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American country artist and television host Trisha Yearwood has received more than 58 award nominations and 10 wins. Yearwood has been nominated a total of 27 times from the Grammy Awards. Her first award from the association came in 1994 for her performance of "I Fall to Pieces", which won her and Aaron Neville the Best Country Collaboration with Vocals accolade. In 1997, she won both the award for Best Female Country Vocal Performance and the Best Country Collaboration with Vocals award. In addition, Yearwood has been given accolades from the Academy of Country Music. She won her first award in 1991 for Top Female Vocalist. She later won in both 1997 and 1998 for Top Female Vocalist. Yearwood has also won three accolades from the Country Music Association, including Female Vocalist of the Year. As a host of the Food Network television show Trisha's Southern Kitchen, Yearwood has been nominated for (and won) the Daytime Emmy Award for Outstanding Culinary Program.
American Music Awards
!
|-
| rowspan="2"| 1992
| Trisha Yearwood
| Favorite New Country Artist
|
| style="text-align:center;"|
|-
| "She's in Love with the Boy"
| Favorite Country Song
|
| style="text-align:center;"|
|-
| 1998
| (Songbook) A Collection of Hits
| Favorite Country Album
|
| style="text-align:center;"|
|-
|}
Academy of Country Music Awards
!
|-
| rowspan="2"| 1991
| Trisha Yearwood
| Top New Female Vocalist
|
| style="text-align:center;" rowspan="13"|
|-
| "She's in Love with the Boy"
| Single Record of the Year
|
|-
| 1993
| Common Thread: The Songs of the Eagles
| Album of the Year
|
|-
| 1994
| Trisha Yearwood & Aaron Neville
| Top Vocal Duet
|
|-
| 1996
| Trisha Yearwood
| Top Female Vocalist
|
|-
| rowspan="4"| 1997
| "In Another's Eyes"
| Vocal Event of the Year
|
|-
| Trisha Yearwood
| Top Female Vocalist
|
|-
| rowspan="2"| "How Do I Live"
| Song of the Year
|
|-
| Single Record of the Year
|
|-
| 1998
| rowspan="2"| Trisha Yearwood
| rowspan="2"| Top Female Vocalist
|
|-
| 2001
|
|-
| 2008
| "Another Try"
| Vocal Event of the Year
|
|-
| 2016
| "Forever Country"
| Video of the Year
|
|}
Country Music Association Awards
!
|-
| rowspan="2"| 1992
| Trisha Yearwood
| Female Vocalist of the Year
|
| style="text-align:center;" rowspan="19"|
|-
| rowspan="2"| Trisha Yearwood
| rowspan="2"| Horizon Award
|
|-
| rowspan="2"| 1993
|
|-
| "Walkaway Joe"
| Vocal Event of the Year
|
|-
| rowspan="3"| 1994
| Common Thread: The Songs of the Eagles
| Album of the Year
|
|-
| Trisha Yearwood
| Female Vocalist of the Year
|
|-
| "I Fall to Pieces"
| Vocal Event of the Year
|
|-
| 1996
| "On My Own"
| Vocal Event of the Year
|
|-
| rowspan="2"| 1997
| Everybody Knows
| Album of the Year
|
|-
| rowspan="2"| Trisha Yearwood
| rowspan="2"| Female Vocalist of the Year
|
|-
| rowspan="2"| 1998
|
|-
| "In Another's Eyes"
| Vocal Event of the Year
|
|-
| rowspan="2"| 1999
| Where Your Road Leads
| Album of the Year
|
|-
| rowspan="3"| Trisha Yearwood
| rowspan="3"| Female Vocalist of the Year
|
|-
| 2000
|
|-
| rowspan="2"| 2001
|
|-
| "I Would've Loved You Anyway"
| Music Video of the Year
|
|-
| 2002
| Trisha Yearwood
| Female Vocalist of the Year
|
|-
| 2008
| "Another Try"
| Vocal Event of the Year
|
|-
|}
Daytime Emmy Awards
!
|-
| 2013
| rowspan="2"| Trisha's Southern Kitchen
| rowspan="2"| Outstanding Culinary Program
|
| style="text-align:center;"|
|-
| 2017
|
| style="text-align:center;"|
|-
|}
Grammy Awards
!
|-
| 1991
| "She's in Love with the Boy"
| rowspan="2"| Best Female Country Vocal Performance
|
| style="text-align:center;" rowspan="25"|
|-
| 1993
| "Walkaway Joe"
|
|-
| rowspan="2"| 1994
| "I Fall to Pieces"
| Best Country Collaboration with Vocals
|
|-
| The Song Remembers When
| rowspan="2"| Best Country Album
|
|-
| rowspan="2"| 1995
| Thinkin' About You
|
|-
| "On My Own"
| Best Country Collaboration with Vocals
|
|-
| rowspan="3"| 1996
| "Believe Me Baby (I Lied)"
| Best Female Country Vocal Performance
|
|-
| Everybody Knows
| Best Country Album
|
|-
| "Hope: Country Music's Quest for a Cure"
| Best Country Collaboration with Vocals
|
|-
| rowspan="2"| 1997
| "How Do I Live"
| Best Female Country Vocal Performance
|
|-
| "In Another's Eyes"
| Best Country Collaboration with Vocals
|
|-
| rowspan="3"| 1998
| "There Goes My Baby"
| Best Female Country Vocal Performance
|
|-
| Where Your Road Leads
| Best Country Album
|
|-
| "Where Your Road Leads"
| Best Country Collaboration with Vocals
|
|-
| rowspan="2"| 2000
| "Real Live Woman
| Best Female Country Vocal Performance
|
|-
| Real Live Woman
| Best Country Album
|
|-
| rowspan="3"| 2001
| "I Would've Loved You Anyway"
| Best Female Country Vocal Performance
|
|-
| Inside Out
| Best Country Album
|
|-
| "Inside Out"
| Best Country Collaboration with Vocals
|
|-
| 2002
| "Squeeze Me In"
| Best Country Collaboration with Vocals
|
|-
| rowspan="2"| 2005
| "Georgia Rain"
| Best Female Country Vocal Performance
|
|-
| Jasper County
| Best Country Album
|
|-
| 2006
| "Love Will Always Win"
| Best Country Collaboration with Vocals
|
|-
| 2007
| "Heaven, Heartache and the Power of Love"
| rowspan="2"| Best Female Country Vocal Performance
|
|-
| rowspan="3"| 2008
| "This Is Me You're Talking To"
|
|-
| "Let the Wind Chase You"
| Best Country Collaboration with Vocals
|
|-
| Heaven, Heartache and the Power of Love
| Best Country Album
|
|-
|}
References
Yearwood, Trisha
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,863
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 349
|
using namespace Shiny;
TEST_CASE_METHOD(EvaluatorFixture, "Test is number", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(number? 0)"));
REQUIRE(Value::True == evaluate("(number? 1)"));
REQUIRE(Value::True == evaluate("(number? -1)"));
REQUIRE(Value::True == evaluate("(number? 1000)"));
REQUIRE(Value::True == evaluate("(number? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(number? \"0\")"));
REQUIRE(Value::False == evaluate("(number? #\\0)"));
REQUIRE(Value::False == evaluate("(number? '())"));
REQUIRE(Value::False == evaluate("(number? #t)"));
REQUIRE(Value::False == evaluate("(number? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(number?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(number? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is complex", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(complex? 0)"));
REQUIRE(Value::True == evaluate("(complex? 1)"));
REQUIRE(Value::True == evaluate("(complex? -1)"));
REQUIRE(Value::True == evaluate("(complex? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(complex? \"0\")"));
REQUIRE(Value::False == evaluate("(complex? #\\0)"));
REQUIRE(Value::False == evaluate("(complex? '())"));
REQUIRE(Value::False == evaluate("(complex? #t)"));
REQUIRE(Value::False == evaluate("(complex? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(complex?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(complex? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is real", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(real? 0)"));
REQUIRE(Value::True == evaluate("(real? 1)"));
REQUIRE(Value::True == evaluate("(real? -1)"));
REQUIRE(Value::True == evaluate("(real? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(real? \"0\")"));
REQUIRE(Value::False == evaluate("(real? #\\0)"));
REQUIRE(Value::False == evaluate("(real? '())"));
REQUIRE(Value::False == evaluate("(real? #t)"));
REQUIRE(Value::False == evaluate("(real? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(real?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(real? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is rational", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(rational? 0)"));
REQUIRE(Value::True == evaluate("(rational? 1)"));
REQUIRE(Value::True == evaluate("(rational? -1)"));
REQUIRE(Value::True == evaluate("(rational? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(rational? \"0\")"));
REQUIRE(Value::False == evaluate("(rational? #\\0)"));
REQUIRE(Value::False == evaluate("(rational? '())"));
REQUIRE(Value::False == evaluate("(rational? #t)"));
REQUIRE(Value::False == evaluate("(rational? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(rational?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(rational? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is integer", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(integer? 0)"));
REQUIRE(Value::True == evaluate("(integer? 1)"));
REQUIRE(Value::True == evaluate("(integer? -1)"));
REQUIRE(Value::True == evaluate("(integer? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(integer? \"0\")"));
REQUIRE(Value::False == evaluate("(integer? #\\0)"));
REQUIRE(Value::False == evaluate("(integer? '())"));
REQUIRE(Value::False == evaluate("(integer? #t)"));
REQUIRE(Value::False == evaluate("(integer? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(integer?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(integer? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is Exact", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(exact? 0)"));
REQUIRE(Value::True == evaluate("(exact? 1000000)"));
REQUIRE(Value::True == evaluate("(exact? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(exact? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(exact? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(exact?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(exact? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is Inexact", "[Procedures]") {
SECTION("returns false for fixnums") {
REQUIRE(Value::False == evaluate("(inexact? 0)"));
REQUIRE(Value::False == evaluate("(inexact? 1000000)"));
REQUIRE(Value::False == evaluate("(inexact? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(inexact? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(inexact? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(inexact?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(inexact? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is Exact Integer", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(exact-integer? 0)"));
REQUIRE(Value::True == evaluate("(exact-integer? 1000000)"));
REQUIRE(Value::True == evaluate("(exact-integer? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(exact-integer? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(exact-integer? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(exact-integer?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(exact-integer? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is Finite", "[Procedures]") {
SECTION("returns true for fixnums") {
REQUIRE(Value::True == evaluate("(finite? 0)"));
REQUIRE(Value::True == evaluate("(finite? 1000000)"));
REQUIRE(Value::True == evaluate("(finite? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(finite? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(finite? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(finite?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(finite? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is Infinite", "[Procedures]") {
SECTION("returns false for fixnums") {
REQUIRE(Value::False == evaluate("(infinite? 0)"));
REQUIRE(Value::False == evaluate("(infinite? 1000000)"));
REQUIRE(Value::False == evaluate("(infinite? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(infinite? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(infinite? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(infinite?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(infinite? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Is NaN", "[Procedures]") {
SECTION("returns false for fixnums") {
REQUIRE(Value::False == evaluate("(nan? 0)"));
REQUIRE(Value::False == evaluate("(nan? 1000000)"));
REQUIRE(Value::False == evaluate("(nan? -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(nan? #\\3)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(nan? \"12\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(nan?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(nan? 1 2)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is zero", "[Procedures]") {
SECTION("returns true for zero fixnums") {
REQUIRE(Value::True == evaluate("(zero? 0)"));
}
SECTION("returns false for non-zero fixnums") {
REQUIRE(Value::False == evaluate("(zero? 1)"));
REQUIRE(Value::False == evaluate("(zero? -10000)"));
REQUIRE(Value::False == evaluate("(zero? 512)"));
}
SECTION("returns false for any non fixnum type") {
REQUIRE(Value::False == evaluate("(zero? \"0\")"));
REQUIRE(Value::False == evaluate("(zero? #\\0)"));
REQUIRE(Value::False == evaluate("(zero? '())"));
REQUIRE(Value::False == evaluate("(zero? #t)"));
REQUIRE(Value::False == evaluate("(zero? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(zero?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(zero? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is positive", "[Procedures]") {
SECTION("returns true for positive fixnums") {
REQUIRE(Value::True == evaluate("(positive? 1)"));
REQUIRE(Value::True == evaluate("(positive? 2)"));
REQUIRE(Value::True == evaluate("(positive? 1000000)"));
}
SECTION("returns false for non positive fixnums") {
REQUIRE(Value::False == evaluate("(positive? 0)"));
REQUIRE(Value::False == evaluate("(positive? -1)"));
REQUIRE(Value::False == evaluate("(positive? -1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(positive? \"0\")"));
REQUIRE(Value::False == evaluate("(positive? #\\0)"));
REQUIRE(Value::False == evaluate("(positive? '())"));
REQUIRE(Value::False == evaluate("(positive? #t)"));
REQUIRE(Value::False == evaluate("(positive? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(positive?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(positive? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is negative", "[Procedures]") {
SECTION("returns true for negative fixnums") {
REQUIRE(Value::True == evaluate("(negative? -1)"));
REQUIRE(Value::True == evaluate("(negative? -2)"));
REQUIRE(Value::True == evaluate("(negative? -1000000)"));
}
SECTION("returns true for non negative fixnums") {
REQUIRE(Value::False == evaluate("(negative? 0)"));
REQUIRE(Value::False == evaluate("(negative? 1)"));
REQUIRE(Value::False == evaluate("(negative? 2)"));
REQUIRE(Value::False == evaluate("(negative? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(negative? \"0\")"));
REQUIRE(Value::False == evaluate("(negative? #\\0)"));
REQUIRE(Value::False == evaluate("(negative? '())"));
REQUIRE(Value::False == evaluate("(negative? #t)"));
REQUIRE(Value::False == evaluate("(negative? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(negative?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(negative? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is even", "[Procedures]") {
SECTION("returns true for even fixnums") {
REQUIRE(Value::True == evaluate("(even? 0)"));
REQUIRE(Value::True == evaluate("(even? 2)"));
REQUIRE(Value::True == evaluate("(even? -2)"));
REQUIRE(Value::True == evaluate("(even? -36)"));
REQUIRE(Value::True == evaluate("(even? 1000000)"));
}
SECTION("returns true for odd fixnums") {
REQUIRE(Value::False == evaluate("(even? 1)"));
REQUIRE(Value::False == evaluate("(even? 3)"));
REQUIRE(Value::False == evaluate("(even? -1)"));
REQUIRE(Value::False == evaluate("(even? -5)"));
REQUIRE(Value::False == evaluate("(even? 1000007)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(even? \"0\")"));
REQUIRE(Value::False == evaluate("(even? #\\0)"));
REQUIRE(Value::False == evaluate("(even? '())"));
REQUIRE(Value::False == evaluate("(even? #t)"));
REQUIRE(Value::False == evaluate("(even? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(even?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(even? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test is odd", "[Procedures]") {
SECTION("returns true for odd fixnums") {
REQUIRE(Value::True == evaluate("(odd? 1)"));
REQUIRE(Value::True == evaluate("(odd? 3)"));
REQUIRE(Value::True == evaluate("(odd? -1)"));
REQUIRE(Value::True == evaluate("(odd? -5)"));
REQUIRE(Value::True == evaluate("(odd? 1000007)"));
}
SECTION("returns true for even fixnums") {
REQUIRE(Value::False == evaluate("(odd? 0)"));
REQUIRE(Value::False == evaluate("(odd? 2)"));
REQUIRE(Value::False == evaluate("(odd? -2)"));
REQUIRE(Value::False == evaluate("(odd? -36)"));
REQUIRE(Value::False == evaluate("(odd? 1000000)"));
}
SECTION("returns false for any non numeric type") {
REQUIRE(Value::False == evaluate("(odd? \"0\")"));
REQUIRE(Value::False == evaluate("(odd? #\\0)"));
REQUIRE(Value::False == evaluate("(odd? '())"));
REQUIRE(Value::False == evaluate("(odd? #t)"));
REQUIRE(Value::False == evaluate("(odd? #f)"));
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(odd?)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(odd? 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test number equal", "[Procedures]") {
SECTION("returns true for fixnums equal in value") {
REQUIRE(Value::True == evaluate("(=)"));
REQUIRE(Value::True == evaluate("(= 1)"));
REQUIRE(Value::True == evaluate("(= 1)"));
REQUIRE(Value::True == evaluate("(= 5 5)"));
REQUIRE(Value::True == evaluate("(= 4 4 (* 2 2))"));
}
SECTION("returns false for fixnums not equal in value") {
REQUIRE(Value::False == evaluate("(= 5 2)"));
REQUIRE(Value::False == evaluate("(= 8 8 8 8 1)"));
}
SECTION("throws exception for non numeric types") {
auto fn1 = [this]() { evaluate("(= \"0\")"); };
auto fn2 = [this]() { evaluate("(= #\\0 0)"); };
auto fn3 = [this]() { evaluate("(= #f #f)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn3(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test number less", "[Procedures]") {
SECTION("returns true for fixnums less in value") {
REQUIRE(Value::True == evaluate("(<)"));
REQUIRE(Value::True == evaluate("(< 1)"));
REQUIRE(Value::True == evaluate("(< 3 5)"));
REQUIRE(Value::True == evaluate("(< -2 4 9)"));
}
SECTION("returns false for fixnums not less in value") {
REQUIRE(Value::False == evaluate("(< 5 3)"));
REQUIRE(Value::False == evaluate("(< 4 4 5)"));
REQUIRE(Value::False == evaluate("(< 4 3 4)"));
}
SECTION("throws exception for non numeric types") {
auto fn1 = [this]() { evaluate("(< \"0\")"); };
auto fn2 = [this]() { evaluate("(< #\\0 0)"); };
auto fn3 = [this]() { evaluate("(< #f #f)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn3(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test number less equal", "[Procedures]") {
SECTION("returns true for fixnums less equal in value") {
REQUIRE(Value::True == evaluate("(<=)"));
REQUIRE(Value::True == evaluate("(<= 1)"));
REQUIRE(Value::True == evaluate("(<= 3 3)"));
REQUIRE(Value::True == evaluate("(<= -2 1 1)"));
}
SECTION("returns false for fixnums not less equal in value") {
REQUIRE(Value::False == evaluate("(<= 6 0)"));
REQUIRE(Value::False == evaluate("(<= 3 4 3)"));
REQUIRE(Value::False == evaluate("(<= 3 6 5)"));
}
SECTION("throws exception for non numeric types") {
auto fn1 = [this]() { evaluate("(<= \"0\")"); };
auto fn2 = [this]() { evaluate("(<= #\\0 0)"); };
auto fn3 = [this]() { evaluate("(<= #f #f)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn3(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Test number greater", "[Procedures]") {
SECTION("returns true for fixnums greater in value") {
REQUIRE(Value::True == evaluate("(>)"));
REQUIRE(Value::True == evaluate("(> 1)"));
REQUIRE(Value::True == evaluate("(> 10 5)"));
REQUIRE(Value::True == evaluate("(> 10 3 -1)"));
}
SECTION("returns false for fixnums not greater in value") {
REQUIRE(Value::False == evaluate("(> 2 3)"));
REQUIRE(Value::False == evaluate("(> 3 2 2)"));
REQUIRE(Value::False == evaluate("(> 3 1 2)"));
}
SECTION("throws exception for non numeric types") {
auto fn1 = [this]() { evaluate("(> \"0\")"); };
auto fn2 = [this]() { evaluate("(> #\\0 0)"); };
auto fn3 = [this]() { evaluate("(> #f #f)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn3(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(
EvaluatorFixture,
"Test number greater equal",
"[Procedures]") {
SECTION("returns true for fixnums greater or equal in value") {
REQUIRE(Value::True == evaluate("(>=)"));
REQUIRE(Value::True == evaluate("(>= 1)"));
REQUIRE(Value::True == evaluate("(>= 10 5)"));
REQUIRE(Value::True == evaluate("(>= 8 8 3)"));
}
SECTION("returns false for fixnums not greater or equal in value") {
REQUIRE(Value::False == evaluate("(>= 2 3)"));
REQUIRE(Value::False == evaluate("(>= 3 2 3)"));
REQUIRE(Value::False == evaluate("(>= 3 1 2)"));
}
SECTION("throws exception for non numeric types") {
auto fn1 = [this]() { evaluate("(>= \"0\")"); };
auto fn2 = [this]() { evaluate("(>= #\\0 0)"); };
auto fn3 = [this]() { evaluate("(>= #f #f)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
REQUIRE_THROWS_AS(fn3(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Addition procedure", "[Procedures]") {
SECTION("sums all passed fixnums") {
REQUIRE(Value{0} == evaluate("(+)"));
REQUIRE(Value{4} == evaluate("(+ 1 3)"));
REQUIRE(Value{12} == evaluate("(+ 10 5 1 -4)"));
}
SECTION("error if any argument is not a fixnum") {
auto fn1 = [this]() { evaluate("(+ 10 1 #t)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(+ 10 1 #\\c)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Subtract procedure", "[Procedures]") {
SECTION("error with no arguments") {
auto fn = [this]() { evaluate("(-)"); };
REQUIRE_THROWS_AS(fn(), ArgumentMissingException);
}
SECTION("with one argument negates") {
REQUIRE(Value{-25} == evaluate("(- 25)"));
REQUIRE(Value{3} == evaluate("(- -3)"));
REQUIRE(Value{0} == evaluate("(- 0)"));
}
SECTION("with two or more arguments subtracts all") {
REQUIRE(Value{2} == evaluate("(- 5 3)"));
REQUIRE(Value{3} == evaluate("(- 10 4 3)"));
REQUIRE(Value{-2} == evaluate("(- 5 6 -2 3)"));
}
SECTION("error if any argument is not a fixnum") {
auto fn1 = [this]() { evaluate("(- \"3\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(- 2 \"3\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if zero arguments") {
auto fn1 = [this]() { evaluate("(-)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Multiply procedure", "[Procedures]") {
SECTION("multiplies all passed fixnums") {
REQUIRE(Value{1} == evaluate("(*)"));
REQUIRE(Value{3} == evaluate("(* 1 3)"));
REQUIRE(Value{-6} == evaluate("(* 2 -3 1)"));
}
SECTION("error if any argument is not a fixnum") {
auto fn1 = [this]() { evaluate("(* \"33\" 1 2)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(* 10 1 #\\1)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Divide procedure", "[Procedures]") {
SECTION("error with no arguments") {
auto fn = [this]() { evaluate("(/)"); };
REQUIRE_THROWS_AS(fn(), ArgumentMissingException);
}
SECTION("with one argument inverts") {
REQUIRE(Value{1} == evaluate("(/ 1)"));
REQUIRE(Value{1 / 2} == evaluate("(/ 2)"));
REQUIRE(Value{1 / -5} == evaluate("(/ -5)"));
}
SECTION("with two or more arguments divides all") {
REQUIRE(Value{2} == evaluate("(/ 4 2)"));
REQUIRE(Value{-8} == evaluate("(/ 24 -3 1)"));
REQUIRE(Value{5} == evaluate("(/ 16 3)"));
REQUIRE(Value{0} == evaluate("(/ 0 3)"));
}
SECTION("with zero") {
// TODO: Handle exception when its added to div.
REQUIRE(Value{0} == evaluate("(/ 0)"));
REQUIRE(Value{0} == evaluate("(/ 2 0)"));
REQUIRE(Value{0} == evaluate("(/ 3 2 0 1)"));
}
SECTION("error if any argument is not a fixnum") {
auto fn1 = [this]() { evaluate("(/ \"3\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(/ 2 \"3\")"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if zero arguments") {
auto fn1 = [this]() { evaluate("(/)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Absolute value", "[Procedures]") {
SECTION("returns same fixnum for zero and positive integers") {
REQUIRE(Value{0} == evaluate("(abs 0)"));
REQUIRE(Value{1} == evaluate("(abs 1)"));
REQUIRE(Value{2} == evaluate("(abs 2)"));
REQUIRE(Value{1000000} == evaluate("(abs 1000000)"));
}
SECTION("returns positive fixnum for negative integers") {
REQUIRE(Value{1} == evaluate("(abs -1)"));
REQUIRE(Value{2} == evaluate("(abs -2)"));
REQUIRE(Value{1000000} == evaluate("(abs -1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(abs \"33\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(abs #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(abs)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(abs 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Floor quotient", "[Procedures]") {
// TODO: Test error divide by zero.
SECTION("returns floored quotient for fixnums with correct sign") {
REQUIRE(Value{2} == evaluate("(floor-quotient 5 2)"));
REQUIRE(Value{-3} == evaluate("(floor-quotient -5 2)"));
REQUIRE(Value{-3} == evaluate("(floor-quotient 5 -2)"));
REQUIRE(Value{2} == evaluate("(floor-quotient -5 -2)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(floor-quotient #\\3 \"33\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(floor-quotient 4 #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(floor-quotient)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(floor-quotient 1 2)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn3 = [this]() { evaluate("(floor-quotient 0 1 2)"); };
REQUIRE_THROWS_AS(fn3(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Truncate quotient", "[Procedures]") {
// Also tests for aliased procedure "quotient".
// TODO: Test error divide by zero.
SECTION("returns truncated quotient for fixnums with correct sign") {
REQUIRE(Value{2} == evaluate("(truncate-quotient 5 2)"));
REQUIRE(Value{2} == evaluate("(quotient 5 2)"));
REQUIRE(Value{-2} == evaluate("(truncate-quotient -5 2)"));
REQUIRE(Value{-2} == evaluate("(quotient -5 2)"));
REQUIRE(Value{-2} == evaluate("(truncate-quotient 5 -2)"));
REQUIRE(Value{-2} == evaluate("(quotient 5 -2)"));
REQUIRE(Value{2} == evaluate("(truncate-quotient -5 -2)"));
REQUIRE(Value{2} == evaluate("(quotient -5 -2)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(truncate-quotient #\\3 \"33\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(truncate-quotient 4 #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(truncate-quotient)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(truncate-quotient 1 2)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn3 = [this]() { evaluate("(truncate-quotient 0 1 2)"); };
REQUIRE_THROWS_AS(fn3(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Truncate remainder", "[Procedures]") {
// Also tests for aliased procedure "remainder".
// TODO: Test error divide by zero.
SECTION("returns truncated quotient for fixnums with correct sign") {
REQUIRE(Value{1} == evaluate("(truncate-remainder 5 2)"));
REQUIRE(Value{1} == evaluate("(remainder 5 2)"));
REQUIRE(Value{-1} == evaluate("(truncate-remainder -5 2)"));
REQUIRE(Value{-1} == evaluate("(remainder -5 2)"));
REQUIRE(Value{1} == evaluate("(truncate-remainder 5 -2)"));
REQUIRE(Value{1} == evaluate("(remainder 5 -2)"));
REQUIRE(Value{-1} == evaluate("(truncate-remainder -5 -2)"));
REQUIRE(Value{-1} == evaluate("(remainder -5 -2)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(truncate-remainder #\\3 \"33\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(truncate-remainder 4 #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(truncate-remainder)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(truncate-remainder 1 2)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn3 = [this]() { evaluate("(truncate-remainder 0 1 2)"); };
REQUIRE_THROWS_AS(fn3(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Floor", "[Procedures]") {
SECTION("returns the same value for all fixnums") {
REQUIRE(Value{0} == evaluate("(floor 0)"));
REQUIRE(Value{1} == evaluate("(floor 1)"));
REQUIRE(Value{-2} == evaluate("(floor -2)"));
REQUIRE(Value{1000000} == evaluate("(floor 1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(floor \"42\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(floor #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(floor)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(floor 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Ceiling", "[Procedures]") {
SECTION("returns the same value for all fixnums") {
REQUIRE(Value{0} == evaluate("(ceiling 0)"));
REQUIRE(Value{1} == evaluate("(ceiling 1)"));
REQUIRE(Value{-2} == evaluate("(ceiling -2)"));
REQUIRE(Value{1000000} == evaluate("(ceiling 1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(ceiling \"42\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(ceiling #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(ceiling)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(ceiling 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Truncate", "[Procedures]") {
SECTION("returns the same value for all fixnums") {
REQUIRE(Value{0} == evaluate("(truncate 0)"));
REQUIRE(Value{1} == evaluate("(truncate 1)"));
REQUIRE(Value{-2} == evaluate("(truncate -2)"));
REQUIRE(Value{1000000} == evaluate("(truncate 1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(truncate \"42\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(truncate #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(truncate)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(truncate 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "Round", "[Procedures]") {
SECTION("returns the same value for all fixnums") {
REQUIRE(Value{0} == evaluate("(round 0)"));
REQUIRE(Value{1} == evaluate("(round 1)"));
REQUIRE(Value{-2} == evaluate("(round -2)"));
REQUIRE(Value{1000000} == evaluate("(round 1000000)"));
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(round \"42\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(round #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(round)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(round 0 1)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "number to string", "[Procedures]") {
SECTION("returns the equiv printable fixnum") {
REQUIRE("\"0\"" == evaluate("(number->string 0)").toString());
REQUIRE("\"10\"" == evaluate("(number->string 10)").toString());
REQUIRE("\"-3\"" == evaluate("(number->string -3)").toString());
REQUIRE("\"67921\"" == evaluate("(number->string 67921)").toString());
REQUIRE(
"\"2147483647\"" == evaluate("(number->string 2147483647)").toString());
REQUIRE(
"\"-2147483648\"" ==
evaluate("(number->string -2147483648)").toString());
}
SECTION("error if argument is not a numeric type") {
auto fn1 = [this]() { evaluate("(number->string \"42\")"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(number->string #t)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(number->string)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(number->string 4 5 6)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
TEST_CASE_METHOD(EvaluatorFixture, "string to number", "[Procedures]") {
SECTION("returns a fixnum for a integer in a string") {
REQUIRE(Value{0} == evaluate("(string->number \"0\")"));
REQUIRE(Value{10} == evaluate("(string->number \"10\")"));
REQUIRE(Value{-3} == evaluate("(string->number \"-3\")"));
REQUIRE(Value{67921} == evaluate("(string->number \"67921\")"));
REQUIRE(Value{2147483647} == evaluate("(string->number \"2147483647\")"));
REQUIRE(
Value{(fixnum_t)-2147483648} ==
evaluate("(string->number \"-2147483648\")"));
}
SECTION("returns false for strings that are not numbers") {
REQUIRE(Value::False == evaluate("(string->number \"\")"));
REQUIRE(Value::False == evaluate("(string->number \"hello\")"));
REQUIRE(Value::False == evaluate("(string->number \"z3\")"));
REQUIRE(Value::False == evaluate("(string->number \" -4\")"));
}
SECTION("error if argument is not a string type") {
auto fn1 = [this]() { evaluate("(string->number 42)"); };
REQUIRE_THROWS_AS(fn1(), WrongArgTypeException);
auto fn2 = [this]() { evaluate("(string->number #f)"); };
REQUIRE_THROWS_AS(fn2(), WrongArgTypeException);
}
SECTION("error if wrong argument count") {
auto fn1 = [this]() { evaluate("(string->number)"); };
REQUIRE_THROWS_AS(fn1(), ArgumentMissingException);
auto fn2 = [this]() { evaluate("(string->number \"4\" 5 6)"); };
REQUIRE_THROWS_AS(fn2(), ArgCountMismatch);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,847
|
\section{Introduction}
\label{sec:introduction}
The study of maps between shapes, manifolds and point clouds is of great
interest in a wide variety of applications. There are many data types, e.g.
shapes (modeled as surfaces), images, sounds, and many more, where a similarity
between a pair of data points involves computing a map between the points, and
the similarity is a functional of that map. The map between a pair of data
points however contains much more information than the similarity measure
alone, and the study of networks of such maps have been successfully used to
organize, extract functional information and abstractions, and help regularize
estimators in large collections of shapes \citep{bigdata5,bigdata10,bigdata8}.
The family of maps to be considered depends on the type of shape, manifold or
point cloud, as well as on the choice of geometric features to be preserved in
a particular application. These considerations are not restricted to data sets
where each point is naturally a geometric object: high-dimensional data sets of
non-geometric nature, from musical pieces to text documents to trajectories of
high-dimensional stochastic dynamical systems, are often mapped, via feature
sets, to geometric objects. The considerations above therefore apply to a very
wide class of data types.
In this paper we are interested in the problem where each object is a point
cloud -- a set of points in $\mathbb{R}^D$ -- and will develop techniques for
computing maps from one point cloud to another, in particular in the situation
where $D$ is very large, but the point clouds are close to being
low-dimensional, for example they may be samples from a $d$-dimensional smooth
manifold $\mathcal{M}$ ($d\ll D$). The two point clouds may have a different
number of points, and they may arise from a sample of a low-dimensional
manifold $\mathcal{M}$ perturbed by high-dimensional noise (for more general
models see the works by~\citet{LMR:MGM1},~\citet{MMS:NoisyDictionaryLearning}
and~\citet{LiaoMaggioni}). In this setting we have to be particularly careful
in both the choice of maps and in their estimation since sampling and noise
have the potential to cause significant perturbations.
We find optimal transport maps rather well-suited for these purposes. They
automatically handle the situation where the two point clouds have different
cardinality, they handle in a robust fashion noise, and even changes in
dimensionality, which is typically ill-defined, for point clouds arising from
real-world data~\citep{LMR:MGM1}. Optimal transport has a very long history in
a variety of disciplines and arises naturally in a wide variety of contexts,
from optimization problems in economics and resource allocation, to mathematics
and physics, to computer science (e.g. network flow algorithms). Thus,
applications of optimal transport range from logistics and
economics~\citep{beckmann:eco1952,carlier:jcc2008}, geophysical
models~\citep{cullen:book2006}, image
analysis~\citep{rubner:iccv1998,haker:ijcv2004} to machine
learning~\citep{cuturi:arxiv2013,cuturi:jmlr2014}. Despite these widespread
applications, the efficient computation of optimal transport plans remains
challenging, especially in complex geometries and in high dimensions.
\begin{figure}[!b]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{scale-3} &
\includegraphics[width=0.45\linewidth]{scale-5} \\
\includegraphics[width=0.45\linewidth]{scale-7} &
\includegraphics[width=0.45\linewidth]{scale-9}
\end{tabular}
\caption{\label{fig:multiscale-strategy}
Optimal transport between two noisy elliptical shapes in $2$-D. We
visualize coarse-to-fine approximations to the source (purple dots) and target
(yellow dots) point clouds, together with the optimal transportation plan at
each scale (gray edges). The intensity of the points is proportional to mass
-- at coarser scales each point represents an agglomerate of points at finer
scales, and its mass is the sum of the masses of the points it represents.
Similarly the intensity of the lines is proportional to the amount of mass
transported along the line. Notice that the optimal transport formulation
permits multiple lines exiting a source point or entering a target point.}
\end{figure}
For point sets the optimal transport problem can be solved by a specialized
linear program, the minimum network flow
problem~\citep{Ahuja:1993:NFT:137406,Tarjan1997}. The minimum network flow
problem has been extensively studied in the operations research community and
several fast algorithms exist. However, these algorithms, at least on desktop
hardware, do not scale beyond a few thousand source and target points. Our
framework extends the applications of these algorithms to problem instances
several orders of magnitude larger, under suitable assumptions on the geometry
of the data. We exploit a multiscale representation of the source and target
sets to reduce the number of variables in the linear program and quickly find
good initial solutions, as illustrated in Figure~\ref{fig:multiscale-strategy}.
The optimal transport problem is solved (with existing algorithms) at the
coarsest scale and the solution is propagated to the next scale and refined.
This process is repeated until the finest scale is reached. This strategy,
discussed in detail in Section~\ref{sec:mdop}, is adaptable to memory
limitations and speed versus accuracy trade-offs. For some of the refinement
strategies, it is guaranteed to converge to the optimal solution.
Our approach draws from a varied set of work that is briefly summarized in
Section~\ref{sec:background}. The proposed approach generalizes and builds on
previous and concurrently developed hierarchical
methods~\citep{glimm:arxiv2011,schmitzer2013hierarchical,schmitzer2015sparse,oberman2015efficient}.
The work in this paper adds the following contributions:
\begin{compactitem}
\item The description of a general multiscale framework for discrete optimal
transport that can be applied in conjunction with a wide range of optimal
transport algorithms.
\item A set of propagation and refinement heuristics, including approaches
that are similar and/or refine existing ones
\citep{glimm:arxiv2011,oberman2015efficient,schmitzer2015sparse} as well as
novel ones. In particular we propose a novel propagation strategy based on
capacity restrictions of the network flow problem at each scale. This new
approach proves to be very efficient and accurate in practice. Overall, the
heuristics result empirically in a linear increase in computation time with
respect to data set size.
\item An implementation in the R package {\em mop} that allows the combination
of multiple heuristics to tailor speed and accuracy to the requirements of
particular applications.
\end{compactitem}
Compared to other linear programming based approaches, the multiscale approach results
in a speedup of up to multiple orders of magnitude in large problems and
permits to solve approximately transportation problems of several orders of
magnitudes larger than previously possible. Comparing to PDE based approaches
is difficult and PDE based methods are limited to low-dimensional domains and
specific cost metrics. The proposed framework is demonstrated on several
numerical examples and compared to the state-of-the-art approximation algorithm
by~\citet{cuturi:nips2013}.
\section{Background}
\label{sec:background}
Optimal transport is the problem of minimizing the cost of moving a source
probability distribution to a target probability distribution given a function
that assigns costs to moving mass from source to target locations. The
classical formulation by~\citet{monge:book1781} considers the minimization over
mass preserving mappings. Later~\citet{kantorovitch:ms1958} considered the
same problem but as a minimization over couplings between the two probability
measures, which permits to split mass from a single source across multiple
target locations. More precisely, for two probability measures $\mu$ and
$\nu$ on probability spaces ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$ respectively, a coupling of
$\mu$ and $\nu$ is a measure $\coupling$ on ${{\mathbf{X}}}\times{{\mathbf{Y}}}$ such that the
marginals of $\coupling$ are $\mu$ and $\nu$, i.e.
$\coupling(A\times{{\mathbf{Y}}})=\mu(A)$ for all ($\mu$-measurable) sets
$A\subseteq{{\mathbf{X}}}$ and $\coupling({{\mathbf{X}}}\times B)=\nu(B)$ for all
($\nu$-measurable) sets $B\subseteq{{\mathbf{Y}}}$. We denote by $\mathcal{C}(\mu,\nu)$
the set of couplings between $\mu$ and $\nu$. Informally, we may think of
$d\coupling(x,y)$ as the amount of infinitesimal mass to be transported from
source $x$ to destination $y$, with the condition that $\coupling$ is a
coupling guaranteeing that the source mass is distributed according to $\mu$
and the destination mass is distributed according to $\nu$. Such couplings
always exist: we always have the trivial coupling $\coupling=\mu\times\nu$. The
trivial coupling is uninformative, every source mass $d\mu(x)$ is transported
to the same target distribution $\nu$. In Monge's formulation the coupling is
restricted to the special form $\coupling(x,y)=\delta_{T(x)}$ where $\delta_y$
is the Dirac-$\delta$ measure with mass at $y\in{{\mathbf{Y}}}$, and $T$ is a function
${\mathbf{X}}\rightarrow{\mathbf{Y}}$: in this case the coupling is ``maximally informative''
in the sense that there is a function mapping each source $x$ to a single
destination $y$; in particular the mass $d\mu(x)$ at $x$ is not split into
multiple portions that are shipped to different target $y$'s.
To define optimal transport and optimal couplings, we need a cost function
$\cost(x,y)$ on ${{\mathbf{X}}}\times{{\mathbf{Y}}}$ representing the work or cost needed to
move a unit of mass from $x$ to $y$. Then for every coupling $\coupling$ we may
define the cost of $\coupling$ to be
\begin{equation}
\mathbb{E}_{(X,Y)\sim\coupling}[\cost(X,Y)]\,,
\end{equation}
with $(X,Y)$ being a pair of random variables distributed according to $\coupling$.
An optimal coupling $\coupling$ minimizes this cost over all choices of couplings.
When seeking an optimal transportation plan, the above becomes
$$
\mathbb{E}_{X\sim\mu}[\cost(X,T(X))]=\int_{{{\mathbf{X}}}} \cost(x,T(x)) d\mu(x)\,,
$$
and one minimizes over all (measurable) functions $T:{{\mathbf{X}}}\rightarrow{{\mathbf{Y}}}$.
One often designs the cost function $c$ in an application-dependent fashion,
and the above framework is extremely general. When ${{\mathbf{X}}}={{\mathbf{Y}}}$ is a metric
space with respect to a distance $\rho$ (with suitable technical conditions
relating the metrics and the measures $\mu$, $\nu$ that we will not delve into,
referring the reader to \cite{villani:book2009}), then distinguished choices
for the cost are those that are related to the metric structure. The natural
choice of $\cost(x,y)=\rho(x,y)^p$, for some $p>0$ leads to the definition of
the Wasserstein-Kantorovich-Rubinstein metric on the space of probability
measures on ${{\mathbf{X}}}$:
\begin{equation}
W_p(\mu,\nu):=\min_{\coupling\in\mathcal{C}(\mu,\nu)}\left(\int_{{{\mathbf{X}}}}\int_{{{\mathbf{X}}}}
\rho(x,y)^p d\coupling(x,y)\right)^{\frac1p}\,.
\end{equation}
Computational solutions to optimal transport split roughly in two settings:
Approaches based on the solution of partial differential equations derived from
the continuous optimal transport formulation, briefly discussed in
Section~\ref{sec:continuous} and, more relevant to this paper, combinatorial
optimization methods to directly solve for a discrete optimal transport plan
discussed in Section~\ref{sec:lp}.
\subsection{Continuous Optimal Transport}
\label{sec:continuous}
In case that at least the source distribution admits a density, and when the
cost function is the squared Euclidean distance, the optimal coupling is
deterministic, i.e. there exists a transport map, and the optimal solution is
the gradient of a convex function~\citep{brenier:cpam1991}. This has been
exploited to solve the optimal transport problem by numerical partial
differential equation
approaches~\citep{benamou:nm2000,angenent:jma2003,haker:ijcv2004,iollo:jcp2011,papadakis:arxiv2013,benamou2014numerical}.
An alternative formulation proposed by~\citet{aurenhammer:algorithmica1998}
shows that the optimal transport from a source density to a target distribution
of a set of weighted Dirac delta's can be solved through a finite dimensional
unconstrained convex optimization. \citet{merigot:cgf2011} proposes a
multiscale approach for the formulation of~\citet{aurenhammer:algorithmica1998}.
Both the numerical PDE based approaches as well as the unconstrained convex
optimization require a discretization of the full domain which is generally not
feasible for higher dimensional domains. For arbitrary cost functions and
distributions the optimal transport problem does typically not result in a
deterministic coupling and can not be solved through PDE based approaches.
\subsection{Discrete Optimal Transport and Linear Programming}
\label{sec:lp}
For two discrete distributions $\mu = \sum_1^n w(x_i) \delta(x_i)$ and $ \nu =
\sum_1^m v(y_i) \delta(y_i)$ with $\sum w(x_i) = \sum v(y_i) = 1$ the optimal
transport problem is equivalent to the linear program
\begin{equation}
\min_\coupling \sum_{\substack{i=1,\dots,n\\ j=1,\dots,m}}
\cost(x_i, y_j) \coupling(x_i, y_j) \quad \text{s.t.}\quad
\begin{cases}
\sum_j \coupling(x_i, y_j) = \mu(\{x_i\})= w(x_i) & \\
\sum_i \coupling(x_i, y_j) = \nu(\{y_j\}) = v(y_j) & \\
\coupling(x_i,y_j)\ge 0
\end{cases}\,.
\label{e:LPformulation}
\end{equation} The solution is called an optimal coupling $\coupling^*$, and the
minimum value attained at $\coupling^*$, is called the cost of $\pi^*$, or the
optimal cost of the transport problem, and is denoted by
$\couplingcost(\coupling^*)=\sum_{i,j} \cost(x_i,y_j)\pi^*(x_i,y_j)$. The
constraints enforce that $\coupling$ is a coupling. The variables
$\coupling(x_i, y_j)$ correspond to the amount of mass transported from
source $x_i$ to target $y_j$, at cost $\cost(x_i, y_j)$. The linear
constraints are of rank $n+m-1$: when $n+m-1$ of the constraints are satisfied,
either the $n$ constraints of the source density or the $m$ constraints of the
target density are satisfied. Since the sum of outgoing mass is equal to the sum
of incoming mass, i.e. $\mu({\mathbf{X}}) = \nu({\mathbf{Y}})$, it follows that all constraints
must be satisfied. The optimal solution lies, barring degeneracies, on a corner
of the polytope defined by the constraints, i.e. is a basic feasible solution.
This implies that exactly $n+m-1$ entries of the optimal coupling $\coupling$
are non-zero, i.e. $\pi$ is a sparse matrix. The optimal coupling is a Monge
transport map if and only if all the mass of a source $x_i$ is transported to
exactly one target location. A Monge solution does not exist for every optimal
transport problem, in fact a small perturbation of $\mu$ will always suffice to
make a Monge solution impossible.
For source and target data sets with the same cardinality and equally weighted
$\delta$-functions, Kantorovich's optimal transport problem reduces to
an assignment problem, whose solution is a Monge optimal transport map. In this
special case, the optimal transport problem can be efficiently solved by the
Hungarian algorithm~\citep{kuhn:nrlq1955}. The assignment problem results in
a degenerate linear program since only $n$ entries are non-zero (instead of
$2n-1$). We can therefore think of optimal transport as a robust version of
assignments. This can also be seen from the point of view of convexity: in the
assignment problem, $\coupling$ is a permutation matrix deciding to which $y_j$
each $x_i$ is transported. The convex hull of permutation matrices is exactly
the set of doubly-stochastic matrices, to which a coupling $\pi$ belongs as a
consequence of the constraints in \eqref{e:LPformulation}.
For point sets with different cardinalities and/or points with different masses
the optimal transport problem can be solved by a linear program and is a
special case of the minimum cost network flow problem. The minimum cost flow
problem is well studied and a number of
algorithms~\citep{ford:ms1956,klein:ms1967,
cunningham:mp1976,goldberg:stoc1987,bertsekas:or1988,orlin:mp1997} exist for
its solution. This discrete solution approach is not constrained to specific
cost functions and can work with arbitrary cost functions.
However, the linear programming approach neglects possibly useful geometric
properties of the measures and the cost function. Our work makes
assumptions about the underlying geometry of the measure spaces and the
associated cost function, and in this way is a mixing of the low-dimensional
``geometric PDE'' approaches with the discrete non-geometric optimization
approaches. It exploits the geometric assumptions to relieve the shortcomings
of either approach, namely it scales to high-dimensional data, provided that
the intrinsic dimension is low in a suitable sense, and does not require a mesh
data structure. At the same time we use the geometry of the data
to speed up the linear program, which per--se does not leverage geometric
structures.
The refinement strategies of the proposed multiscale approach add subsets of paths among
all pairwise paths at each subsequent scale to improve the optimal transport
plan. This strategy of adding paths, is akin to column generation
approaches~\citep{desrosiers:book2005}. Column generation, first developed
by~\citet{dantzig:informs1960} and~\citet{ford:ms1956}, reduces the number of
variables in a linear program by solving a smaller linear program on a subset
of the original variables and introducing new variables on demand. However, the
proposed approach exploits the geometry of the problem instead of relying on an
auxiliary linear program~\citep{dantzig:informs1960} or shortest path
computations~\citep{ford:ms1956} to detect the entering variables.
\subsection{Approximation Strategies}
In the computer vision literature, the Earth Movers distance or equivalently the
Wasserstein-1 distance, which is simply the cost of the optimal coupling, is
a successful similarity measure for image retrieval~\citep{rubner:iccv1998}. In
this application the transport plan is not of interest but only the final
transport cost. For this
purpose~\citet{indyk2003fast},~\citet{shirdhonkar:cvpr2008} and
~\citet{andoni:soda2008} developed algorithms that compute an approximate cost
but do not yield a transport plan. Some of these approaches are based on the
dual formulation of optimal transport, which involves testing against Lipschitz
functions, and observing that Lipschitz functions may be characterized by decay
properties of their wavelet coefficients. In this sense these approaches
are multiscale as well.
To speed up computations in machine learning
applications~\citet{cuturi:nips2013} proposes to {\em smooth} transport plans by
adding a maximum entropy penalty to the optimal transport formulation. The
resulting optimization problem is efficiently solved through matrix scaling
with Sinkhorn fixed-point iterations. Because of the added regularization
term, the solution will in general be different from the optimal
transportation. It may however be the case that these particular (or
perhaps other) regularized solutions are better suited for certain
applications.
\subsection{Related Work}
Very recently a number of approaches have been proposed to solve the optimal
transport in a multiscale
fashion~\citep{glimm:arxiv2011,schmitzer2013hierarchical,schmitzer2015sparse,oberman2015efficient}.
\citet{glimm:arxiv2011} design an iterative scheme to solve a discrete optimal
transport problem in reflector design and propose a heuristic for the iterative
refinements based on linear programming duality. This iterative scheme can be
interpreted as a multiscale decomposition of the transport problem based on
geometry of the source and target sets. The proposed potential refinement
strategy extends the heuristic proposed by~\citet{glimm:arxiv2011} to guarantee
optimal solutions and adds a more efficient computation strategy: Their
approach requires to check all possible variables at the next scale. In
Section~\ref{sec:potential} we introduce a variation of the approach
by~\citet{glimm:arxiv2011} by adding a branch and bound strategy to avoid
checking all variables, and an iterative procedure that guarantees optimal
solutions.
\citet{schmitzer2013hierarchical} propose a multiscale approach on grids that
uses a refinement strategy based on spatial neighborhoods, akin to the
neighborhood refinement described in Section~\ref{sec:neighborhood}.
\citet{schmitzer2015sparse} uses a multiscale approach to develop a modified
auction algorithm with guaranteed worst case complexity and optimal solutions.
We use data structures that enable us to quickly construct neighborhoods, even
for point clouds that live in high-dimensions, but have low-intrinsic dimension;
we also exploit these structures to not compute all the pairwise possible
costs, as well as the candidate neighborhoods in our propagation steps. This
can result in substantial savings in the scaling of the algorithm, from
$|\mathbf{X}|^3$ to just $|\mathbf{X}|\log|\mathbf{X}|$. We are therefore able
to scale to larger problem, solving on a laptop problems an order of magnitude
larger than those in \citet{oberman2015efficient} for example, and showing
linear scaling on a large range of scales. \citet{schmitzer2015sparse} use the
c--cyclical monotonicity property of optimal transport
plans~\citep[Chapter~5]{villani:book2009} to construct ``shielding
neighborhoods'' that permit to exclude paths from further consideration. The
idea of shielding neighborhoods is combined with a multiscale strategy that
permits to quickly refine initial neighborhood estimates.
Finally, in our work we emphasize that the multiscale construction is not only
motivated by its computational advantage, but also as a way of revealing
possibly important features of the optimal transportation map. As we show in
Section \ref{s:brainMR}, features collected from the multiscale optimal
transportation maps leads to improved predictors for brain conditions. More
generally, we expect multiscale properties of optimal transportation maps to be
useful in a variety of learning tasks; the connections between learning and
optimal transportation are still a very open field, to be explored and
exploited.
\section{Multiscale Optimal Transport}
\label{sec:mdop}
Solving the optimal transport problem for two point sets ${\mathbf{X}}$ and ${\mathbf{Y}}$
directly requires $|{\mathbf{X}}||{\mathbf{Y}}|$ variables, or paths to consider. In other
words, the number of paths along which mass can be transported grows
quadratically in the number of points and quickly yields exceedingly large
problems. The basic premise of the multiscale strategy is to solve a sequence
of transport problems based on increasingly accurate approximations of the
source and target point set. The multiscale strategy helps to reduce the
problem size at each scale by using the solution from the previous scale to
inform which paths to include in the optimization at the next finer scale.
Additionally, the solution at the previous scale helps to find a good
initialization for the current scale which results in fewer iterations to solve
the reduced size linear program.
The multiscale algorithm (see Algorithm~\ref{f:algo} and
Figure~\ref{fig:multiscalepic} for a visual illustration) comprises of three
key elements:
\begin{itemize}
\item[(I)] A way of {\textit{\textbf{coarsening} the sets}} of source points ${{\mathbf{X}}}$ and measure $\mu$ in a multiscale fashion, yielding a chain
\begin{equation}
({{\mathbf{X}}},\mu)=:({{\mathbf{X}}}_J,\mu_J) \rightarrow({{\mathbf{X}}}_{J-1},\mu_{J-1})
\rightarrow\dots\rightarrow({{\mathbf{X}}}_j,\mu_j)
\rightarrow\dots\rightarrow({{\mathbf{X}}}_0,\mu_0)
\end{equation}
connecting the scales from fine to coarse, with ${{\mathbf{X}}}_j$ of decreasing
cardinality as the scale decreases, and the discrete measure $\mu_j$
``representing'' a coarsification of $\mu$ at scale $j$, with
$\mathrm{supp}(\mu_j)={{\mathbf{X}}}_j$ (the support of $\mu_j$ is the set of points
with positive measure). Similarly for ${\mathbf{Y}}$ and $\nu$ we obtain the chain
\begin{equation}
({{\mathbf{Y}}},\mu)=:({{\mathbf{Y}}}_J,\nu_J) \rightarrow({{\mathbf{Y}}}_{J-1},\nu_{J-1})
\rightarrow\dots\rightarrow({{\mathbf{Y}}}_j,\nu_j)
\rightarrow\dots\rightarrow({{\mathbf{Y}}}_0,\nu_0)
\end{equation}
This coarsening step is described in Section~\ref{sec:coarsening} and the
resulting multiscale family of transport problems is discussed in
Section~\ref{sec:multiscaleproblems}.
\item[(II)] A way of {\textit{\textbf{propagating} a coupling}} $\coupling_j$
solving the transport problem $\mu_j\rightarrowtail\nu_j$ at scale $j$ to a
coupling $\coupling_{j+1}$ at
scale $j+1$. This is described in Section~\ref{sec:propagation}.
\item[(III)] A way of {\textit{\textbf{refining} the propagated solution}} to
the optimal coupling at scale $j$. This is described in
Section~\ref{sec:refinement}.
\end{itemize}
\begin{algorithm}[h]
\KwIn{Two discrete measures $\mu$ and $\nu$, and a stopping scale $J_0\le J$}
\KwOut{Multiscale family of transport plans $(\coupling_j: \mu_j\rightarrowtail
\nu_j)_{j=0}^{J_0}$}
Construct multiscale structures $\{\{{{\mathbf{X}}}_j,\mu_j)\}_{j=0}^J$ and
$\{\{{{\mathbf{Y}}}_j,\nu_j)\}_{j=0}^J$.
Let $\tilde \coupling_0$ be an arbitrary coupling $\mu_0\rightarrowtail\nu_0$.
\For{ $j=0\dots J_0-1$ }{
Refine initial guess $\tilde\coupling_j$ to the optimal coupling $\coupling_j: \mu_j\rightarrowtail\nu_j$.
Propagate $\coupling_j$ from scale $j$ to scale $j+1$, obtaining a
coupling $\tilde\coupling_{j+1}$}
\caption{Multiscale Discrete Optimal Transport
\label{f:algo}
}
\end{algorithm}
\subsection{The Coarsening Step: Multiscale Approximations to ${\mathbf{X}}$, $\mu$ and $\cost$}
\label{sec:coarsening}
To derive approximation bounds for the error of the multiscale transport
problem at each scale we rely on the notion of a regular family of multiscale
partitions formally described in Section~\ref{sec:mspartitions}. The multiscale
partition is used to define approximations to $\mu$ and $\cost$ at
all scales. An integral part of the definitions is that the constructions can be
interpreted as a tree, with all nodes at a fixed height corresponding to one scale
of the multiscale partitioning.
We start with some notation needed for the definition of the
multiscale partitions. Let $({{\mathbf{X}}},\rho,\mu)$ be a measure metric space with
metric $\rho$ and finite measure $\mu$. Without loss of generality assume that
$\mu({\mathbf{X}})=1$. The metric ball of center $z$ and radius $r$ is $B_z(r)=\{
x\in{{\mathbf{X}}} : \rho(x,z)<r\}$. We say that ${{\mathbf{X}}}$ has doubling dimension $d$ if
every ball $B_z(r)$ can be covered by at most $2^d$ balls of radius
$r/2$~\citep{assouad1983plongements}. Furthermore, a space has a doubling
measure if $\mu(B_z(r))\asymp r^d$, i.e. if there exist a constant $c_1$ such
that for every $z\in{\mathbf{X}}$ and $r>0$ we have $c_1^{-1}r^d\le\mu(B_z(r))\le
c_1r^d$. Here and in what follows, we say that $f\asymp g$ if there are two
constants $c_1,c_2>0$ such that for every $z$ in the domain of both functions
$f,g$ we have $c_1 f(z)\le g(z)\le c_2 f(z)$ (and therefore a similar set of
inequalities holds with the roles of $f$ and $g$ swapped), and we say that $f$
and $g$ have the same order of magnitude. Having a doubling measure implies
having a doubling metric, and up to changing the metric to an equivalent one,
one may choose the same $d$ in the doubling condition for the metric and in
that for the measure: we assume this has been done from now on. This family of
spaces is rather general, it includes regular domains in $\mathbb{R}^D$, as
well as smooth compact manifolds $\mathcal{M}$ endowed with volume measures.
\subsubsection{Multiscale Approximations to ${\mathbf{X}}$ and $\mu$}
\label{sec:mspartitions}
A regular family of multiscale partitions, with scaling parameter $\theta>1$,
is a family of sets $\{\{C_{j,k}\}_{k=1}^{K_j}\}_{j=0}^J$, where $j$ denotes
the scale and $k$ indexes the sets at scale $j$, such that:
\begin{itemize}
\item[(i)] the sets $\{C_{j,k}\}_{k=1}^{K_j}$ form a partition of ${{\mathbf{X}}}$,
i.e. they are disjoint and $\cup_{k=1}^{K_j}C_{j,k} = {{\mathbf{X}}}$;
\item[(ii)] either $C_{j+1,k'}$ does not intersect a $C_{j,k}$, or it is
completely contained in it;
\item[(iii)] there exists a constant $A>0$ such that for all $j,k$ we have
the diameter $\mathrm{diam}(C_{j,k})\le A\theta^{-j}$;
\item[(iv)] each $C_{j,k}$ contains a ``center'' point $c_{j,k}$ such that
$B_{c_{j,k}}(\theta^{-j})\subseteq C_{j,k}$.
\end{itemize}
To ease the notation we will fix $\theta=2$ in what follows, but the
constructions and results hold, mutatis mutandis, for general $\theta>1$. The
properties (i) and (ii) above imply that there exists a tree $\mathcal{T}$, with nodes at
scale $j$ (i.e. at distance $j$ from the root) in bijection with
$\{C_{j,k}\}_{k=1}^{K_j}$, such that $(j+1,k')$ is a child of $(j,k)$ if and
only if $C_{j+1,k'}$ is contained in $C_{j,k}$. Moreover properties (iii) and
(iv), together with the properties of spaces with a doubling measure, imply
that $\mu(C_{j,k})\asymp 2^{-jd}$ and $K_j\asymp 2^{jd}$. These partitions are
classical in harmonic analysis, mimicking dyadic cubes in Euclidean space, and
they have recently been used to construct multiscale decompositions of data
sets in high-dimensions ~\citep{LMR:MGM1,CM:MGM2,IM:GMRA_CS,6410789}. We say
that $C_{j+1,k'}$, or even $(j+1,k')$, is a child of $C_{j,k}$ (respectively,
of $(j,k)$) if $C_{j+1,k'}\subseteq C_{j,k}$, and that such $C_{j,k}$
(respectively $(j,k)$) is a parent of $C_{j+1,k'}$ (resp. $(j+1,k')$).
Given two discrete sets ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$ in a doubling metric space of
homogeneous type with dimension $d$, we construct the corresponding families of
multiscale partitions $\{\{C^{{{\mathbf{X}}}}_{j,k}\}_{k=1}^{K^{{{\mathbf{X}}}}_j}\}_{j=0}^J$
and $\{\{C^{{{\mathbf{Y}}}}_{j,k}\}_{k=1}^{K^{{{\mathbf{Y}}}}_j}\}_{j=0}^J$ (we assume the same
range of scales to keep the notation simple). The construction of the
multiscale approximations is illustrated in Figure~\ref{fig:multiscalepic}.
\begin{figure}[thb]
\centering
\includegraphics[width=0.99\linewidth]{figures/MultiscalePic.pdf}
\caption{
\label{fig:multiscalepic}
An illustration of the multiscale framework. The coarsening step constructs
from a set of weighted points ${\mathbf{X}}_{j+1}$ at scale $j+1$ a smaller set of
weighted points at scale $j$ (blue bars). An equivalent
coarsening is performed on the target point set ${\mathbf{Y}}_{j+1}$ Neighboring points
at scale $j+1$ are combined into a single representative at scale $j$ with mass
equivalent to the weights of the combined points. The optimal transport plan
(green arrows) is solved at the coarser scale $j$, then propagated to scale
$j+1$ and refined.
}
\end{figure}
The space ${{\mathbf{X}}}_j$ will be the partition at scale $j$, namely the set
$\{C^{{{\mathbf{X}}}}_{j,k}\}_{k=1}^{K^{{{\mathbf{X}}}}_j}$, of cardinality $K^{{{\mathbf{X}}}}_j$.
The measures $\mu$ and $\nu$ may be coarsened in the natural way, by letting
$\mu_j$ be defined recursively on ${{\mathbf{X}}}_j$ by
\begin{equation}
\mu_j( C_{j,k}^{{{\mathbf{X}}}} )= \sum_{(j+1,k')\text{ child of } (j,k)} \mu_{j+1}( C_{j+1,k'}^{{{\mathbf{X}}}} )\,,
\label{e:muj}
\end{equation}
and similarly for $\nu_j$.
These are in fact projections of these measures, and may also be interpreted as
conditional expectations with respect to the $\sigma$-algebra generated by the
multiscale partitions. We can associate a point $\overline c_{j,k}^{{{\mathbf{X}}}}$
to each $C_{j,k}^{{{\mathbf{X}}}}$ in various ways, by ``averaging'' the points in
$C_{j+1,k'}^{{{\mathbf{X}}}}$, for $(j+1,k')$ a child of $(j,k)$. At the finest scale
we may let $\overline{c}^{{{\mathbf{X}}}}_{J,k}={c}^{{{\mathbf{X}}}}_{J,k}$ (this being a
``center'' for $C_{J,k}$ as in item (iv) in the definition of multiscale
partitions), and then recursively we defined the coarser centers step from
scale $j+1$ to scale $j$ in one of the following ways:
\begin{itemize}
\item[(i)] If the metric space is also a vector space a natural definition of
$x_{j,k}:=\overline c_{j,k}^{{{\mathbf{X}}}}$ is a weighted average of the $\overline
c_{j+1,k'}^{{{\mathbf{X}}}}$ corresponding to children:
$$
\overline{c}^{{{\mathbf{X}}}}_{j,k} =
\sum_{(j+1,k')\text{ child of } (j,k)} \mu_{j+1}(\{\overline c_{j+1,k'}^{{{\mathbf{X}}}}\}) \overline c_{j+1,k'}^{{{\mathbf{X}}}}\,.
$$
\item[(ii)] In general we can define $x_{j,k}:=\overline c_{j,k}^{{{\mathbf{X}}}}$ as the point
$$
\overline{c}^{{{\mathbf{X}}}}_{j,k}=\mathrm{argmin}_{c \in{{\mathbf{X}}}}\,\,
\sum_{(j+1,k')\text{ child of } (j,k) } \rho^p( c, \overline c_{j+1,k'}^{{{\mathbf{X}}}} ) \,,
$$
for some $p \geq 1$, typically $p=1$ (median) or $p=2$ (Fr\'echet mean).
\end{itemize}
Of course similar constructions apply to the space ${{\mathbf{Y}}}_j$, yielding points
$y_{j,k}:=\overline c_{j,k}^{{{\mathbf{Y}}}}$.
We discuss algorithms for these constructions in Section \ref{s:multiscalepointsets}.
\subsubsection{Coarsening the cost function $\cost$}
The multiscale partition provides several ways to coarsen the cost function: for
every $x_{j,k}$ and $y_{j,k'}$ we consider
\begin{itemize}
\item[($\cost$-i)] the pointwise value
\begin{equation}
\cost_j(c_{j,k}^{{\mathbf{X}}}, c_{j,k'}^{{\mathbf{Y}}}) :=
\cost(x_{j,k}^{{\mathbf{X}}},y_{j,k'}^{{\mathbf{Y}}})\,,
\label{e:costpointwise}
\end{equation}
where $x_{j,k}$ and $y_{j,k'}$ are defined in any of the ways above;
\item[($\cost$-ii)] the local average
\begin{equation}
\cost_j(c_{j,k}^{{\mathbf{X}}}, c_{j,k'}^{{\mathbf{Y}}}) :=
\mathop{\rm argmin}_{\alpha} \sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k},\,y \in
C^{{{\mathbf{Y}}}}_{j,k'}}}\left(\alpha-\cost(x,y)\right)^2
= \frac{\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k},\,y\in
C^{{{\mathbf{Y}}}}_{j,k'}}}\cost(x,y)}{|C_{j,k}^{{{\mathbf{X}}}}| |C_{j,k'}^{{{\mathbf{Y}}}}|}\,;
\label{e:costlocalave}
\end{equation}
\item[($\cost$-iii)] the local weighted average
\begin{align*}
\cost_j(c_{j,k}^{{\mathbf{X}}},c_{j,k'}^{{\mathbf{Y}}})
&:=\mathop{\rm argmin}_{\alpha}\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k},\, y\in
C^{{{\mathbf{Y}}}}_{j,k'}}}\left(\alpha-\cost(x,y)\right)^2
\coupling^*_{j-1}(x_{j-1,k_1},y_{j-1,k'_1})\\
&=\frac{\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k},\,y\in C^{{{\mathbf{Y}}}}_{j,k'}}}\cost(x,y)\coupling^*_{j-1}(x_{j-1,k_1},y_{j-1,k'_1})}{
\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k},\,y\in C^{{{\mathbf{Y}}}}_{j,k'}}}\coupling^*_{j-1}(x_{j-1,k_1},y_{j-1,k'_1})
}\,,
\end{align*}
where $\pi^*_{j-1}$ is the optimal or approximate transportation plan at scale
$j-1$, defined in \eqref{e:LPformulation_j}; $k_1$ is the unique index for
which $C^{{{\mathbf{X}}}}_{j,k}\subseteq C^{{{\mathbf{X}}}}_{j-1,k_1}$ and $k'_1$ is the unique
index for which $C^{{{\mathbf{Y}}}}_{j,k}\subseteq C^{{{\mathbf{Y}}}}_{j-1,k'_1}\,.$
\end{itemize}
\subsection{Multiscale Family of Optimal Transport Problems}
\label{sec:multiscaleproblems}
With the definitions of the multiscale family of coarser spaces ${{\mathbf{X}}}_j$ and
${{\mathbf{Y}}}_j$, corresponding measures $\mu_j$ and $\nu_j$, and corresponding cost
$\cost_j$, we may consider, for each scale $j$, the following optimal transport
problem:
\begin{equation}
\begin{aligned}
\coupling_j^*&: = \mathop{\rm argmin}_\coupling\!\!\!\sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\ k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}
\!\!\!\!\cost_j(x_{j,k}, y_{j,k'}) \coupling(x_{j,k}, y_{j,k'})
\,\,\text{s.t.}\,\,
\begin{cases}
\sum_{k'} \coupling(x_{j,k}, y_{j,k'}) = \mu_j(\{x_{j,k}\}) & \forall k\in K_j^{\mathbf{X}} \\
\sum_{k} \coupling(x_{j,k}, y_{j,k'}) = \nu_j(\{y_{j,k'}\}) & \forall k'\in K_j^{\mathbf{Y}}\\
\coupling(x_{j,k},y_{j,k'})\ge 0
\end{cases}
\label{e:LPformulation_j}
\end{aligned}
\end{equation}
The problems in this family are related to each other, and to the optimal transportation
problem in the original spaces. We define the cost of a coupling as
\begin{equation}
\couplingcost(\coupling_j) = \sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\
k'=1,\dots,K_j^{{{\mathbf{Y}}}}}} \cost_j(x_{j,k}, y_{j,k'}) \coupling_j(x_{j,k},
y_{j,k'}) \, .
\end{equation}
The cost of the optimal coupling $\coupling_j^*$ at scale $j$ is provably an
approximation to the cost of the optimal coupling $\coupling^*$ (which is equal
to $\coupling_J^*$):
\begin{proposition}
Let $\coupling^*$ be the optimal coupling, i.e. the solution to
\eqref{e:LPformulation}, and $\coupling^*_j$ the optimal coupling at scale $j$,
i.e. the solution to \eqref{e:LPformulation_j}. Define
\begin{equation}
E_j(\coupling^*) := \sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\ k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}
\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k}\\ y\in C^{{{\mathbf{Y}}}}_{j,k'}}}
\left(\cost_j(x_{j,k},y_{j,k'})-\cost(x,y)\right) \coupling^*(x,y)\,.
\end{equation}
Then
\begin{equation}
\couplingcost(\coupling_j^*)\le\couplingcost(\coupling^*)+E_j(\coupling^*)\,,
\label{e:UBcouplingcostpij}
\end{equation}
and if $\cost_j=\cost$ and $\cost$ is Lipschitz with constant $||\nabla\cost||_\infty$, we have
\begin{equation}
\couplingcost(\coupling_j^*)\le\couplingcost(\coupling^*)+2^{-j}A||\nabla \cost||_\infty\,,
\label{e:UBcouplingcostpijboundednabld}
\end{equation}
where $A$ is such that
$\max_{k,k'}\{\mathrm{diam}(C^{{{\mathbf{X}}}}_{j,k}),\mathrm{diam}(C^{{{\mathbf{Y}}}}_{j,k'})\}\le
A\cdot2^{-j}$.
\end{proposition}
\begin{proof}
Consider the coupling $\coupling_j$ induced at scale $j$ by the optimal
coupling $\coupling^*$, defined by
\begin{equation*}
\coupling_j(x_{j,k},y_{j,k'}) = \sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k}\,,
y\in C^{{{\mathbf{Y}}}}_{j,k'}}} \coupling^*(x,y)\,.
\end{equation*}
First of all, since $\{C^{{{\mathbf{X}}}}_{j,k}\}_k$ and $\{C^{{{\mathbf{Y}}}}_{j,k'}\}_{k'}$
are partitions, it is immediately verified that $\coupling_j$ is a coupling. Secondly,
observe that
\begin{align*}
\couplingcost(\coupling_j)
&=\sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\ k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}
\cost_j(x_{j,k}, y_{j,k'}) \coupling_j(x_{j,k}, y_{j,k'})
=\sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\ k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}
\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k}\\ y\in C^{{{\mathbf{Y}}}}_{j,k'}}}
\cost_j(x_{j,k}, y_{j,k'}) \coupling^*(x,y)\\
&=\sum_{\substack{x\in{{\mathbf{X}}}\\ y\in{{\mathbf{Y}}}}} \cost(x,y)\coupling^*(x,y)+
\sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\ k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}
\sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k}\\ y\in C^{{{\mathbf{Y}}}}_{j,k'}}}
\left(\cost_j(x_{j,k},y_{j,k'})-\cost(x,y)\right) \coupling^*(x,y)\\
&= \couplingcost(\coupling^*) + \underbrace{ \sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\
k'=1,\dots,K_j^{{{\mathbf{Y}}}}}} \sum_{\substack{x\in C^{{{\mathbf{X}}}}_{j,k}\\ y\in C^{{{\mathbf{Y}}}}_{j,k'}}}
\left(\cost_j(x_{j,k},y_{j,k'})-\cost(x,y)\right) \coupling^*(x,y)}_{=:E_j(\coupling^*)}\,.
\end{align*}
Since $\couplingcost(\coupling^*_j)\le\couplingcost(\coupling_j)$ (since
$\coupling^*_j$ is optimal), we obtain \eqref{e:UBcouplingcostpij}.
When $\cost_j=\cost$ and $\cost$ is Lipschitz with constant $||\nabla\cost||_\infty$, we have
\begin{align*}
E_j(\coupling^*)
&\le \sum_{\substack{k\,:\,x\in C^{{{\mathbf{X}}}}_{j,k}\\
k'\,:\,y\in C^{{{\mathbf{Y}}}}_{j,k'}}} \sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\
k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}||\nabla\cost||_\infty\cdot ||(x_{j,k},y_{j,k'})-(x,y)||\coupling^*(x,y)\\
&\le \sum_{\substack{k\,:\,x\in C^{{{\mathbf{X}}}}_{j,k}\\
k'\,:\,y\in C^{{{\mathbf{Y}}}}_{j,k'}}} \sum_{\substack{k=1,\dots,K_j^{{{\mathbf{X}}}}\\
k'=1,\dots,K_j^{{{\mathbf{Y}}}}}}||\nabla\cost||_\infty 2^{-j}A\coupling^*(x,y)\\
&\le 2^{-j}A||\nabla\cost||_\infty \,,
\end{align*}
with $A$ as in the claim.
\end{proof}
In the discrete, finite case that we are considering, $\mu_j\rightarrow\mu$ and
$\nu_j\rightarrow\nu$ trivially since $\mu_J=\mu$ and $\nu_J=\nu$ by
construction. If $\mu$ and $\nu$ were continuous, and at least when
$\cost(x,y)=\rho(x,y)^p$ for some $p\ge1$, then if $\mu$ and $\nu$ have finite
$p$-moment (i.e. $\int_{\mathbf{X}} \rho(x,x_0)^pd\mu$ and similarly for $\nu$), we
would obtain convergence of a subsequence of $\cost(\coupling^*_j)$ to $\cost(\coupling^*)$ by
general results (e.g. as a simple consequence of Lemma 4.4. in ~\citep{villani:book2009}).
Note that the approximations do not guarantee that the transport plans are
close in any other sense but their cost. Consider the arrangement in
Figure~\ref{fig:counter-approximation}, the transport plans are
$\epsilon$-close in cost but the distances between the target locations of the
sources are far no matter how small $\epsilon$ gets.
\begin{figure}[thb]
\centering
\begin{tabular}{VV}
\includegraphics[width=0.99\linewidth]{counter-approximation} &
\includegraphics[width=0.99\linewidth]{counter-approximation2} \\
(a) & (b)
\end{tabular}
\vspace{-0.1in}
\caption{
\label{fig:counter-approximation}
An example that illustrates that closeness in cost does not indicate closeness
of the transport plan. The transport plans (green arrows) between the sources A
and B (purple) and the targets Y and Z (orange) in (a) and (b) are
$\epsilon$-close but their respective target locations are very far. }
\end{figure}
\subsection{Propagation Strategies}
\label{sec:propagation}
The approximation bounds in Section~\ref{sec:multiscaleproblems} show that the
optimal solution $\coupling_{j}$ at scale $j$ is $| E_{j+1} - E_j |$ close to
optimal solution $\coupling_{j+1}^*$ at scale $j+1$. This suggests that the
solution at scale $j$ can provide a reasonably close to optimal initialization
for scale $j+1$.
As proposed by~\citet{glimm:arxiv2011} the solution at a given scale can be
interpolated at the next scale (or finer discretization). The most direct
approach to initialize the transport problem at scale $j+1$ given the solution
$\coupling_{j}^*$ at scale $j$ is to distribute the mass $\coupling_{j}^*(x_k,
y_{k'})$ equally to all combinations of paths between $\children(x_k)$ and
$\children(y_{k'})$.
This propagation strategy results in a reduction in the number of iterations
required to find an optimal solution at the subsequent scale. This
warm-starting alone is often not sufficient, however. At the finest scale the
problems still requires the solution of a problem of size $O(n^2)$. This
quickly reaches memory constraints with $\Omega(10^4)$ points, and a single
iteration of Newton's method or a pivot step of a linear program becomes
prohibitively slow. Thus, we consider reducing the number of variables, which
substantially speeds up the algorithm, albeit we may lose guarantees on its
computational complexity and/or its ability to achieve arbitrary accuracy, so
that only numerical experiments will support our constructions. These
reductions are achieved by considering only a subset $\mathbf{R}_{j+1}$ of all
possible paths $\mathbf{A}_{j+1}$ at scale $j+1$.
To distinguish the optimal solution on the reduced set of paths at scale $j$
from the optimal solution over all paths we introduce some notation. Let
$\mathbf{A}_{j+1}$ be the set of all possible paths between sources and targets
at scale $j+1$. Let $\mathbf{R}_{j+1} \subseteq \mathbf{A}_{j+1}$ be the set of
paths propagated from the previous solution (e.g. children of mass-bearing
paths found at scale $j$). Let $\coupling^*_{j+1} |_{\mathbf{P}}$ be the
optimal solution to the transport problem restricted to a set of paths
$\mathbf{P} \subset \mathbf{A}_{j+1}$. With this notation, $ \coupling_{j+1}^*
|_{\mathbf{A}_{j+1}} = \coupling_{j+1}^*$. The optimal coupling
$\coupling_{j+1}^*|_{\mathbf{R}_{j+1}}$ on the reduced set of paths problem does
not need to match the optimal coupling $\coupling_{j+1}^*$ on all paths.
However $ \coupling_{j+1}^* |_\mathbf{R_{j+1}}$ does provide a starting point
for further refinements discussed in Section~\ref{sec:refinement}.
\subsubsection{Simple Propagation}
\label{sec:simple}
The most direct approach to reduce the number of paths considered at subsequent
scales is to include only paths at scale $j+1$ whose endpoints are children of
endpoints of mass-bearing paths at scale $j$. The optimal solution at scale $j$
has exactly $K_j^{{\mathbf{X}}} + K_j^{{\mathbf{Y}}} - 1$ paths with non-zero weight. Thus, the
number of paths at scale $j+1$ reduces to $O( C^2(K_{j}^{{\mathbf{X}}} + K_j^{{\mathbf{Y}}}))$,
where $C$ is the maximal number of children of any node at scale $j$. In
particular, $C\asymp 2^{d}$ for a doubling space of dimension $d$. This reduces
the number of variables from ``quadratic'', $O(K_{j}^{{\mathbf{X}}} K_j^{{\mathbf{Y}}} )$, to
linear, $O(K_{j}^{{\mathbf{X}}}+K_j^{{\mathbf{Y}}} )$. This propagation strategy by itself,
however, often leads to a dramatic loss of accuracy in both the cost the
transportation plan, and the transportation plan itself.
\subsubsection{Capacity Constraint Propagation}
\label{sec:capacity}
This propagation strategy solves a modified minimum flow problem at scale $j$
in order to include additional paths at scale $j+1$ that are likely to be
included in the optimal solution $\coupling_{j+1}^*$. This is achieved by
adding a capacity constraint to the mass bearing paths at scale $j$ in the
optimal coupling $ \coupling_j^* |_{\mathbf{R}_j}$: The amount of mass of a
mass bearing path $ \coupling_j^* |_{\mathbf{R}_j}( x_{j,k}, y_{j,k'} )$ is
constrained to $\lambda \min \left ( \mu_j(x_{j,k}), \nu_j(y_{j,k'}) \right)$
with $\lambda$ random uniform on $[0.1, 0.9]$. The randomness is introduced to
avoid degenerate constraints. The solution of this modified minimum--flow
problem forces the inclusion of $n_c$ additional paths, where $n_c$ is the
number of constraints added. There are various options for adding capacity
constraints, we propose to constrain all mass bearing paths of the optimal
solution at scale $\coupling_j^*|_{\mathbf{R}_j}$. The capacity constrained
problem thus results in a solution with twice the number of paths as in the
coupling $\coupling_j^*|_{\mathbf{R}_j}$. The solution of the capacity
constrained minimum flow problem is propagated as before to the next scale.
To increase the likelihood of including paths required to find an optimal
solution at the next scale, the capacity constrained procedure can be iterated
multiple times. Each time the mass bearing paths in the modified solution are
constrained and a new solution is computed. Each iteration doubles the number
of mass bearing paths and the number of iterations controls how many paths are
propagated to the next scale. Thus, the capacity constraint propagation
strategy bounds the number of paths considered in the linear program. The
optimal transport plan on a source set ${\mathbf{X}}$ and ${\mathbf{Y}}$ results in a linear
program with $|{\mathbf{X}}| + |{\mathbf{Y}}|$ constraints and $|{\mathbf{X}}| |{\mathbf{Y}}|$ variables and
the optimal transport plan has $|{\mathbf{X}}| + |{\mathbf{Y}}| - 1$ mass bearing paths. It
follows that the capacity constraint propagation strategy considers linear
programs with at most $O\left( 2^i (|{\mathbf{X}}| + |{\mathbf{Y}}|) \right)$ constraints,
where $i$ is the number of iterations of the capacity propagation scheme. This
results in a significant reduction in problem size, since at each scale we only
consider a number of paths scaling as $O(|{\mathbf{X}}|+|{\mathbf{Y}}|)$ instead of $O(|{\mathbf{X}}|
|{\mathbf{Y}}|)$.
\subsection{Refinement Strategies}
\label{sec:refinement}
Solving the reduced transport problem at scale $j+1$, propagated from scale
$j$, can not guarantee an optimal solution at scale $j+1$. Propagating a
sub-optimal solution further may lead to an accumulation of errors. This
section describes strategies to refine the reduced transport problem to find
closer approximations or even optimal transport plans at each scale. These
refinement strategies are essentially batch column generation
methods~\citep{desaulniers:book2002}, that take advantage of the multiscale
structure.
\subsubsection{Potential Refinement}
\label{sec:potential}
This refinement strategy exploits the potential functions, or dual solution, to
determine additional paths to include given the currently optimal solution on
the reduced set of paths from the propagation. The dual formulation of the
optimal transport can be written as:
\begin{equation}
\max_{\phi, \psi}
\sum_{i=1,\dots,n} \mu(\{x_i\}) \phi(x_i) -
\sum_{j=1,\dots,m} \nu(\{y_j\}) \psi(y_j)
\quad \text{s.t.} \quad
\phi(x_i) - \psi(y_j) \leq \cost(x_i, y_j) \,.
\label{e:DualLPformulation}
\end{equation}
The functions $\phi$ and $\psi$ are called dual variables or potential
functions. From the dual formulation it follows that at an optimal solution
the {\em reduced cost} $\cost(x, y) - \phi(x) + \psi(y)$ is larger or equal to
zero. This also follows from the Kantorovich duality of optimal
transport~\citep[Chapter~5]{villani:book2009}.
The potential refinement strategy uses the potential functions $\phi$ and
$\psi$ from the solution of the reduced problem to determine which additional
paths to include. If the solution on the reduced problem is not optimal on all
paths, then there exist paths with negative reduced cost. Thus, we check the
reduced cost between all paths and include the ones negative reduced cost. A
direct implementation of this strategy would require to check all possible
paths between the source and target points. To avoid checking all pairwise
paths between source and target point sets at the current scale, we introduce a
branch and bound procedure on the multiscale structure that efficiently
determines all paths with negative reduced cost.
Let $ \phi_j^* |_\mathbf{P}$ and $ \psi_j^*|_\mathbf{P}$ the dual variables,
i.e., the potential functions, at the optimal solution
$\coupling_j^*|_\mathbf{P}$ on the set of paths $\mathbf{P}$ for the source and
target points, respectively. Define $\mathbf{V}_j( \coupling_j^*|_\mathbf{P} )$
as the set of paths with non-positive reduced cost with respect to $ \phi_j^*
|_\mathbf{P}$ and $\psi_j^* |_\mathbf{P}$
$$
\mathbf{V}_j( \coupling_j^* |_\mathbf{P}) = \left \{ \coupling_j(x_k, y_{k'})
\in \mathbf{A}_j : \cost(x_k, y_{k'}) - \phi_j^* |_\mathbf{P}(x_k) - \psi_j^*
|_{\mathbf{P}}(y_{k'}) \le 0 \right \} \, .
$$
The {\em potential} refinement strategy now refines the propagated solution
$\coupling_j^*|_{\mathbf{R}_j}$ by including all negative reduced cost paths
$\mathbf{Q}_j^0 = \mathbf{V}_j( \coupling_j^* |_{\mathbf{R}_j})$.
Let $\coupling_j^*|_{\mathbf{Q}_j^0}$ be the associated optimal transport. This
new solution changes the potential functions which in turn may require to
include additional paths. Thus the potential refinement strategy can be
iterated with $\mathbf{Q}_j^i = \mathbf{V}_j(
\coupling_j^*|_{\mathbf{Q}_j^{i-1}})$ leading to monotonically decreasing
optimal transport plans $\coupling_j^*|_{Q_j^i}$. Since a solution is optimal
if and only if all reduced cost are $\ge 0$ this iterative strategy converges
to the optimal solution.
The set of paths with negative reduced cost given $ \phi_j^*|_\mathbf{P}$ and $
\psi_j^* |_\mathbf{P}$ are determined by descending the tree and excluding
nodes that can not contain any negative reduced cost paths. This requires
bounds on the potential functions for any node at scale smaller than $j$. The
bound is achieved by storing at each target node of the multiscale
decomposition the maximal $ \psi_j^* |_\mathbf{P}$ of any of its descendants
at scale $j$. Now for each source node $x_k$ the target multiscale
decomposition is descended, but only towards nodes at which a potential
negative reduced cost can exist.
Depending on the properties of $ \phi_j^* |_\mathbf{P}$ and $\psi_j^*
|_\mathbf{P}$, this refinement strategy may reduce the number of required cost
function evaluations drastically. In empirical experiments the total number of
paths considered in the iterative potential refinement was typically reduced to
$O( 2^d ( K_j^{\mathbf{X}} + K_i^{\mathbf{X}}) )$, with $d$ being the doubling dimension of the
data set.
The potential refinement strategy is also employed by~\citet{glimm:arxiv2011}
without the branch and bound procedure. The shielding neighborhoods proposed
by~\citet{schmitzer2015sparse} similarly uses the potentials to define which
variables are needed to define a sparse optimal transport problem and also
suggests to iterate the neighborhood shields in order to arrive at an optimal
solution.
\subsubsection{Neighborhood Refinement}
\label{sec:neighborhood}
This section presents a refinement strategy that takes advantage of the
geometry of the data. The approach is based on the heuristic that most paths at
the next scale are sub-optimal due to boundary effects when moving from one
scale to the next, induced by the sharp partitioning of the space at scale $j$.
Such artifacts from the multiscale structure are mitigated by including paths
between spatial neighbors in the source and target locations of the optimal
solution on the propagated paths. This refinement strategy is also employed
by~\citet{oberman2015efficient}.
Let $\mathbf{N}_j\left(\coupling_j, r\right)$ be the set of paths such that the
source and target of any path are within radius $r$ of the source and target of
a path with non-zero mass transfer in $\coupling_j$. The neighborhood
refinement strategy is to expand the reduced set of paths using the union of
paths in the current reduced set and its neighbors:
$$
\mathbf{E}_j = \mathbf{R}_j \cup \mathbf{N}_j \left( \coupling_j^*|_{\mathbf{R}_j}, r\right) \,
$$
When moving from one scale to the next the cost of any path can change at most
two times the radius $r$ of the parent node which suggests to set the radius of
the neighborhood in consideration as two times the node radius.
This heuristics does not guarantee an optimal solution, but does reduce the
number of paths to consider at scale $j$ to $O( q_r^2 (K_j^{{{\mathbf{X}}}} +
K_j^{{{\mathbf{Y}}}}))$ with $q_r$ being the number of neighbors within radius
$2^{-j}r$, for a doubling space with dimension $d$, $q_r\asymp r^d$.
The neighborhood strategy requires to efficiently compute the set of points
within a ball of a given radius and location. Depending on the multiscale
structure there are different ways to compute the set of neighbors. A generic
approach that does not depend on any specific multiscale structure is to use a
branch and bound strategy. For this approach, each node requires an upper bound
$u(c_i)$ on the maximal distance from the representative of node $c_i$ to any of
its descendants. Using these upper bounds the tree is searched, starting from
the root node, while excluding any child nodes for which $\cost(x, c_i) - u(c_i) >
r$ from further consideration. For multiscale structures such as regular grids
more efficient direct computations are possible. For this paper we implemented
the generic branch and bound strategy that works with any multiscale structure.
\subsection{Remark on Errors}
The error induced by this multiscale framework stems from two sources. First, if
$J_0<J$, i.e., the optimal transport problem is only solved up to scale $J_0$,
the solution at scale $J_0$ has a bounded approximation error, as detailed in
Section~\ref{sec:multiscaleproblems}. By solving the transport problem only up
to a certain scale permits to trade-off computation time versus precision with
guaranteed approximation bounds. However, to speed up computation we rely on
heuristics that, depending on the refinement strategy, yield solutions at each
scale that might not be optimal. This second type error is difficult to
quantify; however, for the potential refinement strategy that we introduce
Section~\ref{sec:potential}, an optimal solution can be guaranteed. The
propagation and refinement strategies introduced in
Sections~\ref{sec:propagation} and~\ref{sec:refinement} permit trade-offs
between accuracy and computational cost.
\section{Numerical Results and Comparisons}
\label{sec:results}
We tested the performance of the proposed multiscale transport framework with
respect to computational speed as well as accuracy for different propagation
and refinement strategies on various source and target point sets with
different properties.
The multiscale approach is implemented in C++ with an \textsf{R} front end in
the package {\em mop}\footnote{available on
\url{https://bitbucket.org/suppechasper/optimaltransport}}. For comparisons, we also
implemented the Sinkhorn distance approach~\citep{cuturi:nips2013} in C++ with
an \textsf{R} front end in the package {\em
sinkhorn}\footnotemark[\value{footnote}]. Our C++ implementation uses the
Eigen linear algebra library~\citep{eigen}, which resulted in faster runtimes
than the MATLAB implementation by~\citet{cuturi:nips2013}.
\subsection{Implementation Details}
The \textsf{R} package {\em mop} provides a flexible implementation of the
multiscale framework and is adaptable to any multiscale decomposition that can
be represented by a tree. The package permits to use different optimization
algorithms to solve the individual transport problem. Currently the multiscale
framework implementation supports the open source library GLPK~\citep{glpk} and
the commercial packages MOSEK~\citep{mosek} and CPLEX~\citep{cplex} (both free
for academic use). MOSEK and CPLEX support a specialized network simplex
implementation that runs 10-100 times faster than a typical primal simplex
implementation. Both the MOSEK and CPLEX network simplex run at comparable
times, with CPLEX slightly faster in our experiments. Furthermore CPLEX
supports starting from an advanced initial basis for the network simplex
algorithm which improves the multiscale run-times significantly. Thus, the
numerical test are all run using the CPLEX network simplex algorithm.
\subsubsection{Algorithms for Constructing Multiscale Point Set Representations}
\label{s:multiscalepointsets}
Various approaches exist to build the multiscale structures described
in Section~\ref{sec:coarsening}, such as hierarchical clustering
type algorithms~\citep{ward:jasa1963}, or in low dimensions constructions such
as quad and oct-trees~\citep{finkel:ai197,jackins:cgip1980} or
kd-trees~\citep{bentley:acm1975} are feasible. Data structures developed for
fast nearest neighbor queries, such as navigating
nets~\citep{krauthgamer:soda2004} and cover trees~\citep{beygelzimer:icml2006}
induce a hierarchical structure on the data sets with guarantees on partition
size and geometric regularity of the elements of the partition at each scale,
under rather general assumptions on the distribution of the points. The
complexity of cover trees~\citep{beygelzimer:icml2006} is $O(C^d D n\log n)$,
for some constant $C$, where $n=|{{\mathbf{X}}}|$, $d$ is the doubling dimension of
${{\mathbf{X}}}$, and $D$ is the cost of computing a distance between a pair of points.
Therefore the algorithm is practical only when the intrinsic dimension is
small, in which case they are provably adaptive to such intrinsic dimension.
The optimal transport approach does not rest on a specific multiscale structure
and can be adapted to application-dependent considerations. However, the
properties of the multiscale structure, i.e., depth and partition sizes, do
affect run-time and approximation bounds.
In our experiments we use an iterative $K$-means strategy to recursively split
the data into subsets. The tree is initialized using the mean of the complete
data set as the root node. Then K-means is run resulting in $K$ children. This
procedure is recursively applied for each leaf node, in a breadth first
fashion, until a desired number of of leaf nodes, a maximal leaf radius or the
leaf node contains only a single point. For the examples shown we select $K =
2^d$ with $d$ the dimensionality of the data set. For high-dimensional data $K$
could be set to an estimate of the intrinsic dimensionality. Since in all
experiments we are equipped with a metric structure we use the pointwise
coarsening of the cost function as in Equation~\eqref{e:costpointwise}. The
reported results include the computation time for building the hierarchical
decomposition, which is, however, negligible compared to solving the transport
problem at all scales.
\subsubsection{Multiscale Transport Implementation for Point Sets}
Algorithm~\ref{f:algo-detailed} details the steps for computing multiscale
optimal transport plans using the newtork simplex for solving the optimization
problems at each scale and iterated $K$--means to construct the multiscale
structures.
\begin{algorithm}[thb]
\KwIn{ Source point set ${\mathbf{X}} = \{x\}_{i=0}^N$.
Target point set ${\mathbf{Y}} = \{y\}_{i=0}^M$.
A propagation strategy $p$.
A refinement strategy $r$.
}
\KwOut{Multiscale family of transport plans
$(\coupling_j: {\mathbf{X}}_j\rightarrowtail {\mathbf{Y}}_j)_{j=0}^{J}$}
Construct multiscale point sets $\{ {\mathbf{X}}_j \}_{j=0}^N$ and
$\{ {\mathbf{Y}}_j \}_{j=0}^M$ using iterated K--means.
\If{ $N < M$ }{
Add scales $\{ {\mathbf{X}}_j \}_{j=N+1}^J$ by repeating the last scale.
}
\If{ $N > M$ }{
Add scales $\{ {\mathbf{Y}}_j \}_{j=M+1}^J$ by repeating the last scale.
}
Set $J = \max(N, M)$
Form the measures $\mu_j$ and $\nu_j$ as in equation~\eqref{e:muj}.
Compute optimal transport $\coupling_0$ at coarsest scale with the network simplex.
\For{ $j=1\dots J$ }{
Propagate the $\coupling_{j-1}$ from scale $j-1$ to scale $j$ using the
propagation strategy $p$, obtaining a set of paths $\mathbf{S}_j =
p(\coupling_{j-1}$ at scale $j$
\vspace{0.1in}
Use the network simplex algorithm to solve for optimal transport on the
set of paths $\mathbf{S}_j$ yielding a coupling $\tilde \coupling_j$
Create the refined set of paths $\mathbf{R}_j = r( \tilde \coupling_j )$
using the refinement strategy $r$.
Use the network simplex algorithm to solve for optimal transport on the
set of paths $\mathbf{R}_j$ yielding the optimal coupling $\coupling_j$ on
the paths $\mathbf{R}_j$.
}
\caption{Point Set based Multiscale Optimal Transport Implementation
\label{f:algo-detailed}
}
\end{algorithm}
\subsection{Propagation and Refinement Strategy Performance}
Figure~\ref{fig:strategies} illustrates the behaviour of the different
propagation and refinement strategies on two examples: The ellipses example in
Figure~\ref{fig:multiscale-strategy} and Caffarelli's smoothness counter example
described in~\citet[Chapter~12]{villani:book2009} and illustrated in
Figure~\ref{fig:datasets}. The ellipse example consists of two uniform samples
(source and target data set) of size $n$ from the unit circle with normal
distributed noise added with zero mean and standard deviation $0.1$. The source
data sample is then scaled in the x-Axis by $1.3$ and in the y-Axis by $0.9$,
while the target data set is scaled in the x-Axis
by 0.9 and in the y-Axis by $1.1$.
Caffarelli's example consists of two uniform samples on $[-1,1]^2$ of size $n$.
Any points outside the unit circle are then discarded. Additionally, the target
data sample is split along the x-Axis at $0$ and shifted by $+2$ and $-2$ for
points with positive and negative x-Axis values, respectively.
\begin{figure}[htb]
\centering
\begin{tabular}{VV}
\includegraphics[width=0.9\linewidth]{scale-9} &
\includegraphics[width=0.9\linewidth]{caffarelli-plan}\\
(a) & (b)
\end{tabular}
\vspace{-0.2in}
\caption{
\label{fig:datasets}
Optimal transport plans on the (a) ellipse and (b) Caffarelli data sets with
5000 points for the source and target point set each. The optimal transport
plans are indicated by black lines with transparency indicating the amount of
mass being transported from source to target. }
\end{figure}
\vspace{-0.2in}
We tested 7 different strategies of different combinations of propagation
strategies from Section~\ref{sec:propagation} with refinement strategies from
Section~\ref{sec:refinement}:
\begin{compactenum}
\item (ICP) Iterated capacity propagation (Section~\ref{sec:capacity}) with
$0$ to $5$ iterations with no refinements. Note iterated capacity with $0$
iterations is equivalent to simple propagation (Section~\ref{sec:simple}).
\item (NR) Simple propagation (Section~\ref{sec:simple}) with neighborhood
refinement (Section~\ref{sec:neighborhood}) with radius factor ranging from
$0.5$ to $2.5$ in $0.5$ increments.
\item (ICP + NR) Iterated capacity propagation with $1$ to $5$ iterations
combined with neighborhood refinement with radius factor fixed to $1$.
\item (CP + NR) A single iteration of capacity propagation combined with
neighborhood refinement with radius factor from $0.5$ to $2.5$ with $0.5$
increments.
\item (IPR) Simple propagation with iterated potential refinement
(Section~\ref{sec:potential}) with $1$ to $5$ iterations.
\item (ICP + PR) Iterated capacity propagation with $1$ to $5$ iterations
combined with a single potential refinement step.
\item (CP + IPR) A single iteration of capacity propagation combined with
$1$ to $5$ iterations of potential refinement.
\end{compactenum}
Figure~\ref{fig:strategies} shows that almost all strategies have less than one
percent relative error. The exception is the simple propagation with no
refinements applied (i.e. iterated capacity propagation with no iterations).
The randomness of the iterated capacity constrained algorithm can results in worse
results despite a larger number of iterations. However, it will always perform better than
using the simple propagation strategy. The neighborhood refinement strategy
improves the results significantly, but after a radius factor of one the improvements start to
level out. The potential refinement strategy finds the optimal solution when
iterated a few times. Combining the refinement strategy with capacity
propagation reduces relative error and computation time. The computation time
is reduced because the capacity propagation yields a closer initialization to
the optimal solution. The combination of potential refinement strategy and capacity propagation has
the additional benefit that the branch and bound strategy is more efficient
since fewer comparisons need to be made when checking for paths and smaller
linear programs have to be solved in each refinement step.
\begin{figure}[htb]
\centering
\begin{tabular}{VV}
\includegraphics[width=0.85\linewidth]{ellipses-performance-strategies} &
\includegraphics[width=0.85\linewidth]{caffarelli-performance-strategies} \\
(a) & (b)
\end{tabular}
\vspace{-0.15in}
\caption{
\label{fig:strategies}
A comparison of different multiscale propagation and refinement strategies on
the (a) ellipse and (b) Caffarelli data sets with 5000 points for the source and
target point set each. For a description of the different strategies see text,
the lines in the relative error graphs indicate increasing number of iterations
or radius factor for the different strategies. Both the time and accuracy axes
are in logarithmic scale. All strategies, except capacity propagation with $0$
iterations (ICP) and neighborhood refinement (NR) with radius factor 0.5, find
solution with relative error less than one percent. The capacity propagation
strategies can find optimal solutions up to numerical precision.}
\end{figure}
\vspace{-0.15in}
For very small relative error $( < 10^{-11} )$ the optimal solution is
sometimes not achieved. This is due to tolerance settings in the network
simplex algorithm which are on the order of $10^{-11}$. Thus, depending how the
network simplex approaches the optimum it might stop at different basic
feasible solutions.
