underset{1,1}{M}{}_{AA'}
+ \tfrac{2}{3} \underset{1,1}{M}{}^{BB'} \Phi_{ABA'B'}
+ \tfrac{1}{4} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'}
- \tfrac{5}{12} \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'}\nonumber\\
& - (\sCurlDagger_{2,0} \underset{2,0}{N}{})_{AA'}
- \tfrac{1}{6} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'}
- \tfrac{8}{3} \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{0,0} \Lambda)^{BB'}
+ \tfrac{1}{2} (\sTwist_{0,0} \underset{0,0}{N}{})_{AA'}\nonumber\\
& + \tfrac{1}{2} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'}).\end{aligned}$$ Here, there is no reason to multiply with an arbitrary $T^{A'}$ and do an irreducible decomposition of $T^{A'}\phi^A$ because $T^{A'}\phi^A$ is already irreducible. Still the argument in Section \[sec:indepspinors\] gives that the coefficient of $\phi^A$ will have to vanish. With the substitutions and , the vanishing of this coefficient is equivalent to $$\begin{aligned}
(\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&-6 \Lambda P_{A}{}^{A'}
+ 2 \Phi_{AB}{}^{A'}{}_{B'} P^{BB'}
- \tfrac{1}{2} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDA'}\nonumber\\
& + \tfrac{5}{6} \underset{2,2}{L}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'}
+ \tfrac{1}{3} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}
- (\sCurlDagger_{2,0} \sCurl_{1,1} P)_{A}{}^{A'}\nonumber\\
& + \tfrac{1}{3} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}
+ 4 \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}
- \tfrac{4}{3} \Phi_{AB}{}^{A'}{}_{B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}\nonumber\\
& + \tfrac{16}{3} \underset{2,2}{L}{}_{AB}{}^{A'}{}_{B'} (\sTwist_{0,0} \Lambda)^{BB'}
- \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'}{}_{B'C'}.\label{eq:TwistN00Dirac1a}\end{aligned}$$ To simplify the $\sCurlDagger\sCurl\sDiv$ term, we first commute the innermost operators with . Then the outermost operators are commuted with . After that, we are left with the operator $\sDiv\sCurlDagger\sCurl$, which can be turned into $\sDiv\sTwist\sDiv$ by using and . Finally, the $\sDiv\sTwist\sDiv$ operator can be turned into $\sCurlDagger\sCurl\sDiv$ and $\sTwist\sDiv\sDiv$, again by using , but this time on the outermost operators. In detail $$\begin{aligned}
(\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}={}&- \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'}
+ \tfrac{3}{2} (\sCurlDagger_{2,0} \sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AA'}\\*
={}&3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'}
- \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'}
+ 3 (\sDiv_{2,2} \sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AA'}\\
={}&3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'}
- \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'}\nonumber\\*
& - 6 \nabla^{BC'}\square_{(A'}{}^{B'}\underset{2,2}{L}{}_{|AB|C')B'}
- 3 \nabla^{CB'}\square_{(A}{}^{B}\underset{2,2}{L}{}_{C)BA'B'}\nonumber\\*
& + 4 (\sDiv_{2,2} \sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}\\
={}&2 \square_{AB}(\sDiv_{2,2} \underset{2,2}{L}{})^{B}{}_{A'}
+ 6 \square_{A'B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{B'}
+ 3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'}\nonumber\\*
& - \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'}
- 6 \nabla^{BC'}\square_{(A'}{}^{B'}\underset{2,2}{L}{}_{|AB|C')B'}\nonumber\\*
& - 3 \nabla^{CB'}\square_{(A}{}^{B}\underset{2,2}{L}{}_{C)BA'B'}
- 4 (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}
+ 3 (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}.\end{aligned}$$ Isolating the $\sCurlDagger\sCurl\sDiv$ terms, expanding the commutators and using yield $$\begin{aligned}
(\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}={}&- \tfrac{8}{5} \Phi^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'}
+ \tfrac{6}{5} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'}\nonumber\\
& - 2 \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'}
+ \tfrac{6}{5} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{A}{}^{B'C'D'}\nonumber\\
& - \tfrac{8}{5} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{BA'B'C'}
- 2 \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'}\nonumber\\
& - 12 \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{AA'}
+ \tfrac{44}{15} \Phi_{ABA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}
- \tfrac{64}{5} \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{0,0} \Lambda)^{BB'}\nonumber\\
& + \tfrac{3}{5} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'}
+ \tfrac{3}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}.\label{eq:CurlDaggerCurlDivLL22Dirac}\end{aligned}$$ Using this in , and using combined with gives $$\begin{aligned}
(\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&- \tfrac{8}{15} \Phi^{BCA'B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'}
+ \tfrac{3}{20} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDA'}\nonumber\\
& - \tfrac{1}{12} \underset{2,2}{L}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'}
+ \tfrac{3}{20} \bar\Psi^{A'}{}_{B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{A}{}^{B'C'D'}\nonumber\\
& - \tfrac{8}{15} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{A'}{}_{B'C'}
- \tfrac{1}{12} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}\nonumber\\
& - \tfrac{16}{45} \Phi_{AB}{}^{A'}{}_{B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}
+ \tfrac{16}{15} \underset{2,2}{L}{}_{AB}{}^{A'}{}_{B'} (\sTwist_{0,0} \Lambda)^{BB'}\nonumber\\
& - \tfrac{4}{5} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'}{}_{B'C'}
- \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{A}{}^{A'}
+ \tfrac{1}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}.\label{eq:TwistN00Dirac1c}\end{aligned}$$ To simplify the remaining terms, we define $$\Upsilon\equiv\underset{2,2}{L}{}_{ABA'B'} \Phi^{ABA'B'}.\label{eq:UpsilonDef}$$ Using the gradient of $\Upsilon$ reduces to $$\begin{aligned}
(\sTwist_{0,0} \Upsilon)_{AA'}={}&- \tfrac{4}{3} \underset{2,2}{L}{}_{A}{}^{B}{}_{A'}{}^{B'} (\sTwist_{0,0}\Lambda)_{BB'}
+ \tfrac{2}{3} \Phi^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'}\nonumber\\
& + \tfrac{2}{3} \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'}
+ \tfrac{2}{3} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{BA'B'C'}\nonumber\\
& + \tfrac{2}{3} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'}
+ \tfrac{4}{9} \Phi_{ABA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}\nonumber\\
& + \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'}.\label{eq:GradUpsilon}\end{aligned}$$ This can be used to eliminate most of the terms in . Together with the definition of the operator $\ObstrZero$, we find that reduces to $$\begin{aligned}
(\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&\tfrac{9}{20} (\ObstrZero \underset{2,2}{L}{})_{A}{}^{A'}
- \tfrac{4}{5} (\sTwist_{0,0} \Upsilon)_{A}{}^{A'}
- \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{A}{}^{A'}
+ \tfrac{1}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}.\end{aligned}$$ It is now clear that the ansatz $$\begin{aligned}
\underset{0,0}{N}{}={}&-2 Q
- \tfrac{4}{5} \Upsilon
- \tfrac{3}{4} (\sDiv_{1,1} P)
+ \tfrac{1}{5} (\sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{}),\end{aligned}$$ with $Q$ undetermined gives $$\begin{aligned}
(\sTwist_{0,0} Q)_{A}{}^{A'}={}&- \tfrac{9}{40} (\ObstrZero \underset{2,2}{L}{})_{A}{}^{A'}.\label{eq:TwistQ00Dirac1}\end{aligned}$$
We can now conclude that the only restrictive equations are , and . The other equations give expressions for the remaining coefficients in terms of $\underset{2,2}{L}{}_{AB}{}^{A'B'}$, $P_{AA'}$, and $Q$. For convenience we make the replacement $\underset{2,2}{L}{}_{AB}{}^{A'B'} \rightarrow - \tfrac{4}{3} L_{AB}{}^{A'B'}$.
Second kind of symmetry operator for the Dirac-Weyl equation
------------------------------------------------------------
The general second order differential operator, mapping a Dirac-Weyl field $\phi_{A}$ to $\mathcal{S}_{0,1}$ is equivalent to $\phi_{A}\rightarrow \omega_{A'}$, where $$\begin{aligned}
\omega_{A'}={}&N^{B}{}_{A'} \phi_{B}
+ M_{A'}{}^{BCB'} (\sTwist_{1,0} \phi)_{BCB'}
+ L_{A'}{}^{BCDB'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDB'C'},\end{aligned}$$ where $$\begin{aligned}
L_{A'ABCB'C'}={}&L_{A'(ABC)(B'C')},&
M_{A'ABB'}={}&M_{A'(AB)B'}.\label{eq:SymLMDirac2}\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed above. The symmetries comes from the symmetries of $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$. As we did above, we will decompose the coefficients into irreducible parts to more clearly see which parts are independent. The irreducible decompositions of $L^{A'}{}_{ABC}{}^{B'C'}$ and $M^{A'}{}_{AB}{}^{B'}$ are $$\begin{aligned}
L^{A'}{}_{ABC}{}^{B'C'}={}&\underset{3,3}{L}{}_{ABC}{}^{A'B'C'}
+ \tfrac{2}{3} \underset{3,1}{L}{}_{ABC}{}^{(B'}\bar\epsilon^{C')A'},\\
M^{A'}{}_{AB}{}^{B'}={}&\underset{2,2}{M}{}_{AB}{}^{A'B'}
- \tfrac{1}{2} \underset{2,0}{M}{}_{AB} \bar\epsilon^{A'B'},\end{aligned}$$ where $$\begin{aligned}
\underset{3,1}{L}{}_{ABC}{}^{A'}\equiv{}&L^{B'}{}_{ABC}{}^{A'}{}_{B'},&
\underset{2,0}{M}{}_{AB}\equiv{}&M^{A'}{}_{ABA'},\\
\underset{3,3}{L}{}_{ABC}{}^{A'B'C'}\equiv{}&L^{(A'}{}_{ABC}{}^{B'C')},&
\underset{2,2}{M}{}_{AB}{}^{A'B'}\equiv{}&M^{(A'}{}_{AB}{}^{B')}.\end{aligned}$$ With these irreducible decompositions, we get $$\begin{aligned}
\omega_{A'}={}&N^{B}{}_{A'} \phi_{B}
- \tfrac{1}{2} \underset{2,0}{M}{}^{BC} (\sTwist_{1,0} \phi)_{BCA'}
- \underset{2,2}{M}{}_{BCA'B'} (\sTwist_{1,0} \phi)^{BCB'}
- \tfrac{2}{3} \underset{3,1}{L}{}^{BCDB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'}\nonumber\\
& - \underset{3,3}{L}{}_{BCDA'B'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDB'C'}.\end{aligned}$$
The condition for the operator $\phi_{A}\rightarrow \omega_{A'}$ to be a symmetry operator is $$(\sCurl_{0,1} \omega)_{A}=0.\label{eq:omegaDirac}$$ Using the results from Section \[sec:ReductionDirac\], we see that this equation can be reduced to a linear combination of the spinors $\phi_A$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$ and $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$. As above, we can treat these as independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the field to the $\sTwist$ operator, we can therefore study the different order derivatives in separately. We begin with the highest order, and work our way down to order zero.
### Third order part
The third order part of is $$\begin{aligned}
0={}&- \underset{3,3}{L}{}^{BCDA'B'C'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ Using the argument from Section \[sec:indepspinors\], we see that this implies $$\begin{aligned}
\underset{3,3}{L}{}_{ABC}{}^{A'B'C'}={}&0.\end{aligned}$$
### Second order part
The second order part of now takes the form $$\begin{aligned}
0={}&- \underset{2,2}{M}{}^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}
+ \tfrac{1}{2} (\sCurlDagger_{3,1} \underset{3,1}{L}{})^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}\nonumber\\
& + \tfrac{2}{3} (\sTwist_{3,1} \underset{3,1}{L}{})_{ABCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDA'B'}.\end{aligned}$$ Here we multiply with an arbitrary spinor field $T^{A}$ and decompose $(\sTwist_{2,1}\sTwist_{1,0} \phi)^{BCDC'D'}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives
$$\begin{aligned}
(\sTwist_{3,1} \underset{3,1}{L}{})_{ABCD}{}^{A'B'}={}&0,\label{eq:TwistLL31}\\
\underset{2,2}{M}{}_{AB}{}^{A'B'}={}&\tfrac{1}{2} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'}\label{eq:M22Eq1}.\end{aligned}$$
### First order part
The first order part of can now be reduced to $$\begin{aligned}
0={}&- N^{BA'} (\sTwist_{1,0} \phi)_{ABA'}
+ \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})^{BA'} (\sTwist_{1,0} \phi)_{ABA'}
- \tfrac{2}{3} (\sDiv_{2,2} \underset{2,2}{M}{})^{BA'} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\
& + \tfrac{1}{3} \underset{3,1}{L}{}_{BCDB'} \Phi^{CD}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)_{A}{}^{BA'}
- \tfrac{5}{12} \underset{3,1}{L}{}^{CDF}{}_{A'} \Psi_{BCDF} (\sTwist_{1,0} \phi)_{A}{}^{BA'}\nonumber\\
& - 6 \Lambda \underset{3,1}{L}{}_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}
+ \tfrac{1}{3} \underset{3,1}{L}{}_{BCDB'} \Phi_{A}{}^{D}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\
& + \tfrac{5}{3} \underset{3,1}{L}{}_{ACDB'} \Phi_{B}{}^{D}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)^{BCA'}
+ \tfrac{5}{4} \underset{3,1}{L}{}_{B}{}^{DF}{}_{A'} \Psi_{ACDF} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\
& + \tfrac{3}{4} \underset{3,1}{L}{}_{A}{}^{DF}{}_{A'} \Psi_{BCDF} (\sTwist_{1,0} \phi)^{BCA'}
- (\sCurl_{2,2} \underset{2,2}{M}{})_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\
& + \tfrac{1}{2} (\sTwist_{2,0} \underset{2,0}{M}{})_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A}$ and decompose $(\sTwist_{1,0} \phi)^{BCC'}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned}
0={}&- N_{A}{}^{A'}
- \tfrac{1}{3} \underset{3,1}{L}{}^{BCDA'} \Psi_{ABCD}
+ \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})_{A}{}^{A'}
- \tfrac{2}{3} (\sDiv_{2,2} \underset{2,2}{M}{})_{A}{}^{A'},\\
0={}&-6 \Lambda \underset{3,1}{L}{}_{ABC}{}^{A'}
- (\sCurl_{2,2} \underset{2,2}{M}{})_{ABC}{}^{A'}
+ 2 \underset{3,1}{L}{}_{(BC|DB'|}\Phi_{A)}{}^{DA'B'}
+ 2 \underset{3,1}{L}{}_{(A}{}^{DFA'}\Psi_{BC)DF}\nonumber\\
& + \tfrac{1}{2} (\sTwist_{2,0} \underset{2,0}{M}{})_{ABC}{}^{A'}.\end{aligned}$$ By , the commutator and these reduce to
$$\begin{aligned}
N_{A}{}^{A'}={}&- \tfrac{1}{3} \underset{3,1}{L}{}_{ABCB'} \Phi^{BCA'B'}
+ \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})_{A}{}^{A'}
- \tfrac{1}{6} (\sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{A}{}^{A'},\\
(\sTwist_{2,0} \underset{2,0}{M}{})_{ABC}{}^{A'}={}&(\sTwist_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{ABC}{}^{A'}.\end{aligned}$$
If we make the ansatz $$\begin{aligned}
\underset{2,0}{M}{}_{AB}={}&-2 P_{AB}
+ (\sDiv_{3,1} \underset{3,1}{L}{})_{AB},\end{aligned}$$ these equations reduce to
$$\begin{aligned}
N_{A}{}^{A'}={}&- \tfrac{1}{3} \underset{3,1}{L}{}_{ABCB'} \Phi^{BCA'B'}
- \tfrac{2}{3} (\sCurlDagger_{2,0} P)_{A}{}^{A'}
+ \tfrac{1}{6} (\sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{A}{}^{A'},\label{eq:N11Dirac2a}\\
(\sTwist_{2,0} P)_{ABC}{}^{A'}={}&0.\label{eq:TwistPDirac2a}\end{aligned}$$
### Zeroth order part {#zeroth-order-part}
The zeroth order part of can now be reduced to $$\begin{aligned}
0={}&-2 \Lambda \underset{2,0}{M}{}_{AB} \phi^{B}
+ \tfrac{1}{2} \underset{2,0}{M}{}^{CD} \Psi_{ABCD} \phi^{B}
- \phi^{B} (\sCurl_{1,1} N)_{AB}
- \tfrac{1}{20} \underset{3,1}{L}{}_{B}{}^{CDA'} \phi^{B} (\sCurl_{2,2} \Phi)_{ACDA'}\nonumber\\
& + \tfrac{1}{60} \underset{3,1}{L}{}^{BCDA'} \phi_{A} (\sCurl_{2,2} \Phi)_{BCDA'}
- \tfrac{5}{12} \underset{3,1}{L}{}_{A}{}^{CDA'} \phi^{B} (\sCurl_{2,2} \Phi)_{BCDA'}\nonumber\\
& - \tfrac{1}{3} \Phi_{B}{}^{CA'B'} \phi^{B} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ACA'B'}
- \tfrac{1}{2} \phi_{A} (\sDiv_{1,1} N)
- \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} \phi^{B} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\
& + \tfrac{1}{3} \underset{3,1}{L}{}^{CDFA'} \phi^{B} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A}$ and decompose $\phi^{B}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives
$$\begin{aligned}
0={}&- \tfrac{1}{6} \underset{3,1}{L}{}^{ABCA'} (\sCurl_{2,2} \Phi)_{ABCA'}
+ \tfrac{1}{6} \Phi^{ABA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ABA'B'}
- \tfrac{1}{2} (\sDiv_{1,1} N),\label{eq:ZerithOrderDirac2a}\\
0={}&-2 \Lambda \underset{2,0}{M}{}_{AB}
+ \tfrac{1}{2} \underset{2,0}{M}{}^{CD} \Psi_{ABCD}
- (\sCurl_{1,1} N)_{AB}
- \tfrac{7}{15} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\
& - \tfrac{1}{3} \Phi_{(A}{}^{CA'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{B)CA'B'}
- \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'}
+ \tfrac{1}{3} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:ZerithOrderDirac2b}\end{aligned}$$
The equation together with the commutator gives . If we substitute in , we get a term with the third order operator $\sCurl\sCurlDagger\sDiv$. To handle this we use the same technique as in Section \[sec:zerothorderDirac1\]. We first commute the innermost operators with . Then the outermost operators are commuted with . After that, we are left with the operator $\sDiv\sCurl\sCurlDagger$, which can be turned into $\sDiv\sTwist\sDiv$ by using and . Finally, the $\sDiv\sTwist\sDiv$ operator can be turned into $\sCurl\sCurlDagger\sDiv$ and $\sTwist\sDiv\sDiv$, again by using , but this time on the outermost operators. $$\begin{aligned}
(\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}={}&2 (\sCurl_{1,1} \sDiv_{2,2} \sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}
+ 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\
={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'}
+ 3 (\sDiv_{3,1} \sCurl_{2,2} \sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}
+ 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\
={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'}
+ 2 \nabla_{CB'}\square_{A'}{}^{B'}\underset{3,1}{L}{}_{AB}{}^{CA'}
- 6 \nabla^{DA'}\square_{(A}{}^{C}\underset{3,1}{L}{}_{BD)CA'}\nonumber\\
& + 3 (\sDiv_{3,1} \sTwist_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}
+ 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\
={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'}
+ 2 \nabla_{CB'}\square_{A'}{}^{B'}\underset{3,1}{L}{}_{AB}{}^{CA'}
- 6 \nabla^{DA'}\square_{(A}{}^{C}\underset{3,1}{L}{}_{BD)CA'}\nonumber\\
& - 4 (\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}
- 6 \square_{(A}{}^{C}(\sDiv_{3,1} \underset{3,1}{L}{})_{B)C}
+ 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}.\end{aligned}$$ Isolating the $\sCurl\sCurlDagger\sDiv$ terms and expanding the commutators and using yield $$\begin{aligned}
(\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}={}&- \Psi_{B}{}^{CDF} (\sCurl_{3,1} \underset{3,1}{L}{})_{ACDF}
- \Psi_{A}{}^{CDF} (\sCurl_{3,1} \underset{3,1}{L}{})_{BCDF}\nonumber\\
& - \tfrac{42}{25} \underset{3,1}{L}{}_{B}{}^{CDA'} (\sCurl_{2,2} \Phi)_{ACDA'}
- \tfrac{42}{25} \underset{3,1}{L}{}_{A}{}^{CDA'} (\sCurl_{2,2} \Phi)_{BCDA'}\nonumber\\
& - \tfrac{3}{2} \Phi_{B}{}^{CA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ACA'B'}
- \tfrac{3}{2} \Phi_{A}{}^{CA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{BCA'B'}\nonumber\\
& - 12 \Lambda (\sDiv_{3,1} \underset{3,1}{L}{})_{AB}
+ \tfrac{21}{10} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD}
- 12 \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\
& + \tfrac{2}{5} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ The equation together with the equation above, the commutator and gives $$\begin{aligned}
(\sCurl_{1,1} N)_{AB}={}&4 \Lambda P_{AB}
- \Psi_{ABCD} P^{CD}
- 2 \Lambda (\sDiv_{3,1} \underset{3,1}{L}{})_{AB}
+ \tfrac{7}{20} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD}\nonumber\\
& - \tfrac{17}{75} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}
- \tfrac{1}{3} \Phi_{(A}{}^{CA'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{B)CA'B'}\nonumber\\
& - \tfrac{1}{3} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} \underset{3,1}{L}{})_{B)CDF}
- \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\
& + \tfrac{1}{15} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ Due to this, the equation reduces to the auxiliary condition $$\begin{aligned}
0={}&\tfrac{3}{4} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD}
- \tfrac{6}{5} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\
& + \tfrac{5}{3} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} \underset{3,1}{L}{})_{B)CDF}
+ \tfrac{4}{3} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:auxcondDirac2}\end{aligned}$$ We can conclude that the only restrictive equations are , and . The other equations express the remaining coefficients in terms of $\underset{3,1}{L}{}_{ABCA'}$ and $P_{AB}$. For convenience we make the replacement $\underset{3,1}{L}{}_{ABC}{}^{A'} \rightarrow - \tfrac{3}{2} L_{ABC}{}^{A'}$.