Figure~\ref{fig:strategies} shows that the computation time and relative
error for problem of equal sizes depends on the type of problem. An important
aspect of the problem type is the ratio of the transport distance to the diameters
of the source and target point sets. To illustrate this effect we computed
optimal transport plans for two data sets with source and target
distributions uniform on $[0,1]^2$. In the first case the distributions are
perfectly overlapping and in the second case the target distributions shifted
by 2 units in the x direction. Figure~\ref{fig:distance-strategies} shows that
for large ratios the relative error is typically much smaller. This is
expected since variations in the transport plan only change the cost
marginally. For small ratios, i.e. source and target distributions that are
almost identical, a small variation in the transport plan leads to a large
relative error.
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\vspace{-0.3in} \\
\includegraphics[width=0.47\linewidth]{uniform-d0-performance-strategies} &
\includegraphics[width=0.47\linewidth]{uniform-d2-performance-strategies} \\
\vspace{-0.2in} \\
(a) & (b)
\end{tabular}
\vspace{-0.15in}
\caption{
\label{fig:distance-strategies}
The effect of distance to support size ratio on computation time and relative
error for the different multiscale propagation and refinement strategies source
and target point sets of $5000$ points each sampled uniformly from a square of
side length one and ground truth transport distance (a) $0$ and (b) $2$. Both
the time and accuracy axes are in logarithmic scale. }
\end{figure}
Another observation is the following: the potential strategy is less effective
for transport problems where most mass is transported very far, relative to the
distances within source and target point set. This is because a small change in
the length of a path can include many possible source and target locations and
the transport polytope has many suboptimal solutions with similar cost. If on
the other hand most mass is transported on the order of the nearest neighbor
distances within the source and target point set, there are many fewer possible
paths within a small change in path length and the transport polytope is
``steep'' in the direction of the cost function.
\subsection{Comparison to Network Simplex and Sinkhorn Transport}
In this section we compare the CPLEX network simplex algorithm~\citep{cplex}
and the Sinkhorn approach~\citep{cuturi:nips2013} to three different
multiscale strategies:
\begin{compactenum}
\item (CP) Capacity propagation strategy using a single iteration with no
further refinements.
\item (CP + NR) Capacity propagation combined with neighborhood refinement
with radius factor fixed to one and a single capacity constraint iteration.
\item (CP + PR) Capacity propagation combined with potential refinement with a
single iteration each.
\end{compactenum}
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.65\linewidth]{ellipses-performance-npoints} &
\includegraphics[width=0.28\linewidth]{ellipses-performance-npoints-used} \\
(a) & (b)
\end{tabular}
\caption{
\label{fig:npoints}
(a) A comparison of computation time (solid) and relative error ( dashed ) with
respect to the network simplex solution for the CPLEX network simplex, Sinkhorn
distance and the proposed multiscale strategy with increasing number of points.
(b) The number of total paths considered is roughly constant with four times
the number of points in the problem, i.e., the multiscale approach results in
only a linear increase in problem size instead of quadratic for a direct
approach. The number of points on the x-axis denotes the number of source
points $|{\mathbf{X}}|$ which is approximately equal to the number of target points
$|{\mathbf{Y}}|$. The Sinkhorn approach is competitive in computation time for smaller
problems. For larger problems only the multiscale strategies outperform the
Sinkhorn approach quickly, and are the only algorithms that remain viable.
}
\end{figure}
Figure~\ref{fig:npoints} shows computation time, relative error and problem
size for increasing sample size on the ellipse data set. The Sinkhorn transport
employs a regularization term and is thus not expected to converge to the
actual transport distance. The comparison of the relative error provides an
indication of how far the regularization strays from the true transport
distance. For small size problems the Sinkhorn transport is as comparable in
speed to the multiscale approach.. The multiscale approach outperforms both
competitors for moderate data sizes of a few thousand source and target points.
The multiscale approach scales roughly linear in the problem size while both
the Sinkhorn and network simplex scale approximately quadratically. The
capacity propagation without refinement runs an order of magnitude faster than
including refinements and results in relative errors less than one percent for
more than a few hundred points. The network simplex and Sinkhorn approach run
out of memory on a 16GB computer for problems larger than around $2\cdot10^4$
constraints and about $10^8$ variables. The proposed multiscale framework
results in a linear increase in problem size for the propagation and refinement
strategies tested and, on the same 16GB computer, can solve instances with
$2\cdot10^6$ constraints (source and target points) in a few hours, which would
result in about $10^{12}$ variables (paths) for the full optimal transport
problem. The computation times are comparable to the results reported
by~\citet{oberman2015efficient,schmitzer2015sparse} on examples on regular grids
that result in similar sized problems.
The relative error and computation time depend again on the type of problem.
Figure~\ref{fig:dimensions}(a) shows computation time and relative error
on transport problems on source and target data sets sampled uniformly from
$[0,1]^2$ with different shifts in the target distribution. If required, the
relative error can be reduced using multiple capacity propagation and potential
refinement iterations as illustrated in Figure~\ref{fig:distance-strategies}.
In these experiments we set the tolerance parameter for Sinkhorn to $1e^{-5}$, and
tried also $1e^{-2}$ to see if this would result in significant speed ups, with
minimal accuracy loss, but it resulted in only approximately a $10\%$ speedup,
and no fundamental change in the behavior for large problem sizes. The
tolerance parameter for CPLEX is set to $1e^{-8}$. The approach by
~\citet{schmitzer2015sparse} is similar to the refinement property strategy, we
expect that computation time grows similarly as for the potential strategy for
problems with large transport distances compared to the source and target
support size.
The final experiment tests the performance with respect to the dimensionality of
the source and target distributions. We used two uniform distributions $[0,1]^d$
with $d$ varying from $2$ to $5$ and the target shifted such that the actual
transport distance is 0 and 2.
\begin{figure}[htb]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{uniform-performance-distance} &
\includegraphics[width=0.32\linewidth]{uniform-d0-performance-dimensions} &
\includegraphics[width=0.32\linewidth]{uniform-d2-performance-dimensions} \\
\vspace{-0.2in} \\
(a) & (b) & (c)
\end{tabular}
\vspace{-0.1in}
\caption{
\label{fig:dimensions}
Computation time (solid) and relative error (dashed) with respect to (a)
changes transport distance to support size ratio in two dimensions and (b,c)
dimensionality with ground truth transport distance (b) 0 and (c) 2. The source
and target distributions are uniform on a square of side length 1 with
approximately 5000 points each. The potential refinement increases proportional
to the transport distance to support ratio. The neighborhood strategy is less
effective in higher dimensions, due to the curse of dimensionality, but
performs better than the Sinkhorn approach. The capacity propagation strategy
is less affected by the dimensionality of the problem. }
\end{figure}
\section{Application to Brain MR Images}
\label{s:brainMR}
An important building block for the analysis of brain MRI populations is the
definition of a metric that measures how different two brain MRI are. A
mathematically well motivated and popular approach for distance computations
between brain images is based on large deformation diffeomorphic metric mappings
(LDDMM~\citep{miller2002metrics}). Here we explore optimal transport distance
as an alternative metric for comparing brain images.
To solve for the optimal transport map between two 3D brain images we extract
for each image a point cloud from the intensity volumes. Each point represents
a voxel as a point in 3-dimensional space, the location of the voxel. The mass
of the point is equal to the intensity value of the voxel, normalized to sum to
one over all points. For illustration, Figure~\ref{fig:brains} shows a single
slice extracted from the original volumes and optimal transport maps between
the two slices. This 2D problem resulted in point set of approximately $20,000$
points.
\begin{figure}[bth]
\centering
\begin{tabular}{cc}
\vspace{-0.1in} \\
\includegraphics[height=0.375\linewidth,angle=0]{slice00000} &
\includegraphics[height=0.375\linewidth,angle=0]{slice00001} \\
\vspace{-0.1in} \\
(a) & (b) \\
\includegraphics[width=0.375\linewidth,angle=90]{brains-p1-map-v2} &
\includegraphics[width=0.375\linewidth,angle=90]{brains-p2-map-v2} \\
\vspace{-0.1in} \\
(c) & (d)
\end{tabular}
\vspace{-0.1in}
\caption{
\label{fig:brains}
Slice of (a) source and (b) target brain image and optimal transport map for
(c) Euclidean and (d) squared Euclidean cost function. For the squared
Euclidean cost the optimal transport solutions typically prefer to move many
locations small distances, i.e. shift mass among nearest neighbors. For images
the neighborhoods are on regular grids resulting in a staircase appearance of
the transport plan. }
\end{figure}
To compare the optimal transport distance to the LDDMM distance we compare how
well the distances can be used to predict clinical parameters. Using the pairwise
distances, we employ classical multidimensional scaling (MDS) to embed the pairwise
distances into Euclidean space. In this embedding each point corresponds to a 3D
brain image and thus provides a coordinate system for the relative locations
between the brain MRI's. The Euclidean structure of the embedding permits to
use standard statistical tools to form prediction models, in our case linear
regression.
Within this framework we compare Euclidean, LDDMM and optimal transport
distances as the input to the multidimensional scaling step. Note, for the
Euclidean distance, i.e. treating each brain MRI as a point in Euclidean space,
classical multidimensional scaling is equivalent to a principal component
analysis.
For the comparisons we used the OASIS brain database~\citep{oasis-brains}. The
OASIS database consists of T1 weighted MRI of 416 subjects aged 18 to 96. One
hundred of the subjects over the age of 60 are diagnosed with mild to moderate
dementia. The images in the OASIS data set are already skull-stripped,
gain-field corrected and registered to the atlas space
of~\citet{talaraich:book88} with a 12-parameter affine transform. Associated
with the data are several clinical parameters. For the linear regression from
the MDS embeddings we restrict our attention to the prediction of age, mini
mental state examination (MMSE) and clinical dementia rating (CDR).
\begin{table}[tb]
\centering
\begin{footnotesize}
\begin{tabular}{l|c|c|c|c}
Model &
Residual & $R^2$ & $F$--statistic & $p$--value \\
\hline
${\rm age} = a_0 + \sum_{i=1}^5 a_i l_i$ &
{\bf 10.5} & {\bf 0.82} & {\bf 404.9} & $ <\epsilon$ \\
${\rm age} = a_0 + \sum_{i=1}^3 a_i x_i$ &
10.87 & 0.82 & 639.5 & $ <\epsilon$ \\
${\rm age} = a_0 + \sum_{i=1}^5 a_i z_i$ &
12.04 & 0.78 & 297 & $ <\epsilon$ \\
${\rm age} = a_0 + a_1 z^1_1 + a_2 z^1_2 + a_3 z^2_1 + a_4 z^2_2 + a_5 z^3_1 +
a_6 z^4_2 + a_7 z^4_5 + a_8 z^5_1$ &
10.9 & 0.82 & 239 & $<\epsilon$ \\
\hline
${\rm MMSE} = a_0 + a_1 {\rm age} $ &
3.59 & 0.06 & 15.82 & 9.3e-05 \\
${\rm MMSE} = a_0 + a_1 l_1 $ &
3.40 & 0.16 & 43.13 & 3.3e-10 \\
${\rm MMSE} = a_0 +a_1 x_1$ &
3.36 & 0.18 & 50.30 & 1.6e-11 \\
${\rm MMSE} = a_0 + a_1 z_1 + a_2 z_3 + a_3 z_5$ &
3.38 & 0.17 & 16.31 & 1.2e-09 \\
${\rm MMSE} = a_0 + a_1 z^2_2 + a_2 z^3_2 + a_3 z^3_4 + a_4 z^4_5 +
a_5 z^5_1 + a_6 z^5_3 + a_7 z^6_3$ &
{\bf 3.14 } & {\bf 0.30 } & {\bf 14.03} & {\bf 4.3e-15 }\\
\hline
${\rm CDR} = a_0 + a_1 {\rm age} $ &
0.27 & 0.25 & 144.5 & $ <\epsilon$ \\
${\rm CDR} = a_0 + a_1 l_1$ &
0.26 & 0.34 & 223.9 & $ <\epsilon$ \\
${\rm CDR} = a_0 + a_1 x_1$ &
0.25 & 0.36 & 248.5 & $ <\epsilon$ \\
${\rm CDR} = a_0 + a_1 z_1 + a_2 z_3 + a_3 z_5$ &
0.26 & 0.35 & 77.5 & $ < \epsilon$ \\
${\rm CDR} = a_0 + a_1 z^3_4 + a_2 z^5_1 + a_3 z^5_3 + a_4 z^5_5 + a_5 a^6_3$ &
{\bf 0.25} & {\bf 0.38} & {\bf 53.1} & $ < \epsilon$ \\
\end{tabular}
\end{footnotesize}
\caption{Optimal linear regression models from the OASIS data set, for age,
mini mental state examination (MMSE) and clinical dementia rating (CDR). The
PCA coordinates from the Euclidean distances are denoted with $l_i$, the
diffeomorphic manifold coordinates with $x_i$, the transport coordinates by
$z_i$ and the multiscale transport coordinates with $z_i^j$ from coarsest
$j=1$ to finest $j=6$ scale. An entry with ``$ <\epsilon$'' denotes
quantities smaller than machine precision. The best results are indicated in bold. }
\label{tab:regression_oasis}
\end{table}
The MDS computation requires pairwise distances. To speed up computations we
reduce the number of points by downsampling the volumes to size $44 \times 52
\times 44$. This results, discarding zero intensity voxels, in point clouds of
approximately $40,000$ points for each brain MRI. A single distance computation
with capacity propagation takes on the order of 10 seconds resulting in a total
computation time of around 2 weeks for all pairwise distances. For embedding
the optimal transport distance we consider two approaches:
\begin{compactenum}
\item A five dimensional MDS embedding based on the transport cost at the finest
scale.
\item A multiscale embedding using multiple five dimensional MDS embeddings, one
for the transport cost at each scale.
\end{compactenum}
From the embeddings we build linear regression models, using the embedding
coordinates as independent variables and the clinical variables as dependent
variables. As in~\citet{gerber:media10} we use the Bayesian information
criterion (BIC) on all regression subsets to extract a models that trade-off
complexity, i.e. number of independent variables, with the quality of fit.
Table~\ref{tab:regression_oasis} shows the results of the optimal transport,
with squared Euclidean cost, distances compared to the results reported
in~\citet{gerber:media10}. The transport based approach shows some interesting
behaviours. The single scale model performs worse on age while performing
similar on MMSE and CDR. The multiscale transport models perform similar to the
LDDMM approach except for MMSE where it almost doubles the explained variance
$R^2$. This suggests that the multiscale approach captures information about
MMSE not contained in a single scale and prompts further research of multiscale
based models to predict clinical parameters.
\section{Conclusion}
The multiscale framework extends the size of transport problems that can be
solved by linear programming by one to two orders of magnitude in the size of
the point sets.
The framework is flexible, the linear program at each scale can be solved by
any method. Depending on the refinement strategy a dual solution would need to
be constructed as well. The method can also be applied to solve the linear
assignment problem. The solution at the finest scale will be binary if $n=m$,
however, at intermediate steps a binary solution is not guaranteed unless at
each scale the source and target sets have the same number of points. This
could be enforced during the construction of the multiscale structures. As
currently defined the method requires point set inputs for the multiscale
constructing of the transport problems. However, one could design methods that
construct such a multiscale structure from a cost matrix only. However, for
large problems pairwise computations of all costs is typically prohibitively
expensive.
The multiscale transport framework provides several options for further
research. The framework can be combined with various regularizations, e.g.
depending on scale and location and induces a natural multiscale decomposition
of the transport map. This decomposition can be used to extract information
about relevant scales in a wide variety of application domains. We are
currently investigating the use of such multiscale decompositions for
representation and analysis of sets of brain MRI.
The capacity constraint propagation approach suggests a different venue for
further exploration. The capacity propagation strategy does not hinge on a
geometrical notion of neighborhoods and is a suitable candidate to extend the
multiscale framework to non-geometric problems. Studying the interdependency
between the cost function, hierarchical decomposition of the transport problem
(or possibly more generic linear programs) and the efficiency of the capacity
constraint propagation is a challenging but interesting problem.
\acks{"We thank the referees for their constructive feedback. This work was
supported, in part, by NIH/NIBIB and NIH/NIGMS via 1R01EB021396-01A1:
Slicer+PLUS: Point-of-Care Ultrasound." }
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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|
\section{Introduction}
So far, the Laser Interferometer Gravitational-Wave Observatory (LIGO) \citep{aligo} and Virgo \citep{avirgo}, have reported detections of a handful of gravitational-wave (GW) signals from coalescence of compact binary systems \citep{AbEA2016a,AbEA2016e,AbEA2017a,AbEA2017b,AbEA2017c,AbEA2017d}. Isolated compact objects may also emit detectable GWs, though they are predicted to be much weaker than compact binary coalescences \citep{Sathyaprakash2009}. Because of the high energies and mass densities required to generate detectable GWs, neutron stars and supernovae are among the main targets of non-binary searches. This paper focuses on a type of neutron star: magnetars.
Magnetars are highly magnetized isolated neutron stars \citep{Mereghetti2015,woods_thompson_2006}. Originally classified as Anomalous X-ray Pulsars (AXPs), or Soft Gamma Repeaters (SGRs), some AXPs were observed acting like SGRs and vice versa. They are now considered to be a single class of objects defined by their power source: the star's magnetic field, which, at $10^{13}-10^{15}\,\text{G}$, is about 100$\times$ stronger than a typical neutron star. Magnetars occasionally emit short bursts of soft $\gamma$-rays, but the exact mechanism responsible for the bursts is unclear. There are currently 23 known magnetars (and an additional 6 candidates) \citep{mcgillpaper},\footnote{See the catalog at \url{http://www.physics.mcgill.ca/\~pulsar/magnetar/main.html}} which were identified based on observations across wavelengths of bursts, continuous pulsating emission, spin-down rates, and glitches in their rotational frequency. The bursts last $\sim 0.1 \text{s}$ with luminosity of up to $\sim 2 \times 10^{42} \, \text{erg} \, \text{s}^{-1}$, and can usually be localized well enough to allow identification of the source magnetar. Many magnetars also emit pulsed X-rays and some are visible in radio.
The large energies involved originally led to the belief that magnetar bursts could be promising sources of detectable gravitational waves, e.g. \cite{ioka01, corsi11}. Further theoretical investigation indicates that most mechanisms are likely too weak to be detectable by current detectors \citep{LevHov2011, ZinLas2012}. Nevertheless, due to the large amount of energy stored in their magnetic fields and known transient activity, magnetars remain a promising source of GW detections for ground-based detectors with rich underlying physics.
The search presented in this paper is triggered following identification of magnetar bursts by $\gamma$-ray telescopes. The methodology is similar to one done during initial LIGO's sixth science run \citep{QuBr2017, RQJthesis} with a few improvements and the use of an additional pipeline targeted toward shorter-duration signals (X-Pipeline) \citep{SuJo2010}. This pipeline has been used to look for GWs coincident with $\gamma$-ray bursts (GRBs) (see \cite{xgrbo1} for such searches during advanced LIGO's first observing run).
The first searches for GW counterparts from magnetar activity targeted the 2004 hyperflare of SGR 1806-20. Initial LIGO data was used to constrain the GW emission associated with the quasi-periodic seismic oscillations(QPOs) of the magnetar following this catastrophic cosmic event \citep{Matone_2007, PhysRevD.76.062003} as well as the instantaneous gravitational emission \citep{Kalmus_2007, 2008_sgr_search}. \cite{2008_sgr_search} and \cite{AbAb2011} report on GW emission limits associated with additional magnetar activity observed during the initial detector era until June 2009. LIGO data coinciding with the 2006 SGR 1900+14 storm was additionally analyzed by "stacking" GW data \cite{kalmus_2009} corresponding to individual bursts in the storm's EM light curve \citep{stacked_flares}. Additionally, a magnetar was considered as a possible source for a GRB during initial LIGO (GRB 051103), and a search using X-Pipeline and the Flare pipeline placed upper limits on GW emission from the star's fundamental ringing mode \citep{GRB051103}.
The rest of this paper is laid out as follows: in Section \ref{magbursts}, we provide a brief overview of the astrophysics of magnetars as is relevant to gravitational-wave astronomy and the short bursts used in this analysis. Next, in Section \ref{method} we describe the methodology of the GW search. Section \ref{results} describes the results and upper limits on possible gravitational radiation from the studied bursts. The Appendix contains a discussion of the effect of GW polarization on the sensitivity of the intermediate duration search.
\section{Magnetar Bursts} \label{magbursts}
Magnetars are currently not well understood. Their magnetic fields are strong and complex \citep{braithwaite}, and power the star's activity. Occasionally and unpredictably, magnetars give off short bursts of $\gamma$-rays whose exact mechanism is unknown, but may be caused by seismic events, Alfv\'en waves in the star's atmosphere, magnetic reconnection events, or some combination of these, e.g. \cite{thompson1995}. After some of the brighter bursts (giant flares, which have been seen only three times), there is a soft X-ray tail which lasts for hundreds of seconds. Quasi-periodic oscillations (QPOs) have been observed in the tail of giant flares \citep{IsBe2005,StrWat2005} and some short bursts \citep{short_burst_qpo_a, short_burst_qpo_b}, during which various frequencies appear, stay for hundreds of seconds, and then disappear again, indicating a resonance within the magnetar. Many possible resonant modes in the core and crust of the magnetar have been suggested to cause the QPOs, although it is unclear which modes actually produce them. Some of these modes, such as f-modes and r-modes, couple well to GWs, and so, if sufficiently excited, could produce detectable GWs, though current models indicate that they will be too weak \citep{LevHov2011,ZinLas2012}. Other modes, such as the lowest order torsional mode, do not create the time changing quadrupole moment needed for GW emission. None of these models provide precise predictions for emitted GW waveforms.
This search was performed on data coincident with the four short bursts from magnetars during advanced LIGO's second observing run for which there was sufficient data (we require data from two detectors) for both short (less than a second long) and intermediate (hundreds of seconds long) duration signals. Table \ref{burst_data} describes the four bursts. In addition to the four studied bursts, there were five bursts that occurred during times when at least one detector was offline. No GW analysis was done on them. All GW detector data comes from the two LIGO detectors because Virgo was not taking data during any of these bursts.
Three bursts come from the magnetar SGR 1806-20. They were all identified by the Burst Alert Telescope (BAT) aboard NASA's Swift satellite \citep{swift2004}. These were sub-threshold events that were found in BAT data (D. M. Palmer, personal communication, June 6, 2017), an example of which is shown in Figure \ref{SWIFT_data}, with the data from the other two found in Appendix \ref{swiftplots}. The fourth was a short GRB with a soft spectrum observed by the \textit{Fermi} Gamma-ray Burst Monitor \citep{fermi2009}, and named GRB 170304A.
\begin{table}[h!]
\centering
\begin{tabular}{|c c c c c c||}
\hline
Source & Date & Time & Duration & Fluence & Distance \\
& & (UTC) & (s) & (erg $\text{cm}^{-2}$) & (kpc) \\[0.5ex]
\hline\hline
SGR 1806-20 & Feb 11, 2017 & 21:51:58 & 0.256 & $8.9 \times 10^{-11}$ & 8.7 \\
\hline
SGR 1806-20 & Feb 25, 2017 & 06:15:07 & 0.016 & $1.2 \times 10^{-11}$ & 8.7 \\
\hline
GRB170304A & March 4, 2017 & 00:04:26 & 0.16 & $3.1 \times 10^{-10}$ & --\\
\hline
SGR 1806-20 & April 29, 2017 & 17:00:44 & 0.008 & $1.4 \times 10^{-11}$ & 8.7\\[1ex]
\hline
\end{tabular}
\caption{List of magnetar bursts considered in this GW search. GRB170304A is described in GCN circular 20813; data on SGR 1806-20 burst activity is courtesy of David M. Palmer.}
\label{burst_data}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth, angle=90]{swift_data_170211.pdf}
\caption{\label{SWIFT_data}SWIFT BAT's data for the February 11 burst from SGR 1806-20. Image courtesy of David M. Palmer.}
\end{figure}
\section{Method} \label{method}
\subsection{Excess power searches}
Fundamentally, all multi-detector GW searches seek to identify GW signals that are consistent with the data collected at both detectors. Some searches identify candidate signals in each detector separately, then later consider only the candidates that occur in all detectors within the light-travel time and with the same signal parameters. This approach is disfavored in searches that do not rely on templates. We cannot perform a templated search here because there is no current model which can produce templates for magnetar GW bursts. Instead, we first combine the two data streams to create a time-frequency map where the value in each time-frequency pixel represents some measure of the GWs (often energy) consistent with the observations from the detectors.
The next step is to identify GW signals in the time-frequency map. This is done by clustering together groups of pixels, calculating the significance of each cluster with a metric, and searching for the most significant cluster. In order to cover a broader range of frequencies and time scales, we use two different analysis pipelines which use different clustering algorithms.
The short-duration search uses seed-based clustering implemented by X-Pipeline, which focuses on groups of bright pixels (the seed) \citep{SuJo2010}. Specifically, the clusters considered by X-Pipeline are groups of neighboring pixels that are all louder than a chosen threshold. This approach works well for short-duration searches, but fails for longer-duration signals for two reasons: random noise will tend to break up the signal into multiple clusters, and each pixel is closer to the background, so fewer of them will be above the threshold.
We rely on STAMP \citep{ThKa2011} for the intermediate-duration search. STAMP offers a seedless method whose clustering algorithm integrates over many, randomly chosen, B\'ezier curves \citep{ThCo2013, ThCo2014}. Because of this, it can jump over gaps in clusters caused by noise, and thus it is better suited for longer-duration signals. Additionally, it can build up signal-to-noise ratio (SNR) over many pixels of only slightly elevated SNR. This method was previously used to search for signals from magnetars during initial LIGO \citep{QuBr2017, RQJthesis}.
\subsection{X-Pipeline}
X-Pipeline is a software package designed to search for short-duration gravitational wave signals in multiple detectors, and includes automatic glitch rejection, background calculation, and software injection processing (for details, see \cite{SuJo2010}). It forms coherent combinations from multiple detectors, thus making it relatively insensitive to non-GW signals, such as instrumental artifacts. X-Pipeline is used primarily to search for GWs coincident with $\gamma$-ray bursts (GRBs), but is suitable for any short-duration coherent search.
X-Pipeline takes a likelihood approach to estimating the GW energy found in each time-frequency pixel. It models the data collected at the detectors as a combination of signal and detector noise, then uses a maximum likelihood technique to calculate the estimated GW signal power in each time-frequency pixel.
For clustering, X-Pipeline selects the loudest 1\% of pixels and connects neighboring pixels. Each connected group is a cluster, and the clusters are scored based on the likelihood described in the previous paragraph. We want to pick the time length of the pixels in the time-frequency map so that signal is present in the smallest number of time-frequency pixels, as this will recover the signal with the highest likelihood. Since we do not have a model for the waveform we are searching for, we use multiple pixel lengths and run the clustering algorithm on all of them. After clusters are identified, X-Pipeline identifies which candidate clusters are likely glitches by comparing three measurements of signal energy: coherent energy consistent with GWs, coherent energy inconsistent with GWs, and sum of the signal energy in all detectors (referred to as incoherent energy). GW signals can be differentiated from noise by the ratio of coherent energy inconsistent with GWs to the incoherent energy (see Sections 2.6 and 3.4 of \cite{SuJo2010} for full details).
The primary target of this search are GWs produced from the excitation of the magnetar's fundamental mode, which are primarily dampened by the emission of GWs \citep{Det1975,AndKok1998}. We have chosen parameters for X-Pipeline to search for signals a few hundred milliseconds long. The search window begins 4 seconds before the $\gamma$-rays arrive and ends 4 seconds after. The frequency range for the short-duration search is 64--4000\,Hz, and the pixel lengths are every factor of 2 between 2\,s and 1/128\,s, inclusive.
\subsection{STAMP}
STAMP is an unmodeled, coherent, directed excess power search suitable for longer-duration signals, described in more detail in \cite{ThKa2011}. In short, the pipeline calculates the cross power between the two detectors, accounting for the time delay due to the light travel time between detectors. It then makes this into a time-frequency SNR map, where pixel SNR is estimated from the variable $\big \langle \hat{Y} (t;f,\hat{\Omega}, )\big \rangle = \text{Re} \Big[ \tilde{Q}_{IJ}(t;f,\hat{\Omega}) \big(2 \, \tilde{s}_I^*(t;f) \tilde{s}_J(t;f) \big) \Big]$. Here, $\tilde{s}_I(t;f)$ is the Fourier transformed data from detector $I$ and $\tilde{Q}_{IJ}(t; f, \hat{\Omega})$ is the filter function required given the locations and orientations of the pair of detectors and the sky position $\hat{\Omega}$ of the source \citep{ThKa2011}. The ideal filter function also depends on the polarization of the incoming GWs, which is unknown. Thus the best we can do is to use the unpolarized filter function which causes a loss of signal power (though this loss is nearly zero for optimal sky locations). This is more fully discussed in Appendix \ref{polarization}.
To identify signals, STAMP uses seedless clustering and searches over a large number of clusters (30 million in this search, see section III of \citep{ThCo2013}). For clusters, we use B\'ezier curves, which are parameterized by three points \citep{ThCo2013}.
GWs radiated through the mechanisms related to QPOs would be monochromatic, or close to it. So, it makes sense to only search for such signals. Through STAMP, this is easily accomplished by restricting the search to clusters whose frequencies change by only a small amount. This reduces the number of possible clusters, which means we get all of the benefits of searching more clusters without the additional computational cost. Restricting the frequency change too much may cause signals to be missed completely. Compromising between these, we restrict the searched clusters to those with a frequency change less than 10\%.
To estimate the background for this search, we use approximately 15 hours of data from each detector collected around the time of each burst, excluding the data, coincident with the burst, that was searched for GWs (the on-source). We can then perform background experiments free of any possible coincident GW signal by pairing data taken at different times as if they were coincident. Then, breaking up the background data into 33 segments, we generate 1,056 background experiments. Each of these is run in exactly the same way as the on-source analysis, giving an estimate of the SNR that can be expected from detector noise. The resulting distributions for the background data for each burst, along with the on-source results, are shown in Figure \ref{background}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.7\textwidth]{backgrounds_and_onsource.png}
\caption{\label{background}SNR distribution of the background (lines) and on-source result (open circles) for each burst for the intermediate-duration search. As expected, the background distributions are similar; since many background analyses give louder SNR than the on-source, we conclude that no signal has been detected. Inset: a detailed view of the on-source results.}
\end{figure}
With unlimited computing power, we would calculate upper limits by adding software injections of increasing amplitude until the desired fraction of injections are recovered. However, this is prohibitively expensive in computing time.
Instead, at each amplitude for the injection, we search only over a few previously identified clusters. To pick those clusters we do a full run using the seedless algorithm with fifteen injections at varied amplitudes around the expected recovery threshold and identify clusters which recover the injection, setting the same random seed as was used for on-source recovery. We then analyze a large number of injections using only the pre-identified clusters, and calculate the maximum SNR of those clusters. The injections at all amplitudes are done with the same time-frequency parameters. This allows us to efficiently recover the injection by searching a small fraction of the total clusters. This is shown to work as expected in Appendix \ref{singletrack}.
Since choosing a random seed also chooses which clusters will be searched, this value can affect the final upper limit values. This effect is limited by using a large number of clusters (30 million), and analysis performed with different seeds shows this effect leads to about a 10\% uncertainty in the resulting upper limit value. However, this is not a completely new source of error - changing the seed is functionally equivalent to changing the time-frequency location of the software injections. In addition, the on-source analysis uses only one seed since it is only run once, so this uncertainty is a manifestation of the random nature of the search.
\section{Results and Discussion} \label{results}
No signals were found by either the short- or intermediate-duration searches. We present the results and upper limits on GW strain and energy for each analysis below.
\subsection{Short-Duration Search Upper Limits}
No significant signal was found by X-Pipeline. After glitch rejection, the most significant cluster for the February 25 burst had a p-value of 0.63.
Following the previous f-mode search \citep{GRB051103}, we injected white noise bursts (frequencies: 100--200\,Hz and 100--1000\,Hz; durations: 11\,ms and 100\,ms), and ringdowns (damped sinusoids, at frequencies: 1500\,Hz and 2500\,Hz; time constants: 100\,ms and 200\,ms), and chirplets (chirping sine-Gaussians; this differs from the prior search, which used sine-Gaussians). The best limits for the white noise bursts were for the 11 ms long bursts in the 100--200 Hz band, at $2.1 \times 10^{44} \text{ erg}$ in total isotropic energy and $h_{\text{rss}}$ of $5.6 \times 10^{-23}$ at the detectors. We are most sensitive to ringdowns at 1500 Hz and a time constant of 100 ms, with an upper limit of $2.3 \times 10^{47} \text{ erg}$ and $h_{\text{rss}}$ of $1.9 \times 10^{-22}$. Directly comparing the $h_{\text{rss}}$ limits to \cite{AbAb2011}, we see that limits have improved by roughly a factor of 10, though the ringdowns we used had slightly different parameters. Comparing to \cite{GRB051103}, which provided only energy upper limits assuming a distance of 3.6 Mpc, we see an improvement of factor of 60 after correcting for the larger distance. This corresponds to roughly a factor of 8 improvement in $h_{\text{rss}}$ limits. A full list of upper limits for the waveforms tested is found in Table \ref{x_lims}.
\begin{table}[ht!]
\centering
\begin{tabular}{||c c c c c||}
\hline
Injection Type & Frequency (Hz) & Duration/ & $h_{\text{rss}}$ & Energy (erg) \\
& & $\tau$ (ms)& & \\[0.5ex]
\hline\hline
chirplet & 100 & 10 & $5.42\times 10^{-23}$ & $8.49\times 10^{43}$ \\
\hline
chirplet & 150 & 6.667 & $4.93\times 10^{-23}$ & $1.58\times 10^{44}$ \\
\hline
chirplet & 300 & 3.333 & $5.29\times 10^{-23}$ & $7.27\times 10^{44}$ \\
\hline
chirplet & 1000 & 1 & $1.15\times 10^{-22}$ & $3.82\times 10^{46}$ \\
\hline
chirplet & 1500 & 0.6667 & $1.69\times 10^{-22}$ & $1.81\times 10^{47}$ \\
\hline
chirplet & 2000 & 0.5 & $2.32\times 10^{-22}$ & $5.92\times 10^{47}$ \\
\hline
chirplet & 2500 & 0.4 & $3.06\times 10^{-22}$ & $1.56\times 10^{48}$ \\
\hline
chirplet & 3000 & 0.3333 & $3.96\times 10^{-22}$ & $3.65\times 10^{48}$ \\
\hline
chirplet & 3500 & 0.2857 & $5.30\times 10^{-22}$ & $8.51\times 10^{48}$ \\
\hline
white noise burst & 100--200 & 11 & $5.57\times 10^{-23}$ & $2.09\times 10^{44}$ \\
\hline
white noise burst & 100--200 & 100 & $7.88\times 10^{-23}$ & $4.15\times 10^{44}$ \\
\hline
white noise burst & 100--1000 & 11 & $1.00\times 10^{-22}$ & $1.04\times 10^{46}$ \\
\hline
white noise burst & 100--1000 & 100 & $1.83\times 10^{-22}$ & $3.55\times 10^{46}$ \\
\hline
ringdown & 1500 & 200 & $1.89\times 10^{-22}$ & $2.25\times 10^{47}$ \\
\hline
ringdown & 2500 & 200 & $2.87\times 10^{-22}$ & $1.37\times 10^{48}$ \\
\hline
ringdown & 1500 & 100 & $1.89\times 10^{-22}$ & $2.25\times 10^{47}$ \\
\hline
ringdown & 2500 & 100 & $2.80\times 10^{-22}$ & $1.30\times 10^{48}$ \\
\hline
\end{tabular}
\caption{Upper limits on isotropic energy from the short-duration search for the February 25 burst from SGR 1806-20. For white noise bursts, we give the duration of the injection; for the other waveforms, the characteristic time. All limits are given at 50\% detection efficiency, meaning that a signal with the given parameters would be detected 50\% of the time.}
\label{x_lims}
\end{table}
\subsection{Intermediate Duration Search Upper Limits}
To calculate upper limits, we add software injections of two waveforms (half-sine Gaussians and exponentially decaying sinusoids) at five frequencies (55, 150, 450, 750, and 1550 Hz) and at two timescales (150 seconds and 400 seconds). Reported upper limits are for 50\% recovery efficiency, where recovery is defined as finding a cluster, at the same time and frequency as the injection, with SNR greater than that of the on-source (for the February 25 event, it was 6.09). Full results are shown in Table \ref{stamp_lims}.
Due to the improved sensitivity of Advanced LIGO, we are able to set strain upper limits about a factor of 10 lower than the previous search during initial LIGO \citep{QuBr2017}, see Figure \ref{sensitivity}. Unlike the previous search, this search showed little difference in $h_{\text{rss}}$ sensitivity between the two injection lengths. STAMP has been refined to improve PSD estimation, which explains the small gap between the injection timescales for this search.
\begin{table}[ht!]
\centering
\begin{tabular}{||c c c c c c||}
\hline
Frequency (Hz) & Tau (sec) & \multicolumn{2}{c}{$h_{\text{rss}}$} & \multicolumn{2}{c||}{Energy (erg)}\\
& & Half Sine-Gaussian & Ringdown & Half Sine-Gaussian & Ringdown \\[0.5ex]
\hline\hline
55 & 400 & $2.29\times 10^{-22}$ & $2.43\times 10^{-22}$ & $1.82\times 10^{44}$ & $2.06\times 10^{44}$ \\
\hline
55 & 150 & $1.97\times 10^{-22}$ & $2.11\times 10^{-22}$ & $1.35\times 10^{44}$ & $1.55\times 10^{44}$ \\
\hline
150 & 400 & $1.32\times 10^{-22}$ & $1.37\times 10^{-22}$ & $4.52\times 10^{44}$ & $4.86\times 10^{44}$ \\
\hline
150 & 150 & $1.14\times 10^{-22}$ & $1.22\times 10^{-22}$ & $3.37\times 10^{44}$ & $3.89\times 10^{44}$ \\
\hline
450 & 400 & $1.69\times 10^{-22}$ & $1.79\times 10^{-22}$ & $6.62\times 10^{45}$ & $7.47\times 10^{45}$ \\
\hline
450 & 150 & $1.78\times 10^{-22}$ & $1.83\times 10^{-22}$ & $7.43\times 10^{45}$ & $7.83\times 10^{45}$ \\
\hline
750 & 400 & $2.56\times 10^{-22}$ & $2.70\times 10^{-22}$ & $4.21\times 10^{46}$ & $4.69\times 10^{46}$ \\
\hline
750 & 150 & $2.11\times 10^{-22}$ & $2.37\times 10^{-22}$ & $2.87\times 10^{46}$ & $3.61\times 10^{46}$ \\
\hline
1550 & 400 & $5.86\times 10^{-22}$ & $6.22\times 10^{-22}$ & $9.21\times 10^{47}$ & $1.03\times 10^{48}$ \\
\hline
1550 & 150 & $4.38\times 10^{-22}$ & $4.58\times 10^{-22}$ & $5.16\times 10^{47}$ & $5.62\times 10^{47}$ \\
\hline
\end{tabular}
\caption{Upper limits on GW strain and energy from the intermediate-duration search for the February 25 burst from SGR 1806-20. All limits are at 50\% detection efficiency.}
\label{stamp_lims}
\end{table}
\begin{figure*}[ht!]
\centering
\includegraphics[width=\textwidth]{strain_plot_STAMP.pdf}
\includegraphics[width=\textwidth]{strain_plot_xpipeline.pdf}
\caption{\label{sensitivity}Upper limits for the the intermediate-duration search (above) and short-duration search (below), along with the sensitivity of the detectors. We plot $h_{\text{rss}}$ at 90\% detection efficiency for the intermediate-duration search here to allow direct comparison to published figures for the previous search in initial LIGO \citep{QuBr2017}. Short-duration limits are for 50\% efficiency as before. The advanced LIGO search limits are for the February 25 burst from SGR 1806-20 during the second observing run, and detector sensitivity is calculated from data during the analysis window.}
\end{figure*}
\subsection{Discussion}
This search has set the strongest upper limits on short- and intermediate-duration GW emission associated with magnetar bursts. The energy limits, which are as low as $10^{44}$--$10^{47} \text{erg}$, are now well below the EM energy scale of magnetar giant flares ($10^{46} \text{erg}$). The short bursts analyzed here were much weaker than a giant flare (see Table \ref{burst_data}), so for these bursts the limit is much larger than the observed electromagnetic energy. In addition, these limits assume ideal orientation of the magnetar (both sky position and polarization of produced GWs). The impact of other polarizations on the intermediate-duration search are discussed in Appendix \ref{polarization}, and plotted in Figure \ref{distance}.
\begin{figure}[ht!]
\centering
\includegraphics[width=\textwidth]{horizon_distance_STAMP.png}
\caption{\label{distance} Minimum detectable energy for the intermediate duration search vs distance for SGR 1806-20 for varied sky locations and GW polarizations at 55 Hz. The lines show how the variation in sky position (caused by the earth's rotation) and polarization (assumed to be random) affects the sensitivity; the purple 95\textsuperscript{th} percentile line indicates that the network sensitivity will be better than indicated by that line only 5\% of the time. The shaded region indicates the sensitivity to GW energy from the burst on February 25. Here, the uncertainty is only due to the unknown polarization.}
\end{figure}
The upper limits set by this search are still far above the GW energy from f-mode excitation during a giant flare according to \cite{ZinLas2012}, unless the magnetic field strength is far higher than currently accepted value of $2 \times 10^{15}\ \text{G}$ \citep{mcgillpaper}. Using Equation 2 from \cite{ZinLas2012}, f-mode GW emission from a giant flare would be about $1.4 \times 10^{38}\ \text{erg}$. A surface magnetic field of $1.8 \times 10^{16}\ \text{G}$ would be required to reach the best upper limit found with the short-duration source.
As the LIGO detectors increase in sensitivity, these upper limits will improve, and will be well-positioned to place meaningful limits on emitted GW energy in the event of a future nearby magnetar giant flare.
\acknowledgements
\section*{Acknowledgements}
The authors gratefully acknowledge the support of the United States
National Science Foundation (NSF) for the construction and operation of the
LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO
and construction and operation of the GEO600 detector.
Additional support for Advanced LIGO was provided by the Australian Research Council.
The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN),
the French Centre National de la Recherche Scientifique (CNRS) and
the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research,
for the construction and operation of the Virgo detector
and the creation and support of the EGO consortium.
The authors also gratefully acknowledge research support from these agencies as well as by
the Council of Scientific and Industrial Research of India,
the Department of Science and Technology, India,
the Science \& Engineering Research Board (SERB), India,
the Ministry of Human Resource Development, India,
the Spanish Agencia Estatal de Investigaci\'on,
the Vicepresid\`encia i Conselleria d'Innovaci\'o, Recerca i Turisme and the Conselleria d'Educaci\'o i Universitat del Govern de les Illes Balears,
the Conselleria d'Educaci\'o, Investigaci\'o, Cultura i Esport de la Generalitat Valenciana,
the National Science Centre of Poland,
the Swiss National Science Foundation (SNSF),
the Russian Foundation for Basic Research,
the Russian Science Foundation,
the European Commission,
the European Regional Development Funds (ERDF),
the Royal Society,
the Scottish Funding Council,
the Scottish Universities Physics Alliance,
the Hungarian Scientific Research Fund (OTKA),
the Lyon Institute of Origins (LIO),
the Paris \^{I}le-de-France Region,
the National Research, Development and Innovation Office Hungary (NKFIH),
the National Research Foundation of Korea,
Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation,
the Natural Science and Engineering Research Council Canada,
the Canadian Institute for Advanced Research,
the Brazilian Ministry of Science, Technology, Innovations, and Communications,
the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR),
the Research Grants Council of Hong Kong,
the National Natural Science Foundation of China (NSFC),
the Leverhulme Trust,
the Research Corporation,
the Ministry of Science and Technology (MOST), Taiwan
and
the Kavli Foundation.
The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the
State of Niedersachsen/Germany for provision of computational resources.
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Shigeru Miyamoto is a famous game designer, and the creator of such games as Donkey Kong, Mario, The Legend Of Zelda, and Pikmin. In his spare time he likes to jump up and down like a crazy man. As the screenshot on the right demonstrates, after Nintendo split with Rare, some gamers accused Miyamoto of having a God complex.
This page was last modified on 10 July 2008, at 19:07.
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Home » My Views » Our caregivers raise $446,000 for hurricane, wildfire relief to date
Our caregivers raise $446,000 for hurricane, wildfire relief to date
September 19, 2017 Rod Hochman
Update: Due to the dire situation in Puerto Rico and other islands affected by Hurricane Maria, we have extended the deadline for our matching donations to Thursday, Oct. 12.
When you work in health care and see entire communities in a state of emergency, your first instinct is to want to help in any way you can, whether it's in your own backyard or on the other side of the country. It's the first responder in all of us.
Caregivers across Providence St. Joseph Health have done just that in response to the hurricanes that have battered the Southeastern U.S. and the Caribbean, as well as the wildfires that have engulfed some of the most beautiful public and private lands in the West.
When we announced that we would match caregiver donations dollar for dollar, the people of Providence St. Joseph Health generously stepped up. Together, we have raised more than $446,000 to support Red Cross relief efforts in the affected communities.
A breakdown of our fundraising efforts to date
As of Sept. 18, 2017
Caregiver donations
PSJH match
PSJH system donation
Wildfires (disaster relief)
Where it is needed most
$446, 340
Deadline for match: Sept. 28
There is still time to give. We will continue to match contributions to support wildfire and Hurricane Irma relief efforts through 5 p.m. Pacific Thursday, Sept. 28 up to $100,000. Please note: Matching donations to Hurricane Harvey relief are now closed. For those of our caregivers who want to contribute but have not yet done so, I encourage you to make your gift before the deadline so you can take advantage of the match. You can find the link to our dedicated Red Cross page here.
PTO donations: Another way to contribute
I am also pleased to announce that we are implementing a disaster PTO (vacation/annual leave) donation program, which allows our caregivers to donate the value of unused, accrued PTO to the Red Cross through Sept. 28.
To learn more about how this program works, you can find more details here:
About our disaster PTO donation program
PTO donation form
IRS announcement on tax exemption for Hurricane Harvey PTO donations
Thank you again to the caregivers of Providence St. Joseph Health. Your compassion and generosity embody the very spirit of our organization.
Responding to a season of cataclysmic disasters – Sept. 8, 2017
Standing in solidarity with Houston – Aug. 29, 2017
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We hold everyone affected by this storm close in our hearts. We also offer our support in the form of matching donations to the American Red Cross and Catholic Relief Services up to $100,000.
Shootings: We can't be complacent
PSJH recommits to speaking up and taking action to reduce deaths from gun violence. Our Mission calls us to be a steadfast source of hope and healing in the world.
Responding to the call at U.S. Southern border
Our caregivers are serving at the Humanitarian Respite Center in McAllen, Texas where refugees are being taken in droves. PSJH offers matching contributions up to $100,000 for Catholic Charities.
Microsoft and Providence St. Joseph Health forge strategic alliance
Together, we'll accelerate the future of care delivery.
The power of love in action: Our latest community benefit report
Thank you to our caregivers and community partners for responding to the call to serve with love. Last year, we provided $1.6 billion in free and uncompensated care and other community programs.
Homelessness: Our $15 million investment in affordable housing for the poor and vulnerable
This gift will go toward the development of three apartment complexes for low-income seniors and veterans including Seattle's first affordable high rise in 50 years.
Grateful for our nurses
Nurses have made a difference in my life, my family's life and everyone we serve. I am humbled and inspired by the sacred work they do day in, day out.
Celebrating our founders on International Women's Day
Founded by empowered women, PSJH continues to empower women today. Our founders never gave up on the dream of a more balanced, equal and loving world and neither will we.
Back from Guatemala ... for now
Providence St. Joseph Health is kicking off a new four-year plan with Medical Teams International to reduce infant mortality in the Central Highlands of Guatemala.
Coming together to end the measles outbreak
Washington state has the highest number of reported measles infections since 1996. It will take all of us coming together to fight this dangerous outbreak.
Supporting federal workers
PSJH is offering federal workers a grace period on out-of-pocket medical costs, suspending collections and encouraging our caregivers to sign up for food drives and other service activities.
Predictions: 2019 will be "the year of coalitions in health care"
What key industry trends will define health care in 2019? Here are our top 10 New Year's predictions.
It's time to redouble efforts to prevent suicide and addiction-related deaths
Providence St. Joseph Health and Well Being Trust respond to the nation's mental health and opioid crisis, which is responsible for an alarming 117,000 deaths annually.
Chicago shooting reminds us we can't be complacent about hospital violence
Our hearts go out to our colleagues at Mercy Hospital in Chicago. The recent tragedy remind us we can't be complacent about workplace violence in the health care setting.
Living our values in times of change
In this rapidly changing world
Protecting the health of legal immigrants
A new U.S. immigration policy could jeopardize the health of legal immigrants and our communities at large. The Department of Homeland Security is now accepting public comment.
PSJH offers matching contributions for hurricane relief
To support Hurricane Florence relief efforts, Providence St. Joseph Health will match caregiver contributions to the American Red Cross dollar for dollar up to $100,000.
6 ways PSJH is organizing for the future
Our vision - health for a better world - is driven by a belief that health is a human right. That's why we're innovating to increase access for all and responding to the call to operate in new ways.
PSJH joins Pearl Jam to end homelessness
As part of the Pearl Jam campaign to end homelessness in Seattle, PSJH is supporting a modular shelter project that will accommodate 72 people per night, helping those in unauthorized encampments.
Celebrating two years as PSJH
This month marks our second anniversary as Providence St. Joseph Health, and we have much to celebrate. We'll also continue to evolve with the times and lead with our values in these six ways.
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Q: Calculate resistance of coil/inductor made of PCB traces This is to be a NFC coil/inductor of 1.1uH according do ST's eDesignSuite tool, with 7 turns.
The drawing above follows these parameters:
ST's tool give inductance value as 1.1uH, but also I need to know the resistance of entire track, in ohms.
I have measured the antenna total lenght on the layout software and it is 418 mm long.
So I have these parameters:
*
*0.25mm width
*418mm length
*0.5 oz (17.5 micrometers) copper thickness
Then I entered these value of this calculator: https://www.allaboutcircuits.com/tools/trace-resistance-calculator/
And the result given by the site is 1.66 Ohms. Can I trust this value? I need the know correct value of resistance to calculate the components of RF portion of NFC filter and matching circuits.
The board will have 0.5 oz copper thickness, I can guarantee this.
Help please. Regards.
A: Resistivity of Annealed copper: 1.72x10-8 Ωm
Resistivity is defined as:
\$ \rho = R\cdot \frac{A}{l}\$
therefore \$ R = \frac{\rho \cdot l}{A} \$
your length is 0.418m your area is \$0.25\,\mathrm{mm}\times 17.5\,\mu \mathrm{m}\$
therefore your resistance = \$ \frac{ 1.72\times10^{-8} \,\Omega \mathrm{m} \,\times\, 0.418\, \mathrm{m}}{ 0.25\,\mathrm{mm} \,\times\, 17.5\,\mu \mathrm{m}} = 1.64\,\Omega\$ at 20C
A: You can trust the value of 1.66 ohms given by your calacuator, at DC, at 25 C, if the foil thinkness and etched width are as you've specified.
You would be surprised how much in error the etched width could be in practice.
The temperature variation is fairly small, +10% for 25 C change.
The main error will be if you want the resistance at NFC frequencies rather than DC, as the skin effect will increase it significantly.
A: Standard copper foil is 0.000498 ohms per square of foil, for any size square!!!
Just to be very clear, this is 500 microOhms per square, whether 1 square micron or 10 square centimeterS.
Notice this resistance is substantially lower than typical ESR values in capacitors. BUT THE VALUE IS NOT ZERO
This fact is useful for copper resistor design, or for examining hot spots where current crowds, etc.
===================
This value is for 1 ounce of copper per square foot, which is the 1.4 mil thickness or 35 microns.
And 35 microns used in skin_depth formula will produce 1 neper attenuation to penetrating waves at about 4MHz.
Or 10 nepers at 100X the frequency. Which is 88 dB.
The 88 dB attenuation is good explanation of why ground planes or power planes are so useful in creating clean systems, despite the presence of MCUs with 400MHz edge rates (1.25 nanosecond trise).
A: You could not expect the width and the thickness of the track to be very precise.
There are tolerances depending on manufacture methods.
A track width of 250 µm may vary at least by +- 5 µm or more depending on fotoplotter resolution, resist exposure time, resist development time, under etching, etching time, etching temperature and a lot more.
Copper thickness of 18 µm may vary about +- 2 to 5 µm depending on manufacture.
So ask your PCB manufacturer about the tolerances for copper thickness and track width.
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| 6,593
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{"url":"https:\/\/phys.libretexts.org\/Bookshelves\/Nuclear_and_Particle_Physics\/Book%3A_Nuclear_and_Particle_Physics_(Walet)\/05%3A_Basic_Concepts_of_Theoretical_Particle_Physics\/5.02%3A_Antiparticles","text":"$$\\require{cancel}$$\nIn Figure $$\\PageIndex{1}$$ we have sketched the solutions for the Dirac equation for a free particle. It has a positive energy spectrum starting at $$m c^2$$ (you cannot have a particle at lower energy), but also a negative energy spectrum below $$-mc^2$$. The interpretation of the positive energy states is natural \u2013 each state describes a particle moving at an energy above $$mc^2$$. Since we cannot have negative energy states, their interpretation must be very different. The solution is simple: We assume that in an empty vacuum all negative energy states are filled (the \u201cDirac sea\u201d). Excitations relative to the vacuum can now be obtained by adding particles at positive energies, or creating holes at negative energies. Creating a hole takes energy, so the hole states appear at positive energies. They do have opposite charge to the particle states, and thus would correspond to positrons! This shows a great similarity to the behaviour of semiconductors, as you may well know. The situation is explained in Figure $$\\PageIndex{2}$$.","date":"2022-01-24 00:51:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8859526515007019, \"perplexity\": 268.97023939644873}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304345.92\/warc\/CC-MAIN-20220123232910-20220124022910-00233.warc.gz\"}"}
| null | null |
Q: Why must UIKit operations be performed on the main thread? I am trying to understand why UI operations can't be performed using multiple threads. Is this also a requirement in other frameworks like OpenGL or cocos2d?
How about other languages like C# and javascript? I tried looking in google but people mention something about POSIX threads which I don't understand.
A: from : https://www.objc.io/issues/2-concurrency/thread-safe-class-design/
It's a conscious design decision from Apple's side to not have UIKit be thread-safe. Making it thread-safe wouldn't buy you much in terms of performance; it would in fact make many things slower. And the fact that UIKit is tied to the main thread makes it very easy to write concurrent programs and use UIKit. All you have to do is make sure that calls into UIKit are always made on the main thread.
So according to this the fact that UIKit objects must be accessed on the main thread is a design decision by apple to favor performance.
A: In Cocoa Touch, the UIApplication i.e. the instance of your application is attached to the main thread because this thread is created by UIApplicatioMain(), the entry point function of Cocoa Touch. It sets up main event loop, including the application's run loop, and begins processing events. Application's main event loop receives all the UI events i.e. touch, gestures etc.
From docs UIApplicationMain(),
This function instantiates the application object from the principal class and instantiates the delegate (if any) from the given class and sets the delegate for the application. It also sets up the main event loop, including the application's run loop, and begins processing events. If the application's Info.plist file specifies a main nib file to be loaded, by including the NSMainNibFile key and a valid nib file name for the value, this function loads that nib file.
These application UI events are further forwarded to UIResponder's following the chain of responders usually like UIApplication->UIWindow->UIViewController->UIView->subviews(UIButton,etc.)
Responders handle events like button press, tap, pinch zoom, swipe etc. which get translated as change in the UI. Hence as you can see these chain of events occur on main thread which is why UIKit, the framework which contains the responders should operate on main thread.
From docs again UIKit,
For the most part, UIKit classes should be used only from an application's main thread. This is particularly true for classes derived from UIResponder or that involve manipulating your application's user interface in any way.
EDIT
Why drawRect needs to be on main thread?
drawRect: is called by UIKit as part of UIView's lifecycle. So drawRect: is bound to main thread. Drawing in this way is expensive because it is done using the CPU on the main thread. The hardware accelerate graphics is provided by using the CALayer technique (Core Animation).
CALayer on the other hand acts as a backing store for the view. The view will then just display cached bitmap of its current state. Any change to the view properties will result in changes in the backing store which get performed by GPU on the backed copy. However, the view still needs to provide the initial content and periodically update view. I have not really worked on OpenGL but I think it also uses layers(I could be wrong).
I have tried to answer this to the best of my knowledge. Hope that helps!
A: C# behaves the same (see eg here: Keep the UI thread responsive). UI updates have to be done in the UI thread - most other things should be done in the background hen possible.
If that wouldn't be the case there would probably be a synchronization hell between all updates that have to be done in the UI ...
A: Every system, every library, needs to be concerned about thread safety and must do things to ensure thread safety, while at the same time looking after correctness and performance as well.
In the case of the iOS and MacOS X user interface, the decision was made to make the UI thread safe by only allowing UI methods to be called and executed on the main thread. And that's it.
Since there are lots of complicated things going on that would need at least serialisation to prevent total chaos from happening, I don't see very much gained from allowing UI on a background thread.
A: Because you want the user to be able to see the UI changes as they happen. If you were to be able to perform UI changes in a background thread and display them when complete, it would seem the app doesn't behave right.
All non-UI operations (or at least the ones that are very costly, like downloading stuff or making database queries) should take place on a background thread, whereas all UI changes must always happen on the main thread to provide as smooth of a user experience possible.
I don't know what it's like in C# for Windows Phone apps, but I would expect it to be the same. On Android the system won't even let you do things like downloading on the main thread, making you create a background thread directly.
As a rule of thumb - when you think main thread, think "what the user sees".
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| 5,379
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{"url":"http:\/\/www.math.iisc.ac.in\/seminars\/2014\/2014-04-11-mr-jaikrishnan-janardhanan-iisc.html","text":"#### PhD Thesis colloquium\n\n##### Venue: Lecture Hall I, Department of Mathematics\n\nThe aim of this thesis is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for domains in $C^n$. We present results for special classes of manifolds. We study, broadly, two types of structure results:\n\nI. Descriptive: Our first result is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result of Remmert and Stein adapted to the above setting. However, the presence of factors that have no boundary, or boundaries that consist of a discrete set of points, necessitates the use of alternative techniques. Specifically: we make use of a finiteness theorem of Imayoshi.\n\nII. Rigidity: A famous theorem of H. Alexander proves the non-existence of non-injective proper holomorphic self-maps of the unit ball in $C^n,\\ n > 1$. Several extensions of this result for various classes of domains have been established since the appearance of Alexander\u2019s result. Our first rigidity result establishes the non-existence of non-injective proper holomorphic self-maps of bounded, balanced pseudoconvex domains of finite type (in the sense of D\u2019Angelo) in $C^n,\\ n > 1$. This generalizes a result in $C^2$ due to Coupet, Pan and Sukhov to higher dimensions. In higher dimensions, several aspects of their argument do not work. Instead, we exploit the circular symmetry and a recent result in complex dynamics by Opshtein.\n\nOur next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply some of our techniques to more general classes of domains. We illustrate this through a rigidity result for certain convex balanced domains whose automorphism groups are only assumed to be non-compact. For the bounded symmetric domains, our key tool is that of Jordan triple systems.\n\nContact: +91 (80) 2293 2711, +91 (80) 2293 2265 ; \u00a0\u00a0\u00a0 E-mail: chair.math[at]iisc[dot]ac[dot]in\nLast updated: 05 Dec 2019","date":"2019-12-12 11:56:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7500845789909363, \"perplexity\": 277.5489131718549}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540543252.46\/warc\/CC-MAIN-20191212102302-20191212130302-00500.warc.gz\"}"}
| null | null |
Q: HEROKU: ('08001', '[08001] [Microsoft][ODBC Driver 17 for SQL Server]TCP Provider: Error code 0x2746 (10054) (SQLDriverConnect)') I am having an issue trying to use SQL Server with Pyobdc in Django on Heroku. I'm getting this error:
('08001', '[08001] [Microsoft][ODBC Driver 17 for SQL Server]TCP Provider: Error code 0x2746 (10054) (SQLDriverConnect)')
A: I didn't have any chance to test it with less information but I guess it is related to connectivity configurations. You can read the issue here, some of the users solved the problem by overriding the OpenSSL configuration. I'm not sure how you can handle this on Heroku. You can read the comments to learn how to get trace logs. They might help you to find the solution.
|
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<title>クラス jp.ac.maslab.ando.aiwolf.client.data.definition.TalkTagの使用</title>
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<h2 title="クラスの使用 jp.ac.maslab.ando.aiwolf.client.data.definition.TalkTag" class="title">クラスの使用<br>jp.ac.maslab.ando.aiwolf.client.data.definition.TalkTag</h2>
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<caption><span><a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html" title="jp.ac.maslab.ando.aiwolf.client.data.definition内の列挙型">TalkTag</a>を使用しているパッケージ</span><span class="tabEnd"> </span></caption>
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<div class="block">各データのやそれを扱うテーブルなどの要素を管理するための名前やIDの列挙を含むパッケージです。</div>
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<h3><a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/package-summary.html">jp.ac.maslab.ando.aiwolf.client.data.definition</a>での<a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html" title="jp.ac.maslab.ando.aiwolf.client.data.definition内の列挙型">TalkTag</a>の使用</h3>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="表、メソッドのリストおよび説明の使用">
<caption><span><a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html" title="jp.ac.maslab.ando.aiwolf.client.data.definition内の列挙型">TalkTag</a>を返す<a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/package-summary.html">jp.ac.maslab.ando.aiwolf.client.data.definition</a>のメソッド</span><span class="tabEnd"> </span></caption>
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<th class="colFirst" scope="col">修飾子とタイプ</th>
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<td class="colFirst"><code>static <a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html" title="jp.ac.maslab.ando.aiwolf.client.data.definition内の列挙型">TalkTag</a></code></td>
<td class="colLast"><span class="typeNameLabel">TalkTag.</span><code><span class="memberNameLink"><a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html#valueOf-java.lang.String-">valueOf</a></span>(java.lang.String name)</code>
<div class="block">指定した名前を持つこの型の列挙型定数を返します。</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static <a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html" title="jp.ac.maslab.ando.aiwolf.client.data.definition内の列挙型">TalkTag</a>[]</code></td>
<td class="colLast"><span class="typeNameLabel">TalkTag.</span><code><span class="memberNameLink"><a href="../../../../../../../../../jp/ac/maslab/ando/aiwolf/client/data/definition/TalkTag.html#values--">values</a></span>()</code>
<div class="block">この列挙型の定数を含む配列を宣言されている順序で返します。</div>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,383
|
Discussion in 'Meet Up' started by Ali Miller, Oct 15, 2017.
I'm going to be in LA this January, and I'm hoping to get some canyoneering in. I'm also hoping I don't have to do it alone. Anyone interested in meeting up? Or at least, does anyone have canyon suggestions for me to check out?
My dates: 1/2 to 1/7. I might be able to get out earlier, but I can't stay later.
I really want Death Valley, but I'm open to suggestion.
I've got gear, experience, and beer.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,072
|
{"url":"https:\/\/cran.pau.edu.tr\/web\/packages\/itsadug\/vignettes\/test.html","text":"# Testing for significance\n\n#### 15 March 2016\n\nGenerally, there are three methods to test whether a certain predictor or interaction is significantly contributing to the model\u2019s account of the data:\n\n1. Model comparison procedure\n\n2. Inspection of the summary\n\n3. Visual inspection of the model\u2019s estimates\n\n## Example\n\nlibrary(itsadug)\nlibrary(mgcv)\ndata(simdat)\n# select subset of data to reduce processing time:\nselect <- 1:18\nselect <- select[select %% 3 ==0]\n\n## 1. Model comparison\n\nTo test whether the three-way interaction between Time, Condition and Group is significant, we can compare the model with a model that does not include this three-way interaction:\n\nm2Rho <- bam(Y ~ Group + s(Time, by=Group) + s(Condition, by=Group, k=5) + ti(Time, Condition) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, method=\"fREML\", AR.start=simdat$start.event, rho=r1) #### Function compareML The function compareML compares two models on the basis of the minimized smoothing parameter selection score specified in the model, and performes a $$\\chi^2$$ test on the difference in scores and the difference in degrees of freedom. # make sure that info messages are printed to the screen: infoMessages('on') compareML(m1Rho, m2Rho) ## m1Rho: Y ~ Group + s(Time, by = Group) + s(Condition, by = Group, k = 5) + ## ti(Time, Condition, by = Group) + s(Time, Subject, bs = \"fs\", ## m = 1, k = 5) + s(Event, bs = \"re\") ## ## m2Rho: Y ~ Group + s(Time, by = Group) + s(Condition, by = Group, k = 5) + ## ti(Time, Condition) + s(Time, Subject, bs = \"fs\", m = 1, ## k = 5) + s(Event, bs = \"re\") ## ## Chi-square test of fREML scores ## ----- ## Model Score Edf Difference Df p.value Sig. ## 1 m2Rho 43415.70 16 ## 2 m1Rho 43393.79 19 21.901 3.000 1.663e-09 *** ## ## AIC difference: -10.54, model m1Rho has lower AIC. The following conclusions can be derived from the output: \u2022 Model m1Rho has a lower fREML score (lower indicates better fit). \u2022 But model m1Rho is also more complex: it uses more degrees of freedom (Edf). Note that Edf in the model comparison are different from the edf that are presented in the model summary. The first are reflecting the complexity of the model (number of model terms, complexity of model terms), and the second are reflecting the complexity of the smooth or surface pattern (i.e., number of knots or underlying base functions used). \u2022 Model m1Rho is preferred, because the difference in fREML is significant given the difference in degrees of freedom: $$\\chi^2$$(3)=21.836, p < .001. #### Some notes on model comparison Model comparison procedure provides an indication for the best fitting model, but can rarely used on it\u2019s own for determining significance. \u2022 For testing the difference in fixed effects predictors the method fREML does not provide the most reliable test. Rather use ML. However, ML takes longer to run (that is why it is not included here), and penalizes wigglyness more. \u2022 An alternative test is AIC, but when an AR1 model is included, AIC does not provide a reliable test. (Like here!) AIC(m1Rho, m2Rho) ## df AIC ## m1Rho 232.8363 86485.24 ## m2Rho 244.0608 86495.78 ## 2. Inspection of the model summary Beside model comparison the model summary (e.g., summary(m1Rho)) could provide useful information on whether or not a model term is significantly contributing to the model. To include the summary in a R markdown or knitr report use the function gamtabs: gamtabs(m1Rho, type=\"HTML\") A. parametric coefficients Estimate Std. Error t-value p-value (Intercept) 2.0815 0.0782 26.6229 < 0.0001 GroupAdults 3.1233 0.1106 28.2475 < 0.0001 B. smooth terms edf Ref.df F-value p-value s(Time):GroupChildren 8.1701 8.8251 1883.6351 < 0.0001 s(Time):GroupAdults 8.5790 8.9528 3696.9128 < 0.0001 s(Condition):GroupChildren 3.5242 3.6217 226.3898 < 0.0001 s(Condition):GroupAdults 3.7460 3.8068 481.0384 < 0.0001 ti(Time,Condition):GroupChildren 15.4474 15.9501 1275.5365 < 0.0001 ti(Time,Condition):GroupAdults 15.3403 15.9282 845.9445 < 0.0001 s(Time,Subject) 0.0010 56.0000 0.0000 0.8756 s(Event) 174.3143 248.0000 2.4927 < 0.0001 The summary provides the following information: \u2022 There is an overall difference in Y for children and adults (parametric terms) \u2022 The F values \/ p-values of the \u2018fixed\u2019 effects smooth terms indicate that all these smooth terms are significantly different from 0, so each line or surface is significantly wiggly. \u2022 However, we can NOT conclude that the lines or surfaces are different from each other. This is only possible when we would use difference smooths or tensors, with ordered factors or binomial predictors. See below for an example. \u2022 For the random effects the statistics indicates whether or not these terms contribute to the model (s(Event)) or not (s(Time,Subject)). #### Using ordered factors (advanced) It is possible to change the contrasts for grouping predictors in mgcv so that the smooth terms represent differences with the reference level, similar to the treatment coding used in lmer or in the summary of parametric terms in GAMMs. The trick is to first convert the factors to ordered factors so that gam() and bam() won\u2019t use the default contrast coding. Here\u2019s an example: simdat$OFGroup <- as.ordered(simdat$Group) contrasts(simdat$OFGroup) <- \"contr.treatment\"\ncontrasts(simdat$OFGroup) ## Adults ## Children 0 ## Adults 1 Note that in the case of using ordered factors we need to include the reference curves or surfaces as well. m1Rho.OF <- bam(Y ~ OFGroup + s(Time) + s(Time, by=OFGroup) + s(Condition, k=5) + s(Condition, by=OFGroup, k=5) + ti(Time, Condition) + ti(Time, Condition, by=OFGroup) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, method=\"fREML\", AR.start=simdat$start.event, rho=r1)\n\nWith the ordered factors suddenly the lines s(Time):OFGroupAdults and similar lines represent the difference between the adults and the reference group, the children. When the smooth term is significant, the difference smooth is is significantly different from zero. So that means that the two groups are different from each other:\n\ngamtabs(m1Rho.OF, type=\"HTML\")\n A. parametric coefficients Estimate Std. Error t-value p-value (Intercept) 2.0813 0.0781 26.6581 < 0.0001 OFGroupAdults 3.1236 0.1104 28.2922 < 0.0001 B. smooth terms edf Ref.df F-value p-value s(Time) 8.2883 8.7974 1891.3882 < 0.0001 s(Time):OFGroupAdults 7.3870 8.3331 409.8352 < 0.0001 s(Condition) 3.6481 3.7167 224.6983 < 0.0001 s(Condition):OFGroupAdults 2.8349 2.9713 34.0186 < 0.0001 ti(Time,Condition) 15.5467 15.9140 1280.8817 < 0.0001 ti(Time,Condition):OFGroupAdults 13.1446 14.9445 129.8732 < 0.0001 s(Time,Subject) 0.0015 56.0000 0.0000 0.7355 s(Event) 174.7125 248.0000 2.4914 < 0.0001\n\nIn summary, with continuous predictors or ordered factors we can use the summary startistics to determine the difference of smooth terms.\n\nThe function report_stats describes how one could report the smooth terms in the text of an article:\n\nreport_stats(m1Rho.OF)\n\n## 3. Visual inspection of the model\u2019s estimates\n\n#### Function plot_diff\n\nThe function plot_diff allows to plot the (1 dimensional) estimated difference between two conditions. The argument rm.ranef=TRUE indicates that random effects should be excluded first, and the argument cond can be used to specify values for other predictors.\n\nThe plots below visualize the difference between adults and children.\n\npar(mfrow=c(1,2))\n\n# PLOT 1:\nplot_diff(m1Rho, view=\"Time\", comp=list(Group=c(\"Adults\", \"Children\")), cond=list(Condition=1), rm.ranef=TRUE, ylim=c(-15,15))\nlegend('bottom', legend=c(\"Condition=1\", \"Condition=4\"), col=c(1,2), lwd=1, cex=.75, bty='n')\n\n# PLOT 2:\nplot_diff(m1Rho, view=\"Condition\", comp=list(Group=c(\"Adults\", \"Children\")), cond=list(Time=1000), rm.ranef=TRUE, ylim=c(-15,15))\nlegend('bottom', legend=c(\"Time=1000\", \"Time=2000\"), col=c(1,2), lwd=1, cex=.75, bty='n')\n\n#### Function plot_diff2\n\nThe function plot_diff allows to plot the (2 dimensional) estimated difference between two conditions. The argument rm.ranef=TRUE indicates that random effects should be excluded first, and the argument cond can be used to specify values for other predictors.\n\nThe plots below visualize the difference between adults and children.\n\npar(mfrow=c(1,2), cex=1.1)\nplot_diff2(m1Rho, view=c(\"Time\", \"Condition\"), comp=list(Group=c(\"Adults\", \"Children\")), zlim=c(-15,15), se=0, rm.ranef=TRUE)\n## Warning in gradientLegend(zlim, n.seg = 3, pos = 0.875, dec = dec, color =\n## color, : Increase right margin to fit labels or decrease the number of decimals,\n## see help(gradientLegend).\n# with CI:\nplot_diff2(m1Rho, view=c(\"Time\", \"Condition\"), comp=list(Group=c(\"Adults\", \"Children\")), zlim=c(-15,15), se=1.96, rm.ranef=TRUE, show.diff = TRUE)\n## Warning in gradientLegend(zlim, n.seg = 3, pos = 0.875, dec = dec, color =\n## color, : Increase right margin to fit labels or decrease the number of decimals,\n## see help(gradientLegend).","date":"2021-06-24 18:35:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7590119242668152, \"perplexity\": 12701.490318612365}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488556482.89\/warc\/CC-MAIN-20210624171713-20210624201713-00229.warc.gz\"}"}
| null | null |
Fudgy, decadent, and vegan, this chocolate date frosting is naturally sweetened. With a velvety smooth consistency, it spreads beautifully over a homemade cake or cupcakes. After one bite, you'll wonder why you ever thought icing had to be made with butter. Treat yourself and head over to Foodal now to get the recipe!
Calling all chocolate lovers! Made without eggs, butter, or milk, this one-pan fudgy chocolate cake is a no-frills dessert that also happens to be vegan. Perfect for birthday parties or potlucks, it can be topped with powdered sugar or made more indulgent with chocolate frosting. Get the recipe on Foodal now!
No need to sort through hundreds of recipes to find the perfect vegetable side dish – this go-to roasted broccoli with garlic and lemon is simple to make and turns out tender, yet crisp every time. Family-friendly and incredibly versatile, this roasted broccoli will quickly become a dinner staple! Get the recipe now.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,512
|
export class PrismaClientRustPanicError extends Error {
clientVersion: string
constructor(message: string, clientVersion: string) {
super(message)
this.clientVersion = clientVersion
}
get [Symbol.toStringTag]() {
return 'PrismaClientRustPanicError'
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,803
|
Q: Определене времени обработки MySql запросов в .net приложении? Можно ли в Windows при помощи стандартных счетчиков производительности определить время выполнения запроса в базе данных MySql, т.е. время ожидания между вызовом IDbCommand.ExecuteReader и получением данных?
A: Стандартный коннектор поддерживает всего два счетчика - HardProcedureQueries и HardProcedureQueries, в категории .NET Data Provider for MySQL.
Счетчики влючаются опцией Use Performance Monitor=true в connection string (или ее ализами - UsePerformanceMonitor, userperfmon, perfmon).
Но оба стандартных счетчика типа NumberOfItems, поэтому время выполнения на их основе получить нельзя. Максимум - количество запросов в секунду за некоторый интервал времени.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,803
|
Q: why am i not getting print message? while running:
screen.fill((255,255,255))
screen.blit(background,(0,0))
for event in pygame.event.get():
if event.type == pygame.QUIT:
running = False
# keystroke controlling i am not getting this print message
if event.type == pygame.KEYDOWN:
if event.type == pygame.K_SPACE:
print("space is pressed ")
if event.type == pygame.KEYUP:
if event.type == pygame.K_SPACE:
print("released")
playerx += playerx_change
player(playerx , playery)
cactus(cactusx , cactusy)
pygame.display.update()
game()
A: This is because event.type cannot be KEYUP AND K_SPACE at the same time .
Rewrite the code as :
if event.type == pygame.KEYDOWN:
if event.key == pygame.K_SPACE:
print("space is pressed ")
if event.type == pygame.KEYUP:
if event.key == pygame.K_SPACE:
print("released")
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,388
|
{"url":"https:\/\/math.stackexchange.com\/questions\/2843141\/maximum-volume-ball-in-tetrahedron","text":"# Maximum volume ball in tetrahedron\n\nConsider the problem : \"Find the maximum volume of the three dimensionnal ball inscribed to a tetraedron.\"\n\nThe tetraedron have the following summit (just to have a concrete case): $(0,0,0), (0,\\pi,0),(\\pi,0,0),(\\pi,\\pi,\\pi)$.\n\nGeometrically, from the optimization point of view, the problem can be formulated as\n\n\\begin{align} &\\max_{a,b,c,r\\in \\mathbb{R}}\\qquad r^3\\\\ &r-|c|<0\\\\ &r-|a-b|\/\\sqrt{2}<0\\\\ &r-|b-\\pi|<0\\\\ &r-|a-c|\/\\sqrt{2}\\\\ &a-\\pi<0\\\\ &b-\\pi<0\\\\ &c-\\pi<0\\\\ &a>0\\\\ &b>0\\\\ &c>0\\\\ &r>0\\\\ \\end{align}\n\n($r$ stands for the radius of the ball of center $(a,b,c)$). The optimal volume is obtained for $r\\simeq 0.65$.\n\nNow, I have difficulty in formulating the problem with a ball of dimension $4$ in a tetraedron of dimension $4$ because I don't have the geometric point of view and I don't know how to write the constraints.\n\n\u2022 Smilia, usually \"circumscribed\" is reserved for the sphere going around something. I will look it up, for a sphere tangent to the faces of the tetrahedron, i would probably say inscribed \u2013\u00a0Will Jagy Jul 6 '18 at 19:17\n\u2022 \u2013\u00a0Will Jagy Jul 6 '18 at 19:19\n\u2022 What is a tetrahedron of dimension 4? \u2013\u00a0MPW Jul 6 '18 at 19:22\n\u2022 @MPW pretty sure it is just a simplex of full dimension; I also think that the max volume problem for inscribed sphere is the same as the max radius.... trying to think of a compass and straightedge type of construction, don't have it yet \u2013\u00a0Will Jagy Jul 6 '18 at 19:25\n\u2022 @MPW naively it is a tetraedron whose volume varies to make the fourth dimension, but it doesn't help. As I said I am looking for a generalization that I don't even know how to formulate it. \u2013\u00a0Smilia Jul 6 '18 at 19:29\n\nIn $\\mathbb R^3,$ for each pair of planes bounding the tetrahedron, draw the plane that bisects the angle between the two planes. The six new planes meet in the center of the inscribed sphere.\nThere is a little linear algebra\/geometry involved in describing these bisecting hyperplanes... In $\\mathbb R^n,$ a plane containing a face of the simplex is an $(n-1)$-plane. Two of them meet in an $(n-2)$-plane, call it $P.$ The orthogonal complement of $P$ (through a favorite point) is a $2$-plane, call it $O.$ The intersection of $O$ with the two bounding hyperplanes is just two intersecting lines; we construct the angle bisector between these two lines, and take the sum of that with $P,$ getting us back up to a bisecting $(n-1)$-plane.","date":"2019-06-18 07:21:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5894432663917542, \"perplexity\": 688.078606251982}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998690.87\/warc\/CC-MAIN-20190618063322-20190618085322-00139.warc.gz\"}"}
| null | null |
<?php
namespace Omnipay\BillPay\Message\RequestData;
use Omnipay\BillPay\Message\AuthorizeRequest;
use Omnipay\Common\Exception\InvalidRequestException;
use Omnipay\Common\Message\AbstractRequest;
use SimpleXMLElement;
trait DataTrait
{
private static $paymentTypes = [
AuthorizeRequest::PAYMENT_TYPE_INVOICE => 1,
AuthorizeRequest::PAYMENT_TYPE_DIRECT_DEBIT => 2,
AuthorizeRequest::PAYMENT_TYPE_TRANSACTION_CREDIT => 3,
AuthorizeRequest::PAYMENT_TYPE_PAY_LATER => 4,
AuthorizeRequest::PAYMENT_TYPE_COLLATERAL_PROMISE => 7,
];
/**
* @return int
*/
public function getCaptureRequestNecessary()
{
return (int)$this->getParameter('captureRequestNecessary');
}
/**
* Gets the expected delay in shipping
*
* @return int
*/
public function getExpectedDaysTillShipping()
{
return (int)$this->getParameter('expectedDaysTillShipping');
}
/**
* Get the payment issuer.
*
* @return string
*
* @codeCoverageIgnore
*/
abstract public function getPaymentMethod();
/**
* @param int|bool $value
*
* @return AuthorizeRequest
*/
public function setCaptureRequestNecessary($value)
{
return $this->setParameter('captureRequestNecessary', $value ? 1 : 0);
}
/**
* Sets the expected delay in shipping, required for authorize and pay
*
* @param int $value the expected delay in shipping
*
* @return AuthorizeRequest
*/
public function setExpectedDaysTillShipping($value)
{
return $this->setParameter('expectedDaysTillShipping', $value);
}
/**
* @param SimpleXMLElement $data
*
* @throws InvalidRequestException
*/
protected function appendData(SimpleXMLElement $data)
{
if (!$this->getPaymentMethod()) {
throw new InvalidRequestException('This request requires a payment method.');
}
// the customer has accepted the BillPay terms of service / the data protection policy, we assume that the
// gateway is only used after acceptance
$data['tcaccepted'] = 1;
$data['expecteddaystillshipping'] = $this->getExpectedDaysTillShipping();
$data['capturerequestnecessary'] = $this->getCaptureRequestNecessary();
$data['paymenttype'] = self::$paymentTypes[$this->getPaymentMethod()];
}
/**
* Get a single parameter.
*
* @param string $key The parameter key
*
* @return mixed
*
* @codeCoverageIgnore
*/
abstract protected function getParameter($key);
/**
* Set a single parameter
*
* @param string $key The parameter key
* @param mixed $value The value to set
*
* @return AbstractRequest Provides a fluent interface
*
* @codeCoverageIgnore
*/
abstract protected function setParameter($key, $value);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,748
|
Witness: The Tour was the fourth concert tour by American singer Katy Perry, in support of her fifth studio album, Witness (2017). The tour began on September 19, 2017, in Montreal, Canada, and concluded on August 21, 2018, in Auckland, New Zealand. Perry visited North America, South America, Asia, Europe, Africa and Oceania.
At the end of 2017, the tour placed at number 77 on Pollstars "2017 Year-End Top 100 Worldwide Tours" list, estimating that it grossed $28.1 million and that 266,300 people attended throughout the year. In July 2018, Pollstar ranked the tour on number 14 at the Mid Year Top 100 Worldwide Tours 2018 with $48.8 million and 577,617 of tickets sold in 54 shows. In December 2018, Pollstar ranked the tour at number 29 with $55.3 million gross and 633,827 tickets sold in 52 shows. According to Billboard, the full tour grossed an average of $1.08 million while selling 11,560 tickets per show.
Development
Perry first announced on May 15, 2017, that her fifth album Witness would be released on June 9, 2017, and that she would embark on Witness: The Tour in support of it. Copies of the album are included along with the purchased tickets. During the North American leg of the tour, one dollar from each ticket purchased will go to the Boys & Girls Clubs of America, and fans will also have a chance to win free tickets by doing volunteer work for that organization via Global Citizen. The tour was originally scheduled to begin in Columbus on September 7, 2017. However, Perry revealed on August 17, that due to production delays, the tour had been rescheduled to begin on September 19, 2017. She also announced that Noah Cyrus, Purity Ring and Carly Rae Jepsen would respectively serve as opening acts from September 19 to November 1; November 7 to December 20; and January 5, 2018, to February 5, 2018. The singer later added a final Canadian date for February 6 with Jepsen.
On June 2, 2017, a week prior the release of Witness, European dates were announced. An extra date for London, Amsterdam, Paris as well as dates in Barcelona and Lisbon (this one part of the Rock In Rio festival) were later added. On March 6, 2018, Perry announced that Tove Styrke and Hailee Steinfeld will be the opening acts in Europe. Styrke will open the shows from May 23 until June 10, while Steinfeld from June 14 to June 28.
In July 2017, it was revealed that the tour will then visit Oceania from July 2018 to August 2018. In July 2017, three additional Australian dates were added following popular demand, resulting in 14 shows scheduled during the leg. On May 9, 2018, Perry announced two more shows, one in Adelaide and one in Sydney.
In October 2017, she announced three concerts in Mexico the following May taking place on the 3rd, 8th, and 11th. An additional concert for the country in Monterrey was added for May 9 after the first one sold out, and another show in Mexico City was added the following month for Mexico City on May 4. Other dates in Latin America were later added to the itinerary, including shows in Brazil, Chile, Argentina and Peru.
Perry also visited Asia between March and April 2018 and Africa in July 2018.
Witness: Coming Home
On March 13, 2018, Perry announced Witness: Coming Home, a benefit concert that will be held in her hometown of Santa Barbara, California, which will benefit those who are still recovering from the aftermath of the 2017 California wildfires and 2018 Southern California mudflows. Perry partnered with the Santa Barbara Foundation, the 93108 Fund and The 805 UndocuFund, which all help in assisting members of the community in the Santa Barbara area through grants and various philanthropic efforts.
Concert synopsis
The show was divided into five segments—Manifesto, Retrospective, Sexual Discovery, Introspective, Emergence—and concluded with an encore. Shortly before the show begins, the stage's eye-shaped screen displays a video of Perry's eye for a few moments. The eye transforms into a galaxy and a journey through outer space is shown, leading to a red-colored planet. The screen opens to reveal Perry riding upon a star-shaped structure, wearing a bejeweled and hooded red outfit with sunglasses. She performs the chorus of "Witness", followed by "Roulette" which she sings on top of giant climbable dice. "Dark Horse" is then sung, with Perry and her dancers atop of moving platforms. Perry performs "Chained to the Rhythm" next, with large puppets dressed in suits and sporting televisions for heads moving around the stage. A video interlude is played that shows a clock running backward alongside childhood photos of Perry, as her backing singers perform portions of "Act My Age". Perry emerges wearing a pantsuit with a grid design, while her dancers wear shape-themed outfits. "Teenage Dream" is performed, which culminates in Perry and her dancers performing synchronized choreography while sitting on a floating cuboid. Perry removes her jacket to reveal an LED top that displays the lyrics of "Hot n Cold" as Perry performs the song with an electric guitar. "Hot n Cold" transitions into "Last Friday Night (T.G.I.F.)", which Perry and her backing singers perform at the front of the stage. Puppets in the shape of flamingos are guided around the stage by dancers during this number. Perry's female dancers wear paper doll cut-outs of her most memorable outfits during "California Gurls", and she is also joined by the Left Shark that featured in her Super Bowl XLIX halftime show. Perry and the Left Shark then typically bring out an oversized telephone, with which Perry calls someone close to her, such as her parents. Giant lips emerge from behind the stage, which Perry is lifted inside while performing "I Kissed a Girl".
An interlude is played, which features Perry floating through space. The stage is adorned with huge roses, and Perry emerges wearing a black-and-white, polka dot latex outfit. She performs "Déjà Vu", followed by "Tsunami", during which she dances on a rotating pole. Venus flytrap props are added to the stage, and Perry sings "E.T." as she is joined onstage by a dancer in stilts, dressed as an alien insect. "Bon Appétit" is performed next. Towards the end of the song, jeweled flies are attached to Perry's costume and she lies on a leaf, while dancers pour glitter on her from oversized salt and pepper shakers. The performance samples a portion of Janet Jackson's song "What Have You Done for Me Lately", which Perry performs with her backing singers. Perry exits the stage for a short time, while multi-colored planets descend from the venue's ceiling, and Perry emerges from a planet wearing a crystal-covered gown and wig. She sings "Thinking of You" / "Wide Awake" with an acoustic guitar as the planet she is sitting on flies around the venue. After performing "Save as Draft", Perry then moves to the Drop Zone, an area of the stage in which she is closest to fans. She invites a fan to join her on stage before singing "Power", at the end of which Perry is given angel wings.
The next segment of the show is introduced with a video interlude showing Perry wearing a motorcycle helmet. She appears from behind the stage riding a motorcycle and wearing a striped blue outfit, and performs "Hey Hey Hey", followed by "Part of Me". A huge basketball net appears on stage for "Swish Swish", which Perry sings alongside dancers holding inflatable basketballs. Perry interrupts the song to hold an impromptu basketball game, during which she competes against a father of a young attendee from the audience. She finishes the song joined by her dancers on trampolines. "Roar" is performed next, while a large lion's head emerges from behind the screen. Perry returns to sing "Firework" for the encore, which she performs wearing a sequined purple gown on top of a hand that rises from beneath the stage. Fireworks erupt on stage during the number. At the end of the song, the hand closes up with Perry inside as she ends the show.
Critical reception
Jon Caramanica from The New York Times said that the show was "game, wacky, mildly overcompensatory" with an "impeccable" set design, although he thought that the "new material" Perry performed was "among her weakest". He praised Perry's onstage charisma and despite he thought that her live vocals were often "low in the mix", he complimented the acoustic performances when the singer sat down and played on a guitar. Jeffrey Lee Puckett from USA Today praised the visuals of the show, including the "non-stop blur of explosive lights, video, huge puppets, dancers, giant basketball goals, confetti and robot Venus flytraps". However, he felt the setlist was a "less successful" aspect of the show, adding "Perry's new songs from her Witness album don't have the sharpness or pop exuberance of her older material, instead just kind of sliding into a generic beat-per-minute muddle." Jordan Zivitz from the Montreal Gazette said that "extra production time could have been used to adjust some visuals whose juxtapositions and statements were as confused as they were colourful", but complimented Perry's stage presence, adding that "the energy was certainly there, with the singer staying in constant motion on a multi-stage set for two hours."
Commercial performance
At the end of 2017, the tour placed at number 77 on Pollstars "2017 Year-End Top 100 Worldwide Tours" list, estimating that it grossed $28.1 million and that 266,300 people attended throughout the year. The tour is the third best-selling female tour in the United Kingdom in 2018, according to StubHub, only behind Taylor Swift and Britney Spears.
Set list
This set list is from the show on January 19, 2018, in Glendale. It is not intended to represent all concerts for the tour.
"Witness" / "Roulette"
"Dark Horse"
"Chained to the Rhythm"
"Act My Age" (interlude)
"Teenage Dream"
"Hot n Cold" / "Last Friday Night (T.G.I.F.)"
"California Gurls"
"I Kissed a Girl"
"Déjà Vu"
"Tsunami"
"E.T."
"Bon Appétit" (contains elements of "What Have You Done for Me Lately")
"Mind Maze" (interlude)
"Wide Awake"
"Thinking of You"
"Power"
"Part of Me"
"Swish Swish"
"Roar"
Encore
"Firework"
Notes
"Wide Awake" was not performed during the first shows. Moreover, on selected dates, Perry performed "Save as Draft" and "Hey Hey Hey" before "Power" and "Part of Me" respectively.
On October 6, 2017 in New York City, Perry performed an acoustic version of "Part of Me" before "Power", rather than later in the show as a dedication to her fans for coming after the 2017 Las Vegas shooting.
On selected dates in South America, Perry performed "Unconditionally" in place of "Thinking of You". Moreover, during the concert in Rio de Janeiro, Perry dedicated the performance of "Power" to Marielle Franco, who died the week before, and called Franco's relatives onstage to express her condolences.
From the first show in Saitama onwards, Perry performed "Into Me You See" in place of "Thinking of You".
On selected dates, Perry performed "Pendulum" before "Firework", as part of the encore.
On June 22, 2018 in Manchester, Perry performed "By the Grace of God" in place of "Into Me You See" as a dedication to the city due to the Manchester Arena bombing. She also dedicated "Firework" to the 22 victims of the attack.
On June 28, 2018 in Barcelona, Perry performed a cover of "One of Us" by Joan Osborne in place of "Into Me You See".
Tour dates
Cancelled shows
References
Notes
Citations
External links
#WitnessTheFuture with Boys and Girls Clubs of America at KatyPerry.com
Katy Perry concert tours
2017 concert tours
2018 concert tours
Concert tours of North America
Concert tours of the United States
Concert tours of Canada
Concert tours of Europe
Concert tours of the United Kingdom
Concert tours of France
Concert tours of Germany
Concert tours of Oceania
Concert tours of Australia
Concert tours of New Zealand
|
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{"url":"https:\/\/devdude.me\/blog\/section\/physics","text":"Blog Section: Physics\n\nI will usually be on here blogging about Particle Physics, Quantum Physics and the Standard Model. There will sometimes be one-offs here and there as well that are more about Physics in general.\n\nStay awhile and listen!\n\nA Brief Post\n\nWelcome back to another post about the Maxwell Relations. This will be a brief post where I will present a very useful mnemonic for remembering the Maxwell Relations that certainly proved quite helpful during my studies of thermodynamics.\n\nThis post follows directly (but will be much less mathematical) from Read more\n\nDate Posted: 06\/06\/2018\n\nWelcome back to Waves!\n\nI've not put up a post about waves in a while, but I'm back to writing and have a few more posts in this series planned for the next few weeks! If you're new to the series, feel free to start at Waves I, which discusses the wave equation.\n\nThis part will not be following on directly from Read more\n\nIntro and Welcome back to the physics blog!\n\nIt has been a good bit of time since I posted the prelude article to this, so it's about time I write this! Welcome back!!\n\nIn this blog, I will be deriving Maxwell's relations of thermodynamic potentials. These are a set of relations which are useful because they allow us to change certain quantities, which are often hard to... Read more\n\nSomething new!\n\nI had gotten a bit burned out after writing a few posts just about wave physics, so I thought I'd start exploring and writing about something else.\n\nThis post and the next one are inspired (and really sourced) directly from what I'm currently studying in my third year university physics course. As usual, I will be giving my take on things and explaining... Read more\n\nDate Posted: 13\/02\/2018\n\nIntro\n\nIn this brief part of the waves series, we will be defining the concept of impedance of a string. Impedance is an important concept in many physical systems, as it will usually impact how much energy you are able to transmit across certain boundaries. Quite often in physics (and other fields), we're trying to solve problems where we need to do some sort of... Read more\n\nIntro\n\nIn transmission systems, we are often interested in the amount of power that we can transmit given particular conditions. So it would be helpful to derive an expression for the power in terms of the physical conditions of your system. In our case, we will be continuing on our wave on a string discussion, where the physical properties are the tension, $T$, the linear density... Read more\n\nWave Energy\n\nThis post carries on from the discussion in the previous two: Waves I and Waves II. It's recommended to have a look at those if you haven't already.\n\nAny wave on a string (and any in general) will transport energy. As in all physical situations, we can... Read more\n\nWelcome back\n\nThis will be a very brief post, continiuing directly from the previous one, where I will be showing that the speed of waves propagating on a string is given by the relation:\n\n$$c=\\sqrt{\\frac{T}\\rho}$$\n\nIn the the last post, we showed that the wave... Read more\n\nA new series!\n\nWelcome to the first part of a new series about the physics of waves I'm starting. In my post about blog plans for 2018, I mentioned that I want to start writing about physics topics in a more mathematically rigorous manner. This is the start of... Read more\n\nLet's go on!\n\nIn the first part of this series, I introduced the Standard Model and gave a brief introduction about fermions, which are the particles that make up everything we can see, and bosons, which are the particles that mediate the fundamantal forces between these particles.\n\nIn most early particle physics... Read more\n\nWhy is this a prelude?\n\nI was in the final stages of writing a blog post about particle interactions and Feynman Diagrams when I decided to pause for a bit and write this instead. I had structured that article such that I present the fundamental interactions described by the Standard Model in detail and then I would give an introduction to Feynman diagrams and provide... Read more\n\nWhy am I posting this?\n\nI have recieved some brilliant constructive feedback about my previous article, Observers affecting reality? The Double Slit Experiment, and since it has been pointed out that I have misinterpreted some of the results from performing variations of the... Read more\n\nIntroduction\n\nThe modern double slit experiment demonstrates one of the weirdest, and perhaps fundamentally the only weird, phenomenon about quantum mechanics. Do observers affect reality? Are particles \"conscious\" of what's happening around them? In this article, I'll be exploring these concepts, starting off by talking about waves, Young's original double slit experiment... Read more\n\nDate Posted: 30\/04\/2017","date":"2020-04-02 16:03:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5058057904243469, \"perplexity\": 663.5723599166347}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370506988.10\/warc\/CC-MAIN-20200402143006-20200402173006-00464.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathoverflow.net\/questions\/109226\/hahn-banach-theorem-with-real-extended-valued-function\/109231","text":"# Hahn-Banach theorem with real extended valued function\n\nHello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem:\n\nLet V be a vector space and let $p:V\\rightarrow \\mathbb{R}$ be any convex function. Let $W$ be a vector subspace of $V$ and let $L:W\\rightarrow \\mathbb{R}$ be a linear functional dominated by $p$ on $W$. Then there is a (not generally unique) linear extension $\\hat{L}$ of $L$ to $V$ that is dominated by $p$ on $V$. Furthermore $\\hat{L}_{|U}=L$.\n\nDoes the theorem still hold when $p:V\\rightarrow(-\\infty,+\\infty]$ ? Is someone able to give me a proof or to provide a counter-example that show that the theorem does not hold? And if it does not hold, it is possible to add some conditions that make it still true?\n\nTo put it in another way, is the same true if we further relax the hypothesis on $p$ and allow it to be real extended with nontrivial domain, i.e. $\\lbrace x\\in V: p(x) \\in \\mathbb{R} \\rbrace \\neq \\emptyset$ ?\n\nThanks to everyone in advance for helping me.\n\n-\n\"And if it does not hold, it is possible to add some conditions that make it still true?\" -- of course it is. But it would help if you thought more, and then told us, what kind of extra conditions you are prepared to impose. \u2013\u00a0Yemon Choi Oct 9 '12 at 22:27\n\nThe Hahn-Banach theorem is wrong for extended real $p$: For $V=\\mathbb R^2$ let $p$ be the Minkowski functional of $A= \\mathbb R \\times (0,\\infty)$ (so that $p(x,y)=0$ if $y>0$ and $p(x,y)=\\infty$ if $y\\le 0$), $W= \\mathbb R \\times \\lbrace 0 \\rbrace$, and $L: W\\to\\mathbb R$ defined by $L(x,0)=x$.\nThen $L$ is $p$-dominated but there is no $p$-dominated linear extension.","date":"2016-02-13 19:59:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8776278495788574, \"perplexity\": 89.42209991646601}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-07\/segments\/1454701167599.48\/warc\/CC-MAIN-20160205193927-00036-ip-10-236-182-209.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
For all terms and definitions, not defined specifically in this paper, we refer to \cite{BM}, \cite{FH} and \cite{DBW}. Unless mentioned otherwise, all graphs considered here are simple, finite and have no isolated vertices.
The sum set of two sets $A$ and $B$, denoted $A+B$, is the set defined by $A + B = \{a+b: a \in A, b \in B\}$. If either $A$ or $B$ is countably infinite, then their sum set will also be countably infinite. Hence, all sets we consider in this study are finite sets. The cardinality of a set $A$ is denoted by $|A|$. The power set of a set $A$ is denoted by $\mathcal{P}(A)$.
Let $\mathbb{N}_0$ denote the set of all non-negative integers. An {\em integer additive set-indexer} (IASI, in short) of a graph $G$ is defined in \cite{GA} as an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective.
A \textit{weak IASI} $f$ is (see \cite{GS1}) an IASI such that $|f^+(uv)|= \max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A weak IASI $f$ is said to be {\em weakly uniform IASI} if $|f^+(uv)|=k$, for all $u,v\in V(G)$ and for some positive integer $k$. A graph which admits a weak IASI may be called a {\em weak IASI graph}.
The following result is a necessary and sufficient condition for a given graph to admit a weak IASI.
\begin{lem}
\cite{GS1} A graph $G$ admits a weak integer additive set-indexer if and only if every edge of $G$ has at least one mono-indexed end vertex.
\end{lem}
The following definitions are made in \cite{GS3}. The cardinality of the labeling set of an element (vertex or edge) of a graph $G$ is called the {\em set-indexing number} of that element. An element (a vertex or an edge) of graph which has the set-indexing number $1$ is called a {\em mono-indexed element} of that graph. The {\em sparing number} of a graph $G$ is defined to be the minimum number of mono-indexed edges required for $G$ to admit a weak IASI and is denoted by $\varphi(G)$.
\noindent Certain Studies about weak IASIs have been done already and the following are some major results about weak IASI graphs relevant in this study.
\begin{thm}\label{T-WSG}
\cite{GS3} A subgraph of weak IASI graph is also a weak IASI graph.
\end{thm}
\begin{thm}\label{T-WUC}
\cite{GS3} A graph $G$ admits a weak IASI if and only if $G$ is bipartite or it has at least one mono-indexed edge. Also, the sparing number of a bipartite graph $G$ is $0$.
\end{thm}
\begin{thm}\label{T-WUOC}
\cite{GS3} Let $C_n$ be a cycle of length $n$ which admits a weak IASI, for a positive integer $n$. Then, $C_n$ has an odd number of mono-indexed edges when it is an odd cycle and has even number of mono-indexed edges, when it is an even cycle. An odd cycle $C_n$ has a weak IASI if and only if it has at least one mono-indexed edge.
\end{thm}
\begin{thm}\label{T-WUG}
\cite{GS4} The graph $G_1\cup G_2$ admits a weak IASI if and only if both $G_1$ and $G_2$ are weak IASI graphs. More over, $\varphi(G_1 \cup G_2)=\varphi(G_1)+\varphi(G_2)-\varphi(G_1 \cap G_2)$.
\end{thm}
\begin{thm}\label{T-WKN}
\cite{GS3} A complete graph can have at most one vertex that is not mono-indexed. Also, the sparing number of a complete graph $K_n$ is $\frac{1}{2}(n-1)(n-2)$.
\end{thm}
The admissibility of weak IASI by certain graph products and their sparing numbers have been studied in \cite{CGS1}, \cite{CGS2} and \cite{GS7}. In this paper, our intention is to study about the admissibility of weak IASI by a particular product, called edge corona, of two given graphs and estimate the corresponding sparing number.
\section{The Sparing Number of Edge Corona of Graphs}
\noindent Let us first recall the definition of the edge corona of two graphs.
\begin{defn}\label{D-5.1}{\rm
\cite{HS} Let $G_1$ be a graph with $n_1$ vertices and $m_1$ edges and $G_2$ be a graph with $n_2$ vertices and $m_2$ edges. Then, the {\em edge corona} of $G_1$ and $G_2$, denoted by $G_1\diamond G_2$, is the graph obtained by taking $m_1$ copies of $G_2$ and then joining the end vertices of $i$-th edge of $G_1$ to every vertex in the $i$-th copy of $G_2$.}
\end{defn}
Figure \ref{fig:G-ECor} is an example for the graph which is the edge corona of the cycles $C_5$ and $C_3$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{G-ECor.jpg}
\caption{The edge corona $C_5\diamond C_3$.}
\label{fig:G-ECor}
\end{figure}
The weak IASIs of $G_1$ and $G_2$ may not induce a weak IASI for $G_1\diamond G_2$. Hence, we need to define an IASI independently for a graph product.
We say that a graph $G$ is said to be a \textit{$1$-uniform graph} if the set-labels of all elements (vertices and edges) of $G$ are singleton sets. By the term an \textit{integral multiple of a set $A$}, we mean the set obtained by multiplying every element of $A$ by a same integer.
The following theorem establishes a necessary condition for the edge corona of two weak IASI graphs to admit a weak IASI.
\begin{thm}\label{T-SNECG1}
For two given graphs $G_1$ and $G_2$, if $G_1\diamond G_2$ admits a weak IASI, then either $G_1$ is $1$-uniform or $m_1-m_1'$ copies of $G_2$ are $1$-uniform, where $m_1'$ is the number of mono-indexed edges in $G_1$.
\end{thm}
\begin{proof}
Let $G_1$ be a graph on $n_1$ vertices and and $m_1$ edges and $G_2$ be a graph on $n_2$ vertices and and $m_2$ edges. Let $V(G_1)=\{v_1,v_2,\ldots v_{n_1}\}$ and $V(G_2)=\{u_1,u_2,\ldots, u_{n_2}\}$ be the vertex sets and $E(G_1)=\{e_1,e_2,\ldots, e_{m_1}\}$ and $E(G_2)=\{e'_1,e'_2,\ldots, e'_{m_2}\}$ be the edge sets of $G_1$ and $G_2$ respectively. Let $G_{2,j}$ be the $j$-th copy of $G_2$ corresponding to the $j$-th edge $e_j=v_rv_s$ of $G_1$ in $G_1\diamond G_2$ and $V(G_{2,j})=\{u_{1j},u_{2j},\ldots,u_{n_2j} \}$. Then, the subgraph of $G_1\diamond G_2$ induced by the vertices $\{v_r,v_s, u_{kj}, u_{lj}\}$ is the complete graph $K_4$, for any two adjacent vertices $u_{kj}$ and $u_{lj}$ in $G_{2,j}$. That is, all edges of $G_{2,i}$ are the edges of different complete graphs $K_4$ in $G_1\diamond G_2$, all of these complete graphs have the common edge $e_j=v_rv_s$.
First assume that $G_1\diamond G_2$ admits a weak IASI. Then, we have to consider the following two cases.
\noindent {Case-1:} Assume that $G_1$ is not $1$-uniform. Then, $G_1$ will have some elements which are not mono-indexed. Without loss of generality, assume that the edge $e_j$ is not mono-indexed. Then either $v_r$ or $v_s$ must have a non-singleton set-label. Let $v_r$ be the vertex that is not mono-indexed. Then, by Theorem \ref{T-WKN}, no other vertex $v_{lj}$ can have a non-singleton set-label. Therefore, the copy $G_{2,j}$ is $1$-uniform. This argument is valid for the copies of $G_2$ corresponding to all edges of $G$ that are not mono-indexed. Therefore, at least $m_1-m_1'$ copies of $G_2$ must be $1$-uniform, where $m_1'$ is the number of mono-indexed edges in $G_1$.
\noindent {Case-2:} Assume that no copy of $G_2$ is $1$-uniform. Then, each copy $G_{2,j}$ of $G_2$ has at least one edge that is not mono-indexed. Let the edge $u_{kj}u_{lj}$ of $G_{2,j}$ has the non-singleton set-label. Then, by Theorem \ref{T-WKN}, the end vertices $v_r$ and $v_s$ of the the corresponding edge $e_j$ of $G_1$ can not have non-singleton set-label. Hence, as no copy of $G_2$ are $1$-uniform, no vertex of $G_1$ can have a non-singleton set-label. That is, $G_1$ is $1$-uniform.
\end{proof}
The converse of the theorem is also valid for with respect to the weak IASIs defined on $G_1$ and $G_2$. Let $f_1$ and $f_2$ the weak IASIs defined on $G_1$ and $G_2$, which need not be $1$-uniform. The vertices of the copies of $G_2$ corresponding to the non-mono-indexed edges of $G_1$ need to be re-labeled using distinct singleton sets and the vertices of the copies of $G_2$ corresponding to the mono-indexed edges of $G_1$ can be labeled by distinct integral multiples of the set-labels of the corresponding vertices of $G_2$. Clearly, this new labeling will be a weak IASI of $G_1\diamond G_2$. Hence, we have the following necessary and sufficient condition for the edge corona of two weak IASI graphs to admit a weak IASI.
\begin{thm}\label{T-SNECG2}
For given weak IASI graphs $G_1$ and $G_2$, $G_1\diamond G_2$ admits a weak IASI if and only if $m_1-m_1'$ copies of $G_2$ are $1$-uniform, where $m_1'$ is the number of mono-indexed edges in $G_1$.
\end{thm}
In view of Theorem \ref{T-SNECG2}, we can estimate the number of mono-indexed edges in the edge corona of two given graphs.
\begin{thm}
For given graphs $G_1$ and $G_2$, the number of mono-indexed edges in $G_1\diamond G_2$ is $m_1'(1+m_2'+2n_2')+(m_1-m_1')(m_2+n_2)$, where $m_i$ is the number of edges and $n_i$ is the the number of vertices of $G_i$ for $i=1,2$ and $m_i'$ is the number of mono-indexed edges and $n_i'$ is the the number of mono-indexed vertices of $G_i$ with respect to a weak IASI defined on $G_i$ for $i=1,2$.
\end{thm}
\begin{proof}
Let $G_1$ be a graph on $n_1$ vertices and $m_1$ edges and $G_2$ be a graph on $n_2$ vertices and $m_2$ edges. Let $f_1$ and $f_2$ be the weak IASIs defined on $G_1$ and $G_2$ respectively. Let $n_i'$ and $m_i'$ be the number of vertices and edges of $G_i$ that are mono-indexed under the weak IASI $f_i$ for $i=1,2$.
Let $G=G_1\diamond G_2$ be a weak IASI graph. Assume that $G_1$ is not $1$-uniform. Then, $G_1$ has some elements having non-singleton set-labels. Then, by Theorem \ref{T-SNECG1}, $m_1-m_1'$ copies of $G_2$ must be $1$-uniform. Let $\mathfrak{C}_1$ be the set of all $1$-uniform copies of $G_2$ in $G_1\diamond G_2$. Therefore, the members of $\mathfrak{C}_1$ contributes a total of $(m_1-m_1')m_2$ mono-indexed edges to $G_1\diamond G_2$.
In the remaining $m_1'$ copies of $G_2$, we can label the vertices by the distinct integral multiples of the set-labels of the corresponding vertices of $G_2$ with respect to $f_2$. Let $\mathfrak{C}_2$ be the collection of these copies of $G_2$. Then, each element in $\mathfrak{C}_2$ has $m_2'$ mono-indexed edges. Therefore, the elements of $\mathfrak{C}_2$ contributes a total of $m_1'm_2'$ mono-indexed edges.
It remains to determine the number of mono-indexed edges between $G_1$ and different copies of $G_2$. The mono-indexed vertex of every non-mono-indexed edge of $G_1$ is adjacent to all vertices of the corresponding copy of $G_2$, which is also $i$-uniform. The number of such mono-indexed edges is $(m_1-m_1')n_2$. Both end vertices of each mono-indexed edge of $G$ are adjacent to $n_2'$ mono-indexed vertices of the corresponding copies of $G_2$. The number of such mono-indexed edges is $2m_1'n_2'$.
Therefore, the total number of mono-indexed edges in $G_1\diamond G_2$ is $m_1'+(m_1-m_1')m_2+m_1'm_2'+(m_1-m_1')n_2+2m_1'n_2'=m_1'(1+m_2'+2n_2')+(m_1-m_1')(m_2+n_2)$.
\end{proof}
In view of Theorem \ref{T-SNECG1}, let us now proceed to discuss the sparing number of the edge corona of certain graphs. We shall first consider the edge corona of two path graphs.
\begin{thm}\label{T-EC2P}
Let $P_m$ and $P_n$ be two paths on $m$ and $n$ vertices respectively, for $m,n >1$. Then, the sparing number of the edge corona of $P_m$ and $P_n$ is
\begin{equation*}
\varphi(P_m\diamond P_n)=
\begin{cases}
\frac{1}{2}m(n+2)-1; ~~ n ~{\text is~ even}\\
\frac{1}{2}m(n+1)-1; ~~ n ~{\text is~ odd}.
\end{cases}
\end{equation*}
\end{thm}
\begin{proof}
Let $G=P_m\diamond P_n$. Assume that an internal vertex $v$ of $P_m$ has a non-singleton set-label. Then, the $2+2n$ edges incident on $v$ become non-mono-indexed. But, two copies of $P_n$ whose vertices are adjacent to $v$ become $1$-uniform and $(n-1)$ edges of each of these copies of $P_n$ become mono-indexed. More over, $2n-1$ edges between $P_m$ and each of these two copies of $P_2$ become mono-indexed if $n$ odd and $2n-1$ edges between $P_m$ and each of these two copies of $P_2$ become mono-indexed if $n$ even. Therefore, In both cases, we have more mono-indexed edges than when $G_1$ is $1$-uniform. Therefore, $G$ has minimum number of mono-indexed edges when $G_1$ is $1$-uniform.
If $P_m$ is $1$-uniform, each copy of $P_n$ can be labeled in an injective manner alternately by non-singleton sets and singleton sets. Therefore, no edges in these copies need to be mono-indexed. Then, each vertex of $P_m$ $\lfloor \frac{n}{2} \rfloor$ mono-indexed edges together with the mono-indexed vertices of the corresponding copy of $P_n$. Therefore, if $n$ is even, $G$ has $(m-1)+m.\frac{n}{2}= \frac{1}{2}m(n+2)-1$ mono-indexed edges and if $n$ is odd, $G$ has $(m-1)+m.\frac{n-1}{2}= \frac{1}{2}m(n+1)-1$ mono-indexed edges.
\end{proof}
The sparing number of the edge corona of two graphs in which one is a path and the other is a cycle has been determined in the following theorems.
\begin{thm}\label{T-ECPC}
Let $P_m$ be a path on $m$ vertices and and $C_n$ be a cycle on $n$ vertices, for $m>1$. Then, the sparing number of the edge corona of $P_m$ and $C_n$ is
\begin{equation*}
\varphi(P_m\diamond C_n)=
\begin{cases}
\frac{1}{2}m(n+2)-1; ~~ n ~{\text is~ even}\\
\frac{1}{2}m(n+5)-2; ~~ n ~{\text is~ odd}.
\end{cases}
\end{equation*}
\end{thm}
\begin{proof}
Let $G=P_m\diamond C_n$. As proved in Theorem \ref{T-EC2P}, $G$ has minimum number of mono-indexed edges when $P_m$ is $1$-uniform. Then, each copy of $C_n$ can be labeled in an injective manner alternately by singleton sets and non-singleton sets and hence no edges in these copies are mono-indexed. With respect to this labeling, each copy of $C_n$ contains $lceil\frac{n}{2}\rceil$ mono-indexed vertices and makes $lceil\frac{n}{2}\rceil$ mono-indexed edges with each vertex of $P_m$. Therefore, if $n$ is even, no copy of $C_n$ need to have a mono-indexed edge and hence $G$ has $(m-1)+m.\frac{n}{2}= \frac{1}{2}m(n+2)-1$ mono-indexed edges. If $n$ is odd, then each copy of $C_n$ must have a mono-indexed edge and hence $G$ has $2(m-1)+m.\frac{n+1}{2}= \frac{1}{2}m(n+5)-2$ mono-indexed edges.
\end{proof}
\begin{thm}\label{T-ECCP}
Let $C_m$ be a cycle on $m$ vertices and and $P_n$ be a path on $n$ vertices, for $n>1$. Then, the sparing number of the edge corona of $C_m$ and $P_n$ is
\begin{equation*}
\varphi(C_m\diamond P_n)=
\begin{cases}
\frac{1}{2}m(n+2); ~~ n ~{\text is~ even}\\
\frac{1}{2}m(n+1); ~~ n ~{\text is~ odd}.
\end{cases}
\end{equation*}
\end{thm}
\begin{proof}
Let $G=P_m\diamond C_n$. As we have already proved in Theorem \ref{T-EC2P}, $G$ has minimum number of mono-indexed edges when $C_m$ is $1$-uniform. Then, each copy of $P_n$ can be labeled alternately by non-singleton sets and singleton sets and no edges in them are mono-indexed. Also, each copy of $P_n$ contains $lfloor\frac{n}{2}\rfloor$ mono-indexed vertices and makes the same number of mono-indexed edges with each vertex of $C_m$. Therefore, if $n$ is even, $G$ has $m+m.\frac{n}{2}= \frac{1}{2}m(n+2)$ mono-indexed edges and if $n$ is odd, $G$ has $m+m.\frac{n-1}{2}= \frac{1}{2}m(n+1)$ mono-indexed edges.
\end{proof}
In the following result, we study the sparing number of the edge corona of two cycle graphs.
\begin{thm}\label{T-EC2C}
Let $C_m$ and $C_n$ be two cycles on $m$ and $n$ vertices respectively. Then, the sparing number of the edge corona of $C_m$ and $C_n$ is
\begin{equation*}
\varphi(C_m\diamond C_n)=
\begin{cases}
\frac{1}{2}m(n+2); ~~ n ~{\text is~ even}\\
\frac{1}{2}m(n+5); ~~ n ~{\text is~ odd}.
\end{cases}
\end{equation*}
\end{thm}
\begin{proof}
Let $G=P_m\diamond C_n$. Then, as we have stated in above theorems, $G$ has minimum number of mono-indexed edges when $C_m$ is $1$-uniform and we can label each copy of $C_n$ alternately by singleton sets and non-singleton sets. Hence, no edges in these copies will be mono-indexed. With respect to this labeling, each copy of $C_n$ contains $lceil\frac{n}{2}\rceil$ mono-indexed vertices and makes $lceil\frac{n}{2}\rceil$ mono-indexed edges with each vertex of $P_m$. Therefore, if $n$ is even, $G$ has $m+m.\frac{n}{2}= \frac{1}{2}m(n+2)$ mono-indexed edges. If $n$ is odd, then each copy of $C_n$ has one mono-indexed edge and hence $G$ has $2m+m.\frac{n+1}{2}= \frac{1}{2}m(n+5)$ mono-indexed edges.
\end{proof}
So far, we have discussed about the edge corona of certain regular graphs having same vertex degree $2$. Can we generalise this result to all regular graphs having same vertex degree? The following result provide a solution to this problem.
\begin{thm}\label{T-EC2G}
Let $G_1$ and $G_2$ be two $r$-regular graphs on $m$ and $n$ vertices respectively, for $m,n >1$. Then, the sparing number of the edge corona of $G_1$ and $G_2$ is $m[n'+r(1+\varphi_2)]$, where $n'$ is the minimum number of mono-indexed vertices required in $G_2$.
\end{thm}
\begin{proof}
Let $G=G_1\diamond G_2$. Assume that an internal vertex $v$ of $G_1$ has a non-singleton set-label. Then, the $r+2n$ edges incident on $v$ become non-mono-indexed. But, $r$ copies of $G_2$ whose vertices are adjacent to $v$ become $1$-uniform and $rn-\varphi_2$ more edges of each of these copies of $G_2$ become mono-indexed, where $\varphi_2$ is the mono-indexed edges in $G_2$. Let $n_1$ vertices having non-singleton set-labels in $G_2$ must be relabeled by singleton sets in these $r$ copies of $G_2$. Therefore, $rn_1$ edges between $G_1$ and each of these $r$ copies of $G_2$ become mono-indexed. The total number of new mono-indexed edges in $G$ is $r[rn-\varphi_2+rn_1]$. Therefore, in this case, we have more mono-indexed edges than when $G_1$ is $1$-uniform. Therefore, $G$ has minimum number of mono-indexed edges when $G_1$ is $1$-uniform.
If $G_1$ is $1$-uniform, vertices of each copy of $G_2$ can be labeled in an injective manner alternately by distinct integral multiples of the set-labels of the corresponding vertices of $G_2$. Then, the number of mono-indexed edges in each copy of $G_2$ is $varphi_2$. Then, the total number of mono-indexed edges in $G_1\diamond G_2$ is $rm+ rm\varphi_2 +mn'= m[n'+r(1+\varphi_2)]$, where $n'$ is the minimum number of mono-indexed vertices required in $G_2$.
\end{proof}
We can extend the above theorem for the edge corona of an $r$-regular graph $G_1$ and an $s$-regular graph $G_2$ , where $r<s$ as follows.
\begin{thm}\label{T-EC2G1}
Let $G_1$ be an $r$-regular graph on $m$ and $n$ vertices and $G_2$ be an $s$-regular graph on $n$ vertices, for $m,n >1$ and $r\le s$. Then, the sparing number of the edge corona of $G_1$ and $G_2$ is $m(n'+r(1+\varphi_2))$, where $n'$ is the minimum number of mono-indexed vertices required in $G_2$.
\end{thm}
\begin{proof}
Let $G=G_1\diamond G_2$. Assume that an internal vertex $v$ of $G_1$ has a non-singleton set-label. Then, as mentioned in the previous theorem, $r+2n$ edges incident on $v$ become non-mono-indexed. But, $r$ copies of $G_2$ corresponding to the edges incident on $v$ become $1$-uniform and hence $sn-\varphi_2$ more edges of each of these copies of $G_2$ become mono-indexed, where $\varphi_2$ is the mono-indexed edges in $G_2$. Moreover, $rn_1$ edges between $G_1$ and each of these $r$ copies of $G_2$ also become mono-indexed, where $n_1$ is the number of vertices having non-singleton set-labels in $G_2$. Hence, the number of new mono-indexed edges in $G$ is greater than the new non-mono-indexed edges in $G$. Therefore, in this case also, we have the minimum number of mono-indexed edges when $G_1$ is $1$-uniform.
If $G_1$ is $1$-uniform, vertices of each copy of $G_2$ can be labeled in an injective manner alternately by distinct integral multiples of the set-labels of the corresponding vertices of $G_2$. Then, the number of mono-indexed edges in each copy of $G_2$ is $varphi_2$. Hence, the total number of mono-indexed edges in $G$ is $rm+ m\varphi_2 +mn'= m[n'+r+\varphi_2]$, where $n'$ is the minimum number of mono-indexed vertices required in $G_2$.
\end{proof}
Another important problem in this area is about the edge corona of two graphs in which one graph is a complete graph. First consider the edge corona of a path $P_m$ and a complete graph $K_n$.
\begin{thm}
Let $P_m$ be a path on $m$ vertices and $K_n$ be a complete graph on $n$ vertices. Then, the sparing number of $P_m \diamond K_n$ is $\frac{1}{2}n(n+1)(m-1)$.
\end{thm}
\begin{proof}
Let $G=P_m \diamond K_n$. Then, $G$ can be considered as the one point union of $m-1$ complete graphs on $n+2$ vertices. Therefore, by Theorem \ref{T-WKN}, each $K_{n+2}$ has $\frac{1}{2}n(n+1)$ mono-indexed edges. Since each $K_{n+2}$ are edge disjoint, by Theorem \ref{T-WUG}, the total number of mono-indexed edges is $\frac{1}{2}n(n+1)(m-1)$.
\end{proof}
\noindent Next, let us consider the edge corona of a cycle $C_m$ and a complete graph $K_n$.
\begin{thm}
Let $C_m$ be a cycle on $m$ vertices and $K_n$ be a complete graph on $n$ vertices. Then, the sparing number of $C_m \diamond K_n$ is $\frac{1}{2}mn(n+1)$.
\end{thm}
\begin{proof}
Let $G=C_m \diamond K_n$. Then, $G$ can be considered as the one point union of $m$ complete graphs $K_{n+2}$ and by Theorem \ref{T-WKN}, each of these $K_{n+2}$ has $\frac{1}{2}n(n+1)$ mono-indexed edges. Since each $K_{n+2}$ are edge disjoint, by Theorem \ref{T-WUG}, the total number of mono-indexed edges is $\frac{1}{2}mn(n+1)$.
\end{proof}
The above two results can be generalised for the edge corona of an $r$-regular graph $G$ on $m$ vertices and a complete graph $K_n$ as follows.
\begin{thm}
Let $G$ be an $r$-regular graph on $m$ vertices and $K_n$ be a complete graph on $n$ vertices, where $r\le n-1$. Then, the sparing number of $G \diamond K_n$ is $\frac{1}{4}rmn(n+1)$.
\end{thm}
\begin{proof}
Since $G$ is an $r$-regular graph on $m$ vertices, then the number of edges in $G$is $\frac{1}{2}rm$. Then, $G\diamond K_n$ can be considered as a one point union of $\frac{1}{2}rm$ complete graphs on $n+2$ vertices. Then, the total number of mono-indexed edges in $G\diamond K_n$ is $\frac{1}{2}rm. \frac{1}{2}n(n+1)= \frac{1}{4}rmn(n+1)$.
\end{proof}
\section{Conclusion}
In this paper, we have discussed about the sparing number of the edge corona of certain graphs. Some problems in this area are still open. For an $r$-regular graph $G_1$ and an $s$-regular graph $G_2$, with $r>s$, estimation of the sparing number of their edge corona is very complex. For some values $r$ and $s$, we get minimum number of mono-indexed edges when $G_1$ is $1$-uniform and for some other values of $r$ and $s$, we have minimum number of mono-indexed edges when $G_1$ is not $1$-uniform. Hence, determining the sparing number of $G_1\diamond G_2$ in such a situation still remains as an open problem.
In this paper, we have not addressed the problem of determining the sparing number of the edge corona of two graphs in which the first graph is a complete graph. The case when both the graphs are complete graphs are also not attempted.
For two arbitrary graphs, determining the sparing number of their edge corona is more complicated. The uncertainty in the adjacency and incidence pattern of arbitrary graphs makes this study complex. Hence, determining the sparing number of $G_1\diamond G_2$ for arbitrary graphs $G_1$ and $G_2$ is also an open problem.
The problems related to verifying the admissibility of weak IASIs by other graph products of two arbitrary graphs and determining the corresponding sparing numbers are also open.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,406
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\section{Introduction}\label{intro}
It has been known for decades that the broad-band energy spectra of black hole X-ray binaries (BHXBs), which harbour accreting stellar-mass black hole candidates (hereafter, we will drop `candidate'), can be broadly divided into optically thick and optically thin spectral components \citep{Remillard, Done2007, gilfanov2010}. The optically thick black body-like component is well described as originating from a geometrically thin, optically thick accretion disc \citep{Sunyaev73} and it typically has a colour temperature in the range of $kT_{\rm bb}\sim 0.1 - 1$ keV \citep{Remillard, gilfanov2010} ($k$ is the Boltzmann's constant). The optically thin component exhibits roughly a power law shape from a few keV to $\sim$ hundred keV followed by a roughly exponential cut-off \citep{Syunyaev91, Remillard,gilfanov2010}, and is produced due to the Compton up-scattering of seed photons by a hot optically thin electron cloud of temperature $kT_{\rm e}$ and optical depth $\tau$, located somewhere near the black hole \citep{Sunyaev79,Sunyaev80}. Depending upon the relative dominance of these two components, the X-ray emission states of BHXBs can be broadly divided into two classes: soft (or high) state, when the thermal disc component is dominant and hard (or low) state, when the hard Comptonised component is dominant \citep{Remillard,gilfanov2010}. The X-ray spectrum of BHXBs also exhibits reflection features, which are generated as a fraction of the Comtonised photons get scattered or re-emitted into the line of sight by the accretion disc. The most prominent features of this spectral component are an iron K$\alpha$ emission line near $6.4$ keV, formed due to fluorescence, and a Compton reflection hump peaked around $30$ keV and produced due to the inelastic scattering of the Comptonised photons by free electrons in the disc \citep{Basko1973, Ross2005}.
A subclass of accreting neutron stars, with low magnetic field ($B \sim 10^{8-9}$~G), also exhibits the above mentioned spectral properties \citep{barret2001, Done,gilfanov2010}. These are the atoll sources (including millisecond pulsars), which show a transition between the hard `island' state and the soft `banana' state \citep{klis89}, equivalent to spectral state transitions in black holes. The neutron star radii are of the order of $\sim3R_S$ ($R_S=2GM/c^2$ is the Schwarzschild radius; $M$ is the mass of the compact object, and $c$ is the speed of light in vacuum), which is comparable to the radius of the last marginally stable orbit around a black hole \citep{Done2007}. Outside a few Schwarzschild radii, black holes and neutron stars have similar shapes of the gravitational potential well. As a consequence, one may expect that at the same values of $\dot{M}/\dot{M}_{\rm Edd}$ ($\dot{M}$ is the mass accretion rate and $\dot{M}_{\rm Edd}$ is the critical Eddington mass accretion rate), the physical structure of accretion flow around a black hole should be similar to that for a neutron star.
However, there exists a fundamental difference between a black hole and a neutron star: the lack of a physical surface in the case of the former, while the latter has a surface. Such a difference gives rise to prominent observable effects \citep{Syunyaev91} like thermonuclear bursts due to the unstable nuclear burning of the accreted material on the surface of neutron stars \citep{narayan2002}, or differences in the spectral evolution of neutron star and black hole transients \citep{Done}, or pulsation from the residual magnetic field \citep{Done2007}. On the other hand, a boundary or spreading layer \citep{Sunyaev88, Inogamov1999} is formed between the inner part of the accretion flow moving with the Keplerian velocity (or some fraction of it) and the surface of the more slowly rotating neutron star. In this layer, the accreting material is decelerated down to the velocity of the neutron star surface, and its kinetic energy is deposited \citep{gilfanov2014}. In Newtonian geometry, half of the gravitational potential energy of the accreting matter is released in the boundary layer (the other half being radiated away from the disc), while this fraction is even greater in General relativity \citep{Sibgatullin2000}. Thus, the X-ray emission from a neutron star has two constituents of comparable luminosity originating in the accretion disc and boundary layer.
It was shown using the Fourier resolved spectroscopy that at $\dot{M}/\dot{M}_{\rm Edd}>0.1$ the boundary layer emission spectrum is same in different sources, weakly depends on the mass accretion rate and can be approximately described with a Wien spectrum with a colour temperature of about $2.4$ keV \citep{Gilfanov}. Note that additional turbulence expected in the boundary layer, and the resulting variability, make the Fourier resolved spectroscopy a workable tool for neutron stars. For a black hole, on the contrary, the kinetic energy of the accreting material is advected inside the event horizon; therefore, such a luminous soft emission component is absent in black hole X-ray binaries.
In the hard spectral state of black holes and neutron stars, the main spectral formation mechanism is unsaturated Comptonisation \citep{Sunyaev79,Sunyaev80} of soft seed photons in the hot corona located in the vicinity and, presumably, around the compact object.
Additional soft X-ray photons emitted by the surface of neutron stars can interact with the hot electrons in the corona through Comptonisation, significantly changing its energy budget as compared with the black hole case, and leave an imprint on the X-ray spectra of neutron stars in the hard state \citep{Sunyaev89}. The shape of the Comptonised spectrum is determined by three quantities: the temperature ($kT_{\rm e}$) of the hot electron cloud or corona, the Compton $y$-parameter of the electron cloud and the temperature of seed soft photons, while the energy balance in the corona is characterized by the Compton amplification factor $A$ which can be derived from the above quantities \citep{Burke1}. The Compton $y$-parameter (defined as $4kT_e/m_e c^2\cdot Max(\tau,\tau^2)$ with $m_e$ as the rest mass of an electron) describes the average change in energy a collection of photons will suffer as it traverses the Comptonising region. On the other hand, the amplification factor $A$ is determined by the ratio of the energy deposited in the hot electrons to the luminosity of the soft seed photons. Thus, it should be possible to decode the signature of a physical surface or lack of it in the X-ray spectra by studying the distributions of $kT_{\rm e}$, $y$, and $A$ parameter values in the hard state for black hole and neutron star populations.
Motivated by these arguments, \citet{Burke1} analysed 59 spectra of 7 BH and 5 NS binaries. They found in their sample a dichotomy in the distribution of $y$-parameter and Compton amplification factor $A$ between black holes and neutron stars, however the electron temperature distributions, although different, spanned an almost similar range of values. In the follow-up paper \citep{Burke2}, analysing a significantly larger number of NS spectra, they found that marginalised one-dimensional distributions of $y$ and $A$ of NS systems may overlap with those for black holes, depending on the NS spin and critical Eddington ratio. In this paper, we investigate this further, aiming to formulate a comprehensive observational picture of the effect of the NS surface on spectral formation in X-ray binaries in the hard spectral state. To this end, we significantly increased the number of sources and spectra. We doubled the sample of objects to include 11 accreting stellar-mass black holes and 13 atoll neutron stars in the hard state and dramatically increased the number of observations included in our analysis, to nearly
$5000$. Overall, our strategy is similar to that used in \citet{Burke1}, albeit with some important modifications detailed below.
The paper is organized as follows. In section \ref{data}, we briefly describe the data reduction schemes, the criteria we set for selecting the \textit{RXTE} observations, and the spectral models we considered for fitting the data. We report our findings and discuss our results in section \ref{result}. We summarise our results and conclude in section \ref{summary}.
\section{Data Reduction and Analysis}\label{data}
We collect data ($\sim5000$ observations) for 11 accreting black holes and 13 accreting neutron stars (atoll sources and millisecond pulsars) from the archival \textit{RXTE} database. In the case of neutron stars, we do not consider Z sources \citep{klis89} in this study as we require sources to exhibit classic hard state (`island' state). We use data from both the Proportional Counter Array (PCA) and High-Energy X-ray Timing Experiment (HEXTE) onboard \textit{RXTE}, and those data are reduced according to the procedure mentioned in the \textit{RXTE} cookbook
\footnote{\label{foot:itas1}\url{https://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook_book.html}}. Our data analysis procedure follows, with some modifications, the approaches proposed in \cite{Burke1}.
\begin{table*}
{
\centering
\caption{Details of the sources used in this study. The list of sources (the upper ones are black holes and the lower ones are neutron stars) used in this work, along with the no. of observations analysed for each of them $(N_{\rm obs})$, their absorption column densities $(N_{H})$, distances $(D)$ and masses $(M)$ are mentioned in the table. The black hole SAX 1711.6-3808 does not have a well determined mass or distance. For this source, we used ad hoc values of $10M_{\odot}$ and $5\ \rm kpc$ respectively. We assume all the neutron stars have the mass of $1.4M_{\odot}$. References mentioned in the table are: (1) \citep{Steiner}, (2) \citep{Dunn}, (3) \citep{Gladstone}, (4) \citep{Burke1}, (5) \citep{Burke2}, (6) \citep{Munoz}, and (7) \citep{Nathalie}.}.
\medskip
\begin{tabular}{lllllll}
\hline
No. & Source & $N_{\rm obs}$ &\hspace{0.28cm} $N_{H}$ & Distance & Mass & References\\
& & & $(10^{22}\ \rm cm^{-2})$ &\ \rm Kpc & $M_{\odot}$\\
\hline
1 & Cyg X-1 & 169 &\hspace{0.28cm} 0.7 & 1.86 & 14.8 &\hspace{0.28cm} (1), (2)\\
2 & XTE J1752-223 & 52 &\hspace{0.28cm} 0.6 & 3.5 & 9.6 &\hspace{0.28cm} (1), (2)\\
3 & GX 339-4 & 68 &\hspace{0.28cm} 0.3 & 8.0 & 5.8 &\hspace{0.28cm} (1), (2)\\
4 & XTE J1550-564 & 24 &\hspace{0.28cm} 0.8 & 4.4 & 10.39 &\hspace{0.28cm} (1), (2)\\
5 & SWIFT J1753.5-0127 & 107 &\hspace{0.28cm} 0.15 & 5.0 & 10.0 &\hspace{0.28cm} (1), (2)\\
6 & GS 1354-64 & 4 &\hspace{0.28cm} 2.0 & 26.0 & 7.47 &\hspace{0.28cm} (1), (2)\\
7 & XTE J1650-500 & 2 &\hspace{0.28cm} 0.5 & 2.6 & 4.72 &\hspace{0.28cm} (1), (2)\\
8 & GRO J1655-40 & 6 &\hspace{0.28cm} 0.7 & 3.2 & 6.6 &\hspace{0.28cm} (1), (2)\\
9 & SAX 1711.6-3808 & 1 &\hspace{0.28cm} 2.8 & 5.0 & 10.0 &\hspace{0.28cm} (2)\\
10 & H 1743-322 & 1 &\hspace{0.28cm} 2.2 & 10.4 & 13.3 &\hspace{0.28cm} (1), (2)\\
11 & XTE J1748-288 & 6 &\hspace{0.28cm} 7.5 & 10.0 & 10.0 &\hspace{0.28cm} (1), (2)\\
\hline
1 & 4U 1636-536 & 44 &\hspace{0.28cm} 0.37 & 6.0 & &\hspace{0.28cm} (3), (4)\\
2 & Aql X-1 & 54 &\hspace{0.28cm} 0.28 & 5.2 & &\hspace{0.28cm} (4)\\
3 & 4U 1728-34 & 51 &\hspace{0.28cm} 1.24 & 4.6 & &\hspace{0.28cm} (4)\\
4 & GS 1826-238 & 57 &\hspace{0.28cm} 0.17 & 6.0 & &\hspace{0.28cm} (3), (5)\\
5 & 4U 1608-52 & 90 &\hspace{0.28cm} 1.6 & 3.3 & &\hspace{0.28cm} (4), (6)\\
6 & 4U 1724-30 & 45 &\hspace{0.28cm} 1.0 & 6.6 & &\hspace{0.28cm} (3)\\
7 & HETE J1900.1-2455 & 23 &\hspace{0.28cm} 0.2 & 4.7 & &\hspace{0.28cm} (7)\\
8 & 4U 0614+09 & 22 &\hspace{0.28cm} 0.45& 3.0 & &\hspace{0.28cm} (3), (5)\\
9 & 4U 1705-44 & 4 &\hspace{0.28cm} 0.67& 8.4 & &\hspace{0.28cm} (5)\\
10 & 4U 1820-303 & 1 &\hspace{0.28cm} 0.28 & 5.8 & &\hspace{0.28cm} (3)\\
11 & Ara X-1 & 4 &\hspace{0.28cm} 1.1 & 7.3 & &\hspace{0.28cm} (3), (5)\\
12 & SAX J1808.4-3658 & 10 &\hspace{0.28cm} 0.12& 3.15 & &\hspace{0.28cm} (3)\\
13 & XTE J1751-305 & 10 &\hspace{0.28cm} 1.1 & 8.0 & &\hspace{0.28cm} (3)\\
\hline
\label{tab1}
\end{tabular}\\
}
\end{table*}
Using \verb HEASOFT (version \verb 6.24), we first extract 16 s binned light curves in three energy bands: $4.00-18.50$, $6.00-7.50$ and $7.50-18.50$ keV from \textit{RXTE}/PCA standard 2 data for each observation. We define an ObsID \textit{hard} for which hardness ratio $H=I_{7.50-18.50}/I_{6.00-7.50}$ ($I_{7.50-18.50}$ and $I_{6.00-7.50}$ are the background uncorrected intensities in the energy bands $7.50-18.50$ keV and $6.00-7.50$ keV respectively) is $\geq 2$ for at least $60\%$ time bins and mean $H\geq2$ \citep{Burke1}. We also choose only those observations in our work for which the background uncorrected intensity $I_{4.00-18.50}>40$ counts/s for $90\%$ time bins for achieving good statistics in the spectral analyses. We exclusively use data from PCU2, as it was always turned on. For finding the good time intervals (GTI), we exclude those time periods for which the elevation angle is less than $10^{\circ}$, offset is greater than $0.02^{\circ}$, the time since the peak of the last South Atlantic Anomaly passage is less than 30 minutes, electron contamination is greater than $0.1$ and intervals of 600s after PCA breakdown events. We also identify and exclude data contaminated by flares, dips, or thermonuclear bursts while producing the GTI files. For obtaining the light curves in three energy bands, we convert these energy bands to absolute channel number bands \footnote{\label{foot:itas2}\url{https://heasarc.gsfc.nasa.gov/docs/xte/e-c_table.html}}, which depend upon the mission epoch.
We extract background corrected spectra (using the bright background model) for those observations, which pass the criteria mentioned above, from the right and left anode chains of top xenon layer of PCU2 using the same GTI files obtained previously. Using the ftools \verb grppha, we add 0.5 percent systematic uncertainty to each bin, and the spectra are re-binned such that each bin contains a minimum of 40 counts.
We obtain HEXTE source and background spectra using the ftools \verb hxtlcurv and create corresponding response files using \verb hxtrsp. We use data from both the clusters A and B up to December 2005, after which the cluster A has been staring in an on-source position. Thus, we consider only cluster B for the observations done afterward.
In order to achieve a reliable and quick model fitting in \verb XSPEC (version \verb 12.10.1), we consider only those observations for which the background corrected PCU2 counts $>35000$, HEXTE A counts $>8000$, and HEXTE B counts $>8000$.
As our goal is to study the properties of Comptonised emission in X-ray binaries, we fit the spectra with a Comptonisation model including a reflection continuum and an additional Gaussian component to represent the iron K-$\alpha$ line emission. Our spectral model also includes low energy absorption by the interstellar medium. To this end, we use the \verb XSPEC models \verb compPS \citep{Poutanen}, which also includes the Compton up-scattered photons reflected by the accretion disc, and \texttt{ gauss}.
\footnote{There is a number of advanced reflection models currently available for spectral fitting, such as relxill family of models \citep{Dauser2014,Garcia2014} which include accurate account for ionization effects and relativistic smearing of the reflection features. However, the rather high low energy boundary of the \textit{RXTE}/PCA data ($\approx 3-4$ keV) and its relatively coarse energy resolution do not permit to utilise advantages of these sophisticated models. On the other hand, the complexity of these models combined with above mentioned limitations of the data, often makes the fitting procedure unstable. For this reason we choose to keep the reflection prescription based on the \texttt{ireflect} model, implemented in \texttt{compPS}, which catches the main physical aspects of the reflection component.}
The model \verb phabs is used to describe the absorption column with its parameter $N_H$ being set to the Galactic value listed in the Table~\ref{tab1}. Thus, our total spectral model is \verb constant*(phabs*(gauss+compPS)) ~\citep{Burke1}. The multiplicative constant is used to take care of the calibration difference between the PCA and HEXTE instruments. We fit the PCA and HEXTE spectra over the energy range $3-20$ keV and $20-200$ keV respectively.
While fitting the spectra with the above model, we keep the electron temperature $kT_{\rm e}$, seed photon temperature $kT_{\rm bb}$ as emitted from a multicolour disc, Compton-$y$ parameter, reflection strength $R(=\Omega/2\pi)$ ($\Omega$ is the solid angle subtended by the disc as observed from the corona) and the normalisation as free parameters in \verb compPS. We assume Maxwellian electron distribution in the Comptonising medium with a spherical geometry and a binary inclination of $45^{\circ}$ \citep{Burke1}. We do not consider those observations for spectral analysis for which $\chi^2/\nu>1.5$. Visual inspection of these spectra showed that majority of them do not belong to the classical hard state. The list of sources studied in this work, along with the number of observations considered for each of them after applying all the screening criteria mentioned above, their hydrogen column densities, distances and masses are given in the Table~\ref{tab1}.
Several of our sources (e.g., XTE J$1748-288$) lie in the close vicinity of the Galactic centre. As \textit{RXTE} has a low spatial resolution of $1^{\circ}$, spectra of some of these sources can suffer appreciable contamination due to the Galactic ridge (GR) emission. Such an emission may result in softening the spectra along with a significant contribution to the iron K$\alpha$ line. We model $3-20$ keV spectrum of the GR emission by a power law with index $\Gamma=2.15$ and a zero width Gaussian line with peak energy of 6.6 keV such that an equivalent width of $0.8$ keV is obtained \citep{Rev2003}. We also fix the flux value in this energy band using the 3.5 $\mu$m near infrared (NIR)$-$GR X-ray flux relation, as mentioned in \citet{Rev2006}. Following \citet{Burke1}, we make use of \textit{COBE}/DIRBE zodiacal subtracted mission average map for measuring the 3.5 $\mu$m flux near the source location \citep{Bennet} and also subsequently make the correction for extinction using the map of interstellar HI gas \citep{Dickey} and the extinction law by \cite{Rieke}. The contribution of GR emission is found to be appreciable (flux $>10^{-11}\ \rm erg\ \rm cm^{-2}\ \rm s^{-1}$) for the black holes: XTE J$1752-223$, XTE J$1748-288$, SAX $1711.6-3808$ and H $1743-322$, and the neutron stars: 4U $1728-34$, 4U $1724-30$, XTE J$1751-305$ and Ara X$-1$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{kte_y_errbar}
\end{center}
\begin{center}
\includegraphics[width=0.45\textwidth]{kte_y_bh}
\includegraphics[width=0.45\textwidth]{kte_y_ns}
\end{center}
\caption{The distribution of electron temperature $kT_{\rm e}$ values and the $y$ parameter values for black holes and neutron stars. In the top panel, the distribution of these parameters for all the sources considered in this study are depicted. The distribution of data on the $y$ -- $kT_{\rm e}$ plane for multiple observations of different sources -- black holes (middle panel) and neutron stars (bottom panel) are also depicted here. The sources are colour coded, as explained in the legend. Typical error bars for black hole and neutron star systems are shown in the top panel. As one can see, different sources tend to occupy different sub-regions in the diagram; however taken collectively, black holes and neutron stars occupy well separated distinct regions on this plane (top panel). See section \ref{res1} for more details. \label{fig1}}
\end{figure}
\section{Results and Discussion}\label{result}
\subsection{Correlations between parameters of Comptonised spectra}\label{res1}
The best fit values of the Compton $y$-parameter and electron temperature $kT_{\rm e}$ are shown in Fig.\ref{fig1}. We find that the black holes (BHs) and neutron stars (NSs) occupy distinctly different regions in the $y-kT_{\rm e}$ plane, with the BHs tending to have larger $y$-parameter and higher electron temperature $kT_{\rm e}$.
As we see in Fig. \ref{fig1}, the boundary between BH and NS systems has a complex shape, which is why their marginalised one-dimensional distributions over $y$-parameter and $kT_{\rm e}$ overlap, especially for the latter (see Fig. \ref{fig5} and discussion in section \ref{res3}). However, when considered on the two-dimensional $y-kT_{\rm e}$ plane, there is virtually no overlap between the two types of compact object. For example, the electron temperature of the Comptonising media in NSs 4U $1608-52$ and 4U $0614+09$ reach values greater than $100$ keV, unlike the other NSs in our sample. However, their corresponding $y$-parameter values are in the $y\sim 0.7$ range, i.e., significantly less than that of BHs with a similar electron temperature. Similarly, although the $y$-parameter of the spectra for the NS GS $1826-238$ can reach values of $y\sim 1.2$, their corresponding electron temperatures are less than $20\ \rm keV$ (see Fig. \ref{fig1}). We note that the anomalous hardness of 4U $1608-52$ was noticed in \citet{Burke1}; in our analysis, due to a larger sample, it reveals itself yet more graphically. For the same reason, while \citet{Burke2} found 4U $0614+09$ as a quite normal NS system, in our sample, it shows the highest $kT_{\rm e}$ among NSs. (We note, however, that the two rightmost points for this source in the bottom panel in Fig. \ref{fig1} have statistical errors of $\sim 40$ keV.) This shows that the size of the sample is an important factor in a comprehensive sampling of the parameter space of Comptonisation in X-ray binaries.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{lx_y_new}
\end{center}
\begin{center}
\includegraphics[width=0.45\textwidth]{lx_y_bh_new}
\includegraphics[width=0.45\textwidth]{lx_y_ns_new}
\end{center}
\caption{The distribution of 3--200 keV luminosity and the $y$ parameter values for black holes and neutron stars. In the top panel, the distribution of these parameters for all the sources considered in this study are depicted. The two lower panels show respective distributions for black holes and neutron stars with the data for different sources colour coded, as explained in the legend. Note that the three panels have different axes scales. See section \ref{res1} for more details. \label{fig_lx}}
\end{figure}
Interestingly, different spectra of a given source tend to group together rather than uniformly fill the entire BH or NS domain on the $y-kT_{\rm e}$ plane (Fig. \ref{fig1}, two lower panels). Partly, this may be possibly explained by the fact that we fixed the binary inclination on the same fiducial value of $45\degr$, while due to asymmetries in the disk-corona configuration, there is some mild viewing angle dependence in the Comptonised spectra. However, this is unlikely to explain the full range of values in Fig. \ref{fig1}. The reason (or parameter responsible) for such behaviour is unclear and needs further investigation \citep[cf.][]{Gladstone, heil2015}. We note, however, that this parameter is not the mass accretion rate (as traced by X-ray luminosity) as it is confirmed with the plots in Fig.\ref{fig_lx}.
The spectral model includes reflected component accounting for the result of reprocessing of the Comptonised spectrum by the accretion disc and other optically thick media (if any) located in the vicinity of the Comptonising region. The amount of reprocessed emission in the total spectrum is characterised by the reflection strength $R=\Omega/2\pi$, where $\Omega$ is the solid angle subtended by the disc as observed from the corona. With the exception of a small number of spectra, the best fit values of $R$ are all smaller than unity, as it should be expected in commonly considered geometries of the accretion flow, where the main reprocessing site is the accretion disc.
It was earlier reported \citep{Gladstone} that atoll sources and millisecond pulsars exhibit some differences in their behaviour on the colour-colour diagram during the hard/soft transition, on basis of which they can be divided into two groups: the sources (e.g., 4U $1636-536$, 4U $1728-34$) which make a diagonal track on the colour-colour diagram and the sources (e.g., 4U $1608-52$, Aql X-1) which make a vertical track. Our sample includes NSs from each subclass, however, we do not find any difference in their Comptonisation parameters.
\subsection{Compton amplification factor}
\label{res2}
The Compton amplification factor is defined as a ratio of the luminosity of the Comptonised component to the luminosity of the soft seed photons. As a proxy to the former, we used \verb compPS model luminosity in the $3-200$ keV band, corresponding to the energy range where X-ray data was fit.
We calculate the seed photon flux as flux of the \verb compPS model, for each best-fit model setting $\rm cosIncl=-0.5$ and $\tau=10^{-4}$. With these parameters, \verb compPS model spectrum equals to the input spectrum of seed photons. Its flux is then computed with \verb XSPEC flux command in an energy range from 1 eV to 10 keV.
We note that due to the limited energy range used for spectral fitting, the so defined $A$ is some approximation to the true value of the Compton amplification factor.
The Compton amplification factor $A$ is not an independent parameter of the spectral fit and is derived from the best fit parameters. It characterizes the energy balance in the Comptonising region and is closely related to the parameters of the Comptonised spectrum, with larger $A$ corresponding to larger $y$ and flatter spectra \citep[e.g.][]{gilfanov2000}. As discussed in the following section, one should expect that in BH systems, $A$ is larger than in NS systems. The distribution of BH and NS systems in the $A-y$ and $A-kT_{\rm e}$ planes is shown in Fig.\ref{fig4} confirming this expectation.
It is interesting to note that the distribution of points on the $A-y$ plane is essentially one-dimensional. In the low-left corner of the plot (low $A$ and $y$), this is a consequence of some degeneracy between these parameters in the low $A$ regime (see, for example, section 2.8 in \citealt{gilfanov2010}). In the upper-right end of the dependence, however, the range of permitted values of $A$ for the given $y$ is broader than it is observed. Therefore, the narrowness of the distribution may, in principle, reflect some real non-trivial correlations between Comptonisation parameters in BHs. Further investigation of this behaviour is hampered by the limited low energy coverage of the data used in this work.
\begin{figure}
\begin{center}
\includegraphics[width=0.42\textwidth]{A_y_new}
\includegraphics[width=0.42\textwidth]{A_kte_new}
\caption{Relation between the Compton amplification factor and $y$-parameter (top panel) and $kT_{\rm e}$ (bottom panel) for black holes and neutron stars. See section \ref{res2} for more details.\label{fig4}.}
\end{center}
\end{figure}
\subsection{Comptonisation in BH and NS systems -- evidence for the BH event horizon and other implications}
\label{res3}
As one can see from Fig.~\ref{fig1} and \ref{fig4}, there is a clear dichotomy between BHs and NSs in the Comptonisation parameter space with virtually no overlap between the two types of compact object. BHs have larger values of $y$-parameter and their electron temperature distribution is shifted towards higher $kT_{\rm e}$ values. These results confirm and expand, with a much larger sample of sources and higher statistics, earlier findings of \citet{Burke1, Burke2}. An observational manifestation of the dichotomy in Comptonisation parameters of BHs and NSs is the fact that the hard state spectra of the NSs are generally softer than spectra of BHs, as it was initially pointed out in \citet{Syunyaev91} based on the experience of MIR-KVANT and GRANAT observations of Galactic compact sources.
The difference in Comptonisation parameters in BH and NS systems can be understood as a result of more efficient cooling of electrons near a NS due to the presence of the more abundant cooling agent -- soft photons produced by the NS surface. The NS emission is powered by the remaining kinetic energy of the accreting material decelerated upon reaching the slowly rotating NS, and may surpass other sources of seed photons for Comptonisation, such as the accretion disc. Such an additional source of soft radiation is not available for BHs, as in their case, the remaining kinetic energy of the accreting material is advected inside the event horizon, without being converted to radiation. The additional cooling for the Comptonising region present in the case of accretion onto an NS leads to a decrease of the Compton $y$-parameter, electron temperature, and Compton amplification factor, cf. Fig.\ref{fig1},\ref{fig4}.
The marginalised one-dimensional probability distributions of $kT_{\rm e}$, $y$-parameter, and $A$ are shown in Fig. \ref{fig5}. They all have apparently different shapes for BHs and NSs. To quantify this, we performed the Kolmogorov-Smirnov (KS) tests on BH and NS distributions for each parameter. We obtained the KS-statistic value (p-value) of $0.80$ ($p\approx1.1\times10^{-121}$), of $0.5$ ($p\approx8.1\times10^{-47}$) and $0.92$ ($p\approx2.2\times10^{-160}$) for $y$-parameter, $kT_{\rm e}$ and the amplification factor $A$. These results confirm the visual impression from Fig. \ref{fig5} that the BH and NS distributions differ with an extremely high statistical significance (e.g. for $y$ and $A$ it is equivalent to $> 20\sigma$ for a Gaussian distribution).
Because of the curved shape of the boundary between BHs and NSs in the $y-kT_{\rm e}$ plane, there is a considerable overlap between their $y$ and, especially $kT_{\rm e}$ distributions. Nevertheless, their mean values and quantiles are quite different for BHs and NSs. The mean of $kT_{\rm e}$ for our NS sample is $\approx47$ keV, and its 80\% quantile is $16-93$ keV, while BHs are characterised by the mean of $\approx80$ keV and $80\%$ quantile of $45-117$ keV. For Compton $y$-parameter, mean values are $\approx 0.83$ and $\approx 1.3$ for NSs and BHs, their 80\% quantiles : $0.64-1.09$ and $1.03-1.55$ respectively.
If one needed to devise a one-parameter test however, the Compton amplification factor $A$ would be the primary parameter discriminating between BHs and NSs (Fig. \ref{fig4},\ref{fig5}). Indeed, the $A$ distributions have mean values of $\approx 2.7$ and $\approx 5.8$ and 80\% quantiles of $2.08-3.32$ and $3.94-8.17$ for NSs and BHs respectively. This should have been expected, as by definition, $A= L_{\rm tot}/L_{\rm seed}$ where $L_{\rm tot}$ is the total X-ray luminosity of the source and $L_{\rm seed}$ is the luminosity of seed photons for Comptonisation. Following the notation of \citet{Burke1} (their section 5.2), one can write for an NS: $L_{\rm seed}=f_1\,L_{\rm disc}+f_2\,L_{\rm NS}$. Thus, at the same total luminosity $L_{\rm tot}$, in an NS X-ray binary $L_{\rm seed}$ should be larger due to the additional soft radiation $L_{\rm NS}$ from the surface of the NS, which drops the value of $A$. For BHs, having no hard surface, seed photons, in the simplest scenario, are generated by the accretion disc only, $L_{\rm seed}=f_1\,L_{\rm disc}$, and $A$ is correspondingly larger.
One can use these simple considerations to roughly estimate the fractions of the kinetic energy of accreting material released in the Comptonising corona and dumped onto the surface of the NS. Using the mean value of $A\approx 2.7$ determined for NSs in our sample and eq.(7) from \citet{Burke1} we obtain $W_{\rm corona}/W_{\rm NS}\approx 1.7$. Therefore about $\sim 2/3$ of the energy of accreting material is released in the Comptonising corona around the NS, with the remaining $\sim 1/3$ being released on the NS surface. These numbers are consistent with those derived in \citet{Burke1}, who obtained the corona fraction of $\sim 1/2-2/3$.
The general conclusions from both calculations are that (i) the corona is radiatively highly efficient and (ii) the NS makes significant or dominant contribution to the seed photon supply for Comptonisation.
\begin{figure}
\begin{center}
\includegraphics[width=0.42\textwidth]{density_kte_new}
\end{center}
\begin{center}
\includegraphics[width=0.42\textwidth]{density_y_new}
\includegraphics[width=0.42\textwidth]{den_a_new}
\caption{Marginalised probability density distributions of Comptonisation parameters. The marginalised probability density distributions of electron temperature $kT_{\rm e}$ (top panel), Compton $y$ parameter (middle) and Compton amplification factor $A$ (bottom) for black holes and neutron stars are depicted here. See section \ref{res3} for more details.\label{fig5}}
\end{center}
\end{figure}
\section{Summary and Conclusion}\label{summary}
A black hole lacks a hard surface and is confined by an invisible boundary, an event horizon, a definitive proof of which is a holy grail of modern physics and astronomy. A signature of this horizon could be identified, if the X-ray radiation from accreting stellar-mass black holes is compared with that from accreting low magnetised neutron stars, which have hard surfaces. Here, we investigate how the additional X-ray photons emanating from the surface/boundary layer of neutron stars can affect the Comptonisation process in the hard state, and how such interaction is manifested on the distribution of the properties of Comtonising components: the corona temperature $kT_{\rm e}$, the Compton $y$ parameter, and the Compton amplification factor $A$. We analyse the hard X-ray spectra from 11 accreting black holes and 13 atoll neutron stars as observed with \textit{RXTE}/PCA and \textit{RXTE}/HEXTE instruments. We find a clear dichotomy in the distributions of black holes and neutron stars on the $y-kT_{\rm e}$ plane (Fig. \ref{fig1}) and in the distributions of the Compton amplification factor (Fig. \ref{fig4}). The values of the Compton $y$-parameter and the amplification factor $A$, in general, take higher values for black holes than neutron stars. However, there is a significant overlap between the marginalised one-dimensional distributions of corona temperatures for black holes and neutron stars, the neutron stars, on average, have lower values of corona temperature than the black holes.
Thus, our study reveals a clear separation between Comptonisation characteristics of accreting black holes and neutron stars, which is a consequence of the lack of the hard surface in black holes. Our findings establish a new method of determining the nature of the compact object in X-ray binaries through the broad-band X-ray spectroscopy. Previously, several techniques for identifying stellar-mass black holes in X-ray binaries have been proposed, based on dynamical measurements of the mass of the accretor via binary system mass function determination \citep{Remillard} or measurement of characteristic time scales in the vicinity of the compact object through aperiodic variability of its X-ray emission \citep{mikej}, or through the shape of the correlation between the QPO frequencies and the power law slope of X-ray spectrum \citep{tit2006}. It was also demonstrated that black hole X-ray novae typically have a 1--2 orders of magnitude lower quiescent luminosity than neutron star transients \citep{narayan1997}, and it was proposed on theoretical grounds that black hole X-ray binaries never exhibit Type I X-ray bursts \citep{narayan2002}. The latter two suggestions pioneered the use of the signatures of the black hole event horizon for the compact object diagnostics. In this paper, we propose a method that can be employed for the compact object diagnostics given a single broad-band X-ray spectrum of a source in the hard state.
Although the first supermassive black hole has already been imaged with the Event Horizon Telescope using electromagnetic radiation \citep{Event}, such observations are still far from becoming a routine. Moreover, it may not be possible to image a stellar-mass black hole in the foreseeable future, because of its several orders of magnitude smaller angular size. Therefore, the proposed method should be useful to detect the signatures of event horizon of stellar-mass black holes using the observations of their electromagnetic emission. This would be particularly important to probe the regime of strong gravity, as stellar-mass black holes produce several orders of magnitude larger space-time curvature compared to those of supermassive black holes. Although merging stellar-mass black holes have been detected via their gravitational wave emission \citep{Ligo}, those are very short transient events and can be studied only once during the merger. Accreting black holes in low mass binaries, on the contrary, are stable, long-lived systems, for which the results can be refined in the future with better data and more advanced techniques.
\section*{Acknowledgements}
This research has made use of data, software, and/or web tools obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), a service of the Astrophysics Science Division at NASA/GSFC. This work was partially supported by the Russian Science Foundation, project \#19-12-00369. S. Banerjee is thankful to Lavneet Janagal, Dattaraj Dhuri and Prasanta Kumar Nayak for downloading a significant part of the data used in this work.
We thank the referee Prof. Chris Done for her insightful comments and suggestions.
\section*{Data Availability}
The observational data used in this paper are publicly available at HEASARC (\url{https://heasarc.gsfc.nasa.gov/}). Any additional information will be available upon reasonable request.
\bibliographystyle{mnras}
\nocite{*}
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/intermediate-algebra-6th-edition\/chapter-5-section-5-6-factoring-trinomials-exercise-set-page-304\/101","text":"## Intermediate Algebra (6th Edition)\n\n$(x^n+8)(x^n+2)$\nThe given expression can be written as: $\\\\(x^n)^2+10x^n+16$ The constant term of 16 has the factors 8 and 2, the sum of which, 10, is equal to the coefficient of the middle term Thus, the factored form of the given trinomial is: $\\\\=(x^n+8)(x^n+2)$","date":"2018-10-23 14:45:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6705548763275146, \"perplexity\": 267.0921138704976}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583516194.98\/warc\/CC-MAIN-20181023132213-20181023153713-00496.warc.gz\"}"}
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{"url":"https:\/\/cosmologyquestionoftheweek.blogspot.com\/2013_07_01_archive.html","text":"Wednesday, July 31, 2013\n\nPlanck's summer: varying electron masses\n\nfifth post by Youness in the \"Planck's summer\"-series:\n\nHow can we test whether the electron mass changes with time (as compared to the other particle masses)? Give a list of astrophysical or cosmological observations that are sensitive to the electron mass (from the early to the late Universe). Do you have an idea how to measure the present time variation of the electron mass in the lab?\n\nWednesday, July 24, 2013\n\nPlanck's summer: observation of the CNB\n\nfourth post in the \"Planck's summer\"-series by Youness\n\nWhen did the cosmic neutrino background form, why is its temperature lower than that of the CMB and how could an experiment to detect it look like?\n\nWednesday, July 17, 2013\n\nPlanck's summer: gauge independence of cosmological surveys\n\nthird post in the \"Planck's summer\"-series by Youness:\n\nPower spectra in linear perturbation theory depend on the gauge chosen. The differences are small on scales well within the horizon, but they grow as we go to very large scales. However, observers apparently measure galaxy positions unambiguously and calculate power spectra from this. How would you interpret their data on the largest scales?\n\nWednesday, July 10, 2013\n\nPlanck's summer: particle size matters!\n\nsecond post in the \"Planck's summer\"-series by Youness:\n\nLet us assume that dark matter is made of approximately collisionless particles evolving under the laws of gravity (just one type, for simplicity). Is there a minimum mass $m_\\text{min}$ these particles need to have in order to fit observational constraints? Conversely, is there an upper limit $m_\\text{max}$?\n\nWednesday, July 3, 2013\n\nPlanck's summer: CMB observations without CMB photons\n\nCQW presents \"Planck's summer\", a series of posts composed by recent graduate Youness!\n\nImagine a planet with an atmosphere that is, for some reason, completely opaque for low-energy photons such that a direct ground-based detection of the CMB is impossible. Furthermore, the inhabitants of the planet do not yet possess the technology to build spacecraft that leaves the atmosphere. \u00a0However, they are very good experimentalists on the ground. How could they test whether there is a CMB confirming a basic prediction of big bang cosmology?","date":"2017-09-25 18:46:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7394164800643921, \"perplexity\": 1475.3276022847488}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818693240.90\/warc\/CC-MAIN-20170925182814-20170925202814-00216.warc.gz\"}"}
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Iván Cuéllar Sacristán, conegut futbolísticament com a Pichu (nascut el 27 de maig de 1984 a Mèrida), és un futbolista que juga de porter al Sporting de Gijón.
Trajectòria
Va començar a jugar a les categories inferiors del Mérida UD, a la seva localitat natal, ingressant al planter de l'Atlètic de Madrid l'1 de juliol de 2001.
Va debutar amb el primer equip del conjunt matalasser el 20 de gener de 2005 a un partit de Copa del Rei davant el Lorca Deportiva CF. A primera divisió debutà el 29 de maig d'aquell any, a casa i davant el Getafe CF, amb un resultat d'empat a dos gols.
Va gaudir de molt poques oportunitats a l'equip madrileny, sobretot a la temporada 2005-06 on no va jugar cap partit ni a la lliga ni a la copa.
Durant la temporada 2006-07 va jugar set partits a la lliga, i dos a la copa, a causa de les absències del porter titular, Leo Franco.
La temporada 2007-08 va jugar cedit a la SD Eibar de Segona divisió.
En finalitzar el contracte amb l'Atlético de Madrid va signar per quatre temporades amb l'Sporting de Gijón a la temporada 2008-09.
Enllaços externs
Estadístiques a la Primera Divisió espanyola
Fitxa al web de l'Sporting
Estadístiques al web d'ESPN
Persones de Mèrida
Futbolistes de la província de Badajoz
Porters de futbol extremenys
Futbolistes de l'Atlètic de Madrid B
Futbolistes de l'Atlètic de Madrid de la dècada de 2000
Futbolistes de la SD Eibar
Futbolistes de l'Sporting de Gijón
Futbolistes del CD Leganés
Medallistes espanyols als Jocs Mediterranis
|
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"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/mathoverflow.net\/questions\/196635\/classifying-1-cycle-permutation-matrices","text":"# Classifying 1 cycle permutation matrices\n\nGiven a permutation matrix that is not full rank, is there a linear algebraic and corresponding algebraic criterion to tell if matrix contains more than one disjoint non-trivial cycle or exactly one non-trivial cycle?\n\nIs there a criteria to tell $X,Y$ have unique cycles of length $d_x,d_y$ respectively by looking at $X\\otimes Y$?\n\nExample: Consider $$\\begin{pmatrix} 0& 0& 0& 0& 1& 0& 0& 0\\\\ 1& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 1\\\\ 0& 1& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 1& 0& 0 \\end{pmatrix}$$\n\nThis matrix embeds 1st row->2nd row->6th row->8th row->5th row->1st row cycle (essentially a 5-cycle in $8\\times 8$ matrix). Hence we have a non-trivial $5$-cycle.\n\nIn above matrix, if (3,3),(4,4),(7,7) entry is 1, we still get a 5 cycle with 3 fixed points.\n\nOn other hand if (4,3),(3,4) entry is 1 and (7,7) entry is 0, we obtain disjoint 2 and 5 cycles with no fixed points (turning (7,7) to 1 gets you a fixed point).\n\nWhen previous example acts on left on \\begin{pmatrix} a1& a2& a3& a4& a5& a6& a7& a8\\\\ b1& b2& b3& b4& b5& b6& b7& b8\\\\ c1& c2& c3& c4& c5& c6& c7& c8\\\\ d1& d2& d3& d4& d5& d6& d7& d8\\\\ e1& e2& e3& e4& e5& e6& e7& e8\\\\ f1& f2& f3& f4& f5& f6& f7& f8\\\\ g1& g2& g3& g4& g5& g6& g7& g8\\\\ h1& h2& h3& h4& h5& h6& h7& h8 \\end{pmatrix}\n\nwe get \\begin{pmatrix} e1& e2& e3& e4& e5& e6& e7& e8\\\\ a1& a2& a3& a4& a5& a6& a7& a8\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ h1& h2& h3& h4& h5& h6& h7& h8\\\\ b1& b2& b3& b4& b5& b6& b7& b8\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ f1& f2& f3& f4& f5& f6& f7& f8 \\end{pmatrix}\n\nActing on left $5$ times gives\\begin{pmatrix} a1& a2& a3& a4& a5& a6& a7& a8\\\\ b1& b2& b3& b4& b5& b6& b7& b8\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ e1& e2& e3& e4& e5& e6& e7& e8\\\\ f1& f2& f3& f4& f5& f6& f7& f8\\\\ 0& 0& 0& 0& 0& 0& 0& 0\\\\ h1& h2& h3& h4& h5& h6& h7& h8 \\end{pmatrix}\n\n\u2022 You can just see if the characteristic polynomial is of the form $x^r(x^s - 1)$, where $r + s = n$. I think this criterion should be necessary and sufficient. \u2013\u00a0John Binder Feb 16 '15 at 1:45\n\u2022 so we cannot say anything linear algebraically? \u2013\u00a0Brout Feb 16 '15 at 1:52\n\u2022 @JohnBinder your first condition on the characteristic polynomial is not \"if and only if\", since it allows to have nontrivial blocks with the zero eigenvalue in the Jordan normal form... \u2013\u00a0Vladimir Dotsenko Feb 16 '15 at 8:18\n\u2022 IMHO such matrices are not even called permutational, to begin with. \u2013\u00a0Dima Pasechnik Feb 16 '15 at 10:52\n\u2022 @VladimirDotsenko I was operating under the assumption that all such permutation matrices were semisimple, from the statement of the problem, that the matrices in question were semisimple. I should have stated this to begin with, but if this assumption is relaxed then you are correct. \u2013\u00a0John Binder Feb 16 '15 at 13:31","date":"2019-04-22 13:05:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.986092746257782, \"perplexity\": 82.01097618045532}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578553595.61\/warc\/CC-MAIN-20190422115451-20190422141451-00096.warc.gz\"}"}
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## Shadow Command
## Dale Brown
This novel is dedicated to all who make the often difficult decision to do one simple thing: Go For It. When you see it happen, it's more exhilarating than a space launch, and twice as powerful.
## Contents
Cast of Characters
Weapons and Acronyms
Real-World News Excerpts
Prologue
"Stand by...ready...ready...begin climb, now," the ground controller...
Chapter One
"Okay, suckers, c'mon and poke your head out—just a little...
Chapter Two
The command module was the center of activity aboard Armstrong...
Chapter Three
"No bread, no peace! No bread, no peace!" the protesters...
Chapter Four
"Joining us live from Armstrong Space Station, orbiting two hundred...
Chapter Five
"It's ten times more boring than playing video games," Wayne...
Chapter Six
"Two minutes to re-entry initiation, crew," Major Jim Terranova announced.
Chapter Seven
"McLanahan here, secure."
Chapter Eight
"Frankly, Brit, I don't care what the Russians say," Senate...
Chapter Nine
"Here's the latest, guys, so listen up," the SEAL team...
Chapter Ten
The city of Mashhad—"City of Martyrs" in English—in northeastern Iran...
Epilogue
The young boy cast a fishing line into Lake Mojave...
Acknowledgments
Author's Note
About the Author
Books by Dale Brown
Credits
Copyright
About the Publisher
## CAST OF CHARACTERS
AMERICANS:
JOSEPH GARDNER, President of the United States
KEN T. PHOENIX, Vice President
CONRAD F. CARLYLE, President's National Security Adviser
MILLER H. TURNER, Secretary of Defense
GERALD VISTA, Director of National Intelligence
WALTER KORDUS, White House Chief of Staff
STACY ANNE BARBEAU, senior U.S. senator from Louisiana, Senate majority leader; Colleen Morna, her aide
GENERAL TAYLOR J. BAIN, USMC, chairman of the Joint Chiefs of Staff
GENERAL CHARLES A. HUFFMAN, Air Force chief of staff
AIR FORCE GENERAL BRADFORD CANNON, commander of U.S. Strategic Command (STRATCOM)
ARMY GENERAL KENNETH LEPERS, commander of U.S. Central Command (CENTCOM)
MAJOR GENERAL HAROLD BACKMAN, commander of the Fourteenth Air Force; also commander of Joint Functional Component Command-Space (JFCC-S) of U.S. Strategic Command
LIEUTENANT GENERAL PATRICK MCLANAHAN, commander of the High-Technology Aerospace Weapons Center (HAWC), Elliott AFB, Nevada
BRIGADIER GENERAL DAVID LUGER, deputy commander of HAWC
COLONEL MARTIN TEHAMA, incoming commander of HAWC
MAJOR GENERAL REBECCA FURNESS, commander of the First Air Battle Force (air operations), Battle Mountain Air Reserve Base (ARB), Nevada
BRIGADIER GENERAL DAREN MACE, Air Battle Force operations officer, 111th Bomb Wing commander, and EB-1C mission commander
MAJOR WAYNE MACOMBER, deputy commander (ground operations) of the First Air Battle Force, Battle Mountain Air Reserve Base, Nevada
MARINE CORPS MASTER SERGEANT CHRIS WOHL, NCOIC, First Air Battle Force
U.S. ARMY NATIONAL GUARD CAPTAIN CHARLIE TURLOCK, CID pilot
CAPTAIN Hunter "Boomer" NOBLE, XR-A9 Black Stallion spacecraft commander, Elliott Air Force Base, Groom Lake
U.S. NAVY LIEUTENANT COMMANDER LISETTE "FRENCHY" MOULAIN, XR-A9 spacecraft commander
U.S. MARINE CORPS MAJOR JIM TERRANOVA, XR-A9 mission commander
ANN PAGE, PH.D., former U.S. senator, astronaut, and space weapon engineer
AIR FORCE MASTER SERGEANT VALERIE "SEEKER" LUKAS, Armstrong Space Station sensor operator
IRANIANS:
GENERAL HESARAK AL-KAN BUZHAZI, leader of the Persian military coup
AZAR ASSIYEH QAGEV, heir presumptive of the Peacock Throne of Persia
LIEUTENANT COLONEL PARVIZ NAJAR AND MAJOR MARA SAIDI, Azar Qagev's aides-de-camp
COLONEL MOSTAFA RAHMATI, commander of the Fourth Infantry Brigade, Tehran-Mehrabad Airport
MAJOR QOLOM HADDAD, leader of Buzhazi's personal security team
MASOUD NOSHAHR, Lord High Chancellor of the Qagev royal court and marshal of the court's council of war
AYATOLLAH HASSAN MOHTAZ, supreme leader in exile of the Islamic Republic of Iran
RUSSIANS:
LEONID ZEVITIN, president of the Russian Federation
PETER ORLEV, president's chief of staff
ALEXANDRA HEDROV, minister of foreign affairs
IGOR TRUZNYEV, chief of the Federal Security Bureau
ANATOLI VLASOV, secretary of the Russian security council
MIKHAIL OSTENKOV, minister of national defense
GENERAL KUZMA FURZYENKO, Russian chief of the general staff
GENERAL NIKOLAI OSTANKO, chief of staff of the Russian army
GENERAL ANDREI DARZOV, chief of staff of the Russian air force
WOLFGANG ZYPRIES, German laser engineer working with the Russian air force
## WEAPONS AND ACRONYMS
9K89—small Russian surface-to-surface missile
ARB—Air Reserve Base
ATO—air tasking order
BDU-58 Meteor—precision-guided vehicle designed to protect payloads from the heat of re-entry through the atmosphere; can carry approximately 4,000 pounds
CIC—Combat Information Center
coonass—a person of Cajun ethnicity
E-4B—National Airborne Operations Center
E-6B Mercury—U.S. Navy airborne communications and command post aircraft
EB-1D—B-1 Lancer bomber modified as an unmanned long-range supersonic attack plane
ETE—estimated time en route
FAA Part 91—regulations governing private pilots and aircraft
FSB—Russian Federal Security Bureau, follow-on to the KGB
HAWC—High-Technology Aerospace Weapons Center
ICD—implantable cardioverter-defibrillator
Ilyushin—Russian inflight refueling tanker aircraft
MiG—Mikoyan-Gureyvich, Russian military aircraft maker
OSO—offensive systems officer
RQ-4 Global Hawk—high-altitude long-range unmanned reconnaissance aircraft
SAR—synthetic aperture radar; also search and rescue
Skybolt—space-based anti-ballistic missile laser
SPEAR—Self-Protection Electronically Agile Reaction network intrusion defense system
sun-synchronous—an Earth orbit on which a satellite passes over the same spot at the same time of day
Tupolev—twin-engine Russian jet bomber
USAFE—U.S. Air Forces in Europe
VFR—Visual Flight Rules
Vomit Comet—aircraft used to fly parabolic flights to simulate weightlessness
XAGM-279A SkySTREAK (Scramjet Tactical Rapid Employment Attack, or "Streaker")—air-launched hypersonic attack missile, 4,000 pounds, 12 feet long, 24 inches in diameter; uses a solid rocket motor to boost the missile to Mach 3, then switches to a JP-7 jet fuel and compressed atmospheric oxygen scramjet to cruise at Mach 10; inertial and precision GPS navigation; satellite datalink operator mid-course reprogramming; ballistic flight profile max range 600 miles; after accelerating to Mach 10, releases precision-guided warhead with millimeter-wave radar and imaging infrared terminal guidance with auto-target discrimination or satellite datalink remote operator target selection; no warhead; two can be carried aboard EB-1C Vampire bomber in aft bomb bay; four carried internally or four externally by EB-52 Megafortress; four carried internally by B-2 stealth bomber
XR-A9—single-stage to orbit "Black Stallion" spaceplane
## REAL-WORLD NEWS EXCERPTS
STRATFOR MORNING INTELLIGENCE BRIEF, 18 JANUARY 2007—1216 GMT—CHINA, UNITED STATES—U.S. intelligence agencies believe China destroyed the aging Feng Yun 1C polar orbit weather satellite in a successful anti-satellite (ASAT) weapons test Jan. 11, China Daily reported Jan. 18, citing an article to appear in the Jan. 22 issue of Aviation Week & Space Technology. U.S. intelligence agencies are still attempting to verify the ASAT test, which would signify that China has a major new military capability...
...The new cloud of debris orbiting the Earth is an indication of things to come should two space-faring nations face off in a conflict. Especially in the case of the United States, space-based assets have become too essential an operational tool to be ignored any longer in times of war.
STRATFOR DAILY INTELLIGENCE SUMMARY, 3 APRIL 2007—U.S./IRAN: U.S. attacks against Iran would not lead to a decisive military defeat of Tehran and would be a political mistake, Russia's Chief of the General Staff Gen. Yuri Baluyevsky said. He added that it is possible for the United States to damage Iran's military, but not to win a conflict outright.
STRATFOR INTELLIGENCE BRIEF, 7 SEPTEMBER 2007—Cooperation between the Russian Federal Security Service and Iran's Interior Ministry will enhance Iran's border security, First Deputy Director-General of Russian Federal Security and Border Services Viktor Shlyakhtin said, according to an IRNA report. Shlyakhtin is in Iran to inspect Iranian-Russian projects in areas of Iran's Sistan-Balochistan province that border Afghanistan and Pakistan.
RED OCTOBER: RUSSIA, IRAN, AND IRAQ—STRATFOR Geopolitical Intelligence Report, 17 September 2007—Copyright © Strategic Forecasting Inc.—...The Americans need the Russians not to provide fighter aircraft, modern command-and-control systems, or any of the other war-making systems that the Russians have been developing. Above all else, they want the Russians not to provide the Iranians any nuclear-linked technology.
Therefore, it is no accident that the Iranians claimed over the weekend that the Russians told them they would do precisely that.
...[Russian president Vladimir] Putin can align with the Iranians and place the United States in a far more complex situation than it otherwise would be in. He could achieve this by supporting Syria, arming militias in Lebanon, or even causing significant problems in Afghanistan, where Russia retains a degree of influence in the North...
STRATFOR INTELLIGENCE SUMMARY, 25 OCTOBER 2007, © STRATFOR INC.—During Russian President Vladimir Putin's Oct. 16 visit to Tehran, Iranian Supreme Leader Ayatollah Ali Khamenei asked him to order Russian experts to help Iran figure out how Israel jammed Syrian radars prior to the Sept. 6 air raid, a Stratfor source in Hezbollah said. Iran wants to rectify the problem associated with the failure of Syrian radars because Iran uses similar equipment, the source added.
RUSSIA, IRAN: THE NEXT STEP IN THE DIPLOMATIC TANGO—STRATFOR Global Intelligence Brief, 30 October 2007, © 2007 Stratfor, Inc.—...Russia has a fine-tuned strategy of exploiting its Middle Eastern allies' interests for its own political purposes. Iran is the perfect candidate. It is a powerful Islamic state that is locked into a showdown with the United States over its nuclear program and Iraq. Though Washington and Tehran are constantly battling in the public sphere with war rhetoric, they need to deal with each other for the sake of their strategic interests.
Russia, meanwhile, has its own turf war with the United States that involves a range of hot issues, including National Missile Defense, renegotiating Cold War–era treaties, and Western interference in Russia's periphery. By demonstrating that Moscow has some real sway over the Iranians, Russia gains a useful bargaining chip to use in its dealings with the United States...
ALTAY OPTICAL-LASER SOURCEBOOK, 28 December 2007—The Scientific Research Institute of Precision Instrument Engineering [of the Russian Federation] has established a branch satellite tracking facility called the Altay Optical-Laser Center (AOLS) near the small Siberian town of Savvushka. The center consists of two sites, one of which is now operational and the other of which is intended to go into operation in or after 2010.
The present site has a laser rangefinder for precision orbit determination, and, for the first time in Russia, a telescope (60 cm aperture) there has been equipped with an adaptive optics system for high-resolution imaging of satellites. The second site will be equipped with a 3.12-meter satellite-imaging telescope generally similar to the one the United States operates in Hawaii.
...Successful implementation of the AOLS 3.12-meter system would allow satellites to be imaged with a resolution of 25 cm [9.8 inches] or better out to a range of 1,000 km [621 miles].
## PROLOGUE
Don't be too timid and squeamish about your actions. All life is an experiment. The more experiments you make the better.
—RALPH WALDO EMERSON
OVER EASTERN SIBERIA
FEBRUARY 2009
"Stand by...ready...ready...begin climb, now," the ground controller radioed.
"Acknowledged," the pilot of the Russian Federation's Mikoyan-Gurevich-31BM long-range interceptor responded. He gently eased back on his control stick and began feeding in power. The twin Tumanski R15-BD-300 engines, the most powerful engines ever put on a jet fighter, barked once as the afterburners ignited, then quickly roared to life as the engines' fuel turbopumps caught up with the massive streams of air flooding inside, turning air and fuel into raw power and acceleration.
The pilot's eyes darted back and forth from the power gauges to the heads-up display, which showed two crossed needles with a circle in the middle, similar to an Instrument Landing System. He made gentle, almost imperceptible control inputs to keep the crossed needles centered in the circle. His inputs had to be tiny because the tiniest slip or skid now, with his nose almost forty degrees above the horizon and climbing, could result in a disruption of the smooth airflow into the engine intakes, causing a blowout or compressor stall. The MiG-31, known as "Foxhound" in the West, was not a forgiving machine—it regularly killed sloppy or inattentive crewmembers. Built for speed, it required precise handling at the outer edges of its impressive flight envelope.
"Passing ten thousand meters...Mach two point five...fifteen thousand...forty degrees nose-high...airspeed dropping off slightly," the pilot intoned. The MiG-31 was one of the few planes that could accelerate while in a steep climb, but for this test flight they were going to take it higher than its service ceiling of twenty thousand meters, and its performance dropped off significantly then. "Passing twenty K, airspeed below Mach two...passing twenty-two K...stand by...approaching release speed and altitude..."
"Keep it centered, Yuri," the MiG's backseater said over intercom. The needles had drifted slightly to the edge of the circle. The circle represented their target tonight, transmitted to them not by the MiG-31's powerful phased-array radar but by a network of space tracking radars around the Russian Federation and fed to them by a nearby data relay aircraft. They would never see their target and would probably never know if their mission was a success or failure.
"It's getting less responsive...harder to correct," the pilot breathed. Both crewmembers were wearing space suits and full-face sealed helmets, like astronauts, and as the cabin altitude climbed, the pressure in the suit climbed to compensate, making it harder to move and breathe. "How...much...longer?"
"Ten seconds...nine...eight..."
"Come on, you old pig, climb," the pilot grunted.
"Five seconds...missile ready...tree, dva, adeen...pazhar! Launch!"
The MiG-31 was at twenty-five thousand meters above Earth, one thousand kilometers per hour airspeed, with the nose fifty degrees above the horizon, when the ship's computer issued the launch command, and a single large missile was ejected clear of the fighter. Seconds after ejection, the missile's first-stage rocket motor ignited, a tremendous plume of fire erupted from the nozzles, and the missile disappeared from view in the blink of an eye.
Now it was time to fly for himself and not the mission, the pilot reminded himself. He brought back the throttles slowly, carefully, and at the same time started a slight left bank. The bank helped decrease lift and bleed off excessive speed, and would also help bring the nose down without subjecting the crew to negative G-forces. The pressure began to subside, making it a bit easier to breathe—or was it just because their part of the mission was...?
The pilot lost concentration just for a split second, but that was enough. At the moment he let a single degree of sideslip creep in, the fighter flew through the disrupted supersonic air created by the big missile's exhaust tail, and airflow through the left engine was nearly cut off. One engine coughed, sputtered, and then began to scream as fuel continued to pour into the burner cans but the hot exhaust gases were no longer being pushed out.
With one engine running and the other on fire, with not enough air to restart the stalled engine, the MiG-31 launch aircraft was doomed. But the missile it fired performed flawlessly.
Fifteen seconds after the first-stage motor ignited, it separated from the missile and the second-stage motor fired. Speed and altitude climbed quickly. Soon the missile was at five hundred miles above Earth, flying at over three thousand miles per hour, and the second-stage motor separated. Now the third stage remained. High above the atmosphere, it needed no control surfaces to maneuver, instead relying on tiny nitrogen-gas thrusters for maneuvering. A radar in the nose of the third stage activated and began looking at a precise spot in space, and a second later it locked onto its quarry.
The missile didn't have enough speed to begin orbiting the Earth, so as soon as the second stage separated it began its long fall, but it didn't need to orbit: like an atmospheric anti-tank missile, it was falling in a ballistic path toward a computed point in space where its quarry would be in mere seconds. The predicted path, programmed well before launch by ground controllers, was soon verified by on-board targeting computers: the target's orbit had not changed. The intercept was just as planned.
Twenty seconds before impact, the third stage deployed a fifty-yard-wide circular composite net—well above the atmosphere, the net was unaffected by air pressure and stayed round and solid even though traveling several thousand miles an hour. The net was an insurance policy against a near-miss...but this time, it didn't need it. With the third stage solidly locked onto the target, and with very little need for hard jarring maneuvering because of the precision of the launch and flight path, the third stage made a direct hit on its intended target.
"Impact, sir," the technician reported. "No telemetry received from the test article."
The commanding general in charge, Russian Air Forces chief of staff Andrei Darzov, nodded. "But what about the flight path? Was it affected by the improper launch parameters?"
The technician looked confused. "Uh...no, sir, I do not believe so," he said. "The launch seemed to go perfectly."
"I disagree, Sergeant," Darzov said. He turned to the technician and affixed him with an angry glare. An angry look was bad enough, but Darzov kept his head shaved to best reveal his extensive combat injuries and burns across his head and body, and he looked even more fearsome. "That missile went far off-course, and it may have locked onto an errant satellite by mistake and attacked it."
"Sir?" the technician asked, confused. "The target...uh, the American Pathfinder space-based surveillance satellite? That was—"
"Was that what we hit, Sergeant?" Darzov asked. "Why, that was not in the flight test plan at all. There has been a horrible mistake, and I will be sure it is investigated fully." His features softened, he smiled, then clasped the technician's shoulder. "Be sure to write in your report that the missile went off-course because of a sideslip in the launch aircraft—I will take care of the rest. And the target was not the American SBSS, but our Soyuz target spacecraft inserted into orbit last month. Is that clear, Sergeant?"
## CHAPTER ONE
It is better to be violent, if there is violence in our hearts, than to put on the cloak of nonviolence to cover impotence.
—MAHATMA GANDHI
ARMSTRONG SPACE STATION
THAT SAME TIME
"Okay, suckers, c'mon and poke your head out—just a little bit," Captain Hunter "Boomer" Noble muttered. "Don't be afraid—this won't hurt a bit." This was day two of their new patrol, and so far they had squat to show for it except for a persistent headache from watching the sensor monitors for hours at a stretch.
"Hang in there, sir," Air Force Master Sergeant Valerie "Seeker" Lukas said gaily. "You're anticipating, and that negative energy only keeps their heads down."
"It's not negative energy, Seeker, whatever that is," Boomer said, rubbing his eyes. "It's that TV picture—it's killing me." Hunter rubbed his eyes. They were staring at a wide-screen high-definition image of a suburban section of the southeast side of Tehran, in what used to be called the Islamic Republic of Iran but was now referred to by many in the world as the Democratic Republic of Persia. The image, shot from a telescopic electro-optical camera mounted aboard a U.S. Air Force RQ-4 Global Hawk unmanned reconnaissance aircraft orbiting at sixty thousand feet above the city, was fairly steady, but every shake, no matter how occasional, felt like another pinch of sand thrown into Boomer's eyes.
The two were not sitting at a console in a normal terrestrial combat control center, but in the main battle management module of Armstrong Space Station, positioned two hundred and seventy-five miles above Earth in a forty-seven-degree inclination easterly orbit. Noble and Lukas were among four additional personnel brought aboard to run the U.S. Air Force's Air Battle Force monitoring and command mission over the Democratic Republic of Persia. Although Boomer was a space veteran with several dozen orbital flights and even a spacewalk to his credit, floating in zero-G staring at a monitor was not what he joined the Air Force for. "How much longer are we on station?"
"Just five more hours, sir," Lukas said, smiling and shaking her head in mock disbelief when Noble groaned at her reply. Seeker was an eighteen-year U.S. Air Force veteran, but she still looked barely older than she did the day she enlisted in January 1991 when Operation Desert Storm kicked off, and she loved her profession just as much now as she did back then. The images of laser- and TV-guided bombs flying through windows and down ventilator shafts fascinated and excited her, and she started basic training two days after graduating from high school. She joined every high-tech optronic sensor school and course she could find, quickly becoming an all-around expert at remote sensing and targeting systems. "Besides the power plant, environmental, and electronic systems, the most important systems in strategic reconnaissance are patience and an iron butt."
"I'd rather be out there flying myself," Boomer said petulantly, readjusting himself yet again on his attachment spot on the bulkhead in front of the large monitor. He was a little taller than the average American astronaut that most of the instruments on the space station were obviously designed for, so he found almost everything on the station just enough of the wrong size, height, or orientation to irk him. Although the twenty-five-year-old test pilot, engineer, and astronaut was a space veteran, most of his time in space had been spent strapped into a nice secure spaceplane seat at the controls, not floating around in zero-G. "All this remote-control stuff is for the birds."
"You calling me a 'bird,' sir?" she asked with mock disapproval.
"I'm not calling anyone anything, Master Sergeant—I'm giving this particular procedure my own personal opinion," Boomer said. He motioned to the screen. "The picture is really good, but it's the radar aiming thingy that's driving me nuts."
"That's the SAR aiming reticle, sir," Seeker said. "It's slaved to the synthetic aperture radar and highlights any large vehicle or device that appears in the sensor field of view that matches our search parameters. If we didn't have it, we'd have to manually scan every vehicle in the city—that would really drive you nuts."
"I know what it is, Master Sergeant," Boomer said, "but can't you make it stop darting and flitting and shaking around the screen so much?" The monitor showed a rectangular box that appeared and disappeared frequently in the scene. When it appeared, the box surrounded a vehicle, adjusted its size to match the vehicle, and then if it matched the preprogrammed size parameters, a tone would sound and the camera would zoom in so the humans could see what the computers had found. But it would only stay focused on one vehicle for five seconds before starting the wide-area scan again, so Boomer and Seeker had to almost constantly watch the screen and be prepared to hit the HOLD button to study the image before the computer jumped out again. "It's giving me a damned headache."
"I think it's incredible it's doing what it's doing, sir," Seeker said, "and I'm more than willing to put up with a few jiggles if it helps us spot a—" And at that moment the computer locked onto another vehicle, which had just appeared atop a parking structure beside a cluster of apartment buildings. Seeker slapped the HOLD button a second later. "Hey, we got one!" she shouted. "It's a Katyusha...no, I think it's a Ra'ad rocket! We got them setting up a Ra'ad!"
"You're mine, suckers," Boomer said, instantly forgetting all about his purported headache. He glanced at the monitor, but he was already busy making sure the target coordinates obtained by the Global Hawk were being uploaded properly. The live image was incredibly detailed. They watched as four men carried a large rocket, resembling a large artillery shell with fins, out of the parking garage to the back of a Toyota pickup truck—it must've been very heavy, because it appeared they were having difficulty carrying it. The pickup had a large steel skeletal pedestal mounted in the pickup frame, with a circular cradle atop it. The men rested the rocket on the back of the truck, then two of them hopped up and they began struggling to lift the rocket up to the launcher.
"Don't drop it, boys," Seeker said. "You wouldn't want to spoil our fun, would you?" She turned to Boomer. "How much longer, sir?"
"Target coordinates uploaded," Boomer said. "Counting down now. How long do we have?"
"Once they get it up into the launcher, it could be fired in less than a minute."
Boomer glanced up and watched the monitor. Several children ran up to the truck to watch the terrorists at work—at first they were shooed away, but after a few moments they were allowed to get a closer look. "Looks like 'Career Day' is on in Tehran," he said gloomily.
"Get out of there, kids," Seeker murmured. "It's not safe for you there."
"Not because of us," Boomer said coldly. He hit a transmitter button on his console. "Ripper to Genesis."
"I'm right here, Boomer," responded Lieutenant General Patrick McLanahan, "standing" on the bulkhead behind Boomer and looking over his shoulder. The twenty-one-year Air Force veteran and three-star general was the commanding officer at Elliott Air Force Base, Groom Lake, Nevada, the home of the High-Technology Aerospace Weapons Center, or HAWC. HAWC developed the XR-A9 Black Stallion spaceplane, along with countless other air weapons and aircraft, but it was leaders like Patrick McLanahan who saw the capabilities and possibilities of those experimental devices and brought them to bear in crisis situations where America or her allies would otherwise suffer tremendous losses or even defeat. Short, husky but not large, with disarming blue eyes and a quick smile, Patrick McLanahan did not at all resemble the hard-charging, determined, audacious globe-crossing aerial bombardment expert and master tactician portrayed by his reputation. Like Boomer and Seeker, McLanahan was becoming a veteran astronaut—this was his third trip to Armstrong Space Station in as many months.
"We've got a good one, sir," Boomer said, nodding at his monitor. "Not a little homemade Qassam or Katyusha this time, either." Boomer studied the young three-star Air Force general's face carefully, noticing his eyes flicking back and forth across his monitor—not just looking at the rocket, Boomer thought, but at the kids clustered around the makeshift terror weapon launcher. "The master sergeant thinks it's a Ra'ad rocket."
It appeared as if Patrick hadn't heard him, but a few moments later he nodded excitedly. "I agree, Seeker," he said. "A Hezbollah weapon, based on a Russian battalion-level battlefield attack missile. Two-hundred-pound warhead, simple but usually effective barometric fuse, airburst with a backup impact detonation, killing radius a hundred yards or more, usually loaded up with glass, ball bearings, and pieces of metal along with high explosives to increase the injury toll. A real terror weapon." He shook his head. "But there are too many civilians around. Our ROE says no noncombatant casualties and minimal collateral damage. Pick a different target, Boomer, one with fewer bystanders. There will be plenty of opportunities..."
"We don't see many Ra'ad rockets, sir," Seeker said. "That's not a homemade rocket—that's a military-grade short-range ballistic attack missile."
"I know, Master Sergeant, but our orders are specific and—" At that moment the insurgents shooed the children away again, more forcefully this time, as another insurgent fitted ignition wires to the tail end of the rocket in final preparation for launch. "Now," Patrick snapped. "Take it down."
"Yes, sir," Boomer said enthusiastically. He issued commands on his computer, checked the computer's responses, then nodded. "Here we go...missile counting down...doors coming open...ready...ready...now, missile away." He checked a countdown timer. "Don't anyone blink, 'cuz this won't take long."
Over the Caspian Sea two hundred and twenty miles north of Tehran, an unmanned EB-1D Vampire bomber opened its combined forward and center bomb bay doors and released a single large missile. The D-model Vampire was a modified U.S. Air Force B-1B strategic bomber, converted by the High-Technology Aerospace Weapons Center to a long-range unmanned flying battleship. It was capable of autonomously flying itself from takeoff to final parking with an inflight-reprogrammable flight plan, or could be flown by satellite remote control like a large multimillion-dollar video game from a laptop computer located almost anywhere.
The missile the Vampire had just released was an even more sophisticated weapon developed by the engineers at HAWC. Its unclassified designator was the XAGM-279A SkySTREAK, but anyone who knew anything about this missile—and there were only a handful of persons on the entire planet who did—called it the "Streaker." It resembled a cross between a bullet and a manta ray, with a pointed carbon-carbon nosecap and bullet-shaped forebody splaying out into a thin, flat fuselage and pointed tail section. After stabilizing itself in the atmosphere, four solid-fuel rocket motors ignited, pushing the weapon to well past Mach 3 and one hundred thousand feet of altitude in just a few seconds.
Within eight seconds the motors had burned out, and a wide, flat oval air intake popped open underneath the missile. Supersonic air was ingested and compressed by the shape of the now-empty rocket motor casings, mixed with jet fuel, and ignited by high-energy pulses of laser energy. The resultant energy propelled the missile to over ten times the speed of sound in just a few more seconds, and the missile ate up the distance between its launch point and its target in no time, climbing to two hundred thousand feet as it raced downrange. The missile burned all of its jet fuel in just a few seconds, and it quickly decelerated and began descending back through the atmosphere. Once the outside skin temperature was within safe limits, the bullet-shaped forebody detached from the spent propulsion section, which automatically blew itself to bits moments later.
Small stabilizer fins popped out of the forebody, and it became a supersonic re-entry vehicle, guiding itself to its target with its on-board navigation computer refined by Global Positioning System signals. Fifteen seconds to impact, the protective nosecap detached, revealing a combination millimeter-wave radar and imaging infrared scanner, and the warhead began uploading video signals via satellite to Boomer and Seeker back in Dreamland. The steering cue in the video image was several yards off, but Seeker used a trackball and rolled the steering rectangle back on the pickup truck, which sent steering correction signals to the warhead.
The video image from the warhead was sharp and clear all the way to impact. Patrick had a brief glimpse of a young man, no more than fifteen or sixteen years old, wearing a mask and carrying an AK-47 assault rifle that looked almost as big as he was, who looked right up at the incoming weapon milliseconds before the image vanished. Patrick knew that the warhead was programmed to explode a tenth of a second before impact, splitting the warhead apart into thousands of small hypervelocity fragments, increasing the killing radius of the weapon out to about forty to fifty yards.
"Direct hit!" Boomer shouted happily. He looked at the control monitor and slapped his hands together. "Total time from detection to impact: forty-eight point nine seconds. Less than a friggin' minute!"