The Maxwell equation {#sec:spin1}
====================
\[Thm::SymOpFirstKind\] There exists a symmetry operator of the first kind $\phi_{AB}\rightarrow \chi_{AB}$, with order less or equal to two, if and only if there are spinor fields $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that
$$\begin{aligned}
(\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\label{eq:Lfirstkind}\\
(\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{2}{3} (\ObstrOne L)_{AB}{}^{A'B'},\label{eq:Qcondfirstkind}\\
(\sTwist_{0,0} Q)_{BA'}={}&0.\end{aligned}$$
The symmetry operator then takes the form $$\begin{aligned}
\chi_{AB}={}&Q \phi_{AB}+(\sCurl_{1,1} A)_{AB},\\
\intertext{where}
A_{AA'}={}&- P^{B}{}_{A'} \phi_{AB}
+ \tfrac{1}{3} \phi^{BC} (\sCurl_{2,2} L)_{ABCA'}
- \tfrac{4}{9} \phi_{AB} (\sDiv_{2,2} L)^{B}{}_{A'}
- L^{BC}{}_{A'}{}^{B'} (\sTwist_{2,0} \phi)_{ABCB'}.\end{aligned}$$ We also note that $$\begin{aligned}
(\sCurlDagger_{1,1} A)_{A'B'} ={}& 0.\label{eq:CurlDaggerA}\end{aligned}$$
1. [ Observe that one can add a gradient of a scalar to the potential $A_{AA'}$ without changing the symmetry operator. Hence, adding $\nabla_{AA'}(\Lambda^{BC}\phi_{BC})$ to $A_{AA'}$ with an arbitrary field $\Lambda_{AB}$ is possible.]{}
2. [ With $L_{ABA'B'}=0$, the first order operator takes the form $$\begin{aligned}
\chi_{AB}={}&
\hat{\mathcal{L}}_{P}\phi_{AB}+ Q \phi_{AB}.\end{aligned}$$ ]{}
\[Thm::SymOpSecondKind\] There exists second order a symmetry operator of the second kind $\phi_{AB}\rightarrow \omega_{A'B'}$, with order less or equal to two, if and only if there is a spinor field $L_{ABCD}=L_{(ABCD)}$ such that $$\begin{aligned}
(\sTwist_{4,0} L)_{ABCDF}{}^{A'}={}&0.\label{eq:LSecondKind}\end{aligned}$$ The symmetry operator then takes the form $$\begin{aligned}
\omega_{A'B'}={}&(\sCurlDagger_{1,1} B)_{A'B'},\\
\intertext{where}
B_{AA'}={}&\tfrac{3}{5} \phi^{BC} (\sCurlDagger_{4,0} L)_{ABCA'}
+ L_{ABCD} (\sTwist_{2,0} \phi)^{BCD}{}_{A'}.\end{aligned}$$ We also note that $$\begin{aligned}
(\sCurl_{1,1} B)_{AB}={}&0.\label{eq:CurlB}\end{aligned}$$
1. [ Observe that also here we can add a gradient of a scalar to the potential $B_{AA'}$ without changing the symmetry operator. Hence, adding $\nabla_{AA'}(\Lambda^{BC}\phi_{BC})$ to $B_{AA'}$ with an arbitrary field $\Lambda_{AB}$ is possible.]{}
2. [ Due to the equations and , we can use $A_{AA'}+B_{AA'}$ as a potential for both $\chi_{AB}$ and $\omega_{A'B'}$. ]{}
Reduction of derivatives of the field {#sec:ReductionMaxwell}
-------------------------------------
In our notation, the Maxwell equation $\nabla^{A}{}_{A'}\phi_{AB}=0$, takes the form $(\sCurlDagger_{2,0} \phi)_{A'}=0$. From this we see that the only irreducible part of $\nabla_{A}{}^{A'}\phi_{BC}$ is $(\sTwist_{2,0} \phi)_{ABC}{}^{A'}$. By commuting derivatives we see that all higher order derivatives of $\phi_{AB}$ can be reduced to totally symmetrized derivatives and lower order terms consisting of curvature times lower order symmetrized derivatives.
Together with the Maxwell equation, the commutators , , gives
$$\begin{aligned}
(\sDiv_{3,1} \sTwist_{2,0} \phi)_{AB}={}&
-8 \Lambda \phi_{AB}
+ 2 \Psi_{ABCD} \phi^{CD},\\
(\sCurl_{3,1} \sTwist_{2,0} \phi)_{ABCD}={}&2 \Psi_{(ABC}{}^{F}\phi_{D)F},\\
(\sCurlDagger_{3,1} \sTwist_{2,0} \phi)_{ABA'B'}={}&2 \Phi_{(A}{}^{C}{}_{|A'B'|}\phi_{B)C}\end{aligned}$$
The higher order derivatives can be computed using the commutators , , together with the equations above and the Bianchi system to get
$$\begin{aligned}
(\sDiv_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCA'}={}&\tfrac{9}{2} \Phi_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sTwist_{2,0} \phi)_{BC)DB'}
+ \tfrac{9}{2} \Psi_{(AB}{}^{DF}(\sTwist_{2,0} \phi)_{C)DFA'}\nonumber\\
& - \tfrac{15}{2} \phi_{(AB}(\sTwist_{0,0}\Lambda)_{C)A'}
+ \tfrac{21}{10} \phi_{(A}{}^{D}(\sCurl_{2,2} \Phi)_{BC)DA'}\nonumber\\
& + \tfrac{3}{2} \phi^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'}
- 15 \Lambda (\sTwist_{2,0} \phi)_{ABCA'},\\
(\sCurl_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDFB'}={}&\Phi_{(AB|B'|}{}^{A'}(\sTwist_{2,0} \phi)_{CDF)A'}
+ 4 \Psi_{(ABC}{}^{H}(\sTwist_{2,0} \phi)_{DF)HB'}\nonumber\\
& - \tfrac{1}{5} \phi_{(AB}(\sCurl_{2,2} \Phi)_{CDF)B'}
- \phi_{(A}{}^{H}(\sTwist_{4,0} \Psi)_{BCDF)HB'},\\
(\sCurlDagger_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABC}{}^{A'B'C'}={}&\tfrac{9}{2} \Phi^{D}{}_{(A}{}^{(A'B'}(\sTwist_{2,0} \phi)_{BC)D}{}^{C')}
- \tfrac{1}{2} \phi_{(AB}(\sCurlDagger_{2,2} \Phi)_{C)}{}^{A'B'C'}\nonumber\\
& - \tfrac{3}{2} \phi_{(A}{}^{D}(\sTwist_{2,2} \Phi)_{BC)D}{}^{A'B'C'}
- \bar\Psi^{A'B'C'}{}_{D'} (\sTwist_{2,0} \phi)_{ABC}{}^{D'}.\end{aligned}$$
These can in a systematic way be used to reduce any derivative up to third order of $\phi_{AB}$ in terms of $\phi_{AB}$, $(\sTwist_{2,0} \phi)_{ABC}{}^{A'}$, $(\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCD}{}^{A'B'}$ and $(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDF}{}^{A'B'C'}$.
First kind of symmetry operator for the Maxwell equation
--------------------------------------------------------
The general second order differential operator, mapping a Maxwell field $\phi_{AB}$ to $\mathcal{S}_{2,0}$ is equivalent to $\phi_{AB}\rightarrow \chi_{AB}$, where $$\begin{aligned}
\chi_{AB}={}&N_{ABCD} \phi^{CD}
+ M_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'}
+ L_{ABCDFHA'B'} (\sTwist_{3,1} \sTwist_{2,0} \phi)^{CDFHA'B'},\end{aligned}$$ and $$\begin{aligned}
L_{AB}{}^{CDFHA'B'}={}&L_{(AB)}{}^{(CDFH)(A'B')},&
M_{AB}{}^{CDFA'}={}&M_{(AB)}{}^{(CDF)A'},&
N_{AB}{}^{CD}={}&N_{(AB)}{}^{(CD)}.\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed in Section \[sec:ReductionMaxwell\]. The symmetries comes from the symmetries of $ (\sTwist_{2,0} \phi)_{ABC}{}^{A'}$ and $(\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCD}{}^{A'B'}$. To be able to make a systematic treatment of the dependence of different components of the coefficients, we will use the irreducible decompositions
$$\begin{aligned}
L_{AB}{}^{CDFHA'B'}={}&\underset{6,2}{L}{}_{AB}{}^{CDFHA'B'}
- \tfrac{4}{3} \epsilon_{(A}{}^{(C}\underset{4,2}{L}{}^{DFH)}{}_{B)}{}^{A'B'}
- \tfrac{3}{5} \epsilon^{(C}{}_{(A}\epsilon_{B)}{}^{D}\underset{2,2}{L}{}^{FH)A'B'},\\
M_{AB}{}^{CDFA'}={}&\underset{5,1}{M}{}_{AB}{}^{CDFA'}
- \tfrac{6}{5} \epsilon_{(A}{}^{(C}\underset{3,1}{M}{}^{DF)}{}_{B)}{}^{A'}
- \tfrac{1}{2} \epsilon^{(C}{}_{(A}\epsilon_{B)}{}^{D}\underset{1,1}{M}{}^{F)A'},\\
N_{AB}{}^{CD}={}&\underset{4,0}{N}{}_{AB}{}^{CD}
- \epsilon_{(A}{}^{(C}\underset{2,0}{N}{}^{D)}{}_{B)}
- \tfrac{1}{3} \underset{0,0}{N}{} \epsilon_{(A}{}^{(C}\epsilon^{D)}{}_{B)}.\end{aligned}$$
where the different irreducible parts are $$\begin{aligned}
\underset{2,2}{L}{}_{AB}{}^{A'B'}\equiv{}&L^{CD}{}_{ABCD}{}^{A'B'},&
\underset{1,1}{M}{}_{A}{}^{A'}\equiv{}&M^{BC}{}_{ABC}{}^{A'},&
\underset{0,0}{N}{}\equiv{}&N^{AB}{}_{AB},\\
\underset{4,2}{L}{}_{ABCD}{}^{A'B'}\equiv{}&L_{(A}{}^{F}{}_{BCD)F}{}^{A'B'},&
\underset{3,1}{M}{}_{ABC}{}^{A'}\equiv{}&M_{(A}{}^{D}{}_{BC)D}{}^{A'},&
\underset{2,0}{N}{}_{AB}\equiv{}&N_{(A}{}^{C}{}_{B)C},\\
\underset{6,2}{L}{}_{ABCDFH}{}^{A'B'}\equiv{}&L_{(ABCDFH)}{}^{A'B'},&
\underset{5,1}{M}{}_{ABCDF}{}^{A'}\equiv{}&M_{(ABCDF)}{}^{A'},&
\underset{4,0}{N}{}_{ABCD}\equiv{}&N_{(ABCD)}.\end{aligned}$$
Now, we want the operator to be a symmetry operator, which means that $$(\sCurlDagger_{2,0}\chi)_{AA'}=0.\label{eq:chiMaxwell}$$ Using the results from the previous subsection, we see that this equation can be reduced to a linear combination of the spinors $(\sTwist_{2,4} \sTwist_{1,3} \sTwist_{0,2} \phi)^{A'B'C'}{}_{ABCDF}$, $(\sTwist_{1,3} \sTwist_{0,2} \phi)^{A'B'}{}_{ABCD}$, $(\sTwist_{0,2} \phi)^{A'}{}_{ABC}$ and $\phi_{AB}$. For a general Maxwell field and an arbitrary point on the manifold, there are no relations between these spinors. Hence, they are independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the Maxwell field to the $\sTwist$ operator, we can therefore study the different order derivatives of $\phi_{AB}$ in separately.
### Third order part
The third order derivative terms of are $$\begin{aligned}
\label{eq:thirdorderpartmaxwell}
0={}&\tfrac{2}{3} \underset{4,2}{L}{}_{BCDFB'C'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{A}{}^{BCDF}{}_{A'}{}^{B'C'}\nonumber\\
& + \underset{6,2}{L}{}_{ABCDFHB'C'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH}{}_{A'}{}^{B'C'}.\end{aligned}$$ We can multiply this with an arbitrary vector field $T^{AA'}$ and split $(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}T^{H}{}_{A'}$ into irreducible parts. Then we get $$\begin{aligned}
0={}&\underset{6,2}{L}{}_{ABCDFHB'C'} T^{(A|A'|}(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH)}{}_{A'}{}^{B'C'}\nonumber\\
& + \tfrac{2}{3} \underset{4,2}{L}{}_{BCDFB'C'} T_{AA'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}.\end{aligned}$$ The argument in Section \[sec:indepspinors\] gives that the symmetrized coefficients of the irreducible parts $T^{(A|A'|}(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH)}{}_{A'}{}^{B'C'}$ and $T_{AA'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}$ must vanish. This means that is equivalent to the system
\[eq:ThirdOrderSystem\] $$\begin{aligned}
\underset{6,2}{L}{}_{ABCDFHB'C'}={}&0,\label{eq:L62}\\
\underset{4,2}{L}{}_{BCDFB'C'}={}&0.\label{eq:L42}\end{aligned}$$
The only remaining irreducible component of $L_{AB}{}^{CDFHA'B'}$ is $\underset{2,2}{L}{}_{AB}{}^{A'B'}$.
### Second order part
If we use everything above we find that the second order part of reduces to $$\begin{aligned}
0={}&\tfrac{3}{5} \underset{3,1}{M}{}^{BCDB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}
- \tfrac{2}{5} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}\nonumber\\
& + \tfrac{3}{5} (\sTwist_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'}{}^{B'C'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDB'C'}
+ \underset{5,1}{M}{}_{ABCDFB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDF}{}_{A'}{}^{B'}.\end{aligned}$$ Again contracting with an arbitrary vector $T^{AA'}$ and splitting $(\sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDA'B'}T^{FC'}$ into irreducible parts we find $$\begin{aligned}
0={}&\underset{5,1}{M}{}_{ABCDFB'} T^{(A|A'|}(\sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDF)}{}_{A'}{}^{B'}\nonumber\\
& - \tfrac{3}{5} T^{A(A'}(\sTwist_{3,1} \sTwist_{2,0} \phi)_{A}{}^{|BCD|B'C')} (\sTwist_{2,2} \underset{2,2}{L}{})_{BCDA'B'C'}\nonumber\\
& + T_{AA'} (\tfrac{3}{5} \underset{3,1}{M}{}_{BCDB'}
- \tfrac{2}{5} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}) (\sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDA'B'}.\end{aligned}$$ Again using the argument in Section \[sec:indepspinors\] we find
\[eq:SecondOrderSystem\] $$\begin{aligned}
(\sTwist_{2,2} \underset{2,2}{L}{})_{BCD}{}^{A'B'C'}={}&0,\label{eq:TwistL22}\\
\underset{5,1}{M}{}_{ABCDFB'}={}&0,\label{eq:M51}\\
\underset{3,1}{M}{}_{BCDB'}={}&\tfrac{2}{3} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}.\label{eq:M31ToL22}\end{aligned}$$
### First order part
Now, after contracting the first order part of with an arbitrary tensor $T^{A}{}_{A'}$, splitting $(\sTwist_{2,0} \phi)_{ABCA'}T_{DB'}$ into irreducible parts, and using the argument in Section \[sec:indepspinors\], we find that the first order part of is equivalent to the system
$$\begin{aligned}
\underset{2,0}{N}{}_{BC}={}&\tfrac{1}{2} (\sCurl_{1,1} \underset{1,1}{M}{})_{BC}
- \tfrac{1}{2} (\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{BC}
- \tfrac{3}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq1}\\
\underset{4,0}{N}{}_{ABCD}={}&\tfrac{1}{5} (\sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD}
+ \tfrac{3}{10} \underset{2,2}{L}{}_{(AB}{}^{B'C'}\Phi_{CD)B'C'},\label{eq:N40eq1}\\
(\sTwist_{1,1} \underset{1,1}{M}{})_{BC}{}^{A'B'}={}&12 \Lambda \underset{2,2}{L}{}_{BC}{}^{A'B'}
- \tfrac{9}{5} \underset{2,2}{L}{}^{DFA'B'} \Psi_{BCDF}
- \tfrac{6}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\
& + (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{BC}{}^{A'B'}
- 6 \underset{2,2}{L}{}^{D}{}_{(B}{}^{C'(A'}\Phi_{C)D}{}^{B')}{}_{C'},\label{eq:TwistM11Eq1}\\
(\sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'}={}&3 \underset{2,2}{L}{}_{(AB}{}^{C'(A'}\Phi_{CD)}{}^{B')}{}_{C'}
+ 3 \underset{2,2}{L}{}_{(A}{}^{FA'B'}\Psi_{BCD)F}.\label{eq:TwistCurlFirstOrder}\end{aligned}$$
The commutators , and applied to $\underset{2,2}{L}{}{}_{AB}{}^{A'B'}$ yield
$$\begin{aligned}
(\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}={}&\tfrac{2}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}
- 2 \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{B)CA'B'},\label{eq:CommDivCurlL22}\\
(\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}={}&-12 \Lambda \underset{2,2}{L}{}_{AB}{}^{A'B'}
+ \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD}
+ 2 \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\
& - \tfrac{3}{2} (\sDiv_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}
+ 6 \underset{2,2}{L}{}^{C}{}_{(A}{}^{C'(A'}\Phi_{B)C}{}^{B')}{}_{C'}\nonumber\\
& + \tfrac{4}{3} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'},\label{eq:CommCurlDaggerCurlL22}\\
(\sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'}={}&\tfrac{3}{2} (\sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'}
+ 3 \underset{2,2}{L}{}_{(AB}{}^{C'(A'}\Phi_{CD)}{}^{B')}{}_{C'}\nonumber\\
& + 3 \underset{2,2}{L}{}_{(A}{}^{FA'B'}\Psi_{BCD)F}.\label{eq:CommTwistCurlL22}\end{aligned}$$
It is now clear that is a consequence of and . The commutators and together with can be used to reduce and to
$$\begin{aligned}
\underset{2,0}{N}{}_{BC}={}&\tfrac{1}{2} (\sCurl_{1,1} \underset{1,1}{M}{})_{BC}
- \tfrac{1}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BC}
- \tfrac{1}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq2}\\
(\sTwist_{1,1} \underset{1,1}{M}{})_{BCA'B'}={}&- \tfrac{4}{5} \underset{2,2}{L}{}^{DF}{}_{A'B'} \Psi_{BCDF}
+ \tfrac{4}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi_{A'B'C'D'}
+ \tfrac{4}{3} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BCA'B'}.\label{eq:TwistM11Eq2}\end{aligned}$$
Now, in view of the form of we make the ansatz $$\begin{aligned}
\underset{1,1}{M}{}_{AA'}={}&2 P_{AA'}
+ \tfrac{4}{3} (\sDiv_{2,2} \underset{2,2}{L}{})_{AA'},\label{eq:M11Ansatz}\end{aligned}$$ where $P_{AA'}$ is a new spinor field. With this choice and reduce to
$$\begin{aligned}
\underset{2,0}{N}{}_{BC}={}&(\sCurl_{1,1} P)_{BC}
+ \tfrac{1}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BC}
- \tfrac{1}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq3}\\
(\sTwist_{1,1} P)_{BCA'B'}={}&- \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'B'} \Psi_{BCDF}
+ \tfrac{2}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi_{A'B'C'D'}.\label{eq:TwistPeq1}\end{aligned}$$
In conclusion, the third, second and first order parts of vanishes if and only if , , , , and are satisfied.