"It's more like a Maverick missile—or a sniper's bullet—only fired from two hundred miles away!" Seeker exclaimed. She had switched back to the Global Hawk's image of the target area and zoomed in to take a close look at the Streaker warhead's impact spot. "Pretty good urban weapon effects, sir, exactly what you were hoping for. A really good-sized hole, about fifteen to twenty feet in diameter—looks like the center punched through the concrete parking garage roof into the floor below—but no damage to the nearby buildings that I can see except for a few broken windows. Even a two-hundred-and-fifty-pound Small-Diameter Bomb might have caved in the sides of the building facing the blast."
"With no explosive warhead on the Streaker, there's nothing there to create any collateral damage," Boomer said. "We put just enough shaped explosive charges in the warhead to break it apart milliseconds before impact, and that was both to increase the weapon effect slightly as well as to destroy as much of the evidence as we could. All they should find are tiny pieces of—"
"Oh...my...God," Seeker breathed. She had zoomed out to survey a little more of the surrounding area. There were clusters of people, perhaps two dozen or so, just outside the apartment complex area lying on the sidewalk and street, with others attending to them, waving frantically for help. "What in hell happened here? Where did those people come from, and why are they lying on the ground like that? Are they from inside the apartment complex...?"
"The Streaker must've set off the Ra'ad rocket's warhead," Boomer said. They all carefully studied the image as Seeker took manual control of the camera and zoomed in. "But what's going on? Those people over there weren't anywhere near the blast, but they're staggering around like they were hit. Was it shrapnel from the Ra'ad warhead? The Streaker doesn't have an explosive payload—it's all kinetic energy. Is the Persian army moving in? What's going...?"
"A chemical weapon cloud," Patrick said.
"What...?"
"It looks like some sort of chemical weapon cloud, spreading out from the target area," Patrick said. He pointed to the monitor. "Not more than thirty feet away. Here's a little bit of the cloud...see, it's not rising like a cloud from an explosion or from heat, but traveling horizontally, blown around by air currents." He looked closer. "Not twitching...it's hard to tell, but it looks like he's rubbing his eyes and face and is having trouble breathing. I'll bet it's a blister agent...lewisite or phosgene. Mustard agents would take longer to incapacitate someone, even in high concentrations...look, now someone collapsing across the street. Jesus, the warhead must've had several liters of CW in it."
"My God," Seeker gasped. "I've been operating remote sensors for almost twenty years, and I've never seen anyone die from a chemical weapon attack."
"I have a feeling the powers that be aren't going to like this," Patrick said.
"Should we recall the Vampire, sir?"
"Hell no," Patrick said. "We still have three more Streakers on board, and another Vampire loaded and waiting to go at Mosul. Keep on scanning for more insurgents. Congratulations, Boomer. The SkySTREAK worked perfectly. Nail a few more insurgents for us."
"You got it, sir," Boomer said happily.
ARMSTRONG SPACE STATION
A SHORT TIME LATER
Unfortunately, Patrick turned out to be exactly correct. The Global Hawk images were being beamed to several terrestrial locations as well as to Silver Tower, including the Joint Chiefs of Staff Operations Center in Washington, and it was there that he received his first call just moments later: "Genesis, this is Rook." That was from the duty officer at the JCS Operations Center. "Stand by, please." A moment later, the chief of staff of the Air Force, General Charles A. Huffman, appeared on the video teleconference channel, looking a little pale himself but still very angry as well.
Huffman, a tall, dark-haired, and very young man with husky, athletic features—more like a linebacker than a running back, Boomer thought—was typical of the new breed of leaders in the American military. In the five years since the Russian nuclear cruise missile air strike on the continental United States, known as the "American Holocaust," which left several thousand dead, hundreds of thousands injured, several Air Force bases destroyed, and almost all of America's long-range bombers wiped out, the military ranks had filled with eager young men and women wishing to protect their country, and many officers were promoted well below their primary zones and placed into important command positions years before it was ever thought possible. Also, since senior leaders with extensive combat experience were kept in charge of tactical units or major commands, often officers with less direct combat experience were placed in more administrative and training billets—and since the office of the chief of staff was mostly concerned with equipping and training their forces, not leading them into combat, it seemed a good match.
That was true of Huffman as well: Patrick knew he came from the logistics field, a command pilot, wing, and numbered Air Force commander, and former Air Force Materiel Command commander with over fifteen thousand hours flying time in a variety of cargo, transport, and liaison aircraft in two conflicts, and extensive experience in supply, resource management, and test and evaluation. As former head of Materiel Command, Huffman had been notional commander of activities at the top-secret High-Technology Aerospace Weapons Center at Elliott Air Force Base, although that link was mostly administrative and logistical—operationally, the commanders at HAWC reported to the chairman of the Joint Chiefs of Staff or the Secretary of Defense at the Pentagon, the President's National Security Adviser at the White House, or—at least under former President Kevin Martindale—directly to the President himself.
Patrick had never spent any time in logistics, but he knew that logistics officers liked their world as neat, orderly, and organized as possible. Although they learned to expect the unexpected, they very much preferred to anticipate, predict, and manage the unexpected, and therefore anything unexpected was not welcome. He knew Huffman, however, and he knew that's precisely the way Huffman liked it: no surprises. "McLanahan, what in hell happened out there?"
"Calling Genesis, say again, please," Patrick said, trying to remind the general that although the connection was encrypted and as secure as they could make it, it was still a wide-open satellite-based network and prone to eavesdropping.
"We're secure here, McLanahan," Huffman thundered. "What in hell is going on? What happened?"
"We hit an insurgent rocket launcher, and apparently detonated its explosive chemical-weapon warhead, sir."
"What did you hit it with?"
"A XAGM-279 with a kinetic warhead, sir," Patrick responded, using the SkySTREAK's experimental model number instead of its name to confuse any eavesdroppers. "Almost no explosives in it—just enough to fragment the warhead."
"What is a XAGM-279? An experimental precision-guided missile?"
So much for communications security, Patrick thought, shaking his head. It was five years after the American Holocaust and seven years since 9/11, and many folks had forgotten or abandoned the tight security measures that had been put in place after those two devastating attacks. "Yes, sir" was all Patrick said.
"Launched from that unmanned B-1?"
"Yes, sir." Anyone listening to this conversation—and Patrick didn't delude himself that any number of agencies or units around the world could've done so easily—could piece together their entire operation by now. "I briefed the staff two days ago on the operation."
"Dammit, McLanahan, you briefed minimal collateral damage, not dozens of dead women and children lying in the street!" Huffman cried. "That was the only way we could sell your idea to the President."
"The weapon produced virtually no collateral damage, sir. It was the chemical warhead on the insurgents' rocket that caused all those civilian casualties."
"Do you believe anyone is gong to care about that one bit?" Huffman said. "This is a major fuckup, McLanahan. The press is going to have a field day with this." Patrick remained silent. "Well?"
"I don't feel it's my task force's or my responsibility to worry about what the enemy's weapons do to the civilian population, sir," Patrick said. "Our job is to hunt for insurgents firing rockets into population centers in Tehran and destroy them."
"The Qagev members inside the Turkmeni insurgent network and Buzhazi's spies inside Mohtaz's security staff briefed us that the insurgents could use weapons of mass destruction at any time, McLanahan," Huffman said. Patrick suppressed another irritated breath: Huffman had just revealed two highly classified intelligence sources—if anyone was eavesdropping, those sources were dead meat in just a matter of days, perhaps hours. "You should have adjusted tactics accordingly."
"Tactics were adjusted, sir—I was ordered to reduce the number of bombers on station from three to one," Patrick responded—by you, he added to himself. "But we don't have enough coverage of the city to effectively deal with the number of launchers being reported. I recommend we launch two more bombers so we can hunt down more launchers before the insurgents actually start firing live chemical warhead munitions into the city."
"Are you crazy, McLanahan?" Huffman retorted. "The President will probably order the entire program shut down because of this! The last thing he will do is put more bombers up there. As it is, we'll spend a week defending ourselves from being accused of releasing those chemical warheads. You will recall your aircraft immediately, then prepare to debrief the JCS and likely the entire national security staff. I want a full report on the incident on my desk in one hour. Understood?"
"Yes, sir."
"And after the briefing is complete, get your ass off that damned space station," Huffman said. "I don't know why my predecessor allowed you to go up there, but you've got no business traipsing up to that floating pile of tubes every time you feel like it. I need you down here—if for no other reason than to have you personally answer to the national command authority regarding another lapse in judgment."
"Yes, sir," Patrick replied, but the transmission had already been ended by the time he spoke. He terminated the videoconference link, thought for a moment, then spoke, "McLanahan to Mace."
Another window popped open on the opposite lower corner of Boomer's large multifunction screen, and he saw the image of Brigadier General Daren Mace, the operations officer and second-in-command of the Air Battle Force attack wing at Battle Mountain Air Reserve Base in northern Nevada. The air wing at Battle Mountain was the home base and central control facility for the unmanned long-range bombers, although commanders at HAWC could also issue instructions to the bombers as well.
"Yes, General?" Mace responded. Older than Patrick by just a few years, Daren Mace was a veteran B-1B Lancer strategic bomber OSO, or offensive systems officer, and bomb wing commander. His expertise on the B-1's attack systems and capabilities led him to be chosen to head the Air Battle Force's long-range supersonic attack fleet.
"Recall the damned Vampires," Patrick ordered tonelessly.
"But sir, we've still got three more Streakers on board the Vampire, and it's got at least two more hours' endurance before it has to head back to Batman Air Base in Turkey," Boomer interjected. "Intel briefed us that—"
"The operational test was successful, Boomer—that's what we needed to find out," Patrick said, rubbing his temples. He shook his head resignedly. "Recall the Vampire now, General Mace," he said quietly, his head lowered, his voice sounding utterly exhausted.
"Yes, sir," the veteran bomber navigator responded. He entered keyboard instructions on his computer console. "The Vampire's on the way back to Batman Air Base in Turkey, sir, ETE forty-five minutes. What about the follow-on sorties?"
"Hold them in their hangars until I give the word," Patrick replied.
"And what about our shadow, sir?" Daren asked.
Patrick looked at another monitor. Yep, it was still there: a Russian MiG-29 Fulcrum jet fighter, one of several that had been hanging near the bomber since it started its patrol, always within one or two miles of the Vampire, not making any threatening moves but certainly able to attack at any second. It certainly had a front-row seat for the SkySTREAK launch. The Vampire bomber had taken several photographs of the fighter with its high-resolution digital camera so detailed that they could practically read the pilot's name stenciled on the front of his flight suit.
"If it locks onto the Vampire, shoot it down immediately," Patrick said. "Otherwise we'll let it—"
And at that moment they heard a computer-synthesized voice announce, "Warning, warning, missile launch! SPEAR system activated!"
Patrick shook his head and sighed audibly. "The game's afoot, crew," he said. "The battle begins today, and it has little to do with Persia." He turned to the computer screen of the command center at Battle Mountain. "Shut that bastard down, Daren," Patrick radioed.
"He's down, sir," Daren said.
As soon as the Vampire bomber detected the missile launch, its newest and most powerful self-defense system activated: the ALQ-293 SPEAR, or Self-Protection Electronically Agile Reaction system. Large sections of the composite skin of the EB-1D Vampire had been redesigned to act as an electronically scalable antenna that could transmit and receive many different electromagnetic signals, including radar, laser, radio, and even computer data code.
As soon as the MiG's radar was detected, the SPEAR system immediately classified the radar, examined its software, and devised a method to not just jam its frequency but to interface with the radar's digital controls themselves. As soon as the missile launch was detected, SPEAR sent commands to the MiG's fire control system to send a command to the missile to switch immediately to infrared seeker mode, then shut down the digital guidance uplink from the fighter. The missiles automatically deactivated their on-board radars and activated its infrared seeker, but they were too far away from the Vampire bomber to lock on using its heat-seeking sensor, and the missiles harmlessly plummeted to the Caspian Sea without acquiring a target.
But SPEAR wasn't done. After defeating the missiles, SPEAR sent digital instructions to the MiG-29 via the fire control system to start shutting down aircraft systems controlled by computer. One by one, the navigation, engine controls, flight controls, and communications all turned themselves off.
In the blink of an eye, the pilot found himself sitting in a completely silent and dark glider, as if he were sitting on the ramp back at his home base.
To his credit, the veteran pilot didn't panic and eject—he was not out of control, not yet, but just...well, turned off. There was only one thing to do: turn all switches off to reset the computers, then turn everything back on, and hope he could get his stricken jet running again before he crashed into the Caspian Sea. He flipped his checklist to the BEFORE POWER ON pages and started shutting every system in the plane off. His last image out his canopy was watching the big American B-1 bomber bank sharply left, as if giving the Russian a farewell wing-wag, and fly off toward the northwest, speeding quickly out of sight.
No one in the Russian air force had ever run a series of checklists faster than he. He had descended from forty-two thousand feet all the way down to four thousand feet above the Caspian Sea before he was able to get his jet shut down, turned back on, and the engines started again. Thankfully, whatever evil spirits had entered his MiG-29 were no longer present.
For a brief instant the Russian MiG pilot considered pursuing the American bomber completely radar-silent and putting a load of cannon shells into his tail—he was going to be blamed for almost crashing his plane anyway, so why not go out in a blaze of glory?—but after briefly considering it, he decided that was a foolish notion. He didn't know what caused the mysterious shutdown—was it an American weapon of some kind, or a glitch in his own plane? Besides, the American bomber was not launching any more missiles that could be "mistaken" for an attack against him. This was not a war between the Americans and the Russians...
...although he felt it certainly could blossom into one at any moment.
"Let's put together a debrief, then get ready to head back to HAWC, Boomer," Patrick said after they were assured that the EB-1C Vampire bomber was safely on its way back to Batman Air Base in Turkey. His voice sounded very tired, and his facial expression appeared even more so. "Good job. The system seems to be working fine. We've proven we can control unmanned aircraft from Silver Tower. That should get us some sustainment funding for another year at least."
"General, it wasn't your fault that the damned insurgents had a bunch of kids around when the SkySTREAK attacked, or that they loaded up that Ra'ad missile with poison gas," Hunter Noble responded, looking worriedly at Master Sergeant Lukas.
"I know, Boomer," Patrick said, "but it still doesn't make watching innocent men, women, and children die like that any easier."
"Sir, we're on station, the Vampire is loaded, the SkySTREAKs are running cool, and no doubt there are more of those Ra'ads out there with poison gas warheads," Boomer said. "I think we should stay and—"
"I hear you, Boomer, but we've validated the system—that was the mission objective," Patrick said.
"Our other objective was to try to control multiple bombers and multiple engagements," Boomer reminded him. "We had enough trouble getting authorization and funding to fly this mission—getting approval for another mission to do what we could have done on this flight will be even more difficult."
"I know, I know," Patrick said wearily. "I'll ask, Boomer, but I'm not counting on it. We've got to analyze the data, prepare a summary report, and brief the chief. Let's get to it."
"But sir—"
"Meet you back here in ten, Boomer," Patrick said finally, detaching himself from his anchor position and floating his way toward the sleeping module.
"Looks like he took that one hard," Seeker said after the general had left the control module. Boomer didn't respond. "It kind of shook me up too. Is the general feeling okay?"
"He had a rough trip up here," Boomer said. "Every push into orbit has been hard on him, but he keeps on flying up here. The last push took a lot out of him, I think. He probably shouldn't be making these trips anymore."
"It could be watching those people getting killed like that," Seeker said. "I've seen the aftereffects of a guided missile attack plenty of times, but somehow a biochem weapon attack is...different, you know? More violent." She looked at Boomer curiously, unable to read his rather flat expression. "Did it shake you up too, Boomer?"
"Well..." And then he shook his head and added, "No, it didn't, Seeker. All I want to do now is hunt down more bad guys. I don't understand why the general wanted to wrap this up so soon."
"You heard the chief, sir," Seeker said. "The general wanted to send the other two bombers."
"I know, I know." Boomer looked around the module. "The things we can do on board this station are amazing, Sergeant, really amazing—we should be allowed to do them. We need to convince the powers that be that we can turn the Air Force on its ear. We can't do that if we pull our planes out when a little kid ten thousand miles away gets caught in the crossfire. Can't believe the general got all misty-eyed like that."
Master Sergeant Lukas looked at Boomer sternly. "Do you mind if I say something, sir?" she finally asked.
"Go right ahead, Seeker...or is it 'Master Sergeant' now?"
"I haven't been working at HAWC that long—not as long as you," Lukas said, ignoring the sarcastic remark, "and I don't know General McLanahan that well, but the guy is a friggin' hero in my book. He's spent almost twenty years laying his ass on the line fighting battles all over the world. He's been kicked out of the Air Force twice, but he came back because he's dedicated to his country and the service."
"Hey, I'm not bad-mouthing the guy—"
"The 'guy' you're referring to, sir, is a three-star general in the U.S. Air Force and commands the largest and most highly classified aerospace research facility in the U.S. armed forces," Lukas interrupted hotly. "General McLanahan is nothing short of a legend. He's been shot up, shot down, blown up, beat up, ridiculed, busted, demoted, and called every name in the book. He's lost his wife, a close friend, and dozens of crewmembers under his command. You, sir, on the other hand, have been in the force now...seven years? Eight? You're a talented engineer and a skillful pilot and astronaut—"
"But?"
"—but you're not in the general's league, sir—far, far from it," Lukas went on. "You don't have the experience and haven't shown the same level of commitment and dedication as the general. You're not qualified to pass judgment on the general—in fact, in my opinion, sir, you haven't earned the right to be talking about him the way you are."
"Like you're talking to me now?"
"Write me up if you want, sir, but I don't appreciate you second-guessing the general like that," Lukas said flatly. She logged herself off from her console and detached herself from the bulkhead with a perturbed jerk and a loud riip! of Velcro. "I'll help you download the sensor data and prepare your debrief for the general, and then I'll be happy to help you prepare the Black Stallion for undocking...so you can go home as soon as possible, sir." She said the word "sir" more like the word "cur," and that jab wasn't lost on Boomer.
With Seeker's exasperated and irate help—not to mention they didn't do very much chatting as they worked—Boomer was indeed done quickly. He uploaded his data and findings to the general. "Thanks, Boomer," McLanahan radioed back. "We're scheduled to do the videoconference in about ninety minutes. I found out the Joint Chiefs chairman and National Security Adviser are going to sit in. Kick back for a while and get some rest."
"I'm fine, sir," Boomer responded. "I'll go hide out in Skybolt, get my e-mail, and check in on my girlfriends."
"'Girlfriends'...plural?"
"I don't know—we'll see what the e-mails say," Boomer said. "None of them like me disappearing for days and weeks, and I certainly can't tell them I've been blasting terrorists to hell from space."
"They probably wouldn't believe you if you did tell them."
"The ladies I hang out with don't know a space station from a gas station—and that's the way I like it," Boomer admitted. "They don't know, or care, what I do for a living. All they want is attention and a good time on the town, and if they don't get it, they split."
"Sounds lonely."
"That's why I always like to have more than one on the hook, sir," Boomer said.
"Could be fireworks if they ever run into each other, eh?"
"We hook up together all the time, sir," Boomer said. "No brag, just fact. Like I said, all they want is attention, and they get even more attention if folks see them arm in arm with another hot babe. Besides, if there's ever any conversation—"
"Wait, wait, I know this one, Boomer: 'If there's any conversation, you don't have to get involved,'" Patrick interjected with a laugh. "Okay, go say hi to your girlfriends, and don't tell me how many you got waiting for you to get back. Meet me in the command module in sixty minutes so we can rehearse our dog and pony show."
"Yes, sir," Boomer replied. Before McLanahan clicked off, he asked, "Uh, General?"
"Go ahead."
"I'm sorry if I got out of line earlier."
"I expect you to give me your professional opinion and point of view anytime, Boomer, especially on a mission," Patrick said. "If you were out of line, I wouldn't hesitate to let you know."
"It got me pretty steamed, watching those bastards setting up a rocket with a damned chemical warhead on it. All I wanted to do was blast a few more."
"I hear you. But it's more important we get this program off and running. We both know we're going to catch some flak for what happened in Tehran—shooting more missiles wouldn't have helped us."
"Maybe offing a few more terrorists would compel them to keep their heads down and hide in their ratholes for a few days more."
"We have some incredible weapons at our disposal, Boomer—let's not let the power go to our heads," Patrick said patiently. "It was an operational test, not an actual mission. I know the temptation to play Zeus with a few SkySTREAK missiles is powerful, but that's not what we're here for. Meet back here in sixty."
"Yes, sir," he responded. Just before the general logged off, Boomer remarked to himself that the general looked even wearier than any other time since embarking on this sortie to the space station—maybe the combination of witnessing the chemical weapon release and the monthly trips into space were starting to get to him. Boomer was half his age, and sometimes the stress of the trips, especially the recent quick-turn, high-G re-entry profiles, and multiple sorties they had been flying, wore him down fast.
Boomer floated back to the crew quarters module, retrieved his wireless headphones and video goggles, and floated to the Skybolt laser module at the "bottom" of the station. Skybolt was the station's most powerful and so most controversial piece of technology, a multi-gigawatt free-electron laser powerful enough to shoot through Earth's atmosphere and melt steel in seconds. Tied to Silver Tower's radars and other sensors, Skybolt could attack targets as small as an automobile and burn through the top armor of all but the most modern main battle tanks. Classified as a "weapon of mass destruction" by all of America's adversaries, the United Nations had been calling for the weapon's deactivation for many years, and only America's veto power in the Security Council kept it alive.
Ann Page, Skybolt's designer, operator, and chief advocate, was on Earth preparing to testify to Congress on why funding for the weapon should be continued, and Boomer knew that very few others on the station ever went near the thing—Skybolt was powered by an MHDG, or magnetohydrodynamic generator, which used two small nuclear reactors to rapidly shoot a slug of molten metal back and forth through a magnetic field to produce the enormous amount of power required by the laser, and no amount of shielding and assurances by Ann could assuage anyone's fears—so he often went into the module to get some peace and quiet. The Skybolt module was about a fourth of the size of the main modules on the station, so it was relatively cramped inside, and it was crammed with pipes, wire conduits, and a myriad of computers and other components, but the gentle hum of the MHDG drive's circulating pumps and the excellent computers and communications gear there made it Boomer's favorite place to get away from the others for a while.
Boomer connected his headphones and video goggles to the module's computers, logged in, and began downloading e-mail. Even though the headphones and goggles were a pain, there was precious little privacy on Silver Tower, even in the huge modules, so the only semblance of privacy had to come down to the space between one's ears. Everyone assumed that if personnel from the super-secret High-Technology Aerospace Weapons Center were on board the space station that all incoming and outgoing transmissions of any kind were being recorded and monitored, so "privacy" was a vacuous idea at best.
It was a good thing he had bothered to put on the gear, because the video e-mails from his girlfriends were definitely not for public viewing. Chloe's video was typical: "Boomer, where the hell are you?" it began, with Chloe sitting in front of her videophone photographing herself. "I'm getting tired of you disappearing like this. Nobody at your unit would tell me a goddamned thing. That sergeant that answers the phone should be booted out of the service, the fag." Chloe called any man who didn't immediately hit on her a "fag," believing being gay was the only reason that any normal male wouldn't want to screw her right away.
She paused for a moment, her features softening a bit, and Boomer knew the show was about to begin: "You'd better not be with that blond spiky-haired bitch, Tammy or Teresa or whatever the hell her name is. You're over at her place, aren't you, or you two have jetted off to Mexico or Hawaii, haven't you? You two just fucked and you're checking mail while she takes a shower, right?" Chloe set the videophone down on her desk, unbuttoned her blouse, and slipped her large, firm breasts out from under her brassiere. "Let me just remind you what you're missing here, Boomer." She put a finger sensuously in her mouth, then circled her nipples with it. "Get your ass back here and stop screwing around with those skanky bottle-blond hos." She smiled seductively, then hung up.
"Crazy bitch," Boomer muttered as he continued to scroll through the messages, but resolved to look her up as soon as he got back. After previewing more messages he stopped and immediately entered the code to access the satellite Internet server. Another benefit of the new American space initiative, of which Armstrong Space Station was the hub, was the coming availability of almost universal Internet access via a constellation of over a hundred low-Earth-orbit satellites that provided global low-speed Internet access, plus ten geostationary satellites that provided high-speed broadband Internet access to most of the Northern Hemisphere.
"No IP address, no extensions, no open active server identification code—this has got to be a call from outer space," came the reply from Jon Masters a few moments later after establishing a videophone connection to the designated secure address. Jon Masters was the vice president of a small high-tech research and development company called Sky Masters Inc. that designed and licensed many different emerging aerospace technologies, from microsatellites to space boosters. Masters, a multidegree, multidoctorate scientist and engineer regarded as one of the world's most innovative aerospace designers and thinkers, had formed his company at the ripe old age of twenty-five, and he still looked and acted the part of the geeky, eccentric, and flippant child prodigy. "Thanks for returning my call, Boomer."
"No problem, Jon."
"How are things up there?"
"Fine. Good."
"I know you can't talk about it on a satellite server, even if it is encrypted. Just wanted to be sure you're okay."
"Thanks. I'm fine."
There was a slight pause; then: "You sound a little down, my friend."
"No."
"Okay." Another pause. "So. What do you think of my offer?"
"It's extremely generous, Jon," Boomer said. "I'm not sure if I deserve it."
"I wouldn't offer it if I didn't think you did."
"And I get to work on whatever I want?"
"Well, we hope we can entice you to help out on other projects," Masters said, "but I want you to do what you do best: think outside the box and come up with fresh, innovative, and kick-ass designs. I don't try to game or anticipate the aerospace market, Boomer—I try to shape it. That's what I want you to do. You won't answer to anyone else but me, and you get to pick your team, your protocols, your design approach, and your timelines—within reason, of course. You knock my socks off with your ideas, and I'll back you all the way."
"And this estimated budget figure for my lab...?"
"Yes?"
"Is this for real, Jon?"
"That's just the starting point, Boomer—that's the minimum," Masters chuckled. "You want that in writing, just say so, but I'm guaranteeing you that you'll have a generous budget to build the team to research and evaluate your designs."
"Even so, it's not enough for the entire division. I'll need—"
"You don't understand, Boomer," Masters interjected excitedly. "That money is just for you and your team, not split up between everyone in your division, existing projects, or specific company-mandated programs or technology."
"You're kidding!"
"I'm serious as a heart attack, brother," Masters said. "And it's not for stuff like company-wide expenses, compliance mandates, or security, but for your team- and project-specific costs. I believe in giving our top engineers the tools they need to do their job."
"I can't believe it. I've never even heard of that kind of money being invested by a small company like this."
"Believe it, Boomer," Masters said. "We may be small, but we've got investors and a board of directors who think big and expect big things to happen."
"Investors? A board of directors...?"
"We all answer to someone, Boomer," Masters said. "I ran my company by myself with a handpicked board of directors, which was okay until the projects got smaller and the money got tight. There were a lot of investors out there who wanted to be part of what we were doing here, but no one wants to dump hundreds of millions of dollars into a one-man show. We're public, and I'm not president anymore, but everyone knows I'm the guy who makes the magic."
"I don't know..."
"You don't worry about the board, Boomer. You report to me. Be advised, I'm going to make you work for every dime. I'm going to expect big things from you, and I'll be putting bugs in your ear about what I know or discover about government requests for proposals, but like I said, I don't want you waiting around for some weenie in the Pentagon to tell us what they might want—I want us to tell them what they want. So, what do you say? Are you in?"
"I'm thinking about it, Jon."
"Okay. No problem. I know your commitments to the Air Force are up in eight months, correct?" Boomer guessed that Jon Masters knew to the day when his educational commitments to the Air Force for pilot training were up. "I guarantee they'll offer you a regular commission before that, along with a big fat bonus. They might try to stop-loss you, claiming you're in a critical specialty, but we'll deal with that when and if we have to. I have enough contracts with the Air Force, and enough buddies in the Pentagon, to put a little pressure on them to respect your decisions. After all, you're not getting out to go work for the airlines or be a consultant or lobbyist—you'll be working for the company that builds them the next generation of hardware."
"That sounds good."
"You bet it does, Boomer," Jon Masters said. "Don't worry about a thing. One more thing, buddy. I know I'm older than you, probably old enough to be your dad if I started real early, so I get to give you a little heads-up."
"What's that, Jon?"
"I know trying to tell you to take it easy, be safe, and maybe don't fly so many missions is like trying to tell my golden retriever to stay out of the lake, but I wouldn't want to have the company's future vice president of R&D become a shooting star, so take it easy, okay?"
"Vice president?"
"Oh, did I say that out loud?" Masters deadpanned. "You weren't supposed to hear that. Forget I said that. Forget the board was considering it but didn't want me to reveal that. Gotta go before I tell you about the other thing the board was kicking around...oops, almost did it again. Later, Boomer."
OFFICE OF THE PRESIDENT, THE KREMLIN, MOSCOW, RUSSIAN FEDERATION
A SHORT TIME LATER
The room was loudly called to attention as Russian Federation president Leonid Zevitin quickly strode into the conference room, followed by his chief of staff Peter Orlev, the secretary of the security council, Anatoli Vlasov; the minister of foreign affairs, Alexandra Hedrov; and the chief of the Federal Security Bureau, Igor Truznyev. "Take seats," Zevitin ordered, and the officers already in the room—General Kuzma Furzyenko, the chief of staff; General Nikolai Ostanko, chief of staff of the army; and General Andrei Darzov, the chief of staff of the air force—shuffled to their chairs. "So. I gave the command for our fighter to attack the unmanned American bomber if it fired a missile, and since we're meeting like this so quickly, I assume it did, and we did. What happened?"
"The American B-1 bomber successfully launched a missile from over the Caspian Sea that reportedly destroyed a Hezbollah squad preparing to launch a rocket from an apartment complex in southeast Tehran," General Darzov replied. "The missile made a direct hit on the launch squad's location, killing the entire crew..." He paused, then added, "including our Special Forces adviser. The bomber then—"
"Hold on, General, hold on a sec," Zevitin said impatiently, holding up a hand. "They launched a missile from over the Caspian Sea? You mean a cruise missile, and not a laser-guided bomb or TV-guided missile?" Many of those around the table narrowed their eyes, not because they disliked Zevitin's tone or question but because they were unaccustomed to someone with such a distinct Western accent at a classified meeting in the Kremlin.
Leonid Zevitin, one of Russia's youngest leaders since the fall of the czars, was born outside St. Petersburg but was educated and had spent most of his life in Europe and the United States, and so had almost no Russian accent unless he wanted or needed one, such as when speaking before Russian citizens at a political rally. Frequently seen all over the world with starlets and royalty, Zevitin came from the world of international banking and finance, not from politics or the military. After decades of old, stodgy political bosses or bureaucratic henchmen as president, the election of Leonid Zevitin was seen by most Russians as a breath of fresh air.
But behind the secretive walls of the Kremlin, he was something altogether different than just expensive silk suits, impeccable hair, jet-setter style, and a million-dollar smile—he was the puppet master in the grand old Russian tradition, every bit as cold, calculating, and devoid of any warm personality traits as the worst of his predecessors. Because he had no political, apparatchik, military, or intelligence background, no one knew how Zevitin thought, what he desired, or who his allies or captains in government were—his henchmen could be anyone, anywhere. That kept most of the Kremlin off-guard, suspicious, tight-lipped, and at least overtly loyal.
"No, sir—the missile went faster than Mach four, which is the fastest speed our fighter's radar can track a target. I would describe it as a very high-speed guided rocket."
"I assume, then, that you compared the time of launch and the time of impact and came up with a number?"
"Yes, sir." His eyes looked pained—no one could tell whether it was because the general was afraid of telling the president the bad news, or because he was being lectured to by this foreign-sounding young playboy.
"But you don't believe the number you computed," Zevitin said for the air force chief of staff. "Obviously this weapon was something we did not expect. What was the speed, General?"
"Average speed, Mach five point seven."
"Almost six times the speed of sound?" That news rocked every member of the security staff back in their chairs. "And that was the average speed, which means the top speed was Mach...ten? The Americans have an attack missile that can fly at Mach ten? Why didn't we know of this?"
"We know now, sir," General Furzyenko said. "The Americans made the mistake of using their new toy with one of our fighters on his wingtip."
"Obviously they were not concerned enough about our fighter to cancel their patrol or their attack," Zevitin offered.
"It was what the Americans call an 'operational test,' sir," air force chief of staff General Andrei Darzov said. A short, battle-worn air force bomber pilot, Darzov preferred his head shaved bald because he knew how it intimidated a lot of people, especially politicians and bureaucrats. He had visible burn scars on the left side of his neck and on his left hand, and the fourth and fifth fingers of his left hand were missing, all a result of injuries sustained in the bombing of Engels Air Base, Russia's main bomber base, several years earlier, when he served as Forces of Long-Range Aviation division commander.
Darzov had wanted nothing short of bloody payback for the utter devastation wreaked on his headquarters during the sneak attack on Engels, and swore revenge on the American air commander who had planned and executed it...Lieutenant General Patrick McLanahan.
Under former military chief of staff turned president Anatoliy Gryzlov, who wanted revenge on the United States as badly as Darzov, he soon got his opportunity. Andrei Darzov was the architect of the plan just a year later to modify Russia's long-range Tu-95 Bear, Tu-26 Backfire, and Tu-160 Blackjack bombers with aerial refueling probes to allow them the range to attack the United States. It was an audacious, ambitious plan that succeeded in destroying most of the United States' long-range bombers and the control centers for over half of their land-based intercontinental ballistic nuclear-tipped missiles. The devastating assault killed over thirty thousand people and injured or sickened thousands more, and soon became known as the "American Holocaust."
But Darzov hadn't heard the last of his archenemy, Patrick McLanahan. When McLanahan's counterattack destroyed almost an equivalent number of Russia's most powerful silo-based and mobile intercontinental ballistic missiles, someone had to take the blame—other than the then-president of Russia, General Gryzlov, who had been killed during an American air strike on his Ryazan underground command center—and Darzov was it. He was blamed for making the decision to stage all of the Ilyushin-78 and Tupolev-16 tanker aircraft at one isolated air base in Siberia, Yakutsk, and for not providing enough security there, which allowed McLanahan and his Air Battle Force to take over the base and use the enormous amount of fuel stored there to be used by McLanahan's bombers to hunt down and destroy Russia's land-based nuclear deterrent force.
Darzov was demoted to one-star general and sent to Yakutsk to oversee the cleanup and eventual closing of that once-vital Siberian base—because in an attempt to destroy McLanahan's bombers on the ground, Gryzlov had ordered Yakutsk attacked by low-yield nuclear weapons. While only four of the dozens of nuclear warheads penetrated McLanahan's anti-missile shield around the base, and they were all high-altitude airbursts designed to minimize radioactive fallout, most of the base had been severely damaged, and the heart of it had been flattened and rendered uninhabitable. There was much speculation that the general staff hoped Darzov would become sick from the lingering radioactivity so they would be spared the chore of eliminating the popular, intelligent young general officer.
But not only did Darzov not die, he didn't stay long in virtual exile in Siberia. Health-wise, Darzov and his loyal senior staff members survived by using the radioactivity decontamination equipment left behind by the Americans when they evacuated their personnel from Yakutsk. Career- and prestige-wise, he survived by not giving in to despair when it seemed like the entire world was against him.
With the financial and moral support of a young investment banker named Leonid Zevitin, Darzov rebuilt the base and soon made it operational again instead of preparing it for demolition and abandonment. The move revitalized Russia's Siberian oil and gas industry, which relied on the base for much-needed support and supply, and the government raked in enormous amounts of revenue from Siberian oil, most sold to Japan and China through new pipelines. The young base commander garnered the attention and gratitude of Russia's wealthiest and most successful investment banker, Leonid Zevitin. Thanks to Zevitin's sponsorship, Darzov was brought back to Moscow, promoted to four-star general, and eventually picked as chief of staff of the air forces by newly elected president Zevitin.
"The Americans have tipped their hand and revealed a new hypersonic air-to-ground weapon," Furzyenko said. "It shows how overconfident they are, and that will be their weakness. Not only that, but they wasted a multimillion-dollar missile destroying a truck and homemade rocket worth a few dollars."
"Seems to me they have every right to be overconfident, General—they can quickly and accurately destroy any target from two hundred miles away as easy as a child plinking a can with a .22 rifle from twenty meters away," Zevitin said. Many of the generals knitted their eyebrows, as much in confusion at some of Zevitin's Western terms as in struggling to understand his heavily accented Russian. "Plus, they did it right before our eyes, knowing we'd be watching and measuring the weapon's performance. It was a demonstration for our benefit, as well as a very effective terror weapon against the Islamists." Zevitin turned to Darzov. "What happened to the fighter that was shadowing the B-1 bomber, Andrei?"
"The pilot landed safely but with most of his plane's electronic equipment completely disabled," the air force chief of staff responded.
"How? Their terahertz weapon again?"
"Possibly, but the American so-called T-Ray weapon is a subatomic wide-area weapon that destroys electronic circuits at ranges exceeding six hundred kilometers," Darzov replied. "No other stations reported any disruption. The pilot reported that as soon as he launched his missiles his fighter...simply shut itself down."
"You mean, the missile shut itself down."
"No, sir. The entire airplane shut itself down, as if the pilot had turned everything off all at once."
"How is that possible?"
"The terahertz weapon may have been able to do it," Darzov said. "We will not know until we look at the fighter computer's error logs. But my guess would be that McLanahan has deployed his 'netrusion' system on the Dreamland bombers, and possibly all of his aircraft and spacecraft."
"'Netrusion'? What's that?"
"The ability to 'hack' into an opponent's computer systems through any sensor or antenna that receives digital signals," Darzov explained. "We do not completely understand the process, but the bombers can transmit a signal that is picked up and processed like any other digital instruction or message. The enemy signal can be false radar targets, confusing coded messages, flight control inputs, or even electronic commands to aircraft systems..."
"Such as a shutdown order," Zevitin said. He shook his head. "He conceivably could have commanded the MiG to fly straight down or around in circles—luckily he only ordered it to shut down. Must be nice to be so rich that you can build such wonderful toys to load up on your planes." He nodded. "Looks like your old friend is still in the game, eh, General?"
"Yes, sir," Darzov said. "Patrick McLanahan." He smiled. "I will welcome a chance to take him on again and repay him for imprisoning my men and women, taking my base, and stealing my fuel. However, from what I understand, he may not be around much longer. The new administration does not like him at all."
"If McLanahan had any political savvy, he'd have resigned the moment the new president took the oath of office," Zevitin said. "Obviously that has not happened. Either McLanahan is more dedicated—or dumber—than we thought, or Gardner isn't going to fire him, which means he might not be the buffoon we think he is." He looked at the generals around him. "Forget about McLanahan and his high-tech toys that never get built—he's the best they've got, but he's only one man, and he's squirreled away in that awful desert base in Nevada instead of in the White House now, which means no one has the opportunity to listen to him anymore." To Truznyev, chief of the Federal Security Bureau, successor organization to the KGB, he asked, "What about your 'adviser' in Iran? Did you get him out?"
"What was left of him, yes, sir," the FSB chief replied.
"Good. The last thing we need is some enterprising American or Persian investigator finding Russian clothing or weapons mixed in with a lot of Iranian body parts."
"He was replaced with another agent," Truznyev said. He turned angrily to Alexandra Hedrov, the foreign minister. "Giving those Hezbollah bastards weapons like the 9K89 is a waste of time and money, and hurt us in the long run. We should stop supplying them with such advanced missiles and let them go back to firing homemade Katyushas and mortars at the Persian collaborators."
"You agreed to General Furzyenko's recommendation to send the 'Hornet' missile to Iran, Director," Zevitin pointed out.
"I agreed that the Hornet missile should be used to attack Persian army and air force bases with high-explosive and mine-laying warheads, sir," Truznyev said, "not to just fire them indiscriminately into the city. The launch point was at the very edge of the rocket's maximum range to hit the Doshan Tappeh air base, which was the target they told us they were going to strike. The Hezbollah crew also reportedly dragged their feet launching the missile—they even let children come around and watch the launch. This has been reported many times."
"We will obviously have to instruct the insurgents to adjust tactics now that we know about this new American weapon," General Darzov said.
"Will you also instruct them not to put their own homemade poison brews in the warhead?" Truznyev asked.
"What are you talking about, Director?"
"The Hezbollah insurgents loaded the Hornet missile's warhead up with some sort of chemical weapon concoction, similar to mustard gas but much more effective," the FSB chief said perturbedly. "The gas killed a dozen people on the street and injured several dozen others."
"They cooked up their own mustard gas?"
"I do not know where the hell they got it, sir—Iran has a lot of chemical munitions, so maybe they stole it or had it secretly stored away," Truznyev said. "The stuff went off when the American missile hit. But the point is, they violated our directives and attacked an unauthorized target with an unauthorized warhead. There are only a few truck-launched missiles that have the fusing necessary to carry out a chemical weapon attack—it will not be hard for the Americans to discover we supplied the Iranians with the Hornet missiles."
"Get Mohtaz on the phone, now," Zevitin ordered. Chief of staff Orlev was on the phone in an instant.
"Now that the Pasdaran has brought in foreign fighters from all over the world to join this damned jihad against Buzhazi's coup," Truznyev said, "I do not think the clerics have very tight control over their forces." The Ayatollah Hassan Mohtaz, the former Iranian national defense adviser—and the most senior member of the former Iranian government to survive Buzhazi's bloody purge of Islamists—had been proclaimed president-in-exile, and he called upon all the Muslims of the world to come to Iran and fight against the new military-monarchist government. The anti-Persia insurgency grew quickly, spurred on by tens of thousands of Shi'a Muslim fighters from all over the world who answered the fatwa against Buzhazi. Many of the insurgents had been trained by Iran's Revolutionary Guards Corps, the Pasdaran, so their fighting effectiveness was even greater. Within days after Mohtaz's call to arms went out, most of the cities of the new Persia were embroiled in bitter fighting.
But part of the chaos in Persia was due to the fact that the coup leader, General Hesarak al-Kan Buzhazi, inexplicably refused to form a new government. Buzhazi, the past chief of staff and former commander of the paramilitary Internal Defense Forces that battled the Revolutionary Guards Corps, had led a stunningly successful coup, killing most of Iran's theocratic rulers and sending the rest fleeing to neighboring Turkmenistan. It had been assumed that Buzhazi, together with former chief of staff Hoseyn Yassini, the officers of the regular armed forces, and supporters of one of Iran's past royal families, the Qagevs, would take control of the capital city of Tehran and form a government. A name had even been chosen—the Democratic Republic of Persia, indicating a clear direction the people wanted to take—and the country was now referred to by its historic name, "Persia," instead of the name "Iran," which was the name decreed to be used by Reza Shah Pahlavi in 1935. Only supporters of the theocracy still used the name "Iran."
"But I do not think we should stop arming the insurgents," General Darzov said. "Every successful attack against the Persians will weaken them. We need patience."
"And every time the jihadis launch another missile into the city and kill innocent women and children, the insurgency suffers the same fate—it gets weakened, as does Russia, General," foreign minister Alexandra Hedrov said. Tall, dark-haired, and as alluring as any woman in the senior echelons of Russian government could be, Alexandra Hedrov was the highest-ranking woman to ever serve in the Kremlin. Like Zevitin, she came from an international finance background, but as a lifelong resident of Moscow and a married mother of two, she didn't have the jet-setting reputation of her superior. Serious and sharp and without extensive political connections, Hedrov was widely considered the brains behind the presidency. "We look even worse if we are seen supporting baby-killers."
She turned to Zevitin. "Mohtaz has got to find a way to tone down the jihadis, Mr. President, without relieving the pressure on Buzhazi and Qagev to give up and evacuate the country. We cannot be seen supporting mass murder and instability—that makes us look unstable ourselves. If Mohtaz continues on this path, the only recourse we have is to support Buzhazi."
"Buzhazi?" Zevitin asked, confused. "Why support Buzhazi? He turned to the Americans for help."
"That was our fault—he acted out of desperation, and we were not there for him when he needed us, so he turned to McLanahan," Hedrov explained. "But Washington inexplicably has not thrown its support behind Buzhazi, and this creates an opportunity for Russia. We secretly support Mohtaz because Russia benefits from the instability in the region with higher oil prices and greatly increased arms sales. But if we end up backing a loser, we should reverse course and support whom I believe will be the eventual winner: Buzhazi."
"I disagree, Minister," Darzov said. "Buzhazi is not strong enough to destroy Mohtaz."
"Then I suggest you get out of your airplanes and laboratories and take a look at the world as it really is, General," Hedrov said. "Here is the real question, Mr. President: Whom do you want to win, Buzhazi or Mohtaz? That is who we should be supporting. We support Mohtaz because the chaos in the Middle East keeps America from meddling in our affairs in our own spheres of influence. But is a theocratic Iran a better choice for Russia? We know Buzhazi. You and I have both met with him; we supported him for many years, before, during, and after his removal as chief of staff. We still supply each other with intelligence information, although he is keeping information about the American presence in Iran closely guarded and more expensive to obtain. Maybe it is time to increase the level of contact with him."
The phone vibrated beside Orlev, and he picked it up and moments later put it on hold. "Mohtaz on the line, sir."
"Where is he?"
"Iranian embassy in Ashkhabad, Turkmenistan," Orlev replied, anticipating the question.
"Good." When the Ayatollah Mohtaz and his advisers fled Iran, he unexpectedly holed up in the Russian embassy in Ashkhabad, demanding protection from Buzhazi's forces and the so-called monarchist death squads. That created a lot of curiosity and questions from most of the rest of the world. It was well known that Moscow was an ally of Iran, but would they go so far as to protect the old regime? What if elections were held and the theocrats were voted out? Would the clerics and Islamists become an albatross around Russia's neck?
As a concession to the rest of the world, Zevitin had Mohtaz leave the embassy, but quietly guaranteed his safety with Russian FSB units stationed in and around the Iranian compound. At first he thought the Islamist wouldn't leave the embassy—or, worse, threaten to expose Russia's involvement in Iran if he was forced out—but thankfully things didn't reach that stage. He knew Mohtaz could always produce that card in the future, and he needed to decide what to do if he tried to play it.
Zevitin picked up his phone. "President Mohtaz, this is Leonid Zevitin."
"Please stand by for His Excellency, sir," a heavily Persian-accented voice said in Russian. Zevitin rolled his eyes impatiently. It was always a game with weak men like Mohtaz, he thought—it was always so damned important to try to gain the smallest advantage by making the other party wait, even over something as simple as a phone call.
A few moments later, the voice of a young translator said, "The Imam Mohtaz is on the line. Identify yourself please."
"Mr. President, this is Leonid Zevitin calling. I hope you are well."
"Praise be to God for his mercy, it is so."
No attempt to return pleasantries, Zevitin noted—again, typical of Mohtaz. "I wanted to discuss the recent air attack by the Americans in Tehran against a suspected Hezbollah rocket launcher."
"I know nothing of this."
"Mr. President, I warned you against allowing the insurgents to arm the rockets with weapons of mass destruction," Zevitin said. "We specifically chose the Hornet rocket because it is in use all over the world and would be harder to trace back to Russia. The only rocket force known to have the technology to put chemical warheads on them was Russia."
"I know no details of what the freedom fighters do in their struggle against the crusaders, nonbelievers, and Zionists," the translator said. "All I know is that God will reward all who have answered the call of holy retribution. They will earn a place at His right hand."
"Mr. President, I urge you to keep your forces in check," Zevitin said. "Armed resistance to foreign occupation is acceptable to all nations, even with unguided rockets against suspected sympathizers, but using poison gas is not. Your insurgency risks a popular backlash if—"
Zevitin could hear Mohtaz shouting in the background even before the translator finished speaking, and then the flustered young man had to scramble to keep up with the Iranian cleric's sudden tirade: "This is not an insurgency, damn your eyes," the translator said in a much calmer voice than Mohtaz's. "Proud Iranians and their brothers are taking back the nation that has been illegally and immorally taken from us. That is not an insurgency—it is a holy war of freedom against oppression. And in such a struggle, all weapons and all tactics are justified in the eyes of God." And the connection was broken.
"Fucking bastard," Zevitin swore—not realizing until it was too late that he had done so in English—as he slammed the receiver down.
"Why bother with that insane zealot, sir?" foreign affairs minister Hedrov asked. "The man is crazy. He cares for nothing else but retaking power—he does not care how many innocent people he must kill to do it. He is bringing in foreign jihadis from all over the world, and most of them are crazier than he is."
"Do you think I care about Mohtaz or anyone in that damned country, Minister?" Zevitin asked heatedly. "For the time being, it is better for Russia with Mohtaz alive and stirring up the Islamists, calling for them to go to Iran and fight. I hope that country tears itself apart, which is almost a certainty if the insurgency grows."
"I wish Buzhazi had called on us rather than McLanahan when he wanted support for his insurgency—Mohtaz and that monarchist bitch Qagev would be dead by now, and Buzhazi would be firmly in command, with us at his side," Hedrov said, casting a disapproving glare at Federal Security Bureau chief Truznyev. "We should have recruited him the moment he surfaced in the Iranian People's Militia."
"Buzhazi was completely off our radar screens, Minister," Truznyev said dismissively. "He was disgraced and all but condemned to die. Iran had drifted into the Chinese sphere of influence..."
"We sold them plenty of weapons."
"After oil prices rose, yes—they bought Chinese crap because it was cheaper," Truznyev said. "But then we found many of those weapons in the hands of Chechen separatists and drug runners within our own borders in short order. China stopped their support for Iran long ago because they support Islamists in Xinjiang and East Turkestan—Chinese Islamic insurgents were fighting government troops with their own damned weapons! The theocrats in Iran are completely out of control. They do not deserve our support."
"All right, all right," Zevitin said wearily, shaking his hand at his advisers. "These endless arguments are getting us nowhere." To Truznyev, he said, "Igor, get me all the data on that American hypersonic missile you can get your hands on, and get it fast. I don't need to know how to counter it—yet. I need enough information so that I can make Gardner believe that I know all about it. I want to argue that it's a threat to world peace, regional stability, the arms balance, blah, blah, blah. Same with their damned Armstrong Space Station. And I'd like an update on all the new American military technology. I'm tired of hearing about it after we encounter it in the field."
"Argue with the Americans, eh, Mr. President?" chief of the general staff Furzyenko asked sarcastically. "Perhaps we can go in front of the Security Council and argue that the sunlight reflecting off their station's radar arrays keeps us up at night."
"I don't need snide remarks from you today, General—I need results," Zevitin said acidly. "The Americans are settled in Iraq, and they may have gained a foothold in Iran if Buzhazi and the Qagev successfully forms a government friendly to the West. Along with American bases in central Asia, the Baltics, and eastern Europe, Iran adds yet another section of fence with which to pen us in. Now they have this damned space station, which passes over Russia ten times a day! Russia is virtually surrounded—" And at that, Zevitin slapped his hand down hard on the table. "—and that is completely unacceptable!" He looked each of his advisers in the eye, his gaze pausing momentarily on Truznyev and Darzov before sitting back in his seat and irritably running a hand over his forehead.
"That hypersonic missile surprised us all, sir," Truznyev said.
"Bullshit," Zevitin retorted. "They need to test-fire the thing, don't they? They can't do that in an underground laboratory. Why can't we be observing their missile tests? We know exactly where their high-speed instrumented test ranges are for hypersonic missile development—we should be all over those sites."
"Good espionage costs money, Mr. President. Why spy for the Russians when the Israelis and Chinese can offer ten times the price?"
"Then perhaps it's time to cut some salaries and expensive retirement benefits of our so-called leaders and put the money back into getting quality intelligence data," Zevitin said acidly. "Back when Russian oil was only a few dollars a barrel, Russia once had hundreds of spies deep inside every nook and cranny of American weapons development—we once had almost unfettered access to Dreamland, their most highly classified facility. And what places we didn't infiltrate ourselves, we were able to buy information from hundreds more, including Americans. The FSB's and military intelligence's task is to get that information, and since Gryzlov's administration we haven't done a damned thing but whine and moan about being surrounded and possibly attacked again by the Americans." He paused again, then looked at the armed forces chief of staff. "Give us a status report on Fanar, General Furzyenko."
"One unit fully operational, sir," the chief of staff replied. "The mobile anti-satellite laser system proved very successful in downing one of the American spaceplanes over Iran."
"What?" chief of staff Orlev exclaimed. "Then, what the Americans said was true? One of their spaceplanes was downed by us?"
Zevitin nodded to Furzyenko as he pulled a cigarette from his desk drawer and lit up, wordlessly giving him permission to explain. "The Fanar project is a top-secret mobile anti-satellite laser system, Mr. Orlev," the military chief of staff explained. "It is based on the Kavaznya anti-satellite laser system developed in the 1980s, but greatly modified, enhanced, and improved."
"Kavaznya was a massive facility powered by a nuclear reactor, if I remember correctly," Orlev observed. He was only in high school when he learned about it—at the time the government had said there was an accident and the plant had been shut down for safety upgrades. It was only when he assumed his post as chief of staff that he learned that Kavaznya had actually been bombed by a single American B-52 Stratofortress bomber, a highly modified experimental "test-bed" model known as the "Megafortress"—crewed by none other than Patrick McLanahan, who was then just an Air Force captain and crew bombardier. The name McLanahan had popped up many times in relation to dozens of events around the world in the two decades since that attack, to the point that Darzov and even Zevitin seemed obsessed with the man, his high-tech machines, and his schemes. "How can such a system be mobile?"
"Twenty years of research and engineering, billions of rubles, and a lot of espionage—good espionage, not like today," Zevitin said. "Continue, General."
"Yes, sir," Furzyenko said. "Fanar's design is based on the Israeli Tactical High-Energy Laser program and the American airborne laser program, which puts a chemical laser on a large aircraft such as a Boeing 747 or B-52 bomber. It is capable of destroying a ballistic missile at ranges as far as five hundred kilometers. It is not as powerful as Kavaznya was, but it is portable, easily transported and maintained, is durable and reliable, extremely accurate, and if locked onto target long enough, can destroy even heavily shielded spacecraft hundreds of kilometers in space...like the Americans' new Black Stallion spaceplane."
Orlev's mouth dropped. "Then the rumors are true?" Zevitin smiled, nodded, then took another deep drag of his cigarette. "But we denied we had anything to do with the loss of the American spaceplane! The Americans must realize we have such a weapon!"
"And thus the game begins," Zevitin said, smiling as he finished the last of his cigarette. He ground the butt into the ashtray as if demonstrating what he intended to do to anyone who dared oppose him. "We'll see who is willing to play, and who is not. Continue, General."
"Yes, sir. The system can be disguised as a standard twelve-meter tractor-trailer rig and can be driven almost anywhere and mixed in with normal commercial traffic. It can be set up and readied to fire in less than an hour, can fire about a dozen bursts on one refueling, depending on how long the laser is firing at one target—and, most importantly, it can be broken down and moved within minutes after firing."
"Only a dozen bursts? That does not sound like very many engagements."
"We can bring along more fuel, of course," Furzyenko said, "but Fanar was never designed to counter large numbers of spacecraft or aircraft. The system can only fire for up to thirty seconds at a time due to heat, and one load of fuel can fire the laser for approximately sixty seconds total. The next barrage can be fired thirty to forty minutes later after refueling, depending on if the fuel comes from the fire vehicle or a separate support vehicle. Most spacecraft in low-Earth orbit would be well beyond the horizon before another barrage, so we decided it would be best not to try to fire too many barrages at once.
"In addition, everything else in the convoy increases in size as well—security, provisions, spare parts, power generators—so we decided to limit the extra laser fuel to one truck. With one command and fire vehicle, one power and control vehicle, one refueling and supply vehicle, and one support and crew vehicle, it can still travel anonymously enough on open highways anywhere without drawing attention. We brought it back to Moscow for additional tests and upgrades. That will take some time to accomplish."
"I think you've had enough time, General," Zevitin said. "The Americans need to see how vulnerable their precious space station and spaceplanes can be. I want that system up and running now."
"If I had more engineers and more money, sir, I could finish the three that are in the construction pipeline within a year," Furzyenko said. He glanced at General Darzov. "But there seems to be a lot of attention being paid to General Darzov's Molnija project, and I'm afraid our resources are being unduly diverted."
"Darzov has made some good arguments for Molnija, General Furzyenko," Zevitin said.
"I'm afraid I do not know what Molnija is, Mr. President," Alexandra Hedrov said. "I assume it's not a fine watch maker. Is this a new secret weapon program?"
Zevitin nodded to Andrei Darzov, who stood and began: "Molnija is an air-launched anti-satellite weapon, Madam Minister. It is a prototype weapon only, a combination of the Kh-90 hypersonic cruise missile reprogrammed for extreme high-altitude flight with a combination of rocket-ramjet-rocket propulsion to allow it to fly up to five hundred kilometers above Earth. The system was first developed by the Americans in the 1980s; we had a similar system but canceled it many years ago. The technology has improved greatly since then."
"Molnija is a big step backward," Furzyenko said. "The laser system has proven its worth. Air-launched anti-satellite weapons were rejected years ago because it was unreliable and too easily detected."
"With respect, sir, I disagree," Darzov said. Furzyenko turned to glare at his subordinate, but it was difficult to stare at the man's rather disturbing wounds, and he was forced to look away. "The problem with a fixed anti-satellite weapon, as was found with the Kavaznya anti-satellite laser, is that it is too easy to attack it, even with numerous and sophisticated anti-aircraft weapon systems protecting it. Even the mobile laser system we developed is vulnerable to attack since it takes so much support and takes so long to set up, fuel, and aim. We saw how quickly the Americans were able to attack the laser site in Iran—luckily, we had time to move the real system and construct a decoy in its place. Molnija can be carried to many air bases in the target's path and can attack from multiple angles.
"A single Molnija missile is carried aloft by a MiG-29 fighter or Tupolev-16 light bomber, or two missiles can be carried by a Tupolev-95 or Tupolev-160 heavy bomber," Darzov went on. "The launch aircraft are maneuvered into position by ground-based or airborne radars and then the missiles are released. Molnija uses a solid rocket motor to boost it to supersonic speed, where it then uses a ramjet engine to accelerate to eight times the speed of sound and climb to target altitude. Once in range of the target, it uses its on-board sensors to track the target and ignites its third-stage rocket motor to begin the intercept. It uses precision thrusters to get within range, then detonates a high-explosive fragmentary warhead. We can also place a nuclear or X-ray laser warhead on the weapon, depending on the size of the target."
"X-ray laser? What is that?"
"An X-ray laser is a device that collects and focuses X-rays from a small nuclear explosion and produces extremely powerful long-range energy beams that can penetrate even heavily shielded spacecraft as far as two hundred kilometers away," Darzov said. "It is designed to disable spacecraft by scrambling its electronics and guidance systems."
"Using nuclear weapons in space will create problems in the international community, General," Hedrov pointed out.
"The Americans have had a nuclear reactor flying over Russia for decades, and no one seemed to notice, Alexandra," Zevitin said bitterly. "The X-ray laser is just one option—we'll use it only if it's deemed absolutely necessary."
"The nuclear reactor on board the American space station is just for generating power, sir," Hedrov pointed out. "Yes, the laser has been used as an offensive weapon, but the reactor is thought of differently..."
"It is still an atomic device," Zevitin argued, "which is expressly prohibited by treaty—a treaty the Americans casually ignore!"
"I am in agreement with you, sir," Hedrov said, "but after the air attacks against the United States using nuclear weapons by President Gryzlov—"
"Yes, yes, I know...America gets a pass, and the world waits in fear to see what Russia will do next," Zevitin said, the frustration thick in his voice. "I'm sick of the double standard." He shook his head, then turned to General Darzov again. "What is the status of the anti-satellite missile program, General? Can we deploy the system or not?"
"Additional underground tests with the prototype Molnija unit were highly successful," Darzov went on. "The technicians and engineers want more tests done, but I believe it is ready for battle now, sir. We can make improvements, upgrades, and enhancements for years and make it better, but I think it is ready as is, and I recommend deployment immediately."
"Excuse me, sir," Furzyenko interjected, looking at Minister of National Defense Ostenkov in confusion, "but General Darzov isn't in charge of Molnija. It's a secret project that is still being overseen by my research and development bureau."
"Not anymore, General," Zevitin said. "I have tasked General Darzov to develop strategies for dealing with the American space station and spaceplanes. He will report to me and Minister Ostenkov directly."
Furzyenko's mouth opened and closed in confusion, then hardened in sheer anger. "This is an outrage, sir!" he blurted out. "This is an insult! The chief of staff is responsible for organizing, training, and equipping the armed forces, and I should have been informed of this!"
"You are being informed now, General," Zevitin said. "Fanar and Molnija belong to Darzov. He will keep me informed of his actions and will make recommendations to the national security bureau, but he takes his orders only from me. The farther outside your chain of command he operates, the better." Zevitin smiled and nodded knowingly. "A little lesson we've learned from our friend General Patrick Shane McLanahan over the years, yes?"
"I believe the man is obsessive, compulsive, paranoid, and probably schizophrenic, sir," Darzov said, "but he is also courageous and intelligent—two traits I admire. His unit is extremely effective because it operates with speed and daring with small numbers of highly motivated and energetic forces in command of the latest technological innovations. McLanahan also seems to completely disregard most regulations, normal conventions, and chains of command, and acts precipitously, perhaps even recklessly. Some say he is crazy. All I know is, he gets the job done."
"As long as you don't go off the deep end yourself," Zevitin warned.
"Unfortunately I agree with Minister Hedrov, sir: nuclear weapons in space will not be seen as a defensive weapon by the world community," Minister of National Defense Ostenkov said.
"The world community looks the other way and shuts its eyes and ears while the Americans orbit a nuclear reactor over their heads and fill the skies with satellites and spaceplanes—I really don't give a shit about their opinions," Zevitin said angrily. "The Americans can't be allowed to freely go in and out of space as they please. Our mobile ground-based laser got one and almost got another of their spaceplanes—we almost took out their entire active fleet. If we can bring down whatever they have left, we can cripple their military space program and possibly give us a chance to catch up again." He glared at Ostenkov. "Your job is to support the development and fielding of Fanar and Molnija, Ostenkov, not tell me what you think the world will say. Understood?"
"Yes, sir," Ostenkov said. "The anti-satellite missile is ready for operational testing. It could be the most feared weapon in our arsenal since the Kh-90 hypersonic cruise missile which Gryzlov used successfully to attack the United States. It can be deployed quickly and easily anywhere in the world, faster than a spacecraft can be launched or repositioned in an orbit. We can transport Molnija anywhere and run only a small risk of discovery until it's fired."
"And then what?" Orlev asked. "The Americans will retaliate with everything they have. You know they consider space part of their sovereign territory."
"That's why we need to employ Fanar and Molnija carefully—very, very carefully," Zevitin said. "Their usefulness as weapons depend more on quietly degrading the Americans' space assets, not trying to outright destroy them. If it's possible to make it look like their space station, spaceplanes, and satellites are unreliable or wasteful, the Americans will shut them down on their own. This is not an attack plan or a cat-and-mouse game—it's a game of irritation, of quiet degradation and growing uncertainty. I want to bug the shit out of the Americans."
"'Bug the shit,' sir?" Orlev asked. "What does this mean?"
"It means attack the Americans with mosquito bites, not swords," Zevitin said in Russian this time, not realizing until just then that in his excitement he had switched to English again. "Americans have no tolerance for failure. If it doesn't work, they'll scrap it and replace it with something better, even if the malfunction is no fault of theirs. Not only will they scrap something that doesn't work, but they'll blame the failure on everyone else, waste billions of dollars indicting someone to take the blame, then spend billions more to try to come up with a solution that is oftentimes inferior to the first." He smiled, then added, "And the key to this working is President Joseph Gardner."
"Naturally, sir—he is the President of the United States," Orlev remarked, confused.
"I'm not talking about the office, but of the man himself," Zevitin said. "He may be the commander-in-chief of the most powerful military force in the world, but the thing he is not in command of is the most important path to success: control of himself." He looked at the advisers around him and saw mostly blank expressions. "Thank you, all, thank you, that's all for now," he said dismissively, reaching for another cigarette.
Chief of Staff Orlev and Minister of Foreign Affairs Hedrov remained behind; Orlev didn't even try to suggest to Hedrov that he and the president be allowed to talk privately. "Sir, my impression, one that I share, is that the staff is confused about your intentions," Orlev said pointedly. "Half of them see you surrendering power to the Americans; the others think you are ready to start a war with them."
"Good...that's good," Zevitin said, taking a deep drag of his cigarette, then exhaling noisily. "If my advisers leave my office guessing—especially in opposite directions—they don't have an opportunity to formulate a counterstrategy. Besides, if they're confused, the Americans certainly should be as well." Orlev looked worried. "Peter, we can't yet beat the Americans in a military confrontation—we'd bankrupt this country trying. But we have lots of opportunities to stand in opposition to them and deny them a victory. Gardner is the weak link. He needs to be niggled. Irritate him enough and he'll turn on even his most trusted advisers and loyal countrymen." Zevitin thought for a moment, then added, "He needs to be irritated right now. The attack on our fighter...he needs to know how angry we are that they downed our fighter with a low-yield nuclear device."
"But...the fighter was not downed," Orlev reminded him, "and the general said the weapon was not a nuclear T-Ray weapon, but a—"
"For God's sake, Peter, we're not going to tell the Americans what we know, but what we believe," Zevitin said, irritation in his voice but a smile on his face. "My reports state that they shot down our fighter with a T-Ray nuclear device, without provocation. That is an act of war. Get Gardner on the phone immediately."
"Should Minister Hedrov make contact and—?"
"No, I will make the protest directly with Gardner," Zevitin said. Orlev nodded and picked up the phone on Zevitin's desk. "Not the regular phone, Peter. Use the 'hot line.' Voice and data both." The emergency "hot line" between Washington and Moscow had been upgraded after the conflicts of 2004 to allow voice, data, and video communications between the two capitals, as well as teletype and facsimile, and also allowed for more satellite circuits that gave the leaders easier access to one another. "Minister Hedrov, you will file a formal complaint with the United Nations Security Council and the American State Department as well. And I want every media outlet on the planet given a report of the incident immediately."
Orlev made the call to the foreign ministry first, then contacted the Kremlin signal officer to open the "hot line" for the president. "Sir, this could backfire," Orlev warned as he waited for the connection. "Our pilot certainly initiated the attack by firing on the American bomber—"
"But only after the bomber launched their hypersonic missile," Zevitin said. "That missile could've been headed anywhere. The Americans were clearly the aggressors. The pilot was fully justified in firing his missiles. It turns out he was correct, because the missile the Americans fired into Tehran carried a chemical warhead."
"But—"
"The first reports may be proved inaccurate, Peter," Zevitin said, "but that doesn't mean we can't protest this incident now. I believe Gardner will act first and then check out the facts. You wait and see."
Alexandra Hedrov looked at Zevitin silently for a long moment; then: "What is this all about, Leonid? Do you just want to harass Gardner? What for? He is not worth the effort. He will more likely self-destruct without you constantly...how did you say, 'niggled' him. And certainly you cannot want Russia to align with and support the Iranians. As I said before, they are just as likely to turn on us after they retake their country."
"This has absolutely nothing to do with Iran, Alexandra, and everything to do with Russia," Zevitin said. "Russia will not be encircled and isolated any longer. Gryzlov was a megalomaniac, sure, but because of his insane ideas Russia was feared once more. But in their absolute fear, or pity, the world began to give the United States all it wanted, and that was to encircle and try to squash Russia again. I will not allow that to happen."
"But how will deploying these anti-spacecraft weapons accomplish this?"
"You don't understand, Alexandra—threatening war against the Americans will only serve to increase their resolve," Zevitin explained. "Even a spineless fop like Gardner will fight if his back is forced against the wall—at the very least, he'll turn his junkyard dog McLanahan loose on us, as much as he resents his power and determination.
"No, we must make the Americans themselves believe they are weak, that they must cooperate and negotiate with Russia to avoid war and disaster," Zevitin went on. "Gardner's hatred—and fear—of McLanahan is the key. To make himself look like the brave leader he can never be, I'm hoping Gardner will sacrifice his greatest general, dismantle his most advanced weapon systems, and retreat from important alliances and defensive commitments, all on the altar of international cooperation and world peace."
"But why? To what end, Mr. President? Why risk war with the Americans like this?"
"Because I won't stand to see Russia encircled," Zevitin said sharply. "Just look at a damned map, Minister! Every former Warsaw Pact country is a member of the North Atlantic Treaty Organization; almost every former Soviet republic has a NATO or American base of some kind on it."
Zevitin went to light up another cigarette, but threw them across his desk in blind anger. "We are wealthy beyond the dreams of our fathers, Alexandra, and yet we can't spit without the Americans complaining, measuring, analyzing, or intercepting it," he cried. "If I wake up and see that damned space station shooting across the sky—my Russian sky!—once more, I am going to scream! And if I see another youngster on the streets of Moscow watching an American TV show or listening to Western music because he or she has free Internet access courtesy of the American domination of space, I will kill someone! No more! No more! Russia will not be encircled, and we will not be smothered into submission by their space toys!
"I want Russian skies cleared of American spacecraft, and I want our airwaves cleansed of American transmissions, and I don't care if I have to start a war in Iran, Turkmenistan, Europe, or in space to do it!"
ABOARD ARMSTRONG SPACE STATION
A SHORT TIME LATER
"Stud Zero-Seven is ready to depart, sir," Master Sergeant Lukas reported.
"Thanks, Master Sergeant," Patrick McLanahan responded. He flipped a switch on his console: "Have a good trip home, Boomer. Let me know how the module release experiments and new re-entry procedure works."
"Will do, sir," Hunter Noble responded. "Feels weird not having you on board flying the jet."
"At least you get to pilot it this time, right?"
"I had to arm-wrestle Frenchy for it, and it was close—but yes, I won," Boomer said. He got an exasperated glance in his rear-cockpit camera from U.S. Navy Lieutenant Commander Lisette "Frenchy" Moulain, an experienced F/A-18 Hornet combat pilot and NASA space shuttle mission commander and pilot. She had recently qualified to be spacecraft commander of the XR-A9 Black Stallion spaceplane and was always looking for another chance to pilot the bird, but none of her arguments worked this time on Boomer. When Patrick flew to and from the station—which was quite often recently—he usually picked Boomer to be his backseater.
Minutes later the Black Stallion detached from the docking bay aboard Armstrong Space Station, and Boomer carefully maneuvered the craft away from the station. When they were far enough away, he maneuvered into retrorocket firing position, flying tailfirst. "Countdown checklists complete, we're in the final automatic countdown hold," he announced over intercom. "We're about six hundred miles to touchdown. Ready for this one, Frenchy?"
"I've already reported my checklists are complete, Captain," Moulain responded.
Boomer rolled his eyes in mock exasperation. "Frenchy, when we get back home, we need to sit down at a nice bar somewhere on the Strip, have an expensive champagne drink, and talk about your attitude—toward me, toward the service, toward life."
"Captain, you know very well that I'm engaged, I don't drink, and I love my work and my life," Moulain said in that same grinding hair-pulling monotone that Boomer absolutely hated. "I might also add, if you haven't realized it by now, that I hate that call-sign, and I don't particularly care for you, so even if I was unattached, drank alcohol, and you were the last man on earth with the biggest cock and longest tongue this side of Vegas, I wouldn't be seen dead in a bar or anywhere else with you."
"Ouch, Frenchy. That's harsh."
"I think you're an outstanding spacecraft commander and engineer and a competent test pilot," she added, "but I find you a disgrace to the uniform and I often wonder why you are still at Dreamland and still a member of the United States Air Force. I think your skill as an engineer seems to overshadow the partying, hanging out at casinos, and the constant stream of women in and out of your life—mostly out—and frankly I resent that."
"Don't hold back, Commander. Tell me how you really feel."
"Now when I report 'checklist complete,' Captain, as you fully well know, that indicates that my station is squared away, that I have examined and checked everything I can in your station and the rest of the craft and found it optimal, and that I am prepared for the next evolution."
"Oooh. I love it when you talk Navy talk. 'Squared away' and 'evolution' sound so nautical. Kinda kinky too, coming from a woman."
"You know, Captain, I put up with your nonsense because you're Air Force and this is an Air Force unit, and I know Air Force officers always act casually around each other, even if there's a great difference in rank," Moulain pointed out. "You're also the spacecraft commander, which puts you in charge despite the fact that I outrank you. So I'm going to ignore your sexist remarks during this mission. But it certainly doesn't change my opinion of you as a person and as an Air Force officer—in fact, it verifies it."
"Sorry. I didn't catch all that. I was busy sticking pencils in my ears to keep from listening to you."
"Can we follow the test flight plan and just do this, Captain, without all the male macho bullshit nonsense? We're already thirty seconds past the planned commencement time."
"All right, all right, Frenchy," Boomer said. "I was just trying to act like we're part of a crew and not serving on separate decks of a ship in the nineteenth-century Navy. Pardon me for trying." He pressed a control stud on his flight control stick. "Get me out of this, Stud Seven. Begin powered descent."
"Commencing powered descent, stop powered descent..." When the computer did not receive a countermanding order, it began: "Commencing deorbit burn in three, two, one, now." The Laser Pulse Detonation Rocket System engines, or LPDRS, pronounced "leopards," activated and went to full power. Burning JP-7 jet fuel and hydrogen peroxide oxidizer with other chemicals and superheated pulses from lasers to increase the specific impulse, the Black Stallion's four LPDRS engines produced twice as much thrust as all of the engines aboard the space shuttle orbiters combined.
As the spacecraft slowed, it began to descend. Normally at a certain velocity Boomer would shut down the main engines and then turn the spacecraft using its thrusters to a forward-flying nose-high attitude and prepare for "entry interface," or the first encounter with the atmosphere, and then use aerobraking—scraping the shielded underside against the atmosphere—to slow down for landing. This time, however, Boomer kept it flying tailfirst and the LPDRS engines running at full power.
Most spacecraft could not do this for long because they didn't carry enough fuel, but the Black Stallion spaceplane was different: because it refueled while on Armstrong Space Station, it had as much fuel as it would have when blasting into orbit, which meant it could keep its engines running for much longer periods during re-entry. Although aerobraking was much more fuel-efficient, it had its own set of hazards—namely, the intense heat of friction that built up on the underside of the spacecraft—so the crew was trying a different re-entry method.
As the Black Stallion slowed even more, the descent angle got steeper, until it seemed as if they were pointed straight up. The flight and engine control computers adjusted power to maintain a steady 3-G deceleration force. "I hate to ask," Boomer grunted through the G-forces pressing his body back into his seat, "but how are you doing back there, Frenchy? Still optimal?"
"In the green, Captain," Frenchy responded, forcing her breath through constricted throat muscles in order to keep her abdominal muscles tight, which increased blood pressure in her head. "All systems in the green, station check complete."
"A very squared-away report, thank you, M. Moulain," Boomer said. "I'm optimal up here too."
Passing through Mach 5, or five times the speed of sound, and just before reaching the atmosphere at approximately sixty miles' altitude, Boomer said, "Ready to initiate payload separation." His voice was much more serious now because this was a much more critical phase of the mission.
"Roger, payload separation coming up...program initiated," Moulain responded. The cargo bay doors on top of the Black Stallion's fuselage opened, and powerful thrusters pushed a BDU-58 container out of the bay. The BDU-58 "Meteor" container was designed to protect up to four thousand pounds of payload as it descended through the atmosphere. Once through the atmosphere the Meteor could glide up to three hundred miles to a landing spot, or release its payload before impacting the ground.
This mission was designed to show that the Black Stallion spaceplanes could quickly and accurately insert a long-duration reconnaissance aircraft anywhere on planet Earth. The Meteor would release a single AQ-11 Night Owl unmanned reconnaissance aircraft about thirty thousand feet altitude near the Iran-Afghanistan border. For the next month, the Night Owl would monitor the area with imaging infrared and millimeter-wave radars for signs of Muslim insurgents crossing the border, or Iranian Revolutionary Guards Corps or al-Quds convoys smuggling in weapons or supplies from neighboring countries.
After the Meteor container was away, Boomer and Frenchy continued their powered descent. The atmosphere made the spaceplane slow down much more quickly, and soon the LPDRS engines were throttling back to maintain the maximum 3-G deceleration. "Hull temperatures well within the green," Moulain reported. "I sure like these powered descents."
Boomer fought off the G-forces, reached out, and patted the top of the instrument panel. "Good spaceship, nice spaceship," he cooed lovingly. "She likes these powered descents too—all that heat on the belly is not nice, is it, sweetie? Did I tell you, Frenchy, that those 'leopards' engines were my idea?"
"Only about a million times, Captain."
"Oh yeah."
"Air pressure on the surface is up to green...computers are securing the reaction control system," Moulain reported. "Mission-adaptive control surfaces are in test mode...tests complete, MAW system responding to computer commands." The MAW, or Mission Adaptive Wing, system was a series of tiny actuators on the fuselage that in essence turned the entire body of the spaceplane into a lift or drag device—computers shaped the skin as needed to maneuver, climb or descend, make the craft slipperier, or slow down quickly. Even flying backward, the MAW system allowed complete control over the spaceplane. With the atmospheric controls active, Boomer took control of the Black Stallion himself, turned so they were flying forward like a normal aircraft, then hand-flew the ship through a series of steep, high angle-of-attack turns to help bleed off more speed while keeping the descent rate and hull temperatures under control.
At the same time, he was maneuvering to get into position for landing. This landing was going to be a bit trickier than most, because their landing spot was in southeast Turkey at a joint Turkey-NATO military base at a city named Batman. Batman Air Base was a Special Operations Joint Task Force base during the 1991 Gulf War, with American Army Special Forces and Air Force pararescue troops running clandestine missions throughout Iraq. It was returned to Turkish civil control after the war. In a bid for greater cooperation and better relations with its fellow Muslim nations in the Middle East, Turkey forbade NATO offensive military operations to be staged from Batman, but America had convinced the Turks to allow reconnaissance and some strike aircraft to fly from Batman to hunt down and destroy insurgents in Iran. It was now one of the most vital forward air bases for American and NATO forces in the Middle East, eastern Europe, and central Asia.
"Passing sixty thousand feet, atmospheric pressure in the green, ready to secure the 'leopards,'" Moulain said. Boomer chuckled—securing the "leopards" and transitioning to air-breathing turbojet mode was done automatically, as were most operations on the spaceplane, but Moulain always tried to pre-guess when the computer would initiate the procedure. Cute, yes—but she was generally correct, too. Sure enough, the computer notified him that the LPDRS engines were secure. "We're still in 'manual' mode, Captain," Moulain reminded him. "The system won't restart the engines automatically."
"You're really on top of this stuff, aren't you, Frenchy?" Boomer quipped.
"That's my job, Captain."
"You're never going to call me 'Boomer,' are you?"
"Unlikely, Captain."
"You don't know what you're missing, Frenchy."
"I'll survive. Ready for restart."
Part of her allure was definitely the chase. Maybe she was all businesslike in bed too—but that was going to have to wait for a time when they weren't seated in tandem. "Unspiking the engines, turbojets coming alive." They had enough oxygen in the atmosphere now to stop using hydrogen peroxide to burn jet fuel, so Boomer reopened the movable spikes in the engine inlets and initiated the engine start sequence. In moments the turbojets were idling and ready to fly. Their route of flight was taking them over central Europe and Ukraine, and now they were over the Black Sea, heading southeast toward Turkey. Along with keeping hull temperatures low, the powered descent procedures allowed them to descend out of orbit much quicker—they could come down from two hundred miles' altitude into initial approach position, called the "high gate," in less than a thousand miles, where a normal aerobraking descent might take almost five thousand miles.
Below sixty thousand feet they were in Class A positive control airspace, so now they had to follow all normal air traffic control procedures. The computer had already entered the proper frequency in the number one UHF radio: "Ankara Center, this is Stud Seven, due regard, one hundred twenty miles northwest of Ankara, passing flight level five-four-zero, requesting activation of our flight plan. We will be MARSA with Chevron Four-One."
"Stud Seven, Ankara Center, remain outside Turkish Air Defense Identification Zone until radar identified, squawk one-four-one-seven normal." Boomer read back all the instructions.
At that moment, on their secondary encrypted radio, they heard: "Stud Seven, Chevron Four-One on Blue Two."
Boomer had Frenchy monitor the air traffic control frequency, then switched to the secondary radio: "Four-One, this is Stud Seven." They performed a challenge-and-response code exchange to verify each other's identity, even though they were on an encrypted channel. "We launched out of Batman because we heard from Ankara ATC that they are not letting any aircraft cross their ADIZ, even ones with established flight plans. We don't know what's going on, but usually it's because an unidentified aircraft or vessel drifted into their airspace or waters, or some Kurds fired some mortars across the border, and they shut everything down until they sort it out. We're coming up on rendezvous point 'Fishtail.' Suggest we do a point-parallel there, then head out to MK."
"Thank you for staying heads-up, Four-One," Boomer said, the relief obvious in his voice. Using the powered descent profile grossly depleted their fuel reserves—they were almost bingo fuel right now, and by the time they reached the initial approach fix at Batman Air Base they'd be in an emergency fuel status, and they would have no fuel to go anywhere else. Their closest alternate landing site was Mi-hail Kogălniceanu Airport near Constanţa, Romania, or simply "MK" for short, the first U.S. military base established in a former Warsaw Pact country.
With the two aircraft linked via the secure transceiver, their multi-function displays showed them each other's position, the track they had to follow to rendezvous, and the turnpoints they'd need to get into position. The Black Stallion reached the Air Refueling Initial Point fifteen minutes early, four hundred knots too fast, and thirty thousand feet too high, so Boomer started a series of high-bank turns to bleed off the excess airspeed. "I love it—boring holes in the sky, flying around in the fastest manned aircraft on the planet."
"Odin to Stud Seven," Boomer heard on his encrypted satellite transceiver.
"It's God on GUARD," he quipped. "Go ahead, Odin."
"You're cleared to proceed to MK," Patrick McLanahan said from Armstrong Space Station. He was monitoring the spaceplane's progress from the command module. "Crews are standing by to secure the Black Stallion."
"Do I have to have someone back home looking over my shoulder from now on?" he asked.
"That's affirmative, Boomer," Patrick responded. "Get used to it."
"Roger that."
"Any idea why Ankara wasn't letting anyone in, sir?"
"This is Genesis. Still negative," David Luger chimed in. "We're still checking."
Eventually the Black Stallion was able to slow down and descend to get into proper position, five hundred feet below and a half mile behind the tanker. "Stud Seven is established, checklist complete, got you in sight, ready," Boomer reported.
"Roger, Seven, this is Chevron Four-One," the boom operator in the tanker's tail pod responded. "I read you loud and clear, how me."
"Loud and clear."
"Roger that. I have a visual on you too." On intercom, he said, "Boom's lowering to contact position, crew," and he motored the refueling book into position, its own steerable fly-by-wire wings stabilizing it in the big tanker's slipstream. Back on the radio: "Seven is cleared to precontact position, Four-One is ready."
"Seven's moving up," Boomer said. He opened the slipway doors atop the fuselage behind the cockpit, then smoothly maneuvered the spaceplane to the precontact position: aligned with the tanker's centerline, the top of the windscreen on the center seam of the director light panel. The immense belly of the converted Boeing 777 filled the windscreen. "Seven's in precontact position, stabilized and ready, JP-7 only this time," he said.
"Copy precontact and ready, JP-7 only, cleared to contact position, Four-One ready," the boom operator said. He extended the nozzle and set the "maneuver" light blinking, the signal for the receiver to move into position. Boomer barely had to move the controls because the plane was so light—almost as if just by thought, he carefully maneuvered the Black Stallion forward and up. When the maneuver light turned steady, Boomer held his position, again as if by thought only, and the boom operator slid the nozzle into the receptacle. "Contact, Four-One."
"Seven has contact and shows fuel flow," Boomer acknowledged. "You're a very welcome sight, boys."
"We're a Cabernet crew, sir," the tanker pilot said.
It took the KC-77 ten minutes to transfer thirty thousand pounds of jet fuel to the Black Stallion. "Let's start heading west, Four-One," Boomer said. "We're starting to get too close to Krasnodar." Krasnodar on the east coast of the Black Sea was the location of a major Russian air base, and although they were well outside theirs or anyone else's airspace, it was best not to fly around such areas unannounced. Along with their big air defense radar and numerous long-range surface-to-air missile batteries, Krasnodar was one of the largest fighter bases in the entire world, with no less than three full air defense fighter wings based there, including one with the Mikoyan-Gurevich MiG-29 "Fulcrum," considered one of the best interceptors in the world.
Even four years after the American retaliatory attacks in Russia, nerves were still frayed throughout the entire region, and operators were on a hair trigger to scramble fighters and activate air defense systems. Luckily, there were no signs of any air defense activity behind them. "Right turn is best."
"Coming right to two-seven-zero," the tanker's pilot said. Boomer expertly banked behind the modified Boeing 777 aircraft as they started to turn south, maintaining contact in the turn.
They had just rolled out on the new heading when the tanker's boom operator said, "Well well, folks, looks like we have a visitor. Seven, your three o'clock, real damned close."
"What is it, Frenchy?" Boomer asked, concentrating on staying in the refueling envelope.
"Oh shit...it's a Russian MiG-29," Moulain said nervously, "three o'clock, less than a half mile, right on our wingtip."
"See if he has a wingman," Boomer said. "Russkies don't fly around single-ship too often."
Moulain scanned the sky, trying to stay calm, straining to look as far back as she could. "Got him," she said moments later. "Seven o'clock, about a mile." The one at three o'clock slid closer, riveting her attention. In her fifteen-year Navy career she had never seen a MiG-29 except the ones in service in Germany, and that was on a static display, not inflight. It could've been a fixed-wing clone of the F-14 Tomcat Navy carrier fighter, with broad wings, beefy fuselage, and a large nose for its big fire control radar. This one was in green, light blue, and gray camouflage stripes, with the big white, blue, and red flag of Russia on the vertical stabilizer—and she could clearly see one long-range and two short-range air-to-air missiles hanging off the MiG's left wing. "He's loaded for bear, that's for sure," she said nervously. "What are we going to do?"
"I'm going to finish getting my gas," Boomer said, "and then we're going to proceed to landing at MK. This is international airspace; sightseeing is allowed. Let Genesis and Odin know what's out there."
Boomer could hear Frenchy on the number two radio talking to someone, but she stopped a moment later: "That prick at three o'clock's moving closer," she said nervously.
"How are we doing on gas?"
"Three-quarters full."
"We got enough to get to MK with reserves?"
"Plenty."
"I want to top 'em off just in case. How close is the MiG now?"
"He's right on our right wingtip," Frenchy said. "You going to do a disconnect, Captain?"
"Nah. I'm showing him how it's done. No doubt he wants a glimpse of the future too." But the little game wasn't over. The MiG-29 kept on coming closer until shortly Boomer could hear his engine roar and vibration outside his cockpit canopy. "Okay, now he's starting to piss me off. How are we on gas?"
"Almost full."
"Where's the wingman?"
Moulain began to shift in her seat so she could turn all the way around to her left again...but soon found she didn't have to, because the second MiG had zoomed forward and was now sitting right off the tanker pilot's left cockpit window, close enough for his engine exhaust and jet wash to shake the tanker's left wing, barely noticeably at first but soon more violently as the MiG slid closer.
"Seven, this is Four-One. It's getting hard to keep it under control. What do you say?"
"Bastard," Boomer muttered. "Time to call it quits." On the radio he responded, "Four-One, let's do a disconnect and—"
But at that moment the second MiG to the left of the tanker's cockpit stroked its afterburners, its exhausts just yards away from the tanker's left wing's leading edge, causing the wing to shove first violently downward, then upward, causing the tanker to roll right. "Breakaway, breakaway, breakaway!" the boom operator shouted on the radio. Boomer immediately chopped the throttles, hit the voice command button, and spoke, "Speed brakes seventy!" The Mission Adaptive Wing system immediately commanded a maximum drag setting, creating thousands of little speed brakes all over the spaceplane's surface and allowing it to sink quickly...
...and it wasn't a moment too soon, because the tanker pilot, struggling with his plane's controls and at the same time jamming on full military power and a thirty-degree climb angle when he heard the "breakaway" call, had overcorrected and was now violently rolling to his left, in the grip of a full power-on stall and on the verge of a tail-low spin. Boomer could swear he was going to be face-to-face with the boom operator as he saw the tanker's tail swing lower and lower toward him. "C'mon, Chevron, recover, dammit, recover...!"
The KC-77 tanker seemed to be doing a pirouette on the tip of the still-extended refueling boom, rolling left and right as if clawing the sky for a handhold, its wings fluttering like a giant osprey in a climb, except the tanker wasn't climbing but was getting ready to roll over and spin out of control at any second. Just when Boomer thought it was going to roll over on its back and dive uncontrollably into the Black Sea, it stopped its death's oscillations, the left wing stayed down, and the nose started to creep toward the horizon. As the nose dipped below the horizon, the right wing slowly, agonizingly started to come down. When the tanker disappeared from view, it was almost wings-level, steeply nose-low but quickly regaining its lost airspeed.
"Chevron, you guys okay?" Boomer radioed.
A few moments later he heard a high, squeaky, hoarse male voice say, "I got it, I got it, holy shit, I got it...Seven, this is Four-One, we're okay. Man oh man, I thought we were goners. We're at twelve thousand feet. We're okay. One engine flamed out, but we're restarting now."
Boomer scanned the sky and saw the two MiG-29s joined up far above him, heading east. He could almost hear them laughing over their radios on the little scare they put into the Americans. "You motherfuckers!" he shouted into his oxygen visor, and he shoved the throttles forward to max afterburner.
"Noble! What are you doing?" Moulain shouted when she had gotten her breath back after the sudden G-force shove to her chest. But it was soon obvious what he was doing—he was flying right for the middle of the MiG formation. By the time she could cry out, they had blasted past the two MiGs, passing less than a hundred yards above them, traveling more than seven hundred miles an hour! "Jesus, Noble, are you insane?"
Boomer pitched the Black Stallion into a steep sixty-degree climb, still accelerating. "We're going to see if they like scrapping with the other alley cats or if they just pick on the big fat tabbies," he said. The threat warning receiver blared—the MiGs had been running radar-silent until now, which is how they were able to sneak up on their formation so easily, but now they had their big N-019 radar on and searching. Boomer leveled off at forty thousand feet, pulled the throttles back to military power, and switched his multifunction display to the threat depiction, which gave him his best picture of the situation. "Keep an eye on my fuel and let me know when we're getting close to bingo fuel on MK, Frenchy."
"Stud, this is Odin," Patrick McLanahan radioed from Armstrong Space Station. "We just picked up the threat warning. You've got two MiGs behind you! Where are you going?"
"I'm going to drag these guys east as much as possible so they'll stay away from the tanker," Boomer said, "and I'm going to teach them a lesson about screwing with a Black Stallion and especially its tanker."
"Do you know what you're doing, Boomer?" Patrick asked.
"I'm hoping these guys will take a shot at me, General," Boomer said, "and then I'm really going to water their eyes. Any other questions, sir?"
There was a slight pause, during which time Moulain was positive the general would be swearing a blue streak and literally bouncing off the ceiling of the command module in pure anger at Noble's adolescent stunt. To her shock, she heard McLanahan reply: "Negative, Boomer. Just try not to scratch the paint."
"Fifteen minutes to bingo fuel at this rate and course, SC," Moulain reported. "Stop this shit and turn us around!"
"Five more minutes and we'll do a U-turn, Frenchy," Boomer said, then muttered, "C'mon, you chickenshits, shoot already. We're right dead in your sights and we're not jamming—take the—"
At that instant the two "bat-wing" symbols on the threat warning display depicting the MiG's search radars started to blink. "Warning, warning, missile alert, six o'clock, twenty-three miles, MiG-29K..." followed moments later by: "Warning, warning, missile launch, missile launch, AA-12!"
"Here we go, Frenchy—hold on to your bloomers," Boomer said. He jammed the throttles to max afterburner, then spoke, "Leopards online."
"Leopards online, stop leopards...leopards activated," the computer responded, and both crewmembers were shoved back into their seats when the full force of the Laser Pulse Detonation Rocket System engines fired up in full turbojet mode—with the throttles already in full afterburner, rather than moving them up gradually, they got almost full turbojet power in just a few seconds. The airspeed jumped from just below Mach 1 to Mach 2, then 3, then 4 in the blink of an eye. He then started a steep climb, then kept the pitch input in until they were headed straight up, passive fifty, then sixty thousand feet.
"Missiles...still...tracking," Moulain grunted through the nearly seven Gs. "Still...closing..."
"I'm almost...done...with these bozos, Frenchy," Boomer grunted back. He pulled the power back at Mach 4 and kept pulling on the control stick until they were inverted. He rolled upright, his nose now aimed down almost directly vertical, then glanced at the threat display. As he hoped, the two MiGs were still transmitting radar energy, searching for him—the AA-12 missile, a copy of the American AIM-120 Advanced Medium-Range Air-to-Air Missile, was homing in using its own on-board radar.
"Wondering where I went, boys? You'll find out in a sec." Boomer aimed the Black Stallion at a point in space where he thought the MiGs would be in the next heartbeat or two—at his relative speed, the MiGs appeared to be hovering in space, although the threat display said they were flying at almost twice the speed of sound. Just as he caught a glimpse of the black dots below him, he rolled left until he was knifing right between the two Russian jets. He had no idea if he had judged the turn correctly, but it was too late to worry now...
The MiGs were nothing more than imperceptible blurs as he flew directly between them, missing the closest by just fifty yards. As soon as he passed them he pulled the throttles to idle, deactivated the LPDRS engines to conserve fuel, used the MAW system to assist the spaceplane to level off without breaking itself into pieces—at their current rate of speed they would hit the Black Sea in just eight seconds without the Mission Adaptive Wing technology—and started a tight left turn just in case the AA-12 missiles were still tracking...
...but he didn't have to worry about the missiles, because moments later they caught a glimpse of a large flash of light above them, then another. He rolled upright, let the G-forces subside, and scanned the sky. All they could see were two black clouds above them. "Payback's a bitch, huh, comrades?" Boomer said as he headed westbound once again.
They had to chase down the tanker again and refuel because they had reached emergency fuel status in just a couple minutes with the LPDRS engines activated. The tanker crew was jubilant, but Moulain was even more quiet and businesslike than usual—she said nothing else except required call-outs. "You guys okay, Four-One?" Boomer asked.
"We got our dentures loosened big-time," the tanker pilot said, "but it's better than the alternative. Thanks, Stud."
"You can thank us by giving us a little more gas so we can make it to MK."
"As long as we have enough to make it to the nearest runway, you can have the rest," the tanker pilot said. "And don't even think about buying any drinks for any other gas-passer anywhere on the planet—your money's no good with us anymore. Thanks again, Stud Seven."
Less than an hour later the two aircraft made their approach and landing at Constanţa-Mikhail Kogălniceanu Airport in Romania. The airport was located fifteen miles from Constanţa and nine miles from the city's famed Mamaia Beach on the Black Sea, so it was rarely affected by the freezing fog that shrouded the coastal city in winter. The U.S. Air Force had built an aircraft parking ramp, hangars, and maintenance and security facilities on the northeast side of the airfield, as well as upgraded the airport's control tower, radar and communications facilities, and civil airport terminal. Along with NATO and European Union membership, the investments made in Romania by the United States had quickly turned this area known before only for its busy seaport and historic sites into a major international business, technology, and tourist destination.
The two aircraft were escorted to the security area by a small convoy of armored Humvees and parked together in the largest hangar. There was a lot of hugging and handshakes between the crews as they deplaned. They debriefed their mission together and then separately, with promises to meet up for dinner and drinks later in Constanţa.
Noble and Moulain's mission debriefing took considerably longer than the tanker crew's. It took nine grueling hours to debrief the maintenance and intelligence crews, Patrick McLanahan on Armstrong Space Station, Dave Luger at Dreamland, and get their usual post-flight physical exams. When they were finally released, they cleared Romanian customs at the civil airport, then took a shuttle bus to the Best Western Savoy Hotel in Constanţa, where the U.S. military contracted for transient lodging.
The Black Sea coast was not busy at all in winter, so except for a few airline crews from Romania, Germany, and Austria and some surprised businessmen unaccustomed to seeing much partying in Constanţa in winter, the Americans had the bar to themselves. The tanker crew had already been partying hard and was buying drinks for anyone who wore wings, especially the foreign female flight attendants. Boomer was ready as well, but to his surprise he saw Lisette heading for the elevator to her room. He extricated himself from the arms of two beautiful blond flight attendants, with promises he'd be right back, and hurried to follow her.
He barely squeezed himself past the closing elevator doors. "Hey, Frenchy, turning in so soon? The party's just getting started, and we haven't had dinner yet."
"I'm beat. I'm done for the day."
He looked at her with concern. "You haven't said much since our little run-in with the Russkies," he said. "I'm a little—"
Suddenly Moulain whirled toward him and smacked him across the jaw with a closed right fist. It wasn't that hard a blow, but it was still a fist—he was smarting, but mostly from the surprise. "Hey, what'd you do that for?"
"You bastard! You prick!" she shouted. "You could've gotten us both killed today out there!"
Boomer rubbed his jaw, still looking at her with concern; then he nodded and said, "Yeah, I could have. But no one pushes around my tanker." He smiled, then added, "Besides, you gotta admit, Frenchy, that it was one helluva ride."
Moulain looked as if she was going to punch him again, and he was determined to let her do it if it made her feel better...but to his surprise, she rushed forward in the elevator, threw her arms around his neck, smothered him with a kiss, and pressed herself against him, pinning him against the wall.
"You're damned right, Boomer, it was one helluva ride," she breathed. "I've flown jets off of carriers in two wars and been shot at dozens of times, and I have never been so turned on as I was today!"
"Jeez, Moulain..."
"Frenchy. Call me Frenchy, dammit," she ordered, then silenced him with another kiss. She didn't let him up for air for a long time.
"You were so quiet on the way back and in debriefing, I was afraid you were going into some kind of shell-shocked fugue state, Frenchy," Boomer said as Moulain started kissing his neck. "You sure have a funny way of showing your excitement."
"I was so excited, so turned on, so friggin' aroused that I was embarrassed to show it," Moulain said in between kisses, her hands quickly finding their way south of his waist. "I mean, two fighter pilots died, but I was so jacked up I thought I was going to come in my damned flight suit!"
"Dang, Frenchy, this is one strange side of you that I never—"
"Shut up, Boomer, just shut up," she said as the elevator slowed on their floor. She had him practically unzipped and unbuttoned by then. "Just take me to my room and fuck my brains out."
"But what about your fiancé and your—?"
"Boomer, I said, shut the hell up and fuck me, and do not stop until it's morning," Moulain said as the elevator doors slid open. "I'll explain it to...to...oh hell, whatever his name is, in the morning. Remember, Captain, I outrank you, so that's an order, mister!" It was obvious that issuing orders was just as much of a turn-on for her as flying the hypersonic spaceplane.
## CHAPTER TWO
One likes people much better when they're battered down by a prodigious siege of misfortune than when they triumph.
—VIRGINIA WOOLF
ARMSTRONG SPACE STATION
THE NEXT MORNING
The command module was the center of activity aboard Armstrong Space Station, and it was here that Patrick McLanahan attended the video teleconference with select members of President Gardner's national security staff: Conrad F. Carlyle, the President's National Security Adviser; Gerald Vista, the Director of Central Intelligence, who had remained in his post from the Martindale administration; Marine Corps General Taylor J. Bain, chairman of the Joint Chiefs of Staff; Charles A. Huffman, Air Force chief of staff; and Air Force General Bradford Cannon, commander of U.S. Strategic Command and—until the details could be worked out by Congress and the Pentagon—the theater commander of all U.S. space operations and responsible for training, equipping, and directing all space combat missions. Hunter Noble—a little bleary-eyed after not very much sleep, both because of the time difference and because of Lisa Moulain—was linked in to the teleconference via satellite from the command post at Constanţa Air Base.
Patrick and Master Sergeant Valerie Lukas were floating in front of the wide-screen high-definition teleconference monitor, secured by Velcro sneakers to the bulkhead of the command module. Patrick kept his hair buzz-cut short, but Lukas's longer hair floated free on either side of her headset's crossband, giving her a weird wolverine-like appearance. "Armstrong Space Station is online and secure, sir," Patrick announced. "This is Lieutenant General Patrick McLanahan, commander, High-Technology Aerospace Weapons Center, Elliott Air Force Base, Nevada. With me is U.S. Air Force Master Sergeant Valerie Lukas, noncommissioned officer in charge of this station and the sensor operator on duty at the time of the attack in Tehran. Joining us via satellite link from Constanţa, Romania, is Air Force Captain Hunter Noble, chief of manned spaceflight operations and hypersonic weapon development, High-Technology Aerospace Weapons Center. He was the officer in charge of the attack mission over Tehran and the designer of the SkySTREAK missile that was used in the attack. He returned to Earth yesterday after completing a reconnaissance aircraft insertion mission over eastern Iran, which we will brief you on later."
"Thank you, General," General Taylor Bain said from the "Gold Room," also known as the "Tank," the Joint Chiefs of Staff conference center on the second floor of the Pentagon. As was the case of most officers in the post–American Holocaust United States, Bain was young for a four-star Marine Corps officer, with dark brown hair trimmed "high and tight," a ready smile, and warm gray eyes that exuded trust and determined sincerity. "Welcome, everyone. I believe you know everyone here. Joining us from the White House is National Security Adviser Conrad Carlyle; and from Langley, the Director of Intelligence, Gerald Vista.
"I first want to say that I'm pleased and frankly more than a little amazed to be talking to you, General McLanahan, aboard a facility that just a few short years ago was considered little more than a Cold War relic at best and a floating money pit at worst," Bain went on. "But now we're considering putting hundreds of billions of dollars into the next five budgets to create a space force centered on that very same weapon system. I'm convinced we're seeing the beginning of a new direction and future for the American armed forces. Captain Noble, I've been briefed on your incident yesterday, and although we need to discuss your judgment skills I'm impressed with how you handled yourself, your crew, your fellow airmen, and your craft. I believe it was yet another example of the amazing capabilities being developed, and the future path we're on looks incredible indeed. But we've got a long way to go before we take that journey, and the events of the past few days will be critical.
"First, we're going to get a briefing from General McLanahan on Armstrong Space Station and his operational tests recently run, and Captain Noble's incident over the Black Sea. We'll discuss a few other matters, and then my staff will prepare our recommendations to SECDEF and the national security staff. I'm sure it will be a long uphill fight, both in the Pentagon and up on Capitol Hill. But regardless of what ensues, Patrick, I'd like to say 'job well done' to you and your fellow airmen—or should I say, fellow 'astronauts.' Please proceed."
"Yes, sir," Patrick began. "On behalf of everyone aboard Armstrong Space Station and our support crews at Battle Mountain Air Reserve Base, Elliott Air Force Base, and Peterson Air Force Base in Colorado, thank you for your kind words and continuing support."
Patrick touched a button that presented photographs and drawings in a separate window to the videoconference audience as he continued: "A brief overview first: Armstrong Space Station was constructed in the late 1980s and early 1990s. It is the military version of the much smaller NASA Skylab space station, built of spent Saturn-I and Saturn-IV rocket fuel tanks joined together on a central keel structure. Four such tanks, each with over thirty thousand cubic feet of space available inside, form the main part of the station. Over the years other modules had been attached to the keel for specialized missions or experiments, along with larger solar panels for increased power generation for the expanding station. We can house as many as twenty-five astronauts on the facility for as long as a month without resupply.
"The station hosts several advanced American military systems, including the first space-based ultra-high-resolution radar, improved space-based global infrared sensors, advanced space-based global communications and high-speed computer networking, and the first space-based anti-missile laser system, code-named 'Skybolt,' designed to shoot down intercontinental ballistic missiles from space. The station's space-based radar is a sophisticated radar system that scans the entire planet once a day and can detect and identify objects as small as a motorcycle, even underground or underwater.
"The destruction of our strategic command and control systems and ballistic missile defense sites by the Russian Federation's air attacks against the United States highlight the need for a capable, secure, and modern base of operations to conduct a wide spectrum of vital defense activities, and Armstrong Space Station is that facility," Patrick continued. "The station is now the central data collection and dissemination hub of a network of satellites in high- and low-Earth orbits linked together to form a global reconnaissance and communications system, continuously feeding a wide array of information to military and government users around the world in real time. The station and its supporting reconnaissance satellites can track and identify targets on the surface, in the sky, on or under water, underground, or in space, and it could direct manned and unmanned defenders against them, like a space-based multifunction combat control system.
"The state-of-the-art systems aboard Armstrong Space Station give it other important capabilities that complement its primary military function," Patrick went on. "In case of war or natural disaster, the station can serve as an alternate national military operations center, similar to the Air Force's E-4B or Navy's E-6B Mercury airborne command posts, and can communicate with ballistic missile submarines even while deeply submerged. It can tie into radio and television airwaves and the Internet worldwide to broadcast information to the public; act as a nationwide air, maritime, or ground traffic control center; or serve as the central coordination facility for the Federal Emergency Management Agency. The station supports the International Space Station, acts as a space rescue and repair service, supports numerous scientific research and education programs, and is, I believe, the inspiration for a general reawakening to the exploration of outer space for young people around the world.
"Currently, Armstrong Space Station hosts twelve systems operators, technicians, and officers, set up very much like the combat crew aboard an airborne command post or sensor operators aboard a radar aircraft. Additional crews are brought aboard as necessary to run specialized missions—the station has accommodations for another dozen personnel, and can be expanded quickly and easily by attaching additional modules brought aloft by the shuttle, the SR-79 Black Stallion spaceplane, the Orion crew expeditionary vehicle, or remotely piloted launch vehicles—"
"Excuse me, General," National Security Adviser Carlyle interjected, "but how is it possible to bring additional modules up to the station on a spaceplane or remotely piloted vehicles?"
"The fastest and easiest way is to use inflatable modules, Mr. Carlyle," Patrick responded.
"'Inflatable'? You mean, not rigid, like a balloon?"
"Like a balloon, only a very high-tech balloon. The technology is based on NASA's 'Transhab' experiments of ten years ago, when inflatable modules were suggested for the International Space Station. The walls of our models are primarily made of electro-reactive material that is flexible like cloth until a current is applied and it's struck, when it hardens into a material that resists impact a thousand times better than steel or Kevlar; this material is backed up with other non-electro-reactive materials that are still many times stronger than steel or Kevlar. Inflatable structures give just enough to absorb energy from impact without damage—you can't ding the walls of these things.
"The stuff is lightweight and easily packed for launch, then easily and remotely inflated in just a few hours. We've already lofted small inflatable modules on the spaceplanes and Orion, and the technology is sound. We haven't lofted a full crew-sized module yet, but that's in the works. Future space stations and perhaps even habitation modules on the Moon or Mars will probably be inflatable." Carlyle didn't look convinced at all, and neither did several other attendees, but he offered no other comments.
Patrick took a sip of water from a squeeze bottle Velcroed to the bulkhead and was amused to find a line of nervous sweat on his upper lip. How many briefings, he thought, had he given during his over two decades of military service? None, he reminded himself wryly, from space before! Briefing four-star generals was nerve-racking enough, but doing it while flying at over seventeen thousand miles an hour over two hundred miles above Earth made it even more challenging.
"Armstrong Space Station is the ultimate expression of taking the 'high ground' and is, I believe, the centerpiece of America's stated goal of maintaining access and control of space," Patrick went on. "It and the Black Stallion spaceplanes constitute the foundation of what I call the U.S. Space Defense Command, an integrated joint services command that manages all space-based offensive and defensive assets and supports terrestrial theater commanders with reliable, high-speed communications, intelligence, reconnaissance, attack, and transportation services from space. Our mission will be to—"
"That's very interesting, General McLanahan," National Security Adviser Carlyle interjected with a wry and rather bemused expression, "and as interesting as the idea was when you first proposed it last year, that sort of organization is still many years down the road—we don't have time to bring back Buck Rogers right now. Can we move on to a discussion of the Iran operations, General Bain?"
"Of course, Mr. Adviser. General McLanahan?"
"Yes, sir," Patrick said expressionlessly—he was quite accustomed to being tuned out, interrupted, and ignored whenever he brought up his idea of the U.S. Space Defense Command. "Along with all of the other advanced technological capabilities of this station, my staff has recently added another: the ability to control remotely piloted tactical aircraft and their weapons from space. We demonstrated the capability of controlling an unmanned supersonic EB-1C Vampire bomber completely from this station throughout all phases of flight, including several aerial refuelings and hypersonic precision-guided weapon deployment, in real time and with complete man-in-the-loop control. Our communications and networking abilities are entirely and quickly scalable and expandable, and I envision the capability of controlling entire air task forces of potentially hundreds of unmanned combat aircraft, from small reconnaissance micro-UAVs to giant cruise missile haulers, right from Armstrong—securely, safely, and virtually unassailable."
Patrick stuck his briefing notes on the bulkhead. "I hope all of you have received my after-action report on the employment of the XAGM-279 SkySTREAK hypersonic precision-guided cruise missile in Tehran," he said. "The attack was a complete success. The operational test was terminated due to the unintended and unfortunate casualties caused by detonation of an apparent chemical weapon warhead on the target rocket. The casualties were caused by the unexpected detonation of the chemical weapon warhead on the insurgent attacker's rocket, not by the SkySTREAK missile, and therefore—"
"And as I stated in my comments on McLanahan's report," Air Force chief of staff General Charles Huffman interjected, "I believe the SkySTREAK weapon was not the appropriate weapon to use and could negatively impact our efforts to de-escalate the conflict in Iran and bring about a negotiated settlement between the warring parties. Iran was not the right place to test that weapon, and it appears to me that General McLanahan skewed his proposal and the weapon's potential effects in order to dramatize his system. Firing SkySTREAK on his restricted ranges in Nevada wouldn't have had such a 'wow' factor as watching one slam into an insurgent pickup truck. Unfortunately, his magic show resulted in the deaths of dozens of innocent civilians, including women and children, by poison gas."
Joint Chiefs chairman Bain shook his head, then looked straight ahead at his videoconference camera. "General McLanahan?" His brow furrowed as he looked at Patrick's image on the videoconference screen: Patrick was taking another deep sip from a squeeze bottle, and seemed to have some difficulty re-Velcroing the bottle to the bulkhead. "Care to respond?"
Patrick nodded, placing a hand up to his mouth to corral an errant drop of water. "Sorry, sir. Even simple tasks like drinking water take a little extra concentration up here. Almost everything requires a conscious effort."
"Understood, Patrick. I've ridden the 'Vomit Comet' a couple times so I know what zero-G can do to someone, but it's nothing like living the experience 24/7." The 'Vomit Comet' was a modified C-135 cargo plane flown on a roller-coaster-like flight path that allowed several seconds of weightlessness for the occupants during its steep descent. "Any comment on General Huffman's report?"
"I didn't think it was necessary for me to respond with a strong denial, sir," Patrick said, "but to make myself perfectly clear: General Huffman's analysis is dead wrong. I assembled the SkySTREAK operational test exactly as delineated in the general's air tasking order: a precision-guided aerial standoff attack force to support Persian anti-insurgent operations with minimal collateral casualties or damage. We didn't stray outside the ATO one iota.
"I'd like to point out a few other things as well, if I may, sir." He didn't wait for permission to continue: "SkySTREAK was approved by the general's operations staff, along with eight other task forces and units that are operating over Tehran and other cities in Free Persia. So far SkySTREAK has been the only unit to successfully engage any insurgents, even though all the other units have access to the Global Hawk's sensor imagery, Armstrong Space Station's automated surveillance system, and even to SkySTREAK's sensor downlinks. In short, sir, SkySTREAK is working."
"And the civilian casualties?"
"A result of the detonation of the insurgent warhead, sir—it wasn't caused by SkySTREAK."
"It was caused by your missile, McLanahan," Huffman interjected. "You were briefed about the possibility of the insurgents using weapons of mass destruction in Tehran and directed to withhold and request enhanced analysis of the target before engaging. You failed to do that, which resulted in unnecessary civilian casualties."
"As I see it, sir, we limited the casualty count by taking out that Ra'ad rocket before the insurgents had a chance to launch it."
"Be that as it may, McLanahan, you failed to follow my directives," Huffman said. "The technology's not at fault here. But because of your error in judgment, the whole program might be shut down."
"I'm not quite ready to shut anything down yet, Charlie," General Bain said. "My staff and I have reviewed the report submitted by General McLanahan and your response, with a special emphasis on the issue of collateral civilian casualties. My intelligence directorate looked at all spectrums of the Global Hawk surveillance video and the space station's own network of sensors. Everyone has concluded that it would have been possible to determine with certainty that the rocket indeed carried a chemical warhead, and that nearby innocent civilians were at risk if the rocket was hit and the warhead detonated and activated." Huffman smiled and nodded confidently...
...until Bain glared at the Air Force chief of staff, held up a hand, and continued: "...if General McLanahan had the time to study freeze-frame hi-res imagery for at least ninety seconds, sitting at a desk at Langley, Beale, or Lackland Air Force Bases instead of falling around planet Earth traveling at seventeen thousand five hundred miles per hour, or if he had taken the time to consult with expert analysts on the ground; and if he wasn't a three-star general officer and an Air Force tactical officer and air weapons expert and wasn't expected to make command decisions such as this. However, if he had taken the time to ask or had decided not to attack, we feel that the loss of life would've been far greater if the rocket had dispersed its deadly payload as designed.
"The civilian loss of life is regrettable and is something we wish to avoid at all costs, but in this case we feel General McLanahan made a proper decision in line with his rules of engagement and is not responsible for the loss of life. Therefore, the command staff will not convene an investigation board on the matter, unless other evidence is brought forward, and considers the case closed. General McLanahan is free to continue his patrols over Iran as directed and as originally planned with the extra patrols added back into the package, and the joint staff recommends to the National Command Authority that he be allowed to do so.
"On a personal note, I wish to commend General McLanahan and his crews for a job well done," Bain added. "I have no idea what the difficulties of working and living in space could be like, but I imagine the stress levels to be enormous and the operating conditions to be challenging to say the least. You and your people are doing a great job in tough circumstances."
"Thank you, sir."
"This concludes my portion of the video teleconference. Mr. Carlyle, any remarks or questions?" Patrick looked at the image of the National Security Adviser, but he was busy talking on the telephone. "Well, it looks like Mr. Carlyle is already busy on another matter, so we'll log off. Thank you, every—"
"Hold on a minute, General Bain," Conrad Carlyle interjected. "Stand by." Carlyle shifted his seat sideways, the camera zoomed back, widening the view to three seats at the conference table in the White House...and a moment later, the President of the United States, Joseph Gardner, took his place with Carlyle, along with White House chief of staff Walter Kordus, a tall but rather slight man who seemed to wear a perpetual scowl.
Cameras—any kind of camera, even relatively lo-res videoconference ones—loved Joseph Gardner. Dark-haired, thin, and square-jawed, he possessed that strange, almost mystical appearance that defied any efforts by anyone to classify him by ethnicity—at the very same time he looked Italian, Iberian, Black Irish, Latin American, even round-eyed Asian—and therefore he appealed to all of them. He exuded immense self-confidence from every pore, and seemed to project authority like laser beams through his dark green eyes. After just a couple years of his two terms in the U.S. Senate, everyone knew he was destined for bigger and better things.
Being from the state of Florida and coming from a long line of Navy veterans, Gardner had always been a big advocate of a strong navy. Nominated for Secretary of the Navy by then-President Kevin Martindale in his first term, Gardner pushed hard for a grand expansion of the Navy, not just in its traditional maritime roles but in a lot of nontraditional ones as well, such as nuclear warfighting, space, tactical aviation, and ballistic missile defense. Just as the Army was America's primary ground fighting service, he argued, with the Marine Corps as a support service, the Navy should be the leader in maritime warfare and tactical aviation, with the Air Force as its support service. His rather radical "out-of-the-box" ideas had many skeptics but nonetheless got a lot of attention and favorable support from Congress and the American people...
...even before the utter devastation of the American Holocaust, in which Russian long-range bombers armed with nuclear-tipped cruise missiles decimated all but a handful of America's intercontinental ballistic missiles and strategic nuclear-capable long-range bombers. In just a few hours, the U.S. Navy suddenly became the one and only service able to project American military power around the globe, and at the same time virtually the sole keeper of America's nuclear deterrent forces, which were seen as absolutely vital for the very survival of the United States of America in its weakened condition.
Joseph Gardner, the "engineer of the American twenty-first century Navy," was suddenly regarded as a true visionary and the nation's savior. In Martindale's second term of office, Gardner was nominated and unanimously confirmed as Secretary of Defense, and he was universally acknowledged as the de facto Vice President and National Security Adviser rolled into one. His popularity soared, and there were few around the world who doubted he would become the next President of the United States.
"Greetings, gents," Gardner said after positioning himself just so before the videoconference camera. "Thought I'd drop in on your little chat here."
"Welcome, Mr. President," Joint Chiefs of Staff chairman Taylor Bain said. He was obviously perturbed at this very unexpected interruption of his meeting, but tried hard not to show it. "We'd be happy to start the briefing over again, sir."
"Not necessary," the President said. "I have information that is pertinent to the purpose of this meeting, and I thought the best and most expeditious way to get it to you was to just break in."
"You're welcome at any time, sir," Bain said. "Please go on. The floor is yours."
"Thanks, Taylor," the President said. "I just got off the phone with Russian president Zevitin. General McLanahan?"
"Yes, sir."
"He claims you fired a missile at one of his reconnaissance planes in international airspace, and when the missile missed you seriously damaged the aircraft with high-powered radioactive beams called T-waves or some such thing. He also claims a missile fired by one of your aircraft killed several dozen innocent civilians in Tehran, including women and children. Care to explain?"
"He's lying, sir," McLanahan replied immediately. "None of that is true."
"Is that so?" He held up a piece of paper. "I have a copy of the Air Force chief of staff's summary of the incident which seems to say pretty much the same thing. So both the president of Russia and the chief of staff are lying, and you're telling me the truth, General? Is that what you want me to believe?"
"We've just discussed the incident and the issues brought forth by General Huffman, sir," Bain said, "and I've ruled that McLanahan acted properly and as directed and was not responsible for the civilian deaths—"
"And as for Zevitin or anyone else at the Kremlin, sir," McLanahan cut in, "I wouldn't believe one word any of them said."
"General McLanahan, scores of innocent Iranians are dead by chemical weapons and a Russian reconnaissance pilot is badly injured by radiation fired at him by one of your bombers," the President retorted. "The world thinks you're starting another shooting war with Russia in the Middle East and is demanding answers and accountability. This is no time for your bigoted attitude." Patrick shook his head and turned away, reaching for his water bottle, and the President's eyes widened in anger. "You have something else to say to me, General?" Patrick turned back to the camera, then looked at his outstretched arm in confusion, as if he had forgotten why he had extended it. "Is something the matter with you, McLanahan?"
"N—no, sir..." Patrick responded in a muted voice. He missed the water bottle, felt for it, grasped it, then used too much force to rip it from its Velcro mooring and sent it spinning across the module.
"What? I can't hear you." Gardner's eyes squinted in confusion as he watched the water bottle fly away out of sight. "What's going on there? Where are you, General? Why are you moving like that?"
"He's on Armstrong Space Station, sir," General Bain said.
"On the space station? He's in orbit? Are you kidding me? What are you doing up there?"
"As the commander of his task force operating from space, I authorized General McLanahan to oversee the operation from the space station," Bain explained, "just as any commanding officer would take charge of his forces from a forward-deployed command ship or—"
"On the bridge or CIC of a destroyer, yes, but not on a damned space station!" President Gardner shot back. "I want him off that thing immediately! He's a three-star general, for God's sake, not Buck Rogers!"
"Sir, if I may, can we address the question of the air strike on the insurgent rocket launcher and the actions against the Russian aircraft?" General Bain said, worriedly looking on as Valerie Lukas checked on Patrick. "We've conducted a review of the reconnaissance data, and we've determined—"
"It couldn't have been a very thorough review if the incident happened just a couple hours ago, General," the President said. He turned to the National Security Adviser seated beside him. "Conrad?"
"It's a preliminary review of the same sensor data from the Global Hawk unmanned recon plane and the space station's radars that General McLanahan and his crew saw before they attacked, sir," Carlyle responded. "General Bain and his experts at the Pentagon reviewed the images as if they had been asked before the attack if the target was legitimate based on the rules of engagement established by us under the attack order, as is required if there is any uncertainty as to the safety to noncombatants due to weapon effects or collateral damage. The videoconference was convened as a preliminary incident review to determine if a more detailed investigation would be warranted."
"And?"
"General Bain has ruled that, although it could have been possible for General McLanahan to anticipate civilian casualties, his order to engage was justified and proper based on the information at hand, the threat of more civilian deaths at the hands of the insurgents, and his authority under the attack plan," Carlyle responded. "He is recommending to the Secretary of Defense and to you that no further investigation is warranted and that McLanahan be allowed to continue his operation as planned, with the full complement of missile launch bombers instead of just one."
"Is that so?" The President paused for a moment, then shook his head. "General Bain, you're telling me that you thought it was proper that McLanahan attack a target knowing that so many civilian noncombatants were nearby, and that such an attack is within the letter and spirit of my executive order authorizing a hunt for insurgents in Iran?" he retorted. "I think you have grossly misinterpreted my orders. I thought I was being very plain and specific: I don't want any noncombatant casualties. Was that not clear to you, General Bain?"
"It was, sir," Bain responded, his jaw hardening and his eyes narrowing under the scolding, "but with the information General McLanahan had at the time, and with the threat posed by these insurgent rockets, I felt he was fully justified in making the decision to—"
"Let's get this straight right here and now, General Bain: I am the commander-in-chief, and I make the decisions," the President said. "Your job is to carry out my orders, and my orders were no civilian casualties. The only proper order in this instance was to withhold because of the numbers of civilians around that launcher. Even if they had been told to leave the immediate area, you should have anticipated that they would be near enough to be hurt or killed by the explosion. They—"
"Sir, there was no explosion, at least not one caused by us," Bain protested. "The SkySTREAK missile is a kinetic-energy weapon only—it was designed to—"
"I don't care what it was designed to do, General—McLanahan knew there were civilians in the immediate area, and according to General Huffman, you were briefed that some rockets might have chemical weapons on them, so he obviously should have withheld. End of discussion. Now what is this about McLanahan firing a missile at the Russian fighter? McLanahan's bombers have air-to-air missiles on them?"
"That's standard defensive armament for the EB-1D Vampire aircraft, sir, but McLanahan didn't—"
"So why did you fire on that Russian reconnaissance plane, General McLanahan?"
"We did not fire any missiles, sir," McLanahan responded as firmly as he could, nodding to Lukas that he was all right, "and it was not a reconnaissance plane: it was a MiG-29 tactical fighter."
"What was it doing up there, McLanahan?"
"Shadowing our bomber over the Caspian Sea, sir."
"I see. Shadowing...as in, performing reconnaissance? Am I interpreting this correctly, General?" Patrick rubbed his eyes and swallowed hard, licking dry lips. "We're not keeping you up, are we, General?"
"No, sir."
"So the Russian aircraft was just performing reconnaissance after all, correct?"
"Not in my judgment, sir. It was—"
"So you fired a missile at it, and it returned fire, and you then hit it with a radioactive beam of some sort, correct?"
"No, sir." But something was wrong. Patrick looked at the camera, but seemed to be having trouble focusing. "It...we didn't..."
"So what happened?"
"Mr. President, the MiG fired on us first," Boomer interjected. "The Vampire just defended itself, nothing more."
"Who is that?" the President asked the National Security Adviser. He turned to the camera, his eyes bulging in anger. "Who are you? Identify yourself!"
"I'm Captain Hunter Noble," Boomer said, getting to his feet, staring in shock at the image of Patrick being helped by Lukas, "and why the hell don't you stop badgering us? We're only doing our jobs!"
"What did you say to me?" the President thundered. "Who the hell are you to talk to me like that? General Bain, I want him fired! I want him discharged!"
"Master Sergeant, what's going on?" Bain shouted, ignoring the President. "What's happening to Patrick?"
"He's having trouble breathing, sir." She found a nearby intercom switch: "Medical detail to the command module! Emergency!" And then she terminated the videoconference with a keypress on the communications control keyboard.
"McLanahan is having a heart attack?" the President exclaimed after the video images from the space station cut off. "I knew he shouldn't be up in that thing! General Bain, what kind of medical facilities do they have up there?"
"Basic, sir: just a medically trained technician and first aid equipment. We've never had anyone have a heart attack on an American military spacecraft."
"Great. Just fucking great." The President passed a hand through his hair in sheer frustration. "Can you get a doctor and some medicine and equipment up there right away?"
"Yes, sir. The Black Stallion spaceplane can rendezvous with the space station in a couple hours."
"Get on it. And terminate those bomber missions over Iran. No more cruise missile shots until I know for sure what happened."
"Yes, sir." Bain's videoconference link cut off.
The President sat back in his chair, loosened his tie, and lit up a cigarette. "What a clusterfuck," he breathed. "We kill a bunch of innocent civilians in Tehran with a hypersonic missile fired from an unmanned bomber controlled from a military space station; Russia is screaming mad at us; and now the hero of the American Holocaust has a damned heart attack in space! What's next?"
"McLanahan's situation might turn out to be a blessing in disguise, Joe," Chief of Staff Walter Kordus said. He and Carlyle had known Joseph Gardner since their years in college and Kordus was one of the few allowed to ever address the President by his first name. "We've been looking for ways to cut funding for the space station despite its popularity in the Pentagon and Capitol Hill, and this might be it."
"But it has to be done delicately—McLanahan is too popular with the people to be used as an excuse to cut his favorite program, especially since he's been touting it all over the world as the next big thing, the impregnable fortress, the ultimate watchtower, yada yada yada," the President said. "We have to get some congressmen to raise the question of safety on that space station, and if it needs to be manned at all in the first place. We'll have to 'leak' this incident to Senator Barbeau, the Armed Services Committee, and a few others."
"That won't be hard," Kordus said. "Barbeau will know how to stir things up without slamming McLanahan."
"Good. After it comes out in the press, I want to meet with Barbeau privately to discuss strategy." Kordus tried hard to control his discomfort at that order. The President noted his friend and chief political adviser's warning tenseness and added quickly, "Everyone's going to have their hand out for the money once we start the idea of killing that space station, and I want to control the begging, whining, and arm-twisting."
"Okay, Joe," Kordus said, not convinced by the President's hasty explanation, but not wanting to press the issue. "I'll set it up."
"You do that." He took a deep drag of his cigarette, crushed it out, then added, "And we need to get our ducks in a row soon, just in case McLanahan kicks the bucket and Congress kills his program before we can divvy up his budget."
## CHAPTER THREE
One does what one is; one becomes what one does.
—ROBERT VON MUSIL
AZADI SQUARE, OUTSIDE MEHRABAD INTERNATIONAL AIRPORT, TEHRAN, DEMOCRATIC REPUBLIC OF PERSIA
DAYS LATER
"No bread, no peace! No bread, no peace!" the protesters chanted over and over again. It seemed the crowd, now numbering around two or three hundred, was growing bigger and exponentially louder by the minute.
"If they have no bread, where do they get all the energy to stand out here and protest?" Colonel Mostafa Rahmati, commander of the Fourth Infantry Brigade, muttered as he studied the security barriers and observed the crowds getting ever closer. Just two weeks earlier, Rahmati, a short, rather round man with bushy dark hair that seemed to grow thickly across every inch of his body except the top of his head, was executive officer of a transportation battalion, but the way commanding officers were disappearing—presumably killed by insurgents, although no one could rule out desertion—promotions came quickly and urgently in the army of the presumptive Democratic Republic of Persia.
"More smoke," one of Rahmati's lookouts reported. "Tear gas, not an explosion." Seconds later, they heard a loud bang! strong enough to rattle the windows of the airport office building he and his senior staff members were seated in. The lookout sheepishly glanced at his commanding officer. "A small explosion, sir."
"So I gather," Rahmati said. He didn't want to show any displeasure or exasperation—two weeks ago he wouldn't have been able to tell a grenade explosion from a loud fart. "Watch the lines carefully—it could be a diversion."
Rahmati and his staff were on the upper floor of an office building that once belonged to the Iranian Ministry of Transportation at Mehrabad International Airport. Since the military coup and the start of the Islamist insurgency against the military government in Iran, the coup leaders had decided to take over Mehrabad Airport and had established a tight security perimeter around the entire area. Although most of the city east of Tehran University had been left to the insurgents, taking over the airport turned out to be a wise decision. The airport was already highly secure; the open spaces around the field were easy to patrol and defend; and the airport could be kept open to receive and send supplies by air.
Besides, it was often pointed out, if the insurgents ever got the upper hand—which could be any day now—it would be that much easier to get the hell out of the country.
The windows rattled again, and heads turned farther southeast along Me'raj Avenue northeast toward Azadi Square, about two kilometers away, where another billow of smoke, this one topped with a crown of orange fire, suddenly rose. Bombings, arson, intentional accidents, mayhem, and frequent suicide bombings were commonplace in Tehran, and none more common than the area between Mehrabad Airport, Azadi Square, and the famous Freedom Tower, the erstwhile "Gateway to Iran." Freedom Tower, first called Shahyad Tower, or the King's Tower, commemorating the two thousand five hundredth anniversary of the Persian Empire, was built in 1971 by Shah Reza Pahlavi as a symbol of the new, modern Iran. The tower was renamed after the Islamic Revolution and, like the U.S. Embassy, was seen more as a symbol of the decadent monarchy and a warning to the people not to embrace the Western enemies of Islam. The square became a popular area for anti-Western demonstrations and speeches and so became a symbol of the Islamic revolution, which was probably why the marble-clad monument to Iran's last monarchy was never torn down.
Because the entire area was heavily fortified and well patrolled by the military, trade and commerce had started to revive here, and even some luxuries like restaurants, cafés, and movie theaters had reopened. Unfortunately these were frequent targets by Islamist insurgents. A few brave pro-theocratic protesters would organize a rally occasionally in Azadi Square. To their credit, the military did not crack down on these rallies and even took steps to protect them against counterprotesters that threatened to get too violent. Buzhazi and most of his officers knew that they had to do everything possible to demonstrate to the people of Persia, and to the world, that they were not going to replace one brand of oppression with another.
"What's happening over there?" Rahmati asked as he continued to scan the avenue for more signs of an organized insurgent offensive. Every insurgent attack of late had been preceded by a smaller innocuous-looking one nearby, which diverted the attention of police and military patrols just enough to allow the insurgents to create even more havoc somewhere else.
"Looks like that new ExxonMobil gasoline station off the Sai-di Highway, across from Meda Azadi Park, sir," a lookout reported. "A large crowd running toward Azadi Avenue. The smoke is getting thicker—perhaps the underground tanks are on fire."
"Damn it all, I thought we had enough security around there," Rahmati cursed. The station was the government's first experiment into allowing foreign investment and part ownership in businesses in Persia. With the world's fourth-largest oil reserves, petroleum companies around the world were eager to move into the newly freed country and tap its wealth, almost untouched for decades since the Western embargoes against the theocratic Iranian government following the takeover of the U.S. Embassy in 1979. It was much, much more than a simple gasoline station—it was a symbol of a reborn, twenty-first-century Persia.
Everyone understood that, even soldiers like Rahmati, whose main goal in life was to look out for number one—himself. He came from a privileged family and joined the military because of its prestige and benefits after it was apparent that he wasn't smart enough to become a doctor, lawyer, or engineer. After Ayatollah Ruhollah Khomeini's revolution, he saved his skin by swearing fealty to the theocrats, informing on his fellow officers and friends to the Pasdaran i-Engelab, the Revolutionary Guards Corps, and by giving up much of his family's hard-earned riches in bribes and tributes. Although he hated the theocracy for taking everything he had, he didn't join the coup until it was obvious that it was going to succeed. "I want a reserve platoon to go in with the firefighters to put out those fires," he went on, "and if any protesters get near, they are to push them back north of Azadi Avenue and northwest of the square, even if they have to crack some skulls. I don't want—"
"If you were going to say, 'I don't want to let this get out of control,' Colonel, cracking skulls is not the way to accomplish that," a voice said behind him. Rahmati turned, then snapped to and called the room to attention as the leader of the military coup, General Hesarak al-Kan Buzhazi, entered the room.
The struggle to free his country from the grip of the theocrats and Islamists had aged Buzhazi well beyond his sixty-two years. Tall and always slender, he now struggled to take time to eat enough to maintain a healthy weight amidst his twenty-hour-a-day duties, infrequent and sparse meals, and the necessity of staying on the move to confuse his enemies—inside his cadre as well as outside—that were relentlessly hunting him. He still wore a closely cropped beard and mustache, but had shaved his head so he didn't have to take the time to keep his former flowing gray locks looking good. Although he had traded his military uniform for a suit and French-styled Gatsby shirt, he did carry a military-style greatcoat without decorations and wore spit-shined paratrooper's boots under his slacks, and he wore a PC9 nine-millimeter automatic pistol in a shoulder rig under his jacket. "As you were," he ordered. The others in the room relaxed. "Report, Colonel."
"Yes, sir." Rahmati quickly ran down the most serious events of the past few hours; then: "Sorry for that outburst, sir. I'm just a little frustrated, that's all. I put extra men on that station just to prevent such an occurrence."
"Your frustration sounded like an order to retaliate against anti-government protesters, Colonel, and that won't help the situation," Buzhazi said. "We'll deal harshly with the perpetrators, not the protesters. Understood?"
"Yes, sir."
Buzhazi looked carefully at his brigade commander. "Looks like you need some rest, Mostafa."
"I'm fine, sir."
Buzhazi nodded, then looked around the room. "Well, you can't run your brigade from here all the time, can you? Let's go see what happened out there." Rahmati gulped, then nodded, reluctantly following the general out the door, wishing he had agreed to take a nap. Traveling the streets of Tehran—even in broad daylight, within the portion of the city Buzhazi controlled, and with a full platoon of battle-hardened security forces—was never a safe or advisable move.
Every block of the two kilometers from the airport to Meda Azari Park was a maze of concrete and steel chicanes designed to slow the heaviest vehicles down; there was a new checkpoint every three blocks, and even Buzhazi's motorcade had to stop and be searched each time. Buzhazi didn't seem to mind one bit, using the opportunity to greet his soldiers and the few citizens out on the street. Rahmati didn't want to get that close to anyone, choosing instead to keep his AK-74 assault rifle at the ready. As they got closer to the park and the crowds got larger, Buzhazi strode down the street, shaking hands with those who offered their hand, waving to others, and shouting a few words of encouragement. His bodyguards had to step lively to keep up with him.
Rahmati had to hand it to the guy: the old warhorse knew how to work a crowd. He waded into the crowds fearlessly, shook hands with those who might just as well be holding a gun or trigger for a bomb vest, spoke to reporters and gave statements in front of TV cameras, had his picture taken with civilians and military men, kissed babies and old toothless women, and even acted as a traffic officer when fire trucks tried to enter the area, urging the crowds back and directing confused motorists away. But now they were just a few blocks from the gas station fire, and the crowds were getting thicker and much more restive. "Sir, I suggest we interview the security patrols and find out if any witnesses saw what happened or if any security cameras were operating," Rahmati said, making it clear that here would be a good place to do that.
Buzhazi didn't seem to hear him. Instead of stopping he kept on walking, heading right for the largest and noisiest gaggle gathering on the northwest side of the park. Rahmati had no choice but to stay with him, rifle at the ready.
Buzhazi didn't turn around, but seemed to sense the brigade commander's anxiety. "Put the weapon away, Mostafa," Buzhazi said.
"But sir—"
"If they wanted a shot at me they could have done it two blocks ago, before we were looking at each other eye to eye," Buzhazi said. "Tell the security detail to shoulder their weapons as well." The team leader, an impossibly young air force major by the name of Haddad, must have heard him, because the bodyguards' weapons had already disappeared by the time Rahmati turned to relay the order.
The crowd visibly tensed as Buzhazi and his bodyguards approached, and the small knot of men, women, and even some kids quickly grew. Rahmati was no policeman or expert on crowd psychology, but he noticed as more onlookers came closer to see what was going on, the others would be pressed farther and farther forward, toward the source of danger, causing them to feel trapped and scared for their life. Once panic started to set in, the crowd would quickly and suddenly turn into a mob; and when some soldier or armed individual felt his life was in danger, the shooting would start and the casualties would quickly mount.
But Buzhazi seemed oblivious to the obvious: he kept on marching forward—not threateningly, but not with any kind of false bravado or friendliness either; all business, but not confrontational like a soldier or glad-handed like a politician. Did he think he was going to drop in on some friends and discuss the issues of the day, or sit down to watch a football match? Or did he think he was invulnerable? Whatever his mental state, he was not reading this crowd correctly. Rahmati began thinking about how he was going to get to his rifle...and at the same time trying to decide which way he could run if this situation completely went to hell.
"Salam aleikom," Buzhazi called when about ten paces from the growing crowd, raising his right hand in greeting as well as to show he was unarmed. "Is anyone hurt here?"
A young man no more than seventeen or eighteen stepped forward and jabbed a finger at the general. "What does a damned soldier care if anyone is—?" And then he stopped, his finger still extended. "You! Hesarak Buzhazi, the new emperor of Persia! The reincarnation of Cyrus and Alexander himself! Are we required to genuflect before you, or is a simple bow sufficient, my lord?"
"I said, is anyone—?"
"What do you think of your empire now, General?" the young man asked, motioning to the clouds of acrid smoke swirling not too far away. "Or is it 'Emperor' Buzhazi now?"
"If no one is in need of assistance, I need volunteers to keep others away from the blast site, locate witnesses, and gather evidence until the police arrive," Buzhazi said, turning his attention away—but not completely away—from the loud firebrand. He sought out the eldest person in the crowd. "You, sir. I need you to ask for volunteers and to secure this crime scene. Then I need—"
"Why should we help you, lord and master sir?" the first young man shouted. "You were the one who brought this violence upon us! Iran was a peaceful and secure country until you came in, slaughtered everyone who didn't agree with your totalitarian ideas, and took over. Why should we cooperate with you?"
"Peaceful and secure, yes—under the heel of the clerics, Islamists, and crazies who killed or imprisoned anyone who didn't comply with their edicts," Buzhazi said, unable to avoid being drawn into a debate he knew was not going to be won. "They betrayed the people like they betrayed myself and everyone in the army. They—"
"That's what this is about, isn't it, Mr. Emperor: you?" the man said. "You don't like the way you were treated by your former friends, the clerics, so you slaughtered them and took over. Why do we care what you say now? You'll tell us anything to stay in power until you're done raping the country, and then you'll fly off right from your very conveniently located new headquarters at Mehrabad Airport."
Buzhazi was silent for a few moments, then nodded, which surprised everyone around. "You're right, young man. I was angry at the deaths of my soldiers, who had worked so hard to get rid of the radicals and nutcases in the Basij and make something of themselves, their unit, and their lives." After Buzhazi had been dismissed as chief of staff, following the American stealth bomber attacks against their Russian-made aircraft carrier years earlier, he had been demoted to commander of the Basij-i-Mostazefin, or "Mobilization of the Oppressed," a group of civilian volunteers who informed on neighbors, acted as lookouts and spies, and roamed the streets terrorizing others to conform and cooperate with the Revolutionary Guards Corps.
Buzhazi purged the Basij of the gangsters and rabble-rousers and transformed the remainder into the Internal Defense Force, a true military reserve force. But their success challenged the domination of the Revolutionary Guards Corps, and they acted to try to discredit—or preferably destroy—Buzhazi's fledgling national guard force. "When I learned it was the Pasdaran that had staged the attack against my first operational reserve unit, making it look like a Kurdish insurgent action, just to hurt and discredit the Internal Defense Force, I got angry and lashed out.
"But the Islamists and the terrorists the clerics have brought into our country are the real problem, son, not the Pasdaran," Buzhazi went on. "They have gutted the minds of this nation, emptied them of all common sense and decency, and filled it with nothing but fear, contempt, and blind obedience."
"So what is the difference between you and the clerics, Buzhazi?" another young man shouted. Rahmati could see the crowd was getting bolder, more vocal, and not afraid to get closer every second. "You kill off the clerics and take down the government—our government, the one we elected!—and replace it with your junta. We see your troops breaking down doors, burning buildings, stealing, and raping every day!"
The crowd noisily voiced its agreement, and Buzhazi had to raise his hands and voice to be heard: "First of all, I promise you, if you bring me evidence of a theft or rape by any soldier under my command, I will personally put a bullet in his head," he shouted. "No tribunal, no secret trial, no hearing—bring me the evidence, convince me, and I will drag the man responsible before you and execute him myself.
"Second, I am not forming a government in Persia, and I am not a president or emperor—I am commander of the resistance forces temporarily in place to quell violence and establish order. I will stay in command long enough to root out the insurgents and terrorists and supervise the formation of some form of government that will draft a constitution and enact laws governing the people, and then I will step down. That is why I set up my headquarters at Mehrabad—not for a quick escape, but to show that I'm not going to occupy legitimate government offices and call myself a president."
"That's what Musharraf, Castro, Chávez, and hundreds of other dictators and despots said when they engineered their coups and took over the government," the young man said. "They said they fought for the people and would leave as soon as order was established, and before you knew it they had installed themselves in office for life, placed their friends and thugs in positions of power, suspended or tossed out the constitution, taken over the banks, nationalized all the businesses, taken away land and wealth from the rich, and closed any media outlets that opposed them. You will do the same in Iran."
Buzhazi studied the young man for a moment, then carefully scanned the others around him. Those, he observed, were some very good points—this guy was very intelligent and well read for such an age, and he suspected most of the others were too. He was not among normal street kids here.
"I judge a man by his actions, not his words—friend as well as enemy," Buzhazi said. "I could promise you peace, happiness, security, and prosperity, just like any politician, or I could promise you a place in heaven like the clerics, but I won't. All I can promise is that I will fight as hard as I can to stop the insurgents from tearing our country apart before we've had a chance to form a government of the people, whatever that government will be. I will use all my skills, training, and experience to make this country secure until a government by the people stands up."
"Those sound like pretty words to me, Mr. Emperor, the kind you just pledged you wouldn't use."
Buzhazi smiled and nodded, looking at those who seemed the angriest or most distrustful directly in the eye. "I see many of you have cell phone cameras, so you have video proof of what I say. If I was the dictator you think I am, I'd confiscate all those phones and have you tossed in prison."
"You could do that tonight after you break into our homes and roust us out of bed."
"But I won't," Buzhazi said. "You are free to send the videos out to anyone on the planet, post it on YouTube, sell it to the media. The video will be documentation of my promise to you, but my actions will be the final proof."
"How do we send out any videos, old man," a young woman asked, "when power is only on for three hours a day? We are lucky if the phones work for a few minutes each day."
"I read the postings, I surf the Internet, and I lurk on the blogs, just like you," Buzhazi said. "The American satellite global wireless Internet system works well even in Persia—may I remind you that it was jammed by the clerics in order to try to prevent you from receiving contrary news from the outside world—and I know many of you enterprising young people have built pedal-powered generators to recharge your laptops when the power goes out. I may be an old man, young lady, but I'm not completely out of touch." He was pleased to see a few smiles appear on the faces of those around him—finally, he thought, he was starting to speak their language.
"But I remind you that the power goes out because of insurgent attacks on our power generators and distribution networks," he went on. "There's an enemy out there who doesn't care about the people of Persia—all they want is to regain power for themselves, and they'll do it any way they can think of, even if it hurts or kills innocent citizens. I took power away from them and allowed the citizens of this country to communicate with the outside world again. I allowed foreign investment and aid to return to Persia, while the clerics shut out the rest of the world for over thirty years and hoarded the wealth and power of this nation. That's the action I'm talking about, my friends. I can say absolutely nothing, and those actions would speak louder than a thousand thunderstorms."
"So when will the attacks stop, General?" the first man asked. "How long will it take to drive the insurgents out?"
"Long after I'm dead and buried, I think," Buzhazi said. "So then it'll be up to you. How long do you want it to take, son?"
"Hey, you started this war, not me!" the man thundered, shaking his fist. "Do not lay this at my feet! You say you'll be dead long before this is over—well, why don't you just go to hell now and save us all a lot of time!" A few in the crowd blinked at the man's violent outburst, but said or did nothing. "And I am not your son, old man. My father was killed in the street outside the shop my family has owned for three generations, during a gun battle between your troops and the Pasdaran, right before my eyes, my mother's, and my baby sister's."
Buzhazi nodded. "I am sorry. Then tell me your name."
"I don't feel like telling you my name, old man," the young man said bitterly, "because I see you and your forces just as capable of arresting me or shooting me in the head as the Pasdaran reportedly were."
"'Reportedly?' You doubt that the Pasdaran are killing anyone who opposes the clerics?"
"I saw plenty of violence and bloodthirstiness on both sides in the gun battle in which my father was killed," the young man went on, "and I see very little difference between you and the clerics except perhaps the clothes you wear. Are you correct or justified in your actions just because the Americans swooped in and helped you drive the Pasdaran temporarily out of the capital? When you are driven out, will you be the new insurgents then? Will you make war on the innocent because you think you are correct?"
"If you truly believe that I'm no better or worse than the Islamic Revolutionary Guards, then no amount of words will ever convince you otherwise," Buzhazi said, "and you will blame your father's death on any convenient target. I am sorry for your loss." He turned and scanned the others around him. "I see a lot of angry faces out here in the street, but I hear some extremely intelligent voices as well. My question to you is: If you're so smart, what are you doing out here just standing around? Your fellow citizens are dying, and you do nothing but shuffle from attack to attack shaking your fists at my soldiers while the insurgents move to the next target."
"What are we supposed to do, old man?" another man asked.
"Follow your head, follow your heart, and take action," Buzhazi said. "If it's the clerics you truly believe have the best interests of the nation in mind, join the insurgency and fight to drive me and my men out of the country. If you believe in the monarchists, join them and create your own insurgency in the name of the Qagev, battling both the Islamists and my soldiers, and bring the monarchy back to power. If you think what I say and do makes sense, put on a uniform, pick up a rifle, and join me. If you don't want to join anyone, at least keep your damn eyes open and when you see an attack against your family or your neighbors, take action...any action. Fight, inform, assist, protect—do something rather than just stand around and complain about it."
He scanned their faces once again, letting them look directly into his eyes and he into theirs. Most of them did. He saw some real strength in this bunch, and it gave him hope. They were worth fighting for, he decided. No matter which side they chose, they were the future of this land. "It's your country, dammit...it's our country. If it's not worth fighting for, go somewhere else before you become another casualty." He fell silent, letting his words sink in; then: "Now I need your help policing this crime scene. My soldiers will set up a perimeter and secure the area, but I need some of you to help the rescuers recover the victims and the police gather evidence and interview witnesses. Who will help?"
The crowd paused, waiting for someone to move first. Then the first young man stepped forward and said to Buzhazi, "Not for you, Emperor. You think you are any different than the insurgents roaming the streets? You're worse. You're nothing but a pretentious old man with a gun. That doesn't make you right." And he turned and walked away, followed by the rest.
"Shit, I thought I had gotten through to them," Buzhazi said to Colonel Rahmati.
"They're just a bunch of losers, sir," the brigade commander said. "You asked what they're doing out here on the streets? They're stirring up trouble, that's all. For all we know, they are the ones who blew up that gas station. How do we know they're not insurgents?"
"They are insurgents, Mostafa," Buzhazi said.
Rahmati looked stunned. "They...are? How do you know...I mean, we should arrest all of them right now!"
"They're insurgents, but not Islamists," Buzhazi said. "If I had a choice as to which I'd want out on the streets right now, it's definitely them. I still think they'll help, but not the way I might want them to." He looked in the direction of the still-burning gasoline station to the remnants of a smoldering delivery truck that had been blown several dozen meters across the street. "Stay here and keep your weapons out of sight. Get the perimeter set up. I want no more than two soldiers positioned at any intersection, and they should be stationed on opposite corners, not together."
"Why, sir?"
"Because if there are more, informants will not approach them—and we need information, fast," Buzhazi said. He started walking toward the smoking truck. Rahmati started to follow, not wanting to appear any more frightened than he was already, but Buzhazi turned and growled, "I said stay here and get that perimeter set up." Rahmati was only too glad to comply.
A fire truck had approached the burning hulk and two very young-looking firefighters—probably children of dead or injured real firemen, a common practice in this part of the world—started to fight the fire, using a weak stream of water from the old surplus fire truck. It was going to be a long and smoky job. Buzhazi stepped around the fire truck, just far enough from the smoke so he wouldn't be choked by it, but was mostly screened from view. Now that the cleanup job had started, the crowds had started to disperse. Another, larger fire crew was attacking the blaze at the gas station itself, which was still very hot and fierce, rapidly driving huge columns of black smoke skyward. It was unbelievable to Buzhazi that the flames seemed to drink in even that huge volume of water—the fire was so intense that the fire actually seemed to—
"Quite a speech back there, General," he heard a voice say behind him.
Buzhazi nodded and smiled—he had guessed correctly. He turned and nodded formally to Her Highness Azar Assiyeh Qagev, the heir presumptive of the Peacock Throne of Persia. He glanced behind the young woman and spotted Captain Mara Saidi, one of Azar's royal bodyguards, standing discreetly near a lamppost, expertly blending in with the chaos around them. Her jacket was open and her hands clasped before her, obviously shielding a weapon from sight. "I thought I saw the captain there in the crowd, and I knew you'd be nearby. I assume the major is nearby with a sniper rifle or RPG, correct?"
"I believe he's armed with both weapons today—you know how he likes to come prepared," Azar said, bowing in return, not bothering to point out her chief of internal security Parviz Najar's hiding place—just in case Buzhazi's little rendezvous here was really a trap. She couldn't afford to trust this man—alliances changed so quickly in Persia. "I have promoted Najar to lieutenant colonel and Saidi to major for their bravery in getting me out of America and back home."
Buzhazi nodded approvingly. Azar Assiyeh Qagev, the youngest daughter of the pretender to the Peacock Throne, Mohammed Hassan Qagev—still missing since the beginning of Buzhazi's coup against the theocratic regime of Iran—had just turned seventeen years old, but she had the self-confidence of an adult twice her age, not to mention the courage, martial skills, and tactical foresight of an infantry company commander. She was also turning into a woman very nicely, Buzhazi couldn't help but notice, with long shiny black hair, graceful curves starting to bud on her slender figure, and dark, dancing, almost mischievous eyes. Her arms and legs were covered but with a white blouse and "chocolate chip" desert fatigue trousers, not a burka, to protect herself from the sun; her head was covered but with a TeamMelli World Cup Football team "doo-rag," not a hijab.
But his eyes were also automatically drawn to her hands. Every other generation of men of the Qagev dynasty—possibly the women too, but they were probably discarded as newborns rather than have them grow up with any sort of deficiency—had suffered from a genetic defect called bilateral hypoplastic thumb, or missing a thumb on both hands. She had pollicization surgery as a young child, which makes the index fingers function as thumbs, and left her with only four fingers on both hands.
But rather than becoming a handicap, Azar had made the deformity a source of strength, toughening her up beginning at a very young age. She had more than made up for her perceived deficiency: rumor was that she could outshoot most men twice her age and was an accomplished pianist and martial artist. Azar reportedly rarely wore gloves, letting others see her hands both as a symbol of her legacy and as a distraction to her adversaries.
Azar had secretly lived in the United States of America since she was two years old, under the protection of her bodyguards Najar and Saidi, who posed as her parents, separated from her real parents for security reasons, who had also lived in hiding as guests of the U.S. Department of State. When Buzhazi's coup erupted, the Qagevs immediately activated their war council and headed back to Iran. The king and queen—who were supposed to be in hiding yet ran a Web site, regularly appeared in the media blasting the theocratic regime in Iran, and openly vowing to someday return and take control of the country—were still missing and presumed killed by the Iranian Revolutionary Guards Corps or al-Quds terrorist forces, with the help of the Russians and Turkmenis. But Azar did make it into Iran, using her wits, natural-born leadership skills—and a lot of help from the American Battle Force and a small army of armored commandos—and joined up with the royal war council and their thousands of jubilant followers.
"I'm impressed, Highness," Buzhazi said, taking off his helmet and pouring a bit of water on his face before taking a deep drink. "I was looking specifically for you, but you blended into the crowd perfectly. Obviously the others had no idea who you were, because no one tried to form a defensive shield around you when I approached. You hid your mun well."
"I've been hanging around the city trying to listen to these young people to find out what they want and what they expect," Azar said. Her American accent was still thick, making her Farsi hard to understand. She removed the Iranian national soccer team headband to reveal the long waist-length ponytail, the mun, typical of Persian royalty for centuries. She shook her hair, glad to be free from the self-imposed but traditional bonds. Major Saidi, a horrified look on her face, stepped toward her, silently urging her to hide her mun before anyone on the streets noticed. Azar rolled her eyes in mock exasperation and tied the ponytail up again under the doo-rag. "They know me as one of the displaced, that's all—like them."
"Except with a hundred armed bodyguards, a council of war, a secret war chest bigger than the gross national product of most of central Asia, and several hundred thousand followers who would gladly step in front of a line of machine guns to see you back on the Takht-e-Tavous, the Peacock Throne."
"I'd trade all that I control to convince you and your brigades to join me, Hesarak," she said. "My followers are loyal and dedicated, but we are still far too few, and my followers are loyalists, not fighters."
"What do you think is the difference between a so-called loyalist and a soldier, Highness?" Buzhazi asked. "When your country's in danger, there is no difference. In times of war, citizens become fighters, or they become slaves."
"They need a general...they need you."
"They need a leader, Highness, and that person is you," Buzhazi said. "If half your loyalists are as smart, fearless, and daring as that bunch that you were hanging around with back there, they can easily take control of this country."
"They won't follow a girl."
"Probably not...but they'll follow a leader."
"I want you to lead them."
"I'm not taking sides here, Highness—I'm not in the business of forming governments," Buzhazi said. "I'm here because the Pasdaran and the insurgents they sponsor are still a threat to this country, and I will hunt them down until every last one of them is dead. But I'm not going to be the president. John Alton said, 'Power corrupts, and absolute power corrupts absolutely.' I know my power comes from my army, and I don't want the people to be ruled by its military. It should be the other way around."
"If you won't be their president, be their general," Azar said. "Lead your army under the Qagev banner, train our loyalists, draft more fighters from the civilian population, and let us put our nation back together."
Buzhazi looked seriously at the young woman. "What of your parents, Highness?" he asked.
Azar swallowed at the unexpected question, but the steel quickly returned to her eyes. "Still no word, General," she replied firmly. "They are alive—I know it."
"Of course, Highness," Buzhazi said softly. "I have heard your council of war won't approve of you leading your forces until you reach the age of majority."
Azar sneered and shook her head. "The age of majority was fourteen for centuries—Alexander was fourteen when he led his first army into battle," she spat. "When projectile warfare became more advanced and weapons and armor got thicker and heavier, the age of majority—the word comes from majour, the leader of a regiment—was raised to eighteen because anyone younger could not lift a sword or wear the armor. What relevance does that have in today's world? Nowadays a five-year-old can use a computer, read a map, talk on a radio, and understand patterns and trends. But my esteemed council of stuffed-shirt old men and cluck-clucking old women won't let anyone younger than eighteen lead the army—especially one that is female."
"I recommend someone get your battalion commanders together, nominate a commander, get it approved by your war council, and get organized...soon," Buzhazi warned. "Your raids are completely uncoordinated and don't seem to have any purpose other than random killings and mayhem that keep the population on edge."
"I've already said that to the council, but they're not listening to a little girl," Azar complained. "I'm just a figurehead, a symbol. They would rather quibble over who has seniority, who has more followers, or who can bring in more recruits or cash. All they want out of me is a male heir. Without a king, the council will make no decisions."
"Then be the Malika."
"I don't like being called 'Queen,' General, and you know it, I'm sure," Azar said hotly. "My parents are not dead." She said those last words angrily, defiantly, as if attempting to convince herself as well as the general.
"It's been almost two years since they've disappeared, Highness—how much longer are you going to wait? Until you turn eighteen? Where will Persia be in fifteen months? Or until a rival dynasty asserts its claim to the Peacock Throne, or some strongman takes over and has all the Qagevs back on the run?"
Obviously Azar had asked herself all these questions already, because it pained her that she didn't have any answers. "I know, General, I know," she said in a tiny voice, the saddest one he had ever heard her use. "That's why I need you to go before the council of war, join us, take command of our loyalists, and unite the anti-Islamist forces against Mohtaz and his bloodthirsty jihadis. You are the most powerful man in Persia. They would not hesitate to approve."
"I'm not sure if I'm ready to be the commanding general in a monarchist army, Highness," Buzhazi said. "I need to know what the Qagev stand for before I'll throw my support behind them." He looked at Azar somberly. "And until your parents appear, or until you turn eighteen—maybe not even then—the council of war speaks for the Qagev..."
"And they cannot even decide if the royal flag should be raised before or after morning prayers," Azar said disgustedly. "They argue about court protocol, rank, and petty procedures rather than tactics, strategies, and objectives."
"And you want me to take my orders from them? No thank you, Highness."
"But if there was a way to convince them to support you if you announced that you would form a government, Hesarak—"
"I told you, I'm not in the business of forming governments," Buzhazi snapped. "I took down the clerics, the corrupt Islamist leadership, and their hired goons the Pasdaran because they are the true obstacles to freedom and law in this country. But may I remind you that we still have a Majlis-i-Shura that we elected that supposedly have the constitutional authority to take control and form a representative government? Where are they? Hiding, that's what. They're afraid they'll be targeted for assassination if they poke their little heads out, so they'd rather watch in their comfortable villas with their bodyguards surrounding them while their country tears itself apart."
"So it sounds like you just want someone to ask you to help them, is that it, General? You crave the honor and respect of having a politician or princess beg for help?"
"What I crave, Highness, is for the persons who supposedly lead this country to get off their fat asses and lead," Buzhazi said hotly. "Until the Majlis, your so-called war council, or someone else decides it has the stomach to squash the Islamist insurgency, take charge, and form a government, I'll keep doing what I do best—hunting down and killing as many of the enemies of Persia as I can to save innocent lives. At least I have an objective."
"My followers share your vision, General..."
"Then prove it. Help me do my job until you can talk some sense into your war council."
Azar wanted to argue, for her people and their struggle as well as for her own legitimacy, but she knew she had run out of answers. Buzhazi was right: they had the will to resist the Islamists, but they just couldn't get the job done. She nodded resignedly. "All right, General, I'm listening. How can we help you?"
"Tell your loyalists to join my army and pledge to follow my orders for two years. I'll train and equip them. After two years they are free to return to you, with all the equipment and weapons they can carry on their backs."
Azar's eyebrows raised in surprise. "A very generous offer."
"But they must swear during their two-year enlistment to obey my commands and fight for me, all the way and then some, upon penalty of death—not by any war council, court, or tribunal, but by me. If they are caught passing information to anyone outside my ranks, including you, they'll die in humiliation and disgrace."
Azar nodded. "What else?"
"If they will not join my army, they must agree to pass on clear, timely, and actionable information to me, on a constant basis or on demand, and to support my army with everything they have to give—food, clothing, shelter, water, money, supplies, anything," Buzhazi went on. "I've ordered my security details spread out to make it easier for your people to pass notes, photos, or other information to them, and I will provide you with blind drops and secure voice and e-mail addresses for you to use to leave us information.
"But you must help us, all of you. Your loyalists can follow the Qagev, such as you are, but they will help me, or they will stand out of the way while my men and I fight. They either agree that I fight for Persia and I am deserving of their complete and total support, or they will lay down their weapons and stay off the streets—no more raids or bombings, no more roving gangs, and no more assassinations that serve only to terrorize the innocents and cause the Pasdaran and Islamists to increase their attacks against the civilian population."
"That will be...difficult," Azar admitted. "I simply don't know all of the resistance leaders out there. I frankly doubt if anyone on the council knows all of the cells and their leaders."
"You attend the war council meetings, don't you?"
"I'm allowed to attend general meetings of the war council, but I'm not allowed to vote, and I'm discouraged from attending strategy meetings."
Buzhazi shook his head in exasperation. "You're probably the smartest person in that council meeting—why you're not allowed to participate is a damned mystery to me. Well, it's your problem, Highness. I'm telling you that your loyalists are part of the problem, not part of the solution. I don't know if the person with the gun at the other end of the block is an Islamist or one of your loyalists, so I'm going to blow his head off regardless before he tries to do the same to me. That's not the way I want it, but that's the way I'll play it if I have to."
"I'm sorry I can't be of more help, General."
"You can, Highness, if you just drag yourself back into the twenty-first century like I know you can," Buzhazi said, donning his helmet again and pulling the straps tight.
"What?"
"Come on now, Highness—you know exactly what I'm talking about," Buzhazi said irritably. "You're a smart woman as well as a natural-born leader. You've lived in America most of your life and you've obviously learned that the old ways won't work anymore. You know as well as I that this court of yours and this so-called council of war is what's hamstringing you. You've voluntarily imprisoned yourself in this six-hundred-year-old cage called your 'court' and you've committed to cede authority to a bunch of spineless cowards—half of which aren't even in this country right now, am I correct?" He could tell by her expression that he was.
Buzhazi shook his head in disappointment quickly turning to disgust. "Pardon me for saying this, Highness, but get your royal head out of your pretty little ass and get with the program before we all die and our country becomes a mass graveyard," he said angrily. "You're the one out here on the streets, Azar. You can see the problems and are smart enough to formulate a response, but you won't take charge. Why? Because you don't want your parents to think you're taking over their thrones? This is the twenty-first century, Azar, for God's sake, not the fourteenth. Besides, your parents are either dead or cowards themselves if they haven't shown themselves in almost two—"
"Shut up!" Azar screamed, and before Buzhazi could react, she had spun around and planted her right foot solidly in his solar plexus, knocking the wind out of him. Buzhazi went down on one knee, more embarrassed at being taken by surprise than hurt. By the time he got back on his feet and was able to take at least a half of a normal breath, Mara Saidi was shielding Azar, an automatic machine pistol pointed at him.
"Good kick, Highness," Buzhazi grunted, rubbing his abdomen. Obviously, he guessed, one of her accommodations for having defects of the hands was her ability to fight with her feet. "The rumors said you could take care of yourself—I see that's true."
"The meeting is over, General," he heard a man say behind him. Buzhazi turned and nodded at Parviz Najar, who had run out of hiding in the blink of an eye and had another machine pistol pointed at him. "Go quickly."
"After you both lower your weapons," they heard another voice shout. They all turned to see Major Qolom Haddad hidden behind the rear end of the smoldering truck, an AK-74 rifle leveled at Najar. "I'm not going to repeat myself!"
"Everyone, lower your weapons," Buzhazi said. "I think we've both said what we needed to say here." No one moved. "Major, you and your men, stand down."
"Sir—"
"Colonel, Captain, stand down as well," Azar ordered. Slowly, reluctantly, Najar and Saida complied, and when their weapons were out of sight, Haddad lowered his. "There are no enemies here."
Buzhazi took his first full deep breath, smiled, nodded again respectfully, then extended his hand. "Highness, it was good to speak with you. I hope we can work together, but I assure you, I'm going to keep fighting."
Azar took his hand and bowed her head as well. "It was good to speak with you too, General. I have much to think about."
"Don't take too long, Highness. Salam aleikom." Buzhazi turned and headed back to his men, with Haddad and two more soldiers who had been carefully hidden nearby covering his back.
"Peace be unto you as well, General," Azar called after him.
Buzhazi turned halfway to her, smiled, and called out, "Unlikely, Highness. But thanks anyway."
THE WHITE HOUSE RESIDENCE
THAT SAME TIME
Chief of Staff Walter Kordus knocked on the door of the President's sitting room on the third-floor family residence of the White House. "Sir? She's here."
President Gardner looked up over his reading glasses and set down the papers he was reviewing. He had a large flat-screen TV on to a boxing match but with the sound muted. He wore a white shirt and business slacks, with his tie loosened—he rarely wore anything else but business attire until moments before bed. "Good. Where?"
"You said you didn't want to meet in the West Wing, so I had her brought up to the Red Room—I thought that was appropriate."
"Cute. But she asked to see the Treaty Room. Have her brought up."
Kordus took a step into the sitting room. "Joe, are you sure you want to do this? She's the chairwoman of the Senate Armed Services Committee, probably the most powerful woman in the country besides Angelina Jolie. It's got to remain business..."
"This is business, Walt," Gardner said. "I'll be there in a few minutes. Got those notes I asked you for?"
"They're on the way."
"Good." Gardner went back to studying his papers. The chief of staff shook his head and departed.
A few minutes later, Gardner made his way down the Center Hall, now wearing his suit jacket, straightening his tie as he walked. Kordus intercepted him and passed him a folder. "Hot off the press. Want me to—?"
"Nope. I think we're done for the night. Thanks, Walt." He breezed past the chief of staff and entered the Treaty Room. "Hello, Senator. Thanks for meeting me at this ungodly hour."
She was standing beside the immense mahogany U. S. Grant Cabinet table, lovingly running her long fingers across the inlaid cherry features. The steward had placed a tray of tea on the coffee table on the other side of the room. Her eyes widened and that camera-magnetizing smile appeared when she saw Gardner enter the room. "Mr. President, it is certainly my honor and privilege to be with you tonight," Senator Stacy Anne Barbeau said in her famous silky Louisiana accent. "Thank you so much for the invitation." She stood, embraced the President, and exchanged polite kisses on the cheek. Barbeau wore a white low-cut business suit which subtly but effectively displayed her breasts and cleavage, accented for the evening by a shimmering platinum necklace and dangling diamond earrings. Her red hair bounced as if motorized in tune with her smile and batting eyelashes, and her green eyes flashed with energy. "You know that you may call upon me at any time, sir."
"Thank you, Senator. Please." He motioned to a Victorian couch and took her hand as he led her to it, then took an ornate chair to her right, facing the fireplace.
"I hope you give my best to the First Lady," Barbeau said as she arranged herself just so on the couch. "She's in Damascus, if I'm not mistaken, attending the international women's rights conference?"
"Exactly, Senator," the President said.
"I wish my duties in the Senate would have allowed me to attend," Barbeau said. "I sent my senior staffer Colleen to attend, and she's bringing a resolution of support from the full Senate for the First Lady to present to the delegates."
"Very thoughtful of you, Senator."
"Please, sir, will you not call me 'Stacy,' here in the privacy of the residence?" Barbeau asked, giving him one of her mind-blowing smiles. "I think we've both earned the right to a little downtime and relief from the formalities of our offices."
"Of course, Stacy," Gardner said. He did not offer to let her call him "Joe," and she knew enough not to ask. "But the pressure is never really off, is it? Not in our lines of work."
"I've never considered what I do 'work,' Mr. President," Barbeau said. She poured him a cup of tea, then sat back and crossed her legs as she sipped hers. "It's not always pleasurable, to be sure, but doing the people's business is never a chore. I suppose the stress is part of what makes one feel alive, don't you agree?"
"It always seemed to me you thrive on the pressure, Senator," Gardner commented. He suppressed a grimace after he sipped the tea. "In fact, if I may say so, it looks to me like you enjoy creating a bit of it."
"My responsibilities many times dictate that I do things above and beyond what most folks might call 'politic,'" Barbeau said. "We do whatever we need to do in the best interest of our constituents and our country, isn't that right, Mr. President?"
"Call me Joe. Please."
Barbeau's green eyes flashed, and her head bowed without her eyes leaving his. "Why, thank you for the honor...Joe."
"Not at all, Stacy," Gardner said with a smile. "You're right, of course. No one likes to admit it, but the end often justifies the means, as long as the end is a safer and more secure nation." He picked up a telephone sitting on the Monroe desk. "Could you have the libation table brought to the Treaty Room, please?" He hung up the phone. "It's after nine P.M., Stacy, and I'm not in the mood for tea. Hope you don't mind."
"Not at all, Joe." The smile was back, but it was more introspective, more reserved. "I may just join you."
"I know what might convince you." A steward brought a rolling table with several crystal decanters. Gardner poured himself a glass of Bacardi Dark on ice and fixed Barbeau a drink. "I thought I read in People magazine that you preferred a 'Creole Mama,' correct? I hope I got it right...bourbon, Madeira, and a splash of grenadine, topped with a cherry, right? Sorry, we only have red cherries, not green."
"You are a real surprise sometimes, Joe," she said. They touched glasses, their eyes locked together. She tasted hers, her eyes glistened again, and she took a deeper sip. "My my, Mr. President, a little intelligence work, even after hours, and a skilled hand at the bar. I'm again impressed."
"Thank you." Gardner took a deep sip of his drink as well. "Not as sophisticated as a Creole Mama, I'm sure, but when you're a politician from Florida, you'd better know your rum. Cheers." They touched glasses and sipped their drinks once more. "Do you know the origin of touching glasses, Stacy?"
"I'm sure I don't," Barbeau replied. "I didn't even realize there was an origin to it. It's not just a cute little noisemaker then?"
"In medieval times, when adversaries met to discuss terms of treaties or alliances, when they drank after negotiations were concluded they tipped a bit of the contents of their cups into the other's to show neither was poisoned. The custom evolved into a sign of friendship and camaraderie."
"Why, how fascinating," Barbeau said, taking another sip, then letting her tongue run across her full lips. "But I certainly hope you don't see me as an adversary, Joe. I'm anything but. I have been an admirer of yours for years, as was my father. Your political skills are exceeded only by your intelligence, charm, and true dedication to the service of the nation."
"Thank you, Stacy." He let his eyes drift across Barbeau's body as she took another sip. Even as it appeared that she was concentrating on enjoying her drink, she noticed he was looking her over...again. "I knew your father when we were in the Senate together. He was one powerful man, very strong-willed and passionate in his pursuits."
"He counted you among his most trusted friends, even though you and he were on opposite sides of the political and ideological aisle then," Barbeau said. "After I was elected to the Senate, he often reminded me that if I wanted some straight talk from the other side, I shouldn't hesitate to come to you." She paused, adopting a rather wistful expression. "I wish he was still here now. I could use his strength and wisdom. I love him so much."
"He was a fighter. A tough opponent. You knew where he stood and he wasn't afraid to tell you. He was one hell of a man."
Barbeau put her hand on Gardner's and pressed it. "Thank you, Joe. You're a sweet man." She took an instant to look at him deeply, then let her lips part slightly. "You...look very much like I remember him in his younger, more fiery years, Joe. We had a dining room very much like this in Shreveport, and we used to spend endless hours together, just like this. I wanted to talk politics and he wanted to find out about who I was dating."
"Daddies and daughters always stay close, eh?"
"He made me tell him my most intimate secrets," she said, a mischievous smile spreading across her face. "I couldn't deny him anything. He made me tell him everything—and I was a very naughty girl growing up. I dated all the politicians' boys. I wanted to learn everything about politics: strategies, planning, fund-raising, candidates, issues, alliances. They wanted..." She paused, giving him another sly smile and a bat of her eyes. "...well, you know what they wanted." Gardner swallowed hard as he imagined what they got from her. "It was a mutually beneficial relationship. Sometimes I think my daddy set me up on some of those dates just so I could be his spy—the Cajun political version of turning your daughter out, I suppose."
Gardner chuckled, and unconsciously let his eyes roam her body again, and this time Barbeau allowed herself to show that she noticed, smiled, and blushed—she was one of those women who could blush anytime, anywhere, in any situation, at will. He sat back in his chair, wanting to get this meeting under way so they could concentrate on other things, if the opportunity presented itself. "So, Stacy, we both know the issue before us. Where does the White House stand with the Armed Services Committee? Are we going to have a fight over the military budget, or can we come to an agreement and form a united front?"
"Unfortunately we're more confused than ever, I'm afraid, Joe," Barbeau replied. She took her hand away, watching a sudden pang of loss cloud his face. "This is all confidential, Mr. President?"
"Of course." He touched her hand, and her eyes fluttered. "On both sides. Strictly confidential."
"My lips are sealed." Barbeau smiled, then put her red lips together, made a locking motion with her long fingers, and tucked the invisible key in the ample cleavage between her breasts. Gardner took that as open permission to look at her chest this time, and he did so liberally. "The committee is in an uproar, Joe. They're concerned about General McLanahan's health and well-being, of course. Have you heard anything more about him?"
"Not much. The doctors originally told me not to expect him back to duty for several months. Some kind of heart thing."
That jibed with what her sources at Walter Reed National Military Medical Center told her, she thought—so far, Gardner wasn't lying to her. That was a good sign. "For such a strong young man to be suddenly rendered unconscious like that, the stresses of living on that space station and making repeated trips back and forth in the Black Stallion spaceplane must be enormous, far more than anyone could have possibly anticipated."
"McLanahan's a tough guy, but you're right—although he is over fifty and has a family history of heart disease, he was incredibly fit. Shuttle astronauts usually get several days between liftoff and re-entry—McLanahan has taken five round-trips to the space station in the past four weeks. That's unprecedented, but for the past few months it's been the norm. We're restricting travel to the space station and are in the process of doing extensive physicals on everyone involved. We need answers as to what's happened."
"But that's exactly my point, Joe. McLanahan is tough and strong, especially for a middle-aged man, and he's a combat veteran and national military figure—my God, he's a hero!—who I'm sure gets regular fitness checkups. Yet he was still incapacitated and God knows what sort of injury he has sustained. It calls into question the safety and utility of the proposed military space plan. For heaven's sake, Joe, why are we risking good men on such a project? I grant you it's modern and exotic and exciting, but it's technology that just hasn't been perfected and probably won't be for another ten years—not to mention the fact that it's four-fifths fewer aircraft and one-tenth the payload for the same money. If a strong guy like General McLanahan is knocked senseless by flying the thing, is it safe for other crewmembers?"
"What does the committee think, Stacy?"
"It's simple and logical, Joe," Barbeau said. "It's not about impressing the folks with global Internet access or half-meter resolution photographs of everyone's backyards—it's about creating value and benefit for our country's defense. As far as I can see, the spaceplanes benefit only the handful of contractors assigned to the project, namely Sky Masters and their subcompanies. We have a dozen different space booster systems with proven track records that can do a better job than the Black Stallion." She rolled her eyes. "For God's sake, Joe, who else is McLanahan in bed with?"
"Certainly not Maureen Hershel anymore," Gardner chuckled.
Barbeau rolled her eyes in dramatized disbelief. "Oh, that dreadful woman—I'll never understand why President Martindale chose her of all people to be his Vice President," Barbeau retorted. She looked inquisitively, then playfully at Gardner over the rim of her glass, then asked, "Or was the cold-fish routine just for public consumption, Joe?"
"We became close friends because of the demands of the job, Stacy, just business. All the rumors floating around about us are completely bogus."
Now he was lying, Barbeau thought, but she expected nothing less than a complete and outright denial. "I completely understand how the working conditions in Washington thrust two people together, especially ones who seem complete opposites," Barbeau said. "Combine power politics with a brewing war in the Middle East and long nights attending briefings and planning sessions, and sparks can fly."
"Not to mention McLanahan was obviously not getting business done back at home," Gardner added. They both laughed, and Gardner used that opportunity to clasp Barbeau's hand again. "He was too busy playing space cadet to pay any attention to her." He affixed Barbeau with a deep, serious stare. "Look, Stacy, let's get right down to it, okay? I know what you want—you've been lobbying for it since you set foot inside the Beltway. With most of the rest of the Air Force bomber bases destroyed by the Russians in the '04 Holocaust nuclear attacks, Barksdale Air Force Base is the natural home for a new long-range bomber fleet—"
"If the Pentagon doesn't keep on dumping money into that dust-bowl desert base in Battle Mountain, black programs in Dreamland—another Nevada base that mostly falls outside congressional oversight, I might point out—or the space station."
"It's no secret McLanahan's stock rose to all-time highs after his actions in the counterattacks against Russia," Gardner said, "and his pet projects were the unmanned bombers at Battle Mountain, his high-tech laser gizmos at Dreamland, and now the space station. It gave Martindale something to point at and brag to the American people that he devised and supported—"
"Even though President Thomas Thorn was the one who authorized their construction, not Martindale," Barbeau pointed out.
"Unfortunately, President Thorn will always and forever be known as the president who allowed the Russians to pull off a sneak attack against the United States that killed thirty thousand men, women, and children and injured another quarter million," Gardner said. "It won't matter that he was just as interested in high-tech toys as Martindale: Thorn will always be thought of as the weaker president.
"But the question is, what do we think is in the best interest of the American people and national defense, Stacy—these fancy spaceplanes that can't carry as much as the Secret Service's Suburbans, or proven technology like stealth bombers, unmanned combat aerial vehicles, and aircraft carriers? McLanahan has convinced Martindale that spaceplanes are better, even though he used unmanned bombers almost exclusively in his attacks on Russia—"
"And as you've pointed out many times, Joe," Barbeau added, "we can't afford to put all our eggs in one basket again. The Russian attack was so successful because all the bombers were located at a small handful of undefended bases, and unless they're all in the air, they're vulnerable to attack. But aircraft carrier battle groups deployed to bases all around the world, or far out at sea, are heavily equipped for self-protection and are far less vulnerable to sneak attack."
"Exactly," Gardner said, nodding with pleasure that Barbeau had brought up the aircraft carriers. "That's the point I've been trying to make for all these years. We need a mix of forces—we can't dump all the money for new weapon systems on one unproven technology. An aircraft carrier battle group is no more expensive that what McLanahan is proposing we spend on these spaceplanes, but they are far more versatile and battle-proven."
"The Senate Armed Services Committee needs to hear that argument from you and your administration, Joe," Barbeau said, giving his hand another caress and leaning forward toward him sympathetically, exposing more of her ample cleavage. "McLanahan was the hero of the war to avenge the American Holocaust, but that was in the past. A lot of senators may be afraid to cross McLanahan for fear there will be a backlash against them if the American people wonder why they're not supporting America's most famous general. But with McLanahan silenced, if they get the direct support of the President, they'll be more inclined to break ranks. Now is the time to act. We must do something, and it has to be now, while McLanahan is...well, with all due respect, while the general is out of the picture. Undoubtedly the committee's confidence in the spaceplane program is rattled. They are much more amenable to a compromise."
"I think we need to get together on this, Stacy," Gardner said. "Let's hammer out a plan that both the committee and the Pentagon will support. We should present a united front."
"That sounds marvelous, Mr. President, really marvelous."
"Then I have the full support of the Senate Armed Services Committee?" Gardner asked. "I have allies in the House I can call on too, but the backing of the Senate is crucial. Together, united, we can go before the American people and Congress and make a convincing argument."
"What if McLanahan pulls out of this? He and that ex-senator astronaut science geek Ann Page are a formidable team."
"McLanahan is out—he'll surely retire, or be forced to retire."
"That man is a bulldog. If he recovers, he won't retire."
"If he won't do it for his own good, he'll do it because I'll order him to do it," Gardner said. "And if he still fights it, I'll make sure the world understands how dangerous the man has been over the years. He is a loose cannon—the world just doesn't know about it. The man killed dozens of innocent civilians in Tehran, for Christ's sake."
"He did?" She hated to let it slip that the majority leader of the U.S. Senate didn't know something, but she couldn't help it. It was a surprise, and she didn't like surprises. Would Gardner fill her in? "When?"
"On the very mission we were discussing when he had his episode, the operational test mission he was running from the Armstrong Space Station," Gardner replied. "He set off a missile that released chemical weapons outside an apartment building in Tehran, killing dozens including women and children, and then he attacked a Russian reconnaissance plane with some kind of death ray—probably to cover up the attack on Tehran."
Thank God Gardner was a blabbermouth. "I had no idea...!"
"That's not the half of what this joker does, Stacy. I know a dozen different criminal infractions and outright acts of war he's responsible for over the years—including an attack that probably made Russian president Gryzlov plan the atomic attacks against the United States."
"What?"
"McLanahan is a loose cannon, a complete wild card," Gardner said bitterly. "He attacked Russia with absolutely no authorization; he bombed a Russian bomber base simply for personal revenge. Gryzlov was a former Russian bomber pilot—he knew it was an attack against him, a personal attack." Gardner was on a roll—this was better than the Congressional Research Service, Barbeau thought. "That's why Gryzlov went after bomber bases in the United States—not because our bombers were any great strategic threat to Russia, but because he was trying to get McLanahan."
Barbeau's mouth was open in shock...but at the same time, she was tantalized, even aroused. Damn, she thought, McLanahan seemed like such a milquetoast, a Boy Scout—who the hell knew he was some kind of maverick action hero? That made him more appealing than ever. What else lurked underneath that impossibly quiet, unassuming frame? She had to shake herself out of her sudden reverie. "Wow..."
"The Russians are scared of him, that's for sure," Gardner went on. "Zevitin wants me to have him arrested. He demands to know what he's been doing and what he intends to do with the space station and those spaceplanes. He's madder than hell, and I don't blame him."
"Zevitin sees the space station as a threat."
"Of course he does. But is that the only damned benefit of the thing? It's costing us as much as two aircraft carrier battle groups to keep that thing up there...for what? I've got to reassure Zevitin that the space stuff is no direct offensive threat to Russia, and I don't know exactly what the thing can do! I didn't even know McLanahan was on board the thing!"
"If it's only a defensive system, I don't see any reason not to tell Zevitin all there is to tell about the space station, if it'll help defuse tensions between us," Barbeau said. "The McLanahan situation may have solved itself."
"Thank God," Gardner grunted. "I'm sure for every crime I know McLanahan is guilty of, there are ten more I don't know about...yet," Gardner went on. "He's got weapons at his disposal from dozens of different black research programs that I don't even fully know about, and I was the damned Secretary of Defense!"
She looked at Gardner carefully. "McLanahan will certainly retire on his own, or you can have him medically retire," she said. "But he could be even more dangerous to us on the outside."
"I know, I know. That's why Zevitin wants him put away."
"If I can help you put pressure on McLanahan, Joe, just tell me," Barbeau said sincerely. "I'll do whatever I can to turn him, or at least make him think about what his opinions mean to others in the government and around the world. I'll make him realize it's personal, not just business. I'll ruin him if he persists, but I'm sure I can convince him to see it our way."
"If anyone can convince him, Stacy, it's you."
They looked into each other's eyes for a long moment, each silently asking and answering the questions they dared not verbalize. "So, Stacy, I know this isn't your first time in the residence. I assume you've seen the Lincoln Bedroom before?"
Barbeau's smile was as hot as a bonfire, and she unabashedly looked Gardner up and down hungrily as if sizing him up in a pickup bar. She slowly rose from her seat. "Yes, I've seen it," she said in a low, breathy voice. "I played there as a young girl when my father was in the Senate. It was a children's playroom back then. Of course it has an entirely different connotation now—still a playroom, but not for children."
"It's still the best fund-raiser in town—twenty-five grand a night per person is the going rate."
"It's too bad we've been reduced to such tawdry acts, isn't it?" Barbeau asked. "It spoils the feel of this place."
"The White House is still a house," Gardner said distractedly. "It's impossible for me to see it as more than just a workplace. I haven't seen a tenth of the rooms in here yet. They tell me there are thirty-five bathrooms here—I've seen three. Frankly I don't have much desire to explore the place."
"Oh, but you should, Joe," Barbeau said. "I think you will, when you get over the tumultuous first few months in office and get a chance to relax."
"If McLanahan can stop stirring the shit, maybe I could."
She turned, her arms outstretched, looking around the room. "I asked Mr. Kordus if we could meet here, in the Treaty Room, because I don't recall ever being in here although it's right next door to the Lincoln Bedroom. But the history in this place is so strong you can feel it. The Treaty Room has been used as a Cabinet meeting room, reception and waiting room, and as the President's study. It's historically been the place in the White House where the real political business gets done, even more so than the Oval Office."
"I've had a few informal meetings in here, but mostly the staff uses it."
"The staff is usually too busy to appreciate the energy that flows through this room, Joe," Barbeau said. "You should take the time to sense it." With her arms still outstretched, she closed her eyes. "Imagine: Ulysses S. Grant conducting his half-drunken Cabinet meetings here, followed by a card game and arm-wrestling matches with his friends; Teddy Roosevelt nailing animal hides to the walls; Kennedy signing the Nuclear Test Ban Treaty here, then days later seducing Marilyn Monroe in the same place, right down the hall from where his wife and children slept."
Gardner stepped behind her and lightly put his hands on her waist. "I never heard that story before, Stacy."
She took his hands and pulled them around her waist, drawing him closer. "I just made that last one up, Joe," she said in a whisper, so quiet that he moved his cheek to hers and pulled her tightly to him to hear. "But I'll bet it happened. And who knows what a man like Kevin Martindale did in here after his divorce—the divorce that should have wrecked his political career but only enhanced it—with all his Hollywood starlets flitting in and out of here all the time at all hours?" She took his hands, swirled them around her belly, then took his fingers and gently lifted them to her breasts, encircling her nipples. She could feel his body stiffen and could practically hear his mind whirring as he tried to decide what to do about her sudden advance. "He probably had a different bitch in here every night of the year."
"Stacy..." She could feel Gardner's breath on her neck, his hands gently caressing her breasts, barely touching...
Barbeau whirled around toward him and roughly pushed him away. "Martindale was an imbecile, Joe, but he spent two terms as president and two terms as vice president and became a damned fixture in the White House—and he got to fuck Hollywood starlets in here! What are you going to do to beat that, Joe?"
Gardner was frozen in shock. "What the hell is wrong with you, Stacy?" he finally managed to blurt out.
"What is it you want, Mr. President?" Barbeau asked loudly. "What is your game plan? Why are you here?"
"What are you talking about?"
"You're the President of the United States of America. You live in the White House...but you've only used three bathrooms? You don't know what's been done in this room, this house, the enormous history of this place? You have a three-star general under your command that has twice the voter approval rating you do, with a heart condition no less, and he's still in uniform? There's a space station orbiting the planet that you don't want and it's still up there? You have a woman in your arms but you touch her like some sweaty lovestruck adolescent on his first date trying to get to second base? Maybe all you really did with Maureen Hershel is 'business,' is that it?"
Gardner was flustered, then angry, then indignant. "Listen, Senator, this is no damned game. You're hot as hell, but I came here to discuss business."
"You've been honest with me since I called you for this meeting, Joe—don't fucking lie to me now," Barbeau snapped, taking one step away from him and letting her green eyes bore into him. Her sudden change in persona, from seductress to barracuda, startled him. "I didn't have to threaten you to invite me to the residence; I didn't drag you down that hallway and into this room. We're not children here. We're talking about joining forces to get an important job done, even if it means siding with the Russians and ruining a distinguished military career. What did you think we'd do—shake hands on it? Sign a contract? Cross our hearts and hope to die? Not on your life. Now, if you don't want to do this, you let me know right now, and we'll both go back to our offices and responsibilities and forget this meeting ever took place."
"What is this shit—?"
"Don't give me the innocent-waif routine either, Gardner. I know this is the way politics is played in Louisiana—don't tell me you've never played it like this in Florida or Washington. We're going to do it, right here, right now, or you can just tuck your tail between your legs and crawl back to your nice safe cozy apartment down the hall. What's it going to be?" When he didn't answer, she sighed, shook her head, and tried to step around him...
...but when she felt his arm across her chest and his hand on her breast, she knew she had him. He pulled her close, grasped her behind the head with his other hand, and pulled her lips to his, kissing her deeply, roughly. She returned the kiss just as forcibly, her hand finding his crotch, massaging him impatiently. Their lips parted, and she smiled at him confidently, assuredly. "That's not going to be enough, Mr. President, and you know it," she said. She smiled at his quizzical expression, darkly this time, confidently, and his mouth opened when he realized what she meant, what she wanted. "Well?"
He scowled at her, then moved his hands back to her breasts, then to her shoulders, pushing her down. "Let's seal the deal, Senator," he said, leaning back against the Grant conference table, steadying himself.
"Good boy. Get over here." She dropped to her knees and quickly began to undo his belt and pants. "My, my, look what we have here. Are you sure you don't have a little coonass in you, Mr. President?" He didn't reply as she began her vigorous, rhythmic ministrations.
## CHAPTER FOUR
A man who has to be convinced to act before he acts is not a man of action.... You must act as you breathe.
—GEORGE CLEMENCEAU
ABOARD ARMSTRONG SPACE STATION
THE NEXT MORNING, EAST COAST TIME
"Joining us live from Armstrong Space Station, orbiting two hundred some odd miles above Earth, is a man that needs no introduction: Air Force Lieutenant General Patrick McLanahan," the cable news morning show host began. "General, thanks for joining us today. The question everyone wants an answer to, of course, is: How are you, sir?"
There was a second or two delay because of the satellite relay, but Patrick was accustomed to waiting those few seconds to make sure he wasn't talking over the host. "It's nice to be with you, Megyn," Patrick responded. He was Velcroed as usual to the station commander's console, wearing his trademark black flight suit with black insignia. "Thanks for having me on the show again. I'm doing fine, thank you. I feel pretty good."
"All of America is relieved to see you up and around, General. Have they determined what exactly happened?"
"According to Navy Captain George Summers at Walter Reed National Medical Center, who reviewed all my tests remotely from up here, it's called long-QT syndrome, Megyn," Patrick replied. "That's an infrequent prolongation of the electrical activation and inactivation of the heart's ventricles, caused by stress or shock. Apparently, other than eyesight, it's one of the most common disqualifying conditions in the astronaut corps."
"So you've been disqualified from flying ever again?"
"Well, I hope I won't be," Patrick said. "Officially I'm not really an astronaut in the conventional sense. I'm hoping that the docs will determine that incapacitation due to long-QT syndrome is most likely to occur just while traveling in space and won't stop me from all other flying activities."
"You do have a history of heart disease, is that correct?"
"My dad did die of heart problems, yes," Patrick replied somberly. "Dad suffered from what they used to call 'heart flutters' and was treated for anxiety and stress. Long-QT is hereditary. Apparently in my dad's case it was the police department and running a family business that triggered it; in my case, it was flying in space."
"And he died around the same age as you are now?"
A cloud passed briefly over Patrick's face that was clearly visible to millions of viewers around the world. "Yes, a couple years after retiring from the Sacramento Police Department and opening up McLanahan's in Old Town Sacramento."
"A shameless plug for your family tavern, eh, General?" the host asked, trying to liven up the conversation.
"I'm not ashamed of McLanahan's in Old Town Sacramento at all, Megyn."
"Another plug. Good. Okay, that's enough, General, you did your job fantastically," the host said, laughing. "Was this heart condition already noted on your records, and if so what were you doing flying repeatedly to Armstrong Space Station?"
"I did report the family history on my medical records," Patrick replied, "and I get a Class One Air Force flight physical twice a year, plus pre- and post-space flight checkups, and no problems have ever been detected before. Even though long-QT syndrome is a common disqualifying condition in the astronaut corps, I wasn't specifically tested for it because, as I said, technically I'm not an astronaut—I'm a unit commander and engineer who just happens to get to ride on his unit's research vehicles whenever I feel it's necessary."
"So do you feel that your lack of astronaut training and screening contributed to onset of this medical condition?"
"One of the things we're trying to prove with the Black Stallion spaceplane and Armstrong Space Station program, Megyn, is to make space more accessible to everyday folks."
"And it appears that the answer might be, 'No, they can't,' is that right?"
"I don't know all there is to know about long-QT syndrome, Megyn, but if it's commonly found only in combat aviators over the age of fifty who have to go into space frequently, perhaps we can test for it and exclude only those who show a proclivity for that disease," Patrick said. "I don't see why it has to disqualify everyone."
"But it is disqualifying for you?"
"I'm not ready to throw in the towel yet," Patrick said with a confident smile. "We have some incredible technology at our disposal, and new and better technologies being developed every day. If I can, I'll keep on flying, believe me."
"You haven't seen enough combat and orbited the Earth enough times already, General?" the host said with an amused laugh. "As I understand it, you've been on the station several times just in the past few months. That's more than a NASA astronaut goes into space in his entire career, isn't that true? John Glenn only flew in space twice."
"Pioneers like Senator John Glenn will always be the inspiration our future astronauts need to summon the courage and fortitude to undergo the rigorous preparation for space," Patrick replied, "but as I said, one goal of our military space program is to gain greater access to space. I don't consider episodes like mine a setback. It's all part of the learning experience."
"But you have to think of yourself and your family too, don't you, General?"
"Of course—my son sees me on TV more than he does in person," Patrick said gamely. "But no aviator likes to lose his wings, Megyn—we have an inbred aversion to doctors, hospitals, weight scales, eye charts, sphygmomanometers, and anything else that can keep us from flying..."
"Okay, General, you lost me there. Sphygmo...sphygmo...what is that, one of your high-tech laser ray guns?"
"A blood pressure tester."
"Oh."
"It'll be up to the flight docs, but you can bet I'll be fighting disqualification the whole way," Patrick said. A beep in his communications earset got his attention, and he turned and briefly activated his command monitor and read the display. "Sorry, Megyn, I have to go. Thanks for having me on this morning." The host was able to get out a confused and startled "But General, we're live around the—!" before Patrick terminated the link. "What do you have, Master Sergeant?" he asked on the command module intercom.
"COMPSCAN alert in the target region, sir, and it says it's a big one, although we might have nothing but a big glitch on our hands," Master Sergeant Valerie "Seeker" Lukas replied. The COMPSCAN, or Comparison Scans, collected and compared radar and imaging infrared data during sensor sweeps and alerted the crew whenever there was a significant buildup of personnel or equipment in a particular target region—thanks to the power and resolution of Armstrong's space-based radar and other satellites and unmanned aircraft, the target region could be as large as continent, and the change between comparison scans could be as small as four or five vehicles.
"What's the target?"
"Soltanabad, a highway airfield about a hundred miles west of Mashhad. Imaged recently by the new Night Owl unmanned reconnaissance plane Captain Noble just launched." Seeker studied the reconnaissance file on the area before continuing: "Attacked once by the Air Battle Force with a Vampire bomber with runway-cratering munitions last year because it was suspected of being used to fly in weapons and supplies to the Islamists operating out of Mashhad. The highway portion of the base was reopened by the Revolutionary Guards Corps, reportedly for relief and humanitarian supply shipments. We put the entire base on the 'watch' list and launched the Night Owl over the area to be sure they weren't repairing the ramps and taxiways or flying military stuff in there."
"Let's see what they're doing," Patrick said. A few moments later an incredibly detailed overhead image of the spot came up on his monitor. It clearly showed the four-lane highway with aircraft distance marks, taxi lines, and touchdown zone designations—it looked like a typical military runway, only with cars and trucks running on it. On both the north and south sides of the highway/airstrip were wide paved areas with aircraft taxiways, large aircraft parking areas, and the remnants of bombed-out buildings. Many of the destroyed buildings had been razed and a number of tents of various sizes put in their place, some with the seal of the Red Crescent humanitarian relief organization on them. "Do those tents look like they have open sides to you, Master Sergeant?" Patrick asked.
Seeker peered closer at the image, then magnified it until it started to lose resolution. "Yes, sir," she replied, unsure of why the general had asked—it was fairly plain to her. Per agreement between the United Nations, Buzhazi's Persian occupying force, and the Iranian government-in-exile, large tents set up in certain combat areas servicing refugees or others traveling through the Iranian deserts had to have open sides during reconnaissance flyover time periods so all sides could see inside, or they could be designated as hostile emplacements and attacked.
"Looks like a big shadow on that side, that's all," Patrick said. "This photo was taken during nighttime, correct?" Lukas nodded. "The sides look open, but the shadows on the ground from the nearby floodlights are making it look...I don't know, they just don't look right to me, that's all." He zoomed in again on the former aircraft parking ramps. Both paved areas were dotted with dozens of bomb craters, from several yards to over a hundred feet wide, with huge chunks of concrete heaved up around the edges. "Still looks busted up to me. How old is this image?"
"Just two hours, sir. No way they could have repaired all those craters and brought in aircraft in two hours."
"Let's see the scans compared by the computer." The image split first into two, then four, then sixteen shots of the same spot taken over a period of several days. The pictures appeared identical.
"Looks like a glitch—false alarm," Seeker said. "I'll reset the images and take a look at the comparison parameters for—"
"Wait a minute," Patrick said. "What is the computer saying has changed?" A moment later, the computer had drawn rectangles around several of the craters. The craters were precisely the same—the only difference was that the rectangles were not exactly oriented the same in all the images. "I still don't get what COMPSCAN is flagging."
"Me neither, sir," Seeker admitted. "Could be just a looking-angle computation error."
"But we're sun-synchronous on this part of the world, right?"
"Yes, sir. We're precisely over Tehran at the same time—approximately two A.M. local—every day."
"So the looking angle should be the same except for minor station or sensor attitude changes, which the computer should be correcting for," Patrick said.
"Obviously something's screwed up in the adjustment routine, sir," Seeker said apologetically, anchoring herself at her terminal to begin work. "Don't worry, I'll get it straightened out. Sorry about that, sir. These things need recalibrating—obviously a bit more often that I thought. I should probably look at the station attitude gyro compensation readouts and fuel consumption figures to see if there's a major shift taking place—we might have to make a gross alignment change, or just throw out all the old attitude adjustment figures and come up with new ones. Sorry, sir."
"No problem, Master Sergeant," Patrick said. "We'll know to look for things like that more often from now on." But he continued staring at the images and the computer's comparison boxes. The boxes disappeared as Lukas erased the old comparison data, leaving very clear images of the bomb craters on the ramps and taxiways. He shook his head. "The space-based radar's pictures are stunning, Seeker—it's like I can measure the thickness of those concrete blocks heaved up by the bombs. Amazing. I can even see the colors of the different layers of concrete, and where the steel reinforcing mesh was applied. Cool."
"The SBR is incredible, sir—it's hard to believe it's almost twenty-year-old technology."
"You can clearly see where the concrete ends and the road base begins. It's—" Patrick looked closely at the images, then put on a pair of reading glasses and peered closer. "Can you enlarge that image for me, Seeker?" he asked, pointing at a large crater on the south side of the highway.
"Yes, sir. Stand by."
A moment later the crater filled the monitor. "Fantastic detail, all right." But now something was niggling at him. "My son loves those 'I Spy' and 'Where's Waldo?' books—maybe he'll be an imagery analyst someday."
"Or he'll design the computers that will do it for us."
Patrick chuckled, but he still felt uneasy. "What is wrong with this picture? Why did the computer ring the bell?"
"I'm still checking, sir."
"I spent a short but insightful period of time as a detachment commander in the U.S. Air Force's Air Intelligence Agency," Patrick said, "and the one thing I learned about interpreting multispectral overhead imagery was not to let the mind fill in too many blanks."