### Zeroth order part {#zeroth-order-part-1}
After making irreducible decompositions of the derivatives, using and contracting the remaining part of with an arbitrary tensor $T^{AA'}$, splitting $T_{AA'} \phi_{CD}$ into irreducible parts, and using the argument in Section \[sec:indepspinors\], we find that the order zero part of is equivalent to the system
$$\begin{aligned}
0={}&4 \Lambda P_{BA'}
- \tfrac{4}{3} \Phi_{BCA'B'} P^{CB'}
+ \tfrac{2}{9} \Phi^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}
- \tfrac{8}{15} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'}\nonumber\\*
& + \tfrac{26}{45} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'}
+ \tfrac{2}{9} \Phi_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{CA'B'C'}
+ \tfrac{2}{45} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}\nonumber\\*
& + \tfrac{2}{3} (\sCurlDagger_{2,0} \sCurl_{1,1} P)_{BA'}
+ \tfrac{2}{9} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'}
+ \tfrac{8}{3} \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{BA'}
- \tfrac{20}{27} \Phi_{BCA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{CB'}\nonumber\\*
& + \tfrac{28}{9} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{0,0} \Lambda)^{CB'}
- \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'}
- \tfrac{1}{3} \underset{2,2}{L}{}^{CDB'C'} (\sTwist_{2,2} \Phi)_{BCDA'B'C'},\label{eq:OrderZeroEq1}\\
0={}&P^{D}{}_{A'} \Psi_{ABCD}
+ 2 \Lambda (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'}
+ \tfrac{1}{5} (\sCurlDagger_{4,0} \sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'}
+ \tfrac{2}{3} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'}\nonumber\\*
& - \tfrac{37}{75} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'}
+ \tfrac{7}{30} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'}
- \tfrac{5}{3} \underset{2,2}{L}{}_{(AB|A'|}{}^{B'}(\sTwist_{0,0} \Lambda)_{C)B'}\nonumber\\*
& + \tfrac{7}{10} \underset{2,2}{L}{}_{(A}{}^{DB'C'}(\sTwist_{2,2} \Phi)_{BC)DA'B'C'}
- \Phi_{(AB|A'|}{}^{B'}P_{C)B'}
- \tfrac{19}{15} \Phi_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)DB'}\nonumber\\*
& + \tfrac{1}{6} \Phi_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \underset{2,2}{L}{})_{C)A'B'C'}
- \tfrac{5}{9} \Phi_{(AB|A'|}{}^{B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{C)B'}
- \tfrac{1}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'}\nonumber\\*
& + \tfrac{3}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'}
- \tfrac{1}{2} (\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'}
- \tfrac{1}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABCA'}.\label{eq:OrderZeroEq2}\end{aligned}$$
Applying the commutators repeatedly we have in general that $$\begin{aligned}
(\sCurlDagger_{4,0} &\sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\
={}&\tfrac{5}{3} \square^{A'}{}_{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{B'}
- \tfrac{4}{3} (\sDiv_{4,2} \sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}
- \square_{(A}{}^{D}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}\nonumber\\
& + \tfrac{5}{4} (\sTwist_{2,0} \sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\
={}&\tfrac{5}{3} \square^{A'}{}_{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{B'}
- 2 (\sDiv_{4,2} \sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}
- \square_{(A}{}^{D}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}\nonumber\\
& - \nabla^{DB'}\square_{(AB}\underset{2,2}{L}{}_{C)D}{}^{A'}{}_{B'}
- \nabla^{DB'}\square_{(A|D|}\underset{2,2}{L}{}_{BC)}{}^{A'}{}_{B'}
- \tfrac{5}{4} \nabla_{(A}{}^{A'}\square^{B'C'}\underset{2,2}{L}{}_{BC)B'C'}\nonumber\\
& + \tfrac{5}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\
={}&-10 \Lambda (\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}
- 2 \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{DA'}
- 2 (\sDiv_{4,2} \sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\
& + \tfrac{25}{3} \underset{2,2}{L}{}_{(AB}{}^{A'B'}(\sTwist_{0,0}\Lambda)_{C)B'}
+ \tfrac{49}{15} \underset{2,2}{L}{}_{(A}{}^{DA'B'}(\sCurl_{2,2} \Phi)_{BC)DB'}
+ \tfrac{5}{6} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)}{}^{A'}{}_{B'C'}\nonumber\\
& - \tfrac{7}{2} \underset{2,2}{L}{}_{(A}{}^{DB'C'}(\sTwist_{2,2} \Phi)_{BC)D}{}^{A'}{}_{B'C'}
+ \tfrac{19}{3} \Phi_{(A}{}^{DA'B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)DB'}\nonumber\\
& - \tfrac{5}{6} \Phi_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \underset{2,2}{L}{})_{C)}{}^{A'}{}_{B'C'}
+ \tfrac{25}{9} \Phi_{(AB}{}^{A'B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{C)B'}\nonumber\\
& + \tfrac{7}{2} \Phi_{(A}{}^{DB'C'}(\sTwist_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}{}_{B'C'}
- \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DF}{}^{A'}\nonumber\\
& - \underset{2,2}{L}{}^{DFA'B'} (\sTwist_{4,0} \Psi)_{ABCDFB'}
+ \tfrac{5}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}.\end{aligned}$$ With this and , the equation reduces to $$\begin{aligned}
0={}&P^{D}{}_{A'} \Psi_{ABCD}
+ \tfrac{4}{15} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'}
+ \tfrac{4}{25} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'}\nonumber\\*
& + \tfrac{2}{5} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'}
- \Phi_{(AB|A'|}{}^{B'}P_{C)B'}
- \tfrac{2}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'}\nonumber\\*
& + \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'}
- \tfrac{1}{2} (\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'}.\end{aligned}$$ Using the commutator $$\begin{aligned}
(\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'}={}&2 P^{D}{}_{A'} \Psi_{ABCD}
+ 2 (\sCurl_{2,2} \sTwist_{1,1} P)_{ABCA'}
- 2 \Phi_{(AB|A'|}{}^{B'}P_{C)B'}.\end{aligned}$$ this becomes $$\begin{aligned}
0={}&- (\sCurl_{2,2} \sTwist_{1,1} P)_{ABCA'}
+ \tfrac{4}{15} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'}
+ \tfrac{4}{25} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'}\nonumber\\
& + \tfrac{2}{5} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'}
- \tfrac{2}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'}
+ \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'}.\end{aligned}$$ However, after substituting in this equation, decomposing the derivatives into irreducible parts and using , this equation actually becomes trivial.
Doing the same calculations as for the Dirac-Weyl case we see that also holds for the Maxwell case. Directly from the commutators we find $$\begin{aligned}
(\sCurlDagger_{2,0} \sCurl_{1,1} P)_{AA'}={}&-6 \Lambda P_{AA'}
+ 2 \Phi_{ABA'B'} P^{BB'}
- (\sDiv_{2,2} \sTwist_{1,1} P)_{AA'}
+ \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{AA'}.\end{aligned}$$ With this, and we can reduce to $$\begin{aligned}
0={}&- \tfrac{2}{15} \Phi^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}
- \tfrac{4}{15} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'}
+ \tfrac{2}{15} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'}\nonumber\\
& + \tfrac{4}{15} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{B'C'D'}
- \tfrac{2}{15} \Phi_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{CA'B'C'}
- \tfrac{2}{5} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}\nonumber\\
& - \tfrac{4}{45} \Phi_{BCA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{CB'}
- \tfrac{2}{3} (\sDiv_{2,2} \sTwist_{1,1} P)_{BA'}
+ \tfrac{4}{15} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{0,0} \Lambda)^{CB'}
- \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'}\nonumber\\
& - \tfrac{1}{5} \underset{2,2}{L}{}^{CDB'C'} (\sTwist_{2,2} \Phi)_{BCDA'B'C'}
+ \tfrac{1}{2} (\sTwist_{0,0} \sDiv_{1,1} P)_{BA'}
+ \tfrac{2}{15} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'}.\end{aligned}$$ Using and the irreducible decompositions, we find $$\begin{aligned}
(\sDiv_{2,2} \sTwist_{1,1} P)_{BA'}={}&- \tfrac{2}{5} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'}
+ \tfrac{2}{5} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'}\nonumber\\
& + \tfrac{2}{5} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{B'C'D'}
- \tfrac{2}{5} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}.\label{eq:DivTwistQ2}\end{aligned}$$ To simplify the remaining terms, we use the same trick as for the Dirac-Weyl case. The definition and the equation can be used together with , to reduce the equation to $$\begin{aligned}
0={}&- \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'}
- \tfrac{1}{5} (\sTwist_{0,0} \Upsilon)_{BA'}
+ \tfrac{1}{2} (\sTwist_{0,0} \sDiv_{1,1} P)_{BA'}
+ \tfrac{2}{15} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'}.\label{eq:OrderZeroEq3}\end{aligned}$$ We therefore make the ansatz $$\begin{aligned}
\underset{0,0}{N}{}={}&3 Q
- \tfrac{3}{5} \Upsilon
+ \tfrac{3}{2} (\sDiv_{1,1} P)
+ \tfrac{2}{5} (\sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{}).\label{eq:N00Ansatz}\end{aligned}$$ Now, becomes $$\begin{aligned}
0={}&(\sTwist_{0,0} Q)_{AA'}.\label{eq:OrderZeroEq4}\end{aligned}$$
### Potential representation
From all this we can conclude that the only equations that restrict the geometry are and . Now, the operator takes the form $$\begin{aligned}
\chi_{AB}={}&\tfrac{1}{3} \underset{0,0}{N}{} \phi_{AB}
+ \underset{4,0}{N}{}_{ABCD} \phi^{CD}
- \underset{2,0}{N}{}_{(A}{}^{C}\phi_{B)C}
- \tfrac{4}{5} (\sCurl_{2,2} \underset{2,2}{L}{})_{(A}{}^{CDA'}(\sTwist_{2,0} \phi)_{B)CDA'}\nonumber\\
& + \tfrac{1}{2} \underset{1,1}{M}{}_{CA'} (\sTwist_{2,0} \phi)_{AB}{}^{CA'}
+ \tfrac{3}{5} \underset{2,2}{L}{}^{CDA'B'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}.
\label{eq:chiform1}\end{aligned}$$ where $\underset{0,0}{N}$, $\underset{2,0}{N}{}_{AB}$, $\underset{4,0}{N}{}_{ABCD}$, $\underset{1,1}{M}{}_{AA'}$ are given by , , and respectively.
We can in fact simplify this expression by defining the following spinor $$\begin{aligned}
A_{AA'}\equiv{}&P_{BA'} \phi_{A}{}^{B}
+ \tfrac{1}{5} \phi^{BC} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'}
+ \tfrac{4}{15} \phi_{A}{}^{B} (\sDiv_{2,2} \underset{2,2}{L}{})_{BA'}
+ \tfrac{3}{5} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{2,0} \phi)_{A}{}^{BCB'}.\end{aligned}$$ Substituting this onto the following, and comparing with , we find $$\begin{aligned}
(\sCurl_{1,1} A)_{AB}={}&- Q \phi_{AB}
+ \chi_{AB}
- \tfrac{1}{15} \phi_{(A}{}^{C}(\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{B)C}
+ \tfrac{1}{10} \phi_{(A}{}^{C}(\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{B)C}\nonumber\\
& - \tfrac{1}{10} \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{|C}{}^{D}{}_{A'B'|}\phi_{B)D}
- \tfrac{1}{10} \underset{2,2}{L}{}^{CDA'B'}\Phi_{(A|CA'B'|}\phi_{B)D}\nonumber\\
={}&- Q \phi_{AB}
+ \chi_{AB},\label{eq:CurlAeq1}\end{aligned}$$ where the last equality follows from a commutator relation. In fact the coefficients in $A_{AA'}$ were initially left free, and then chosen so all first and second order derivatives of $\phi_{AB}$ where eliminated in .
We also get $$\begin{aligned}
(\sCurlDagger_{1,1} A)_{A'B'}={}&\tfrac{12}{5} \Lambda \underset{2,2}{L}{}_{ABA'B'} \phi^{AB}
- \tfrac{3}{5} \underset{2,2}{L}{}^{CD}{}_{A'B'} \Psi_{ABCD} \phi^{AB}
+ \tfrac{1}{5} \phi^{AB} (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABA'B'}\nonumber\\
& - \tfrac{6}{5} \underset{2,2}{L}{}^{AB}{}_{(A'}{}^{C'}\Phi_{|A|}{}^{C}{}_{B')C'}\phi_{BC}
- \phi^{AB} (\sTwist_{1,1} P)_{ABA'B'}\nonumber\\
& - \tfrac{3}{5} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'} (\sTwist_{2,0} \phi)^{ABCC'}
- \tfrac{4}{15} \phi^{AB} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABA'B'}\nonumber\\
={}&0.\end{aligned}$$ where we in the last step used , a commutator and
To get the highest order coefficient equal to 1 in $A_{AA'}$ and in $\chi_{AB}$, we define a new symmetric spinor, which is just a rescaling of $\underset{2,2}{L}{}_{ABA'B'}$ $$\begin{aligned}
L_{ABA'B'}\equiv{}&\tfrac{3}{5} \underset{2,2}{L}{}_{ABA'B'}.\end{aligned}$$ Now, the only equations we have left are
$$\begin{aligned}
(\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\
(\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{2}{3} (\ObstrOne L)_{AB}{}^{A'B'},\\
(\sTwist_{0,0} Q)_{BA'}={}&0,\\
A_{AA'}={}&P_{BA'} \phi_{A}{}^{B}
+ \tfrac{1}{3} \phi^{BC} (\sCurl_{2,2} L)_{ABCA'}
+ \tfrac{4}{9} \phi_{A}{}^{B} (\sDiv_{2,2} L)_{BA'}\nonumber\\
& - L^{BC}{}_{A'}{}^{B'} (\sTwist_{2,0} \phi)_{ABCB'}.\end{aligned}$$
Second kind of symmetry operator for the Maxwell equation {#subsection:second_kind}
---------------------------------------------------------
For the symmetry operators of the second kind, one can follow the same procedure as above. However, this case was completely handled in [@KalMcLWil92a]. In that paper it was shown that a symmetry operator of the second kind always has the form $\phi_{AB}\rightarrow \omega_{A'B'}$, $$\begin{aligned}
\omega_{A'B'}={}&\tfrac{3}{5} \phi^{CD} (\sCurlDagger_{3,1} \sCurlDagger_{4,0} L)_{CDA'B'}
- \tfrac{8}{5} (\sCurlDagger_{4,0} L)^{CDF}{}_{(A'}(\sTwist_{2,0} \phi)_{|CDF|B')}\nonumber\\
& + L^{CDFH} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{CDFHA'B'},\end{aligned}$$ where $L_{ABCD}=L_{(ABCD)}$ satisfies $$\begin{aligned}
(\sTwist_{4,0} L)_{ABCDF}{}^{A'}={}&0.\label{eq:LSecKillingSpinor}\end{aligned}$$ Hence, the treatment in [@KalMcLWil92a] is satisfactory. However, it is interesting to see if the operator can be written in terms of a potential. Let $$\begin{aligned}
B_{AA'}\equiv{}&\tfrac{3}{5} \phi^{BC} (\sCurlDagger_{4,0} L)_{ABCA'}
+ L_{ABCD} (\sTwist_{2,0} \phi)^{BCD}{}_{A'}.\label{eq:BDef}\end{aligned}$$ Then, from the definition of $\sCurlDagger$, the irreducible decompositions and we get $$\begin{aligned}
(\sCurlDagger_{1,1} B)_{A'B'}={}&\tfrac{3}{5} \phi^{AB} (\sCurlDagger_{3,1} \sCurlDagger_{4,0} L)_{ABA'B'}
- \tfrac{8}{5} (\sCurlDagger_{4,0} L)^{ABC}{}_{(A'}(\sTwist_{2,0} \phi)_{|ABC|B')}\nonumber\\
& + L^{ABCD} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}\nonumber\\
={}&\omega_{A'B'}.\label{eq:CurlDaggerB}\end{aligned}$$ The coefficients in where initially left free, and then chosen to get .
We also get $$\begin{aligned}
(\sCurl_{1,1} B)_{AB}={}&6 \Lambda L_{ABCD} \phi^{CD}
- \tfrac{3}{2} L_{AB}{}^{FH} \Psi_{CDFH} \phi^{CD}
+ \tfrac{3}{5} \phi^{CD} (\sCurl_{3,1} \sCurlDagger_{4,0} L)_{ABCD}\nonumber\\
& + \tfrac{3}{10} \phi_{(A}{}^{C}(\sDiv_{3,1} \sCurlDagger_{4,0} L)_{B)C}
- \tfrac{3}{2} L_{(A}{}^{CDF}\Psi_{B)CD}{}^{H}\phi_{FH}
- \tfrac{1}{2} L_{(A}{}^{CDF}\Psi_{|CDF|}{}^{H}\phi_{B)H}\nonumber\\
& - (\sTwist_{4,0} L)_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'}\nonumber\\
={}&- \tfrac{1}{2} \phi^{CD} (\sDiv_{5,1} \sTwist_{4,0} L)_{ABCD}
- \tfrac{4}{5} \phi_{B}{}^{C} L_{(A}{}^{DFH}\Psi_{C)DFH}
- \tfrac{4}{5} \phi_{A}{}^{C} L_{(B}{}^{DFH}\Psi_{C)DFH}\nonumber\\
& - (\sTwist_{4,0} L)_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'}\nonumber\\
={}&0.\end{aligned}$$ Here, we have used together with the irreducible decomposition of $L_{AB}{}^{FH} \Psi_{CDFH}$ and the relations
$$\begin{aligned}
(\sDiv_{3,1} \sCurlDagger_{4,0} L)_{AB}={}&-2 L_{(A}{}^{CDF}\Psi_{B)CDF},\\
(\sCurl_{3,1} \sCurlDagger_{4,0} L)_{ABCD}={}&-10 \Lambda L_{ABCD}
- \tfrac{5}{6} (\sDiv_{5,1} \sTwist_{4,0} L)_{ABCD}
+ 5 L_{(AB}{}^{FH}\Psi_{CD)FH},\\
L_{(B}{}^{DFH}\Psi_{C)DFH}={}&0.\end{aligned}$$
The last equation follows from the integrability condition (cf. Section \[sec:integrabilitycond\]) $$\begin{aligned}
L_{(ABC}{}^{L}\Psi_{DFH)L} ={}& - \tfrac{1}{4} (\sCurl_{5,1} \sTwist_{4,0} L)_{ABCDFH}=0,\label{eq:intcondKS4}\end{aligned}$$ as explained in [@KalMcLWil92a].
Factorizations {#sec:factorizations}
==============
In this section we will consider special cases for which the auxiliary conditions will always have a solution. We will now prove Proposition \[prop:factorize\], considering each case in turn.
The case when $L_{ABA'B'}$ factors in terms of conformal Killing vectors
------------------------------------------------------------------------
If $\xi_{AA'}$ and $\zeta_{AA'}$ are conformal Killing vectors, i.e. $$\begin{aligned}
(\sTwist_{1,1}\xi)_{AB}{}^{A'B'}&=0,&
(\sTwist_{1,1}\zeta)_{AB}{}^{A'B'}&=0,\label{eq:xizetaCKV}\end{aligned}$$ then we have a solution $$\begin{aligned}
\mathfrak{L}_{\xi\zeta AB}{}^{A'B'}\equiv{}&\zeta_{(A}{}^{(A'}\xi_{B)}{}^{B')}\label{eq:Lxizetadef}\end{aligned}$$ to the equation $$\begin{aligned}
(\sTwist_{2,2}\mathfrak{L}_{\xi\zeta})_{ABC}{}^{A'B'C'}&=0.\end{aligned}$$ Let
$$\begin{aligned}
\mathfrak{Q}_{\xi\zeta}\equiv{}&\Lambda \zeta^{AA'} \xi_{AA'}
+ \tfrac{1}{3} \Phi_{ABA'B'} \zeta^{AA'} \xi^{BB'}
+ \tfrac{1}{8} (\sCurl_{1,1} \zeta)^{AB} (\sCurl_{1,1} \xi)_{AB}\nonumber\\
& + \tfrac{1}{6} \xi^{AA'} (\sCurl_{0,2} \sCurlDagger_{1,1} \zeta)_{AA'}
+ \tfrac{1}{6} \zeta^{AA'} (\sCurl_{0,2} \sCurlDagger_{1,1} \xi)_{AA'}
+ \tfrac{1}{8} (\sCurlDagger_{1,1} \zeta)^{A'B'} (\sCurlDagger_{1,1} \xi)_{A'B'}\nonumber\\
& - \tfrac{1}{32} (\sDiv_{1,1} \zeta) (\sDiv_{1,1} \xi),\label{eq:Qxizetadef}\\
\mathfrak{P}_{\xi\zeta AA'}\equiv{}&\tfrac{1}{4} \xi^{B}{}_{A'} (\sCurl_{1,1} \zeta)_{AB}
+ \tfrac{1}{4} \zeta^{B}{}_{A'} (\sCurl_{1,1} \xi)_{AB}
- \tfrac{1}{4} \xi_{A}{}^{B'} (\sCurlDagger_{1,1} \zeta)_{A'B'}
- \tfrac{1}{4} \zeta_{A}{}^{B'} (\sCurlDagger_{1,1} \xi)_{A'B'}.\label{eq:Pxizetadef}\end{aligned}$$
Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a long expression with the operators $\sDiv$, $\sCurl$, $\sCurlDagger$, $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sDiv$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$, $\sDiv\sCurl\sCurlDagger$, $\sCurl\sCurl\sCurlDagger$ and $\sCurlDagger\sCurl\sCurlDagger$ operating on $\xi^{AA'}$ and $\zeta^{AA'}$. Using the commutators , , , and on the outermost operators and using , the list of operators appearing can be reduced to the set $\sDiv$, $\sCurl$, $\sCurlDagger$, $\sDiv\sTwist\sCurlDagger$, $\sCurl\sTwist\sCurlDagger$ and $\sCurl\sCurl\sCurlDagger$. Then using the relations and on the innermost operators the list of operators appearing is reduced to $\sCurl$, $\sCurlDagger$, $\sCurl\sTwist\sDiv$, where the latter can be eliminated with on the outer operators. After making an irreducible decomposition of $\xi^{AA'}\zeta^{BB'}$ and identifying the symmetric part though , one is left with $$\begin{aligned}
(\sTwist_{0,0} \mathfrak{Q}_{\xi\zeta})_{A}{}^{A'}={}&\mathfrak{L}_{\xi\zeta}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'}
+ \tfrac{1}{4} \Psi_{ABCD} \xi^{BA'} (\sCurl_{1,1} \zeta)^{CD}
+ \tfrac{1}{4} \Psi_{ABCD} \zeta^{BA'} (\sCurl_{1,1} \xi)^{CD}\nonumber\\
& + \mathfrak{L}_{\xi\zeta}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}
+ \tfrac{1}{4} \bar\Psi^{A'}{}_{B'C'D'} \xi_{A}{}^{B'} (\sCurlDagger_{1,1} \zeta)^{C'D'}\nonumber\\
& + \tfrac{1}{4} \bar\Psi^{A'}{}_{B'C'D'} \zeta_{A}{}^{B'} (\sCurlDagger_{1,1} \xi)^{C'D'}.\label{eq:TwistQxizetaEq1}\end{aligned}$$
Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a expression with the operators $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$ operating on $\xi^{AA'}$ and $\zeta^{AA'}$. Using the commutators , and and using , the entire expression can be reduced to only contain curvature terms. After making an irreducible decomposition of $\xi^{AA'}\zeta^{BB'}$ and identifying the symmetric part though , one is left with $$\begin{aligned}
(\sTwist_{1,1} \mathfrak{P}_{\xi\zeta}{})_{AB}{}^{A'B'}={}&\mathfrak{L}_{\xi\zeta}{}^{CDA'B'} \Psi_{ABCD}
- \mathfrak{L}_{\xi\zeta}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}.\label{eq:TwistPxizetaEq1}\end{aligned}$$ Substituting into the definition of $\ObstrZero$, allows us to see that and reduces to
$$\begin{aligned}
(\sTwist_{0,0} \mathfrak{Q}_{\xi\zeta})_{A}{}^{A'}={}&(\ObstrZero \mathfrak{L}_{\xi\zeta})_{A}{}^{A'},\label{eq:Qxizetaaux}\\
(\sTwist_{1,1} \mathfrak{P}_{\xi\zeta})_{AB}{}^{A'B'}={}&(\ObstrOne \mathfrak{L}_{\xi\zeta})_{AB}{}^{A'B'}.\label{eq:Pxizetaaux}\end{aligned}$$
The actual form of and was obtained by making sufficiently general symmetric second order bi-linear ansätze. The coefficients where then chosen to eliminate as many extra terms as possible in and .
The case when $L_{ABA'B'}$ factors in terms of Killing spinors
--------------------------------------------------------------
Another way of constructing conformal Killing tensors is to make a product of valence $(2,0)$ and valence $(0,2)$ Killing spinors. It turns out that also this case admits solutions to the auxiliary conditions.
In principle we could construct $L_{ABA'B'}$ from two different Killing spinors, but if the dimension of the space of Killing spinors is greater than one, the spacetime has to be locally isometric to Minkowski space. In these spacetimes the picture is much simpler and has been studied before. The auxiliary conditions will be trivial in these cases. We will therefore only consider one Killing spinor.
Let $\kappa_{AB}$ be a Killing spinor, i.e. a solution to $$\begin{aligned}
(\sTwist_{2,0} \kappa)_{ABCA'}={}&0.\label{eq:kappaKS}\end{aligned}$$ We have a solution $$\begin{aligned}
\label{eq:Lkappadef}
\mathfrak{L}_{\kappa AB}{}^{A'B'}\equiv{}&\kappa_{AB}\bar\kappa^{A'B'},\end{aligned}$$ to the equation $$\begin{aligned}
(\sTwist_{2,2}\mathfrak{L}_{\kappa})_{ABC}{}^{A'B'C'}&=0.\end{aligned}$$ Now, let
$$\begin{aligned}
\mathfrak{Q}_{\kappa}\equiv{}&\tfrac{2}{3} \Phi_{ABA'B'} \kappa^{AB} \bar{\kappa}^{A'B'}
+ \tfrac{1}{9} \kappa^{AB} (\sCurl_{1,1} \sCurl_{0,2} \bar{\kappa})_{AB}
+ \tfrac{4}{27} (\sCurl_{0,2} \bar{\kappa})^{AA'} (\sCurlDagger_{2,0} \kappa)_{AA'}\nonumber\\
& + \tfrac{1}{9} \bar{\kappa}^{A'B'} (\sCurlDagger_{1,1} \sCurlDagger_{2,0} \kappa)_{A'B'},\label{eq:Qkappadef}\\
\mathfrak{P}_{\kappa AA'}\equiv{}&\tfrac{4}{3} \kappa_{AB} (\sCurl_{0,2} \bar{\kappa})^{B}{}_{A'}
- \tfrac{4}{3} \bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \kappa)_{A}{}^{B'}.\label{eq:Pkappadef}\end{aligned}$$
Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a long expression with the operators $\sCurl$, $\sCurlDagger$, $\sDiv\sCurl$, $\sDiv\sCurlDagger$, $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$, $\sCurl\sCurlDagger\sCurlDagger$, $\sCurlDagger\sCurl\sCurl$, $\sTwist\sCurl\sCurl$ and $\sTwist\sCurlDagger\sCurlDagger$ operating on $\kappa_{AB}$ and $\bar\kappa_{A'B'}$. Using the commutators , , , , and on the outermost operators and using , the list of operators appearing can be reduced to the set $\sCurl$, $\sCurlDagger$, $\sCurl\sTwist\sCurl$, $\sCurlDagger\sTwist\sCurlDagger$, $\sDiv\sTwist\sCurl$ and $\sDiv\sTwist\sCurlDagger$. Then using the relations , , and on the innermost operators the expression will only contain the operators $\sCurl$, $\sCurlDagger$. $$\begin{aligned}
(\sTwist_{0,0} \mathfrak{Q}_{\kappa})_{A}{}^{A'}={}&\kappa^{BC} \bar{\kappa}^{A'B'} (\sCurl_{2,2} \Phi)_{ABCB'}
- \tfrac{2}{9} \Phi_{BC}{}^{A'}{}_{B'} \kappa^{BC} (\sCurl_{0,2} \bar{\kappa})_{A}{}^{B'}
+ \tfrac{1}{3} \Psi_{ABCD} \kappa^{CD} (\sCurl_{0,2} \bar{\kappa})^{BA'}\nonumber\\
& + \tfrac{2}{9} \Phi_{BC}{}^{A'}{}_{B'} \kappa_{A}{}^{C} (\sCurl_{0,2} \bar{\kappa})^{BB'}
- \tfrac{2}{9} \Phi_{AC}{}^{A'}{}_{B'} \kappa_{B}{}^{C} (\sCurl_{0,2} \bar{\kappa})^{BB'}\nonumber\\
& + \kappa_{A}{}^{B} \bar{\kappa}^{B'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}
+ \tfrac{1}{3} \bar\Psi^{A'}{}_{B'C'D'} \bar{\kappa}^{C'D'} (\sCurlDagger_{2,0} \kappa)_{A}{}^{B'}\nonumber\\
& - \tfrac{2}{9} \Phi_{ABB'C'} \bar{\kappa}^{B'C'} (\sCurlDagger_{2,0} \kappa)^{BA'}
+ \tfrac{2}{9} \Phi_{ABB'C'} \bar{\kappa}^{A'C'} (\sCurlDagger_{2,0} \kappa)^{BB'}\nonumber\\
& - \tfrac{2}{9} \Phi_{AB}{}^{A'}{}_{C'} \bar{\kappa}_{B'}{}^{C'} (\sCurlDagger_{2,0} \kappa)^{BB'}.\label{eq:TwistQkappaEq1}\end{aligned}$$
Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives an expression with the operators $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$ and $\sTwist\sCurlDagger$ operating on $\kappa_{AB}$ and $\bar\kappa_{A'B'}$. Using the commutators , , and and using , the expression reduces to $$\begin{aligned}
(\sTwist_{1,1} \mathfrak{P}_{\kappa}{})_{AB}{}^{A'B'}={}&\Psi_{ABCD} \kappa^{CD} \bar{\kappa}^{A'B'}
- \bar\Psi^{A'B'}{}_{C'D'} \kappa_{AB} \bar{\kappa}^{C'D'}.\label{eq:TwistPkappaEq1}\end{aligned}$$
Substituting into the definition of $\ObstrZero$, and making an irreducible decomposition of $\kappa_{AB} (\sCurl_{0,2} \bar{\kappa})_{C}{}^{B'}$ and $\bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \kappa)_{AC'}$, allows us to see that and reduces to
$$\begin{aligned}
(\sTwist_{0,0} \mathfrak{Q}_{\kappa})_{A}{}^{A'}={}&(\ObstrZero \mathfrak{L}_{\kappa})_{A}{}^{A'},\\
(\sTwist_{1,1} \mathfrak{P}_{\kappa})_{AB}{}^{A'B'}={}&(\ObstrOne \mathfrak{L}_{\kappa})_{AB}{}^{A'B'}.\end{aligned}$$
Example of a conformal Killing tensor that does not factor
----------------------------------------------------------
The following shows that the condition \[point:A0\] is non-trivial. We also see that \[point:A1\] does not imply \[point:A0\]. Unfortunately, we have not found any example of a valence $(1,1)$ Killing spinor which does not satisfy \[point:A1\].
Consider the following Stäckel metric (see [@michel:radoux:silhan:2013arXiv1308.1046M] and [@kalnins:mclenaghan:BelgAcad1984] for more general examples.) $$g_{ab} = dt^2- dz^2 - (x + y)(dx^2 + dy^2)$$ with the tetrad $$\begin{aligned}
l^{a}={}&\tfrac{1}{\sqrt{2}}(\partial_t)^{a}
+ \tfrac{1}{\sqrt{2}}(\partial_z)^{a},&
n^{a}={}&\tfrac{1}{\sqrt{2}}(\partial_t)^{a}
- \tfrac{1}{\sqrt{2}}(\partial_z)^{a},&
m^{a}={}&\frac{(\partial_x)^{a}}{\sqrt{2} (x + y)^{1/2}}
+ \frac{i (\partial_y)^{a}}{\sqrt{2} (x + y)^{1/2}}.\end{aligned}$$ Expressed in the corresponding dyad $(o_A,\iota_A)$, the curvature takes the form $$\begin{aligned}
\Psi_{ABCD}={}&-12 \Lambda o_{(A}o_{B}\iota_{C}\iota_{D)},&
\Phi_{ABA'B'}={}&12 \Lambda o_{(A}\iota_{B)} \bar o_{(A'}\bar\iota_{B')},&
\Lambda={}& \frac{1}{12 (x + y)^3}.\end{aligned}$$ We can see that the spinor $$\begin{aligned}
L_{AB}{}^{A'B'}={}&\tfrac{1}{2} (x + y)
(\bar o^{A'} \bar o^{B'} \iota_{A} \iota_{B}
+ o_{A} o_{B} \bar\iota^{A'} \bar\iota^{B'})
- (x - y) o_{(A}\iota_{B)} \bar o^{(A'}\bar\iota^{B')}\end{aligned}$$ is a trace-free conformal Killing tensor. We trivially have solutions to the auxiliary condition \[point:A1\] because $$\begin{aligned}
(\ObstrOne L)_{AB}{}^{A'B'} = {}&L{}^{CDA'B'} \Psi_{ABCD}
- L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} = 0.\end{aligned}$$ If there is a solution to we will automatically have $(\sCurl_{1,1}\ObstrZero L)_{AB}=0$ because $\sCurl_{1,1}\sTwist_{0,0}=0$. However, with the current $L_{AB}{}^{A'B'}$ we get $$\begin{aligned}
(\sCurl_{1,1} \ObstrZero L)_{AB}={}&\frac{5i o_{(A}\iota_{B)}}{(x + y)^5}.\end{aligned}$$ This is non vanishing, which means that the auxiliary condition \[point:A0\] does not admit a solution. This example shows that the conditions \[point:A0\] and \[point:A1\] are not equivalent. From the previous two sections, we can also conclude that this $L_{AB}{}^{A'B'}$ can not be written as a linear combination of conformal Killing tensors of the form $\zeta_{(A}{}^{(A'}\xi_{B)}{}^{B')}$ or $\kappa_{AB} \bar{\kappa}_{A'B'}$. For the more general metric in [@kalnins:mclenaghan:BelgAcad1984] we can in fact also construct a valence $(2,2)$ Killing spinor which trivially satisfies condition \[point:A1\], but which in general will not satisfy condition \[point:A0\]. It is interesting to note that in general this metric does not admit Killing vectors, but we can still construct symmetry operators for the Maxwell equation.
Auxiliary condition for a symmetry operator of the second kind for the Dirac-Weyl equation
------------------------------------------------------------------------------------------
Let $\kappa_{AB}$ be a Killing spinor, and $\xi^{AA'}$ a conformal Killing vector, i.e. $$\begin{aligned}
(\sTwist_{2,0} \kappa)_{ABCA'}={}&0, &(\sTwist_{1,1}\xi)_{AB}{}^{A'B'}&=0.\label{eq:TwistkappaTwistxi}\end{aligned}$$ then we have a solution $$\begin{aligned}
\mathfrak{L}_{\kappa\xi ABC}{}^{A'}\equiv{}&\kappa_{(AB}\xi_{C)}{}^{A'}\end{aligned}$$ to the equation $$\begin{aligned}
(\sTwist_{3,1}\mathfrak{L}_{\kappa\xi})_{ABCD}{}^{A'B'}&=0.\end{aligned}$$ The auxiliary equation now takes the form $$\begin{aligned}
0={}&\tfrac{3}{4} \Psi_{ABDF} \kappa^{CD} (\sCurl_{1,1} \xi)_{C}{}^{F}
+ \Psi_{ABCD} \xi^{CA'} (\sCurlDagger_{2,0} \kappa)^{D}{}_{A'}
- \tfrac{3}{4} \Psi_{ABCD} \kappa^{CD} (\sDiv_{1,1} \xi)\nonumber\\
& - \tfrac{5}{4} \Psi_{(A}{}^{CDF}\kappa_{B)C}(\sCurl_{1,1} \xi)_{DF}
- \tfrac{5}{4} \Psi_{(A}{}^{CDF}\kappa_{|CD|}(\sCurl_{1,1} \xi)_{B)F}
+ \tfrac{6}{5} \kappa_{(A}{}^{C}\xi^{DA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\
& + \tfrac{3}{5} \kappa^{CD}\xi_{(A}{}^{A'}(\sCurl_{2,2} \Phi)_{B)CDA'}
- 2 \kappa^{DF} \xi^{CA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:auxcondDirac2factored}\end{aligned}$$ Using the technique from Section \[sec:integrabilitycond\] we get that the integrability conditions for are
$$\begin{aligned}
0={}&\Psi_{(ABC}{}^{F}\kappa_{D)F},\label{eq:intcondkappaDirac2}\\
0={}&\tfrac{1}{2} \Psi_{ABCD} (\sDiv_{1,1} \xi)
+ 2 \Psi_{(ABC}{}^{F}(\sCurl_{1,1} \xi)_{D)F}
- \tfrac{4}{5} \xi_{(A}{}^{A'}(\sCurl_{2,2} \Phi)_{BCD)A'}
+ \xi^{FA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:intcondxiDirac2}\end{aligned}$$
Applying the operator $\sCurlDagger$ on the condition gives $$\begin{aligned}
0={}&- \tfrac{1}{2} \Psi_{ABCD} (\sCurlDagger_{2,0} \kappa)^{D}{}_{A'}
- \tfrac{9}{10} \kappa_{(A}{}^{D}(\sCurl_{2,2} \Phi)_{BC)DA'}
+ \tfrac{1}{4} \kappa^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:curlintcondkappaDirac2}\end{aligned}$$ Using to elliminate $\Psi_{ABCD} (\sDiv_{1,1} \xi)$ and to elliminate $\kappa^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'}$, and doing an irreducible decomposition of $\Psi_{ABCF} \kappa_{D}{}^{F}$ we see that reduces to $$\begin{aligned}
0={}&-2 (\sCurl_{1,1} \xi)^{CD} \Psi_{(ABC}{}^{F}\kappa_{D)F},\end{aligned}$$ which is trivially satisfied due to .
Factorization of valence $(4,0)$ Killing spinors with aligned matter {#sec:factorValence4}
--------------------------------------------------------------------
Assume that the matter field and the curvature are aligned, that is $$\begin{aligned}
0={}&\Psi_{(ABC}{}^{F}\Phi_{D)FA'B'}.\label{eq:AlignmentPsiPhi}\end{aligned}$$ Furthermore, assume that $\Psi_{ABCD}$ does not vanish, and assume that there is a solution $L_{ABCD}$ to $$(\sTwist_{4,0}L)_{ABCDEA'} = 0.\label{eq:KS4}$$ The integrability condition for this equation together with the non-vanishing of the Weyl spinor, gives that $L_{ABCD}$ and $\Psi_{ABCD}$ are proportional (c.f. [@KalMcLWil92a]). This means that
$$\begin{aligned}
0={}&L_{(ABC}{}^{F}\Phi_{D)FA'B'},\label{eq:LPhiAlign}\\
0={}&- L_{(ABCD}(\sTwist_{0,0}\Lambda)_{F)A'} + L_{(ABC}{}^{H}(\sCurl_{2,2} \Phi)_{DF)HA'}
+ \tfrac{1}{5} \Phi_{(AB|A'|}{}^{B'}(\sCurlDagger_{4,0} L)_{CDF)B'},\label{eq:DerLPhiAlign}\end{aligned}$$
where the second equation is obtained by taking a derivative of the first, decomposing the derivatives into irreducible parts, using the Killing spinor equation, and symmetrizing over all unprimed indices.
Split $L_{ABCD}$ into principal spinors $L_{ABCD}=\alpha_{(A}\beta_B\gamma_C\delta_{D)}$. Now, the Killing spinor equation , and the alignment equation gives
$$\begin{aligned}
0={}&\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \alpha^{F} (\sTwist_{4,0} L)_{ABCDFA'}=
\alpha^{A} \beta_{A} \alpha^{B} \gamma_{B} \alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \nabla_{FA'}\alpha_{D},\\
0={}&\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} L_{(ABC}{}^{F}\Phi_{D)FA'B'}=
\tfrac{1}{4} \alpha^{A} \beta_{A} \alpha^{B} \gamma_{B} \alpha^{C} \delta_{C} \alpha^{D} \alpha^{F}\Phi_{DFA'B'}.\end{aligned}$$
We will first assume that $\alpha^A$ is not a repeated principal spinor of $L_{ABCD}$. This means that $\alpha^A\beta_A\alpha^B\gamma_B\alpha^C\delta_{C} \neq 0$ and hence $\alpha^A\alpha^B\nabla_{A'A}\alpha_{B}=0$, that is $\alpha_A$ is a shear-free geodesic null congruence. We also get $\alpha^{D} \alpha^{F}\Phi_{DFA'B'}=0$. Contracting with $\alpha^A\alpha^B\alpha^C\alpha^D\alpha^F$ we get $$\begin{aligned}
0 ={}& \tfrac{1}{4} \alpha^{A} \beta_{A}\alpha^{B} \gamma_{B}\alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \alpha^{H} (\sCurl_{2,2} \Phi)_{DFHA'} + \tfrac{1}{5} \Phi_{ABA'}{}^{B'} \alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \alpha^{F} (\sCurlDagger_{4,0} L)_{CDFB'}\nonumber\\
={}&\tfrac{1}{4} \alpha^{A} \beta_{A}\alpha^{B} \gamma_{B}\alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \alpha^{H} (\sCurl_{2,2} \Phi)_{DFHA'}.\end{aligned}$$ Hence, $\alpha^{A} \alpha^{B} \alpha^{C} (\sCurl_{2,2} \Phi)_{ABCA'}=0$. But the Bianchi equations give $$\begin{aligned}
\alpha^{A} \alpha^{B} \alpha^{C} \nabla^{DD'}\Psi_{ABCD}={}&\alpha^{A} \alpha^{B} \alpha^{C} (\sCurl_{2,2} \Phi)_{ABCA'}=0.\end{aligned}$$ It follows from the generalized Goldberg-Sachs theorem that $\alpha^A$ is a repeated principal spinor of $\Psi_{ABCD}$, see for instance [@PenRin86 Proposition 7.3.35]. But $L_{ABCD}$ and $\Psi_{ABCD}$ are proportional, so $\alpha^A$ is a repeated principal spinor of $L_{ABCD}$ after all. Without loss of generality, we can assume that $\gamma^A=\alpha^A$, a relabelling and rescaling of $\beta^{A}$, $\gamma^{A}$ and $\delta^{A}$ can achieve this. Repeating the argument with $\beta^A$, we find that also $\beta^A$ is a repeated principal spinor of $L_{ABCD}$. If $\beta^A\alpha_A=0$, we can repeat the argument again with $\delta^A$ and see that all principal spinors are repeated, i.e. Petrov type N. Otherwise, we have Petrov type D. In conclusion, we have after rescaling $L_{ABCD}=\alpha_{(A}\alpha_{B}\beta_{C}\beta_{D)}$. Now, let $\kappa_{AB}=\alpha_{(A}\beta_{B)}$.
First assume that $\alpha^A\beta_A\neq 0$. Contracting with $\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \beta^{F}$, $\alpha^{A} \alpha^{B} \alpha^{C} \beta^{D} \beta^{F}$, $\alpha^{A} \alpha^{B} \beta^{C} \beta^{D} \beta^{F}$, $\alpha^{A} \beta^{B} \beta^{C} \beta^{D} \beta^{F}$ we find $$\begin{aligned}
0={}&\alpha^{A} \alpha^{B} \alpha^{C} (\sTwist_{2,0} \kappa)_{ABCA'},&
0={}&\alpha^{A} \alpha^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'},\\
0={}&\alpha^{A} \beta^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'},&
0={}&\beta^{A} \beta^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'}.\end{aligned}$$ Hence, $(\sTwist_{2,0} \kappa)_{ABCA'}=0$.
If $\alpha^A\beta_A= 0$, we can find a dyad $(o^A, \iota^A)$ so that $\alpha^A=o^A$. Then we have $L_{ABCD}=\upsilon^2 o_{A}o_{B}o_{C}o_{D}$ and $\kappa_{AB}=\upsilon o_{A}o_{B}$. Contracting with $o^{A} o^{B} o^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $o^{A} o^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $o^{A} \iota^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $\iota^{A} \iota^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$ we find $$\begin{aligned}
0={}&o^{A} o^{B} o^{C} (\sTwist_{2,0} \kappa)_{ABCA'},&
0={}&o^{A} o^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'},\\
0={}&o^{A} \iota^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'},&
0={}&\iota^{A} \iota^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'}.\end{aligned}$$ Hence, $(\sTwist_{2,0} \kappa)_{ABCA'}=0$.
We can therefore conclude that if the curvature satisfies , $\Psi_{ABCD}$ does not vanish, and we have a valence $(4,0)$ Killing spinor $L_{ABCD}$, then we have a valence $(2,0)$ Killing spinor $\kappa_{AB}$ such that $L_{ABCD}=\kappa_{(AB}\kappa_{CD)}$.
The symmetry operators with factorized Killing spinor {#sec:symopfactored}
=====================================================
Symmetry operators for the conformal wave equation
--------------------------------------------------
Let us now consider special cases of symmetry operators for the conformal wave equation. If we choose $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},&
P_{AA'}={}&0,&
Q ={}& \tfrac{2}{5} \mathfrak{Q}_{\xi\zeta}.\end{aligned}$$ Then the operator takes the form $$\begin{aligned}
\chi={}&\tfrac{1}{2} \hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi
+ \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi.\end{aligned}$$ One can also add an arbitrary first order symmetry operator to this.
We can also choose $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},&
P_{AA'}={}&0,&
Q ={}& \tfrac{2}{5} \mathfrak{Q}_{\kappa}.\end{aligned}$$ Substituting these expressions into gives a symmetry operator, but we have not found any simpler form than the one given by .
Apart from factorizations, one can in special cases get symmetry operators from Killing tensors. If $K_{AB}{}^{A'B'}$ is a Killing tensor, then we have $$\begin{aligned}
(\sTwist_{2,2} L)_{ABCA'B'C'}={}&0, &
(\sDiv_{2,2} L)_{AA'} ={}& - \tfrac{3}{4} (\sTwist_{0,0} S)_{AA'},&
K^{ABA'B'}={}&L^{ABA'B'} + \tfrac{1}{4} S \epsilon^{AB} \bar\epsilon^{A'B'}\end{aligned}$$ where $L_{AB}{}^{A'B'}=K_{(AB)}{}^{(A'B')}$ and $S=K_{A}{}^{A}{}_{A'}{}^{A'}$. The commutator gives $(\sCurl_{1,1} \sDiv_{2,2} L)_{AB} = 0$. If we also assume vacuum, then the equation gives $$\begin{aligned}
(\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} L)_{AA'}={}&-2 \Psi_{ABCD} (\sCurl_{2,2} L)^{BCD}{}_{A'}
- 2 \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} L)_{A}{}^{B'C'D'}.\end{aligned}$$ Hence, we can choose $$\begin{aligned}
Q ={}& - \tfrac{1}{15} (\sDiv_{1,1} \sDiv_{2,2} L),\end{aligned}$$ to satisfy condition \[point:A0\], and get the well known symmetry operator $$\begin{aligned}
\chi={}&-\tfrac{1}{2} (\sTwist_{0,0} S)^{AA'} (\sTwist_{0,0} \phi)_{AA'}
+ L^{ABA'B'} (\sTwist_{1,1} \sTwist_{0,0} \phi)_{ABA'B'}&
={}&\nabla_{AA'}(K^{ABA'B'} \nabla_{BB'}\phi),\end{aligned}$$ which is valid for vacuum spacetimes.