"Analysis 101, sir: Don't see what isn't there," Seeker said.
"But never ignore what is there but isn't right," Patrick said, "and there is something not right about the position of those craters. They're different...but how?" He looked at them again. "They look to me like they're turned, and the computer said they moved, but—"
"That's not possible for a crater."
"No...unless they're not craters," Patrick said. He zoomed in again. "I might be seeing something that's not there, but those craters look too perfect, too uniform. I think they're decoys."
"Decoy craters? I've never heard of such a thing, sir."
"I've heard of every other kind of decoy—planes, armored vehicles, troops, buildings, even runways—so why not?" Patrick remarked. "That might explain why COMPSCAN flags them—if they're moved and not placed in exactly the same spot, COMPSCAN flags it as a new target."
"So you think they've rebuilt that base and are secretly using it, right under our noses?" Lukas asked, still unconvinced. "If that's true, sir, then the space-based radar and our other sensors should have picked up other signs of activity—vehicles, tire tracks, storage piles, security personnel patrolling the area..."
"If you know exactly when a satellite is going to pass overhead, it's relatively easy to fool it—just cover the gear with radar-absorbent camouflage, erase the tracks, or disguise them with other targets," Patrick said. "All those tents, trucks, and buses out there could be housing an entire battalion and hundreds of tons of supplies. As long as they offload the planes, get the men and vehicles out of the area, and sweep up the area within the two-to-three-hour span between our overflights, they're safe."
"So all our gear is practically useless."
"Against whoever is doing this, yes—and I'll bet it's not the Islamist clerics or even the remnants of the Revolutionary Guards Corps," Patrick said. "There's only one way to find out: we need eyes on the ground. Let's get a report ready for STRATCOM and I'll append my recommendations for action...but first I want to get Rascal working on a plan." While Lukas began downloading sensor data and adding her observations—and reservations—about the activity at Soltanabad, Patrick selected the command channel on his encrypted satellite communications system. "Odin to Rascal."
A moment later the image of a large, blond-haired, blue-eyed, powerful-looking man appeared on Patrick's monitor: "Rascal here, sir," replied Air Force Major Wayne Macomber rather testily. Macomber was the new commander of the Battle Force ground forces based at Elliott Air Force Base in Nevada, replacing Hal Briggs, who had been killed while hunting down mobile medium-range ballistic missiles in Iran a year earlier. Macomber was only the second person ever to take charge of the Battle Force. He had big shoes to fill, and that, in Patrick's mind, would never happen.
Macomber was not Patrick's first choice to lead "Rascal" (which had been Hal's call-sign and was now the new unclassified call-sign of the Battle Force). To put it mildly, Macomber had serious problems dealing with authority. But he had somehow managed to use that personality glitch to propel himself into more and more challenging situations in which he was ultimately able to adapt, overcome, and succeed.
He was kicked out of public middle school in Spokane, Washington, because of "behavioral incompatibilities" and was sent off to the New Mexico Military Institute in Roswell in hopes of having round-the-clock military discipline straighten him out. Sure enough—after a difficult first year—it worked. He graduated near the top of his class both academically and athletically and won a nomination to attend the Air Force Academy in Colorado Springs, Colorado.
Although he was a nationally ranked linebacker for the Falcons football team, where he earned his nickname "Whack," he was kicked off the squad in his senior year for aggressive play and "personality conflicts" with several coaches and teammates. He used the extra time—and probationary period—to improve his grades and again graduated with honors with a bachelor of science degree in physics and a pilot training slot. Once again he dominated in his undergraduate pilot training class, graduating top of his class, and won one of only six F-15E Strike Eagle pilot slots awarded straight out of flight school—almost unheard of for a first lieutenant at the time.
But again, he couldn't keep his drive and determination in check. An F-15 Eagle air superiority fighter is a completely different bird with an offensive systems operator, big radar, conformal long-range fuel tanks, and ten thousand pounds of ordnance on board, and for some reason Wayne Macomber couldn't figure out that airframes bend in unnatural directions when an F-15E Strike Eagle pilot loaded up with bombs tries to dogfight with another fighter. It didn't matter that he was almost always the winner—he was racking up victories at the expense of bending expensive airframes, and was eventually...ultimately...asked to leave.
But he was not orphaned for long. One organization in the Air Force welcomed and even encouraged aggressive action, out-of-the-box thinking, and virulent leadership: Air Force Special Operations. To his dismay, however, the unit that wanted rude and crude "Whack" the most was the Tenth Combat Weather Squadron at Hurlburt Field, Florida: because of his physics education, the Air Force quickly made him a combat weather parachutist. He got to wear the coveted green beret and parachutist wings of an Air Force commando, but it still grated on him to be known as a "weatherman."
Although he and his squadron mates always took a lot of ribbing from other commando units for being "combat weather-guessers" or "groundhogs," Macomber soon learned to like the specialty not only because he happened to like the science of meteorology but also because he got to parachute out of perfectly good planes and helicopters, carry lots of guns and explosives, learn how to set up airfields and observation posts behind enemy lines, and how to kill the enemy at close quarters. Whack performed more than a hundred and twenty combat jumps in the next eight years and rose quickly through the ranks, eventually taking command of the squadron.
When Brigadier General Hal Briggs was planning the assault and occupation of Yakutsk Air Base in Siberia in Patrick McLanahan's retaliatory operation against Russia following the American Holocaust, he turned to the one nationally recognized expert in the field to assist in mission planning for operations behind enemy lines: Wayne Macomber. At first Whack didn't like taking orders from a kid eight years younger than he, especially one who outranked him, but he quickly recognized Briggs' skill, intelligence, and guts, and they made a good team. The operation was a complete success. Macomber won a Silver Star for saving dozens of personnel, Russians as well as Americans, by getting them into fallout shelters before Russian president Gryzlov's bombers attacked Yakutsk with nuclear-tipped cruise missiles.
"I'm sending you the most recent shots of a highway airbase in northeastern Iran, Wayne," Patrick said. "I think it's being secretly repaired, and I'm going to ask permission for you to go in, recon it, and render it unusable again—permanently."
"A ground op? About time," Macomber responded gruffly. "Almost all I've been doing since you brought me here is sweating—either out doing PT or tryin' to squeeze into one of those damned Tin Man union suits."
"And complaining."
"The sergeant major been yakkin' about me again?" Marine Corps Sergeant Major Chris Wohl was the noncommissioned officer in charge of Rascal, the Air Battle Force ground team, and one of the most senior members of the unit. Although Macomber was commander of Rascal, everyone fully knew and understood that Chris Wohl was in charge—including Macomber, a fact which really rankled him. "I wish that sumbitch would retire like I thought he would do so I can pick my own first shirt. He's ready to be put out to pasture."
"I'm the commander of the Air Battle Force, Wayne, and even I wouldn't dare say that to the sergeant major's face," Patrick said, only half jokingly.
"I told you, General, that as long as Wohl is around, it'll be his unit and his baggage I'll have to drag around," Whack said. "All he does is mope around after Briggs." Patrick couldn't remotely picture Wohl moping for a second, but he didn't say so. "Guys die in special ops, even in tin can suits like that robot thing he was in—he better get used to that. Retire his ass, or at least reassign him, so I can spin up this unit my way."
"Wayne, you're in charge, so be in charge," Patrick said, not liking the way this conversation was going. "You and Chris can make a great team if you learn to work together, but you're still the man in charge whether you use him or not. I expect you to get your team ready to fly and fight, soonest. If it's not set up the way you want it in time for the next op, put Wohl in charge until—"
"I lead the unit, General, not the no-cock," Macomber retorted, using his own personal term "no-cock" instead of the Air Force acronym NCOIC, or "noncommissioned officer in charge."
"Then lead it, Wayne. Do whatever you need to do to accomplish the mission. Chris Wohl, the Cybernetic Infantry Devices, and the Tin Man armor can all be part of the problem or part of the solution—it's up to you. The men are pros, but they need a leader. They know Chris and will follow him into hell—you have to prove you can lead them along with the NCOIC."
"I'll whip them into line, General, don't worry about that," Macomber said.
"And if you haven't done it already, I'd suggest you not use that term 'no-cock' in front of Wohl, or you two might be standing before me bloody and broken. Fair warning."
Macomber's expression gave absolutely no indication that he understood or agreed with McLanahan's warning. That was unfortunate: Chris Wohl didn't tolerate most officers below flag rank and was not afraid to risk his career and freedom to straighten out an officer who didn't show the proper respect to a veteran noncommissioned officer. If the situation wasn't resolved properly, Patrick knew, those two were heading for a confrontation. "It would be a lot easier if I didn't have to train in that Tin Man getup."
"The 'getup,' as you call it, allows us to go into hot spots no other special ops team would ever consider," Patrick said.
"Excuse me, General, but I can't recall any hot spot I ever considered not going into," Macomber said testily, "and I didn't wear the long undies."
"How many men would you need to go in and take out an airfield, Major?"
"We don't 'take out' airfields, sir—we reconnoiter or disrupt enemy air ops, or we build our own airfields. We call in air strikes if we want it—"
"The Battle Force takes them out, Major," Patrick interjected. "Remember Yakutsk?"
"We didn't destroy that airfield, sir, we occupied it. And we brought in a hundred guys to help us do it."
"The Battle Force was prepared to destroy that base, Major—if we couldn't use it, the Russians weren't going to, either."
"Destroy an airfield?" The skepticism in Macomber's voice was obvious, and Patrick could feel the heat rise up under the collar of his black flight suit. He didn't want to waste time arguing with a subordinate, but Macomber had to be made aware of what was expected of him, not just busted because he was a junior officer. "How can a handful of lightly armed men destroy an airfield?"
"That's what you're here to learn, Wayne," Patrick said. "I told you when we first talked about taking over the command that I needed you to think outside the box, and around there it means not just learning to use the gadgets that you have at your disposal but embracing and expanding the technology and developing new ways to use it. Now I need you up to speed quick, because I've got an airfield in Iran I might want destroyed...tomorrow."
"Tomorrow? How can that happen, General? I just learned about the target location just now—if we hustled, we might make it off the base by tomorrow, and that's with no intel and no rehearsals on how to assault the target! You can't run a successful infiltration on a military base with no intel and no practice runs! I'll need at least a week just to—"
"You're not hearing what I'm telling you, Major: you have to start thinking differently around here," Patrick insisted. "We locate targets and attack them, period—little or no rehearsal, no strategic intel, first-cut organic intel received while en route, no joint support packages, and small but mobile and high-tech ground units with minimal but devastating air support. I told you all this when I first briefed you on Rascal, Wayne..."
"I assumed you got your intel and tasking from higher headquarters, sir," Macomber argued. "You mean you launch on an operation without gathering strategic intel from—?"
"We don't get any help from anyone, and we still launch and get the friggin' job done, Whack," Patrick interjected pointedly. "Are you finally getting the picture?" Patrick waited a heartbeat and got no response—considering Macomber's mercurial, almost rabid personality, the silence was a real stunner. "Now I know you're accustomed to Air Force special ops tactics and methodology, and I know you're a good operator and leader, but you have to get with the program at the Lake. I know PT is important, but knowing the hardware and resources we have is more important. It's a mind-set as well as a job. Understand?"
"Yes, sir," Macomber said—probably the first real hint of acquiescence Patrick had sensed from this guy. "Looks to me like I'll need Wohl's help after all if I'm going out on a mission...tomorrow?"
"Now you're getting the idea, Major."
"When can I get the intel you have, sir?"
"I'm sending it now. I need a game plan drawn up and ready to brief to the powers that be in an hour."
"An hour...?"
"Is there something wrong with this connection, Major?"
"No, sir. I heard you. One hour. One more question?"
"Hurry it up."
"What about my request to change the unit call-sign, sir?"
"Not again, Major..."
"That was Briggs' call-sign, sir, and I need to change that name. Not only do I hate it, but it reminds the guys of their dead former boss, and that detracts from their mission focus."
"Bill Cosby once said if it was up to him he would never have picked a name for his kids—he would just send them out onto the street and let the neighborhood kids name him," Patrick said.
"Bill who?"
"When it's time to change the unit's name, Major, the entire unit will come to me with the request."
"It's my unit, sir."
"Then prove it," Patrick said. "Get them ready to roll immediately, learn how to use the tools I've busted my butt to get you, and show me a plan—drawn up as a unit—that will get the job done and get approved right away. Get on it, Major. Genesis out." He broke the connection with a stab at the button so hard that it almost detached him from his Velcro perch. For Pete's sake, Patrick thought, he never realized how lucky he was to be working with the men and women under his command and not true prima donnas like Macomber. He might be one of America's premier specials ops commandos, but his interpersonal skill set needed some serious re-evaluation.
After taking an exasperated sip of water from a squeeze tube, he reopened the satellite link: "Odin to Condor."
"Condor here, secure," the senior controller at the Joint Functional Component Command-Space (JFCC-Space) command post at Vandenberg Air Force Base, California, responded. "Saw you on the news a bit ago. You looked A-OK, sir. Good to see you're feeling okay. That Megyn is a fox, isn't she?"
"Thanks, Condor, but unfortunately I never saw the host, so I'll have to take your word for it," Patrick responded. "I have an urgent reconnaissance assessment alert and request for ground ops tasking message for the boss."
"Roger that, sir," the senior controller responded. "Ready to copy whenever you're ready."
"I've detected a possible covert re-establishment of an illegal Iranian air base in the Persian Republic, and I need eyes-only confirmation and tasking authority for a shutdown if it's verified." Patrick quickly ran down what he knew and what he surmised about the Soltanabad highway airbase.
"Got it, sir. Sending to JFCC-Space DO now." The DO, or deputy commander of operations for Joint Functional Component Command-Space, would report to his commander after assessing the request, investigating availability of forces, gathering intelligence, and computing an approximate timeline and damage expectancy. It was time-consuming, but probably kept the commander from being inundated with requests for support. "We should get a message back soon if the DO wants to act. How do you feel, sir?"
"Just fine, Condor," Patrick responded. "Sure wish I could upload my requests directly to STRATCOM or even SECDEF," Patrick remarked.
"I hear you, sir," the controller said. "I think they're afraid you'll bury them with data. Besides, no one wants to give up their kingdoms." In a convoluted and rather frustrating mix of responsibilities, tasking and coordination for air missions involving Armstrong Space Station and HAWC's unmanned B-1 and B-52 bombers flying over Iran had to be channeled through two different major commands, who both reported directly to the President through the national security staff: JFCC-Space in California, who upchanneled the information to U.S. Strategic Command (STRATCOM) in temporary headquarters in Colorado and Louisiana; and to U.S. Central Command (CENTCOM) at MacDill Air Force Base in Florida, which handled all military operations in the Middle East and central Asia. CENTCOM and STRATCOM's different intelligence, plans, and operations staffs would go over the data separately, make their own recommendations, and present them to the Secretary of Defense and the President's National Security Adviser, who would then make recommendations to the President.
"I don't understand why these reports should go to STRATCOM at all," Patrick groused. "CENTCOM is the theater commander—they should get reports, draw up a plan of action, get approvals, and then task everyone else for support."
"You don't need to convince me, sir—if you ask me, your reports should go directly to SECDEF," the senior controller said. There was a slight pause; then: "Stand by for Condor, Odin. Good to talk with you again, General."
A moment later: "Condor-One up, secure," came the voice of the Fourteenth Air Force's commanding officer, Air Force Major General Harold Backman. The commander of the U.S. Air Force's Fourteenth Air Force, Backman was "dual-hatted" as Joint Forces Component Command-Space, or JFCC-S, a unit of U.S. Strategic Command (which had been destroyed in the Russian air attacks against the United States and was being reconstituted in various locations around the country).
JFCC-S was responsible for planning, coordinating, equipping, and executing all military operations in space. Before McLanahan, his High-Technology Aerospace Weapons Center, and the XR-A9 Black Stallion spaceplanes, "military operations in space" generally meant the deployment of satellites and monitoring space activities of other nations. No longer. McLanahan had given JFCC-Space a global strike and ultra-rapid mobility capability, and frankly he didn't feel they were yet up to the task.
"Odin here, secure and verified," Patrick said. "How are you doing, Harold?"
"Up to my eyeballs as usual, sir, but better than you, I'm guessing. The duty officer said he saw you on TV but you cut off the interview suddenly without warning. You okay?"
"I got a COMPSCAN warning and got right on it."
"If it scared the piss out of one of my controllers, it's going to panic the brass, you know that, right?"
"They should learn to relax. Did you get my data?"
"I'm looking at it right now, Muck. Give me a sec." A few moments later: "I've got my intel chief looking it over now, but it just looks like a bombed-out highway airbase to me. I take it you don't think so?"
"I think those craters are decoys, Harold, and I'd like some of my guys to go out there and take a look."
Another slight pause. "Khorasan province, just a hundred miles from Mashhad—that area is controlled by Mohtaz and his Revolutionary Guards Corps," Backman said. "Well within armed-response distance from Sabzevar, which certainly has a lot of Pasdaran hiding out there. If Soltanabad is really vacant, you'll still be in the teeth of the storm if the bad guys spot you—and if it's active like you said, it'll be a meat grinder. I assume you want to go in with just a couple of your robots, right?"
"Affirmative."
"Thought so. Your gizmos up there can't give you any more detailed imagery?"
"Our only other option is a direct flyover by a satellite or unmanned aircraft, and that'll alert the bad guys for sure. I'd like to get a peek first before I plan on blowing the place, and a small force would be the fastest and easiest."
"How fast?"
"I haven't looked at the orbital geometry, but I'm hoping we can launch them within four, have them on the ground in seven, airborne again in eight, and home within twelve."
"Days?"
"Hours."
"Shit," Backman cursed. "Pretty friggin' unbelievable, sir."
"If I had my guys based up here, Harold, like I briefed you and STRATCOM I'd like to do, I could possibly be out of there and back home in four hours."
"A-friggin'-mazing. I'm all for that, Muck, but I think that idea is just boggling too many minds down here on plain old planet Earth. You know that we've been directed by the National Command Authority to restrict all spaceplane missions to resupply and emergency only, right?"
"I consider this an emergency, Harold."
"I know you do...but is it really an emergency?"
Patrick swallowed down a flare of anger at being questioned about his judgment, but he was accustomed to everyone second- and third-guessing him, even those who knew and liked him. "I won't know for sure until I get some of my guys out there."
"I don't think it'll be authorized, sir. You still want me to ask the question?"
Patrick didn't hesitate: "Yes."
"O-kay. Stand by." The wait was not very long at all: "Okay, Muck, the DO of STRATCOM says you can get your guys moving in that direction, but no one puts boots—or whatever the hell your robots wear on their feet—on the ground, and no aircraft crosses any lines on any maps, without a go-ahead from CENTCOM."
"Can I load up a few Black Stallion spaceplanes and put them in orbit?"
"How many, and loaded up with what?"
"One or two with operators, staggered and in different orbits until I can get a firm A-hour; one or two cover aircraft, loaded with precision-guided weapons; perhaps one or two decoys that will double as in-orbit retrieval backups; and one or two Vampire bombers airborne from Iraq ready to destroy the base if we find it to be operational."
"That many spacecraft might be a hard sell—and the armed spacecraft might be a deal-breaker."
"The more I can forward-deploy, and the more support stuff I get into orbit, the quicker this will be over, Harold."
"I get it," Backman said. The pause was longer this time: "Okay, approved. No one crosses any political boundaries in the atmosphere without a go-ahead, and keep the re-entry weapons tight until given the green light." He chuckled, then added, "Jeez, I sound like friggin' Battlestar Galactica Commander Adama or something. Never thought I'd be okaying an attack from outer space in my lifetime."
"It's the way things need to be from now on, my friend," Patrick responded. "I'll have the complete package plan out to you within the hour, and the air tasking order for movement of spacecraft will be out to you sooner. Thanks, Harold. Odin out."
Patrick's next videoconference call was to his battle management area at Elliott Air Force Base: "Macomber notified us that you had given him a ground op in Iran and that he was in a time crunch to do some planning, so we've already jumped in," his deputy commander, Brigadier General David Luger, said. The two navigators had been together for over two decades, first as fellow B-52G Stratofortress crewmembers and then assigned to the High-Technology Aerospace Weapons Center as aircraft and weapon flight test engineers. Tall, lean, quiet, and deliberate in personality as well as appearance, Luger's best attribute was acting as Patrick McLanahan's conscience whenever his irascible, determined, single-minded side threatened to obliterate all common sense. "We should have something for you in no time. The guy's fast and pretty well organized."
"I knew you'd be on it, buddy," Patrick said. "Surprised to hear from Whack?"
"Surprised? How about thunderstruck?" Luger deadpanned. "Everyone in the Air Battle Force goes out of their way to avoid the guy. But when he gets down to business, he does okay."
"Any thoughts on Soltanabad?"
"Yeah—I think we should skip the prelims and just put a couple spreads of SkySTREAKs or Meteors with high explosives down there, instead of wasting time inserting a Battle Force team," Luger replied. "If the Iranians are hiding something there, our guys will be landing right on top of them."
"As much as I like blowing things up, Texas," Patrick responded, "I think we should get a look first. If those craters are really decoys, they're the best I've ever seen, which means—"
"They're probably not Iranian," Luger said. "You thinking maybe the Russians?"
"I think Moscow would like nothing better than to help Mohtaz destroy Buzhazi's army and station a few brigades there as his reward," Patrick said.
"You think that's what Zevitin wants to do?"
"An American-friendly state in Iran would be completely unacceptable," Patrick said. "Mohtaz is a nutcase, but if Zevitin can convince him to allow Russian troops into Iran to help defeat Buzhazi's army—or for any other reason such as defending against American aggression—Zevitin will be able to send in troops to counterbalance American domination in the region. At the very least, he can put pressure on President Gardner to back away from supporting former Soviet bloc countries that are drifting into the American sphere of influence."
"All that geopolitical stuff makes my head hurt, Muck," Dave said with mock weariness. Patrick could see Dave's attention diverted away from the videoconference camera. "I have the first draft of the plan ready—I'll upload it to you," he said, entering instructions into his computer.
"Okay, Muck, here's the preliminary status reports," Luger went on moments later. "We have two Black Stallion spaceplanes available within four hours along with their dedicated tankers and enough fuel and supplies for orbital missions, and three available in seven hours if we cancel some training sorties. Macomber says he can get loaded up in time to launch. How do you want to build the air tasking order?"
Patrick made fast mental calculations, working the timing backward from when he wanted the Black Stallion off the ground and out of Persian airspace. "I'd sure like to have decoys, backups, more intel, and more rehearsals for Whack and the ground forces, but my primary concern is getting a good look at that base soon without the Revolutionary Guards being alerted," he said. "I'll see if I can get approval for two Studs to go in right away. If we launch in four hours, we'll be over the objective by midnight to one A.M. local time—let's call it two A.M. to be safe. We recon for one hour max, blast off before civil sunrise, refuel somewhere over western Afghanistan, and head home."
"The 'Duty Officer' is spitting out the preliminary guesstimate for the air tasking order," Luger said. The "Duty Officer" was the central computer system based at the High-Technology Aerospace Weapons Center that tied in all of the various departments and laboratories around the world and could be securely accessed by any member of HAWC anywhere in the world—or, in the case of Armstrong Space Station, around it. "The biggest question mark we have right now is the KC-77 tanker support for the exfiltration aerial refueling. Our closest XR-A9-dedicated tanker is at Al Dhafra Air Base in the United Arab Emirates, which is two hours' flight time to the closest possible refueling point over Afghanistan. If everything worked absolutely perfectly—they loaded the tanker without mishap, got all the diplomatic and air traffic clearances in a timely manner, et cetera—they'd make a possible rendezvous spot over western Afghanistan just as the Black Stallion goes bingo fuel."
"And when was the last time we ever had a mission go completely flawless?"
"I don't recall that ever happening," Luger reassured him. "There are several emergency landing sites in that area we can use, but they are very close to the Iranian border, and we would need a lot of ground support to secure the base until fuel arrived. We can move recovery teams into Afghanistan to assist in case the Stud has to make an emergency landing, or we can push the mission back a couple days..."
"Let's push ahead with this plan," Patrick said. "We'll present it as is and bring in as many contingency assets as we can—hopefully we won't need any of them."
"You got it, Muck," Dave said. "I need to...stand by, Patrick...I have a call from your flight surgeon at Walter Reed. He wants to talk with you."
"Plug me in, and stay on the line."
"Roger that. Stand by..." A moment later the video image split in two, with Dave on the left side and the image of a rather young-looking man in Navy Work Uniform camouflage blue digital fatigues, typical of all military personnel in the United States since the American Holocaust. "Go ahead, Captain, the general is on the line, secure."
"General McLanahan?"
"How are you, Captain Summers?" Patrick asked. U.S. Navy Captain Alfred Summers was the chief of cardiovascular surgery at Walter Reed National Military Medical Center and the man in charge of Patrick's case.
"I saw your interview this morning," the surgeon said testily, "and with all due respect, General, I was wondering where you got your medical degree from?"
"You have some problems with what I told the interviewer, I take it?"
"You made it sound like long-QT syndrome can be cured by taking a couple aspirin, sir," Summers complained. "It's not as easy as that, and I don't want my staff blamed in case your request to remain on flight status is denied."
"Blamed by whom, Captain?"
"Frankly, sir, by the great majority of Americans who think you are a national treasure that should not be sidelined for any reason whatsoever," the physician responded. "I'm sure you know what I mean. In short, sir, long-QT syndrome is an automatic denial of flight privileges—there's no appeal process."
"My staff has been researching the condition, Captain, as well as the medical histories of several astronauts who have been disqualified from space duties but still retained flight status, and they tell me that the condition is not life-threatening and might not be serious enough to warrant a denial of—"
"As your doctor and the leading expert on this condition in the United States, General, let me set it straight for you if I may," Summers interjected. "The syndrome was most likely caused by what we call myocardial stretch, where severe G-forces deform the heart muscles and nerves and create electrical abnormalities. The syndrome has obviously lain dormant for your entire life until you flew into space, and then it hit full force. It's interesting to me that you obviously experienced some symptoms during some or perhaps all of your space flights, but then it lay dormant again until you had a mere videoconference confrontation—I'd guess it was equally as stressing as flying in space, or maybe just stressful enough to provide the trigger for another full-blown episode."
"The White House and Pentagon can do that, Doctor," Patrick said.
"No doubt, sir," Summers agreed. "But do you not see the danger in this condition, General? The stress of that simple videoconference episode, combined with your repeated trips into orbit, sparked electrical interruptions that eventually created an arrhythmia. It was so severe that it created cardiac fibrillation, or irregular heartbeat, a true heat 'flutter,' which like a cavitating pump means that not enough blood gets circulated to the brain even though the heart hasn't stopped. It goes without saying, sir, that any stressor now can bring on another episode, and without constant monitoring we have absolutely no way of knowing when or how severe it would be. Allowing you to stay on flight status would jeopardize every mission and every piece of hardware under your control."
"I assume you were going to add, 'not to mention your life,' eh, Captain?" Patrick added.
"I assume we're all thinking of your welfare first, sir—I could be mistaken about that," Summers said dryly. "Your life is at risk every minute you spend up there. I cannot stress that too strongly."
"I get it, I get it, Doctor," Patrick said. "Let's move on past the dire warnings now. What's the treatment for this condition?"
"'Treatment?' You mean, other than avoiding stress at all costs?" Summers asked with obvious exasperation. He sighed audibly. "Well, we can try beta blockers and careful monitoring to see if any electrical abnormalities crop up again, but this course of treatment is recommended only for non-syncopic patients—someone who has never passed out before from the condition. In your case, sir, I would strongly recommend an ICD—implantable cardioverter-defibrillator."
"You mean, a pacemaker?"
"ICDs are much more than just a pacemaker, sir," Summers said. "In your case, an ICD would perform three functions: carefully monitor your cardiac condition, shock your heart in case of fibrillation, and supply corrective signals to restore normal rhythm in case of any tachycardia, hypocardia, or arrhythmia. Units nowadays are smaller, less obtrusive, more reliable, and can monitor and report on a wide variety of bodily functions. They are extremely effective in correcting and preventing cardiac electrical abnormalities."
"Then it doesn't affect my flight status, right?"
Summers rolled his eyes in exasperation, completely frustrated that this three-star general wouldn't let go of the idea of getting back on flying status. "Sir, as I'm sure you understand, installing an ICD is a disqualifier for all flight duties except under FAA Part 91, and even then you'd be restricted to solo day VFR flights," he said, taken aback simply by the fact that anyone who had an episode like this man did would even think about flying. "It is, after, all an electrical generator and transmitter that can momentarily cause severe cardiac trauma. I can't think of any flight crewmember, military or civilian, who's been allowed to maintain flight status after getting an ICD."
"But if they're so good, what's the problem?" Patrick asked. "If they clear up the abnormalities, I should be good to go."
"They're good, much better than in years past, but they're not foolproof, sir," Summers said. "About one in ten patients suffer pre-syncopic or syncopic episodes—dizziness, drowsiness, or unconsciousness—when the ICD activates. Three in ten experience enough discomfort to make them stop what they're doing—truck drivers, for example, will feel startled or uncomfortable enough that they will pull off to the side of the road, or executives in meetings will get up and leave the room. You can't pull off to the side of the road in a plane, especially a spaceplane. I know how important flying is to you, but it's not worth—"
"Not worth risking my life?" Patrick interrupted. "Again, Doctor, with all due respect, you're wrong. Flying is essential to my job as well as an important skill and a source of personal pleasure. I'd be ineffective in my current position."
"Would you rather be dead, sir?"
Patrick looked away for a moment, but then shook his head determinedly. "What are my other alternatives, Doctor?"
"You don't have any, General," Summers said sternly. "We can put you on beta blockers and constant monitoring, but that's not as effective as an ICD, and you'd still be restricted in flight duties. It's almost guaranteed that within the next six months you'll have another long-QT episode, and the odds are greater that you'll suffer some level of incapacitation, similar or probably more severe than what you experienced before. If you're in space or at the controls of an aircraft, you'd become an instant hazard to yourself, your fellow crewmembers, innocent persons in your flight path, and your mission.
"General McLanahan, in my expert opinion, your current job or just about any military position I can think of is too stressful for a man in your condition, even if we install an ICD. More than any treatment or device, what you need now is rest. If there is no history of drug abuse or injury, long-QT syndrome is almost always triggered by physical, psychological, and emotional stress. The damage done to your heart by your position, duties, and space flights will last the rest of your life, and as we saw, the stress of just one simple videoconference meeting was enough to trigger a syncoptic episode. Take my advice: Get the ICD installed, retire, and enjoy your son and family."
"There have to be other options, other treatments," Patrick said. "I'm not ready to retire. I've got important work to do, and maintaining flying status is a big part of it—no, it's a big part of who I am."
Summers looked at him for a long moment with a stern and exasperated expression. "Bertrand Russell once wrote, 'One symptom of an approaching nervous breakdown is the belief that one's work is terribly important,'" he said, "except in your case, you won't suffer a nervous breakdown—you'll be dead."
"Let's not get too dramatic here, Captain..."
"Listen to me carefully, General McLanahan: I'm not being dramatic—I'm being as honest and open with you as I can," Summers said. "It is my opinion that you have suffered unknown but serious damage to your cardiac muscles and myocardium as a result of your space flight that is triggering long-QT episodes that are causing arrhythmia and tachycardia resulting in pre-syncoptic and syncoptic occurrences. Is that undramatic enough for you, sir?"
"Captain—"
"I'm not finished, sir," Summers interjected. "The likelihood is that even with rest and medication you will suffer another syncoptic event within the next six months, more severe than the last, and without monitoring and immediate medical attention, your chances of survival are twenty percent, at best. With an ICD, your chances of surviving the next six months go up to seventy percent, and after six months you have a ninety percent chance of survival."
He paused, waiting for an argument, and after a few moments of silence he went on: "Now if you were any other officer, one who didn't use to date the Vice President of the United States with the Secret Service in tow, I would simply advise you that I will recommend to your commanding officer that you be confined to the hospital for the next six months. I will—"
"Six months!"
"I will still advise your commanding officer so," Summers went on. "Whether you decide to get an ICD installed is your decision. But if you insist on not getting the ICD installed and you are not on 24/7 monitoring, you have virtually no chance of surviving the next six months. None. Do I make myself clear to you, sir?" Patrick momentarily looked like a rapidly deflating balloon, but Dave Luger could see his dejection quickly being replaced with anger—anger at what, he wasn't quite certain yet. "It appears to me that the final decision is up to you. Good day, General." And Summers logged out of the videoconference with a rueful shake of his head, certain that the three-star general had no intention of complying with his orders.
Once Summers left the conference, Patrick sat back in his chair, took a deep breath, then stared at the conference room table. "Well, shit," he breathed after several long moments in silence.
"You okay, Muck?" Dave Luger asked.
"Yeah, I guess so," Patrick replied, shaking his head in mock puzzlement. "I always thought it was Will Rogers who made that quote about mental breakdowns, not Bertrand Russell."
Dave laughed—this was the guy he was familiar with, making jokes at a time when most sane men would be on the verge of tears. "I guess Mark Twain was right when he said, 'It's not what you know, it's what you know that ain't so.'"
"It wasn't Mark Twain, it was Josh Billings."
"Who?"
"Never mind," Patrick said, turning serious again. "Dave, I need to learn everything about long-QT syndrome and treatment for heart arrhythmias before I can make a decision about what I can handle and what I can't. There are probably a dozen companies doing research on modern ICDs, or whatever the next generation of those things becomes—I should know about the latest advances before I decide to get any old technology installed. Jon Masters probably has an entire lab devoted to treating heart disorders."
"Excuse me for saying so, buddy, but you just had probably the best heart doc in the country on the line, ready to answer any questions you have, and you pretty much blew him off."
"He wasn't ready to help me—he was standing by ready to punch my ticket to a medical retirement," Patrick said. "I need to handle this in my own way."
"I'm worried about how much time you have to make this decision, Patrick," Dave said. "You heard the doc: most patients who have this condition either start continual monitoring and drugs or get an ICD installed, right away. The others die. I don't see what other research you need to do on this."
"I don't know either, Dave, but it's the way I always do things: I check them out for myself, using my own sources and methods," Patrick said. "Summers may be the best heart doc in the military, maybe even the country, but if that's so, then my own research will tell me that too. But riddle me this, bro: What do guys like Summers do with active-duty cardiac victims who are still alive?"
"They retire them, of course."
"They retire them," Patrick echoed, "and then they're cared for by the Veterans Administration or private doctors paid for in part by the government. Summers is doing what he always does: discharging sick guys and pushing them off to the VA. Most of his patients are so thankful to be alive that they never give retirement a second thought."
"Aren't you glad to still be alive, Muck?"
"Of course I am, Dave," Patrick said, giving his longtime friend a scowl, "but if I'm going to punch out, I'm doing it on my terms, not Summers'. In the meantime, maybe I'll learn something more about the condition and possible treatments that these docs don't know, something that will let me keep my flying status. Maybe I'll—"
"Patrick, I understand flying is important to you," Luger said sincerely, "but it's not worth risking your life to—"
"Dave, I risk my life just about every time I go up in a warplane," Patrick interrupted. "I'm not afraid of losing my life to—"
"The enemy...the outside enemy," Dave said. "Hey, Patrick, I'm just playing devil's advocate here—I'm not arguing with you. You do what you want. And I agree: it's worth risking your life using your skills, training, and instincts to battle an adversary who's out to destroy the United States of America. But the enemy we're talking about here is you. You can't outfly, outguess, or outsmart yourself. You're not equipped or trained to handle your own body trying to kill you. You should approach this battle like any battle you've ever prepared for..."
"That's exactly what I intend to do, Dave," Patrick said flatly. "I'm going to study it, analyze it, consult with experts, gather information, and devise a strategy."
"Fine. But take yourself off flight status and check into the hospital for round-the-clock monitoring while you do it. Don't be stupid."
That last comment took Patrick aback, and he blinked in surprise. "You think I'm being stupid?"
"I don't know what you're thinking, man," Luger said. He knew Patrick wasn't stupid, and he was sorry he said it, but the one thing that his longtime friend had taught him was to speak his mind. Patrick was scared, and this was his response to fear, just as it had been in the cockpit of a strategic bomber all these many years: Fight the fear, focus on the objective, and never stop fighting no matter how awful the situation appears.
"Look at it from the doc's point of view, Muck," Luger went on. "I heard the doctors tell you that this thing is like a ticking time bomb with a hair trigger. It might not go off at all, but the odds are it could go off in the next ten seconds as we're standing here arguing. Hell, I'm afraid you could vapor-lock on me as I'm arguing with you right now, and there's not a damned thing I could do from down here but watch you die."
"My chances of dying up here in Earth orbit are just a little bit greater than average with this heart thing—we can be blasted wide open and sucked out into space by a hypersonic piece of debris the size of a pea at any friggin' time, and we'd never know it," Patrick said.
"If you're not sure about an ICD, then go ahead and research it; talk to Jon Masters or the dozen or so brainiacs on our list, and think it over," Dave said. "But do it from the safety of a private hospital room where the docs can keep an eye on you." Patrick's eyes and features remained determined, stoic, impassive. "C'mon, Muck. Think about Bradley. If you continue to fly without the ICD, you might die. If you don't stress yourself out, you'll probably live on. What's the question?"
"I'm not going to give in, Dave, and that's it. I'm up here to do an important job, and I'm—"
"A job? Muck, do you want to risk hurting yourself over a job? It's important, sure, but dozens of younger, stronger guys can do it. Give the job to Boomer, or Raydon, or even Lukas—anyone else. You haven't figured it out yet, Patrick?"
"Figure what out?"
"We're expendable, General McLanahan. We're all disposable. We're nothing but 'politics by other means.' When it comes right down to it, we're just hard-core hard-assed type-A gung-ho military prima donnas in ill-fitting monkey suits, and nobody in Washington cares if we live or die. If you blow a gasket tomorrow there'll be twenty other hard-asses waiting to take your place—or, more likely, Gardner could just as easily order us shut down the day after you croaked and spend the money on more aircraft carriers. But there are those of us who do care, your son being at the top of the list, but you're not paying attention to us because you're focusing on the job—the job that doesn't care one whit about you."
Luger took a deep breath. "I know you, man. You always say that you do it because you don't want to order another flyer to do something you haven't done yourself, even if the flyers are trained test crewmembers, the best of the best. I've always known that's bullshit. You do it because you love it, because you want to be the one to pull the trigger to take down the bad guys. I understand that. But I don't think you should be doing it anymore, Muck. You're unnecessarily risking your life—not by flying a mostly untested machine, but by exposing yourself to stresses that can kill you long before you reach the target area."
Patrick was silent for a long time; then he looked at his old friend. "I guess you do know what it's like to face your own mortality, don't you, Dave?"
"Unfortunately, yes," Luger said. As a young navigator-bombardier flying a secret mission to destroy the old Soviet Union's Kavaznya ground-based laser site, Dave Luger had been captured by the Russians, interrogated, tortured, and imprisoned for several years, then brainwashed into believing he was a Russian aerospace engineer. The effects of that treatment affected him emotionally and psychologically—stress would cause him to unexpectedly enter a detached fugue state that left him nearly incapacitated with fear for minutes, sometimes hours—and he voluntarily took himself off active flight status years ago. "It was a hell of a ride...but there are other rides out there."
"Don't you miss flying?" Patrick asked.
"Hell no," Dave said. "When I want to fly, I pilot one of the unmanned combat air vehicles or my radio-controlled model planes. But I have enough things going on where I don't have the desire anymore."
"I'm just not sure how it would affect me," Patrick said honestly. "I think I'd be okay—no, I'm sure I would—but would I always be demanding one more flight, one more mission?"
"Muck, you and me both know that manned aircraft are going the way of the dinosaur," Dave said. "Are you all of a sudden getting some kind of romantic notion about aviation, some kind of weird 'slip the surly bonds' idea that somehow makes you forget everything else? Since when did flying ever become anything more than 'plan the flight, then fly the plan' for you? Man, if I didn't know you, I'd swear you cared more about flying than you did about Bradley. That's not the Patrick Shane McLanahan I know."
"Let's drop it, okay?" Patrick asked irritably. He hated it when Luger (or his former girlfriend, Vice President Maureen Hershel) brought up his twelve-year-old son Bradley, believing it was a too-oft-used argument to try to get Patrick to change his mind about something. "Everyone's all worried about my heart, but no one stops arguing with me." He made sure to give Luger a smile when he added, "Maybe you're all trying to make me crash. Change the damned subject, Texas. What's going on at the Lake?"
"The rumor mill is churning, Muck," Dave said. "Guess who might be back at HAWC?"
"Martin Tehama," Patrick responded. Dave blinked in surprise—this was a guy who was rarely surprised. "I saw a strange e-mail address on a CC from SECDEF and checked to see who was in that office. I think he's going to be reinstated as HAWC commander."
"With his buddy in the White House? No doubt." Air Force Colonel Martin Tehama was designated the commander of the High-Technology Aerospace Weapons Center after Major General Terrill "Earthmover" Samson's departure, bypassing Patrick McLanahan. A well-respected test pilot and engineer, Tehama wanted to rein in the "extracurricular" activities HAWC often got involved with—such as using experimental aircraft and weapons in "operational test flights" around the world—and get back to the serious business of flight test. When Patrick left his White House adviser position he was awarded command of HAWC, bumping Tehama out. He retaliated by delivering reams of information on HAWC's classified missions to members of Congress. "After Summers files a full report on your condition, he'll reappear and take charge as soon as you announce your retirement—or the President announces that you're being medically retired."
"The President and Senator Barbeau will use my heart thing to cancel the Black Stallion program, citing health concerns, and their errand boy Tehama will promptly shut it down within months."
"Not even that long, Muck," David said. "The word from the Senate is that they're going to push the White House to move quicker to shut us down."
"Barbeau wants her bombers, that's for sure."
"It's not just her, but she's the loudest voice," Dave said. "There are lobbyists for every weapon system imaginable—carriers, ballistic missile subs, special ops, you name it. President Gardner wants another four aircraft carrier battle groups at least, maybe six, and he's likely to get them if the space program is canceled. Everyone's got their own agenda. The spaceplane lobby is practically nonexistent, and your injury just casts a shadow on the program, which delights the other lobbyists no end."
"I hate this political shit."
"Me too. I'm surprised you lasted as long as you did working in the White House. You definitely weren't made for wearing a suit, listening to meaningless speeches while wasting weeks testifying before another congressional committee, and being jerked around by lobbyists and so-called experts."
"Copy that," Patrick said. "Anyway, the heat's been turned up, and Tehama will turn it up even more—right in our faces. All the more reason to accomplish this Soltanabad mission, bring the crew back safely, and get some good intel all before tomorrow morning. The Russians are up to something in Iran—they can't be content to just sit in Moscow or Turkmenistan and watch Iran become democratic, or disintegrate."
"I'm on it," Dave said. "The air tasking order will be ready by the time you get the green light. I'll send you the orbital game plan and the complete force timing schedule right away. Genesis out."
## CHAPTER FIVE
Integrity is praised and starves.
—DECIMUS JUNIUS JUVENALIS
HIGH-TECHNOLOGY AEROSPACE WEAPONS CENTER, ELLIOTT AIR FORCE BASE, NEVADA
A SHORT TIME LATER
"It's ten times more boring than playing video games," Wayne Macomber complained, "because I can't even play the thing."
"Pretty deep wash ahead, Whack," U.S. Army National Guard Captain Charlie Turlock said. "It angles away from the objective, so we'll eventually have to get out. We should—"
"I see it, I see it," Macomber grumbled. "Wohl, clear those railroad tracks again."
"Roger," Marine Corps Sergeant Major Chris Wohl responded in his usual gravelly whisper. A moment later: "Rails are clear, Major. Satellite reports the next train is twenty-seven miles to the east, heading in our direction at twenty-five miles an hour."
"Copy," Macomber responded, "but I keep on seein' a return at my three o'clock, five miles, right in front of you somewhere. It's there for a second and then it disappears. What the hell is it?"
"Negative contact, sir," Wohl radioed.
"This is nuts," Macomber muttered, knowing that both Turlock and Wohl could still hear him but not caring one bit. This was not how he envisioned doing mission planning...although he had to admit it was pretty darned cool.
As incredible as the spaceplane was, even the passenger module was a pretty nifty device. It served to not only carry passengers and cargo inside the Black Stallion but also as a docking adapter between the spaceplane and a space station. In an emergency the module could even be used as a spacecraft crew lifeboat: it had maneuvering thrusters to facilitate retrieval by repair spacecraft while in orbit and to keep it upright during re-entry; little winglets for stability in case it was jettisoned in the atmosphere; enough oxygen to allow six passengers to survive for as long as a week; enough shielding to survive re-entry if the module was jettisoned during re-entry; and parachutes and flotation/impact attenuation bags that would cushion the module and its passengers upon land or water impact. Unfortunately all this protection was only available to the passengers—there was no way for the Black Stallion's flight crew to get inside the module after takeoff except by spacewalking while in orbit and using the transfer tunnel.
Macomber and Wohl were wearing a full Tin Man armor system, a lightweight suit made of BERP, or ballistic electronically reactive process material which was totally flexible like cloth but protected the wearer by instantly hardening to a strength a hundred times greater than steel when struck. The suit was completely sealed, affording excellent protection even in harsh or dangerous environments, and was supplemented with an extensive electronic sensor and communications suite that fed data to the wearer through helmet visor displays. The Tin Man system was further enhanced by a micro-hydraulic exoskeleton that gave the wearer superhuman strength, agility, and speed by amplifying his muscular movements.
Charlie Turlock—"Charlie" was her real name, not a call-sign, a young woman given a boy's name by her father—was not wearing a Tin Man suit, just a flight suit over a thin layer of thermal underwear; her ride was in the cargo compartment behind their seats. She wore a standard HAWC flight helmet, which displayed sensor and computer data on an electronic visor similar to the sophisticated Tin Man displays. Trim, athletic, and of just slightly more than average height, Turlock seemed out of place with a unit full of big, muscular, commandos—but she brought something along from her years at the Army Research Laboratory's Infantry Transformational Battle-lab that more than made up for her smaller physical size.
All three were watching a computer animation of their planned infiltration of the Soltanabad highway airfield in Persia. The animation used real-time satellite sensor images to paint an ultra-realistic view of the terrain and cultural features in the target area, complete with projections of such things as personnel and vehicle movement based on past information, lighting levels, weather predictions, and even soil conditions. The three Battle Force commandos were spread out approximately fifty yards apart, close enough to support one another quickly if necessary but far enough apart to not give one another away if detected or engaged by a single enemy patrol.
"I can see the fence now, range one point six miles," Charlie reported. "Moving over the wash now. The 'Goose' reports thirty minutes of flight time left." The "Goose" was the GUOS, or Grenade-launched Unmanned Observation System, a small powered flying drone about the size of a bowling pin, launched from a backpack launcher, that sent back visual and infrared images to the commandos by a secure datalink.
"That means we're behind," Macomber groused. "Let's pick it up a little."
"We're right on schedule, sir," Wohl whispered.
"I said we're behind, Sergeant Major," Macomber hissed. "The drone will be running out of fuel and we'll still be inside the damned compound."
"I've got another Goose ready," Charlie said. "I can launch it—"
"When? When we get close enough for the Iranians to hear it?" Macomber growled. "How noisy are those things anyway?"
"If you'd show up for my demos, Major, you'd know," Charlie said.
"Don't give me any lip, Captain," Macomber spat. "When I ask you a question, give me an answer."
"Outside a couple hundred yards of engine ignition, they won't hear a thing," Charlie said, not disguising her exasperation at all, "unless they have audio sensors."
"If we had proper intel before starting this mission, we'd know if the Iranians had audio sensors," Macomber groused some more. "We need to plan delaying the drone launch until we're within two miles of the base, not three. You got that, Turlock?"
"Roger," Charlie acknowledged.
"Next I need—" Macomber stopped when he noticed a flicker of a target indicator appearing again in the very periphery of his electronic visor's field of view. "Dammit, there it is again. Wohl, did you see it?"
"I saw it that time, but it's gone," Wohl responded. "I'm scanning that area...negative contact. Probably just a momentary sensor sparkle."
"Wohl, in my book, there's no such thing as 'sensor sparkle,'" Macomber said. "There's something out ahead of you causing that return. Get on it."
"Roger," Wohl responded. "Moving off-track." He used a small thumbwheel mouse to change direction in the animation, waiting every few meters until the computer added available detail and plotted more warnings or cautions regarding whatever lay ahead. The process was slow because of all the wireless computer activity, but it was the only available means they had of rehearsing their operation and getting ready to fly it at the same time.
"We're supposed to be commandos—there's no such thing as a 'track' for us," Macomber said. "We have an objective and a million different ways of getting there. It should be a damned piece of cake with all these pretty pictures floating in front of us—why is this making my head hurt?" Neither Turlock nor Wohl replied—they had grown quite accustomed to Macomber's complaining. "Anything yet, Wohl?"
"Stand by."
"Looks like tire tracks just past the wash," Charlie reported. "Not very deep—Humvee-sized vehicle."
"That's new," Macomber said. He checked the source data tags. "Fresh intel—downloaded in just the past fifteen minutes by a low-altitude SAR. A perimeter patrol, I'd guess."
"No sign of vehicles."
"That's the reason we're doing this, isn't it, kids? Maybe the general was right after all." It sounded to both Wohl and Turlock as if Macomber hated to admit that the general could be right. "Let's proceed and see what—"
"Crew, this is the MC," the mission commander, Marine Corps Major Jim Terranova, cut in over the intercom, "we've commenced our countdown to takeoff, T-minus fifty-six minutes and counting. Run your pre-takeoff checklists and prepare to report in."
"Roger, S-One copies," Macomber responded...except, as he noted himself with not a small bit of shock, that his words came out through an instantly dry, raspy throat and vocal cords, with barely enough breath for the words to escape his lips.
If there was one thing these guys at the High-Technology Aerospace Weapons Center and the Air Battle Force were really good at, Macomber had learned early on, it would definitely be computer simulations. These guys ran simulations on everything—for every hour of real flight time, these guys probably did twenty hours on a computer simulator beforehand. The machines ranged from simple desktop computers with photo-realistic displays to full-scale aircraft mockups that did everything from drip hydraulic fluid to smoke and catch on fire if you did something wrong. Everyone did them: air crews, maintenance, security, battle staff, command post, even administration and support staffs conducted drills and simulations regularly.
A good percentage of all the personnel at both Elliott and Battle Mountain Air Bases, probably one-tenth of the five thousand or so at both locations, were involved solely in computer programming, with other private and military computer centers tied in all around the world contributing the latest codes, routines, subroutines, and devices; and at least a third of all the code these top secret super-geeks wrote 24/7 had to be involved solely with simulations. This was his first real trip into space, but the simulations were so realistic and so numerous that he truly felt as if he had done this dozens of times before...
...until just now, when the mission commander announced they were less than an hour from takeoff. He had been so busy preparing for the approach and infiltration into Soltanabad—just three hours to get ready, when he demanded no less than three days to prepare in the Combat Weather Squadron!—that he had completely forgotten that they were going to be blasted into space to get there!
But now that frightening reality hit home with full force. He was not going to just pile his gear into a C-17 Globemaster II or C-130 Hercules for a multiday trip to some isolated airstrip in the middle of nowhere—he was going to be shot almost a hundred miles into space, then flutter down through the atmosphere through hostile airspace to a landing in a desert in northeastern Iran, where quite possibly an entire brigade of Iranian Revolutionary Guards Corps fighters, the elite of the former theocratic regime's terror army, could be waiting for them.
In the time it would normally take for him to just arrive at his first transition base en route to his destination, this mission would be completed! That simple fact was absolutely astounding, almost unbelievable. The time compression was almost too much to comprehend. And yet, here he was, sitting in the actual spacecraft—not a simulator—and the clock was ticking. By the time the sun rose again, this mission would be over, and he'd be debriefing it. He would have entered low-Earth orbit, traveled halfway across the globe, landed in Iran, scoped it out, blasted off again, re-entered low-Earth orbit, and hopefully landed at a friendly base...
...or he'd be dead. There were a million unforeseen and un-simulatable things that could kill them, along with the hundred or so simulatable things they practiced dealing with day after day, and even when they knew something bad was going to happen, sometimes they couldn't deal with it. It would either work out okay, or they'd be dead...or a hundred other things could happen. Whatever would happen, it was all going to happen now.
Macomber certainly felt the danger and the uncertainty...but as it so often did, the frenetic pace of every activity dealing with McLanahan and everyone at the High-Technology Aerospace Weapons Center and the Air Battle Force quickly pushed every other feeling of dread out of his conscious mind. It seemed a dozen voices—some human, but most computerized—were speaking to him at the same time, and all needed acknowledgment or an action, or the speaking quickly changed to "demanding." If he didn't respond quickly enough, the computer usually ratted on him, and a rather irate human voice—usually the mission commander but sometimes Brigadier General David Luger, the deputy commander himself, if it was critical enough—repeated the demand.
He was accustomed to performing and succeeding under intense pressure—that was the common denominator for any Special Operations commando—but this was something entirely different: because at the end of all the sometimes chaotic preparation, they were going to shoot his ass into space! It seemed Terranova made the announcement just moments earlier when Macomber felt the Black Stallion move as four Laser Pulse Detonation Rocket System engines, or "leopards," in full turbofan propulsion mode, easily propelled the aircraft to Dreamland's four-mile-long dry lake bed runway.
Whack was not afraid of flying, but takeoffs were definitely his most fearsome phase of flight—all that power behind them, the engines running up to full power sucking up tons of fuel per minute, the noise deafening, the vibration its most intense, but the aircraft still moving relatively slowly. He had done many Black Stallion takeoffs in the simulator, and he knew that the performance numbers even with the spacecraft still in the atmosphere were impressive, but for this part he was definitely on pins and needles.
The initial takeoff from the dry lake bed runway at Elliott Air Force Base was indeed spectacular—a massive shove as the LPDRS engines in turbofan mode moved into full military thrust, then a rapid, high-angle climb-out at well over ten thousand feet per minute after a short takeoff roll. The first few seconds of the run-up and takeoff roll seemed normal...but that was it. At full military power in turbofan mode, the four LPDRS engines developed one hundred thousand pounds of thrust each, optimized by solid-state laser igniters that superheated the jet fuel before ignition.
But high-performance takeoffs were nothing new to Whack or to most commandos and others who flew in and out of hostile airstrips. He had been in several huge C-17 Globemaster II and C-130 Hercules transport planes where they had to do max-performance takeoffs to get out of range of hostile shoulder-fired anti-aircraft missiles in the vicinity of the airstrip, and those planes were many times larger and far less high-tech than the Black Stallion. There was nothing more frightening than the feel of a screaming five-hundred-thousand-pound C-17 Globemaster III cargo plane standing on its tail clawing for every foot of lifesaving altitude.
The Tin Man outfit actually helped his body take some of the G-forces and even gave him a little extra shot of pure oxygen when it sensed his heart and breathing rates jumping up a bit. Because the thrust was so powerful and the air so dense at lower altitudes, the laser igniters had to be "pulsed," or rapidly turned off and on again, to avoid blowing up the engines. This created the distinctive "string of pearls" contrails across the Nevada skies that conspiracy theorists and "Lakespotters"—guys who sneaked into the classified test ranges in hopes of photographing a top secret aircraft for the first time—associated with the Air Force's Aurora hypersonic spy plane.
They had a short high subsonic cruise out over the Pacific coast to the refueling area, and then a rendezvous with an Air Battle Force KC-77 tanker. The secret of the Black Stallion spaceplane program was the inflight refueling, where they took on a full load of jet fuel and oxidizer right before blasting into orbit—instead of launching from zero altitude in the thickest part of the atmosphere, they would begin the cruise into space from twenty-five thousand feet and three hundred knots, in far less dense air.
Refueling always seemed to take forever in every aircraft Whack had ever flown in, especially the big intercontinental-range jet transports, but the Black Stallion took even longer because they actually required three consecutive refuelings: the first to top off the jet fuel tanks, since they didn't take off with a full load and needed a refueling right away; the second to top off the large borohydrogen tetroxide oxidizer—BOHM, nicknamed "boom"—tanks; and a third to top off the jet fuel tanks once more right before the boost into space. Filling the JP-7 jet fuel tanks went fairly quickly each time, but filling the large BOHM tanks took well over an hour because the boron and enhanced hydrogen peroxide mixture was thick and soupy. It was easy to feel the XR-A9 get heavier and noticeably more sluggish as the tanks were being filled, and every now and then the pilot needed to stroke the afterburners on the big LPDRS engines to keep up with the tanker.
Macomber spent the time checking intel updates downloaded to his on-board computers on their target area and studying the maps and information, but he was starting to get frustrated because precious little new data seemed to be coming in, and boredom was setting in. That was dangerous. Although they didn't have to prebreathe oxygen before this flight, as they would if they were going to wear a space suit, they couldn't take their helmets off during refueling operations; and unlike Wohl, who could take a combat catnap anywhere and anytime, like right now, Macomber couldn't sleep before a mission. So he reached into his personal kit bag attached to the bulkhead and...
...to Turlock's stunned amazement, pulled out a ball of red yarn and two knitting needles, which already had a section of knitted material strung on them! He found it amazingly easy to manipulate the needles with the Tin Man armored gloves, and before long he was picking up speed and almost at his normal work pace.
"Crew, this is S-Two," Turlock said on intercom, "you guys are not going to believe this."
"What is it?" the spacecraft commander, U.S. Navy Lieutenant Commander Lisette "Frenchy" Moulain asked, the concern thick in her voice. There was normally very little conversation during aerial refueling—anything said on the open ship-wide intercom was usually an emergency. "Do we need a disconnect...?"
"No, no, SC, not an emergency," Charlie said. She leaned forward in her seat to get a better look. Macomber was seated ahead of her and on the opposite side of the passenger module, and she strained in her straps to see all the way into his lap. "But it is definitely a shocker. The major appears to be...knitting."
"Say again?" Jim Terranova asked. The Black Stallion spaceplane burbled momentarily as if the spacecraft commander was momentarily so stunned that she almost flew out of the refueling envelope. "Did you say 'knitting'? Knitting...as in, a ball of yarn, knitting needles...knitting?"
"Affirmative," Charlie said. Chris Wohl, who was seated beside Macomber, woke up and looked over at Macomber for a few seconds, the surprise evident even through his helmet and Tin Man body armor, before he dropped back off to catnap again. "He's got the needles, the red ball of yarn, the 'knit one purl two' thing going, the whole show. Martha friggin' Stewart right over here."
"Are you shitting me?" Terranova exclaimed. "Our resident bad-ass snake-eating commando is knitting?"
"He looks sooo cute, too," Charlie said. Her voice changed to that of a young child's: "I can't tell if he's making a cute widdle doily, or maybe it's a warm and cozy sweater for his widdle French poodle, or maybe it's a—"
In a blur of motion that Turlock never really saw, Macomber withdrew another knitting needle from his kit bag, twisted to his left, and threw it at Turlock. The needle whistled just to the right of her helmet and buried itself three inches deep into her seat's headrest.
"Why, you motherfucker...!" Turlock exclaimed, pulling the needle out. Macomber waved at her with his armored fingers, grinning beneath his bug-eyed helmet, then turned and went back to his knitting.
"What in hell is going on back there?" Moulain asked angrily.
"Just thought since the captain was talking baby talk that maybe she wanted to try knitting too," Whack said. "You want the other one, Turlock?"
"Take off that helmet and I'll give it back to you—right between your eyes!"
"You jerks knock it off—maintain radio discipline," Moulain ordered. "The most critical part of aerial refueling and you bozos are farting around like little snot-nosed kids. Macomber, are you really knitting?"
"What if I am? It relaxes me."
"You didn't get clearance from me to bring knitting stuff on board. Put that shit away."
"Come back here and make me, Frenchy." There was silence. Macomber glanced over at Wohl—the only one on the spacecraft who probably could make him, if he wanted to—but he looked like he was still asleep. Whack was sure he wasn't, but he made no move to intervene.
"You and I are going to have a little talk when we get home, Macomber," Moulain said ominously, "and I'll explain to you in terms I hope you can understand the authority and responsibilities of the spacecraft commander—even if it takes a swift kick in your ass to make it clear."
"Looking forward to it, Frenchy."
"Good. Now knock off the horseplay, put away any nonauthorized equipment in the passenger module, and cut the chatter on the intercom, or this flight is terminated. Everyone got it?" There was no response. Macomber shook his head but put away his knitting stuff as directed, smiling at the feeling of Turlock's angry glare on the back of his helmet. The rest of the refueling was carried out with only normal call-outs and responses.
After refueling was completed, they subsonically cruised northward along the coast for about an hour, flying loose formation with the KC-77—it was now easy for the tanker to keep up with the Black Stallion since the spaceplane was so heavy. They hooked up with the tanker once again to top off the JP-7 tanks, which didn't take long, and then the tanker headed back to base. "Orbital insertion checklist programmed hold, crew," Terranova reported. "Report in when your checklist is complete."
"S-One, wilco," Macomber growled. Yet another checklist. He called up the electronic checklist on his helmet's electronic data visor and used the eye-pointing cursor and voice commands to check off each item, which mostly dealt with securing loose items, checklisting the oxygen panel, cabin pressurization, yada yada yada. It was all busywork that a computer could check easily, so why have humans do it themselves? Probably some touchy-feely human engineering thing to make the passengers feel they were something else other than exactly what they were: passengers. Whack waited until Turlock and Wohl completed their checklists, checked his off as complete, then spoke, "MC, S-One, checklist complete."
"Roger. Checklist complete up here. Stand by for orbital insertion burn, crew."
It all sounded very routine and quite boring, just like the endless simulator sessions they made him take, so Macomber began thinking about the target area in Soltanabad once again. Updated satellite images confirmed the presence of heavy-vehicle tire tracks again but did not reveal what they were—whoever was down there was very good at keeping the vehicles hidden from satellite view. The Goose drones were not much better than the space-based radar network in detecting very small targets, but maybe they needed to stay away from the highway airstrip and send in the Goose drones first to get a real-time look before...
...and suddenly the LPDRS engines kicked in, not in turbojet mode but now in hybrid rocket mode, and Macomber was suddenly and violently thrust back into the here and now. No simulator could prepare you for the shove—it felt like hitting a football tackle training sled except it was completely unexpected, the sled was hitting you instead of the other way around, and the force was not only sustained but increasing every second. Soon it felt like the entire offensive line had piled on top of him, being joined shortly by the defensive line as well. Whack knew he could call up data readouts about their altitude, speed, and G-force levels, but it was all he could do just to concentrate on his breath control to fight off the G-force effects and keep from blacking out.
The G-forces seemed to last an hour, although he knew the boost into orbit only took seven or eight minutes. When the pressure finally eased, he felt exhausted, as if he had just finished running the stadium stairs at the Academy before football season, or jogging across the Iraqi desert with a hundred-pound pack.
Obviously his labored breathing was loud enough to be heard on the intercom, because a few moments later Charlie Turlock asked, "Still feel like farting around with your knitting needles, Macomber?"
"Bite me."
"Get your barf bag ready, Major," Charlie continued gaily, "because I'm not cleaning up after you if you spew in the module. I'll bet the macho commando didn't take his anti-motion-sickness medication."
"Cut the chatter and run your 'After Orbital Insertion Burn' checklists," Moulain said.
Macomber's breathing quickly returned to normal—more from embarrassment than by will. Damn, he thought, that hit him too suddenly, and a lot harder than he'd expected. Getting back into a routine would surely take his mind off his queasiness, and the Air Battle Force was nothing if not driven by checklists and routine. He used his eye-pointing system to call up the proper checklist by looking at a tiny icon in the upper left corner of his electronic visor and speaking...
...but instead of issuing a command, all he could manage was a throatful of bile. Scanning the electronic visor with his eyes suddenly gave him the worst case of vertigo he had ever experienced—he felt as if he was being swung upside down by the ankles on a rope, suspended a hundred feet aboveground. He couldn't stop the spinning sensation; he lost all sense of up and down. His stomach churned as the spinning intensified, a thousand times worse than the worst case of the spins and leans he had ever had on the worst all-night party in his life...
"Better clear the major off-helmet, Frenchy," Charlie said, "'cause it sounds like he's ready to blow lunch."
"Screw you, Turlock," Macomber meant to say, but all that came out was a gurgle.
"You're cleared off-helmet, S-One, module pressurization in the green," Moulain said. "I hope you kept a barf bag handy—vomit in free fall is the most disgusting thing you've ever seen in your life, and you might be too sick to do your job."
"Thanks a bunch," Macomber said through gritted teeth, trying to hold back the inevitable until he got the damned Tin Man helmet off. Somehow he managed to unfasten his helmet—he had no idea where it floated off to. Unfortunately the first bag he could reach was not a motion sickness bag—it was the personal bag containing his knitting stuff. To his shock and dismay, he quickly found that vomit in free fall didn't behave as he expected: instead of filling the bottom of his bag in a disgusting but controllable clump, it curled back into a smelly, chunky cloud right back up into his face, eyes, and nose.
"Don't let it out, Whack!" he heard Turlock yell from behind him. "We'll spend the next hour Dustbustering globs of barf out of the module." That bit of imagery didn't help to settle his stomach one bit, nor did the awful smell and feel of warm vomit wafting across his face inside the bag.
"Relax, big guy," he heard a voice say. It was Turlock. She had unstrapped and was holding his shoulders, steadying his convulsions and helping seal the bag around his head. He tried to shrug her hands off, but she resisted. "I said relax, Whack. It happens to everyone, drugs or no drugs."
"Get away from me, bitch!"
"Shut up and listen to me, asshole," Charlie insisted. "Ignore the smell. The smell is the trigger. Remove it from your consciousness. Do it, or you'll be a vegetable for the next three hours minimum. I know you bad-ass commando types know how to control your senses, your breathing, and even your involuntary muscles so you can endure days of discomfort in the field. Hal Briggs fought on for several minutes after being shot up by the Iranians..."
"Screw Briggs, and screw you, too!"
"Pay attention, Macomber. I know you can do this. Now is the time to turn whatever you got on. Concentrate on the smell, isolate it, and eliminate it from your consciousness."
"You don't know shit..."
"Just do it, Wayne. You know what I'm telling you. Just shut up and do it, or you'll be as wasted as if you've been on a three-day bender."
Macomber was still blindingly angry at Turlock for being right there with him at this most vulnerable moment, taking advantage of him, but what she said made sense—she obviously knew something about the agony he was experiencing. The smell, huh? He never thought about smell that much—he was trained to be hypersensitive to sight, sound, and the indefinable sixth sense that always warned of nearby danger. Smell was usually a confusing factor, something to be disregarded. Shut it down, Whack. Shut it off.
Somehow, it worked. He knew that breathing through his mouth cut off the sense of smell, and when he did that a lot of the nausea went away. His stomach was still doing painful knots and waves of roiling convulsions, as bad as if he had been stabbed in the gut, but now the trigger of those awful spasms was gone, and he was back in control. Sickness was not allowable. He had a team counting on him, a mission to perform—his damned weak stomach was not going to be the thing that let his team and his mission down. A few pounds of muscle and nerve endings were not going to control him. The mind is the master, he reminded himself, and he was the master of the mind.
A few moments later, with his stomach empty and the aroma erased from his consciousness, his stomach quickly started to return to normal. "You okay?" Charlie asked, offering him a towelette.
"Yeah." He accepted the wipe and began to clean up, but stopped and nodded. "Thanks, Turlock."
"Sorry about the shit I gave you about the knitting."
"I get it all the time."
"And you usually bust somebody's head for ragging on you, except it was me and you weren't going to bust my head?"
"I would have if I could've reached you," Whack said. Charlie thought he meant it until he smiled and chuckled. "Knitting relaxes me, and it gives me a chance to see who gets in my shit and who leaves me be."
"Sounds like a screwed-up way to live, boss, if you don't mind me sayin'," Charlie said. He shrugged. "If you're okay, drink some water and stay on pure oxygen for a while. Use the vacuum to clean up any pieces of vomit you see before we re-enter, or we'll never find them and they'll become projectiles. If they stick on our gear the bad guys will smell it yards away."
"You're right, Tur—Charlie," Whack said. As she headed back to her seat, he added, "You're all right, Turlock."
"Yes, I am, boss," she replied. She found his helmet lodged somewhere in the cargo section in the back of the passenger module and handed it back to him. "Just don't you forget it." She then detached the cleanup vacuum from its recharging station and floated it over to him as well. "Now you really look like Martha Stewart, boss."
"Don't push it, Captain," he growled, but he smiled and took the vacuum.
"Yes, sir." She smiled, nodded, and returned to her seat.
PRESIDENT'S RETREAT, BOLTINO, RUSSIA
A SHORT TIME LATER
They didn't always meet like this to make love. Both Russian president Leonid Zevitin and minister of foreign affairs Alexandra Hedrov loved classic black-and-white movies from all over the world, Italian food, and rich red wine, so after a long day of work, especially with a long upcoming trip ready to begin, they often stayed after the rest of the staff had been dismissed and shared some time together. They had become lovers not long after they first met at an international banking conference in Switzerland almost ten years earlier, and even as their responsibilities and public visibility increased they still managed to find the time and opportunity to get together.
If either of them was concerned about the whispered rumors of their affair, they showed no sign of it. Only the tabloids and celebrity blogs spoke of it, and those were all but dismissed by most Russians—certainly no one in the Kremlin would ever wag their tongues about such things and about such powerful people in anything louder than a quiet thought. Hedrov was married and was the mother of two grown children, and they long ago learned that their lives, as well as the life of their wife and mother, belonged to the state now, not to themselves.
The president's dacha was the closest to security and privacy than anything else they could ever expect in the Russian Federation. Unlike the president's official residence in the Senate Building at the Kremlin, which was rather unassuming and utilitarian, Zevitin's dacha outside Moscow was modern and stylish, fit for any international business executive. Like the man itself, the place revolved around work and business, but it was hard to discern that at first glance.
After flying in to Boltino to the president's private airport nearby, visitors were driven to the residence by limousine and escorted through a sweeping grand foyer to the great room and dining room, dominated by three large fireplaces and adorned with sumptuous leather and oak furniture, works of art from all over the world, framed photos of world leaders, and mementos from his many celebrity friends, topped off with a spectacular panoramic view of Pirogovskoje Reservoir outside the floor-to-ceiling windows. Special guests would be invited up the double marble curved staircases to the bedroom suites on the second floor, or down to the large Roman-style baths, indoor pool, thirty-seat high-definition movie theater, and game room on the ground floor. But all that was still only a fraction of the square footage of the place.
A guest being dazzled by the grand view outside the great room would miss the dark, narrow cupola on the right side of the foyer, almost resembling a doorless closet, which had small and unimpressive paintings hanging on the curved walls illuminated by rather dim LED spotlights. But if one stepped into the cupola, he would be instantly but surreptitiously electronically searched by X-ray to locate weapons or listening devices. His facial features would be scanned and the data run through an electronic identification system that was able to detect and filter out disguises or impostors. Once positively identified, the hidden door inside the cupola would be opened from within, and you would be admitted to the main part of the dacha.
Zevitin's office was as large as the great- and dining rooms combined, large enough for a group of generals or ministers to confer with each other on one side and not be heard by a similarly sized meeting of the president's advisers on the other—unheard except for the audio and video recording devices planted everywhere on the grounds, as well as out on the streets, neighborhoods, and roads of the surrounding countryside. Eight persons could expansively dine on Zevitin's walnut and ivory-inlaid desk with elbow room to spare. Video feeds and television reports from hundreds of different sources were fed to a dozen high-definition monitors located in the office, but none were visible unless the president wanted to view them.
The president's bedroom upstairs was the one made up for show: the bedroom adjoining the office suite was the one Zevitin used most of the time; it was also the one Alexandra preferred, the one that she thought best reflected the man himself—still grand, but warmer and perhaps plusher than the rest of the mansion. She liked to think he made it so just for her, but that would be foolish arrogance on her part, and she often reminded herself not to indulge in any of that around this man.
They had slipped beneath the silk sheets and down comforter of his bed after dinner and movies and just held each other, sipping tiny glasses of brandy and talking in low intimate voices about everything but the three things both mostly cared about: government, politics, and finances. Phone calls, official or otherwise, were expressly forbidden; Alexandra couldn't remember ever being interrupted by an aide or a phone call, as if Zevitin could somehow make the rest of the world instantly comatose while they were together. They touched each other occasionally, exploring each other's silent desires, and mutually deciding without a word that tonight was for companionship and rest, not passion. They had known each other a long time, and she never considered that she might not be fulfilling his needs or desires, or he was disregarding hers. They embraced, kissed, and said good night, and there was no hint of tension or displeasure. All was as it should be...
...so it was doubly surprising for Alexandra to be awakened by something she had never heard before in that room: a beeping telephone. The alien sound made her sit bolt upright after the second or third beep; she soon noticed that Leonid was already on his feet, the bedside light on, the receiver to his lips.
"Go ahead," he said, then listened, glancing over to her. His eyes were not angry, quizzical, confused, or fearful, as she was certain hers were. He obviously knew exactly who was calling and what he was going to say; like a playwright watching a rehearsal of his latest work, he was patiently waiting for something he already knew would be said.
"What is it?" she mouthed.
To her surprise, Zevitin reached down to the phone, touched a button, and hung up the receiver, activating the speakerphone. "Repeat that last, General," he said, catching and arresting her gaze with his.
General Andrei Darzov's voice, crackling and occasionally fading with interference as if talking across a vast distance, could still clearly be heard: "Yes, sir. KIK Command and Measurement Command sites have detected an American spaceplane launch over the Pacific Ocean. It crossed over central Canada and was inserted safely into low-Earth orbit while over the Arctic ice pack of Canada. If it stays on its current trajectory, its target area is definitely eastern Iran."
"When?"
"They could be starting their re-entry burn in ten minutes, sir," Darzov replied. "It possibly has enough fuel to fly to the same target area after re-entering the atmosphere after a complete orbit, but it is doubtful without a midair refueling over Iraq or Turkey."
"Do you think they discovered it?" Hedrov didn't know what "it" was, but she assumed, because Zevitin had allowed her to listen in on the conversation, that she would find out soon enough.
"I think we should assume they have, sir," Darzov said, "although if they positively identified the system, I am sure McLanahan would not hesitate to attack it. They may have just detected activity there and are inserting more intelligence-gathering assets to verify."
"Well, I'm surprised they took this long," Zevitin remarked. "They have spacecraft flying over Iran almost every hour."
"And those are just the ones we can positively detect and track," Darzov said. "They could have many more that we can't identify, especially unmanned aircraft."
"When will it be within striking range for us, General?"
Hedrov's mouth opened, but at a warning glare from Zevitin, she said nothing. What in hell were they thinking of...?
"By the time the spaceplane crosses the base's horizon, sir, they'll be less than five minutes from landing."
"Damn, the speed of that thing is mind-boggling," Zevitin muttered. "It's almost impossible to move fast enough against it." He thought quickly; then: "But if the spaceplane stays in orbit instead of re-entering, it will be in perfect position. We have one good shot only."
"Exactly, sir," Darzov said.
"I assume your men are preparing for an assault, General?" Zevitin asked seriously. "Because if the spaceplane successfully lands and deploys its Tin Man ground forces—which we must assume they will have on board—"
"Yes, sir, we must."
"—we will have no time to pack up and get out of Dodge."
"If I understand you correctly, sir—yes, we would undoubtedly lose the system to them," Darzov acknowledged, not knowing what or where "Dodge" was but not bothering to reveal his own ignorance. "The game will be over."
"I see," Zevitin said. "But if it does not re-enter and stays in orbit, how long will you have to engage it?"
"We should acquire it with optronic observation sensors and laser rangefinders as soon as it crosses the horizon, at a range of about eighteen hundred kilometers or about four minutes away," Darzov replied. "However, we need radar for precise tracking, and that is limited to a maximum range of five hundred kilometers. So we will have a maximum of two minutes at its current orbital altitude."
"Two minutes! Is that enough time?"
"Barely," Darzov said. "We will have radar tracking, but we still need to hit the target with an air data laser that will help compute focusing corrections to the main laser's optics. That should take no longer than sixty seconds, assuming the radar stays locked on and the proper computations are made. That will give us a maximum of sixty seconds' exposure time."
"Will it be enough to disable it?"
"It should, at least partially, based on our previous engagements," Darzov replied. "However, the optimum time to attack is when the target is directly overhead. As the target moves toward the horizon the atmosphere grows thicker and more complex, and the laser's optics cannot compensate quickly enough. So—"
"The window is very, very small," Zevitin said. "I understand, General. Well, we must do everything we can to be sure the spaceplane stays in that second orbit."
There was a noticeable pause; then: "If I can help in any way, sir, please do not hesitate to call on me," Darzov said, obviously completely unsure as to what he could do.
"I'll keep you posted, General," Zevitin said. "But for now, you are cleared to engage. Repeat, you are cleared to engage. Written authorization will be sent to your headquarters via secure e-mail. Advise if anything changes. Good luck."
"Luck favors the bold, sir. We cannot lose if we take the fight to the enemy. Out."
As soon as Zevitin hung up the phone, Hedrov asked, "What was that all about, Leonid? What is going on? Was it about Fanar?"
"We are about to create a crisis in space, Alexandra," Zevitin responded. He turned to her, then ran the fingers of both hands through his hair as if wiping his thoughts completely clear so he could start afresh. "The Americans think they have unfettered access to space—we are going to throw some roadblocks up in their faces and see what they do. If I know Joseph Gardner, as I think I do, I think he will stomp on the brakes of McLanahan's vaunted space force, and stomp on them hard. He would destroy one of his own just to keep someone else from having a victory he couldn't claim for himself."
Alexandra rose from the bed, kneeling before him. "Are you so sure of this man, Leonid?"
"I'm positive I've got this guy pegged."
"And what of his generals?" she asked softly. "What of McLanahan?"
Zevitin nodded, silently admitting his own uncertainty about that very factor. "The American attack dog is on his leash, and he is apparently hurt...for now," he said. "I don't know how long I can count on that leash holding. We've got to prompt Gardner to put McLanahan out of commission...or be prepared to do it ourselves." He picked up the phone. "Get me American president Gardner on the 'hot line' immediately."
"It is a dangerous game you are playing, no?" Hedrov asked.
"Sure, Alexandra," Zevitin said, running the fingers of his left hand through her hair as he waited. He felt her hands slip from his chest to below his waist, soon tugging at his underwear and then ministering to him with her hands and mouth, and although he heard the beeps and clicks of the satellite communications system quickly putting the "hot line" call through to Washington, he didn't stop her. "But the stakes are that high. Russia can't allow the Americans to claim the high ground. We need to stop them, and this is our best chance right now."
Alexandra's efforts soon increased both in gentleness and urgency, and Zevitin hoped that Gardner was preoccupied enough to allow him a few more minutes with her. Knowing the American President as he did, he knew he very well might be similarly distracted.
ABOARD AIR FORCE ONE, OVER THE SOUTHEAST UNITED STATES
THAT SAME TIME
Relaxing in his newly reupholstered seat at his desk in the executive office suite aboard Air Force One, on his way to his "southern White House" oceanside compound outside St. Petersburg, Florida, President Gardner was studying the very ample bosom and shapely fanny of the female Air Force staff sergeant who had just brought a pot of coffee and some wheat crackers into the office. He knew she knew he was checking her out, because every now and then she would cast a glance over to him and a tiny smile would appear. He had a newspaper on his lap but was angled over just enough to surreptitiously watch her. Yep, he thought, she was taking her sweet time setting out his stuff. Damn, what an ass...
Just as he was going to make his move and invite her to bring those tits and ass over to his big desk, the phone beeped. He was tempted to push the DO NOT DISTURB button, cursing himself that he hadn't done so after he finished his last meeting with the staff and settled in, but something told him that he should take this call. He reluctantly picked up the receiver. "Yes?"
"President Zevitin of the Russian Federation calling for you on the 'hot line,' sir," the communications officer responded. "He says it's urgent."
He held the MUTE button on the receiver, groaned aloud, then gave the stewardess a wink. "Come back in ten minutes with fresh stuff, okay, Staff Sergeant?"
"Yes, sir," she replied enthusiastically. She stood to attention, thrusting her chest out to him, before glancing at him mischievously, slowly turning on a heel, and departing.
He knew he had her pegged, he thought happily as he released the button. "Give me a minute, Signals," he said, reaching for a cigarette.
"Yes, sir."
Shit, Gardner cursed to himself, what in hell does Zevitin want now? He pressed the buzzer button to summon his chief of staff Walter Kordus. He was going to have to review the policy he'd established of immediately taking calls from Zevitin, he thought—he was starting to speak with him almost on a daily basis. Ninety seconds and a half a cigarette later: "Put him through, Signals," he ordered, stubbing out the cigarette.
"Yes, Mr. President." A moment later: "President Zevitin is on the line, secure, sir."
"Thank you, Signals. Leonid, this is Joe Gardner. How are you?"
"I'm fine, Joe," Zevitin replied in a not-so-pleasant tone. "But I'm concerned, man, real concerned. I thought we had a deal."
Gardner reminded himself to stay on guard while talking to this guy—he sounded so much like an American that he could be talking to someone from the California congressional delegation or some Indiana labor union leader. "What are you talking about, Leonid?" The chief of staff entered the President's office, picked up the dead extension so he could listen in, and turned on his computer to start taking notes and issuing orders if necessary.
"I thought we agreed that we would be notified whenever you'd fly manned spaceplane missions, especially into Iran," Zevitin said. "This is really worrisome, Joe. I'm working hard to try to defuse the situation in the Middle East and keep the hard-liners in my government in check, but your activities with the Black Stallions only serve to—"
"Hold on, Leonid, hold on," Gardner interrupted. "I have no idea what you're talking about. What Black Stallion missions?"
"C'mon, Joe—do you think we can't see it? Do you think it's invisible? We picked it up as soon as it crossed the horizon over the Greenland Sea."
"One of the spaceplanes is flying over Greenland?"
"It's over southwestern China now, Joe, according to our space surveillance and tracking units," Zevitin said. "C'mon, Joe, I know you can't talk about ongoing classified military missions, but it's not hard to guess what they're going to do, even if it is the Black Stallion spaceplane we're talking about. Orbital mechanics are as predictable as sunrise and sunset."
"Leonid, I—"
"I know you can't confirm or deny anything—you don't have to, because we know what's going to happen," Zevitin went on. "It is obvious that in the next orbit, in about ninety minutes, it will be directly over Iran. We expect it to begin deorbit maneuvers in about forty-five minutes, which will put it directly over the Caspian Sea when its atmospheric engines and flight controls will become active. You're obviously flying a mission into Iran, Joe. I thought we had an agreement: hands off Iran while we pursue a diplomatic solution to the military coup and the murder of the elected Iranian officials."
"Hold on, Leonid. Stand by a sec." Gardner hit the MUTE button. "Get Conrad in here," he ordered, but Kordus had already hit the button to page the National Security Adviser. Gardner released the MUTE button. "Leonid, you're right, I can't talk about any ongoing operations. You just have to—"
"Joe, I'm not calling to discuss anything. I'm pointing out to you that we can clearly see one of your spaceplanes in orbit right now, and we had no idea you were going to launch one. After all we've discussed over the past several weeks, I can't believe you'd do this to me. When they find out about this, my Cabinet and the Duma will think I've been duped, and they'll demand I take action, or else I'll lose all the support for our cooperative efforts and rapprochement I've taken months to cultivate. You cut the rug out from under me, Joe."
"Leonid, I'm in the middle of an important meeting, and I need to finish up what I'm doing first," the President lied, impatiently rising to his feet and resisting the urge to yell outside his door for Carlyle and Kordus to tell him what in hell was going on. "I assure you, we don't have any actions under way against Russia anywhere, in any fashion—"
"'Against Russia?' That sounds like an alarming equivocation, Joe. What does that mean? Are you launching an operation against someone else?"
"Let me clear my desk and finish this briefing, Leonid, and I'll fill you in. I'll—"
"I thought we agreed, Joe: essential flights only until we had a treaty governing military travel in space," Zevitin pressed. "As far as we can tell, the spaceplane isn't going to dock with the space station, so this is not a logistical mission. I know things are bad in Iran and Iraq, but bad enough to stir up widespread fear by launching a Black Stallion? I think not. This is a complete disaster, Joe. I'm going to get butchered by the Duma and the generals—"
"Don't panic, Leonid. There's a rational and completely benign explanation. I'll call you back as soon as I can and—"
"Joe, you had better be straight with me, or else I won't be able to rein in the opposition leaders and some of the more powerful generals—they'll all be clamoring for an explanation and a strong response in kind," Zevitin said. "If I can't give them a plausible answer, they'll start searching for one themselves. You know I'm holding on by a shoestring out here. I need your cooperation or everything we've worked for will unravel."
"I'll call you right back, Leonid," Gardner said. "But I assure you, on my honor, that nothing is going on. Absolutely nothing."
"So our ambassadors and observers on the ground in Tehran shouldn't be worried about another hypersonic missile slamming through the ceiling any moment now?"
"Don't even joke about that, Leonid. It's not going to happen. I'll call you back." He impatiently hung up the phone, then wiped the beads of sweat off his upper lip. "Walter!" he shouted. "Where the hell are you? And where's Conrad?"
The two advisers trotted into the executive suite moments later. "Sorry, Mr. President, but I was downloading the latest spacecraft status report from Strategic Command," National Security Adviser Conrad Carlyle said. "It should be on your computer." He accessed the computer on the President's desk, opened a secure file location, and quickly scanned the contents. "Okay, it's right here...yes, General Cannon, commander of U.S. Strategic Command, authorized a spaceplane launch about four hours ago, and the mission was approved by Secretary Turner."
"Why wasn't I notified of this?"
"The mission is described as 'routine,' sir," Carlyle said. "Crew of two, three passengers, six orbits of the Earth and return to Elliott Air Force Base, total mission duration ten hours."
"What is this, a fucking joy ride? Who are the passengers? I ordered essential missions only! What in hell is going on? I thought I grounded all of the spaceplanes."
Carlyle and Kordus exchanged puzzled expressions. "I...I'm not aware of an order grounding the spaceplanes, sir," Carlyle responded feebly. "You did recall the SkySTREAK bombers from their patrols, but not the space—"
"I had a deal with Zevitin, Conrad: No more spaceplane launches without first notifying him," Gardner said. "He's hopping mad about the launch, and so am I!"
Carlyle's brows knitted, and his mouth opened and closed with confusion. "I'm sorry, Joe, but I'm not aware of any agreement we made with Zevitin to inform him of anything dealing with the spaceplanes," he said finally. "I know he's been clamoring for that—he rants and raves to every media outlet in the world that the spaceplanes are a danger to world peace and security because they can be mistaken for an intercontinental ballistic missile, and he's demanding that we notify him before we launch one—but there's been no formal agreement about—"
"Didn't I order Cannon to be sure that those spaceplanes and any space weapons didn't enter sovereign airspace, even if it meant keeping them on the ground?" the President thundered. "They were to stay out of any country's airspace at all times. Didn't I give that order?"
"Well...yes, sir, I believe you did," Kordus replied. "But the spaceplanes can easily fly above a country's airspace. They can—"
"How can they do that?" the President asked. "We have airspace that's restricted from the surface to infinity. Sovereign airspace is all the airspace above a nation."
"Sir, as we've discussed before, under the Outer Space Treaty no nation can restrict access or travel through outer space," Carlyle reminded the President. "Legally space begins one hundred kilometers from Earth's surface. The spaceplane can climb into space quickly enough while over friendly countries, open ocean, or the ice packs, and once up there can fly around without violating anyone's sovereign airspace. They do it—"
"I don't give a shit what it says in an obsolete forty-year-old treaty!" the President thundered. "For many months we have been involved in discussions with Zevitin and the United Nations to come up with a way to alleviate the anxiety felt by many around the world to spaceplane and space station operations without restricting our own access to space or revealing classified information. Until we had something worked out, I made it clear that I didn't want the spaceplanes flitting around unnecessarily making folks nervous and interfering with the negotiations. Essential missions only, and that meant resupply and national emergencies—I had to personally approve all other missions. Am I mistaken, or have I not approved any other spaceplane flights recently?"
"Sir, General Cannon must have felt it important enough to launch this flight without—"
"Without my approval? He thinks he can just blast off into space without anyone's permission? Where's the emergency? Is the spaceplane going to dock with the space station? Who are the three passengers? Do you even know?"
"I'll put in a call to General Cannon, sir," Carlyle said, picking up the phone. "I'll get all the details right away."
"This is a damned nightmare! This is out of control!" the President thundered. "I want to know who's responsible for this, and I want his ass out! Do you hear me? Unless war has been declared or aliens are attacking, I want whoever's responsible for this shit-canned! I want to speak with Cannon myself!"
Carlyle put his hand over the phone's mouthpiece as he waited and said, "Sir, I suggest I speak with General Cannon. Keep an arm's-length distance from this. If it's just a training flight or something, you don't want to be perceived as jumping off the deep end, especially after just speaking with the president of Russia."
"This is serious, Conrad, and I want it clear to my generals that I want those spaceplanes under tight control," the President said.
"Are you sure that's how you want to handle it, Joe?" Kordus asked in a quiet voice. "Reaching down past Secretary Turner to dress down a four-star general is bad form. If you want to beat someone up, pick on Turner—he was the final authority for that spaceplane launch."
"Oh, I'll give Turner a piece of my mind too, you can bet on that," the President said angrily, "but Cannon and that other guy, the three-star—"
"Lieutenant General Backman, commander of CENTAF."
"Whatever. Cannon and Backman have been fighting me too hard and too long over this space defense force idea of McLanahan's, and it's about time to bring them back into line—or, better, get rid of them. They're the last holdouts of Martindale's Pentagon brain trust, and they want the space stuff because it builds up their empires."
"If you want them gone, we'll get rid of them—they all serve at the pleasure of the commander-in-chief," Kordus said. "But they're still very powerful and popular generals, especially with congressmen who are for the space program. They may push their own plans and programs while in uniform, but as disgraced and disgruntled retired generals, they'll attack you openly and personally. Don't give them a reason."
"I know how the game is played, Walter—hell, I made most of the rules," the President said hotly. "I'm not afraid of the generals, and I shouldn't be worried about tiptoeing around them—I'm the damned commander-in-chief. Get Turner on the line right away." He reached over and snatched the phone out of the National Security Adviser's hand. "Signals, what the hell is going on? Where's Cannon?"
"Stand by, sir, he should be connecting any second now." A few moments later: "Cannon here, secure."
"General Cannon, this is the President. Why the hell did you authorize that spaceplane to launch without my authority?"
"Uh...good afternoon, sir," Cannon began, perplexed. "As I explained to the Secretary of Defense, sir, it's a pre-positioning flight only while we await final approval for a mission inside Iran. With the spacecraft in orbit, if we got approval it would be easy to insert the team, do their job, then get them out again. If it was not approved, it would be equally easy to return them to base."
"I specifically ordered no spaceplanes to cross foreign borders without my approval."
"Sir, as you know, once the spaceplane is above the sixty-mile threshold, it's—"
"Don't give me that Outer Space Treaty crap!" the President thundered. "Do I have to spell it out for you? I don't want the spaceplanes in orbit unless it's to support the space station or it's an emergency, and if it's an emergency it had better be a damned serious one! The rest of the world thinks we're getting ready to launch attacks from space...which apparently is exactly what you are planning, behind my back!"
"I'm not hiding anything from anyone, sir," Cannon argued. "Without orders to the contrary, I launched the spaceplanes on my own authority with strict orders that no one crosses into any sovereign airspace. That is my standing general order from SECDEF. Those instructions have been complied with to the letter."
"Well, I'm rescinding your authority, General," the President said. "From now on, all movements of any spacecraft will need my direct permission before execution. Do I make myself clear, General? You had better not put so much as a rat in space without my permission!"
"I understand, sir," Cannon said, "but I don't recommend that course of action."
"Oh? And why not?"
"Sir, keeping that level of control on any military asset is dangerous and wasteful, but it's even more critical with the space launch systems," Cannon said. "Military units need one commander to be effective, and that should be a theater commander with instantaneous and constant access to information from the field. The spaceplanes and all of our space launch systems are designed for maximum speed and flexibility, and in an emergency they'll lose both if final authority remains in Washington. I strongly recommend against taking operational command of those systems. If you're not happy with my decisions, sir, then may I remind you that you can dismiss me and appoint another theater commander to have control of the spaceplanes and other launch systems."
"I'm well aware of my authority, General," Gardner said. "My decision stands."
"Yes, sir."
"Now who the hell is aboard that spaceplane, and why wasn't I informed of this mission?"
"Sir, along with the two flight crewmembers, there are three members of General McLanahan's Air Battle Force ground operations unit aboard the spaceplane," Cannon responded tonelessly.
"McLanahan? I should have known," the President spat. "That guy is the definition of a loose cannon! What was he up to? Why did he want that spaceplane launched?"
"They were being pre-positioned in orbit pending approval for a reconnaissance and interdiction mission inside Iran."
"'Pre-positioned'? You mean, you sent a spaceplane and three commandos over Iran without my permission? On your sole authority?"
"I have the authority to pre-position and forward-deploy forces anywhere in the world to support my standing orders and fulfill my command's responsibilities, sir," Cannon said testily. "The spaceplanes were specifically directed not to enter any foreign airspace without permission, and they have fully complied with that order. If they do not receive authorization to proceed with their plan, they are directed to return to base."
"What kind of nonsense is this, General? This is the spaceplane we're talking about—loaded with McLanahan's armed robots, I assume, correct?"
"It's not nonsense, sir—it's how this command and all major theater commands normally operate," Cannon said, trying mightily to keep his anger and frustration in check. Gardner was the former Secretary of the Navy and Secretary of Defense, for God's sake—he knew this better than anyone...! "As you know, sir, I give orders to pre-position and forward-deploy thousands of men and women all over the world every day, both in support of routine day-to-day operations as well as in preparation for contingency missions. They all operate within standing orders, procedural doctrine, and legal limits. They don't deviate one iota until given a direct execution order by myself, and that order isn't given until I receive a go-ahead from the national command authority—you, or the Secretary of Defense. It doesn't matter if we're talking about one spaceplane and five personnel, or an aircraft carrier battle group with twenty ships, seventy aircraft, and ten thousand personnel."
"You seem to believe that the spaceplanes are simple little windup toy planes that no one notices or cares about, General," the President said. "You may think it's routine to send a spaceplane over Iran or an aircraft carrier battle group off someone's coastline, but I assure you, the entire world is in mortal fear of them. Wars have been started by far less. It's obvious your attitude toward the weapons systems under your command has to change, General, and I mean now." Cannon had no response. "What members of McLanahan's Battle Force are aboard?"
"Two Tin Men and one CID unit, sir."
"Jesus...that's not a recon team, that's a damned strike team! They can take on an entire infantry company! What were you thinking, General? Did you think McLanahan was going to fly that kind of force all that way and not use them? What in hell were McLanahan's robots going to do in Iran?"
"Sensors picked up unusual and suspicious activity at a remote highway airbase in eastern Iran that had previously been used by the Iranian Revolutionary Guards," Cannon said. "General McLanahan believes the base is secretly being reopened either by the Iranians or by the Russians. His satellite imagery can't give him precise enough pictures to tell for sure, so he requested an insertion of a three-person Battle Force squad to take a look and, if necessary, destroy the base."
"Destroy the base?" the President thundered, angrily slapping the handset into an open hand. "My God, he authorized McLanahan to send an armed spaceplane over Iran to destroy a military base, and I didn't know about it? Is he insane?" He raised the receiver: "And when were you going to let the rest of us know about McLanahan's little plan, General—after World War Four was under way?"
"McLanahan's plan has been passed along to us here at Strategic Command, and my operations staff is reviewing it and will be presenting a recommendation to the Secretary of Defense," Cannon replied. "We should be making a decision any moment—"
"I'll make a decision for you right now, General: I want that spaceplane to land as soon as possible back at their home base," the President said. "Do you understand me? I don't want those commandos deployed, or that spaceplane to land, anywhere but back in Nevada or wherever the hell it's from, unless it's a life-or-death emergency. And I don't want one thing to be launched, ejected, or otherwise leave that spacecraft that might be considered an attack on anyone...noth-ing. Am I making myself perfectly clear, General Cannon?"
"Yes, sir."
"And if that spaceplane crosses one political boundary anywhere on the planet under that damned sixty-mile altitude limit, you will lose your stars, General Cannon...all of them!" the President went on hotly. "You overstepped your authority, General, and I hope to hell I don't have to spend the rest of my first term in office explaining, correcting, and apologizing for this monumental blunder. Now get on it."
The President slammed the phone down, then took his seat, fuming. After a few moments of muttering to himself, he barked, "I want Cannon fired."
"Sir, technically he does have the authority to move his assets anywhere he wants to on routine missions," National Security Adviser Carlyle said. "He doesn't need permission from the national defense authority—you or the Secretary of Defense—for day-to-day operations."
"But we usually tell the Russians before we move any weapon systems that might be confused as an attack, correct?"
"Yes, sir—that's always a wise precaution," Carlyle said. "But if the theater commander needed to position his assets in preparation for an actual mission, we aren't obligated to tell the Russians anything. We don't even have to lie to them and tell them it's a training mission or something."
"Part of the problem with these spaceplanes, Conrad, is that they move too quickly," Chief of Staff Kordus said. "Even if this was a routine mission, they're around the world in the blink of an eye. We've got to put stricter controls on those guys."
"If Cannon had something going, something important, he should have told me or Turner before launching that spaceplane," the President said. "Walter's right: those spaceplanes are too fast and too threatening to just launch them anytime, even on a perfectly peaceful, benign, routine mission—which this certainly was not. But I thought I made it clear to everyone that I didn't want the spaceplanes up unless it was an emergency or a war. Am I mistaken about that?"
"No, sir, but apparently General Cannon thought this was a pretty serious indication, because he moved very quickly. He—"
"It doesn't matter," the President insisted. "The Russians spotted him, and I'm sure they're radioing the Iranians, Turkmenis, and half the spies in the Middle East to be on the lookout for the Battle Force. The gig is blown. The Russians are hopping mad, and so will the United Nations, our allies, the media, and the American people be as soon as they find out about this—"
"Which will probably be any minute now," Kordus added, "because we know Zevitin runs and leaks his information to the European press, who can't wait to excoriate us on the most trivial matter. On something this big, they'll have a field day. They'll roast us alive for the next month."
"Just when things were starting to settle down," the President said wearily, lighting another cigarette, "Cannon, Backman, and especially McLanahan have managed to stir it all up again."
"The spaceplane will be on the ground before the press can run with this, Joe," the chief of staff said, "and we'll just refuse to confirm or deny any of the Russians' allegations. The thing will die out soon enough."
"It'd better," Gardner said. "But just to be sure, Conrad, I want the spaceplanes grounded until further notice. I want all of them to stay put. No training, no so-called routine missions, nothing." He looked around the suite and, raising his voice just enough to show his irritation and let anyone outside the suite hear, asked, "Is that clear enough for everyone? No more unauthorized missions! They stay grounded, and that's that!" There was a chorus of muted "Yes, Mr. President" responses.
"Find out exactly when that spaceplane will be on the ground so I can notify Zevitin before someone impeaches or assassinates his ass," the President added. "And find out from the flight docs when McLanahan can get off that space station and be brought back to Earth so I can fire his ass too." He took a deep drag of his cigarette, stubbed it out, then reached for his empty coffee mug. "And on your way out, have that stewardess bring me something hot."