Symmetry operator of the first kind for the Dirac-Weyl equation
---------------------------------------------------------------
Let us now consider special cases of symmetry operators of the first kind for the Dirac-Weyl equation. We can choose $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},&
P^{AA'}={}&- \tfrac{1}{3} \mathfrak{P}_{\xi\zeta}{}^{AA'},&
Q ={}& \tfrac{3}{10} \mathfrak{Q}_{\xi\zeta},\end{aligned}$$ to get a symmetry operator for the Dirac-Weyl equation. The operator then becomes $$\begin{aligned}
\chi_{A}={}&\tfrac{1}{2}\hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{A}+\tfrac{1}{2}\hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi_{A}.\end{aligned}$$ We can add any conformal Killing vector to $P^{AA'}$ and any constant to $Q$. Note that if we add the conformal Killing vector $\tfrac{1}{2} (\xi^{BB'} \nabla_{BB'}\zeta^{AA'} - \zeta^{BB'} \nabla_{BB'}\xi^{AA'})$ to $P^{AA'}$, the operator gets the factored form $$\begin{aligned}
\chi_{A}={}&\hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{A}.\end{aligned}$$
We can also choose $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},&
P^{AA'}={}&- \tfrac{1}{3} \mathfrak{P}_{\kappa}{}^{AA'},&
Q ={}& \tfrac{3}{10} \mathfrak{Q}_{\kappa}.\end{aligned}$$ Substituting these expressions into gives a symmetry operator, but we have not found any simpler form than the one given by .
Symmetry operator of the first kind for the Maxwell equation {#sec:symopfirstmaxwellfact}
------------------------------------------------------------
Let us now consider the symmetry operators of the first kind for the Maxwell equation. Let $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},&
P^{AA'}={}&- \tfrac{2}{3} \mathfrak{P}_{\xi\zeta}{}^{AA'},&
Q ={}&0,\end{aligned}$$ to get a symmetry operator. With this choice the symmetry operator and the potential reduce to
$$\begin{aligned}
\chi_{AB}={}&\tfrac{1}{2} \hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi_{AB}
+ \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{AB},\\
A_{AA'}={}&- \tfrac{1}{2} \zeta^{B}{}_{A'}\hat{\mathcal{L}}_{\xi}\phi_{AB}
- \tfrac{1}{2} \xi^{B}{}_{A'}\hat{\mathcal{L}}_{\zeta}\phi_{AB} .\end{aligned}$$
A general first order operator can be added to this. If we add an the same commutator as above with an appropriate coefficient to $P^{AA'}$, we get the same kind of factorization of the operator as above.
We can also get a solution by setting $$\begin{aligned}
L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},&
P^{AA'}={}&- \tfrac{2}{3} \mathfrak{P}_{\kappa}{}^{AA'},&
Q ={}&0,\end{aligned}$$ With this choice the symmetry operator and the potential reduce to
$$\begin{aligned}
\chi_{AB}={}&(\sCurl_{1,1} A)_{AB},\\
A_{AA'}={}&- \tfrac{1}{3} \Theta_{AB} (\sCurl_{0,2} \bar{\kappa})^{B}{}_{A'}
+ \bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \Theta)_{A}{}^{B'},\\
\Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$
This proves the first part of Theorem \[Thm:SymopMaxwellSimple\].
Symmetry operator of the second kind for the Dirac-Weyl equation {#sec:symopsecondDiracfactored}
----------------------------------------------------------------
Let
$$\begin{aligned}
L_{ABC}{}^{A'}={}&\mathfrak{L}_{\kappa\xi ABC}{}^{A'},\\
P_{AB}
={}&- \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\kappa_{AB}
+ \tfrac{3}{8} \kappa_{AB} (\sDiv_{1,1} \xi)\nonumber\\
={}&\tfrac{1}{8} \kappa_{AB} (\sDiv_{1,1} \xi)
- \tfrac{1}{2} \kappa_{(A}{}^{C}(\sCurl_{1,1} \xi)_{B)C}
+ \tfrac{1}{3} \xi_{(A}{}^{A'}(\sCurlDagger_{2,0} \kappa)_{B)A'}.\end{aligned}$$
Using the equations , the commutators , , , and the irreducible decompositions of $\Psi_{ABCF} \kappa_{D}{}^{F}$ and $\Phi_{ABA'B'} \xi_{C}{}^{B'}$ we get $$\begin{aligned}
(\sTwist_{2,0} P)_{ABCA'}={}&- \tfrac{1}{6} \kappa_{(AB}(\sCurlDagger_{2,0} \sCurl_{1,1} \xi)_{C)A'}
- \tfrac{1}{2} \kappa_{(A}{}^{D}(\sTwist_{2,0} \sCurl_{1,1} \xi)_{BC)DA'}
+ \tfrac{1}{8} \kappa_{(AB}(\sTwist_{0,0} \sDiv_{1,1} \xi)_{C)A'}\nonumber\\
& + \tfrac{1}{6} \xi_{(A|A'|}(\sCurl_{1,1} \sCurlDagger_{2,0} \kappa)_{BC)}
+ \tfrac{1}{3} \xi_{(A}{}^{B'}(\sTwist_{1,1} \sCurlDagger_{2,0} \kappa)_{BC)A'B'}\nonumber\\
={}&\xi^{D}{}_{A'} \Psi_{(ABC}{}^{F}\kappa_{D)F}\nonumber\\
={}&0,\end{aligned}$$ where we in the last step used the integrability condition . Observe that $P_{AB}$ is given by a conformally weighted Lie derivative, but now with a different weight. The operator $\hat{\mathcal{L}}_{\xi}$ has a conformal weight adapted to the weight of the conformally invariant operator $\sCurlDagger$. The operator $\sTwist$ is also conformally invariant, but with a different weight. This explains the extra term in $P_{AB}$.
The symmetry operator of the second kind for the Dirac-Weyl equation now takes the form $$\begin{aligned}
\omega_{A'}={}&\kappa^{BC} (\sTwist_{1,0} \hat{\mathcal{L}}_{\xi}\phi)_{BCA'}-\tfrac{2}{3} \hat{\mathcal{L}}_{\xi}\phi_{B} (\sCurlDagger_{2,0} \kappa)^{B}{}_{A'}.\end{aligned}$$ Hence, we can conclude that if $L_{ABCA'}$ factors, then one can choose a corresponding $P_{AB}$ so that the operator factors as a first order symmetry operator of the first kind followed by a first order symmetry operator of the second kind.
Symmetry operator of the second kind for the Maxwell equation {#sec:symopsecondmaxwellfact}
-------------------------------------------------------------
If we let $L_{ABCD}=\kappa_{(AB}\kappa_{CD)}$ with $$\begin{aligned}
(\sTwist_{2,0} \kappa)_{ABCA'} ={}& 0,\end{aligned}$$ Then the operator if the second kind now takes the form
$$\begin{aligned}
\omega_{A'B'}={}&(\sCurlDagger_{1,1}B)_{A'B'},\\
B_{AA'}={}&\kappa_{AB} (\sCurlDagger_{2,0} \Theta)^{B}{}_{A'}
+ \tfrac{1}{3} \Theta_{AB} (\sCurlDagger_{2,0} \kappa)^{B}{}_{A'},\\
\Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$
This proves the second part of Theorem \[Thm:SymopMaxwellSimple\].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Steffen Aksteiner and Lionel Mason for helpful discussions. We are particularly grateful to Lionel Mason for his ideas concerning Theorem \[thm:Valence4Factorization\]. Furthermore we would like to thank Niky Kamran and J. P. Michel for helpful comments. LA thanks Shing-Tung Yau for generous hospitality and many interesting discussions on symmetry operators and related matters, during a visit to Harvard University, where some initial work on the topic of this paper was done. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the semester of 2013.
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[^1]: The use of the terms left and right is explained by noting that spinors of valence $(k,0)$ represent left-handed particles, while spinors of valence $(0,k)$ represent right-handed particles, cf. [@PenRin84 §5.7]. The Dirac equation is the equation for massive, charged spin-1/2 fields, and couples the left- and right-handed parts of the field, see [@PenRin84 §4.4]. We shall not consider the symmetry operators for the Dirac equation here.
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KZFR
KZFR 90.1 FM is a non-profit, community radio station located in Chico, California. The idea started out as a translator for KPFA in Berkeley in 1981, but when it began transmitting in July 1990 it was locally originated content. KZFR broadcasts local Internet station Radio Paradise from midnight to 6 AM.
KZFR features diverse programming. Music programs include shows devoted to reggae, rock and roll, folk, Celtic, blues, jazz, country, zydeco and spiritual music. Informational programs include Democracy Now, The Real Issue, and the Peace and Social Justice Show. KZFR also occasionally broadcasts concerts live, usually from the Big Room at the Sierra Nevada Brewery. KZFR also has several Americana themed shows. KZFR can also be heard via the internet at www.kzfr.org.
See also
Community Radio
Sacramento Valley
National Federation of Community Broadcasters
List of community radio stations in the United States
External links
KZFR - People Powered Radio
ZFR
Category:Community radio stations in the United States
Category:Public benefit corporations
Category:1990 establishments in California
Category:Radio stations established in 1990
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{
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The Russian beauty queen at the centre of a divorce row with the former King of Malaysia has challenged him to take a DNA test after he claimed their son may not be his.
Rihana Oksana Veovodina married Sultan Muhammad V last year and he dramatically gave up his throne in January amid growing pressure over their relationship.
The Sultan now claims they have divorced, and is said to have 'disinherited' their two-month-old child Leon after raising doubts over the baby's parentage.
His lawyer Koh told a Singapore newspaper there was 'no objective evidence as yet as to the biological father of the child'.
But friends of the former Miss Moscow insist the marriage is still valid and have angrily rejected the Sultan's suggestion that someone else fathered the child - claiming that father and son 'share one face'.
Mother and son: Rihana Oksana Veovodina holds her baby Leon, who is at the centre of a row between the couple after the ex-King of Malaysia 'disinherited' him
Sultan Muhammad V (pictured as King of Malaysia in July 2018) gave up his throne in January this year to be with the Russian model
Oksana, who is now living with her child in a country house near Moscow, is now demanding a 'public apology' from the former head of state.
Her close friend Liliya Nastaeva, 32, said today of the Sultan's fatherhood claims: 'What nonsense.
'We are very sorry that his royal, though former, highness chose such a talkative and unprofessional lawyer.
'All people close to Oksana - her parents, friends, family - are waiting for a public apology from him.
'And if (the former king) has some suspicions, then he can always do a DNA test. Oksana, for her part, is ready for this - I can tell you as I am her closest friend.
Rihana Oksana Veovodina married Sultan Muhammad V (pictured together) last year and he dramatically gave up his throne in January amid growing pressure over their relationship
'If Oksana had ever shown the face of a child, no one would have any doubts, because father and son share one face.'
Asked if the Sultan had seen the child, Liliya - who formerly ran a Moscow beauty parlour with Oksana - replied: 'I don't think I can talk about it.'
Oksana's lawyer in Moscow, Evgeny Tarlo, a former Russian senator, told MailOnline the fatherhood claims were 'lies' and a 'vulgar stupidity'.
He challenged the Singapore lawyer to prove he was acting for the ex-king.
'I do not know who Mr Koh Tien Hua is, and by what right is he commenting on the personal life of Oksana Voevodina,' he said.
'He must show his credentials.'
He insisted to Komsomolskaya Pravda news paper in Moscow that there had been no divorce, despite widespread reports to the contrary.
'There was no divorce, I tell you with all responsibility,' he said.
'As for paternity, what is there to comment on? It is they who must prove that the child is a stranger, and it is not for us to justify ourselves.
Lyudmila Gorbatemko (pictured) is the mother of the bride, and is believed to be helping take care of the two-month-old baby in Moscow
Friends of the former Miss Moscow (pictured in her beauty queen days) insist the marriage is still valid and have rejected the Sultan's suggestion that someone else could have fathered the child
'Oksana is a married woman, and Faris [the ex-king] is the father of her child. It is immutable.'
Royal sources have said that the Sultan intends to 'provide for' the mother and child but they have not been granted royal titles.
The couple married in a secret Islamic ceremony on June 7 last year, when Muhammad V still ruled as King of Malaysia.
A further ceremony then followed in November 2018 in Moscow, details of which leaked out to the media.
Concerns began to grow in Malaysia after lurid reports emerged about the beauty queen's past, including a Russian reality show appearance in which she was reportedly filmed having sex in a swimming pool.
A collection of raunchy modelling pictures also began to circulate, fuelling further discontent in conservative Malaysia.
Facing growing pressure, Sultan Muhammad V dramatically gave up his throne in January this year to be with the Russian model.
The couple married in Moscow last year in a shock ceremony, before Muhammad abdicated his throne in January. She gave birth in May
Oksana Veovodina, the Russian beauty queen who married a Malaysian sultan, released a video of the pair talking about their marriage amid reports that they have gotten divorced
The abdication was the first for the country since its independence from British rule in 1957.
Lawyers for the Sultan insist that the couple have now divorced after Muhammad performed a ceremony in which he rejected his wife three times in the presence of witnesses.
However, the divorce proceedings have been shrouded in secrecy, and Kelantan officials earlier denied any knowledge of them.
Oksana has not commented directly on the latest twists but posted: 'When you are a good person, you don't lose people, people lose you.'
Earlier she insisted she was 'so proud that Malaysian blood flows in the veins of my son', also saying she hoped one day he would be Malaysian king.
The royal bride's father Andrey Gorbatenko refused to comment on the scandal.
'I'm sorry, I won't say anything, I'm on holiday,' he said.
Previously he said that if he doubted the king he would not have given his blessing to their marriage.
After Muhammad's abdication, Sultan Abdullah Sultan Ahmad Shah - a keen athlete who holds a string of positions on sporting bodies - was chosen as Malaysia's new king during a special meeting of the country's Islamic royalty.
Malaysia has unique arrangement where the national throne changes hands every five years between the Islamic royal rulers of the country's nine states.
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Friederich Jeppe
Friederich (Fred) Jeppe (Rostock, 1834 - 1898, Transvaal) was Postmaster General of the South African Republic.
See also
Isaac van Alphen
Postage stamps and postal history of Transvaal
Der Skandal, zwei Mecklenburger Buben erproben die Globalisierung im 19. Jahrhundert
References
Category:1898 deaths
Category:People from Mecklenburg
Category:Postmasters
Category:People from Rostock
Category:1834 births
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"pile_set_name": "Wikipedia (en)"
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1.35.12 First Look!
It's been another exciting week here at Rat HQ!
We're working hard on the next content release for WWII Online and the Dev team was able to create the first alpha build for version 1.35.12 last night! Check out what we've been up to!
This release will include the first set of new content planned for 2018 including the long awaited Italian Troopers!
The Italians will, after a long wait, be in game fighting side by side with their German allies. The Italians will be integrated into the German OrBat and will not have a separate persona for this initial introduction.
The Italians will share support equipment with their German Allies and will be outfitted with unique and iconic equipment including the Carcano rifle and Beretta SMG.
The first of the Close Support vehicles will also be included in this release! We're excited to include another new way to support your team and assault the enemy positions!
The new Close Support vehicle variants that are included with this release will be:
Matilda Mk II CS
Crusader Mk II CS
Vickers Mk VIc with the upgraded Besa 15mm main armament
The CS variants will be equipped with larger caliber main weapons with HE and Smoke loadouts to support the infantry advances and take out soft targets!
There may also be a couple of surprises with this release, in addition to a number of bug fixes that we're continuously working to include!
We know we have lots more work to do but, this is an exciting start to 2018! You can count on our team at Cornered Rat Software to continue what it has demonstrated over the last year, and that is: DELIVER. So consider backing us up for an extended period of time and giving us the proper tools to do the job even better. Upgrade your subscription now by going to http://www.wwiionline.com/account, and if you need any help the support guys are waiting for you at http://support.wwiionline.com.
We'll see you on the battlefield!
S!
Dagger
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Export/Download Printable Text (.txt) CSV Multiverse id (.txt) Markdown/Reddit MTGO (.dek) MTG Salvation
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"pile_set_name": "OpenWebText2"
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Background {#Sec1}
==========
Colorectal cancer (CRC) is one of the most common causes of cancer death, bringing much inconvenience to people's daily life \[[@CR1]\]. Although an increasing number of new therapy methods are applied to CRC treatment, the therapeutic efficacy for advanced CRC patients is not good. Tumor recurrence or metastasis is still the main cause of therapy failure \[[@CR2]\] and poor prognosis. The 5-year survival rates of CRC patients were about 90% for the ones with only primary tumors, 70.4% for the ones with lymph node or peripheral metastasis, and 12.5% for the ones with distal metastasis \[[@CR3]\].
Long non-coding RNAs (lncRNAs) have been discovered to act important roles in tumor development \[[@CR4]\]. Metastasis associated in lung adenocarcinoma transcript 1 (MALAT1) is one of these functional lncRNAs, which is highly expressed in CRC patients. In our previous studies, we have successfully demonstrated that MALAT1 could promote tumor growth and metastasis in CRC by binding to SFPQ (PTB-associated splicing factor) and releasing oncogene PTBP-2 (polypyrimidine tract binding protein) from the SFPQ/PTBP-2 complex \[[@CR5]\], and increase the nuclear translocation of β-catenin from cytoplasm, thereby activating downstream genes of β-catenin signaling pathway \[[@CR6], [@CR7]\]. Ying et al also revealed that, in bladder cancer, MALAT1 promoted EMT by activating Wnt signaling in vitro \[[@CR8]\]. However, the upstream regulatory mechanism of MALAT1 is not well-elucidated.
Jumonji domain-containing protein 2C (JMJD2C), also known as KDM4C, could modulate transcription factors, establish global chromatin environments and regulate gene expression \[[@CR9]--[@CR11]\]. A number of evidences have suggested the association between JMJD2C protein and various tumors \[[@CR12]--[@CR15]\]. Histone demethylase JMJD2C holds great potential of epigenetic regulating mechanism in tumor diseases \[[@CR16], [@CR17]\], especially the regulating effect on the promoter activity of targeted genes. Based on bioinformatics analysis of the MALAT1 promoter and our previous studies, we hypothesized that, JMJD2C might influence MALAT1 promoter activity, thereby activating MALAT1/β-catenin signaling pathway and leading to the promotion of CRC metastasis.
Therefore, in this study, we aimed to investigate the new effect of JMJD2C on the metastasis of CRC cells in vitro and in vivo, and to detect the regulatory effect of JMJD2C on the expression of MALAT1 and β-catenin signaling pathway related genes. Most importantly, the potential epigenetic regulating mechanism of JMJD2C on MALAT1 was the focus of this research.
Materials and methods {#Sec2}
=====================
Human tissues collection and cell culture {#Sec3}
-----------------------------------------
A total of 78 human primary CRC tissues without metastasis and 46 human primary CRC tissues with matched hepatic or lung metastasis were collected between 2006 and 2015 at Shuguang Hospital, Shanghai University of Traditional Chinese Medicine, and Fudan University Shanghai Cancer Center. HCT116 (human colon, CRC) and LoVo (human colon, Dukes' type C, grade IV, colorectal adenocarcinoma) cells were purchased from the ATCC (Manassas, USA), and cultured in RPMI-1640 and F12K medium, respectively. Both mediums were supplemented with 10% heat-inactivated FBS, 100 U ml^− 1^ penicillin, and 100 μg ml^− 1^ streptomycin. These cells were incubated under 37 °C, 5% CO~2~ conditions. All the experimental procedures were approved by the Institutional Review Board of Shuguang Hospital, Shanghai University of Traditional Chinese Medicine.
Plasmids construction and lentivirus infection {#Sec4}
----------------------------------------------
Three shRNA fragments for human JMJD2C (Gene ID:23081) were synthesized by the Sangon Biotech company (Shanghai, China), sub-cloned into the pLKD shRNA vector, and named pLKD-CMV-G&PR-U6-JMJD2C-shRNA1/2/3, respectively. The full length of JMJD2C gene was amplified by PCR, following by gene sequencing, and the right full fragments was cloned into the plenti overexpression vector, named plenti-CMV-EGFP-P2A-JMJD2C-3FLAG. These above plasmids were used to transfect well-growing state HCT116 and LoVo cells by using Lipofectamine 3000 transfection reagent. After 48 h transfection, the transfected cells were selected with neomycin (G418, 1 mg ml^− 1^, Sigma, USA). Recombinant lentiviruses containing pLV4-shRNA/NT, pLV4-shRNA/JMJD2C, pLV4-empty vector, and pLV4-JMJD2C^+/+^ were prepared by GeneChem (Shanghai, China), respectively. HCT116 or LoVo cells were infected with 2 × 10^6^ transducing units of corresponding lentiviruses, and were selected with 2 μg/ml puromycin for 2 weeks. The efficiencies of knockdown or overexpression of JMJD2C were determined by real-time PCR and western blot.
Real-time PCR {#Sec5}
-------------
Total RNA was extracted using the TRIzol reagent (Takara) according to the manufacturer's instructions. The RNA concentrations were determined using a NanoDrop ND-1000 (NanoDrop). cDNA was synthesized with the PrimeScript RT Reagent Kit (TaKaRa) using 500 ng total RNA as template. Real-time PCR analyses were conducted to quantitate KDM4C mRNA and lncRNA-MALAT1 relative expression using SYBR Premix Ex Taq (TaKaRa) with GAPDH as an internal control. The Real-time PCR results were defined from the threshold cycle (Ct), and relative expression levels were calculated by using the 2^-△△Ct^ method. PCR was performed using an ABI 7500 instrument (Applied Biosystems, USA). The primers used for real-time PCR analysis were as follow: KDM4C, forward primer: 5-AGGCCTAAGGCTGATGAGGA-3, reverse primer: 5-TTGGCCATGAAAGCTCGGAT-3; MALAT1, forward primer: 5-GCTCTGTGGTGTGGGATTGA-3, reverse primer: 5-GTGGCAAAATGGCGGACTTT-3, GAPDH, forward primer: 5-GGTGGTCTCCTCTGACTTCAACA-3, reverse primer: 5-CCAAATTCGTTGTCATACCAGGAAATG-3.
Luciferase reporter assay {#Sec6}
-------------------------
To test MALAT1 promoter activity, HCT116 or LoVo cells were co-transfected with the recombinant plasmid pGL3-basic-MALAT1 promoter with a control positive plasmid pRL-SV40 as previously described \[[@CR18]\]. The promoter activity was analyzed using a commercial dual-luciferase assay kit (Promega, USA) according to the manufacturer's instructions.
Immunofluorescence staining {#Sec7}
---------------------------
To observe the expression and location of JMJD2C and β-catenin, HCT116 and LoVo cells after transfection were plated at a density of 2.0 × 10^4^/ml in 6-well plates, fixed with methanol, blocked with 5% BSA. The cells were first stained with β-catenin mouse antibody followed by Cy3-conjugated goat anti-mouse IgG (Millipore). After the cells were washed four times with PBS, the JMJD2C rabbit antibody was added, followed by FITC-conjugated goat anti-rabbit IgG (Millipore). Nuclear staining was done with 4′,6-diamidino-2-phenylindole dihydrochloride (DAPI) solution. Cells were imaged using TCS SP2 spectral confocal system (Leica, Germany). All experiments were conducted according to instructions from the antibody manufacturer.
Western blot analysis {#Sec8}
---------------------
Protein lysates from cells were prepared in lysis buffer and centrifuged at 12000 rpm at 4 °C. The primary antibodies used were JMJD2C (Santa Cruz, USA), β-catenin (CST, USA), c-Myc (CST, USA), ITGBL1 (Abcam, USA), PCNA (CST, USA) and GAPDH (CST, USA). The secondary antibody used was HRP-labeled goat anti-rabbit/mouse IgG (H + L) (Beyotime, China). Each band was quantitatively analyzed using quantity one Software and normalized to the expression of GAPDH in the same lane.
Chromatin immunoprecipitation assay {#Sec9}
-----------------------------------
ChIP assays were performed using a ChIP assay kit (Millipore, USA). Chromatin extracts were immunoprecipitated using 10 μg anti-JMJD2C (Abcam, USA), 4 μg anti-H3K9 (Abcam, USA), and 4 μg anti-H3K36 (Abcam, USA), respectively. IgG (Merck, Germany) were used as mock ChIP controls. Fold enrichments were calculated by determining the ratios of the amount of immunoprecipitated DNA to that of the input sample, and were normalized to the level observed at a control region, which was defined as 1.0. The ChIP primers for MALAT1 promoter are as follow: forward primer: 5-GGTCAGCCTGAGACCACTTC-3, reverse primer: 5-CTGTGCCTGTTCTGGGGAAT-3.
Transwell assay {#Sec10}
---------------
To measure cell migration, the transfected HCT116 and LoVo cells (2 × 10^5^) were seeded to the upper chambers and cultured with 100 μl serum-free F12K or RPMI1640 medium, whereas the lower chamber was filled with 600 μl F12K or RPMI1640 medium containing 15% FBS and 10 μg ml^− 1^ fibronectin. After incubation for 48 h at 37 °C and 5% CO~2~, the chambers were fixed with 4% paraformaldehyde and stained with 0.1% crystal violet. The cells in the upper chamber were carefully removed with a cotton swab. The numbers of cells were analyzed by counting five independent visual fields under a DMI3000 B inverted microscope (Leica, USA) with a 20 × objective. All assays were performed in triplicate and independently repeated three times.
Wound healing assay {#Sec11}
-------------------
Cells (4.5 × 10^5^) were seeded on a six-well plate to form a confluent monolayer in 10% FBS-containing medium. The monolayer cells were scratched by a plastic tip and washed with PBS to remove cell debris; 0.5% FBS-containing F12K or RPMI1640 were then added to each well, and the scratched monolayer was incubated in a 37 °C incubator with 5% CO~2~ for 24 h. Wound closure was measured in five random fields at × 200 magnification using Image J software and a DMI3000B inverted microscope (Leica, USA). Percentage of wound healing was calculated as follows: migrated cell surface area/ total surface area × 100, in which, migrated cell surface area = length of cell migration (mm) × 2 × length of defined areas, total surface area = beginning width × length of defined areas.
MTT assay {#Sec12}
---------
Cells were cultured at a density of 2.5 × 10^3^ cells per well in a flat-bottomed 96-well plate. MTT \[3-(4,5)-dimethylthiahiazo(−z-y1)-3,5-di-phenytetrazoliumromide\] was used to determine the cell viability by measuring the absorbance at 490 nm. All assays were performed in triplicate and independently repeated three times.
Flow cytometry {#Sec13}
--------------
Cells were harvested by trypsinization, washed once with cold PBS. Then, the cells were stained with PI (propidium iodide) and anti-Annexin-V antibody (Becton Dickinson, USA) at 4 °C for 1 h. Subsequently, cells were washed once with PBS and analyzed by FACS (BD, USA).
In vivo analysis {#Sec14}
----------------
All the animal experiments were performed under the approval of the Institutional Committee for Animal Research and were in accordance with national guidelines for the care and use of laboratory animals. HCT116-shRNA/NT, HCT116-shRNA/JMJD2C, HCT116-empty vector, and HCT116-JMJD2C^+/+^ cells (2 × 10^6^ cells in 100 μl) were injected into female BALB/c nude mice (6--8 weeks old, obtained from SLAC Laboratory Lab, Shanghai, China) by tail intravenous to establish the lung metastasis mice model, respectively. Mice were euthanized 42 days post injection, and the lung metastases images were observed and quantified by LB983 NIGHTOWL II system. The numbers of lung metastatic nodes were measured, and the metastatic tumor tissues were collected for immunohistological and western blot analysis with JMJD2C, β-catenin, c-Myc, and ITGBL1 antibody.
HE staining and IHC staining {#Sec15}
----------------------------
Paraffin-embedded tissues were sectioned for HE (hematoxylin and eosin) and IHC (immunohistochemistry) staining. For IHC analysis, the experiments were performed using the first antibody, HRP-conjugated secondary antibody, and DAB (diaminobenzidine) detection reagents. The DMI3000B microscope connected to the digital imaging system was applied for taking photographs and the following analysis. All the data were evaluated and classified blindly by two investigators (ZYW and LLW) from the pathology department of Shuguang Hospital, Shanghai University of Traditional Chinese Medicine.
Statistical analysis {#Sec16}
--------------------
All data are presented as means with standard deviation (SD) or median with 95% confidence interval (95% CI). Statistic comparison was performed using the Student's t-test, one-way ANOVA analysis, Mann-Whitney test, or Kruskal-Wallis test, as appropriate, with the significance level at *P*\<0.05. All statistical analyses were completed with SPSS 22 software.
Results {#Sec17}
=======
Overexpression of JMJD2C in CRC metastatic foci predicted poor prognosis {#Sec18}
------------------------------------------------------------------------
Firstly, the expression levels of KDM4C mRNA (encoding JMJD2C protein) were detected in 124 CRC tumor specimens. The results demonstrated that, the levels of KDM4C mRNA were significantly higher in recurrent tumors and metastatic sites than that of non-recurrent ones (Fig. [1](#Fig1){ref-type="fig"}a), and the mRNA levels were closely associated with TMN stages and distant metastasis (Table [1](#Tab1){ref-type="table"}). Survival analysis concluded that CRC patients with lower KDM4C mRNA expression had longer OS (Overall Survival) (Fig. [1](#Fig1){ref-type="fig"}b). In our previous reports, we have shown in 124 CRC tumor specimens that MALAT1 had higher expression levels in recurrent primary and metastatic sites, relative to non-recurrent primary tumors (7). In the present study, we further found that, highly matched co-expression of KDM4C mRNA and MALAT1 was found in our 124 CRC cases (Fig. [1](#Fig1){ref-type="fig"}c). In addition, a TCGA (The Cancer Genome Atlas)-dataset analysis also showed the elevated mRNA levels of KDM4C in 367 CRC primary tissues, and the KDM4C mRNA expression was inversely correlated with OS of CRC patients (Fig. [1](#Fig1){ref-type="fig"}d). Next, the immunohistochemistry staining further revealed significantly higher protein levels of JMJD2C (encoded by KDM4C) in recurrent tumors and lung/liver metastasis than the ones in non-recurrent tumors (Fig. [1](#Fig1){ref-type="fig"}e), implying that JMJD2C overexpression correlated closely with CRC progression. Fig. 1KDM4C expression in post-surgical, recurrent primary and metastatic sites, comparing to primary sites of non-recurrent CRC patients. **a** Expression levels of KDM4C mRNA in 124 CRC tissues and matched metastatic sites were analyzed by real-time PCR. The significant differences between primary tumor I (without paired metastatic tissues) and primary tumor II (with paired metastatic tissues, Metastasis II) were analyzed using the Wilcoxon signed-rank test. **b** Kaplan-Meier analyses of the correlations between KDM4C mRNA expression levels and overall survival (OS) of 124 CRC patients, and the median expression level was used as the cutoff. **c** The co-expression between KDM4C and MALAT1was showed in 124 cases from our datasets. **d** The mRNA expression of JMJD2C in 367 CRC primary tissues from TCGA dataset, and its association with CRC prognosis, including OS. **e** Immunohistochemical analysis of JMJD2C protein (encoded by KDM4C) in representative CRC and metastatic lung/liver tissues, including primary CRC tumor without paired metastatic tissues, and primary CRC tumor with paired metastatic tissues (scale bars, 100 μm, respectively). \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test) Table 1Association between KDM4C expression and clinicopathological variables of CRC patientsVariablesLow KDM4C expression (*n* = 67)High KDM4C expression (*n* = 57)*P*n (%)n (%)Age \> 6513 (19.40)10 (17.54)0.8210 ≤ 6554 (80.60)47 (82.46)Gender Male39 (58.21)39 (68.42)0.2673 Female28 (41.79)18 (31.58)Tumor site Rectum37 (55.22)36 (63.16)0.4642 Colon30 (44.78)21 (36.84)Tumor differentiation Well20 (29.85)19 (33.33)0.7019 Moderate + Poor47 (70.15)38 (66.67)TNM stage Stage II35 (52.23)12 (21.05)0.0004\*\* Stage III32 (47.77)45 (78.95)Lymph vascular invasion Positive44 (65.67)42 (73.68)0.4347 Negative23 (34.33)15 (26.32)Perineural invasion Positive52 (77.61)46 (80.70)0.8253 Negative15 (22.39)11 (19.30)Liver metastasis Positive9 (13.43)22 (38.60)0.0017\*\* Negative58 (86.57)35 (51.40)Lung metastasis Positive1 (1.49)8 (14.04)0.0114\* Negative66 (98.51)49 (85.96)\*, *P* \< 0.05; \*\*, *P* \< 0.01
JMJD2C promoted the migration of CRC cell lines in vitro {#Sec19}
--------------------------------------------------------
To thoroughly understand the biological function of JMJD2C, we first knocked down and overexpressed JMJD2C expression using shRNA and overexpressing vector, respectively. Real-time PCR assay was carried out to detect the mRNA expression of JMJD2C and showed efficient down-regulation or up-regulation of JMJD2C in HCT116 and LoVo cells (Fig. [2](#Fig2){ref-type="fig"}a, Additional file [1](#MOESM1){ref-type="media"}: Figure S1A). The protein expression of JMJD2C was further confirmed by western blot and quantitative assay (Fig. [2](#Fig2){ref-type="fig"}b, c, Additional file [1](#MOESM1){ref-type="media"}: Figure S1B, C). Subsequently, by transwell and wound healing assays in HCT116 and LoVo cells, we found a positive role of JMJD2C in promoting CRC cells migration (Fig. [2](#Fig2){ref-type="fig"}d-g, Additional file [1](#MOESM1){ref-type="media"}: Figure S1D, E). Additionally, we also found that, JMJD2C could promote the proliferation of CRC cells (Additional file [2](#MOESM2){ref-type="media"}: Figure S2A), but affected little on the apoptosis of CRC cells (Additional file [2](#MOESM2){ref-type="media"}: Figure S2B). Fig. 2JMJD2C promoted the metastasis of CRC cells in vitro. **a-c** Real time PCR and western blotting were performed to confirm the gene silencing and overexpressing efficiency for JMJD2C. HCT116 was transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. **d** Migration assays of HCT116 cells transfected with shRNA/NT, shRNA/JMJD2C, empty vector, or JMJD2C overexpression vector, respectively. **e** Numbers of migrated cells are shown as mean ± SD; *n* = 3. **f-g** Wound healing assay was used to evaluate the effect of JMJD2C on migration of HCT116 cells. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
Nuclear translocation of JMJD2C lowered the histone methylation level of MALAT1 promoter in CRC {#Sec20}
-----------------------------------------------------------------------------------------------
Histone demethylase JMJD2C holds great potential of epigenetic regulating mechanism in tumor diseases \[[@CR19]--[@CR27]\], especially its important regulating effect on the promoter activity of targeted genes \[[@CR28], [@CR29]\]. By immunofluorescent staining assay, we found that, knockdown of JMJD2C could significantly decrease the nuclear accumulation of JMJD2C protein in CRC cells, while overexpression of JMJD2C could effectively elevate the distribution of JMJD2C protein in the nuclei of CRC cells (Fig. [3](#Fig3){ref-type="fig"}a, b). Then, above results were further validated by the western blot detection (Fig. [3](#Fig3){ref-type="fig"}c, d). Fig. 3Translocation of JMJD2C protein from the cytoplasm into the nuclei in CRC cells in vitro. **a**-**b** Immunofluorescence detection of JMJD2C protein in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. **c**-**d** Western blot and quantitative assay of JMJD2C protein (nuclear and whole cell lysates) in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
Urged by above data, we next studied if nuclear JMJD2C could bind to the promoter of MALAT1 to affect the expression of MALAT1 by chromatin immunoprecipitation (ChIP) (Fig. [4](#Fig4){ref-type="fig"}a). Based on our ChIP assay, we found that JMJD2C could bind to the promoter of MALAT1, and regulate lower the histone methylation level of MALAT1 promoter in sites of H3K9m3 and H3K36m3, whether not only in HCT116 or but also in LoVo cells (Fig. [4](#Fig4){ref-type="fig"}b). Next, the luciferase reporter assay further confirmed that, JMJD2C could promote the promoter activity of MALAT1 gene (Fig. [4](#Fig4){ref-type="fig"}c, d). Moreover, the real-time PCR results also demonstrated that, JMJD2C could promote the transcriptional level of MALAT1 (Fig. [4](#Fig4){ref-type="fig"}e, f). All above results suggested that JMJD2C could translocate into the nuclei, regulate lower the histone methylation level of MALAT1 promoter, and elevate the expression of MALAT1. Fig. 4JMJD2C promoted MALAT1 expression by regulating the histone methylation level of MALAT1 promoter. **a** Schematic diagram of the ChIP (Chromatin Immunoprecipitation) procedure for detecting the effect of JMJD2C on the histone methylation level of MALAT1 promoter in sites of H3K9m3 and H3K36m3. Purple ovals: no-targeting protein; Yellow ovals: targeting protein A; Green ovals: protein binding to targeting protein A. **b** JMJD2C, H3K9, H3K36 antibody and MALAT1 special primers were used to investigate the interaction between JMJD2C and MALAT1 promoter, IgG was used as the negative control. **c-d** MALAT1 promoter activities assay in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. **e-f** Real time PCR assay of MALAT1 levels in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
JMJD2C enhanced nuclear translocation of β-catenin and activated β-catenin signaling in CRC {#Sec21}
-------------------------------------------------------------------------------------------
Our previous studies have demonstrated that MALAT1 could enhance the β-catenin signaling pathway \[[@CR6], [@CR7]\]; thus we further investigated the role of JMJD2C on regulating the expressions of β-catenin signaling pathway related proteins including β-catenin, c-Myc, and ITGBL1 (integrin β-like 1). By immunofluorescent staining assay, we found that, knockdown of JMJD2C could significantly decrease the nuclear accumulation of β-catenin protein in CRC cells, while overexpression of JMJD2C could effectively reverse the distribution of β-catenin in the nuclei of CRC cells (Fig. [5](#Fig5){ref-type="fig"}a, b). The following western blot results further confirmed the immunofluorescent staining results (Fig. [5](#Fig5){ref-type="fig"}c, d). Fig. 5JMJD2C activated the β-catenin signaling pathway in CRC cells. **a**-**b** Immunofluorescence detection of β-catenin protein in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. **c**-**d** Western blot and quantitative assay of β-catenin protein (nuclear, cytoplasm and whole cell lysates) and β-catenin signaling downstream targets including c-Myc and ITGBL1 in HCT116 or LoVo cells transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
As we know, the nuclear β-catenin and other transcription factors such as LEF, TCF and BCL9 will function together to trans-activate downstream target genes \[[@CR30]\]. Therefore, the luciferase reporter assay was further performed to observe the effect of JMJD2C on the activity of the LEF/TCF promoter. As expected, the final results confirmed the promoting effect of JMJD2C on the activity of the LEF/TCF promoter (Additional file [3](#MOESM3){ref-type="media"}: Figure S3). In addition, in Fig. [5](#Fig5){ref-type="fig"}c, d we also showed that, JMJD2C could enhance the expression of β-catenin signaling downstream targets including c-Myc and ITGBL1. All together, above data demonstrated JMJD2C enhanced the activity of β-catenin signaling pathway in CRC.
JMJD2C promoted CRC lung metastasis in vivo {#Sec22}
-------------------------------------------
In order to observe the in vivo effect of JMJD2C, HCT116-shRNA/NT, HCT116-shRNA/JMJD2C, HCT116-empty vector, and HCT116- JMJD2C^+/+^ cells were injected into nude mice (into the lateral tail vein), and the lung metastases images were observed by LB983 NIGHTOWL II (IVIS) system (Fig. [6](#Fig6){ref-type="fig"}a). As shown in Fig. [6](#Fig6){ref-type="fig"}b, knockdown of JMJD2C could significantly decrease the luciferase intensity of CRC cells in lung metastasis, while overexpression of JMJD2C could increase the luciferase intensity of CRC cells in lung metastasis. Additionally, the numbers of lung metastatic lesions were in accordance with the luciferase detecting results (Fig. [6](#Fig6){ref-type="fig"}c). Then, we detected the expression of JMJD2C protein in the lung metastatic lesions by immunohistochemical assay. The results showed the positive roles of JMJD2C in promoting CRC cells metastasis in vivo (Fig. [6](#Fig6){ref-type="fig"}d, e). Fig. 6JMJD2C promoted the metastasis of CRC cells in vivo. **a-b** HCT116-shRNA/NT, HCT116-shRNA/JMJD2C, HCT116-empty vector, and HCT116-JMJD2C^+/+^ cells were respectively injected into the tail vein of nude mice (*n* = 6). After 42 days, the lung metastases images were observed and quantified by LB983 NIGHTOWL II system. **c** Quantification of lung metastatic nodules from 6 mice subjected to the indicated treatments. **d-e** Immunohistochemical and quantitative analysis of JMJD2C proteins on consecutive tissue microarray slides of lung metastatic nodules from 6 mice subjected to the indicated treatments. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
JMJD2C elevated the expression of MALAT1 and β-catenin signaling related proteins in CRC lung metastasis mice models {#Sec23}
--------------------------------------------------------------------------------------------------------------------
Since the in vitro study has found the regulatory effect of JMJD2C on the expression of MALAT1 and β-catenin signaling related proteins, we next validated the results in vivo. Real-time PCR showed that, knockdown of JMJD2C could significantly decrease the expression of MALAT1 in the lung metastatic lesions, while overexpression of JMJD2C could significantly elevate the expression of MALAT1 in the lung metastatic lesions (Fig. [7](#Fig7){ref-type="fig"}a). Then, the western blot experiments further demonstrated that, knockdown of JMJD2C could significantly decrease the nuclear accumulation of β-catenin protein in the lung metastatic lesions, while overexpression of JMJD2C could effectively reverse the distribution of β-catenin in the nuclei of lung metastatic lesions (Fig. [7](#Fig7){ref-type="fig"}b, c). By immunohistochemical assay we further found that, knockdown of JMJD2C could significantly decrease the expression of β-catenin signaling downstream targets including c-Myc and ITGBL1 in the lung metastatic lesions, while overexpression of JMJD2C could significantly elevate the expression of c-Myc and ITGBL1 in the lung metastatic lesions (Fig. [7](#Fig7){ref-type="fig"}d, e). Fig. 7JMJD2C elevated the expression of MALAT1 and β-catenin signaling related proteins in CRC lung metastasis mice models. **a** Real-time PCR was performed to detect the expression of MALAT1 in lung metastatic nodules from 6 mice subjected to the indicated treatments. **b-c** Western blot and quantitative assay of β-catenin protein (nuclear, cytoplasm and whole cell lysates) in the lung metastatic tissues from 6 mice subjected to the indicated treatments. **d-e** Immunohistochemical and quantitative analysis of ITGBL1 and c-Myc proteins on consecutive tissue microarray slides of lung metastatic nodules from 6 mice subjected to the indicated treatments.\*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test)
Discussion {#Sec24}
==========
Histone demethylase JMJD2C has been reported to play crucial roles in the progression of colon cancer \[[@CR19], [@CR20]\], breast cancer \[[@CR21]--[@CR23]\], prostate cancer \[[@CR24], [@CR25]\], gastric cancer \[[@CR26]\], lung cancer \[[@CR27]\] and so on, indicating that JMJD2C represents a promising anti-cancer target.
JMJD2C is mapped to human chromosome 9p24.1, and it has been proved to be a demethylase for H3K9 and H3K36 methylation \[[@CR28], [@CR29]\]. More evidences have shown that JMJD2C is a candidate oncogene both in normal biology and tumorigenic processes \[[@CR31]\]. Ye et al \[[@CR23]\] found that knockdown of JMJD2C inhibited the proliferation of breast cancer cells in vitro and in vivo. In addition, Kim et al firstly uncovered that JMJD2C was overexpressed in five colon cancer cell lines and especially associated closely with the growth of HCT116 cells, and JMJD2C might promote the survival of colon cancer cells and stimulate the proliferation of colon cancer cells via up-regulating the levels of Cyclin D1 and FRA1 \[[@CR19]\]. These results revealed the important roles of JMJD2C in tumors through targeting different oncogenes or tumor suppressor genes.