## CHAPTER SIX
It is difficult to overcome one's passions, and impossible to satisfy them.
—MARGUERITE DE LA SABLIÈRE
ABOARD THE XR-A9 BLACK STALLION SPACEPLANE
THAT SAME TIME
"Two minutes to re-entry initiation, crew," Major Jim Terranova announced. "Re-entry countdown initiated. First auto countdown hold in one minute. Report when your checklist is complete."
"S-One, roger," Macomber responded.
"How are you feeling, Whack?" Terranova asked.
"Thanks to copious amounts of pure oxygen, a little Transcendental Meditation, not using the eye-pointing electronic checklists, and the mind-numbing routine of still more damned checklists to perform, I feel pretty good," Macomber responded. "Wish this thing had windows."
"I'll put it on the wish list, but don't count on it anytime soon."
"It's a pretty spectacular sight, guys," Frenchy Moulain said. "This is my eleventh flight in orbit and I never get tired of it."
"It looks pretty much the same after the first orbit," groused Chris Wohl. "I've been on the station three times, and it just feels like you're standing on a really tall TV tower, looking down."
"Only the sergeant major could minimize a sight like this," Moulain said. "Ask to spend a couple nights on the station, Whack. Bring lots of data cards for your camera. It's pretty cool. You'll find yourself waking up at all times of the night and scheduling window time a day in advance just to take a picture."
"I doubt that very much," Macomber said dryly. He received a notification beep in his helmet. "I'm getting another data dump from the NIRTSats, guys." NIRTSats, or Need It Right This Second Satellites, were small "microsatellites," no bigger than a refrigerator, designed to do a specific task such as surveillance or communications relay from low-Earth orbit. Because they were smaller, carried less positioning thruster fuel, and had substantially less solar radiation shielding, the NIRTSats stayed in orbit for very short periods of time, usually less than a month. They were launched from aircraft aboard orbital boosters or inserted into orbit from the Black Stallion spaceplanes. A constellation of four to six NIRTSats had been put into an eccentric orbit designed to maximize coverage of Iran, making multiple passes over Tehran and the major military bases throughout the country since the military coup began. "Finish your checklists and let's go over the new stuff before we get squished again."
"I don't think we'll have time unless we delay re-entry for another orbit," Terranova said. "You'll have to review the data after we land."
"Listen, we have time...we'll make the time, MC," Macomber said. "We already launched on this mission without any proper mission planning, so we need to go over this new data right away."
"Not another argument," Moulain said, exasperated. "Listen, S-One, just run your checklists and get ready for re-entry. You know what happened last time you weren't paying attention to the flight: your stomach gave you a little warning."
"I'll be ready, SC," Macomber said. "Ground team, finish your checklist, report when complete, and let's get on the new data dump. S-One is complete." Turlock and Wohl reported complete moments later, and Macomber reported that the passengers were ready for re-entry. Moulain acknowledged the call and, tired of arguing with the zoomie again right before an important phase of flight, said nothing else.
Cautiously, Macomber opened the new satellite data file using voice commands instead of the faster but vertigo-causing eye-pointing system, allowing the data to flow onto the old imagery so he could see changes to the target area. What he got was a confusing jumble of images. "What the hell...looks like the data's corrupted," he said over private intercom, which allowed him to talk to his Ground Force team members without disturbing the flight crew. "Nothing's in the right place. They'll have to resend."
"Wait one, sir," Wohl said. "I'm looking at the computer frameholders on the two shots, and they're matching up." As Macomber understood them—which meant he didn't understand them hardly at all—the frameholders were computer-derived marks that aligned each image with known, fixed landmarks that compensated for differences in photograph angle and axis and allowed more precise comparisons between images. "Recommend you do not delete the new data yet, sir."
"Make it quick. I'll rattle HQ's cage." Macomber cursed into his helmet, then switched over to the secure satellite communications network: "Rascal to Genesis. Resend the last TacSat images. We got garbage here."
"Stand by, Rascal." Jeez, I really hate that call-sign, Macomber complained to himself. A few moments later: "Rascal, this is Genesis, set code Alpha Nine, repeat, Alpha Nine. Acknowledge."
"What? Is that the abort code?" Macomber thundered. "Are they telling us we're not going in?"
"Shut up, S-One, until we get this figured out," Moulain snapped. "MC, did the authentication come in?"
"Affirmative—got it just now," Terranova said. "The mission's been scrubbed, crew. We're directed to remain in present orbit until we get a flight plan change to a transfer orbit that will bring us back for a refueling and landing ASAP. Canceling re-entry procedure checklist...'leopards' secure, checklist canceled."
Macomber punched a fist into his hand and was instantly sorry he did so—it felt as if he punched a steel wall. "What in hell is going on? Why didn't we get a clearance? This is bull—"
"Rascal, this is Genesis." This time it was David Luger himself, calling from the battle management area at HAWC. "That data dump was valid, Rascal, I repeat, valid. We're looking it over, but it looks like the landing zone is hot."
"Well, that's the reason we're going in, isn't it, Genesis?" Macomber asked. "Let us drop in there and we'll take care of business."
"Your mission was scrubbed by the White House, Whack, not us," Luger said, the tension obvious in his voice. "They want you guys back home right away. We're computing a re-entry schedule now. It's looking like you'll have to stay up for at least another day before we can—"
"Another day! You've gotta be shitting me—!"
"Stand by, Rascal, stand by—"
There was a moment's pause, with a lot of encryption clicking and chattering on the frequency; then a different voice called: "Rascal, Stud, this is Odin." This was from McLanahan, up on Armstrong Space Station. "Recon satellites are picking up strong India-Juliet radar signals coming from your target area. Looks like a long-range search radar. We're analyzing now."
"A radar, eh?" Macomber commented. He started studying the new NIRTSat images again. Sure enough, it was the same Soltanabad highway airbase...but now all the craters were gone, and several semi tractor-trailers, troops and supply trucks, helicopters, and a large fixed-wing aircraft were parked on the ramp. "Looks like you were right, Odin. The bastards are setting the place up again."
"Listen to me, guys," McLanahan said, and the tone of his voice even over the encrypted satellite link was plainly very ominous indeed. "I don't like the smell of this. You'd be safer if you deorbited, but you've been ordered to return to base, so we have to keep you up there."
"What's the problem, sir?" Moulain asked. "Is there something you're not telling us?"
"You cross the target's horizon in eleven minutes. We're trying to compute if we have enough time to deorbit you and have you land in central Asia or the Caucasus instead of overflying Soltanabad."
"Central Asia! You want us to land where...?"
"Button it, Whack!" Moulain shouted. "What's going on, Odin? What do you think is down there?"
There was a long pause; then McLanahan responded simply: "Stud One-One."
He could have not made a more explosive response. Stud One-One was the XR-A9 Black Stallion that was shot down over Iran in the early days of the military coup, when the Air Battle Force was hunting down and destroying Iranian medium- and long-range mobile ballistic missiles that threatened not only the anti-theocratic insurgents but all of Iran's neighbors as well. The spaceplane was downed not by a surface-to-air missile or fighter jet but by an extremely powerful laser similar to the Kavaznya anti-satellite laser built by the Soviet Union over two decades earlier...that had appeared not over Russia, but in Iran.
"What do we do, sir?" Moulain asked, the fear thick in her voice. "What do you want us to do?"
"We're working on it," Patrick said from Armstrong Space Station. "We're trying to see if we can start you down right now in time to stay out of line of sight, or at least out of radar coverage."
"We can translate right now and get ready," Terranova said.
"Do it," Patrick said immediately. He then spoke, "Duty Officer, get me the President of the United States, immediately."
"Yes, General McLanahan," the computer-synthesized female voice of Dreamland's virtual "Duty Officer" responded. A moment later: "General McLanahan, your call is being forwarded to the Secretary of Defense. Please stand by."
"I want to speak with the President of the United States. It's urgent."
"Yes, General McLanahan. Please stand by." Another long moment later: "General McLanahan, your 'urgent' request has been forwarded to the President's chief of staff. Please stand by."
That was probably the best he was going to do, Patrick thought, so he didn't redirect the Duty Officer again. "Inform the chief of staff that it's an emergency."
"The 'urgent' request has been upgraded to an 'emergency' request, General. Please stand by."
Time was running out, Patrick thought. He thought about just having the Black Stallion crew declare an inflight emergency—there were dozens of glitches occurring on every flight that could constitute a real no-shit emergency—but he needed to be sure the Stud had someplace to land before ordering them to drop out of orbit.
"This is Chief of Staff Kordus."
"Mr. Kordus, this is General McLanahan. I'm—"
"I don't like being called by your computerized staffers, General, and neither does the President. If you want to talk to the President, show us the simple courtesy of doing it yourself."
"Yes, sir. I'm on board Armstrong Space Station, and I'm—"
"I know where you are, General—my staff was watching the live broadcast with great interest until you abruptly cut it off," Kordus said. "When we give you permission to do a live interview we expect you to finish it. Mind telling me why you cut it off like that?"
"I believe the Russians have placed an anti-spacecraft weapon of some kind, possibly the same laser that downed the Black Stallion over Iran last year, in an isolated highway airbase in Iran once used by the Revolutionary Guard Corps," Patrick responded. "Our sensors picked up the new activity at the base and alerted us. Now our unmanned reconnaissance aircraft are picking up extremely high-powered radar signals from that very same location that are consistent with the anti-spacecraft laser's acquisition and tracking system. I believe the Russians will attack our Black Stallion spacecraft if it passes overhead still in orbit, and I need permission to deorbit the spacecraft and divert it away from the target area."
"You have positive proof that the Russians are behind this? How do you know this?"
"We have satellite imagery showing the base is now completely active, with fixed-wing aircraft, trucks, and vehicles that appear similar to the vehicles we detected in Iran where we believe the laser that downed the Black Stallion came from. The radar signals confirm it. Sir, I need permission immediately to divert that flight. We can have it come out of orbit and maneuver it as much as possible with all but emergency fuel until it reaches the atmosphere, and then we can fly it away from the target area to an alternate landing site."
"The President has already ordered you to land the spaceplane back in the United States at its home base, General. Did you not copy that order?"
"I did, sir, but complying with that order means flying the spaceplane over the target base, and I believe it will be attacked if we do so. The only way we can protect the crew now is to deorbit the spaceplane to keep it as low as possible on the horizon until we can—"
"General, I don't understand a word of what you just said," Kordus said. "All I understand is that you have a strong hunch that your spaceplane is in danger, and you're asking the President to countermand an order he just issued. Is this correct?"
"Yes, sir, but I need to stress the extreme danger of—"
"I got that part loud and clear, General McLanahan," Kordus said, the exasperation thick in his voice. "If you start bringing the spaceplane down, will you be overflying anyone's airspace, and if so, whose?"
"I don't know precisely, sir, but I'd say countries in eastern Europe, the Middle East—"
"Russia?"
"Possibly, sir. Extreme western Russia."
"Moscow?"
Patrick paused, and when he did he could hear the chief of staff say something under his breath. "I don't know if it will be below the sixty-six-mile limit, sir, but depending on how fast and how successful we are at maneuvering the—"
"I'll take that as a yes. Perfect, just perfect. Your spaceplane coming out of orbit right over the capital of Russia will look like an ICBM attack for damned sure, won't it?" He didn't wait for an answer. "This is precisely the nightmare scenario the President was afraid of. He's going to tear your throat out, McLanahan." He paused for a moment; then: "How long does the President have to decide this, General?"
"About five minutes, sir."
"For God's sake, McLanahan! Five minutes? Everything is a crisis with you!" Kordus shouted. "But poor planning on your part doesn't constitute an emergency on our part!"
"Lives could be at stake, sir."
"I'm well aware of that, General!" Kordus snapped. "But if you had bothered to wait and have this plan approved by the White House and the Pentagon before launching the spaceplane, none of this would be happening!" He muttered something else under his breath; then: "I'll take this request to the President right away. In the meantime, stay on the line because you will have to explain all this to the National Security Adviser so he can properly advise the President, because I doubt if you have the capacity to explain it clearly enough to him to his satisfaction—or that he would even listen to you if you tried. Stand by."
"Crew, be advised, we're doing a y-translation in preparation for deorbit. Stand by." Using her multifunction display and her piloting skills, Moulain used the Black Stallion's hydrazine thrusters to flip the spaceplane around so it was flying tailfirst. The maneuver took almost two minutes—a record for her. Everything felt exactly the same to the crewmembers in the passenger module, and even Macomber's stomach didn't complain. "Maneuver complete, Genesis. When do we start down? When can we fire the 'leopards'?"
"We need to find out if you can reach a safe landing runway if you deorbited right now," Dave Luger interjected. "We're also looking for a tanker that can refuel you in case you can't reach a suitable airport, and we need permission from the White House to bring you down over national boundaries."
"You need what?" Macomber retorted. "You think the Russians are going to shoot at us with a fucking laser, and you need permission to get us the hell out of here?"
"We're running the calculations, Major—put a sock in it and let us do our work," Luger said sternly, unaccustomed to being yelled at by a field-grade officer. Still, the tone in his voice made it obvious he wasn't all that happy about the circumstances either. "Stand by."
"Do it, Frenchy," Macomber said on intercom. "Get us the hell out of here."
"I can't do that without authorization, S-One."
"The hell you can't. You're the spacecraft commander—you made that real clear to me, remember? Exercise some of your authority and get us the hell out of here!"
"I can't just drop us out of the sky without knowing where we go once we re-enter the atmosphere," Moulain said. "I need to know where we'll be when we resume atmospheric flight, what our best range will be, which runway we'll approach, what the terrain is, how long the runway is, what the political, diplomatic, and security situation will—"
"For Christ's sake, Frenchy, stop asking questions and hit the damned button!" Macomber shouted. "Don't wait for some politician to wave his hand or give us the finger—just do it!"
"Shut up and stand by, Macomber!" Moulain shouted. "We can't just pull over and shut off the engine. Just hold your water, will you?"
"We'll be crossing the target area's horizon in about two minutes," Terranova reported.
"We briefed several recovery, alternate, and emergency bases in eastern Europe, India, and the western Pacific," Macomber persisted. "We know we have alternates. Just declare an emergency and land at one of them."
"We've already passed most of the safe emergency bases," Terranova said. "The alternate landing sites we had picked were designed in case of failure to insert into orbit, failure of re-entry burn engines, or alternate landing sites if we started deorbit but weren't authorized to go into the target area. We're past that point now. If we didn't deorbit by now, the plan was to overfly the target area, transfer orbits if we had enough fuel, or stay in orbit until we could land back at Dreamland. We can't just turn on a dime and head back the other way."
"So we're screwed," Turlock said. "We've got to overfly the target area now."
"Not necessarily, but the longer we delay firing the 'leopards,' the fewer options we have," Terranova said. "We can always bleed off more energy and drop faster through the atmosphere to try to stay as low to the horizon as possible, then once we're back in the atmosphere we can use the rest of the available fuel to fly away from the tracking radar."
"Then do it!"
"If we bleed off all our energy and don't have enough fuel to make it to a suitable landing site, we're dead," Moulain said. "This bird glides just a little bit better than a damned brick. I'm not going to throw away all our options unless there's a plan! Besides, we don't even know if there's a Russian anti-satellite laser down there. This could all be just a bad case of paranoia."
"Then there's one more option..."
"No way, MC."
"What's the last option?" Macomber asked.
"Jettisoning the passenger module," Terranova said.
"What?"
"The passenger module is designed to be its own re-entry vehicle and lifeboat..."
"I'm not releasing the module except in an emergency," Moulain insisted. "Absolutely not."
"There's no way we can make it down by ourselves!" Macomber cried.
"The simulations say it can, although we've never tested it for real," Terranova said. "The passenger module has its own reaction control system, high-tech heat shields better than the Stud, parachutes and impact attenuation bags for landing, a pretty good environmental system—"
"'Pretty good' isn't good enough, MC—the captain doesn't have any armor on," Chris Wohl interjected.
"It'll work, Sergeant Major."
"I'm not jettisoning anything, and that's that," Moulain cut in. "That's the last resort only. I'm not even going to consider it unless all this fearmongering comes true. Now everyone shut up for a minute." On the command channel: "Genesis, Odin, what do you got for us?"
"Nothing," Patrick responded. "I've spoken to the chief of staff, and he's going to talk to the President. I'm waiting to talk to SECDEF or the National Security Adviser. You're going to have to—"
"I've got it!" Dave Luger suddenly cut in. "If we deorbit now and use max-G maneuvers to lose altitude, we should have enough energy to make it to Baku on the Caspian coast of Azerbaijan. If not, you can make it to Neftcala, which is an Azerbaijan border and coastal patrol base. Turkey and the United States are expanding an airstrip there and you might have enough runway to make it. The third option—"
"Jettison the passenger module into the Caspian Sea, then ditch the Stud in the Caspian Sea or eject before hitting the water depending on how out of control we become," Moulain intoned.
"Stand by, Stud," Patrick said after a short pause. "Genesis, I'm studying the latest images of the target area, and I'm concluding that the trucks and setup at Soltanabad are virtually identical to the ones we saw in Kabudar Ahang in Iran. I believe the Russians set up their mobile anti-spacecraft laser in Soltanabad. Can you verify?"
"General, are you sure this Russian threat is for real? If we do this, there's no turning back."
"No, I'm not sure of any of this," Patrick admitted. "But the signs are looking just like Stud One-One. Genesis?"
"I'm double-checking, Odin," Dave Luger said. "Remember they faked the setup at Kabudar Ahang to suck in the Battle Force. They could be doing the very same thing again."
"We'll know in about sixty seconds, crew," Terranova said.
"We can't wait," Patrick said finally. "Stud, this is Odin, I'm ordering you to deorbit, do a max-rate re-entry interface profile, and attempt an emergency landing at Baku or Neftcala, Azerbaijan. Genesis, upload the flight plan to the Black Stallion and be sure it's executed. Do you copy?"
"Odin, I copy, but are you sure about this?" Moulain asked. "It doesn't make any sense."
"Just do it, Frenchy," Macomber said. "If he's wrong and everything goes snafu, we might take a swim in the damned polluted Caspian Sea with the caviar. Big deal. Been there, done that. If he's right, we'll still be alive in an hour. Do it."
"Flight plan uploaded," Luger reported. "Awaiting execution."
"Stud, advise when you execute deorbit procedures."
"What are you waiting for, Frenchy?" Macomber shouted. "Start us down! Fire the rockets!"
"I don't want to crash into the Caspian Sea," Moulain said. "If we don't make it, we'll have no option but to ditch—"
"Dammit, Frenchy, get us down now!" Macomber shouted. "What's with you?"
"I don't believe General McLanahan, that's why!" Moulain cried out. "I don't believe any of this!"
"Stud, I'm sure this is a trap," Patrick said. "I think we stumbled onto a Russian anti-spacecraft laser weapon site in Iran. If you don't get out of there, any way you can, their laser will burn through your heat shielding and destroy the spacecraft. I don't want to take that risk. Deorbit the spacecraft and get out of there."
"Crossing the target's horizon, now," Terranova said.
"Stud, that was an order: deorbit the spacecraft," Patrick said. "Your objection is noted. I take full responsibility. Now do it."
"I'm sorry, sir, but I copied valid and authenticated orders to the contrary from the national command authority: stay in orbit until we're in a position to return to Groom Lake," Moulain said. "Those orders supersede yours. We're staying. MC, remove the deorbit flight plan and reload the previous one."
"Frenchy—"
"Do it, MC," Moulain said. "That's an order. I'll stay in this orientation to conserve thruster fuel, but we're staying in orbit, and that's final."
The radios and intercoms got very quiet after that, with Luger and McLanahan feeding the crew and each other a steady stream of radar threat warnings and updated reconnaissance imagery. Time seemed to drag on forever. Finally, Macomber said, "What the hell is going on, Genesis, and how long until we're out of the shit?"
"Four minutes ten seconds until we cross back below the target area horizon," Dave Luger responded.
"I'm sorry, Odin," Moulain said, "but I had to make a decision. I'm following orders."
"I hope I'm wrong, SC," Patrick responded. "You did what you thought was right. We'll talk about it after you're home safe."
"How are we doing on that Baku landing site, Genesis?" Terranova asked.
"You'll lose it in thirty seconds. You won't have enough energy to make it to Forward Operating Base Warrior in Kirkuk, Iraq, after you re-enter the atmosphere—Herat, Afghanistan, is your best option, but you'll still have to overfly Soltanabad. Another option might be the deserts of southern Turkmenistan—we can get a special ops team from Uzbekistan in to help you quickly."
"You suggesting we land in Turkmenistan, sir?"
"I didn't say 'land,' MC."
Terranova gulped. Luger obviously meant for them to "jettison the aircraft"—let it crash-land in the desert. "What's the next abort base?"
"Karachi and Hyderabad beyond that."
"We're ready to fire the 'leopards,'" Terranova said. "Ten-second checklist hold. Should I set the re-entry for maximum deceleration?"
"We're not going to deorbit," Moulain said. "The Russians wouldn't dare take a shot at us. Leonid Zevitin's not crazy. The guy can dance, for God's sake!" The radios sparkled with low chuckles. But she looked at her aft-cockpit camera and nodded to Terranova, silently ordering him to program the computers for a maximum-rate speed and altitude loss. "I mean, think about it, everyone: no male who knows how to dance would be nutty enough to—"
Suddenly they heard, "Warning, warning, laser detected...warning, warning, hull temperature increasing, stations two hundred fifty through two-ninety...warning, hull temperatures approaching operational limits...!"
"The Kavaznya laser!" Patrick McLanahan exclaimed. "They're attacking from extreme range. Stud, get out of there now!"
"Initiate deorbit procedures!" Moulain shouted. "Crew, stand by to deorbit immediately! 'Leopards' engines throttling up!"
"...hull temperature rate warning, stations two-seventy through two-ninety...warning, warning...!"
The crew was slammed back into their seats as the Laser Pulse Detonation Rocket System engines fired at full power. The immense power of the hybrid rocket engines immediately and dramatically decelerated the Black Stallion aircraft, and it quickly began its fall to Earth. Macomber cried out as the G-forces quickly increased, far past anything he had previously experienced. Soon he could no longer muster the strength to make any noise at all—it took all of his concentration to inflate his lungs enough to keep from passing out.
"Passing twenty-eight thousand feet per second," Terranova said amidst the almost-constant warning messages. "Passing ninety miles' altitude...'leopards' at ninety percent power, three point zero Gs..."
"Go to one hundred and ten percent power," Moulain grunted through the pressure.
"That's over five Gs, SC," Terranova said. "We'll have to sustain that for—"
"Do it, MC," Moulain ordered. "Crew, SC, it's going to get real uncomfortable for a few minutes. Keep ahead of it the best you can." A few moments later, her words were cut off by a feeling that her chest was going to implode as the G-forces nearly doubled. Cries of anguish and surprise were evident. "Hang...on...crew..."
"Five point three Gs," Terranova gasped. "Jesus...passing twenty-five K, passing eighty miles..."
"Oh God, how much longer?" someone murmured—it was impossible to tell who was speaking now.
STRATEGIC AIR FORCES ALTERNATE OPERATIONS COMMAND CENTER, POLDOSK, RUSSIAN FEDERATION
THAT SAME TIME
With the destruction of Engels Air Base near Saratov and the bombing of R'azan underground command center by the Americans, air forces chief of staff General Andrei Darzov had reactivated and modernized an old civil defense shelter and reserve forces reconstitution center southwest of Moscow called Poldosk for use as his evacuation and alternate command post. It didn't have an air base or even room for a large helicopter landing pad, but it had underground rail lines adjacent to the facility, plenty of freshwater supplies (as fresh as could be expected in the Greater Moscow area)...
...and—more importantly, Darzov believed—it was sufficiently close to large numbers of city dwellers that even someone as crazy as the American bomber commander Lieutenant General Patrick McLanahan might think twice about bombing the place.
Because of its mostly modern high-speed data and communications upgrades, Poldosk today served yet another purpose: as the monitoring and command center for the Molnija anti-spacecraft air-launched missile and Fanar anti-spacecraft laser systems. From a simple room with a bank of four computers, Darzov maintained contact with his forces in the field via secure high-speed Internet and voice-over-IP connections. The command center was completely mobile, could be packed up in less than an hour and set up elsewhere in about as much time, and in an emergency could be run from a single laptop computer and secure cellular or satellite phone anywhere on the planet.
This evening, the focus was on Soltanabad. It was unfortunate that the Americans found Fanar so quickly—it had to be blind luck, or maybe some Iranian Revolutionary Guards Corps members turned traitor and informed on them to the coup leader Hesarak Buzhazi or to the Americans. But he had set up Fanar at Soltanabad precisely because so many American spacecraft overflew the area so often. It was, as the Americans put it, a "target-rich environment."
Darzov scowled at a new readout and hit the TRANSMIT button on the computer keyboard: "Striker, this is Keeper. Say status. You terminated the attack...why?"
"We had full optronic lock on the target and opened fire as ordered, General," the chief engineer and project officer at Soltanabad, Wolfgang Zypries, replied. "But seconds after we initiated the attack we lost contact." Zypries was a German laser engineer and scientist and formerly a colonel in the German air force. Unknown to him, Zypries' longtime girlfriend was a Russian spy, hacking into his computer at home and transferring volumes of classified material to Moscow. When his girlfriend informed him of who she was and that the German Militärischer Abschirmdienst, or Military Screen Service's counterespionage group, was on his tail, he allowed himself to be whisked off to Russia. Darzov immediately plied him with everything he desired—money, a house, and all the women he could handle—to work on improving and mobilizing the Kavaznya anti-spacecraft laser system. After over five years' work, he was more successful than even Darzov dared to hope.
"The spacecraft appears to be descending rapidly," Zypries went on. "We suspect our optics were blinded when the spacecraft fired its retrorockets."
"You did brief me that might happen, Colonel," Darzov said. To avoid detection they had decided to use a telescopic electro-optical acquisition and tracking system and keep their extreme long-range space tracking radar in standby. They locked onto the American spaceplane seconds after it crossed the horizon and tracked it with ease. As they hoped, it had not started its descent through the atmosphere, although the highly magnified image showed it was indeed turned in the proper direction to begin slowing down, flying tailfirst. It was still in the perfect position, and Darzov ordered the attack to commence.
The next step in the laser engagement was to hit the target with a higher-powered laser to measure the atmosphere and apply corrections to the main laser's optics, allowing it to focus more precisely on the target before firing the main chemical-oxygen-iodine laser. Darzov and Zypries decided, since the spacecraft was turned in position to fire its retrorockets, to use the main laser itself to make its own corrections in order to engage more rapidly.
"The crew was obviously expecting an attack," Zypries said, "because they fired their main engines seconds after our laser hit. We were able to maintain contact for about fifteen seconds, but the optics were still fine-focusing so we were probably only laying sixty percent power on their hull. Then the optronic system broke lock. They must be squishing their crewmembers like bugs inside that thing—they are decelerating at three times the normal rate. I am tracking them on infrared scanners but that's not precise enough for the main laser, so I need permission to use the main radar to reacquire and engage."
"Are they still in range and high enough to engage?"
"They are at one hundred thirty kilometers' altitude, sixteen hundred kilometers downrange, decelerating quickly below seven thousand eight hundred meters per second—they are dropping like a stone, but they are well within the laser's engagement envelope," Zypries assured him. "The structure of that spacecraft must be incredibly strong to withstand that kind of stress. They will be in the atmosphere soon but they will not be able to fly away fast enough now. I will get him for you, General."
"Then permission granted to continue the attack, Colonel," Darzov said immediately. "Good hunting."
"Five point seven Gs...twenty-two K feet per second...seventy-five miles...five point nine Gs..." It seemed to take forever for Terranova to grunt out each readout. "Passing seventy miles...sixty-five miles, reaching entry interface, crew, 'leopards' cutoff." The G-forces suddenly were reduced, followed by a chorus of moans and swearing from throughout the spacecraft. Macomber couldn't believe he hadn't passed out from that sustained pressure. He still felt the deceleration forces as the spaceplane continued to lose energy, but it wasn't nearly as bad as it was when the "leopards" were firing. "Crew, report."
"You guys okay?" Macomber asked the others in the passenger module. "Sing out."
"S-Two, I'm okay," Turlock said weakly.
"S-Three, okay," Wohl responded, sounding as if nothing at all had just happened. The jarhead bastard was probably sound asleep through it, Macomber thought.
"S-One is okay too. SC, passengers are okay, everything back here's in the green. That was some ride."
"Roger that," Moulain said. "The laser looks like it's broken lock for now. We've initiated maneuvering to entry interface attitude." The Black Stallion began to turn so it was nose-forward again, then pitched up to forty degrees above the horizon for atmospheric entry, presenting its bottom heat shields to the onrushing atmosphere to protect the ship against the heat built up by friction. "MC, let's brief the approach."
"Roger," Terranova said. "We've passed the terminal alignment cylinder for Baku, so I've programmed in Herat, Afghanistan, as our landing site. We are still on max-energy descent profile, and Herat is fairly close—around thirteen hundred miles—so we have plenty of energy to reach the base. In sixty seconds the airflow pressure will be great enough for the adaptive surfaces on the Stud to take effect, and we'll shut down the reaction control system, transition to maximum-drag profile, and deviate east over Turkmenistan to stay away from Soltanabad. Once we pass one hundred thousand feet we can transition to atmospheric flight, shut down the 'leopards,' start up the turbojets, and head down on a normal approach profile."
"How much gas do we have, MC?" Macomber asked.
"After we start up the turbojets, we'll have less than an hour of fuel, but we'll be gliding in at around Mach five so we'll have plenty of energy to get rid of before we need the turbojets," Terranova replied. "We'll start securing the thrusters and get ready to secure the 'leopards' so when we—"
"Warning, warning, search radar, twelve o'clock, nine hundred sixty miles, India-Juliet band," the computerized voice of the threat warning receiver suddenly blared. Seconds later: "Warning, warning, target tracking radar, twelve o'clock, nine hundred fifty miles...warning, warning, pulse-Doppler target tracking radar, twelve o'clock, nine hundred forty miles...warning, warning, laser detected, twelve o'clock...warning, warning...!"
"They hit us with radar at almost a thousand miles?" Terranova blurted out. "That's impossible!"
"It's the Kavaznya radar, crew," Patrick McLanahan said. "The range of that thing is incredible, and now it's mobile."
"Warning, warning, emergency cooling system activated...warning, warning, spot hull temperature increasing, station one-ninety..."
"What do we do, Odin?" Lisa Moulain cried on the radio. "What do I do?"
"The only choice you have is to roll the spacecraft to keep the laser energy from focusing on any one spot for too long," Patrick said. "Use the reaction control system to roll. Once your mission adaptive system is effective, you can use max bank angle to fly away from the laser and do heading changes as much as possible to keep the laser off you. Dave, I need you to launch the Vampires from Batman Air Base and knock out that laser site! I want Soltanabad turned into a smoking hole!"
"They're on the way, Odin," Luger responded.
But as the seconds ticked by, it was obvious that nothing Moulain could do was going to work. They were getting almost constant overtemperature warnings from dozens of spots on the hull, and some began reporting leaks and structural integrity losses. Once Moulain accidentally looked directly at the laser light shining through the cockpit windshield and was partially blinded even though they both had their dark visors lowered.
Terranova finally muted the threat warnings—they were doing them no good anymore. "Frenchy, you okay?"
"I can't see, Jim," Moulain said on the "private" intercom setting so the crewmembers in the passenger compartment couldn't hear. "I glanced at the laser beam for a split second, and all I see are big black holes in my vision. I screwed up. I killed us all."
"Keep rolling, Frenchy," Terranova said. "We'll make it."
Moulain began nudging the side control stick back and forth, using the thrusters to turn and roll the spacecraft. Terranova fed her a constant stream of advisories when she was going too far. The temperature warnings were almost constant no matter how hard she tried. "We've got to jettison the passenger module," Moulain said, still on "private" intercom. "They might have a chance."
"We're way over the G-force and speed limits for jettison, Frenchy," Terranova said. "We don't even know if they'll survive even if we slowed down enough—we've never jettisoned the module before."
"There's only one way to find out," Moulain said. "I'm going to initiate a powered descent to try to slow us down enough to jettison the passenger module. We'll use every drop of fuel we have left to slow us down. I'll need your help. Tell me when we're ass-end backward." She gently rolled wings-level, then with Terranova's assistance turned the Black Stallion so they were flying tailfirst again. On full intercom she spoke, "Crew, prepare for max retrorocket fire, powered-descent profile. 'Leopards' coming online."
"What?" Macomber asked. "You're firing the 'leopards' again? What—?"
He didn't get to finish his question. Moulain activated the Laser Pulse Detonation Rocket System engines and immediately pushed them up to powered-descent profile power, then to maximum power, far exceeding the normal G-limits for passengers and crewmembers. Their speed dropped dramatically—they were still flying at over Mach 5, but that was over half of the speed they would normally be flying. Everyone in the passenger module was hit with G-forces so severe and so unexpected that they immediately blacked out. Jim Terranova blacked out too...
...and so did Lisa Moulain, but not before she opened the cargo bay doors on the upper fuselage of the XR-A9 Black Stallion, unlocked the securing bolts holding the module to the cargo bay, lifted a red-guarded switch, and activated it...
...and at the very instant the doors were fully open, the securing bolts were free, and the module's jettison rockets fired, the Black Stallion exhausted every pound of propellant left in its tanks...and it was ripped apart by the Russian laser and exploded.
"Target destroyed, General," Wolfgang Zypries reported from Soltanabad. "Showing massive speed loss, multiple large targets probably debris, and quickly losing radar and visual contact. Definite kill."
"I understand," General Andrei Darzov responded. Many of the technicians and officers in the room triumphantly raised fists and gave low cheers, but he silenced them with a warning glare. "Now I suggest you get out of there as fast as you can—the Americans have certainly sent a strike force out to destroy that base. They could be there in less than an hour if they launch from Iraq."
"We will be out of here in thirty minutes, General," Zypries said. "Out."
Darzov broke the connection, then activated another and spoke: "Mission accomplished, sir."
"Very well, General," Russian president Leonid Zevitin responded. "What do you expect will be their reaction?"
"They are undoubtedly launching unmanned B-1 bombers from Batman Air Base in Turkey, fitted with the hypersonic attack missiles to attack and destroy the base in Iran," Darzov said. "They could be in position to fire in less than an hour—even as quickly as thirty minutes if they had a plane ready to launch. The target will be struck less than a minute later."
"My God, that's incredible—we need to get our hands on that technology," Zevitin muttered. "I assume your people are haulin' ass and getting away from that base."
"They should be well away before the Americans attack—I assure you, they feel those hypersonic missiles on the backs of their necks even now."
"I'll bet they do. Where was the spaceplane when it went down, General?"
"Approximately one thousand kilometers northwest of Soltanabad."
"So by chance does that place it...over Russia?"
There was a short pause as Darzov checked his computerized maps; then: "Yes, sir, it does. One hundred kilometers northwest of Machackala, the capital of Dagestan province, and three hundred kilometers southeast of the Tupolev-95 bomber base at Mozdok."
"And the debris?"
"Impossible to say, sir. It will probably be scattered for thousands of kilometers between the Caspian Sea and the Iran-Afghanistan border."
"Too bad. Track that debris carefully and advise me if any reaches land. Order a search team from the Caspian Sea Flotilla to begin a search immediately. Have our radar stations alerted our air defense systems?"
"No, sir. The normal air defense and air traffic radar systems would not be able to track a target so high and traveling so fast. Only a dedicated space tracking system would be able to do so."
"So without such radar, we wouldn't know anything has happened yet, would we?"
"Unfortunately not, sir."
"When would you expect the debris to be detected by a regular radar system?"
"We are not tracking the debris anymore since we are breaking down the Fanar radar system at Soltanabad," Darzov explained, "but I would guess that within a few minutes we might be able to start picking up the larger pieces as they re-enter the atmosphere. I will have our air defense sites in Dagestan report immediately when debris is detected."
"Very good, General," Zevitin said. "I wouldn't want to complain about the latest American attack against Russia too soon, would I?"
ABOARD AIR FORCE ONE
THAT SAME TIME
"My, my, Mr. President," the female staff sergeant said as she rose from her knees and began rebuttoning her uniform blouse, "you certainly get my vote."
"Thank you, Staff Sergeant," President Gardner said, watching her rearrange herself as he zipped his fly. "I think there's a position available on my...staff for someone as skilled as you." She smiled at the very much intended double entendre. "Interested?"
"Actually, sir, I've been waiting for an opening in Officer Training School," she replied, looking the commander-in-chief up and down hungrily. "I was told a slot might not open up for another eighteen months. I finished my bachelor's degree and put in my application just last semester. I'm very determined to get my commission."
"What was your degree in, sugar?"
"Political science," she replied. "I'm going for a law degree, and then I'd like to get into politics."
"We could sure use someone of your...enthusiasm in Washington, Staff Sergeant," the President said. He noticed the CALL light blinking on the phone—an urgent call, but not urgent enough to override the DO NOT DISTURB order. "But OTS is in Alabama?"
"Yes, sir."
"That's too bad, honey," the President said, acting disappointed—the last thing he wanted was for this one to show up in Washington. Maxwell Air Force Base in Alabama would be perfect—far enough away from Washington to avoid rumors, but close enough to Florida for her to sneak down when he was at his estate in Florida. "I'd sure like to work with you more often, but I admire your dedication to the service. I'm sure I heard of an OTS slot opening up in the next class, and I think you'd fit in perfectly. We'll be in touch."
"Thank you very much, Mr. President," the steward said, smoothing out the rest of her hair and uniform, then departing without even a backward glance.
That's the way he liked them, Gardner thought as he took a sip of juice and started to get his heart rate and thoughts back in order: the ones bold and aggressive enough to do anything necessary to get an advantage over all the others, but wise enough to go back to work and avoid getting emotionally involved—those were the real powerhouses in Washington. Some did it with talent, brains, or political connections—there was nothing wrong, or different, about the ones who did it on their knees. Plus, she understood the same as he that both their careers would be finished if word ever got out about their little rendezvous, so it benefited both of them to do what the other wanted and, more important, keep their mouths shut about it. That one was going to go very far.
Seconds later, his mind quickly refocused on the upcoming events and itinerary, he punched off the DO NOT DISTURB button. Moments later his chief of staff and National Security Adviser knocked, checked the peephole to be sure the President was alone, waited a moment, then entered the suite. Both had cell phones up to their ears. Air Force One could act as its own cellular base station, and unlike passengers on commercial airliners there were no restrictions on the use of cell phones inflight on Air Force One—users could light up as many terrestrial cell towers as they liked. "What's going on?" the President asked.
"Either nothing...or the shit has just hit the fan, Mr. President," Chief of Staff Walter Kordus said. "Air forces in Europe headquarters got a call from the Sixth Combined Air Operations Center in Turkey requesting confirmation for an EB-1C Vampire bomber flight of two scramble launch out of Batman Air Base in southern Turkey...the same ones we grounded after the missile attack in Iran. USAFE called the Pentagon for confirmation since there was no air tasking order for any bomber missions out of Batman."
"You mean, McLanahan's bombers?" Kordus's panicked face had the answer. "McLanahan ordered two of his bombers to launch...after I ordered them grounded? What the hell is going on?"
"I don't know yet, sir," Kordus said. "I told USAFE that no bombers were authorized to launch for any reason, and I ordered them to deny launch clearance. I have a call in to McLanahan and to his deputy Luger out in Nevada, trying to find out what's going on."
"Are the bombers armed?"
"We don't know that yet either, sir. This mission was totally unauthorized."
"Well, we should assume they are—knowing McLanahan, he would keep weapons on his planes even though they're all grounded, unless we specifically ordered him not to, and even then he might do it. Just keep them on the ramp until we find out what's going on. What's the story with the spaceplane? Is it still in orbit?"
"I'll check as soon as McLanahan picks up the phone, sir."
"It'd better be, or I'll nail his hide to my bathroom door," the President said, taking another sip of orange juice. "Listen, about the 'meet-and-greet' thing in Orlando..." And then he heard Carlyle swear into his phone. "What, Conrad?"
"The B-1 bombers launched," the National Security Adviser said. The President's jaw dropped in surprise. "The tower controller at the air base told the crew to hold their position, but there is no crew on those planes—they're remotely controlled from Elliott Air Force Base in Nevada—"
"McLanahan."
"McLanahan is still aboard the space station, so it's his deputy, Brigadier General Luger, in charge of the bombers out of Elliott," Carlyle said. "I've got a call in to Secretary of Defense Turner to order Luger to get those bombers back on the ground. Je-sus...!"
"He is out of control!" the President snapped. "I want him off that space station and in custody immediately! Send a damned U.S. Marshal up there if you have to!"
"Send a U.S. Marshal—into space?" Kordus asked. "I wonder if that's ever been done before...or if we could get a marshal to volunteer to do that?"
"I'm not kidding around, Walter. McLanahan has to be slapped down before he starts another damned war between us and Russia. Find out what in hell is going on, and do it fast. Zevitin will be on the phone before we know it, again, and I want to assure him everything is under control."
BATTLE MANAGEMENT AREA, BATTLE MOUNTAIN AIR RESERVE BASE, NEVADA
THAT SAME TIME
"Headbanger Two-One flight of two is level at flight level three-one-oh, due regard, Mach point nine-one, thirty minutes to launch point," the mission commander reported. "Due regard" meant that they had terminated all normal air traffic control procedures and were flying without official flight-following or civil aviation monitoring...because they were going to war.
Two officers sat side by side in a separate section of the BATMAN, or battle management area, at Battle Mountain Air Reserve Base in northern Nevada, seated at what appeared to be a normal computer workstation that might be used by a security guard or securities day trader...except for the jet-fighter-style joysticks. On each side of the officers were two enlisted technicians with their own bank of computer monitors. The men and women in the room talked into their microphones in muted voices, bodies barely moving, eyes scanning from monitor to monitor. Only an occasional flick of a finger on a keyboard or hand rolling a cursor with a trackball led anyone to believe anything was really happening.
The two officers were piloting two unmanned EB-1C Vampire supersonic "flying battleships" which had launched from their forward operating base in eastern Turkey across northern Iran. Three high-resolution monitors showed the view in front and to the sides of the lead bomber, while other monitors showed performance, systems, and weapons readouts from both planes. Although the two bombers were fully flyable, they were usually flown completely on computer control, reacting autonomously to mission commands entered before the flight and deciding for themselves what to do to accomplish the mission. The ground crew monitored the flight's progress, made changes to the flight plan if necessary, and could take over at any time, but the computers made all the decisions. The technicians watched over the aircraft's systems, monitored the electromagnetic spectrum for threats, and looked over incoming intelligence and reconnaissance data along the route of flight that might affect the mission.
"Genesis copies," David Luger responded. He was back at the battle staff area at Elliott Air Force Base in south-central Nevada, watching the mission unfold on the wall-sized electronic "big boards" before him. Other displays showed enemy threats detected by all High-Technology Aerospace Weapons Center aircraft and satellites and other allied sensors operating in the region. But Luger's attention was drawn to two other displays: the first was the latest satellite imagery of the target area in eastern Iran...
...and the second was of the satellite space tracking data, which at the moment was blank.
"They're taking down the laser stuff in a pretty big damned hurry," Dave commented. "They must have guessed we'd send bombers to blast the hell out of that base. I'm not sure if we'll get there in time, Muck."
"Push 'em up, Dave," Patrick McLanahan said. He was monitoring the mission as well from the command module on Armstrong Space Station. "Get a tanker airborne to meet the bombers on the way back, but I want those missiles on the way before the Russian cockroaches scatter."
"Roger, Muck. Stand by. Headbanger, this is Genesis. Odin wants the bombers to attack before the target scatters. Push up the bombers and say status of the support tankers."
"Already got the alert tankers taxiing out, Dave," the commander of the Air Battle Force's air forces from Battle Mountain, Major General Rebecca Furness, responded. "He'll be airborne in five minutes."
"Roger that. Odin wants the Vampires pushed up as much as you can."
"As soon as the tanker's within max safe range, we'll push the Vampires up to Mach one point two—that's the max launch speed for the SkySTREAKs. Best we can do with the current mission parameters."
"Suggest you erase the one-hour fuel reserve for the tanker and push up the Vampires now," Luger said.
"Negative—I'm not going to do that, Dave," Rebecca said. Rebecca Furness was the U.S. Air Force's first female combat pilot and first female commander of a tactical combat air unit. When Rebecca's Air Force Reserve B-1B Lancer unit at Reno, Nevada, was closed and the bombers transferred to the High-Technology Aerospace Weapons Center for conversion into manned and unmanned "flying battleships," Furness went along. Now she commanded the five tactical squadrons at the new Reserve base at Battle Mountain, Nevada, composed of converted manned and unmanned B-52 and B-1 bombers, unmanned QA-45C stealth attack aircraft, and KC-76 aerial refueling tankers. "We'll get them, don't worry."
Luger glanced again at the latest satellite image of the highway air base at Soltanabad, Iran. It was only five minutes old, but it already showed a few of the larger trucks gone and what appeared like an entire battalion of workers taking down the rest. "We're running out of time, ma'am. The cockaroaches are scattering quick."
"I know, Dave—I see the pictures too," Rebecca said, "but I'm not risking losing my bombers."
"Like we lost the Stud?"
"Don't give me that crap, Dave—I know what's going on here, and I'm just as mad about it as you are," Rebecca snapped. "But may I remind you that our bombers are the only long-range strike aircraft we have now, and I'm not going to risk them on...an unauthorized mission." It was no exaggeration, and Dave Luger knew it: since the American Holocaust, the Russian cruise missile attack on American bomber and intercontinental missile bases four years earlier, the only surviving long-range bombers had been the handful of bombers deployed overseas and the converted B-52 and B-1 bombers based at Battle Mountain.
Furness's bombers soon racked up casualties of their own. All of Battle Mountain's bombers had been sent to a Russian aerial refueling tanker base in Yakutsk, Siberia, from where Patrick McLanahan led attack missions against nuclear ballistic missile bases throughout Russia. When the American bombers were discovered, then–Russian president General Anatoliy Gryzlov attacked the base with more nuclear-tipped cruise missiles. Half the force had been lost in the devastating attack. The remaining bombers successfully attacked dozens of Russian missile bases, destroying the bulk of their strategic nuclear force; McLanahan himself, aboard one of the last EB-52 Megafortress battleships, attacked and killed Gryzlov in his underground bunker southeast of Moscow in a grueling twenty-hour-long mission that took him across the entire breadth of the Russian Federation.
After the conflict, Rebecca Furness had been given command of the Air Force's few remaining bombers; consequently, no one knew better than she the incredible responsibility placed upon her. The surviving planes, and the few unmanned stealth bombers built since the American Holocaust, were the only air-breathing long-range strike aircraft left in the American arsenal—if any bombers were going to be built ever again, it might take decades to build the force back up to credible levels.
"Ma'am, I'm sure the strike mission will be approved once the national command authority gets our report on what happened to our spaceplane," Dave said. "That mobile Kavaznya laser is the biggest threat facing our country right now—not just to our spacecraft, but possibly to anything that flies." He paused, then added, "And the Russians just killed five of our best, ma'am. It's time for some payback."
Rebecca was silent for a long moment; then, shaking her head, she said wryly, "Three 'ma'ams' out of you in one conversation, General Luger—I believe that's a first for you." She punched some instructions into her computer. "I'll authorize a change to thirty minutes' bingo fuel."
"Odin to Headbanger, I said, push them up, General Furness," Patrick interjected from Armstrong Space Station. "Take them up to Vmax, then slow them down to one point two for weapon release."
"What if they don't make it to the air refueling anchor on the way back, General?" she asked. "What if there's a navigation error? What if they can't hook up on the first go? Let's not lose sight of—"
"Push 'em up, General. That's an order."
Rebecca sighed. She could legally ignore his order and be sure her bombers were safe—that was her job—but she certainly understood how badly he wanted retribution. She turned to her Vampire flight crew and said, "Push them up to one point five, recompute bingo fuel at the air refueling control point, and advise."
The crew complied, and a moment later reported: "Headbanger flight of two now at flight level three-one-oh, on course, speed Mach one point five, due regard, in the green, twenty minutes to launch point. Bingo fuel at the ARCP is gone; we're down to ten minutes' emergency fuel. We should make up a few more minutes after we get the tanker's updated ETE."
"That's ten minutes after the second bomber cycles on the boom, right?" Rebecca asked. The grim, ashen expression and silent no on the face of the tech told her that they were in really deep shit.
## CHAPTER SEVEN
In war, there are no unwounded soldiers.
—JOSE NAROSKY
ABOARD ARMSTRONG SPACE STATION
MINUTES LATER
"McLanahan here, secure."
"McLanahan, this is the President of the United States," Joseph Gardner thundered. "What in hell do you think you're doing?"
"Sir, I—"
"This is a direct order, McLanahan: Turn those bombers around right now."
"Sir, I'd like to give you my report before—"
"You're not going to do a damned thing except what I order you to do!" the President snapped. "You've violated a direct order from the commander-in-chief. If you want to avoid life in prison, you'd better do what I tell you. And that spaceplane had better still be in orbit, or by God I'll—"
"The Russians shot down the Black Stallion spaceplane," Patrick quickly interjected. "The spaceplane is missing and presumed lost with all souls."
The President was silent for a long moment; then: "How?"
"A mobile laser, the same one that we think shot down our spaceplane last year over Iran," Patrick replied. "That was what the Russians were hiding at Soltanabad: their mobile anti-spacecraft laser. They brought it into Iran and set it up at an abandoned Revolutionary Guards Corps base, one we thought had been destroyed—they even placed fake bomb craters on it to fool us. The Russians set up the laser in a perfect spot to attack our spacecraft overflying Iran. They got the second-biggest prize of them all: another Black Stallion spaceplane. The positioning suggests their real target was Armstrong Space Station."
Again, silence on the other end of the line...but not for long: "McLanahan, I'm very sorry about your men..."
"There were two women on board too, sir."
"...and we're going to get to the bottom of this," the President went on, "but you violated my orders and launched those bombers without permission. Turn them around immediately."
Patrick glanced up at the time remaining: seven-plus minutes. Could he stall the President that long...? "Sir, I had permission to launch the spaceplane into standard orbit from STRATCOM," he said. "We suspected what the Russians were up to, but we were awaiting permission to go in. Our worst fears were confirmed..."
"I gave you an order, McLanahan."
"Sir, the Russians are packing up and moving the laser and their radar out of Soltanabad as we speak," he said. "If they are allowed to slip away, that laser will be an immense threat to every spacecraft, satellite, and aircraft in our inventory. We're just a few minutes away from launch, and it'll be over in less than a minute. Just four precision-guided missiles with kinetic-kill warheads—no collateral damage. It'll take out the components that haven't been moved yet. The Russians can't complain about the attack because then they'd be admitting moving attack troops into Iran to kill Americans, so there won't be any international backlash. If we can get Buzhazi's troops in there to start a forensic search as soon as possible after the attack, we might uncover evidence that—"
"I said, turn those bombers around, McLanahan," the President said. "That's an order. I'm not going to repeat myself. This conversation is being recorded and witnessed and if you don't comply it'll be used against you in your court-martial."
"Sir, I understand, but I ask you to reconsider," Patrick pleaded. "Five astronauts aboard the spaceplane were killed. They're dead, blasted apart by that laser. It was an act of war. If we don't get direct evidence that Russia has commenced direct offensive military action against the United States of America, they'll get away with murder and we'll never be able to avenge their deaths. And if we don't destroy, damage, or disable that laser, it'll pop up somewhere else and kill again. Sir, we must—"
"You are in violation of a direct order from the commander-in-chief, General McLanahan," the President interrupted. "I'll give you one last chance to comply. Do it, and I'll let you retire quickly and quietly without a public trial. Refuse, and I'll strip you of your rank and throw you in prison at hard labor for life. Do you understand me, General? One last chance...which is it going to—?"
Six minutes left. Could he get away with the "scratchy radio" routine? He decided he was far, far beyond that point now: he had no choice. Patrick cut off the transmission. Ignoring the stunned expressions of the technicians around him, he spoke: "McLanahan to Luger."
"Just got off the phone with the SECDEF, Muck," Dave said from Elliott Air Force Base via their subcutaneous global transceiver system. "He ordered the Vampires recalled immediately."
"My phone call trumps yours, buddy: I just heard from the President," Patrick said. "He ordered the same thing. He offered me a nice quiet retirement or a lifetime breaking big rocks into little ones at Leavenworth."
"I'll get them turned—"
"Negative...they continue," Patrick said. "Bomb the crap out of that base."
"Muck, I know what you're thinking," Dave Luger said, "but it might already be too late. The latest satellite image shows at least a fourth of the vehicles already gone, and that was over ten minutes ago. Plus we're already past bingo fuel on the Vampires and well into an emergency fuel situation—they might not reach the tanker before they flame out. It's a no-win scenario, Muck. It's not worth risking your career and your freedom. We lost this one. Let's pull back and get ready to fight the next one."
"The 'next one' could be an attack against another spaceplane, a satellite, a reconnaissance aircraft over Iran, or Armstrong Space Station itself," Patrick said. "We've got to stop it, now."
"It's too late," Luger insisted. "I think we've missed it."
"Then we'll leave 'em with a little calling card in their rearview mirrors, if that's the best we can do," Patrick said. "Nail it."
"He's going to what?"
"You heard me, Leonid," the President of the United States said on the "hot line" from Air Force One, just minutes after the connection was broken to the space station—he had to let loose a string of epithets for a full sixty seconds after the line went dead before he could speak with anyone else. "I think McLanahan is going to launch an air strike on a place called Soltanabad in northeastern Iran. He insists you have set up a mobile anti-spacecraft laser there and you used it to shoot down his Black Stallion spaceplane just a few moments ago."
Russian president Leonid Zevitin furiously typed instructions on a computer keyboard to Russian air forces chief of staff Darzov while he spoke, warning him of the impending attack and ordering him to get fighters airborne to try to stop the American bombers. "This is unbelievable, Joe, simply unbelievable," he said in his most convincing, sincere, outraged tone of voice. "Soltanabad? In Iran? I've never heard of the place! We don't have any troops anywhere in Iran except the ones guarding our temporary embassy in Mashhad, and it's there because our embassy in Tehran has been blasted to hell and Mashhad is the only secure place in the entire country right now, thanks to Buzhazi."
"I'm just as flabbergasted as you are, Leonid," Gardner said. "McLanahan must have flipped. He must've suffered some kind of brain injury when he had that heart flutter episode. He's unstable!"
"But why does an unstable officer have control of supersonic bombers and hypersonic missiles, Joe? Maybe you can't get your hands on McLanahan, but you can shut down his operation, can't you?"
"Of course I can, Leonid. It's being done as we speak. But those bombers may get off a few missiles. If you have any forces on the ground out there, I suggest you get them out pronto."
"I thank you for the call, Joe, but we don't have forces in Iran, period." Still no reply from Darzov, he noticed—damn, he'd better get that laser out of there, or else their game was going to be over. "And we certainly don't have some kind of magic super-laser that can shoot down a spacecraft orbiting Earth at seventeen thousand miles per hour and can then disappear like smoke. The United Nations investigated those reports last year and came up with nothing, remember?"
"I believe they said there were inconclusive results because—"
"Because President Martindale didn't allowed them to interview anyone at Dreamland, and Buzhazi and his insane rebel insurgents didn't allow them access to debris or the suspected site where the laser was supposedly set up," Zevitin said. "The bottom line is that there is not one scrap of evidence out there pointing to some damned super-laser. McLanahan is obviously whipping up a lot of fear in Congress, in the media, and with the American public in order to keep his expensive and dangerous secret programs afloat."
"Well, that's going to be put to a halt real quick," Gardner said. "McLanahan is finished. The bastard hung up on me and ordered that attack to continue."
"Hung up on you?" That was perfect, Zevitin thought happily. Not only was McLanahan going to be removed, but he was going to be portrayed as a lunatic...by his own commander-in-chief! No way his supporters in the military or Congress were going to support him now! He choked down his glee and went on in a low, ominous voice, "That is insane! Is he crazy? You can't allow this to continue! This unstable, insubordinate man has got to be stopped, Joe. You're making a lot of folks real scared out here. Wait until the Duma and the Cabinet hears about another hypersonic missile attack in Iran. They're going to shit their pants."
"Convince them not to worry, Leonid," Gardner said. "McLanahan is done for, and so is his private military force."
"Shut it down, Joe," Zevitin urged. "Shut it all down—the space station, those hypersonic missiles, the unmanned bombers with their EMP death rays—before it's too late. Then let's get together and present the world with a unified, peaceful, cooperative front. That's the only way we're going to ratchet down the tension around here."
"Don't worry about a thing," Gardner insisted. "In case your Caspian Sea ships are in the vicinity, you might tell them that the bombers might launch high-speed missiles."
"Joe, I'm concerned about the backlash in Iran if those missiles hit that area," Zevitin said. "The last I recall, that base was being used by the Red Crescent to fly in relief supplies, and by United Nations monitors—"
"Oh no," Gardner moaned. "This is a damned nightmare."
"If McLanahan blasts that base, he'll be killing dozens, perhaps hundreds of innocent civilians."
"Damn," Gardner said. "Well, I'm sorry, Leonid, but McLanahan's out of control for the time being. There's nothing else I can do."
"I have one radical suggestion, my friend—I hope you don't think I'm crazy," Zevitin said.
"What's your—?" And then Gardner stopped, because he soon figured it out for himself. "You mean, you're asking my permission to—?"
"It's the only way, Joe," Zevitin said, almost unable to contain his amazement at the direction this conversation was taking. "You know it, and I know it. I don't believe even a stressed-out schizoid like McLanahan would ever dare launch missiles against a humanitarian relief airfield, but I can't think of any other way to stop this madness, can you?" There was no response, so Zevitin quickly went on: "Besides, Joe, the bombers are unmanned, correct? No one will get hurt on your side, and we'll be saving many lives." There was a very long pause. Zevitin added, "I'm sorry, Joe, I shouldn't have brought up such a crazy idea. Forget I said—"
"Hold on, Leonid," Gardner interrupted. A few moments later: "Do you have jets nearby, Leonid?" he heard the President of the United States ask.
Zevitin almost doubled over with disbelief. He swallowed his shock, quickly composed himself, then said, "I don't know, Joe. I'll have to ask my air force chief of staff. We normally patrol this area, of course, but since our MiG was shot down by McLanahan's bomber with the EMP nuclear T-wave thing we've pulled back quite a bit."
"I understand," Gardner said. "Listen to me. My National Security Adviser tells me that the bombers launched from Batman Air Base in Turkey and are undoubtedly heading directly to a launch point over the southern Caspian Sea. We can't tell you any more because we simply don't know."
"I understand," Zevitin said. He could scarcely believe this—Gardner was actually telling him where the bombers had launched from and where they were going!
"We don't know their weapons either, but we'll assume they have the same hypersonic cruise missiles they used before, so the launch point is a couple hundred miles from Soltanabad."
"I agree with your assumptions, Joe," Zevitin said, trying to disguise the surprise in his voice and stay calm and serious. "We can search for them where you suggest. But if we do find them...Joe, should I proceed? I think it's the only way to avoid a disaster. But it's got to be your call, Mr. President. Tell me what you'd like me to do."
Another pause, but this one shorter: "Yes, Leonid," Gardner said, obviously racked with great anger. "I hate to do it, but that bastard McLanahan has left me no choice."
"Yes, Joe, I understand and agree," Zevitin said. "What about the T-wave weapon? Will they use it again to attack our fighters?"
"You must assume they will, and launch your attack from maximum range," Gardner said. "I'm sorry, but I don't have any control over that, either."
"I know it's not your doing, my friend," Zevitin said as solemnly as he could muster through his glee. Hell, now the guy was giving him suggestions on how to successfully attack his own people! "We'll do everything possible to avert a disaster. I'll be in touch shortly with an update."
"Thank you so much, my friend."
"No, thank you for the responsible notification, my friend. I don't know if I can be in time, but I'll do everything I can to avoid an embarrassing situation from getting worse. Wish me luck. Goodbye." Zevitin hung up the phone...then resisted the impulse to take a little victory dance around the desk. He snatched up the phone again and asked to be connected immediately to Darzov. "Status, General?"
"We are moving as fast as we can," Darzov said. "We are prioritizing the main components first—the radar, laser chamber, and adaptive optics. The fuel tanks and power generators will have to wait."
"Do you have any fighters on patrol over the Caspian, General?"
"Of course, sir."
"Are you shadowing the American B-1 bombers?"
"I have an entire squadron of MiG-29s airborne to try to keep up with them," Darzov said. "The unmanned Vampires are much faster than a regular B-1 Lancer, so we've loaded a few of the fighters up with Molnija missiles adapted to work at reduced range with the MiG-29's fire control radar. They might be able to take down their hypersonic attack missiles if they can be fired—"
"I've just received permission from the President of the United States for you to shoot the bombers down," Zevitin said happily.
"The President of the United States told us to shoot down his own bombers?"
"He doesn't consider them his bombers—to him they're McLanahan's bombers now, and they might as well be invading Martians," Zevitin said. "Do it. Shoot them down...but after they launch their missiles."
"After?" Darzov asked incredulously. "Sir, if we cannot move our equipment out in time, or if they target the main Fanar components, we could lose billions of rubles of precious equipment!"
"Do the best you can, General," Zevitin said, "but let those missiles launch and hit the base. You do have the screening implements in place, as we discussed earlier?"
"Yes, sir, of course," Darzov replied. "But we also have—"
"If any part of Fanar gets hit, your first priority is to get it out of there while you continue to set the stage as planned," Zevitin went on breathlessly, "because minutes after the missiles hit, I'm going to tell the whole world about it. The world media will want to see for themselves, and it's important that they see it right away. Do you understand me, General?"
"Yes, sir," Darzov replied. "I will do as you ask. But I hope we are not sacrificing our most important assets for mere public relations purposes."
"You'll do as I tell you for whatever reason I devise, General, whether you understand it or not," Zevitin snapped. "Just make sure when the media descends on Soltanabad—which I am going to work very hard to see happen—they see nothing but senseless ruin and destruction, or I'll have your ass. Do I make myself clear?"
"Sir, we're picking up a locator beacon signal!" Master Sergeant Lukas shouted from her station in the command module of Armstrong Space Station. "It's from the passenger module."
"My God, they made it," Patrick said breathlessly. "Any data yet?"
"Nothing yet...yes, sir, yes, we're receiving location and environmental readouts!" Lukas said. "It's intact! Stabilizers have deployed and it is under computer guidance! Telemetry says the passenger module is still pressurized!"
"Good God, it's a miracle," Patrick said. "Moulain and Terranova must have ejected the module just before the Black Stallion was destroyed. Rebecca—"
"We're readying two more Vampires for launch to provide air cover for the recovery," Rebecca Furness said. "They'll be airborne in twenty minutes."
"Dave—"
"We're talking to Special Operations Command right now about launching a CSAR mission from Afghanistan, Muck," Dave Luger said. "As soon as we know where they might come down, they'll launch. We're hoping they'll land in western Afghanistan. A Pave Hawk is standing by at Herat Air Base. We're trying to get a couple Predators and Reapers retasked to fly over the area." The MQ-1 Predator and MQ-9 Reaper were unmanned reconnaissance aircraft, each configured to carry air-to-surface attack missiles; both were controlled via satellite from control stations in the United States.
"Sixty seconds to the launch point," Dave Luger reported. "Airspeed coming back to one point two Mach." He was by himself at the command console in the Batman, but he still lowered his voice as if not wanting anyone else to hear as he went on: "Muck, now would be a good time to turn them around."
"Continue," Patrick McLanahan responded.
He sounded every bit as resolute and confident as when he first made the decision to attack—that, at least, made him feel a little better. If Patrick showed the slightest hesitation in his decision, Dave vowed he would've turned the bombers around on his own authority to make sure the planes made it back to the refueling control point—as well as to save Patrick's career.
In seconds, it was going to be too late...
On the command-wide net he spoke, "Roger, Odin, copy, continue. Forty-five seconds. No threats, no surveillance radar. Airspeed steady at one point two Mach. Thirty seconds...twenty...ten, doors coming open on Headbanger Two-One...missile one away...doors coming open on Two-Two...missile two away, doors coming closed...missile one away from Two-Two...missile two away, doors coming closed, the flight is secure, heading westbound to the ARIP."
"How are the Vampires doing on fuel, Dave?" Patrick asked.
"We'll make it—barely," Luger responded. "If the hookups go smoothly, Two-One will be able to get on the boom, take on emergency fuel, cycle off, and Two-Two will start to take on fuel with ten minutes left to dry tanks."
"Good going, Headbanger," Patrick breathed with audible relief. No reply from Rebecca Furness—this was not over, not by a long shot, and he knew she was still angry about her decision being overruled.
"Thirty seconds to impact...SkySTREAK speed Mach ten point seven, all in the green...scramjet motor burnout, warhead coasting...flight controls active and responding, steering control good...twenty TG, datalink active." They all watched as the composite millimeter-wave radar and imaging infrared picture flared to life, revealing Russian transport planes and helicopters on the runway, several lines of men handing boxes and packages from various parts of the base to waiting trucks, several large unidentifiable buildings on trailers...
...and several large tents with clearly identifiable Red Cross and Red Crescent logos on the tops. "Jesus!" Dave Luger gasped. "They look like relief worker tents!"
"Target the large trailers and portable buildings!" Patrick shouted. "Stay away from those tents!"
"We got it, Odin," Rebecca said. She had commander's override authority and could take over targeting from the weapons officer, but she didn't need to—the weapons officer smoothly centered the aiming reticle over the four largest trailers. The SkySTREAK's millimeter-wave radar was able to look through the outer steel shell of each truck, and it verified that the trailers under the aiming reticle were indeed dense and not hollow or less densely packed, like a partially empty cargo trailer might be. Otherwise, the trailers all looked the same and were being attended to by equivalent-looking numbers of workers.
"Five seconds...targeting locked...breakapart charge initiated." The final image from the SkySTREAK missiles showed nearly direct hits on the center of each trailer...all except one, which had skittered off-target to land in a clear area somewhere beside the targeted trailer. The computer's estimate of the area of damage, approximately fifty feet in diameter, showed nothing except some soldiers carrying rifles and boxes and perhaps one lone individual standing nearby, probably a supervisor—it didn't hit any of the relief tents. "Looks like one missed, but it hit in a clearing beside the trailer."
"Good shooting, Headbanger," Patrick said. "Those trailers looked identical to the ones that attacked Stud One-One."
"They looked like a billion other trailers around the world—there's no way of knowing what we got, sir," Rebecca Furness said, the exasperation obvious in her voice. "We didn't see any radar arrays or anything that looked like laser fuel storage tanks or laser optics. We could've hit anything...or nothing."
"Our first priority is to set up a rescue and recovery operation for the passenger module and a search for any debris and remains of the Black Stallion and its crew," Patrick said, ignoring Furness's exasperated remarks. "I want a Battle Force team sent out immediately to Afghanistan, along with every support aircraft we have available. I want unmanned vehicles and NIRTSats set up for immediate deployment to search along all possible trajectories for survivors or debris. Recall every asset we have for the search. I want a progress update in one hour. Do you copy, Headbanger?"
"Stand by, Odin," Rebecca responded, concern thick in her voice. Patrick immediately turned his attention back to the mission status monitors...and immediately saw the new threat: a swarm of missiles barreling down on the Vampire bombers. "We did a post-turn long-range LADAR sweep and spotted them," she said. The LADAR, or laser radar, was a system of electronically agile laser emitters embedded throughout the fuselage of the Vampire bombers that instantaneously "drew" a high-resolution image of everything around the plane for a hundred miles, then compared the three-dimensional picture to a catalog of images for immediate identification. "Look at the speed of those things—they have to be traveling at greater than Mach seven!"
"Countermeasures!" Dave Luger shouted. "Knock them out of the sky!"
But it was soon clear that it was too late. Traveling at more than fourteen miles per second, the Russian missiles ate up the distance long before the Vampire bombers' microwave emitters could activate, lock on, and disrupt their guidance systems. Three of the four hypersonic missiles scored direct hits, quickly sending both bombers spiraling into the Caspian Sea.
"Damn it," Dave swore. "Looks like the Russians have a new toy for their MiGs. Well, I guess we won't have to worry if the bombers will make their tanker, will we, Rebecca?"
"We just lost one-fourth of our remaining B-1 bomber inventory, Dave," Rebecca Furness radioed from Battle Mountain Air Reserve Base. "It's not a laughing matter. We only have two Vampires at Batman now."
"Get 'em airborne to provide air cover for the CSAR guys out of Herat, Rebecca," Patrick ordered. "Use active LADAR to scan for intruders. If anyone comes within a hundred miles of your planes, fry 'em."
"With pleasure, Muck," Rebecca said. "I'm ready for a little payback. They'll be ready to taxi in about fifteen." But just a few minutes later she called back: "Odin, this is Headbanger, we have a problem. Security Forces are parked in front of the hangar and preventing the Vampire from taxiing. They're ordering us to shut down or they'll disable the plane."
Patrick was on the secure videoconference line in a heartbeat, but he was beaten to the punch by an incoming call: "General McLanahan, you are either deranged or suffering from some sort of mental breakdown," Secretary of Defense Miller Turner said. "This is an order directly from the commander-in-chief: stand down all your forces immediately. You are relieved of command. Do I make myself clear?"
"Sir, one of my Black Stallion spaceplanes has been shot down by a Russian anti-satellite laser based in eastern Iran," Patrick said. "We have indications that the passengers may have survived. I want air cover..."
"General, I'm sympathetic, but the President is pissed and he's not listening to any arguments," Turner said. "You hung up on him, for God's sake! Do you expect him to listen to you now?"
"Sir, the passenger module is intact, and it'll be on the ground in less than fifteen minutes," Patrick said.
"What? You mean, someone ejected from the spaceplane...?"
"The passenger module is jettisonable and is designed to act as a lifeboat for the space station crewmembers," Patrick explained. "It can withstand re-entry, fly itself to a landing spot, safely glide in for a landing, and save the crew. The module is intact, sir, and we're hoping the crew is safe. We're zeroing in on the possible landing zone right now, and as soon as we compute the exact landing spot we can deploy a rescue team there right away—that's the only advantage we'll have over the enemy. But it'll take at least ninety minutes for a rescue team and air cover to arrive in the recovery area. We have to launch right away."
"General, you have already disobeyed direct orders from the President," Turner said. "You're already on your way to prison, do you understand that? Don't compound it by arguing anymore. For the last time: Stand down. I'm directing General Backman to take command of all of your forces. I'm telling you—"
"And I'm telling you, sir," Patrick interrupted, "that most of the Middle East and central Asia will have seen the Black Stallion fall to Earth, and the Iranian Revolutionary Guards Corps, the al-Quds forces, all of the terrorists that have flooded into Iran since the military coup, and probably the Russians will be on their way to the crash site to retrieve whatever they can find. We must get every aircraft and combat search and rescue team possible airborne to find the survivors before the enemy does."
"Central Command will coordinate that, McLanahan, not you. You are ordered to stand down. Take no further actions whatsoever. You will do or say nothing to anyone. You are relieved of your command and will be placed under arrest as soon as you can be brought off that station."
For the second time that day, Patrick hung up on a civilian military leader. His next call was directly to General Kenneth Lepers, the four-star Army general in charge of U.S. Central Command, the major combat command overseeing all military operations in the Middle East and central Asia, to try to convince him to allow the bombers to take off.
"General McLanahan, your ass is in a really big sling right now," Lepers' deputy said. "The general has been directed not to speak with you, and this call will be reported to SECDEF. I advise you to straighten this thing out with SECDEF before the whole world cuts you off." And he hung up.
Patrick's next call was back to Rebecca Furness at Battle Mountain Air Reserve Base. "I was just going to call you, sir," Rebecca said. "I'm sorry about the Black Stallion. I wish we could've done more."
"Thanks, Rebecca. I'm sorry about your Vampires."
"Not your fault, sir." It was, she reminded herself: if he hadn't ordered to launch on this unauthorized mission, she'd still have her bombers. But the Vampires were unmanned, and the Black Stallion wasn't, so she didn't feel the need to rub salt on a wound. "We should have been scanning for bandits—I made the call to go in completely silent. I don't know how the Russians knew we were coming or when, but they are going to get it back in spades, I guarantee that."
"Are you still being stopped by the sky cops?"
"Affirmative. We've shut down as ordered and are holding our position inside the hangar."
Patrick thought for a moment; then: "Rebecca, I tried calling General Lepers at CENTCOM to get his permission to launch the Vampires, and he's not talking to me. I would guess if I tried to call STRATCOM I'd get the same response."
"Cannon's an okay guy," Rebecca commented. "The others think you're gunning for their jobs." Or nuts, she silently added.
"If we don't launch some air cover, our guys and maybe the CSAR troops will get chewed apart by the Pasdaran," Patrick said. "I'm going to clear those Security Forces away from the hangar. I want you ready to launch as soon as they're away."
"But you said Lepers won't talk to you, and you haven't spoken to CENTAF yet, so who's going to—?" Furness paused for a moment, then said simply, "That's crazy. Sir."
"The question is, Rebecca: Will you launch?"
The pause was very, very long; just when Patrick was going to repeat himself, or was wondering if Furness was dialing SECDEF's number on another line, she said, "Get 'em out of my ships' way, General, and I'll launch."
"Thank you, General." Patrick hung up the phone, then spoke, "Odin to Genesis."
"Go ahead, Muck," Dave Luger responded via their subcutaneous global transceiver.
"Move those security guys away from the bombers."
"They're moved, Muck. Out." Luger turned to his command radio: "Saber, this is Genesis."
BATMAN AIR BASE, REPUBLIC OF TURKEY
THAT SAME TIME
"Saber copies, go ahead, Genesis," Air Force First Lieutenant James "JD" Daniels, commander of the Battle Force ground operations team code-named "Saber," responded. Daniels had been sent to Batman Air Base to provide security for the EB-1C Vampire bombers, but also because the base was an isolated, well-equipped place to train with new CID pilots in real-world scenarios. As a technical sergeant the thirty-year-old tall, brown-eyed, brown-haired rancher's son from Arkansas was one of the first of the Battle Force commandos to check out as a Cybernetic Infantry Device pilot. After being injured from radiation sickness after fighting in Yakutsk Air Base in Russia following the American Holocaust, Daniels used his recovery time to get a bachelor's degree, then attended Officer Training School and earned his commission. Now he was the senior training officer and, except for Charlie Turlock herself, the resident expert in the CID weapon system.
"I have a task for you, Saber, but you might not like it," Dave Luger said. "Odin wants to launch the Vampire bombers."
"Yes, sir. We were ready to go a moment ago, but the Security Forces guys showed up at the hangar, and the planes shut themselves down. The base commander ordered us to assist and protect the Security Forces from any remote-controlled actions by you regarding the aircraft. We verified the orders. Sorry, sir. What is it I won't like?"
"One of our spaceplanes has been shot down in eastern Iran, and there are survivors. We need air cover for a rescue operation. The NCA still says no. We want to launch the Vampires anyway."
"Why won't the NCA approve the mission, sir?"
"I don't know why, Saber, but we believe the NCA is worried that our actions over Iran are inciting fear and intimidating everyone in the region."
"Sir, I received authenticated orders to stand down—us as well as the Vampires. The base commander ordered us to help secure you. You're asking me to violate those orders."
"I know, Saber. I can't order you to violate valid orders. But I'm telling you that the survivors of the spaceplane will be caught and captured or killed if we don't do something."
"Who shot down the spaceplane, sir?"
"We believe the Russians did, Saber."
"Yes, sir," Daniels said. That was enough for him. Daniels had spent a year in the hospital recovering from radiation poisoning which occurred when the Russian air force used tactical nuclear weapons to destroy their own air base, Yakutsk, that was being used by McLanahan and the Air Battle Force to hunt down and destroy Russian mobile intercontinental ballistic missiles that were being readied to launch a second nuclear attack on the United States. He endured severe dehydration, nausea for days on end, incredible pain, and eventually a liver transplant—but he survived, won the right to go back on active duty, requalified for field operations, rejoined the Battle Force, and took command of a CID team.
He had won, then lost, then won back all the things he ever wanted to do in his life, except one: get some payback for what the Russians did to him, his comrades, and to their own people in Yakutsk.
"You still there, Saber?"
"I'm sorry, sir, but I have my orders," Daniels said in a deep monotone voice, quite different from his normally energetic, upbeat tone. "If those planes were to move, I and my team would do everything in our power to protect the Security Forces from harm. Good night, sir."
"Genesis to Headbanger."
"Go ahead, Dave," Rebecca Furness replied.
"Get ready."
"Can't. My grounds crews say the sky cops are still blocking the hangar and taxiways."
"Get ready anyway."
"Did you order your guys to take out the sky cops?"
"No, ma'am, I did not. The base commander ordered the Battle Force team to assist and protect the Security Forces from unauthorized aircraft movement, and that's what they will do."
This is crazy, Rebecca told herself for the umpteenth time, utterly crazy. She turned to her operations officer, Brigadier General Daren Mace: "Daren, start 'em up and launch the Vampires immediately." She closed her eyes and saw herself standing in front of a court-martial, being sentenced to prison for the rest of the best years of her life; then, thinking about her fellow airmen on the ground in Iran being chased by Pasdaran and Muslim insurgents, opened her eyes and said, "Stop for nothing."
"Yes, ma'am," Mace said. He adjusted the mike on his headset and spoke: "Headbanger, start 'em up and launch without delay. Stop for nothing. Repeat, stop for nothing."
"Affirmative, Panther, the APUs are still on, both planes," the Air Force Security Forces detail team leader reported to NATO base headquarters. It was creepy enough that the APU started and stopped by itself, but ten times more so when the engines did the same. The crew chiefs and assistants for each plane were outside the hangars, per the base commander's orders.
"This is Panther. Put the fucking senior crew chief on," the base commander, a Turkish army colonel, ordered in very good English.
"Stand by, Panther." The SF officer handed his radio to the head crew chief, an Air Force technical sergeant. "It's the base commander, and he's steamed."
"Tech Sergeant Booker here, sir."
"I ordered those planes shut down, and I mean completely shut down—APUs also."
"Yes, sir, I know, but you ordered us not to hook the ground power units up either, and without power the command center at Battle Mountain can't talk to the planes, so I think that's why the APUs are—"
"Sergeant, I am giving you a direct order: I want those planes completely shut down, immediately, or I will have you arrested!" the base commander screamed. "I do not care if no one can talk to the planes—I do not want anyone to talk to the planes! Now turn off those APUs, and do it now!"
"Yes, sir," Booker said, and he handed the radio back to the SF officer.
"Detail One here, Panther."
"I just ordered that tech sergeant to completely shut down those planes, including the APUs—the power units in the tail," the base commander said. If they do not comply right away, place them all under arrest." Mallory swallowed hard, then made a gesture to his team members, a sign that said "Get ready for action." "Do you understand me, Detail One?"
"Yes, sir, I do."
"What is that tech sergeant doing right now?"
"He's going over to the other crew chiefs...he's gesturing to the planes...they're putting on gloves, like they're getting ready to go to work."
They were sure taking their sweet time, the Security Forces officer thought—the colonel's going to have a shit fit if they don't get their rears in gear. Sure enough, moments later the base commander called: "What are they doing, dammit? Are those planes shut down yet?"
"Negative, sir. They're just standing there talking right now, sir," Mallory replied. "One of them has a radio, and another one has a checklist. Maybe they're discussing shutting down the APUs from here."
"Well, go find out what is taking them so damned long."
"Roger, Panther. Stand by." He holstered his radio and started walking toward the crew chiefs. The three men and one woman crew chiefs saw him coming...and then, without a backward glance, they started walking toward their end unit hangar which served as the Air Battle Force's headquarters. "Hey, you jerkoffs, get back here and shut those power units off, colonel's orders." Just as he was about to yell at them again, to his complete surprise, they started running toward the hangar! "Where the hell are you going?" he shouted. He pulled his radio out of its holster. "Panther, the crew chiefs are running away toward their headquarters building!"
"They are what?" the base commander shouted. "Arrest those sons of bitches!"
"Roger that, sir. Break. Detail One to Control, signal Alert Red, Alpha Seven ramp area, repeat, Alert Red, Alpha—" Then Mallory heard a sound, much louder than the APUs, and realized moments later what it was. His hand shaking, he raised his radio again: "Control, Detail One, be advised, the articles in the Alpha Seven hangars are starting engines, repeat, starting engines! Requesting a Code Niner-Niner alert, full response, repeat, full—"
And then he saw them, emerging from the hangar the crew chiefs had just run toward, sprinting like linebackers from hell...and he nearly fell over backward in shock, surprise, and a mad scramble to get the hell out of there. He had seen them before, of course, but usually just walking around or being folded or unfolded near a truck or helicopter—never running right at him!
"Saber Four and Five responding!" one of the Cybernetic Infantry Device manned robots said in a loud computer-synthesized voice. "Say status!" Mallory was still on his hands and knees cowering in terror as the first robot ran right up to him. Both had him surrounded within moments. They were wearing huge backpacks, with what appeared to be grenade launchers deployed over their shoulders aimed right at him. "Team leader, I say again: say status!"
"I...uh...the bombers...they've started engines!" Mallory stammered. The muzzle of the grenade launcher was just a few feet from his nose. "Get that weapon out of my face!"
The robot ignored the order. "Have they taxied yet?" the robot blared at him. Mallory couldn't respond. "Five, report to Alpha-Seven-Two, I'll take Alpha-Seven-One. Protect the Security Force units." The second robot nodded and ran off, just like a football player breaking from a huddle except it was gone literally in the blink of an eye. "Are you hurt, Team Leader?"
"I...no," Mallory said. He scrambled to his feet. "Get in those hangars and find some way to disable those—"
At that instant they heard an impossibly loud roar of aircraft engines and a tremendous blast of jet exhaust from the open rear of both occupied shelters. "The bombers are taxiing!" the robot said. "Five, bombers are moving! Protect the Security Force units!"
"No! Stop the bombers! Find some way to—!" But the robot had sped off toward the hangar entrance. Well, he thought, the bombers weren't going anywhere, and if for some reason the Humvees didn't stop them, the robots certainly could. "Detail One units, the CID units are headed inside the hangars. Assist them if possible, but monitor and report if—"
At that instant, Mallory saw an object fly out of the near hangar. At first he thought it was a cloud of smoke or perhaps an explosion of some kind...and then seconds later realized it was the Humvee that had been stationed inside blocking the hangar! Moments later the robot ran out of the hangar clutching a Security Forces officer in each hand, carrying him out as easily as someone might carry a beach towel. Directly behind him, the B-1 bomber careened out of the hangar and sped up the throat toward the main taxiway.
"What in hell is going on?" Mallory shouted. "What happened? What are you...?" But the robot kept coming. It scooped up the Security Forces team leader with a bone-jarring tackle and ran him a hundred yards away in the blink of an eye, finally depositing the three stunned officers in a heap near the security fence surrounding the detachment area. The robot huddled over them as if shielding them from something. "What the hell are you doing? Get off me!"
"The bomber is transmitting its microwave weapon system," the robot said. "I had to get the Humvee out of the hangar before it exploded, and then I evacuated you. At close range the MPW can be lethal, and I had to get away or else it could have disabled my electronics too."
"What are you talking about?" Mallory struggled to get a better look. "The second bomber is moving too! They're taxiing for takeoff!" He fumbled for his radio, realizing he'd dropped it when the robot tackled him. "Call security control!" he told the robot. "Alert the base commander! Get units on the taxiways and runways before those things can get into takeoff position!"
"Roger," the robot responded. "I'll call it in, then see what I can do to stop them." And the robot stood up and was gone, running away with amazing speed, the muzzle of the grenade launcher swiveling back and forth searching for targets. It cleared the twelve-foot fence surrounding the detachment area—he just noticed that the gate across the throat was wide open—and was out of sight within seconds.
"What the fuck are those things doing? Who's in control of those things—ten-year-olds?" Mallory ran back to the first hangar and found his radio. "Control, Detail One, the bombers are taxiing out. There are two CID units in pursuit. They said the bombers were transmitting some kind of microwave weapon."
"Control, Knifepoint West, the bombers are crossing Taxiway Foxtrot on the way to Runway One-Niner," another Security Force unit radioed. "I'm parking my vehicle in the middle of Taxiway Alpha at the intersection of Hotel taxiway. I'm going to dismount. Those fuckers are coming this way awfully damned fast!" Mallory and the other Security Forces officers ran up the throat to the main taxiway to see what was going on...
...and just as they reached Taxiway Alpha they saw a Humvee fly into the air to the north, and the B-1 bombers roar past it! "Knifepoint West, Knifepoint West, do you copy?" Mallory radioed as he watched the nearly five-thousand-pound Humvee hit and tumble across the ground like a child's toy. "What happened? Say status!"
"Those robots threw my Humvee off the taxiway!" the officer radioed a few moments later. "They're not trying to stop them—they're helping them escape!"
"Those bastards!" Mallory swore. "I knew something screwy was going on! Control, Detail One, those robots are engaging our security units!"
"Detail One, this is Panther," the base commander cut in. "I do not care what you have to do, but stop those bombers from leaving the ground! Do you read me? Stop those bombers! Then place that entire Headbanger contingent under arrest! I want some butts, and I want them now!"
But as he listened, Mallory saw the first unmanned B-1 bomber leave the ground and streak across the night sky, trailing four long afterburner flames behind it, followed just a few short seconds later by the second. "Ho-lee shit," he cried aloud as the twin afterburner booms rolled over him. "What in hell is going on?"
It took almost a minute for the noise to subside enough so he could talk on the radio: "Control, Panther, Detail One, the bombers have launched, repeat, they've launched. All available patrol and response units, report to the Alpha Seven special detachment area with restraints and transport. Control, notify the base hospital and all command units that a special security enforcement operation has commenced." His ears were buzzing and his head felt as if it was going to explode from the tension and sheer disbelief over what had just happened. "Notify all responding units that there are two of those CID robot units that assisted the bombers to launch and are armed and dangerous. Do not approach the CID units, only report and observe. Do you copy?"
The two bombers were just bright dots in the night sky, and soon those telltales winked out as the afterburners were cut off. This was unbelievable, Mallory told himself over and over again, simply unbelievable. Those Saber guys had to be nuts or on drugs, he thought, wiping sweat from his forehead. The robot guys had to be crazy...or maybe the robots had been hijacked by terrorists? Maybe they weren't Air Force after all, but fucking Muslim terrorists, or maybe Kurdish terrorists, or maybe...?
And then he realized he wasn't thinking all this, but screaming it at the top of his lungs! His skin felt as if it was going to burst into flames, and his head felt ready to explode! What in God's name was happening? He turned...
...and then he saw the shape of one of the robots, about thirty yards away, slowly heading toward him. He raised his radio to his suddenly sweat-stained lips: "Control, Detail One, one of the CID units is heading toward me, and I am engaging," he said, wiping yet another rivulet of sweat away from his eyes. "Request backup, Alpha Seven and Taxiway Alpha, get backup out here now." He unholstered his sidearm, but he couldn't summon enough strength to lift it. The burning sensation increased, completely disrupting his vision and creating an intense headache, the pain finally forcing him to his knees. "Control...Control, how do you copy?"
"I'm sorry, Sergeant Mallory, but no one is here to take your call right now," he heard a strange voice say. "But don't worry. You and your friends will wake up in a nice cozy cell, and you won't have a care in the world." The robot advanced toward him menacingly, the muzzle of the grenade launcher aimed right between his eyes...but then, just before his vision completely shut down in a cloud of stars, he saw the robot wave "bye-bye" to him with his huge armored but incredibly lifelike fingers. "Nightie-night, Sergeant Mallory," he heard over the radio lying somewhere on the ground, and then everything went blank.
"Odin, Headbanger, Genesis, this is Saber, we have control of the base," Lieutenant Daniels reported a few minutes later. "Those new microwave emitters built into the CID units worked great out to thirty yards or so." The nonlethal microwave emitters broadcasted an intense feeling of heat, pain, disorientation, and eventually unconsciousness but did no actual injury to a human target. "The bombers are away and we're securing the perimeter. The base commander is pretty sore at us but he opened up his hidden liquor cabinet so he's not quite as verbal as before."
"Roger that," Patrick McLanahan responded from Armstrong Space Station. "Thank you, Saber."
"Our pleasure, sir," Daniels responded. "Maybe we can all share a cell in Leavenworth together."
"Or Supermax, if we're not so lucky," Rebecca added.
"We received a coded locator beacon and status data dump from the Black Stallion's passenger module," Luger said. "It's intact, its parachute and impact attenuation bags have deployed, and it's coming down in eastern Iran, about a hundred and twenty miles northwest of Herat, Afghanistan."
"Thank God."
"No indications if anyone inside made it yet, but the module is intact and still pressurized. We've got an Army Special Forces team in Herat gearing up for a rescue mission."
"The bombers will be in maximum SkySTREAK launch position in sixty minutes, and overhead in ninety—if they're not jumped by Russian fighters again," Rebecca Furness said. "We'll be on the lookout for them this time."
"That's probably the same amount of time it'll take the Special Forces team to chopper in—if they get permission to launch," Luger added.
"I'll speak to the commander myself," Patrick said. "I don't have much pull with the Army, but I'll see what I can do."
"Wait a minute, wait a minute—are you boys forgetting something?" Rebecca Furness interjected. "We just took over a Turkish-NATO military base by force and ignored direct orders from the commander-in-chief. You guys are acting as if that's no big deal. They are going to come after us, all of us—even the general, even though he's up on a space station—and they are going to haul us off to prison. What do you propose we do about this?"
"I propose we rescue our crewmembers on the ground in Iran, then hunt down any parts of that anti-spacecraft laser the Russians fired at us, General Furness," Patrick said immediately. "Anything else is background noise at this point."
"'Background noise'? Do you call the Turkish and U.S. governments—possibly our own military—coming after us just 'background noise'? We'll be lucky if they just send in an infantry battalion to drag us out of here. Do you intend on continuing to disregard orders and take down anyone who gets in your way, General? Are we going to war against our own people now?"
"Rebecca, I'm not ordering you to do anything—I'm asking," Patrick said. "We have crewmembers down in Iran, the Russians blasting away with a laser, and the President doing nothing about any of it except telling us to stand down. Now if you don't want to help, just say so, recall the Vampires, and call the Pentagon."
"And tell them what, Patrick—that you forced me to launch those planes? You're two hundred miles up on the space station, probably on the other side of the planet. I'm already committed, General. I'm screwed. My career is over."
"Rebecca, you did what you did because we have friends and fellow warriors on the ground in Iran, and we wanted to save and protect them if possible," Patrick said. "You did it because you had the forces standing by and ready to respond. If we'd followed orders, the survivors would be captured, tortured, then killed—you know it, and I know it. You acted. That's more than I can say for the Pentagon and our commander-in-chief. If we're going to lose our freedom, I'd rather it be because we tried to make sure our fellow airmen kept theirs."
Rebecca fell silent for a long moment, then shook her head ruefully. "I hate it when you're right, General," she said. "Maybe I can tell them that you threatened to blast me with Skybolt if I didn't do as you ordered."
"Maybe they'll laugh so hard they'll forget what we did."
"We need a plan, General," Rebecca said. "The Turks are going to send a force to retake Batman Air Base, and if they don't there's an entire U.S. airborne division in Germany that could be dropping on our heads within half a day. We've only got three CID units and four Tin Men at Batman, plus the security and maintenance troops. And we all know that Battle Mountain and probably Elliott will be next."
"We should move the Air Battle Force units to Dreamland," Patrick said. "We can hold that base a lot easier than Battle Mountain."
"Do you hear what you're saying, Patrick?" Rebecca asked incredulously. "You're conspiring to organize and direct U.S. military forces against the orders of the commander-in-chief, illegally marshal them under your own command without any authority, and directly oppose and engage with American military forces. That's sedition! That's treason! You won't go to prison, Patrick—you could be executed!"
"Thanks for the legal primer, Rebecca," Patrick said. "I'm hoping it won't come to this. After the survivors are rescued and the Russian anti-spacecraft laser is destroyed or at least discovered, all of this will be over. I understand if you don't want to do as I suggest, Rebecca. But if you want to take the Air Battle Force and assist, you can't stay at Battle Mountain. They could be rolling up outside to take you down as we speak."
Everyone on the secure video teleconference could see the tortured expression on Rebecca Furness's face. Out of all of them, she probably had the most to lose in this, and it was obvious she didn't want this. But just a moment later, she nodded. "All right. In for a dime, in for a dollar—in for twenty to life. Maybe the court-martial will take pity on me because I'm a woman. I'll get the planes moving right away, Dave. Make room for me."
"Yes, ma'am," Dave Luger responded from Elliott Air Force Base. Then: "What about the personnel and equipment at Batman Air Base, Muck? The Turks and our own guys could be waiting for them to return...if Turkey doesn't try to shoot them down when they cross back into Turkish airspace."
"I've got an idea for them, Dave," Patrick said. "It's going to be risky, but it's our only chance..."
PRIVATE RESIDENCE OF LEONID ZEVITIN, BOLTINO, RUSSIA
THAT SAME TIME
"Calm yourself, Excellency," Leonid Zevitin said. He was in his private study with Foreign Minister Alexandra Hedrov, making phone calls and sending secure e-mails to military and diplomatic units around the world alerting them to the events unfolding over Iran. The phone call from Iranian supreme leader Hassan Mohtaz happened much later than expected, but that was undoubtedly because it was probably very hazardous for anyone to wake the guy up with bad news.
"Calm myself? We are under attack—and it is because of you!" Mohtaz cried. "I allowed you to put your weapons on my soil because you said it would protect my country. It has done just the opposite! Four bombs have destroyed one of my Revolutionary Guards Corps bases, and now my air defense forces tell me that American bombers are flitting freely across our skies!"
"There are no bombers over Iran, Excellency—we have seen to that," Zevitin said. "As far as your base: remember that Russia paid to refurbish and disguise that base so we could use it temporarily, and we agreed that it would be turned over to you after we were done with it..."
"And now you are done with it because the Americans have destroyed it!" Mohtaz said. "Will you leave us a smoking hole in the ground now?"
"Calm yourself, Mr. President!"
"I want anti-aircraft weapons, and I want them now!" Mohtaz screamed. "You told me six units of the S-300 and another dozen Tor-M1 missile vehicles were waiting for pre-delivery checkout in Turkmenistan. How long ago was that, Zevitin? Eight, ten weeks? How long does it take to unpack a few missile launchers, turn them on, and see if all the pretty lights come on? When are you going to deliver on your promises?"
"They will be delivered, Mr. President, do not worry," Zevitin said. He didn't want to deliver missiles, especially the advanced S-300 strategic anti-aircraft and anti-ballistic missile system, until he was sure he could not get any more concessions from American President Joseph Gardner in exchange. Zevitin was perfectly willing to let Mohtaz rant and rave if he could get the Americans to agree not to put troops in Poland or the Czech Republic, or agree to veto any resolution in the United Nations that might allow Kosovo to break away from Serbia, in return. Those negotiations were in a critical stage, and he wasn't going to let Mohtaz screw them up.
"I want them now, Zevitin, or you can take all of your planes and tanks and radars back to Russia!" Mohtaz said. "I want the S-300 and Tor protecting Mashhad tomorrow. I want an impenetrable shield of missiles around that city when I return in triumph with my exiled government."
"That is impossible, Excellency. It takes time to test those advanced weapon systems properly before deployment. I will have Minister Ostenkov and chief of staff General Furzyenko brief your military advisers on—"
"No! No! No more briefings and wasted time!" Mohtaz shouted. "I want them deployed immediately or I will see to it that the entire world knows of your duplicity! What would your American friends say if they learn that you agreed to sell Iran anti-aircraft missiles, chemical weapons, and anti-personnel rockets?"
"You agreed not to share any information..."
"And you agreed to give me anti-aircraft missiles, Zevitin," Mohtaz interjected. "Break your promises further, and we are finished. Your infantry and tanks can rot in Turkmenistan for all I care." And at that the connection was broken.
TORBAT-E-JAM UNITED NATIONS REFUGEE CAMP, IRAN
A SHORT TIME LATER
"Easy now, lass, you're hurt. Don't move, eh?"
Captain Charlie Turlock opened her eyes...and immediately what little vision she had was shattered in a cloud of stars as the pain shot through her lower back, up through her spine, and into her brain. She gasped, the pain doubled, and she cried aloud. She felt a cool hand hold her forehead. "My God, my God...!"
"Believe it or not, lass, you shouting in pain is music to me ears," the man said, his thick Irish brogue slowly becoming clearer and soothing in a way, "because if you were'na cryin' out so, I'd believe your spine was broken. Where does it hurt, lass?"
"My back...my lower back," Charlie gasped. "It feels like...like my whole back is on fire."
"On fire...that's funny, lass," the man said. "I'm na surprised." Charlie looked at the man in confusion. She could see the stethoscope dangling around his neck now. He was very young, like an older teenager, with closely cut reddish-blond hair, bright green eyes, and an ever-present smile—but his eyes showed deep concern. The glare of a single overhead lightbulb hurt her eyes, but she was thankful that at least her eyes were working. "You might say you're an angel from heaven...or maybe a fallen angel?"
"I don't understand, Doctor...Doctor..."
"Miles. Miles McNulty," the man replied. "I'm na a doctor, but everyone out here believes I am, and that's good enough for all of us for now."
Charlie nodded. The pain was still there, but she was starting to get accustomed to it, and found that it even subsided a bit if she moved just so. "Where are we, Mr. McNulty?" she asked.
"Och, c'mon, lass, you're makin' me feel old callin' me by what they call me old man," Miles said. "Call me Miles, or Wooz if you like."
"Wooz?"
"Some of the docs gave me the nickname after I got here—I guess I'd get a little woozy seein' some of the shit that goes on around here: the blood, the putrid water, the injuries, the infant deaths, the starvation, the damned evil that someone can do to another human bein' in the name of God," Miles said, his young features momentarily turning hard and gray.
Charlie chuckled. "Sorry." She was pleased when his smile returned. "I'll call you Miles. I'm Charlie."
"Charlie? I know I've been here in the desert for a while, lass, but you na look like a 'Charlie' to me."
"Long story. I'll tell it to you sometime."
"Love to hear it, Charlie." He found a bottle in his jacket pocket and shook out some tablets. "Here. It's just over-the-counter NSAIDs—all the pain medication I dare give you until I do some more tests to find out if you're bleeding internally or if anything's broken."
A large armored hand reached out and completely surrounded the man's hand—Charlie couldn't turn her head, but she knew who it was. "I'll have a look at those first," he heard Chris Wohl's electronically synthesized voice say.
"Ah, it speaks," Miles said. He took his hand and the pills back. Wohl undid his helmet, exercising a kink out of his neck. "Pardon me for saying, bub, but ye looked better with the helmet on," he quipped, smiling broadly until he saw Wohl's warning glare. He put the tablets back into the bottle, shook it up, took one out, and popped it in his mouth. "I'm tryin' to help the lady, na hurt her." Wohl allowed him to give Charlie three tablets and a sip of water.
"How do you feel?" Wohl asked.
"Not bad if I don't...move," she said, gasping through a surge of pain. "I can't believe we made it." Wohl's warning glance reminded her not to talk any more about what they had just experienced. "How long have we been here?"