Our previous studies have successfully demonstrated that MALAT1 could promote CRC metastasis through regulating β-catenin signaling pathway \[[@CR6], [@CR7]\]. However, the upstream regulatory mechanism of MALAT1 is not well-elucidated. Based on bioinformatics analysis of the MALAT1 promoter, we found the potential binding sites for JMJD2C in the promoter of MALAT1, and hypothesized that, JMJD2C, the important histone demethylase, could influence the activity of MALAT1 promoter, thereby regulating MALAT1/β-catenin signaling pathway and leading to the promotion of CRC metastasis.
In the presented paper, we demonstrated that the mRNA expression levels of JMJD2C in CRC metastatic lesions are significantly higher than the ones in primary lesions, and there is a positive correlation between JMJD2C mRNA expression and MALAT1 expression. Our previous studies have shown the promoting effect of MALAT1 on the proliferation and migration of CRC cells \[[@CR6]\]. Currently, our results further suggested that JMJD2C can also promote the proliferation and migration of CRC cells in vitro. Nevertheless, the direct evidence about the interaction between JMJD2C and MALAT1 is not presented. Various studies have demonstrated the complicated biological function of JMJD2C. Ishimura et al has proved that overexpression of JMJD2C could up-regulate the expression of oncogene Mdm2 and lead to the decreased expression of tumor suppressor gene p53 \[[@CR32]\]. Our data suggested that JMJD2C elevated the expression of MALAT1 and regulated the β-catenin signaling pathway, as well as the downstream gene expression including c-Myc and ITGBL1. Further mechanism studies demonstrated that, JMJD2C could directly bind to the promoter region of MALAT1, lower the histone methylation level of MALAT1 promoter in the sites of H3K9me3 and H3K36me3, and enhance the promoter activity and the final expression of MALAT1.
Ye et al has shown that knockdown of JMJD2C not only inhibited the breast tumor growth but also effectively blocked the lung metastasis in mice model \[[@CR23]\]. In our study, by establishing the lung metastasis mice model, we found that JMJD2C could promote CRC metastasis in vivo. Moreover, in vivo, we also found that, JMJD2C could significantly elevate the expression of MALAT1 in the lung metastatic lesions, increase the nuclear accumulation of β-catenin protein, and elevate the protein expression of downstream targets including c-Myc and ITGBL1 in the lung metastatic lesions. These results further confirmed that JMJD2C could up-regulate the MALAT1 expression and enhance the activity of β-catenin signaling pathway, which were in accordance with the results in vitro.
Conclusion {#Sec25}
==========
In summary, our findings imply that JMJD2C plays a vital role in the epigenetic regulation process. In details, JMJD2C could directly influence the expression of MALAT1 by regulating the histone methylation level of MALAT1 promoter in the sites of H3K9me3 and H3K36me3, enhancing the activity of β-catenin signaling pathway, and promoting CRC metastasis (Fig. [8](#Fig8){ref-type="fig"}). Thus, JMJD2C might be a useful target for the prevention and treatment of CRC metastasis, and the interaction between JMJD2C and MALAT1 could be considered as the points for drug development. Fig. 8A schematic model of JMJD2C promoted CRC metastasis by regulating the histone methylation level of the MALAT1 promoter and enhancing β-catenin signaling pathway
Supplementary information
=========================
{#Sec26}
**Additional file 1: Figure S1.** JMJD2C promoted the metastasis of CRC LoVo cells. a-c Real time PCR and western blotting were performed to confirm the gene silencing and overexpressing efficiency for JMJD2C. LoVo was transiently transfected with shRNA/NT vector, shRNA/JMJD2C vector, empty overexpression vector, or JMJD2C overexpression vector. d Migration assays of LoVo cells transfected with shRNA/NT, shRNA/JMJD2C, empty vector, or JMJD2C overexpression vector, respectively. e Numbers of migrated cells are shown as mean ± SD; *n* = 3. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test). **Additional file 2: Figure S2.** JMJD2C promoted the proliferation of CRC HCT116 cells, but affected little on the apoptosis of the indicated cells. a MTT assay of HCT116 cells transfected with shRNA/NT, shRNA/JMJD2C, empty vector, or JMJD2C overexpression vector, respectively. b Flow cytometry was performed to measure the apoptosis rates of HCT116 cells transfected with shRNA/NT, shRNA/JMJD2C, empty vector, or JMJD2C overexpression vector, respectively. **Additional file 3: Figure S3.** JMJD2C enhanced the activity of the LEF/TCF promoter. a-b LEF/TCF promoter activity assay in HCT116 and LoVo cells transfected with shRNA/NT, shRNA/JMJD2C, empty vector, or JMJD2C overexpression vector, respectively. \*, *P* \< 0.05; \*\*, *P* \< 0.01 (*t* test).
ChIP
: Chromatin immunoprecipitation;
CRC
: Colorectal cancer
DAB
: Diaminobenzidine
HE
: Hematoxylin and eosin;
IHC
: Immunohistochemistry
ITGBL1
: Integrin β-like 1
JMJD2C
: Jumonji domain-containing protein 2C
lncRNAs
: Long non-coding RNAs
MALAT1
: Metastasis associated in lung adenocarcinoma transcript 1
OS
: Overall survival
PTBP-2
: Polypyrimidine tract binding protein
SFPQ
: PTB-associated splicing factor
**Publisher's Note**
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Xinnan Wu, Ruixiao Li and Qing Song contributed equally to this work.
Supplementary information
=========================
**Supplementary information** accompanies this paper at 10.1186/s13046-019-1439-x.
We thank Professor Sizhi Paul Gao from Department of Medicine, Memorial Sloan Kettering Cancer Center.
QJ and QL conceived and designed the study and the experimental setup and wrote the manuscript. XW, CZ, QS, RL, ZH, RJ, YW, LZ, LY, and HS performed the experiments. QJ, QL and HZ analyzed the data. All authors read and approved the final version of the manuscript.
This work was supported by National Natural Science Foundation of China (No. 81830120 to QL), National Natural Science Foundation of China (No. 81573749 to QJ), National Natural Science Foundation of China (No. 81673783, No. 81874405 to XL), National Natural Science Foundation of China (No. 81603457 to ZFH), National Natural Science Foundation of China (No. 81874399 to HRZ), Science Foundation of Shanghai Committee of Science Project (No. 16401970500 to QL), Shanghai Rising-Star Program (No. 18QA1404100 to XL).
All of the data and materials in this paper are available when requested.
All the procedures involving human tumor biopsies were performed with the approval of the ethics committee of the Shuguang Hospital, Shanghai University of Traditional Chinese Medicine on samples from patients who had given written informed consent. All the animal protocols were approved by the Animal Care Commission of Shanghai University of Traditional Chinese Medicine.
Not applicable.
The authors declare that they have no competing interests.
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Friday, February 12, 2010
Goal
As I've mentioned on here many, many, many times, we're going to Hawaii at the end of April. I want to feel more comfortable in my body, I want to be more toned AND I want a new suit. Usually I just swim in whatever Speedo I happen to be wearing at the time for lap swimming, but I want something cute.
Here's what I'm thinking:In the blue. Or maybe black.
It's an LL Bean suit and lovely LL Bean actually has it in long torso, which means a minimum of wedgies and minimum of feeling hunched over because the straps are tugging my shoulders down in order for my bum to stay covered. Unfortunately the number of long torso suits out there is kind of limited (though there are a lot more available now than several years ago). But I thought this one was pretty cute, especially the little twisty detail on the back.
|
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Top DUI News for
the Week of December 27th - January 2nd in the year Two Thousand and
Eleven
Sunday January 2nd, 2010
DUI houseboater to face court
AN ALDINGA man has been reported for driving a houseboat on the River Murray with almost double the legal alcohol limit.
Saturday January 1st, 2011
Fair City's DUI chief fired from St. Tammany
A newly-hired patrolman at the Franklinton Police Department whose duties will include heading up the department's DUI task force was fired earlier year from the St. Tammany Sheriff's Department for his alleged role in a fight at a Slidell bar.
Wedneday December 29th, 2010
DUI driver held on $150K bond after fighting with cop in car
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ..
Monday, December 27th, 2010
Chris Leben Says DUI Won't Be An Issue
UFC 125 will feature Chris Leben (21-6), who was arrested in October on suspicion of DUI. Leben has a February court date, but told ifight365.com he doesn't expect it to pose a problem on Saturday when he faces Brian Stann (9-3). George [W.] Bush got a DUI and he ended up being the president," Leben told the website. "I really feel as though nothing's really going to happen with it, honestly ...
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ...
Chicago Sun-Times - Dec 28 07:02am
Drunk Driver Caught in Parking Lot: Cops - FOX News Chicago
Cop smashes drunken driver's window, gets in - Chicago Sun-Times
all 5 news articles
Omahan sentenced on DUI
Aaron Danoff, who is charged in the accident that killed Jessica Bedient of Omaha, has pleaded guilty and been sentenced in a separate drunken driving case.
Omaha World-Herald - Jan 01 03:47pm
DUI driver charged over police chase crash
A man will face court this morning after he allegedly led police on a pursuit in Sydney's west which ended in a crash.
LAKEWOOD Police charged Louis Cruz, 20, of Conover Street, Freehold Township, with driving while intoxicated after he abandoned his overturned car and one sneaker in a wooded area off Cedar Bridge Avenue early Saturday morning. Brick Police responded to a 6:15 a.m. report of an overturned car on Cedar Bridge Avenue at the Lakewood border. Brick Police officers Thomas Caufield, Scott Smith ...
Asbury Park Press - Jan 01 04:16am
DUI Interlock Device To Become Mandatory Jan. 1
Drivers who get behind the wheel after having too much to drink, and individuals who can't prove they're in the country legally, face tougher consequences in Tennessee with the start...
Eyewitness News Memphis - Dec 31 12:17pm
Suspected DUI Chase, Crash Shuts Down 2 Lanes On 405 Freeway
A suspected drunken driver being pursued by officers crashed into the center divider of the southbound San Diego (405) Freeway early Thursday morning, shutting down two lanes during the morning commute.
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ...
Chicago Sun-Times - Dec 28 07:02am
Drunk Driver Caught in Parking Lot: Cops - FOX News Chicago
Cop smashes drunken driver's window, gets in - Chicago Sun-Times
all 5 news articles
Ohio man charged with DUI in Kane County crash
An Ohio man was charged today with drunk driving, accused of causing a three-vehicle crash in Kane County Wednesday night that left two people in critical condition, the county sheriff said. Eric Barth, 21, has been charged with six counts of aggravated driving under the influence and cited with several traffic violations, including driving without a driver's license. His blood-alcohol level was ...
Chicago Tribune - Dec 30 04:26pm
Man Killed In Possible DUI Crash In Federal Way
A man is killed by a possible DUI driver who ran a red light in Federal Way, officers say.
KIRO 7 Seattle-Tacoma - Dec 29 07:58am
DUI warrant sweep under way in Alameda County
Armed with 1,100 warrants for drunken drivers, eight law enforcement agencies in Alameda County began knocking on doors this morning and making arrests, a spokeswoman for the county's Avoid the 21 program said.
A newly-hired patrolman at the Franklinton Police Department whose duties will include heading up the department's DUI task force was fired earlier year from the St. Tammany Sheriff's Department for his alleged role in a fight at a Slidell bar.
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ...
Chicago Sun-Times - Dec 28 07:02am
Drunk Driver Caught in Parking Lot: Cops - FOX News Chicago
Cop smashes drunken driver's window, gets in - Chicago Sun-Times
all 5 news articles
DUI Processing Increases During the Holidays
The number of rides to the jailhouse garage increase during the holidays. It's the first involuntary stop on the journey of a driver arrested for a DUI.
A suspected drunken driver being pursued by officers crashed into the center divider of the southbound San Diego (405) Freeway early Thursday morning, shutting down two lanes during the morning commute.
CBS 2 Los Angeles - Dec 30 09:46am
DUI Suspect Crashes During Pursuit - FOX 11 News Los Angeles
all 2 news articles
DUI Laws Changing January 1
Beginning Jan. 1, drunken drivers who had a high breath alcohol level will have to blow into a tube in order to start their car.
On January 1, a new DUI law goes into effect. It will require convicted DUI offenders who had a BAC of .15 or higher to install an ignition interlock device in their vehicle in order to have a restricted license.
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ...
Beginning Jan. 1, drunken drivers who had a high breath alcohol level will have to blow into a tube in order to start their car.
WSMV Nashville - Dec 29 07:26pm
Vt. Auditor: 'Don't mess around with DUI'
The Vermont auditor turns his own DUI arrest into a safe driving message this holiday.
WCAX-TV Vermont - Dec 28 05:11am
Vt. auditor to help DUI patrol On New Year's Eve - Boston Globe
Drinking on NYE? Hitch a ride with Auditor Salmon - Bennington Banner
all 6 news articles
340 arrested in holiday DUI crackdown
More than 300 people have been arrested on suspicion of driving under the influence in Orange County during a campaign that went into effect through Christmas Day, according to a multiagency DUI task force.From Dec. 17 to Dec. 26, authorities...
Mr. Ticket, traffic ticket attorney in California, has an important message to give people that have been charged with a DUI. The message is simple: legal representation is needed from the beginning of the process. Waiting can result in losing one's driver's license, receiving prison time, and paying hefty fines in excess of what a lawyer could have gotten for a defendant. For more information ...
Top DUI News, Wedneday December 29th, 2010
DUI driver held on $150K bond after fighting with cop in car
A Riverside police officer was injured after busting through a passenger car window with a baton in an attempt to get a drunk driver to stop reversing through a Walgreens parking lot in Berwyn.Joshua J. Kemper, of Lyons, was ordered held on $150,000 bond in a Monday hearing, Riverside police said. He was charged with felony fleeing and eluding police, two counts of felony DUI, resisting arrest ...
The Washington Redskins say defensive lineman Joe Joseph has expressed remorse for his arrest on charges of driving under the influence and that the team will withhold further comment until the matter is resolved through the courts. The Redskins released a two-sentence statement Tuesday on Joseph's arrest.
If you come across your self in urgent will need of a DUI lawyer to symbolize by yourself in the court as you have currently being arrested with DUI charge, exactly where really should you go about ...
PHOENIX - Over 2,700 Arizona drivers have been arrested for DUI since Thanksgiving this year; over 3,600 drivers were charged during the same period last year.
KVOA Tucson - Dec 27 08:40am
Our View: Keep off the road if you've been drinking - The Sierra Vista Herald
all 2 news articles
Device monitors DUI offenders in Stanislaus
Nearly two dozen Stanislaus County residents previously convicted of driving under the influence had their licenses suspended Monday because they haven't installed a monitor in their vehicles.
Modesto Bee - Dec 27 10:54pm
Raise a pint - and read the DUI message - Bennington Banner
A Safe and Happy New Year's - Daily Sparks Tribune
Super badge - The Plainville Citizen
all 8 news articles
DUI accident can cause hardships at anytime
Last year there were nearly 900 alcohol-related car accidents in Hamilton County. With New Year's a few days away, police say it's a good reminder that alcohol-related crashes can happen anytime, anywhere.
FOX 19 Cincinnati - Dec 27 01:54pm
Driver in fatal Struppler crash arrested for DUI - Morning Times
Police: Driver found asleep behind wheel on Rt. 30 cited for DUI - York Daily Record
A Cocoa police officer arrested and charged with two counts of driving under the influence will remain on paid administrative leave as his department conducts an internal affairs investigation, officials said today.
A Tower City woman who was DUI when she caused a fatal accident in March while driving the wrong way on Interstate 81 faces more than two decades in state prison.Rhonda M. Gettle, 27, was sentenced Wednesday to 3 1/4 years to 22 years in a state correct
The Pottsville Republican & Herald - Dec 23 08:38pm
Woman sentenced to state prison for DUI crash that killed Lebanon man - The Patriot-News
Schuylkill woman sentenced for fatal DUI crash - Lebanon Daily News
all 5 news articles
UFC 125: Chris Leben Says DUI Won't Be An Issue
UFC 125 will feature Chris Leben (21-6), who was arrested in October on suspicion of DUI. Leben has a February court date, but told ifight365.com he doesn't expect it to pose a problem on Saturday when he faces Brian Stann (9-3). George [W.] Bush got a DUI and he ended up being the president," Leben told the website. "I really feel as though nothing's really going to happen with it, honestly ...
Bleacher Report - Dec 26 09:08am
November DUI/DWAI Convictions - The Durango Herald
DUI convictions -- Published Dec. 19, 2010 - The Stockton Record
DUI Convictions - West Hawaii Today
all 115 news articles
Alabama's DUI Rapper a Future Problem for the Team or a Brilliant Pick-Up?
Aaron Douglas is a 6'6" 290 pound offensive lineman with good footwork and lots of muscle. He is considered one of the best junior college prospects in America. He was a freshman All American at Tennessee and his work at Arizona Western Community College was considered outstanding.
CBS Sports - Dec 25 05:29am
Former Vol Aaron Douglas charged with DUI - WVLT-TV Knoxville
Ex-Vol is charged with DUI - The Tennessean
Douglas Hit With DUI - WTVC Chattanooga
all 27 news articles
Kane Co. warns of "no-refusal" DUI sting on New Year's Eve
Kane County law enforcement officials will continue their campaign of cracking down on drunken driving with another "no-refusal" operation on New Year's Eve. Under the "no-refusal" program, police in certain areas of the county will be pulling over suspected drunken drivers and county prosecutors will be on call to draft warrants aimed at forcing the drivers to provide breath or blood samples ...
Chicago Tribune - Dec 22 03:41pm
Kane County to attack DUI on New Year's Eve - Chicago Tribune
Another "no refusal" holiday weekend on tap in Kane - The Courier News
New Year's Eve "no refusal" weekend on tap in Kane - The Courier News
all 8 news articles
Ravens rookie Kindle arrested and charged with DUI
Arrest continues troubled season for linebacker who suffered fractured skull in preseason Ravens rookie Sergio Kindle's troubled first season took another hit Sunday morning when he was arrested and charged with drunken driving in Howard County, an incident the player called a "mistake."
SANTA ANA, Calif., Dec. 22 (UPI) -- A judge in California Wednesday sent the man who killed pro baseball pitcher Nick Adenhart and two others in a drunken driving accident to prison for 51 years.
UPI - Dec 22 01:10pm
Nick Adenhart Case: Drunk Driver Andrew Gallo Sentenced to 51 to Life - The Epoch Times
Andrew Gallo, Drunk Driver in Nick Adenhart Case, Sentenced to 51 to Life - The Epoch Times
all 3 news articles
Councilman De La Fuente arrested, suspected of DUI
Oakland City Councilman Ignacio De La Fuente was arrested on suspicion of drunken driving Thursday night, the California Highway Patrol said. De La Fuente was pulled over for speeding near the Fruitvale Avenue exit of...
|
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|
Playstation Store Update: Sony Unwraps its Holiday Deals
Today via the official Playstation Store Update, Sony unveiled whats to come in the holiday season for the Playstation Store. Starting December 3, a wealth of discounts to full PS3 and Vita games will go into effect, and special digital bundles will also see a price cut.
The discounts range from 50% -70%, depending on whether you’re a Playstation Plus subscriber. You can find the full list below.
Defiance
Sale Price – $4.99
PS Plus Price – $2.50
Defiance Deluxe Edition
Sale Price – $14.99
PS Plus Price – $7.50
Army of Two The Devil’s Cartel
Sale Price – $14.99
PS Plus Price – $7.50
The Bureau: XCOM Decalssified
Sale Price – $29.99
PS Plus Price – $20.99
Remember Me
Sale Price – $23.99
PS Plus Price – $19.19
Terraria
Sale Price – $7.49
PS Plus Price – $3.75
Ni No Kuni: Wrath of the White Witch
Sale Price – $13.99
PS Plus Price – $9.79
Final Fantasy XIV
Sale Price – $19.99
PS Plus Price – $10.00
Final Fantasy XIV Collector’s Edition
Sale Price – $29.99
PS Plus Price – $15.00
Rocksmith 2014
Sale Price – $41.99
PS Plus Price – $29.39
In addition these more drastic price cuts, special “Holiday Edition” bundles for the following EA titles will also see a price cut to $49.99:
Battlefield 4
Madden NFL 25
NCAA Football 14
Need for Speed Rivals
FIFA 14
With the release of last year’s JRPG darling Rainbow Moon on the Vita, the Cross-Buy version of the game will be discounted to $22.49. Interestingly, developer EastAsiaSoft has also instituted a special discount for those buying the Vita version of the game who previously owned the PS3 version. This edition is deemed the “PS3 Player Special Offer” and discounts the game 50% to $7.49.
|
{
"pile_set_name": "Pile-CC"
}
|
Q:
How to add onClick event to an a tag with class
Here's what I have on a page
<a class="download-link" href="http://www.test.com/pages-">Pages 1</a>
Below is the Result I want to achieve with jQuery.
<a class="download-link" href="http://www.test.com/pages-1.asp" onclick="_sz.push(['event', 'PDF', 'download', ('report') ])">Pages 1</a>
jQuery
<script type="text/javascript">/
$('.download-link').each(function(){
this.href += '-1.asp';
this.attr('onclick', '_sz.push(['event', 'PDF', 'download-request', ('report') ])');
})
</script>
A:
try this one.
$('.download-link').bind('click', function (event) {
_sz.push(['event', 'PDF', 'download', ('report')])
});
I hope. It helps you.
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{
"pile_set_name": "StackExchange"
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Pumping nutrients into the water supply at the front end would tend to boost the fertilizing potency of sewer water discharged at the other end. The area’s sewage treatment plant had recently spent millions of dollars to reduce the area’s nutrient discharges, which cause excessive algae and weed growths in lakes and streams, and officials didn’t want to offset those gains, Grande said.
But the alternative, replacing lead pipes, would be costly, and the water utility was legally responsible only for the lead pipe that ran from the iron water mains under the streets to the property line of a home or business. The rest of the lead connection leading to the tap was the property owner’s responsibility.
Grande said the DNR insisted on a guarantee that all the lead would be replaced, so the city passed an ordinance requiring residents in older properties to replace their pipes, setting off a backlash over costs that would run to thousands of dollars for each homeowner.
“We had to figure out a way, particularly for low-income people, how they were going to be able to afford that,” said Sue Bauman, who was mayor from 1997 to 2003.
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{
"pile_set_name": "OpenWebText2"
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This invention relates to a novel method for precision etching of a succession of articles from a moving metal strip.
The method is particularly useful for preparing flat apertured masks which are subsequently formed and installed in color television picture tubes.
Precision etching is employed to produce articles with complex arrays of apertures therein, where the sizes and shapes of the apertures are to be held within very narrow size tolerances. An apertured mask, which is an important part of a shadow-mask-type picture tube used in color television receivers, is one such article. The production of flat apertured masks by photoexposure and precision etching has been described previously. In a typical process, light-sensitive coatings are applied to both major surfaces of a continuous strip of thin sheet metal. In one practice, both major surfaces of a cold-rolled-steel strip about 0.15 mm (6 mils) thick and about 550 mm (22 inches) wide are coated with a dichromate-sensitized casein composition. The light-sensitive coatings are exposed to a succession of actinic light images, as by contact-printing exposure, to render the exposed portions thereof less soluble in water. The exposed coatings are developed to remove the more-soluble unexposed portions, thereby producing a succession of stencils on each surface of the strip, and then baked to render the retained, less-soluble, exposed portions etch resistant. Then, the strip with the etch-resistant stencils thereon is advanced through an etching station where it is selectively etched from both surfaces with an etching solution that is sprayed on the strip. The strip continues to advance beyond the etching station through successive stations where the strip is rinsed, the stencils are removed, the strip is dried and the light transmissions through the masks are monitored. The flat masks produced by the foregoing method have an array of apertures therein, which apertures are usually round holes or rectangular slits, but may be of any desired shape. Round apertures are typically about 0.30 to 0.38 mm (12 to 15 mils) in diameter, and rectangular apertures are typically about 0.13 to 0.20 mm (5 to 8 mils) wide by about 0.76 to 1.27 mm (30 to 50 mils) high. The flat mask is formed to a desired shape and detachably mounted in the faceplate panel of the tube. The formed and mounted mask is then used as an optical master for photographically depositing one or more screen structures of the tube. The mask is also used to shadow the scanning electron beams during the operation of the picture tube.
The sizes and shapes of the apertures are critical towards reliably and reproducibly implementing these functions. Many factors affect the sizes of the apertures in the mask. Some important process variables relating to the photoexposure and etching steps that are now carefully controlled are (1) temperature of the etching solution, (2) density of the etching solution, (3) pressure applied to spray the etching solution, (4) thickness of the stencils, (5) baking temperature of the stencils, (6) size of the apertures in the developed stencils, and (7) conditions for photoexposing light-sensitive coating. Even though these many process controls are applied, there is still a need to reduce the variation in aperture sizes in the etched masks.
We have found that: (a) the prior photoexposure-and-etching process produces mask apertures whose sizes, and therefore light transmissions of the masks, are very dependent on small changes in the thickness of the metal strip, and (b) the ordinary thickness range of the metal strip received from the metal supplier can produce variations in aperture sizes which are greater than can be tolerated by the user of the mask. For example, 0.025 mm (0.1-mil) change in the thickness of a 0.150 mm (6-mil)-thick steel strip can cause a change of 0.3% in the light transmission of the etched mask, or about one third of the allowable etching tolerance. Steel strip, as received from the supplier, may have a thickness variation of .+-.0.0125 mm (.+-.0.5 mil), which would, if uncompensated for, produce masks with a greater than allowable variation in transmission.
Where these wide variations in strip thickness exist, the masks must be more thoroughly inspected and a substantial proportion of the etched masks must be discarded as being out of tolerance. Frequently, masks selected for retention are classified into one of several groups according to light transmission so that wider ranges of aperture sizes can be tolerated through other compensations made later in the manufacturing processes.
|
{
"pile_set_name": "USPTO Backgrounds"
}
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CRGroup is headed to the big-easy for Microsoft Envision 2016. Here are some great reasons to join us there!
Microsoft Envision brings together the most forward-thinking minds in business and technology. Whether you’re a business leader or decision maker, you won’t want to miss the opportunity to hear the latest business trends and discover solutions that can help you, your team and your business achieve more; including the latest products and solutions offered by Corporate Renaissance Group:
Attending? Be sure to stop by the CRGroup Booth #756 to meet the team, pick up some nifty gifts. enter your name into a draw for a $250 Visa® gift card, and to get a first-hand look at the hottest business technology available.
Not Attending? No problem! Stay on top of the latest news and offers from CRGroup by visiting our Blog or by exploring our website.
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{
"pile_set_name": "Pile-CC"
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|
The 6’3, 210-pound defenseman recently completed his senior season at Boston College, recording career highs in points (13), assists (12), games played (37, tied) and penalty minutes (54) and registered a plus-14 rating this season. Wey helped Boston College to a 22-12-4 record and a berth in the NCAA tournament. He was named a 2013 Hockey East Second-Team All-Star and named Hockey East’s 2013 Best Defensive Defenseman.
In 133 career games with Boston College, Wey collected 33 points (four goals, 29 assists) and 147 penalty minutes. He helped Boston College win the NCAA National Championship in 2011-12 and tallied one goal and three assists in four tournament games. Boston College also won the Hockey East Championship and the NCAA regular-season championship last season as Wey was named to the NCAA All-Tournament Team at the Northeast Regional.
The Pittsburgh native helped the United States win the bronze medal at the 2011 Under-20 World Junior Championship, appearing in six tournament games. Wey was drafted by Washington in the fourth round (115th overall) in the 2009 NHL Entry Draft.
|
{
"pile_set_name": "OpenWebText2"
}
|
City of Redding improvements for your trips by bicycle
Have you noticed these City of Redding improvements for your bike commute? These improvements are planned for implementation for this fall, including:
Through bike lane treatment on eastbound Cypress approach to Parkview with some buffered bike lanes on Cypress. On the approach to the intersection there will be a through bike lane, double skip striping with green paint to highlight conflict zones where vehicle cross the bike lane to the right turn pocket. Also westbound bike lanes to just prior to the Pine intersection.
Bike Box and Green Markings – Redding’s First! Northbound Hartnell to Cypress – through bike lane treatment with double skip stripes, green markings to highlight conflict zone and a “two-stage queue bike box” treatment for safety and ease in making left turns. How do you use a bike box? Check out this video for a fun description of how they work.
Segment of Hartnell from Cypress to Bechelli buffered bicycle lanes in some places.
Quartz Hill Road from Benton to Market Street road diet to a single lane in each direction with bike lane buffers and improvements to the bicycle approach to intersections as well as parking.
Do you have input on these improvements for the City of Redding?
Share your thoughts and ideas with us and we will gather this input and share it with the City of Redding through the Active Transportation Advisory Group? Send your thoughts in an email to Anne Thomas.
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{
"pile_set_name": "Pile-CC"
}
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317 F.Supp.2d 220 (2004)
DCMR, d/b/a Diversified Components Manufacturing Representative, Plaintiff,
v.
TRIDENT PRECISION MANUFACTURING, Defendant.
No. 02-CV-6237T.
United States District Court, W.D. New York.
January 28, 2004.
*221 *222 Richard L. Bourland, Bourland Kirkman Seidler Evans Jay & Michel, Fort Worth, TX, Thomas N. Trevett, Trevett, Lenweaver & Salzer, P.C., Rochester, NY, for Plaintiff.
Greta Katrin Kolcon, Woods Oviatt Gilman LLP, Rochester, NY, Jeff A. Cody, Fulbright & Jaworski, Dallas, TX, for Defendant.
DECISION and ORDER
TELESCA, District Judge.
INTRODUCTION
Plaintiff DCMR, d/b/a Diversified Components Manufacturers' Representative ("plaintiff"), brings this diversity action against defendant Trident Precision Manufacturing, Inc. ("defendant"), alleging breach of contract, violation of the Sales Representative Act of Texas, fraud, quantum meruit and breach of the implied covenant of good faith and fair dealing, claiming that defendant failed to honor its contractual obligations to plaintiff. Defendant brings this motion for dismissal pursuant to Federal Rule of Civil Procedure 12 for failure to state a claim, or alternatively, for summary judgment pursuant to Federal Rule of Civil Procedure 56, arguing that no triable issue of fact exists, and thus, it is entitled to judgment as a matter of law because it has fully complied with its obligations under the contract.[1] For the reasons set forth below, defendant's motion requesting summary judgment in its favor is granted in its entirety, and each of plaintiff's claims is dismissed with prejudice.
BACKGROUND
Plaintiff is a corporation duly organized under the laws of the State of Texas, with its principal place of business located in Fort Worth, Texas, and conducts business as a manufacturer's sales representative on behalf of several principals, including defendant. Defendant is a corporation duly organized under the laws of the State of New York, with its principal place of business located in Webster, New York, and is engaged in the business of manufacturing, among other things, electro-mechanical sub-assemblies.
The basis of this action is a contract ("the Contract") entered into by the parties on July 1, 1999, and terminated by defendant on September 27, 2001.[2] By the terms of the contract, plaintiff agreed to perform certain marketing services for defendant in exchange for an agreed upon commission. Section "V(C)" of the contract provides that plaintiff is entitled to a 5% commission "on the net invoice price of TRIDENT PRODUCTS provided by TRIDENT and accepted and paid by the customer...."
Section "VII(B)" provides that either party may terminate the Contract, stating:
either party may terminate this agreement for any reason whatsoever upon thirty (30) days written notice of termination *223 to the other party. Other than a termination by TRIDENT for [DCMR's] material breach of this agreement, TRIDENT shall pay [DCMR] commissions for all orders obtained as a result of [DCMR]'s efforts that were booked, accepted and shipped prior to the end of the one hundred and twenty (120) day period commencing on the date of notice of termination by the terminated party.
The Contract further provides that upon termination, defendant's liability to plaintiff is limited, specifically stating:
It is agreed and understood that TRIDENT will not, by any reason of any termination of this agreement or for any reason whatsoever, be liable to [DCMR] for indirect, incidental or consequential damages of any kind, including, but not limited to, compensation, reimbursement or damages on account of present or prospective loss of profits on sales, goodwill, termination of employees, salaries, expenditures, investment or commitments made in connection herein.
Prior to the termination of the contract on December 27, 2001, plaintiff introduced defendant to Applied Science Fiction ("ASF"), a digital imaging company which manufactured photo kiosks and needed to locate a company to manufacture components for the kiosks. ASF anticipated eventually manufacturing 40,000 kiosks. Defendant informed ASF that it could produce the needed components for a price of $10,000 per unit. Several "test" kiosks were manufactured, but a complete order was never placed. In December 2002, ASF informed defendant that the kiosk project was on hold, and in 2003 ASF was acquired by Eastman Kodak Company. At no point did ASF sign a contract with defendant, nor was an invoice issued for kiosks other than the "test" kiosks.
Plaintiff brings this action claiming that it is entitled to a commission for the 40,000 unit forecast ASF provided defendant, and seeks relief based upon: (1) breach of contract; (2) violation of the Sales Representative Act of Texas; (3) fraud; (4) quantum meruit; (5) breach of implied covenant of good faith and fair dealing. Plaintiff also claims that it is entitled to punitive damages based on defendant's willful and malicious conduct and that it is entitled to attorneys fees incurred in this action.
Defendant moves for summary judgment requesting that the Court find in its favor and dismiss plaintiff's claims, arguing that there exists no genuine issue as to a material fact, claiming it fully satisfied its obligations under the contract with plaintiff.
DISCUSSION
Rule 56 of the Federal Rules of Civil Procedure provides that a party is entitled to summary judgment as a matter of law only where, "the pleadings, depositions, answers to interrogatories and admissions on file, together with the affidavits, if any, show that there is no genuine issue as to any material fact...." F.R.C.P. 56(c) (2003). The party seeking summary judgment bears the burden of demonstrating that no genuine issue of material fact exists, and in making the decision the court must draw all reasonable inferences in favor of the party against whom summary judgment is sought. Ford v. Reynolds, 316 F.3d 351, 354 (2d Cir.2003) (citing Marvel Characters v. Simon, 310 F.3d 280, 285-86(2d Cir.2002)). "Summary judgment is improper if there is any evidence in the record that could reasonably support a jury's verdict for the non-moving party." Id.[3]
*224 A. Breach of Contract Claim
In the Count One of its Amended Complaint (Doc. No. 27), plaintiff alleges that "[i]n breach of the Sales Representative Agreement the Defendant wrongfully and in bad faith terminated the Agreement to the damage of Plaintiff...." (Amended Complaint, Doc. No. 27, p. 3). Defendant, in its motion for summary judgment, argues that dismissal is appropriate because, by contract, the parties had the right to terminate the agreement upon thirty days notice, for any reason or no reason, and thus its choice to exercise that right can not amount to a breach of the Contract.
Under New York law, "an employer has the right to terminate an at will employee at any time for any reason or for no reason, except where that right has been limited by express agreement." Sabetay v. Sterling Drug, Inc., 69 N.Y.2d 329, 334, 514 N.Y.S.2d 209, 506 N.E.2d 919 (1987). Thus, defendant could terminate plaintiff at any time, for any reason or no reason at all, provided the Contract did not limit its broad right to do so.[4]
When a court interprets a contract, the intent of the parties governs, and words and phrases should be given their plain meaning. American Exp. Bank Ltd. v. Uniroyal, Inc., 164 A.D.2d 275, 277, 562 N.Y.S.2d 613 (1st Dep't 1990). "Rather than rewrite an unambiguous agreement, a court should enforce the plain meaning of that agreement." Id. Where, as here, "the intent of the parties can be determined from the face of the agreement, interpretation is a matter of law and the case is ripe for summary judgment." Zolotar v. New York Life Insurance Company, 172 A.D.2d 27, 30, 576 N.Y.S.2d 850 (1st Dep't 1991).
Section "VII", Paragraph "B" of the Contract states "either party may terminate this agreement for any reason whatsoever upon thirty (30) days written notice of termination to the other party. ..." By letter dated September 27, 2001, defendant informed plaintiff that it was terminating the Contract in accordance with Section "VII", Paragraph "B" of the Contract.[5] Therefore, plaintiff's first cause of action for "wrongful" breach of contract is dismissed.
B. Sales Representative Act of Texas Claim
Plaintiff's Second Cause of Action alleges that defendant wrongfully and in bad faith terminated the Contract in violation of the Sales Representative Act of Texas, codified at Tex. Bus. & Comm.Code §§ 35.81-35.86 (2002). Defendant responds claiming that the Contract and any litigation arising from the Contract is governed by New York law, and thus plaintiff's claim under Texas law should be dismissed.
Federal courts sitting in diversity in New York State must apply New York's choice of law rules when determining which law governs a contract. Lehman Bros. Commercial Corp. v. Minmetals Int'l Non-Ferrous Metals Trading *225 Co., 179 F.Supp.2d 118 (S.D.N.Y.2000). New York courts may refuse to enforce a choice of law clause only where: (1) there is no reasonable basis for the choice of law; or (2) the application of the chosen law would violate a fundamental public policy of another jurisdiction with materially greater interests in the dispute. Hartford Fire Ins. Co. v. Orient Overseas Container Lines (UK), Ltd., 230 F.3d 549, 556 (2d Cir.2000).
Here, the parties agreed that New York law would govern the Contract and any litigation arising from it. Section "IX", Paragraph "A" of the Contract directs that "[t]his agreement and the legal relations between the parties, shall be governed by and interpreted solely in accordance with the laws of the State of New York." Moreover, neither exception to New York's choice of law rule applies. First, as defendant has its principal place of business in Webster, New York, there is a reasonable basis for choosing New York law to control. Second, no fundamental public policy will be offended by the application of New York law, particularly since the parties agreed that New York State law would control their "legal relations." Therefore, I conclude that New York law, and not Texas law, governs this dispute. Accordingly, plaintiff's second cause of action, based on Texas State law, is dismissed.
C. Fraud Claim
Plaintiff's Count 3 alleges that defendant fraudulently induced plaintiff, to plaintiff's detriment, into entering a contractual relationship with it. Defendant contends that plaintiff fails to plead a proper fraud claim, and instead simply converts its breach of contract claim into a fraud claim by adding an allegation that defendant never intended to fulfill its obligations under the contract.
Under New York law, a plaintiff asserting a claim for fraud must establish: (1) a material misrepresentation or omission of fact; (2) made with knowledge of its falsity; (3) with an intent to defraud; (4) reasonable reliance on the part of the plaintiff; and (5) injury to plaintiff. Schlaifer Nance & Co. v. Estate of Warhol, 119 F.3d 91, 98 (2d Cir.1997). However, where a plaintiff alleges only that the defendant entered into a contract with no intention of performing, a claim for fraud is not generally established. Grappo v. Alitalia Linee Aeree Italiane, 56 F.3d 427, 434 (2d Cir.1995).
Here, in support of its contention that defendant committed fraud, plaintiff offers nothing more than bald assertions that defendant never intended to fulfill its obligations under the contract. Because plaintiff fails to state a claim for which relief may be granted on its fraud claim, the claim is dismissed.
D. Quantum Meruit Claim
In Count 4, plaintiff alleges that it is entitled to recover for services it provided to defendant on the basis of quantum meruit. Defendant responds, arguing that plaintiff is unable to recover under quantum meruit because a valid contract covers the subject of this dispute.
Quantum meruit is a doctrine of "quasi contract." Zolotar v. New York Life Insurance Company, 172 A.D.2d 27, 576 N.Y.S.2d 850, 852 (1st Dep't 1991). In New York State, "[t]he existence of a valid and enforceable written contract governing a particular subject matter ordinarily precludes recovery in quasi contract for events arising out of the same subject matter. A `quasi contract' only applies in the absence of an express agreement, and is not really a contract at all, but rather a legal obligation imposed in order to prevent unjust enrichment." Clark-Fitzpatrick, *226 Inc. v. Long Island Rail Road Company, 70 N.Y.2d 382, 388, 521 N.Y.S.2d 653, 516 N.E.2d 190 (1987)(internal citations omitted).
Here, there exists a valid contract between plaintiff and defendant which clearly and concisely establishes the circumstances under which plaintiff would be entitled to a commission. Furthermore, plaintiff has failed to demonstrate any issue of material fact which would indicate that defendant had been unjustly enriched as a result of a benefit conferred by plaintiff. As such, plaintiff is unable to recover on the basis of quantum meruit, and its fourth cause of action is dismissed.
E. Breach of Good Faith and Fair Dealing Claim
Plaintiff further alleges that defendant violated the covenant of good faith and fair dealing when it terminated the contract. Defendant argues that it cannot be held liable for a breach of the implied covenant of good faith and fair dealing based on its exercise of expressed rights under the Contract.
"Implicit in every contract is a promise of good faith and fair dealing, which is breached when a party acts in a manner that, although not expressly forbidden by any contractual provision, would deprive the other party of the right to receive the benefits under the agreement." Skillgames, LLC v. Brody, 1 A.D.3d 247, 767 N.Y.S.2d 418, 422 (1st Dep't 2003). A defendant "does not breach its duty of good faith and fair dealing by exercising it rights under the contract[ ]...." Associates Capital Services Corp. of New Jersey v. Fairway Private Cars, Inc., 590 F.Supp. 10, 16 (S.D.N.Y.1982).
Moreover, plaintiff has failed to offer any evidence that it is entitled to additional payments under the contract. Plaintiff's claim for breach of the implied covenant of good faith and fair dealing cannot substitute for an unsustainable breach of contract claim. Accordingly, plaintiff's claim for breach of the covenant of good faith and fair dealing is dismissed.[6]
CONCLUSION
For the reasons set forth above, I find that there exists no material facts in dispute as to any of plaintiff's claims. Accordingly, defendant's motion for summary judgment is granted in its entirety, and each plaintiff's claims is dismissed with prejudice.
ALL OF THE ABOVE IS SO ORDERED.
NOTES
[1] Plaintiff also filed what is purported to be a cross-motion for summary judgment, but in actuality is simply an attorney's affidavit, which raises no additional arguments other than those it used to oppose defendant's motion for summary judgment, and since it supports no triable issue of fact it is dismissed as a matter of law.
[2] A copy of the Contract can be found at "Exhibit A" to plaintiff's Amended Complaint. (Amended Complaint, Doc. No. 27, Exhibit A).
[3] Defendant asserts for the first time in its reply papers that jurisdiction in this Court is improper because plaintiff fails to plead damages in excess of the statutory minimum required under 28 U.S.C. § 1332. However, in its Complaint, plaintiff claims that it is entitled to a 5% commission on a forecasted sale totaling $40,000,000 an amount well in excess of $75,000 the amount necessary to confer federal jurisdiction by diversity of citizenship. Therefore, I find that diversity jurisdiction is proper.
[4] Although Section "IV", Paragraph "B", of the Contract identifies plaintiff as an "independent contractor", and "not an employee", its duties under the Contract are sufficiently similar to those of an employee so as to negate any distinction. See Arledge v. Stratmar Systems, Inc., 948 F.2d 845 (2d Cir.1991) (Independent contractor relationship sufficiently analogous to employment contract so as to bring it within purview of "at-will" rule of discharge.).
[5] See Affidavit of Nicholas Juskiw, Doc. No. 44, Exhibit C.
[6] Plaintiff also pleaded two additional causes of action: (1) that it was entitled to punitive damages; and (2) that it was entitled to an award of attorneys' fees. As each of plaintiff's claims upon which an award could be made is hereby dismissed with prejudice, plaintiff's request for punitive damages and attorneys' fees is also hereby dismissed.
|
{
"pile_set_name": "FreeLaw"
}
|
This invention relates generally to automotive vehicle clutches.
It is well known to those acquainted with the particular field that after a long time of use, a clutch mechanism of a vehicle becomes worn out, and the clutch face plate friction pads become worn thin so that they must be replaced with new friction pads. On conventional clutches now used on automotive vehicles, this necessitates the engine-transmission separation in order to gain access to the pads, and which dismantling operation is accordingly time consuming and costly. This situation is objectionable, and is therefore in need of an improvement.
|
{
"pile_set_name": "USPTO Backgrounds"
}
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