"Not long," Wohl responded. "About an hour."
"Where's Three?" Wohl motioned to Charlie's left. Charlie's mouth instantly turned dry. The pain forgotten, she followed the big Marine's glance beside her...and she saw the other Tin Man, Wayne Macomber, lying on another table beside her as if laid out on a funeral bier. "Is he dead?" she asked.
"No, but he's been unconscious awhile," Wohl said.
"I asked your comrade here if there's an on-off switch or latch or can opener to peel him open and check him out—I'm not even sure if it's a 'him' or a machine."
"We've got to get out of here as soon as possible," Wohl said.
"I think I'd like to give the lass a look, if you don't mind," Miles said to Wohl. "Ten minutes to look you over first, eh?"
"Five minutes."
"All right, all right." He turned to Charlie, smiling confidently. "I hate to do this while you're hurting, lass, but it'll help me isolate the injured areas. Ready?"
"I guess so."
"There's a game lass. I'm going to try not to move you too much myself, so try to move yourself along with me as much as you can—you're the best judge about how much is too much, yes? We'll start with the head and work our way down. Ready? Here we go." With surprising gentleness, McNulty examined her head, turning it ever so carefully, stooping down with a flashlight as low as he could go to look behind her head and neck without her having to turn her head as much.
"Well, I'm na seein' anything sticking out," Miles said after a few minutes. "You have a fun number of bruises and cuts, but so far nothing critical. I've seen much worse around here."
"Where are you from, Miles?"
"I'm from God's back porch: Westport, County Mayo." He didn't have to specify "Ireland." "And you?" Charlie turned her eyes away and down, and Wohl changed position—not very much, just enough for everyone to remember he was present and not let the conversation drift into unwanted territory. "Ah, that's okay, lass, I figured as much anyway. The only whites in these parts are relief workers and spies, and you're na dressed like a nurse."
"Where are we?"
"You're here at Torbat-e-Jam, the United Nations refugee camp, originally set up for the poor bastards fleein' the Taliban in Afghanistan, and now used by the other poor bastards fleein' the Muslim insurgents," Miles said. "I volunteered to help bring in a load of food and supplies about six months ago, but when the doctor's assistant went missing, I stayed. About a month ago, the doctor went missing—if the Taliban or al-Quds forces need a doctor, they don't send fer one, they take one—so I'm fillin' in until the next flight comes in. No tellin' when that will be, so I play the doc and help as best I can. I lose a few more than the doc did, but I'm startin' to get the hang of it, I think."
"Tobat-e-Jam?"
"Iran," Miles said. "Around here they still call it 'Iran'—the insurgency hasn't reached this far yet, so they don't call it 'Persia' yet, although the Revolutionary Guards Corps and al-Quds forces are gettin' pretty nervous, like the rebels are nippin' at their heels a wee bit. We're about sixty klicks from the border."
"Inside Iran?"
"Afraid so, lass," Miles said. "About two hundred kilometers from Mashhad, the capital of Khorasan province."
"God, this is the last place we want to be," Charlie moaned. She attempted to get up off the hard plywood board she was resting on and nearly passed out from a surging wave of pain that eclipsed anything else she had felt since awakening. "I'm not sure if I can make it yet," she told Wohl. "Where's my...briefcase?"
"Right here," Wohl said, without indicating where or what they were really talking about.
"You're in no shape to go anywhere, lass, and neither is your friend—as far as I can tell, at least," Miles said.
"I'll make it," Charlie said. "How far are we from the crash site?"
"About ten klicks," Miles replied. "What is that thing, anyway...Mercury's chariot? It's not exactly an airplane, is it—more like a tin can with balloons on it. It was badly burned but intact."
"How did you find us?"
"That wasn't a problem, lassie—we saw you streak across the sky and fall to Earth like a lightning bolt from Zeus himself!" Miles said, his eyes twinkling as the memory of seeing that sight came back. "Like the biggest meteor ever seen! You must have been trailing a tail of fire fifty kilometers long if it was an inch! It was a miracle to see three human beings still recognized as such in the wreckage, and even more amazing to find you still alive! We nearly shit our pants watchin' you blazin' down right toward us—thought the good Lord was going to end all of our sufferin' right then and there on the spot—but ya missed us. Findin' you alive was nothin' short of a miracle."
"Unfortunately that means that the Pasdaran probably saw us as well."
Miles nodded. "They di'na come around too often, but they're surely be sniffin' around out this way, for sure. The faster we get you folks out of here, the better for all of us. You should be well enough to travel after the painkiller kicks in. It won't be easy, but I think you can do it." He turned to the Tin Man lying beside her. "Now this gent, I'm still not so sure. Can you tell me how to...unlock him, unscrew him, unbolt him, whatever, so I can have a look and check him over?"
"We don't have time, Miles," Charlie said. "We'll carry him." Choking back the pain, she managed to sit up on her cot. "We'll be going now, Miles. I want to thank you for all you've done for us."
"I'll be sad to see you go, Charlie, but frankly I'd rather not have you around when the Pasdaran or al-Quds goons track you down here." He looked carefully at Wohl and the Tin Man suit. "I think I've read about these things lately, haven't I? The American anti-terrorist outfit." Charlie didn't respond. "Oh, I see—you could tell me, but then you'd have to kill me, right?" She laughed, causing a ripple of pain through her back, but she still welcomed the humor. "All right, no more questions, Charlie. I'll go out and see if the coast is clear. Good luck to you, lass."
"Thanks." She grimaced at the pain as she started to pull herself up, but the stuff McNulty gave her must've started working because the pain wasn't debilitating this time. After McNulty departed, Charlie lowered her voice and spoke, "Odin, Stud Four."
"We read you loud and clear, Four," Patrick McLanahan responded via the subcutaneous global transceiver system. Every member of the Air Battle Force had the communications and data transceiver system implanted into their bodies for the rest of their lives, ostensibly for situations like this but realistically to allow the government to monitor each member's whereabouts for life. "Thank God you're alive. We read Five is with you."
"Affirmative—he's alive but still unconscious," Charlie said. Wohl started to put his helmet on, preparing to move out. "I'm going to mount up and we'll—"
Suddenly McNulty ran back into the tent, completely out of breath. "Soldiers, just outside the camp," he said frantically. "Hundreds of them."
"Odin, do we have a ride yet?" Charlie radioed.
"Stud, this is Genesis," Dave Luger cut in. "We have a CSAR team on the way from Herat, ETE ninety minutes. We're launching cover aircraft from Batman Air Base in Turkey, but they'll take about the same amount of time. What's your situation?"
"Getting tense," Charlie said. "We'll give you a call when we're safe. Stud Four, out." Charlie went over to the large box lying on the dirt floor. "Any backpacks or rifles, Five?"
"Negative," Wohl replied. "Sorry."
"That's okay—you had your hands full," Charlie said. "Let's get moving."
Miles motioned to the large box that Wohl had been carrying with him when he entered the camp. "Are those your weapons? Now would be a good time to get them out, lass."
"Not exactly," Charlie said. "CID One, deploy."
As Miles watched in amazement, the box began to move, quickly shifting size and shape like a magician's wand changing into a bouquet of flowers. In seconds, the large but ordinary-looking metal box had transformed into a ten-foot-tall robot, almost bursting out the top of the tent, with smooth black "skin," a bullet-shaped head with no discernible eyes or ears, and large, fully articulating arms, legs, and fingers.
"CID One, pilot up," Charlie spoke. The robot assumed a leaning-forward stance as if on a sprinter's starting block, but with one leg and both arms extended backward. Grimacing from the pain, Charlie stepped around the robot and climbed up the extended leg, using the arms as handrails. She entered a code into a tiny keypad somewhere behind the robot's head, a hatch popped open on its back, and she slipped herself inside. The hatch closed...
...and moments later, to the Irishman's amazement, the robot came to life and stood, resembling a regular person in everything but its appearance—its movements were so smooth, fluid, and lifelike that Miles immediately found himself forgetting it was a machine!
Charlie scooped up the still-unconscious Wayne Macomber. "Now is a very bad time to be out of it, Whack," she said. She activated the Cybernetic Infantry Device's millimeter-wave radar and scanned the area outside the tent. "Looks like they're trying to surround us," she said. "The south side looks like our best escape route—just one truck set up down that way."
"How about a little diversion to the north and west?" Wohl asked, studying the radar image data being transmitted to him from Charlie's CID unit. "Looks like a machine-gun squad getting set up on the north side. I can use one of those."
"Sounds good." She reached a fist out, and he punched it in return with his own. "As a hunky Australian actor said in a movie once: 'Unleash hell.'"
"On the way. Better give him some cover." Wohl sprinted out the front of the tent. Charlie knocked Miles to the ground and covered him just as a hail of automatic gunfire shredded the tent apart.
"Hop on, Miles," Charlie's electronically synthesized voice said. Still bent over, she shifted the inert form in her arms aside, far enough to form a space between her body and the Tin Man. He hesitated, still dumbfounded by what he had just seen. "You can't stay here. The Revolutionary Guards Corps will think you're one of us."
"Can ye carry us both?"
"I can carry twenty of you, Miles. Let's go." He lay across her arms, and she rolled Macomber back on top of him and tightened her grip, sandwiching him in securely. "Hang on."
But when she got up, there was obviously something wrong—Miles felt a high-frequency vibration within the machine, and Charlie's gait was unsteady. "What's wrong?" he shouted.
"The CID unit is damaged," Charlie said. "Must've been from the crash."
"I copy," Wohl radioed. Charlie could see his position in her electronic data visor—he was moving rapidly through the Iranian Revolutionary Guards Corps' positions, stopping briefly at each concentration of troops. "Head out the best you can. I'll be beside you in a moment."
The next few minutes were sheer torture. Wohl had drawn some of their fire away briefly, but it returned full force just moments after Charlie burst from the tent, seemingly all aimed at them. The sounds were deafening. They were consumed with clouds of smoke, occasional blasts of fire, and continuous gunfire. McNulty screamed when a round hit his left leg, and screamed again when a crushing explosion knocked Charlie to the ground. They were up again within moments, but now the smooth running rhythm was replaced by an awkward limping shuffle, like an automobile with a flat tire and bent rim.
Wohl ran beside Charlie, a Chinese Type 67 machine gun in his right hand, a metal can of ammunition in his left. "Can you travel, Captain?"
"Not for long."
"What the hell is going on?" they heard.
"Whack!" Thankfully, Macomber was awake, although he sounded sluggish and doped-up. "Are you okay?"
"My head feels like it's been cracked open," Macomber said thickly. Charlie suspected a concussion. "Am I alive?"
"So far—hopefully it'll stay that way," Charlie said. "Can you walk?"
"Do I still have legs? I can't feel anything down there."
"Stay put and try not to move—you'll squish the other passenger."
"Other passenger?"
Charlie tried to run, but things were definitely getting worse. A rocket-propelled grenade exploded on her back, sending them flying again. "Power is down to forty percent already," Charlie said as Wohl helped them up, "my primary hydraulic system is out, and I can't move my right leg."
"Can you keep moving?"
"Yes, I think so," Charlie said. Using her right leg as a crutch, she limped along, with Wohl laying down suppression fire with his machine gun until he ran out of ammunition. He half supported, half carried Charlie, and they were able to move faster up a low ridgeline. They could easily see their pursuers below them, advancing slowly, with more and more units joining the pursuit.
Charlie set Macomber and McNulty down, then dismounted from the CID unit. "It's getting ready to shut down," she said. "It's done. There's just enough power left to start erasing the firmware. Once we move away, it'll automatically self-destruct."
"It looks like they're not sure where we are," Wohl said, scanning the desert below them with night-vision optics. He zoomed in on a few of the details. "Let's see...infantry...infantry...ah, got one, another machine-gun squad. I'll be right back." He raced off into the darkness.
Macomber struggled to his hands and knees. "Okay, I'm starting to tell up from down," he said. "Who's our guest?"
"Miles McNulty, a UN relief worker," Charlie replied, filling in the details.
A few minutes later, Wohl ran back with an even larger weapon than the first, a Russian DshK heavy machine gun with a huge drum magazine on top, along with a wooden box of more magazines. "Looks like they brought some anti-aircraft weapons with them—they were obviously expecting company. How are you doing, Major?"
"Peachy, Sergeant Major," Macomber replied. He looked at McNulty. Charlie was busy tying a scrap of cloth torn from her uniform around his leg. "The passenger is hurt. Where's the cavalry?"
"At least sixty mike out."
"Where are we headed?"
"East toward the Afghanistan border," Charlie said. "About thirty miles away. Hilly and pretty open. No towns or villages for fifty miles."
"How are you doing on power, Sergeant Major?" Macomber asked.
"Down to thirty percent."
"Here—I can't use it yet." He unclipped one of his circular batteries from his belt and swapped it for one of Wohl's more depleted ones. "Can we use the CID unit to charge our batteries?"
"Not when it's in shutdown mode, Whack," Charlie said.
"Can't we tap into a power or telephone pole?" Macomber asked. Charlie looked at him with astonishment. "Hey, I have been studying these things—I may not like them, but I do read the manuals. We're not going to follow the highway, but if we spot a breaker box or control junction, I think I can rig up a jumper. Let's get—"
"I hear helicopters," Wohl said. He used his night-vision and enhanced hearing systems to sweep the skies, pinpointing the approaching aircraft's position. "Two light scout helicopters, about three miles away," he said, raising the DshK machine gun.
"Let's spread out," Macomber said. But he soon found out that was all but impossible: Charlie was still in pain from her injuries, and McNulty was hurt badly and going into shock, so he had to carry both of them even though he still wasn't a hundred percent himself, so it was slow-going. Wohl moved about ten yards away from them, close enough to support them if they came under attack but not close enough that one explosive round fired from a helicopter could take them all out at once.
They had run up the ridge just a few hundred yards when Wohl shouted, "Take cover!" Macomber found the largest piece of rock nearby and threw his charges and then himself behind it, placing himself between the helicopters and the others to shield them the best he could with his armored body. The Tin Man armor system featured an electronically actuated material that stayed flexible but instantly hardened when struck into a protective shield a hundred times stronger than plate steel.
Macomber could hear the oncoming helicopters through his own enhanced hearing system, but his eyes couldn't focus on his electronic displays. "I can't see them, Wohl."
"Stay down." A moment later he opened fire with the DshK machine gun, the muzzle flash of the big 12.7-millimeter cannon illuminating a ten-yard-diameter area around him. They heard a loud metallic screech as several rounds pierced the first helicopter's turbine engine and seized it solid, then an explosion as the engine blew itself apart. Seconds later they heard more explosions as the second scout helicopter opened fire on Wohl's position. He managed to jump out of the way just in time to avoid the full force of the Iranian 40-millimeter rocket attack.
Wohl opened fire on the second helicopter, but the fire soon cut off. "Jammed...shit, a round stuck in the chamber...won't clear." He was surprised the gun had fired as many rounds as it did—it looked as if it was fifty years old and hadn't been cleaned in half that number of years. He discarded the weapon and scanned the area for more nearby Pasdaran units so he could grab another machine gun, but the three remaining units were hanging back, blindly peppering the ridgeline with occasional rifle and mortar fire and content to let the scout helicopter do some fighting for them.
"The infantry units are hanging back, and there's still one helicopter overhead," Wohl reported. "I'm down to throwing rocks." He wasn't kidding—the microhydraulically actuated exoskeleton on the Tin Man combat system gave him enough power to hurl a five-pound rock almost two hundred yards with enough force to do some damage, which could put him within range of that scout helicopter if he could dash toward it, jump, and throw with perfect timing. He found a softball-sized rock and prepared to do just that...
...but then his sensors picked up another helicopter, and this time it wasn't a little scout. He'd recognize that silhouette anywhere: "We've got more trouble, ma'am," Wohl said. "Looks like a Mi-24 Hind inbound." The Russian-built Mi-24, NATO code name "Hind," was a large attack helicopter which could also carry up to eight fully outfitted soldiers inside. It carried a formidable array of weapons...
...the first of which opened fire seconds later, from over three miles away. Wohl immediately dashed away from the rest of his team, then stopped to make sure the anti-tank guided missile was still tracking him. It was, and he realized that the helicopter itself was following him too, which meant that the helicopter crew had to keep him in sight to keep the missile on him. Good. It had to be an older guided missile, probably an AT-6 line-of-sight radio-controlled missile.
Wohl waited another heartbeat, then dashed toward the nearest group of Pasdaran ground pursuers at top speed. He could no longer see the missile, but he remembered that an AT-6's flight time was somewhere around ten seconds when fired from maximum range. That meant he had just seconds to make it. This Pasdaran unit was an armored vehicle with a heavy machine gun on top, which opened fire as he closed in. A few shells hit, but not enough to slow him down. Now he was between the armored vehicle and the helicopter—certainly, Wohl thought, the Hind's gunner had to turn the missile away. His mental stopwatch ran to zero...
...just as the AT-6 Spiral anti-tank missile slammed into the Pasdaran armored vehicle, setting it afire in a spectacular fireball. Wohl was thrown skyward by the concussion. The damned Pasdaran gunner got so target-fixated that he lined up and hit his own guys!
Wohl rolled unsteadily to his feet, alive and mostly unhurt except his eyes and throat were clogged with oily smoke. The entire left side of his helmet, along with most of his sensors and communications, had been damaged in the blast. He had no choice but to take the helmet off. The blast had also ruined his hearing, and the acrid smoke burned his eyes and throat. He was a sitting duck. His first order of business was to get away from the burning vehicles behind him, which could be highlighting him...
...but before he could move, a line of automatic gunfire stitched the ground in front of him, and the big Mi-24 Hind attack helicopter zoomed before him and stopped, the chin-mounted 30-millimeter cannon trained directly on him. His armor would protect his body, but that would be of no use to him without a head. Wohl had no idea if they would accept a surrender, but if they were distracted long enough it might provide the others a chance to escape, so he raised his hands. The Mi-24 started its descent to touchdown, and he could see the clamshell crew doors open on either side, with soldiers ready to dismount as soon as the big chopper set...
...and at that instant there was a flash of fire on the right side of the attack chopper, followed by a large plume of smoke, more fire, an explosion, and a scream of metal, and then the big chopper spun to the left and hit the ground. Wohl dashed away just as the helicopter began to disintegrate in several more tremendous explosions. He was about to head back toward the others when he saw several vehicles, including an armored personnel carrier, approach. The lead vehicle, a pickup truck with a machine gunner in back, was flying a flag, but he couldn't make it out yet. He thought about running away from where he last left Turlock, Macomber, and the Irishman...until he saw the vehicles veer left away from him and toward the hiding place.
Wohl took off at top speed toward the vehicle at the tail end of the six-vehicle convoy, which had a machine gunner covering the rear of the formation. The other vehicles wouldn't fire toward their own vehicles, and hopefully he could reach the machine gunner, disable him, and take the gun before he could get a shot off. Just a hundred yards to go...
...and then he saw Turlock coming out of her hiding place, with her arms up. Was she surrendering? It might be good timing after all—if they were concentrating on them he had a better chance of reaching the last pickup truck and...
...but then as he got closer Wohl realized that Turlock wasn't raising her hands in surrender, but waving to him, motioning him back! Why was she doing this? Now she was pointing at the lead vehicle, the one with the flag...
...and Wohl finally realized what she was trying to tell him. The flag the vehicle was carrying had the green, white, and red stripes of the Islamic Republic of Iran on it, but the center symbol wasn't the "red tulip" stylized word "Allah," but the profile of a lion carrying a sword with the rising sun behind it—the flag representing the pre-revolutionary era and the opposition to the Islamists.
Chris trotted over to Turlock and Macomber, carefully watching to be sure none of the gunners pointed their weapons at him. "Not answering your phone, Sergeant Major?" Turlock asked, pointing to her ear, indicating his subcutaneous transceiver system.
"Got my bell rung back there," Wohl said. He nodded toward the newcomers. "Who are these guys?"
"These are Buzhazi's men," Charlie said. "General McLanahan actually called Buzhazi and asked for help."
"They came right on time. Good thing they brought Stinger missiles with them."
"They didn't shoot down the Hind, Sergeant Major." Charlie pointed to the sky, and they saw the contrails of a very large aircraft high overhead. "Compliments of the general. They'll be on station for another two hours."
"Outstanding. That should get us enough time to get across the border."
"The general suggests we head back toward Tehran with these guys," Charlie said. "They're bringing in a helicopter to pick us up, and the Vampires will cover for us."
"I don't think that's such a hot idea, ma'am."
"I'll explain." She did...and Wohl couldn't believe what he had just heard.
## CHAPTER EIGHT
You don't hold your own in the world by standing on guard, but by attacking and getting well hammered yourself.
—GEORGE BERNARD SHAW
CAPITOL HILL, WASHINGTON, D.C.
A SHORT TIME LATER
"Frankly, Brit, I don't care what the Russians say," Senate majority leader Stacy Anne Barbeau said. She was in the second-floor area of the Senate normally used by reporters for "staking out" senators for comments on their way to the floor or between committee meetings. "They have been claiming all sorts of things for many months and none of them have been proven. Although I believe Leonid Zevitin to be a capable and forthright leader, the statements made by his foreign minister Alexandra Hedrov seem more shrill and bombastic every time we see her in the news. President Zevitin is certainly not like that at all, which naturally leads me to the obvious question: Who is telling the truth out there at the Kremlin these days, and who is lying, and for what purpose?"
"But tomorrow there is a key vote in the Senate about funding for the U.S. military," the reporter pressed, "and in the midst of all this wrangling about where to spend the money in the military, President Zevitin's Cabinet members seem to be taking great pleasure in stirring up anxiety about another future confrontation. Are the two activities related, and if so to what end?"
"I'm sure I don't know what is in the mind of a Russian, even one as Westernized, worldly, and charming as Leonid Zevitin," Barbeau said. "I would think they would want to avoid rattling sabers at a time where we in the Congress are trying to determine the proper direction for the world's greatest military force."
"But this is more than just saber-rattling, Senator," the reporter went on. "There is definitely something stirring out there, Senator, and I'm not just talking about the turmoil in Iran, but with American military activities, isn't there? To put it plainly, ma'am: We can't seem to get out of our own way. The civil war in Iran is threatening to blow the entire Middle East into an inferno, and yet we're not doing much of anything except sending unmanned reconnaissance aircraft over the region; oil prices are skyrocketing; the economy is sinking like a rock; Russia accuses us daily of killing civilians, bombing a civil relief base in Iran, and causing unrest and chaos around the world, especially with the Armstrong Space Station and our spaceplanes; the space program seems robust and substantial one day, then completely ineffectual the next. We even have a famous and well-loved American three-star general, the hero of the American Holocaust, in essence stranded in space because no one can tell us if he's well enough to be brought back home. My question is, madam: What in the world is happening, what has Congress been told by the White House and the Pentagon, and what are you going to do about it?"
Barbeau gave him her most appealing mind-blowing smile, again defining the phrase "making love to the camera" to millions of viewers as she replied: "Why, sir, what a dreadful picture of doom and gloom you are painting here this morning! Let me assure you, and everyone in your audience around the world, that the Congress of the United States is working very closely with the President and his department officials not only to deal with current and future crises as they rear their ugly heads, but to chart a course for America's armed forces that is second to none, forward-looking, adaptable, scalable, and affordable. It has been less than five years since the American Holocaust, and three different governments have had to deal with the world as it has become since those awful attacks on our soil. We are making progress, but it will take time."
"So tell us how you envision the debate will develop, Senator. What's on the table?"
"The most important question for us right now is simply this: What is the best force to take the place of the land-based long-range strategic bombers and intercontinental ballistic missiles that were destroyed in the Holocaust?" Barbeau replied, still radiant even while wearing a stern, concerned, determined expression. "President Thorn favored land- and sea-based tactical air forces, both manned and unmanned, along with ballistic missile defense systems. President Martindale favored the same but, as advocated by his special adviser General Patrick McLanahan, also sought to 'skip a generation,' as he said, and develop a fleet of spaceplanes that could strike any target anywhere around the world with amazing speed, launch satellites into orbit whenever needed, and fly troops and equipment anywhere around the planet within hours.
"As the former Secretary of Defense, Joseph Gardner supported those ideas and encouraged development of Armstrong Space Station, the entire constellation of space-based assets, and the Black Stallion spaceplane," Barbeau went on. "The space program has taken some amazing strides and has greatly benefited the entire world—the global Internet access provided by our space program has without question truly changed all of our lives and brought our world together—but it has also suffered some serious setbacks. As President, Joseph Gardner has wisely recognized that perhaps the space-based defense force visualized by Patrick McLanahan wasn't mature enough yet to serve America."
"So where does this leave us, Senator?" the host asked.
"President Gardner has met with the leadership and proposed a more reliable, familiar, proven mix of weapon systems," Barbeau said. "He wants to take the best concepts proposed by previous administrations and combine them in a comprehensive program to quickly stand up a credible force to meet the country's needs."
"And which concepts are those, Senator?"
"I can't give you any specifics, Brit, or I'll have a lot of very angry gentlemen nipping at my heels in short order," Barbeau said sweetly. "But in a nutshell, we have the individual services do what the services do best, what has served the nation and the world so well for the past three generations but also recognizes changes in technology and our vision for the future: fully fund and support an expanded and strengthened Army and Marine Corps as the dominant land and special operations forces; fully support the Navy as the dominant sea and air force; and the Air Force as the dominant global support and space defense force."
"The Air Force wouldn't be the dominant air force in the U.S. arsenal? That doesn't seem right."
"Details have yet to be worked out, and of course I'm sure we will adjust and rearrange things as necessary to ensure the absolute best force we can build," Barbeau began, "but it seems to President Gardner and we in the congressional leadership that there is a wasteful and costly overlap between the Air Force and Navy regarding tactical air forces. It all comes down to the basic notion, Brit, that Navy planes can do everything Air Force planes can do, but Air Force planes cannot do everything Navy planes can do—namely, take off and land on an aircraft carrier, which as everyone readily recognizes is the undisputed definition of power projection in the world today."
"And the President as we all know is a big supporter of the Navy, being the former Navy secretary."
"It's a duplication of forces, plain and simple, and now is the time to address this if we want to have a robust, mature, twenty-first century fighting force," Barbeau said. "We're trying to think ahead. The Air Force is the proven expert in long-range strategic attack and rapid resupply, and the Navy has no such equivalent capability—it makes sense to give that mission to the Air Force and let the Navy have the mission of training and equipping tactical fighters for theater commanders around the world."
"Won't your constituents in Louisiana object to this plan, Senator?"
"I represent the finest, most patriotic, and most pro-military folks in the country, Brit: the good people of Barksdale Air Force Base near Bossier City, Louisiana—Bomber Town, USA," Barbeau said. "But even the staunchest bomber supporters, like me, have seen the shift coming for years: the shift from World War Two–era land-based bombers to the importance of global reach, rapid mobility, unmanned aircraft, space technology, and most importantly, information warfare. The Air Force is and will remain the leader in these areas. We've seen this coming for years, and President Gardner and I think it's time to design our twenty-first-century forces around this new reality."
"But the battles are just beginning, aren't they, Senator?"
"With President Gardner's strong leadership and his steadfast pledge to work closely with Congress, I think the battles will be kept to the barest minimum. Together, we'll prevail. The alternative is too awful to consider."
"So does this mean we'll see the end of the Black Stallion spaceplanes and military space stations watching over us 24/7?"
"The Black Stallion is a remarkable technological advancement, to be sure, but as we've seen with a man like General McLanahan, it has its risks and dangers," Barbeau said, a serious look of concern briefly shadowing her features. "My heart sank when I learned of General McLanahan's illness, and we are doing everything we can to bring him safely home. But my concern is this, Brit: Patrick...General McLanahan...is a powerful man. You know the stories as well as I, Brit..."
"The ones about McLanahan being challenged by visiting heads of state and generals to rip their respective capital's phone books in half?" the reporter filled in with a chuckle. "I thought that was a White House Press Corps rumor."
"It's not a rumor, I assure you!" Barbeau exclaimed. "I've seen it with my own eyes—Patrick can rip a D.C. phone book in half as easily as you or I could rip a page out of your little notebook there. And yet he was still brought down by something difficult to detect, diagnose, or treat, something so debilitating that it could put the lives of every space crewman we have in jeopardy. There is great concern that the injury has affected more than just his heart."
The reporter's mouth opened in surprise. "I haven't heard anything about that, Senator. Would you care to elaborate? What exactly do you mean?"
"It's all just speculation and nonsense, I'm sure," Barbeau said dismissively, acting as if she'd said something completely unintended but riveting the attention of every viewer by looking directly into the camera for a brief moment. "But we do need to fully understand what happened to him. We owe it to him because he is truly a national treasure, a hero in every sense of the word.
"But the fundamental question remains: Can we afford to put our nation's military future on hold while we study this awful catastrophe?" Barbeau asked resolutely, first looking at the reporter and then directly at the camera, right into the hearts of the viewers. "As responsible caretakers of our armed forces, sworn to build the best possible force to protect and defend our homeland and way of life, the answer is simple and obvious: the space defense force is not ready, and so we must turn to proven systems that we know will work. That's our job here today, and with the cooperation of the President and the House, we're going to get it done. The American people expect no less from us."
Stacy Anne Barbeau fielded more questions from the gaggle of reporters, until finally the officials of the Senate Press Gallery and Barbeau's aide shooed them away and let her go. On the way to a late-night meeting in a committee conference room, she took a call on her cellular phone: "I thought you laid on the praise for McLanahan a little too thick, Stacy Anne," President Joe Gardner said. "His ass will be grass here shortly."
"All the more reason to sing his praises, Mr. President," Barbeau said, greeting supporters and colleagues as she walked and talked. "I advise you to do similarly, Mr. President: Let your Secretary of Defense, the pundits, the Russians, and the anti-military media trash him, not us."
"You won't be saying that when you hear what just happened, Senator."
Barbeau's mouth instantly turned dry. "What's happened, Mr. President?" she asked, turning a puzzled expression to her aide, Colleen Morna. As they reached the conference room, Morna immediately shooed everyone else out so Barbeau could talk in private.
"McLanahan lost it, and I mean completely," Gardner said. She detected a slight hint of triumph in his voice, like he'd finally gotten something that Barbeau didn't have and expected some quid pro quo for sharing it with her. "His people took over a Turkish air base, captured the base commander and most of the personnel with their manned robots, then launched another air mission over Iran."
Barbeau froze, and her mouth dropped open in complete shock before she exclaimed, "What!" Her expression was so alarming that her aide Colleen Morna thought she was having a heart attack. "I...I don't believe it..."
"What do you say about your knight in shining armor now, Stacy?" the President asked. "But you haven't heard the best part. When the brass sent in some security units from Incirlik Air Base to arrest McLanahan's people, they were gone. The planes and most of their stuff are gone. We have no idea where they are."
"They...they must be on their way back to the States, Mr. President..."
"Not that anyone is aware, Stacy," Gardner said. "McLanahan has stolen about four experimental attack planes and moved them somewhere. We hope they're on their way back to Dreamland, their main base in south-central Nevada north of Vegas. If they are, McLanahan can be charged with conspiracy and sedition against the U.S. government. How about them apples? How's your hero looking now?"
"I...I just cannot believe it, Mr. President," Barbeau breathed. Shit, after what she just said to the media, all the nice things about McLanahan...God, this could ruin her! "We need to meet and discuss this right away, Mr. President. We need to come up with a united stance, both for Congress and for the press."
"We're getting all the information we can, and we'll prepare a briefing for the leadership that we'll give first thing in the morning," the President said. "McLanahan is going down, I promise you, and so is his entire command. He won't be so popular after people find out what he's done. We won't have to look like we're destroying a national hero anymore—he's taking himself down."
"We need all the facts first, Mr. President," Barbeau said, her mind racing, trying to make sense of this explosive news. "Why exactly did he launch those bombers? McLanahan doesn't do something for no reason."
"It doesn't matter one bit to me, Stacy," Gardner said. "He's disobeyed orders, ignored my authority, and now he's launched military strike missions overseas, stolen military property, moved and directed military forces without authority, and opposed our own and allied military forces. For all we know, he could be engineering a military coup against the government or even preparing a military strike against Washington. He has to be stopped!"
"Whatever our response is, Mr. President, I suggest we find out all we can first, carefully discuss it, formulate a plan, and carry it out together," Barbeau repeated. "I know your military forces are an executive responsibility, but it would be easier to do what we have to do if we are together on this beforehand."
"Agreed," the President said. "We should meet and discuss strategy, Senator, after we present our findings. Tonight. Private meeting in the Oval Office."
Barbeau rolled her eyes in exasperation. The man's greatest general just stole some bombers and captured a Turkish air base, and all the man could think about was canoodling with the Senate majority leader. But she had been suddenly thrust onto the defensive, especially after her statements to the press, and the President had the upper hand. If she wanted any chance of retaining her bargaining position for the space force funds that were certainly going to be freed up soon, she had to play his game...for now. "The Senate has a full schedule, Mr. President, but I'm sure I can...squeeze you in," Barbeau said, flipping the phone closed.
"What in the world happened?" her aide, Colleen Morna, asked. "You look as pale as a ghost."
"Possibly the worst thing imaginable...or it could be the best," she said. "Set up a meeting with the President after the last agenda conference tonight."
"Tonight? It's already past five, and you have that meeting with that law firm that represents those defense and technology industry lobbies at seven. That was scheduled to last until nine. What's the President want? What's going on?"
"We all know what the President has got on his mind. Set it up."
"It'll be another late night, and with the Armed Services Committee hearings starting tomorrow, you'll be running ragged. What's so important that the President wants to meet so late? He still wants to take McLanahan to the woodshed?"
"Not just to the woodshed—he wants to bury the whole damned ax in his chest," Barbeau said. She filled her in quickly, and soon Morna's expression was even more flabbergasted than her own. "I don't know precisely what happened, but I think I know McLanahan: he's the definition of a goody two-shoes. If he hit something in Iran, he probably had spot-on intelligence that something bad was going down, and he didn't get the green light to take it out, so he did the deed himself. Gardner should be encouraging him, not taking him on. But the President wants to show he's still in charge and in control, so he's going to destroy McLanahan." She thought for a moment; then: "We need to find out exactly what's happened, but not from Gardner's perspective. We need our own intel on this. McLanahan's not crazy. If we come to his rescue, we might come out on top of this after all."
"Now you want McLanahan to win, Stacy?" Morna asked.
"Of course I want him to win, Colleen, but I want him to win for me, not just for himself or even for the country!" Barbeau said. "He's a genuine hero, a knight in shining armor, as Gardner puts it. Gardner's pride is hurt, and he's not thinking clearly. I need to find out what he has in mind, even if it means doin' the dirty with him whenever the First Lady is on the road, but then we need to find out what really happened and plan our own strategy. I gotta keep my eye on the prize, honey, and that is getting contracts and perks for my buddies in Louisiana."
"What if he's really flipped out?"
"We need to find out what happened to McLanahan and what he did out in Iran, and fast," Barbeau said. "I'm not going to blindly side with the President and oppose McLanahan unless the guy really has flipped out, which I seriously doubt. Get on the horn and find out all you can about what happened. You still in contact with the space playboy buddy of his...what's his name?"
"Hunter Noble."
"Oh yes, the luscious Captain Noble, the young space cowboy. You need to pump him for information, but not make it sound like it. You still screwing him?"
"I'm one of a very long line of Hunter Noble East Coast screwees."
"You can do better than that, child," Barbeau said, giving her a pat on the back and then a discreet one on the butt. "Don't just be another squeeze—be his wingman, his confidante. Tell him the Senate Armed Services Committee is going to look in on goings-on in Dreamland, and you'd like to help. Warn him. Maybe he'll give up some useful information."
"It'll be tough to meet up with the guy if he's flying around in space, stuck in that base out there in the desert...or in prison."
"We might have to plan a fact-finding trip to Vegas soon so you can really put the squeeze on him. Maybe I'll get to join in too." She paused, savoring the thought of a three-way with the Air Force playboy. "Tell him that if he cooperates, we can keep his tight young ass out of prison." She smiled and added, "And if he doesn't cooperate, get me some dirt on the boy that I can use against him. If he won't play nice, we'll use him to start dismantling McLanahan and the rest of those characters at Dreamland."
TEHRAN MEHRABAD AIRPORT, TEHRAN, DEMOCRATIC REPUBLIC OF PERSIA
EARLY THAT EVENING, TEHRAN TIME
The motorcade of armored Mercedes sedans and limousines sped down Me'raj Avenue toward Mehrabad International Airport unhindered by roadblocks. All along the motorcade route, General Buzhazi had his troops take down the checkpoints and barricades just before the motorcade arrived, let it pass, then hurriedly put them back up. The heavy troop presence throughout western Tehran that night kept citizens and insurgents away from the main thoroughfares, so few got to see the extraordinary procedures.
The motorcade bypassed the main terminal, where Buzhazi had set up his headquarters, and instead moved quickly down a taxiway and out to a row of Iran Air hangars. Here security appeared routine, almost invisible—unless you had night-vision goggles and a map showing the locations of dozens of sniper and infantry units scattered throughout the airport grounds.
A lone unmarked plain white Boeing 727 sat in front of one of the hangars, its airstair guarded by two security men in suits and ties. The lead sedan pulled forward just beyond the foot of the airstair, and four men in dark business suits, dark caps similar to chauffeur's hats, white shirts, dark ties, dark slacks and shoes, and carrying submachine pistols exited and took up stations around the stairs and the nose of the aircraft. One by one the two stretch limousines pulled up to the foot of the airstair, with more sedans unleashing eight more similarly attired and armed security agents to guard the tail and right side of the aircraft. Out of each limo several individuals exited, including an older man in a military uniform, a young woman surrounded by bodyguards, and men and women both in Western-style business suits and Iranian-style high-collared jackets.
In moments all the persons had trotted up the stairs and into the jetliner. The security men stayed in their positions until the jet had started its engines, and then they re-entered their sedans. The big armored cars formed a bubble around all sides of the airliner as it taxied down the empty taxiways and to the main runway, and in minutes the jetliner was airborne. The limousines retreated to a secure fenced area behind the Iran Air hangars and were parked outside a battered-looking repair garage. The Mercedes sedans performed a quick patrol of the ramp and hangar perimeters, then were parked in the same fenced area as the limousines. Minutes after the drivers and security men stepped out and locked their cars, workers came out, used towels to wipe dirt off the vehicles, and covered each of them with elastic-bottomed nylon covers. The lights were turned out, and soon the airport returned to the tense quiet it had become since the insurgency began.
The gaggle of security agents walked across the parking ramp to the main terminal building, weapons slung on their shoulders, most smoking, all saying little. They had their ID badges examined by a security guard outside the terminal and were allowed inside. They walked across the passenger concourse to a door marked CREWMEMBERS ONLY, had their ID badges checked once more, and were admitted. Other agents inside took their weapons, unloaded and cleared them, and the group went down a dimly lit hallway and inside to a conference room.
"I think everyone played their part as best as could be expected," the first "security guard," General Hesarak al-Kan Buzhazi, said. "Nice to see how the other half lives, eh, Chancellor?"
"I found it uncomfortable, unconvincing, unnecessary, and if my hearing has been damaged by those aircraft engines, I will hold you personally responsible, General Buzhazi," Masoud Noshahr, the Lord High Chancellor of the Qagev royal court, said indignantly. He was tall and thin, in his late forties, with long and slightly curly gray hair, a salt-and-pepper goatee, and long and delicate-looking fingers. Although he was young and appeared healthy, Noshahr, obviously unaccustomed to much physical exertion, was out of breath from their fast walking pace and from climbing stairs instead of taking elevators. He stripped off the jacket and cap and removed the tie as if they were burning his skin with acid, then snapped his fingers to one of the other men in dark suits, one of his real security guards, who went to fetch his ankle-length fur and leather coat. "It was nothing but a petty parlor game that fooled no one."
"We had better hope it worked, Lord Chancellor," another of the "security guards," Princess Azar Assiyeh Qagev, said. Instead of handing her weapon off to a guard, she unloaded and cleared it herself, then began field-stripping the weapon for inspection and cleaning. "The insurgents penetrate our network deeper and deeper every day."
"And we capture and kill more of them every day as well, Highness," Noshahr reminded her. "God and time are on our side, Princess, have no fear." Finally his attention was drawn to the weapon disassembly going on in front of him. "What in the world are you doing, Highness?" Noshahr asked in amazement as Azar's deformed but obviously skilled fingers worked the seemingly hidden levers and pins of the weapon. He squinted uncomfortably at the princess working with the submachine gun and nodded to a bodyguard, who went over to the princess, bowed politely at the waist, then reached out to take the gun parts from her hands. She gave him a stern expression and a slight shake of her head, and he bowed again and backed away. In seconds the submachine gun lay in pieces before her on the table.
"You don't carry an unknown or unfamiliar weapon into battle, Lord Chancellor," Azar said. "How do you know if the thing will work when you want it to? How do you even know if it was loaded if you don't bother to check?"
"We carried those things for show, to fool any insurgents who may have been watching us," Noshahr said. "I don't care what shape it's in. That's why we have trained guards with us. Princesses are not supposed to be handling dangerous weapons."
"It's not dangerous now, Lord Chancellor—it looks like it's in good shape to me," Azar said. She began to reassemble the weapon. In less than thirty seconds it was back together, loaded, cocked, and safed, and she slung it over her shoulder. "I don't carry weapons for show."
"Very impressive, Highness," Noshahr said, hiding his astonishment with a bored and unimpressed expression. He turned to Buzhazi. "We're wasting time here. Now that we have played along with your charade, General—putting the princes in considerable danger, I will maintain—shall we get down to business?"
"Let's," Buzhazi responded, using the same haughty country-club tone of voice as Noshahr. "I asked you to come here to talk about coordinating our efforts against Mohtaz and his foreign insurgents. Last night's gun battle with what turned out to be your assassination squad must never be repeated. We need to start working together."
"The fault was completely yours, General," Noshahr said. "Your troops did not allow our freedom fighters to identify themselves. They had just come from a successful raid on an insurgent hideout when your men opened fire. My men discovered more than three dozen high-explosive devices ready for the streets, including a dozen suicide bomber vests and explosives disguised to look like everything from telephones to baby carriages."
"I've had that bomb-making factory under surveillance for days, Noshahr," Buzhazi said. "We were waiting for the master bomb-maker to arrive to arm those bombs. What good does it do to kill a bunch of low-level know-nothing worker bees and let the chief bomb-maker himself escape? Now it'll take us another month or more to locate the new factory, and by then they'll have fabricated another three dozen or more bombs to use against us."
"Do not change the subject, Buzhazi," Noshahr snapped. "Your unit's sneak attack cost us the lives of six of our best agents. We demand reparations, and we demand that you withdraw your troops from the slums and alleys and confine your activities to the avenues, highways, and the airport. Or, better yet, place yourself and your troops under the command of the council of war, which is the legitimate and rightful government of Persia, and we shall ensure that you shall not interfere again with our anti-terrorist missions."
"We bear equal responsibility for their deaths, Lord Chancellor," Azar said.
"You don't have to apologize for the war council's mistakes, Azar—"
"You will address Her Highness properly, Buzhazi!" Noshahr ordered. "You dare not speak to the princess as if she is a commoner!"
"She's not my princess, Noshahr," Buzhazi said, "and I don't take orders from pretend generals or defense ministers like you, either!"
"How dare you! The Shahdokht is the rightful heir to the Peacock Throne of Persia, and you will address her as such and show her the proper respect! And I will remind you that I am the appointed chancellor of the Qagev court, royal minister of war, and marshal of the council of war! Have some respect for the office, even if you have no respect for yourself!"
"Noshahr, a year ago you were hanging out in the casinos in Monaco and making up stories about leading freedom fighters against the Pasdaran while trying to boink old rich ladies for their money," Buzhazi said. "In the meantime your loyalists were being captured and tortured because you couldn't keep your drunken mouth shut about their identities and locations—"
"That is preposterous!" Noshahr sputtered.
"The Pasdaran spies in Monaco, Singapore, and Las Vegas were getting a constant stream of information about your network just by sitting near you in the casinos, bars, and whorehouses you frequented, listening to you spin your wild stories about single-handedly freeing Iran."
"You peasant! You insolent pup! How dare you speak to me like this!" Noshahr cried. "I serve a king and his queen, directed twenty million loyalists around the world, equip and organize a fighting force of half a million, and have kept the royal treasury safe and secure for the past twenty years! You are little more than a thief and murderer, disgraced by your own words and actions over two decades, and demoted and humiliated by the government you served and then betrayed. You are spurned by your fellow citizens, and you lead by nothing more than fear of the next murderous rampage you will embark on, like the hideous massacre at Qom. You dare call yourself a Persian—!"
"I don't call myself anything you call yourself, Noshahr!" Buzhazi shouted. He turned to Azar, his eyes blazing. "I won't have anything to do with you or your so-called court, Princess, as long as he's in charge. I'm not in the mood for playing dress-up and kings and castles."
"General—"
"Sorry, Princess, but this is a huge waste of my time," Buzhazi said angrily. "I've got a war to fight. This imbecile who calls himself a marshal and minister of war doesn't know which end of a rifle to point at the enemy. I need fighters, not popinjays. I've got work to do."
"General, please stay."
"I'm leaving. Good luck to you and your pretty little court jesters, Princess."
"General, I said stay!" Azar shouted. She whipped off the dark cap, letting her long mun whip in the air. The Persians in the room were stunned into silence by the sudden appearance by the symbol of royalty in their midst...all except Buzhazi, who was stunned instead by the young woman's commanding tone of voice: part drill sergeant, part disapproving mother, part field general.
"Shahdokht...Highness...my lady..." Noshahr sputtered, his eyes fixed on the dark shining flowing locks as if a golden scepter had just appeared before his eyes, "I think it is time for us to depart and—"
"You will stay and shut your mouth, Chancellor!" Azar snapped. "We have important business to discuss."
"We cannot conduct business with this...this terrorist!" Noshahr said. "He's nothing but an old tottering fool with delusions of grandeur—"
"I said, we have business to discuss with the general," Azar said. This time the word "we" coming from her lips had a different meaning: it no longer referred to him, but clearly indicated the imperial "we," meaning her alone. "Be silent, Chancellor."
"Be...silent...?" Noshahr gurgled, his mouth opening and closing indignantly. "Pardon me, Shahdokht, but I am the Lord High Chancellor of the royal court, the representative of the king in his absence. I have full and sole authority to negotiate and make agreements and alliances with friendly and allied forces."
"Not any longer, Chancellor," Azar said forcefully. "It has been a year since anyone has heard or seen the king and queen. In the meantime the court has been run by appointed servants who, although true and loyal, do not have the interests of the people in mind."
"I beg your pardon, Shahdokht—!"
"It's true, Chancellor, and you know it," Azar said. "Your primary objective has been the organization, security, and placement of the court, in preparation for running the government upon the return of the king and queen. You have done a fine job of that, Chancellor. The court is safe, secure, well run, well financed, and is ready to administer this country when the time comes. But right now the people don't need or want an administrator—they want a leader and a general."
"I am the rightful leader until the king returns, Shahdokht," Noshahr insisted. "And as minister of war and marshal of the council of war, I am the commander-in-chief of our military forces. There are no others permitted."
"You're wrong, Chancellor...I am," Azar said.
"You? But that...that is highly irregular, Shahdokht," Noshahr said. "A proclamation of death or abdication has not yet been made. A council must be convened, composed of myself, the religious leaders, and representatives of the eleven royal houses, to investigate the likely whereabouts of the king and queen and decide what actions to take. That is impossible and unsafe to do in time of war!"
"Then, as heir apparent, I will make the proclamation myself," Azar said.
"You!" Noshahr repeated. "You...that is...pardon me for saying so, Shahdokht, but that is an insult to the memory of your blessed father and mother, our beloved king and queen. They may be still in hiding, or perhaps injured and healing, or even captured. Our enemies could be waiting for you to do such a thing and then reveal that they are still alive, hoping to throw us into confusion and rebellion against the court and royal family. You cannot...I mean, you should not do this, Shahdokht—"
"I am no longer Shahdokht, Chancellor," Azar said. "You will hereby refer to me as Malika."
Noshahr gulped, his eyes bulging. He stole a glance back at his bodyguards, then back at Azar, studying her carefully, trying to decide if she meant what she'd just said and if she would back down or compromise if confronted. "I...I am afraid I cannot allow that, Princess," he said, after finally summoning up enough courage. "I have a responsibility to the king and queen to safeguard and preserve the court. In their absence, and without guidance from a council of the royal houses, I'm afraid I cannot do as you wish."
Azar lowered her eyes, nodded, and seemed to even sigh. "Very well, Chancellor. I see your point of view."
Noshahr was filled with relief. He would certainly have to deal with this young Americanized upstart, and soon—she obviously had aspirations far beyond her years, and that could not be tolerated. But he was willing to act the supportive and protective uncle—all the better to keep an eye on her while he...
"I see it is time to take back the throne," Azar said. In a blur of motion, she suddenly whipped the German-made Heckler & Koch HK-54 submachine gun up and steadied it from her hip...aiming it squarely on Masoud Noshahr's chest. "You are under arrest, Chancellor, for defying my authority." She turned to the Persian bodyguards behind Noshahr. "Guards, place the chancellor under arrest."
"This is preposterous!" Noshahr screamed, more in shock and surprise than anger. "How dare you?"
"I dare because I am the Malika, Chancellor," Azar said confidently, "and the throne has been vacant long enough." She looked past Noshahr to the bodyguards, who still had their guns slung on their shoulders. "Guards, place the chancellor under arrest. He is forbidden to make any communications with the outside."
"They won't follow you, Azar Assiyeh," Noshahr said. "They are loyal to me and to the king and queen, the rightful rulers of Persia. They will not follow a spoiled, bewitched brat from America."
Azar glanced around the conference room, noting that neither Lieutenant Colonel Najar nor Major Saidi, her longtime aides, had raised their weapons—they were unslung, but still pointing at the floor with safeties on. The same with Hesarak Buzhazi and his bodyguard, Major Haddad, and the chief of the infantry brigade based at Mehrabad Airport, Colonel Mostafa Rahmati, both of whom had accompanied them on this diversionary mission. She was the only one with her weapon raised.
"I gave an order, Master Sergeant: Place the chancellor under arrest," Azar commanded. "Allow no outside communications. If he resists, bind and gag him." Still no one moved.
"Master Sergeant...all of you, it is time to make a decision," Azar said, affixing each of them with a steady gaze, hoping to hell her hands wouldn't start shaking. "You may follow Chancellor Noshahr and continue on with this so-called revolution as it has been for the past year, or swear loyalty to me and to the Peacock Throne, and follow me in taking back this country for a free Persian republic."
"Follow you?" Noshahr sneered. "You're just a girl. You may be a princess, but you're not a queen—and you're certainly not a general. The loyalists won't follow a girl into battle. What will you do if no one chooses to accept you as queen?"
"Then I will abdicate my title and join General Buzhazi's forces," Azar replied, to the absolute amazement of all. "It is time to join forces and fight as one nation, and if it won't be done under the Qagev banner, it will be under the general's flag. If you're ready to take me and my followers, General, we're ready to join you."
"That won't be necessary," Hesarak Buzhazi said...and to everyone's great surprise, he unslung his submachine gun, held it before him with arms outstretched...and dropped to one knee before Azar. "Because I am surrendering command of my forces and pledging my loyalty to the Malika Azar Assiyeh Qagev, the rightful queen of Persia and mistress of the Peacock Throne."
Azar smiled, silently praying that she wouldn't keel over from the surprise or burst into tears herself, then nodded. "We are pleased to accept your oath of loyalty, Hesarak al-Kan Buzhazi." She kissed his forehead, then put her hands on his shoulders. "Rise, sir, take your weapon, and assume leadership of the ministry of war and the council of war of the royal court of Qagev, and command of the combined forces of the Democratic Republic of Persia...Marshal Buzhazi."
"Thank you, Malika," Buzhazi said. He turned to Noshahr. "My first official act shall be to offer an appointment to Masoud Noshahr as deputy minister of war, vice marshal of the army, and my representative to the court. Do you accept?"
"You want me to serve under you?" Noshahr asked, even more shocked now than before. "You take my position and then you want me back? Why?"
"The queen is a good and astute judge of character, Noshahr," Buzhazi said. "If she says you have served the court well as chancellor and prepared it to lead the country when the time came, I believe her. I want you to keep on doing your job, the one you're best at. Prepare the court to rule a constitutional monarchy, and keep supplies and equipment flowing to my troops. I need someone to represent me in Tehran, because I'll be in the streets suppressing this insurgency and restoring security to the country. That's what I'm good at. And as vice-marshal, you will answer to me. Screw up, and you'll have to deal with me. Do you accept?"
For a moment Buzhazi thought Noshahr was going to say something crude or insulting; instead, he did something Buzhazi never thought he'd do: he saluted. "Yes, sir, I accept."
"Very good, Vice-Marshal. I want a meeting of the council of war set up immediately." He turned to Azar. "Malika, with your permission, I'd like to appoint Lieutenant Colonel Najar as my chief of staff and promote him to full colonel. Major Saidi will remain as your aide-de-camp."
"Permission granted, Marshal," Azar said.
"Thank you, Malika. Colonel, work with Vice-Marshal Noshahr to set up a meeting of the war council. Major Haddad is hereby promoted to lieutenant colonel and will be in charge of security." To Azar he said, "Malika, I would like you to attend the war council meeting and provide your input on resources and personnel we may be able to recruit from the streets of Tehran and the surrounding towns and villages. We'll need every helping hand we can find to make this work."
"Gladly, Marshal," Azar said.
"Thank you, Malika," Buzhazi said. "If you would, Malika, Vice-Marshal Noshahr, I'd like to show you something first before we proceed that could have a bearing on our planning. Colonel Najar, take over."
Azar walked beside Buzhazi through the airport terminal to the exit. "Very dramatic gesture you made back there, Marshal," she said. "I never thought I'd see you kneeling before anyone, let alone me."
"I had to do something to outdo your grand gesture, Highness," Buzhazi said. "Besides, if all this fancy froufrou court stuff is what your people know and expect, I guess I had to play along. You were really going to give up your throne and join my ragtag force of outlaws?"
"Did you mean what you said about surrendering your forces to me and swearing allegiance?" They smiled together, knowing each other's reply. "Do you think we can pull it off, Hesarak?" she asked.
"Well, before today, I gave us no more than one chance in ten of winning," Buzhazi said honestly. "Since then, things have improved greatly. I give us perhaps one chance in five now."
"Really? A one hundred percent improvement so fast? We haven't done anything yet except perhaps rearrange the deck chairs on a sinking ship! We have the same forces as before, the same resources—perhaps better organization and a little extra motivation. What else has changed other than our names, titles, and allegiances?"
They had walked outside and were escorted by guards to the nearby Iran Air hangar. After their identities were verified, Buzhazi stepped aside to let Azar pass him. "What else has changed?" he asked with a smile. "Let's just say something from above has dropped into our laps."
"What...?" Azar stepped into the hangar......and was immediately confronted by a ten-foot-tall humanoid robot, wearing some sort of cannon on his shoulders. The robot stepped closer to her with amazing speed and agility, examined them all for a moment, then stood at attention and shouted, "Detail, ten-hut!" in a loud computer-synthesized voice, then repeated it again in Farsi. It stepped aside...
...revealing that the hangar had two sleek, jet-black, massive American bombers inside. Azar recognized them as Air Force B-1 bombers, except the cockpit windows appeared sealed closed. The hangar floor was choked with vehicles, cargo containers of every size and description, and perhaps two hundred American airmen in utility uniforms standing at attention.
"As you were," Azar said. The Americans, men and women alike, relaxed. Many came over to the newcomers, introducing themselves with salutes and handshakes.
A few moments later, a tall man in a strange dark gray all-body suit of armor that Buzhazi recognized as the American Tin Man battle system, without his helmet, came over, stood before Qagev and Buzhazi, and saluted. "General Buzhazi?" he said via his Tin Man suit's on-board electronic translator. "Major Wayne Macomber, U.S. Air Force, detail commander."
Buzhazi returned his salute, then shook hands. "Thank you, Major. May I present Her Highness, Azar Assiyeh Qagev..." He paused for effect, giving her a sly wink and nod, then added, "Queen of Persia."
Macomber's eyes widened in surprise, but he recovered quickly enough, snapped to attention again, and saluted. "Nice to meet you, Your Highness." She extended her hand, and he shook it, his armored hand dwarfing hers. "Never met a queen before."
"I have met a Tin Man before, and I take great pleasure and comfort knowing you're here," Azar said in English so perfect, so American that it surprised him. "Welcome to Persia, Major."
"Thanks." He turned his hand and looked down at hers. "Hypoplastic thumb. Nice job fixing it. My youngest sister has it too. Bilateral?"
"Yes, Major," Azar replied rather awkwardly. "I'm surprised at you. Most people I greet look at my hand and then look away, pretending not to notice."
"Ignorance, that's all, ma'am," Macomber said. "Good for you not hiding it. My sister doesn't hide it either. Freaks people out but that's her plan. She still has a wicked tennis backhand."
"You should see me on the rifle range, Major."
The big commando smiled and nodded, his turn to be surprised. "Looking forward to that, ma'am."
"Me too, Major." She looked at another commando in a Tin Man battle armor system approach. "Hello, Sergeant Major Wohl," she said, extending her hand. "Nice to see you again."
"Thank you, Highness," Wohl said. "Nice to see you too." He glanced at Buzhazi. "I hope your new title doesn't mean bad news about your parents."
"I hope so too, Sergeant Major," Azar said, "but the situation has forced my elevation, and so we proceed." Wohl nodded in approval, but still gave Buzhazi a warning glare.
The ten-foot robot came over to them. Macomber motioned to her and said, "Ma'am, I'd like to introduce you to my second-in-command, Captain Charlie Turlock, U.S. Army Reserves, piloting a Cybernetic Infantry Device manned robot battle system she helped develop. She's on patrol now so she can't get out to greet you properly. Captain, meet Queen Azar Qagev of Persia."
"Nice to meet you too, Captain," Azar said, shaking hands with the giant, amazed at her delicate touch despite the size of her mechanical hand. "My minister of war and commander of my armed forces, Marshal Hesarak Buzhazi."
"Nice to meet you, Highness, Marshal," Charlie said from within the CID unit. Macomber's eyes widened at Buzhazi's new title. "All patrols reporting secure, sir. Excuse me, but I'll continue my assignment." The robot saluted and hurried off.
"Incredible, absolutely incredible," Azar remarked. "Thank you so much for the extraordinary job you did in hunting down the Pasdaran's mobile missiles. But now I'm confused. Did Marshal Buzhazi ask you to come to Tehran?"
"We had a little...trouble, you might say, with our accommodations in Turkey," Macomber explained. "My commanding officer Lieutenant General Patrick McLanahan got in contact with General—er, Marshal Buzhazi, and he offered to put us up until we get our situation straightened out."
"McLanahan? The general up in the space station?"
"Let's go somewhere and talk, shall we?" Macomber suggested. They moved through the hangar, greeting more airmen, and took a quick tour of the EB-1 Vampire bombers before entering an office just off the main hangar floor. Macomber spoke as if to thin air; a moment later, a telephone rang right beside him. He picked the receiver up and handed it to Azar. "It's for you, Highness."
Azar took the phone, trying to act like impromptu and mysterious phone calls for her were completely normal. "This is Queen Azar Assiyeh Qagev of Persia," she said in English. "Who is this, please?"
"Highness, this is Lieutenant General Patrick McLanahan. How are you tonight?"
"I'm well, General," she responded, trying to sound official and coherent even though her senses were swimming trying to keep up with the amazing otherworldly technology she was being exposed to here at breakneck speed. "We were just talking about you."
"I was listening in—hope you don't mind," Patrick said. "We keep a close eye on our troops all around the world."
"I understand," Azar said. "I hope you are recovered from your space flight injuries. Are you in Persia?"
"No, right now I'm over southern Chile, aboard Armstrong Space Station," Patrick said. "Highness, I was in a little bit of trouble, and I called on General Buzhazi for help. I apologize for not informing you first, but time was of the essence."
"You and your forces are welcome forever and always in Persia, General," Azar said. "You are a hero and champion to all free Persians, and we consider you our brother-in-arms. But perhaps you can explain what's going on."
"We believe Russia has moved military forces into Iran and is working with the theocratic regime to exert influence in the region."
"Well of course they have, General," Azar said matter-of-factly. "Don't tell me that's a surprise to you?" His rather embarrassed pause gave her all the answer she needed. "The Russians have pledged substantial military and economic assistance over the years to the theocratic regime in exchange for presence and to put pressure on them to stop supporting anti-Russian separatist movements inside the Russian Federation and its near abroad, such as in Kosovo, Albania, and Romania. Russia has enjoyed its most-favored-nation status for decades."
"We knew that Russia was using Iran along with the conflict in Iraq to distract the United States from its other activities around its periphery," Patrick said, "but we didn't know their involvement was so widely known and accepted."
"The aid Iran has received from the Russians is reportedly greater than what the United States gives any other nation in the region except perhaps Israel," Azar said. "That was very important not only to keep the theocrats in power but to sustain the Iranian people. Unfortunately a lot of that aid went to the Revolutionary Guards Corps and their drastic arms buildup, which they used to crack down on any dissent in our country. But has something else changed recently? Is Russia playing a different game?"
"We believe the Russians have brought a new weapon, a powerful mobile anti-spacecraft laser, into Iran and have used it to down one of our spacecraft," Patrick said. "Major Macomber, Captain Turlock, and Sergeant Major Wohl survived such an attack."
"You mean, one of the spaceplanes I've heard so much about?" Azar asked. "They were riding in one in space when it was hit by this laser?"
"Yes, Highness. I would like assistance to hunt down this Russian weapon and neutralize it."
"I don't think that'll be difficult at all," Azar said. She handed the phone to Buzhazi, who put it on a speaker and asked Major Haddad to translate for him.
"Marshal Buzhazi?"
"Greetings, General McLanahan," Buzhazi said through Haddad.
"Hello, Marshal. You got a promotion, I see."
"And I judge by your unexpected call, the sudden appearance of such a large force on my doorstep, and the disturbing lack of information from your military or foreign ministries, that your career has not enjoyed similar success," Buzhazi said. "But you helped me when I was on the run, and I was hoping to one day do the same for you. So. The Russians have shot down your spaceplane?"
"Can you help us find that laser, Buzhazi?"
"Of course. I am sure we can find it quickly, if my men do not already know where it is."
"You sound pretty confident."
"General, we do not automatically distrust the Russians like you do—in fact, we have more reasons to distrust the Americans," Buzhazi said. "We are neighbors with Russia, and our borders have been safe and secure for decades; we have purchased many weapons and received substantial military, economic, industrial, and trade assistance from Russia, which was urgently important to us during all the years of the trade embargo with the West; we even still have a mutual defense treaty that is in full force and effect."
"So you're saying that you have been working with the Russians, Marshal," Patrick asked with surprise, "including supplying them with information on our activities in Iran?"
"General McLanahan, sometimes the depth of the naïveté of the Americans astounds me," Buzhazi said. "We have to live here; you merely influence events here for America's national interests, sometimes from the relative comfort of a battle staff room—or a space station. Of course we supply Russia with information, just as we supply you with information on Russia's activities and assist you when you run into...domestic political problems, shall we say?" Again, no response from Patrick.
"We all have our own necessities, pursuits, and agendas," Buzhazi went on. "We hope such cooperation enriches us all and is mutually beneficial, but in the end it is our own objectives that must be attended to first, no?" Again, silence. "General McLanahan? Are you still there?"
"I'm still here."
"I am sorry to have upset or disillusioned you, General," Buzhazi said. "You did save my life and help me defeat the Pasdaran in Qom and Tehran, and for that I would help you until the last of my days. All you had to do was ask. But you should not be so surprised to learn that I would extend similar courtesies to any other country that helps my cause, including your adversaries. So. You wish to locate this Russian mobile laser system? Very well. I shall contact you immediately through Major Macomber when I have its precise location. Is that agreeable?"
"Yes, it is, Marshal," Patrick said. "Thank you. And what of my men there in Tehran?"
Buzhazi turned to Azar and spoke in low tones for a few moments; then: "The queen wishes to extend all possible aid and comfort to you and your men. In return, she hopes you will assist us when the time comes."
"So do I have to worry about a Russian attack on that location, Buzhazi?" Patrick asked.
"Patrick, I think I have made myself plain to you," Buzhazi said through his translator. "I hope you are not one of those idealistic men who believe that we help each other because we believe it is the right or just thing to do, or because one side is inherently good and the other is evil. You brought your forces to Tehran for reasons that are not entirely clear to me yet, but I know that we did not invite you. We will learn all soon, God willing. Until then, I will do what I must for our nation and our survival. You will do what you must for your men, your cause, and yourself. Hopefully all those things are mutually beneficial." And he hung up the phone without even a departing salutation.
"Everything okay, sir?" Macomber asked via his subcutaneous transceiver after he had excused himself from Buzhazi and Azar.
"Major, I think we need to trust Buzhazi, but I just can't make myself do it," Patrick admitted. "He may be a patriot, but he's first and foremost a survivor. When he was chief of staff and commander of the Pasdaran, he was fully prepared to sink an American aircraft carrier and kill thousands of sailors just to prove how tough he thought he was. I think he wants to get rid of the theocracy and the Pasdaran, but I think he'll do anything he needs to do—include screwing us both—to survive. You're going to have to make the call."
"Yes, sir," Macomber said. "I'll let you know."
"Well, Major?" Buzhazi asked via the electronic translator when Macomber returned. "What does your commanding officer say? Does he trust me yet?"
"No, sir, he doesn't," Macomber said.
"So. What shall we do?"
Macomber thought for a moment; then: "We take a little ride, Marshal."
## CHAPTER NINE
Never contend with a man who has nothing to lose.
—BALTASAR GRACíAN
OVER SOUTH-CENTRAL NEVADA
EARLY THE NEXT MORNING
"Here's the latest, guys, so listen up," the SEAL team leader, U.S. Navy Lieutenant Mike Harden, said. The fifteen members of his SEAL platoon, all pre-breathing oxygen in the cargo compartment of their C-130 Hercules cargo plane, stopped looking at charts and turned their attention to him. "Our guy on the inside tells us that the place is virtually deserted. He counts a total of twenty Security Forces personnel, mostly centered on the main computer center next to the headquarters building. The battle staff area has been deserted and there is just a skeleton security force stationed there, about six guys. The hangars have been locked up for a couple days. This checks with our own overhead surveillance. So our objective remains the four main offices in the headquarters building: one squad each going for the security operations center, the battle management area, the communications center, and the flight operations center. Unit Bravo is right behind us, and his guys will take the hangars and the weapons storage area.
"Our guy on the inside says he's seen just one of those CID manned robot units around the place patrolling the hangars and weapon storage area. We know they had a total of six CIDs. One was deployed to Iran, two deployed to Turkey, and one surrendered when the Rangers assaulted Battle Mountain, so there's two left, and we have to assume they're both at Elliott. There are approximately a dozen Tin Man units unaccounted for as well.
"Remember, use regular ammo only against the Security Forces guys if they open fire on you—don't waste ammo on the CIDs or Tin Man units." He held up a 40-millimeter grenade round. "These are our best hope of putting those things out of commission: microwave pulse generators, like a direct fucking lightning bolt hit. They tell us it should shut down all their systems instantly. Probably lethal for the guy inside, but that's his problem if he chooses to fight. These guys are fast, so stay on your toes and concentrate fire. Questions?" There were none. "All right. We have about five minutes to go. Get ready to kick some zoomie ass." There was a muffled round of "Hoo-ah!" in oxygen masks all around.
It seemed like just a minute later when Harden was notified by the cockpit crew that the jump zone was two minutes out. The SEALs quickly detached themselves from the aircraft oxygen system, hooked up to portable oxygen bottles, got to their feet, and held on tightly to handholds as the rear cargo ramp was lowered. No sooner had the ramp motored down than the red light turned green, and Harden led his platoon out into the frigid darkness. Less than twenty seconds after Harden jumped, all sixteen men deployed parachutes. Harden checked his chute and oxygen, made sure his infrared marker light was operating so the others could follow him in the darkness, then started following the steering indications from his wrist-mounted GPS unit.
This was a HAHO, or High Altitude–High Opening jump. From twenty-seven thousand feet, the team could sail about thirty miles from their jump point to their objective: Elliott Air Force Base, nicknamed "Dreamland." By order of the President of the United States, the two SEAL units had been ordered to assault the base, neutralize the Cybernetic Infantry Devices and Tin Man units patrolling the base, capture all base personnel, and secure the aircraft, weapons, computer center, and laboratories.
The winds were a little squirrelly, definitely different than forecast, which probably explained the hurried jump. Harden found himself steering his canopy in some rather radical maneuvers to get on-course. Each turn soaked up some horizontal speed, so that meant a little more marching once they got on the ground. They would fly for about ten minutes.
Once finally established on-course, Harden started looking for landmarks using his binocular night-vision goggles. He quickly saw that things weren't looking quite as planned. The first visual target was Groom Lake, the big dry lake bed south of the base that had the majority of Elliott's twenty-thousand-foot-long runway embedded in it. It was soon obvious they were too far west—they had jumped way too early. The GPS said they were right on-course, but the landmarks didn't lie. They had planned for this contingency, but Harden was going to give the flight crew a good chewing-out when this mission was over. He had studied the entire surrounding area in his pre-jump target study and was confident he could find a good place to land, even if it had to be on the dry lake bed itself.
He couldn't quite reach the dry lake bed, but he was able to find a flat area about fifty yards north of a dirt road. The landing was a lot harder than he anticipated—again, the GPS was lying about the wind direction and he landed with the wind instead of into it, which increased his ground speed and the force of the landing. Fortunately they were wearing so much cold-weather gear for the long HAHO jump, and the extra impact force was mostly soaked up. He formed up the team in less than three minutes, and it took them less than five to get their parachutes, harnesses, and extra cold-weather gear off and stowed, and their weapons, comm gear, and night-vision systems checked and ready.
Harden checked his GPS and motioned their direction of movement, but the assistant officer in charge, who had the backup GPS, waved his hand and indicated a different direction. They put their GPS receivers side by side, and sure enough, their readouts were completely different...in fact, they were different by about three miles!
That explained them being off-course and landing in the wrong direction based on GPS-derived winds: their GPS receivers were being spoofed. Harden knew that GPS jammers were being developed, but a jammed GPS receiver could be disregarded and alternate navigation methods used right away before significant errors were made. On the other hand, a spoofed GPS receiver would appear to be working properly. Even the C-130's GPS receivers had been spoofed. He had to remember that they were up against a unit that developed and tested next-generation weapons of all kinds, top secret stuff that probably wouldn't be seen by the rest of the world for years but would revolutionize warfare when it did hit the streets.
The platoon chief pulled out a lensatic compass, ready to take some fixes on terrain landmarks and cross-check their position on his map, but it must've taken a hit in the accelerated landing because the compass dial was spinning as if it were attached to an electric motor. Harden wouldn't be surprised if the eggheads here had developed a way to jam or spoof compasses too! He decided that since they landed west of the edge of the dry lake bed they would just head east until they found the lake, then they'd move north until they found the inner perimeter fence. He again signaled their direction of movement, overriding all queries, and headed off at a trot.
They had stripped off the cold-weather gear and left their parachutes behind, greatly lightening their load, but soon Harden found himself wiping sweat from his eyes. Jeez, he thought, it had to be below freezing out here in the high desert, but he was sweating to death! But he ignored it and kept on...
"Windward," he heard in his headset. He dropped to his belly and scanned the area. That was the code word for a team member in trouble. He crawled back along his direction of movement and found the platoon chief on his back, with the AOIC checking him over. "What in hell happened?" he whispered.
"He just collapsed," the assistant officer in charge said. He wiped sweat from his face. "I don't feel too good either, LT. Would they use nerve gas on us?"
"Stay down," someone said on the secure FM tactical radio.
Harden looked down the line of SEALs spread out in the desert. "Radios tight!" he whispered. The AOIC passed the word back to the others. He had briefed to use code words only on the radios on this mission unless they were in a firefight and the whole team was compromised.
The platoon chief sat up. "You feeling okay, Chief?" Harden asked. The chief signaled he was, and they prepared to move out again. But this time it was Harden who felt woozy—the minute he stood up, he was bathed in warm, dry heat, as if he had just opened the door to a red-hot oven. The feeling subsided when he dropped to a knee. What in hell...?
And then he realized what it was. They had been briefed on the incident in Turkey, where the guys from Dreamland used nonlethal microwave weapons to knock out the base security personnel—they reported that it felt like intense heat, like their skin was on fire, and soon their brains got scrambled so bad that they passed out. "Crocodile, crocodile," Harden spoke into his whispermike, the code word for "enemy nearby."
"Just stay down and don't move," they all heard in their headsets.
Shit, the Air Force guys had found their FM frequency, decoded the encryption routine, and were talking over their whispermike channel! He turned and made a hand signal to switch to the secondary frequency, and the word was passed down to the others. In the meantime, Harden pulled out his satellite phone and punched up the other SEAL unit's secure channel: "Silver, this is Opus, crocodile."
"Did you know," they heard in their headsets on the new channel, "that there are no words that rhyme with 'silver' and 'opus,' just like 'orange'?"
Harden wiped a rivulet of sweat out of his eyes. Comm discipline completely forgotten, he angrily switched back to the whispermike: "Who the hell is this?"
"Ah ah ah, Lieutenant, beadwindow, beadwindow," the voice said again, using the old code word warning of inappropriate radio transmissions. "Listen, guys, the exercise is over. We already took down the other unit heading to the flight line and weapon storage area—you guys did much better than they did. We have some nice comfy rooms ready for you. Stand up with your hands in the air and we'll take a little drive back to base. We have a truck on the way to come get you."
"Fuck you!" Harden shouted. He got into a low crouch and scanned the area, ignoring the growing pain radiating throughout his body...and then he saw it, a huge robot, less than twenty meters in front of him. He raised his rifle, flicked off the safety, and fired a grenade round. There was a tremendous flash, the smell of high-tension electricity frying the air, and a feeling of millions of ants crawling across his body...but the sensation of heat had vanished, replaced by bone-chilling cold as his sweat-soaked uniform quickly released body heat to the frigid night air.
He trotted back to his men. "Everybody okay?" he whispered. They all signaled they were fine. He checked his GPS receiver—it was completely dead, but the platoon chief's compass was working properly again, and he quickly plotted their position on his map, got a bearing toward their destination, and headed out.
On the way they passed the robot. It looked as if its limbs, torso, and neck had twisted in different and very unnatural directions all at once, and it smelled of a short-circuited and burned-out power drill. Harden was at first sorry for the guy inside—after all, he was a fellow American and soldier—but he wasn't going to stick around to check on him in case he was just stunned.
It was completely dark as they approached the inner perimeter fence, a double-layered fifteen-foot-high chain-link fence topped with razor wire. No lights around the fence meant either dogs or infrared sensors, Harden knew. He gave the order for the team to break into squads and begin their approach to...
...and at that moment he heard a whirring sound, like a high-speed fan, and he looked up. Through his night-vision goggles he saw an object about the size of a garbage can about twenty feet in the sky and just thirty or forty yards away, with a wide round shroud on the bottom, long legs, and two metallic arms which held white flags—and incredibly it had a lighted LED scrolling display on the top that read DON'T SHOOT JUST TALK WE'RE LISTENING.
"What the hell is this?" Harden asked. He waited until the flying robot got about ten yards away, then shot it down with a single burst from his MP5 submachine gun. He was sure he hit it, but it managed to fly down in a more or less controlled manner, landing awkwardly a few yards away, the scrolling LED message still visible. He repositioned his whispermike to his lips. "Who is this?"
"This is Brigadier General David Luger," the voice on the other end replied. "You know who I am. This has got to end, Lieutenant Harden, before anyone else gets hurt or killed."
"I have orders to take you into custody and secure this base, sir," Harden said. "I'm not leaving until my mission is accomplished. On authority of the President of the United States, I'm ordering you to deactivate all of your base defenses and surrender yourselves immediately."
"Lieutenant, there are a dozen more drones flying overhead right now carrying stun grenades," Luger said. "We can see you and each of your fifteen comrades, and we can hit each one of them with a stun grenade. Watch carefully. In front of you, right near the fence." A moment later he heard a tiny metallic ping! sound from almost directly overhead...and seconds later there was a tremendous flash of light, followed moments later by an impossibly loud craack! of sound and then a wall of pressure like a hurricane-force wind lasting a fraction of a second.
"Now that was about a hundred yards away, Lieutenant," Luger said. The ringing in Harden's ears was so loud he had trouble hearing him over the radio. "Imagine what that'll feel like just five yards away."
"Sir, you're going to have to take me and all my men out, because we're not leaving," Harden said after letting his hearing return somewhat to normal. "Unless you want to be responsible for wounding or killing fellow Americans, I urge you to follow my orders and surrender."
There was a long pause on the line; then, in a sincere fatherly voice, Luger said: "I really admire you, Lieutenant. We were being honest when we said you made it farther than the other SEAL unit. They surrendered the first time we hit them with the microwave emitter, and they even told us your identity when we captured them—that's how we knew who you were. You guys did good. I know you didn't mean to kill Staff Sergeant Henry. He was the NCO piloting the CID."
"Thank you, sir, and no, I didn't mean to kill anyone, sir," Harden said. "We'd been briefed on that microwave weapon your robots carry and we knew we had to knock it out."
"We developed the microwave disruptor grenade because we were afraid the CID technology had fallen into Russian hands," Luger said. "I didn't think it'd be used by our own against our own."
"I'm sorry, sir, and I'll take responsibility of personally informing his next of kin." He had to keep him talking as long as he could. The main occupying force, a Marine security company from Camp Pendleton, was due to arrive in less than thirty minutes, and if this guy Luger had second thoughts about attacking more Marines, maybe he'd hold off long enough for the others to arrive. "Should I go back and help the staff sergeant?"
"No, Lieutenant. We'll handle that."
"Yes, sir. Can you explain how—?"
"There's no time for explanations, Lieutenant."
"Yes, sir." Time was running out. "Listen, sir, no one wants this. Your best bet is to stop fighting, get a lawyer, and do this the right way. There don't have to be any more attacks. This is not who we are supposed to be battling. Let's stop all this right now. You're the unit commander here. You're in charge. Give the order, have your people lay down their weapons, and let us come in. We won't hurt anyone. We're all Americans, sir. We're on the same side. Please, sir, stop this."
There was another long pause. Harden truly believed that Luger was going to back down. All this was insane, he thought. Have some guts and stop this, Luger! he thought. Don't be a hero. Stop this or...
Then he heard a whirring sound overhead—the little trash-can robots returning—and then Luger said: "The pain will be more intense this time, but it won't last very long. Good day, Lieutenant."
Harden leaped to his feet and yelled, "All squads, fire grenades for effect and make for the fence, go, go, go!" He raised his MP5, loaded a disruptor grenade into the launcher breech, racked it home, and raised the weapon to...
...and it felt as if his entire body had instantly burst into flame. He screamed...and then everything quickly, thankfully went dark.
THE WHITE HOUSE CABINET ROOM, WASHINGTON, D.C.
LATER THAT MORNING
"I can't believe this...I fucking can't believe this!" President Joseph Gardner moaned. He and a handful of Senate and congressional leaders were being briefed by Secretary of Defense Miller Turner on their efforts to detain the members of the Air Battle Force and secure their weapons, and the information was not good. "They knocked out and captured two SEAL teams in Dreamland? I don't believe it! What about the other locations?"
"The SEAL team sent to Battle Mountain encountered light resistance and managed to capture one of their manned robots, but the robot had apparently either malfunctioned or was damaged and was abandoned," Turner said. "The aircraft and most of the personnel were gone; the SEALs captured about a hundred personnel without resistance. The FAA couldn't track any of the aircraft because of heavy jamming or netruding and so we don't know where they went."
"'Netruding'? What in hell is that?"
"Apparently the next-generation aircraft based out of Dreamland and Battle Mountain don't simply jam enemy radar, but they actually use the radars and their associated digital electronic systems to insert things like viruses, false or contrary commands, false targets, and even programming code changes into the radar's electronics," National Security Adviser Conrad Carlyle responded. "They call it 'netruding'—network intruding."
"Why wasn't I briefed about this?"
"It was first put into use on McLanahan's planes deployed to the Middle East," Carlyle said. "He disabled a Russian fighter by commanding it to shut itself down. Most digital radar systems in use these days, especially civilian sets, don't have any way to block these intrusions. He can do it with all sorts of systems such as communications, the Internet, wireless networks, even weather radar. Plus, since a lot of the civilian networks are tied into the military's systems, they can insert malicious code into the military network without even directly attacking a military system."
"I thought he shot a missile at the fighter!"
"The Russians claimed he shot a missile, but he used this new 'netrusion' system to force the MiG to turn itself off," Carlyle explained. "McLanahan had his heart thing before he could explain what happened, and we took the Russians' word on the incident after that."
"How can he send a virus through radar?"
"Radar is simply reflected radio energy timed, decoded, digitized, and displayed on a screen," Carlyle said. "Once the frequency of the radio energy is known, any kind of signal can be sent to the receiver, including a signal containing digital code. Nowadays the radio energy is mostly digitally displayed and disseminated, so the digital code enters the system and is treated like any other computer instruction—it can be processed, stored, replicated, sent out over the network, whatever."
"Jee-sus..." Gardner breathed. "You mean, they could already have infected our communications and tracking systems?"
"As soon as McLanahan decided to embark on this conflict, he could have ordered the attacks," Miller said. "Every piece of digital electronic equipment in use that receives data from the airwaves, or is networked into another system that is, could have been infected almost instantly."
"That's every electronic system I know of!" the President exclaimed. "Hell, my daughter's handheld game machine is tied into the Internet! How could this have happened?"
"Because we ordered him to find a way to do it, sir," chairman of the Joint Chiefs of Staff General Taylor Bain replied. "It's an incredible force multiplier, which was important when almost every long-range attack aircraft in our arsenal was destroyed. Every satellite and every aircraft—including his unmanned aircraft and Armstrong Space Station—is capable of electronic netrusion. He can infect computers in Russia from space or simply from a drone flying within range of a Russian radar site. He can prevent a war from happening because the enemy would either never know he was coming or would be powerless to respond."
"The problem is, he can do it to us now too!" the President exclaimed. "You need to find a way to shield our systems from this kind of attack."
"It's in the works, Mr. President," Carlyle said. "Firewalls and anti-virus software can protect computers that already have it, but we're developing ways to plug the security gaps in systems that aren't normally considered vulnerable to network attacks, such as radar, electronic surveillance such as electro-optical cameras, or passive electronic sensors."
"The other problem," Bain added, "is that being the unit that developed and is designing the netrusion systems, the High-Technology Aerospace Weapons Center has been in the forefront of developing countermeasures to it."
"So the guys who are employing the thing are the ones who know how to defeat it," the President said disgustedly. "Swell. That helps." He shook his head in exasperation as he tried to think. Finally he turned to the two congressmen in the Oval Office. "Senator, Representative, I asked you in here because this has become a very serious problem, and I need the advice and support of the leadership. Most of us in this room think McLanahan is unhinged. Senator, you seem to feel differently."
"I do, Mr. President," Senator Stacy Anne Barbeau said. "Let me try talking with him. He knows I support his space program, and I support him."
"It's too dangerous, Senator," the President said. "One man has died, and several more have been injured by McLanahan and his weapons."
"A frontal assault with armed troops won't work unless you're going to attempt a D-Day invasion, Mr. President," Barbeau said, "and we can't pen him up inside Dreamland when he's got spaceplanes, unmanned aerial vehicles, and bombers roaming around inside a thousand square miles of desert, patrolled by gadgets no one's ever heard of before. He won't be expecting me. Besides, I think I might have some folks on the inside who will help. They're just as concerned as I about the general's welfare."
There were no other comments made—no one had any other suggestions, and certainly no one else was going to volunteer to stick their heads in the tiger's jaws like the Navy SEALs had. "Then it's decided," the President said. "Thank you for this undertaking, Senator. I assure all of you, we'll do everything possible to see to your safety. I'd like to speak to the senator in private for a moment. Thank you all." The White House chief of staff escorted them all out of the Cabinet Room, and Gardner and Barbeau moved to the President's private office adjacent to the Oval Office.
No sooner had the door closed than Gardner's arms were around her waist and he was snuggling her neck. "You macho hot bitch," he said. "What kind of crazy idea is this? Why do you want to go to Dreamland? And who is this guy you say you've got on the inside?"
"You'll find out soon enough, Joe," Barbeau said. "You sent in the SEALs and they didn't get it done—the last thing you want to do is start a war out there. Your poll numbers will go down even farther. Let me try it my way first."
"All right, sugar, you got it," Gardner said. He let her turn in his arms, then began to run his hands over her breasts. "But if you're successful—and I have no doubt you will be—what is it you want in return?"
"We have a good deal going already, Mr. President," Barbeau said, pressing his hands even tighter around her nipples. "But I'm interested in one thing Carlyle was talking about: the netrusion thing."
"What about it?"
"I want it," Barbeau said. "Barksdale gets the network warfare mission—not the Navy, not STRATCOM."
"You understand all that stuff?"
"Not all of it, but I will, in very short order," Barbeau said confidently. "But I do know that Furness at Battle Mountain has all the bombers and unmanned combat aircraft that use netrusion technology—I want them at Barksdale, along with all the network warfare stuff. All of it. Downsize or even eliminate the B-52s if you want, but Barksdale runs network warfare for anything that flies—drones, B-2s, satellites, the space-based radar, everything."
The fingers on Barbeau's nipples tensed. "You're not talking about keeping the space station?" Gardner asked. "That's five billion I want to go to two aircraft carriers."
"The space station can fry for all I care—I want the technology behind it, especially the space-based radar," Barbeau said. "The space station is dead anyway—folks consider it McLanahan's orbiting graveyard, and I don't want to be associated with it. But the nuts and bolts behind the station are what I want. I know STRATCOM, and Air Force Space Command will want netrusion aboard their reconnaissance, airborne command posts, and spacecraft, but you have to agree to fight that. I want the Eighth Air Force at Barksdale to control netrusion."
The President's hands began their ministrations once again, and she knew she had him. "Whatever you say, Stacy," Gardner said distractedly. "It's a lot of hocus-pocus gobbledygook to me—what bad guys around the world understand is a fucking aircraft carrier battle group parked off their coastline, in their faces, not network attacks and computer magic. If you want this computer-fucking virus thing, you're welcome to it. Just get Congress to agree to stop funding the space station and give me my two aircraft carriers, minimum, and you can have your cyberwar shit."
She turned toward him, letting her breasts slide tightly across his chest. "Thank you, baby," she said, kissing him deeply. She placed a hand on his crotch, feeling him jump at her touch. "I'd seal our deal in the usual manner, but I have a plane to catch to Vegas. I'll have McLanahan in prison by tomorrow evening...or I'll expose him as a raving lunatic so severely that the American people will be clamoring for you to take him down."
"I'd love to give you a big going-away present too, honey," Gardner said, giving Barbeau a playful pat on her behind, then taking a seat at his desk and lighting up a cigar, "but Zevitin's going to call in a few minutes, and I've got to explain to him that I'm still in control of this McLanahan mess."
"Screw Zevitin," Barbeau said. "I suspect that everything McLanahan said about the Russians putting a super-laser in Iran and firing on the spaceplane is true, Joe. McLanahan might be going off the deep end by ignoring your orders, attacking without authorization, and then battling the SEALs, but Zevitin's up to something here. McLanahan doesn't just fly off the handle."
"Don't worry about a thing, Stacy," Gardner said. "We've got good communications open with Moscow. All they want are assurances that we're not trying to bottle them up. McLanahan is making the whole world, not just the Russians, nervous, and that's bad for business."
"But it's good for getting votes in Congress for new aircraft carrier battle groups, honey."
"Not if we have a rogue general on our hands, Stacy. Take McLanahan down, but do it quietly. He could ruin everything for us."
"Don't worry about a thing, Mr. President," Barbeau said, giving him a wink and a toss of her hair. "He's going down...one way or another."
Barbeau met up with her chief of staff Colleen Morna outside the executive suites, and they walked quickly to her waiting car. "The trip's all set, Senator," Morna said after they were on their way back to her office on Capitol Hill. "I have the billing codes for the whole trip from the White House, and they even gave us authorization for a C-37—a Gulfstream Five. That means we can take eight guests with us to Vegas."
"Perfect. I got a verbal agreement from Gardner about relocating and centralizing all of the DoD network warfare units to Barksdale. Find out which contractors and lobbyists we need to organize to get that done and invite them along with us to Vegas. That should water their eyes."
"You got that right, Senator."
"Good. Now, what about that hard-body boyfriend of yours, Hunter Noble? He's the key to this Las Vegas trip as long as McLanahan is up in that space station. What did you dig up on him?"
"You had him pegged from day one, Senator," Colleen said. "Our Captain Noble seems to be stuck in junior high school. For starters: he got a woman six years older than him pregnant in high school—the school nurse, I think."
"Happens every year where I'm from, sugar. The only virgin in my hometown was an ugly twelve-year-old."
"He was expelled, but it didn't matter because he already had enough credits to graduate two years early from high school and start engineering school," Colleen went on. "Seems his way of celebrating graduation is getting some woman pregnant, because he did it again in both college and grad school. He married the third one, but the marriage was annulled when yet another affair was uncovered."
"McLanahan he definitely isn't," Barbeau said.
"He's an outstanding pilot and engineer, but apparently has a real problem with authority," Morna went on. "He gets high marks on his effectiveness reports for job performance but terrible marks for leadership skills and military bearing."
"That's no help—now he sounds like McLanahan again," Barbeau said dejectedly. "What about the juicy stuff?"
"Plenty of that," Morna said. "Lives in bachelor officers' quarters at Nellis Air Force Base—barely six hundred square feet of living space—and has been written up many times by base security for loud parties and visitors coming and going at all times of the day and night. He's a regular in the Officers' Club at Nellis and piles up a pretty hefty bar tab. Rides a Harley Night Rod motorcycle and has received numerous speeding and exhibitionist driving citations. License just recently returned after a three-month suspension for unsafe driving—apparently decided to race an Air Force T-6A training aircraft down the runway."
"That's good, but I need the real juicy stuff, baby."
"I saved the best for last, Senator. The list of female visitors admitted for on-base visits is as long as my arm. A few are wives of married men, a couple known bisexual women, a few prostitutes—and one was the wife of an Air Force general officer. However, visits on-base seemed to have subsided a bit in the last year...mostly because he has signature credit authority with three very large casinos in Vegas for a total of one hundred thousand dollars."
"What?"
"Senator, the man hasn't paid for a hotel room in Vegas in over two years—he's on a first-name basis with managers, doormen, and concierges all over town, and uses comped rooms and meals almost every week," Colleen said. "He likes blackjack and poker and is invited backstage a lot to hang out with showgirls, boxers, and headliners. Usually has at least one and many times two or three ladies in tow."
"One hundred grand!" Barbeau remarked. "He beats out every Nevada legislator I know!"
"Bottom line, Senator: He works hard and plays hard," Colleen summarized. "He maintains a low profile but has made some fairly high-profile transgressions that have apparently been swept under the rug because of the work he does for the government. He's contacted regularly by defense contractors who want to hire him, some offering incredible salaries, so that probably makes him cocky and contributes to his attitude that he doesn't have to play the Air Force's games."
"Sounds like a guy living on the edge—and that's exactly where I like 'em," Barbeau said. "I think it's time to go pay Captain Noble a little visit—in his native habitat."
## CHAPTER TEN
The deed is everything, the glory nothing.
—JOHANN WOLFGANG VON GOETHE
MASHHAD, ISLAMIC REPUBLIC OF IRAN
THAT NIGHT
The city of Mashhad—"City of Martyrs" in English—in northeastern Iran was the second-largest city in Iran and, as the location of the shrine of the eighth imam, Reza, it was the second-largest Shiite holy city in the world and second only to Qom in importance. Over twenty million pilgrims visited the Imam Reza shrine every year, making it as noteworthy and spiritual as the Haji, the pilgrimage to Mecca. Located in a valley between the Kuh-e-Ma'juni and Azhdar-Kuh mountain ranges, the area had brutally cold winters but was pleasant most of the rest of the year.
Located in the hinterlands of Iran, Mashhad held relatively little military or strategic importance until the rise of the Taliban regime in Afghanistan in the 1980s. Fearing that the Taliban would try to export its brand of Islam westward, Mashhad was turned into a counterinsurgency stronghold, with the Iranian Revolutionary Guards Corps operating several strike teams, intelligence units, counterinsurgency fighter-bomber and helicopter assault and attack units from Imam Reza International Airport.
When Hesarak Buzhazi's military coup hit, Mashhad's importance quickly grew even stronger. The remnants of the Revolutionary Guards Corps was chased all the way from Tehran to Mashhad. However, Buzhazi barely had the resources to maintain his tenuous hold on the capital, so he had no choice but to let the survivors flee without mounting a determined effort to root out the commanders. With the surviving Revolutionary Guards Corps commanders freely moving about the city, and with a very large influx of Shiite pilgrims that continued almost unabated even during the growing violence, the Pasdaran had lots of recruits to choose from in Mashhad. From mosques, the marketplaces and malls, and from every street corner, the call to jihad against Buzhazi and the Qagev pretenders went far and wide and quickly spread.
Spurred on by the powerful spiritual aura of the city and the entrenched power of the Revolutionary Guards Corps, acting Iranian president, chief of the Council of Guardians, and senior member of the Assembly of Experts Ayatollah Hassan Mohtaz was emboldened to return from exile in Turkmenistan, where he had been living under the protection of the Russian government. At first there was talk of all of the eastern provinces of Iran splitting from the rest of the country, with Mashhad as the new capital, but the instability of the coup and the failure of Buzhazi and the Qagevs to form a government postponed such discussions. Perhaps all Mohtaz had to do was encourage the faithful to jihad, continue to raise money to fund his insurgency, and wait—Tehran might drop right back into his hands soon enough all by itself.
Three full divisions of the Revolutionary Guards Corps, over one hundred thousand strong, were based in and around Mashhad, nearly the entire surviving complement of frontline elite troops. Most of the Pasdaran forces, two divisions, were infantry, including two mechanized infantry brigades. There was one aviation brigade with counterinsurgency aircraft, attack and assault helicopters, transports, and air defense battalions; one armored brigade with light tanks, artillery, and mortar battalions; and one special operations and intelligence brigade that conducted demolition, assassination, espionage, surveillance, interrogation, and specialized communications missions such as propaganda broadcasts. In addition, another thirty thousand al-Quds paramilitary forces were deployed within the city itself, acting as spies and informers for the Pasdaran and theocratic government-in-exile.
The Revolutionary Guards Corps' headquarters and strategic center of gravity was Imam Reza International Airport, situated just five miles south of the Imam Reza shrine. However, all of the tactical military units at the airport were relocated to make room for a new arrival: an S-300OMU1 Favorit air defense regiment from the Russian Federation.
The S-300 strategic air defense system was considered one of the finest in the world, equal to the American PAC-3 Patriot missile system. An S-300 battery consisted of a long-range three-dimensional scanning acquisition radar, a target engagement and missile guidance radar, and twelve trailers each loaded with four missiles, along with maintenance, crew support, and security vehicles. One such battery was set up at the airport, with another northwest and a third positioned west of the city. The S-300 missile was effective against targets flying as low as thirty feet aboveground, as high as one hundred thousand feet, as fast as Mach 3, as far out as one hundred and twenty miles, and deadly against even low-flying cruise missiles and theater ballistic missiles.
The S-300s were augmented by the Tor-M1 air defense system, which were tracked armored vehicles that fired eight high-speed, short-range radar-guided anti-aircraft missiles from vertical launch tubes. The Tor-M1 was designed to protect mobile headquarters vehicles, vehicle marshaling areas, refueling areas, and ammunition dumps from attack helicopters, unmanned aerial vehicles, and low-flying subsonic tactical bombers. Although the Tor-M1 had a crew of three, it was designed to be a "set and forget" system, allowing for fully autonomous engagements, or it could be tied into the S-300's fire control system to form an integrated air defense system. Together they formed an almost impenetrable shield around Mashhad.
That day, Mashhad was one of the most heavily defended cities on planet Earth...and it was about to be put to the test.
About two hours before dawn, the first alert was issued from the long-range air defense radar at S-300 battery number two, located thirty miles northwest of Mashhad: "Alarm, alarm, alarm, this is Syeveer battery, high-speed low-altitude target inbound, bearing two-eight-zero, range one-fifty, velocity nine-six-five, altitude nine-zero."
"Syeveer, this is Tsentr, acknowledged," the tactical action officer, Captain Sokolov, responded. His tactical display showed three high-speed, low-altitude targets heading toward Mashhad. "Contact, sir," he reported to the regimental commander. "Looks like a terrain-following bomb run, right where you thought they'd be."
"Completely predictable," Colonel Kundrin, the air defense regimental commander, said confidently. As if sensing that something might happen that morning, he had been dressed and at his post in the regimental air defense command center on the top floor of the administration building at Reza International hours earlier. "The planes may change over the years, but the tactics remain the same. We placed that battery in perfect position—the bomber is trying to terrain-mask down the valley, but the mountains funnel right down to where we placed that battery. A fatal flaw in their mission planning. He can't continue straight ahead, and if he pops up over the ridges he'll be exposing himself even more."
"Too fast and too low for a B-2 stealth bomber—this must be a B-1 bomber," Sokolov surmised. "And they haven't launched their hypersonic cruise missiles either."
"I don't think they have any stealth bombers left after President Gryzlov and General Darzov expertly pounded their bases and caught the fools flat-footed on the ground," Kundrin said. "Besides, this is not the American air force we're up against—it's just McLanahan, the general that went crazy up in space. He's probably fired all his missiles already. Tell Syeveer to engage at optimal range, and be sure to watch for a trailing aircraft. If he's got more than one bomber, he'll either be in close trail or attacking from a different axis. I don't want anyone to slip inside."
Sokolov relayed the order. "Order to engage confirmed, sir, fifteen seconds to go...wait one! Sir, Zapat battery reports new hostile target inbound, bearing two-five-zero, range one hundred, altitude one hundred, speed eight-seventy and increasing!" Zapat was the westernmost battery, situated fifty miles west of Mashhad.
"I knew it! Predictable, all too predictable," Kundrin said happily. "Looks like we placed that number three battery in a perfect place too—covering the Binalud ridgeline west of the city. If I were to plan an attack on the airport, I'd hug the ground along the ridge, then pop around the end of the ridge and launch missiles right at rollout. That's exactly what McLanahan did—and we were in exactly the right spot to nail him! He'll have his bomb doors open and his radar signature will be massive! Tell Zapat to engage when ready!"
Each battery had three missile trailers, separated by several miles but linked to each other via microwave datalink, each carrying four 48N6 vertical-launch interceptor missiles which were already raised to launch position. Once the order to attack was given and the proper attack mode set—launch at optimal range—the engagement was virtually automatic. As soon as the target came within range, a nitrogen gas catapult pushed the missile out of the launch tube to a height of about thirty feet and the rocket motor ignited, accelerating the missile to greater-than-a-mile-per-second velocity in less than twelve seconds. Three seconds later, a second missile automatically fired to assure a kill. The S-300's missiles climbed to an altitude of only twenty thousand feet, guided to a predicted intercept point.
"Status?" the regimental commander asked.
"Batteries engaging targets, four missiles in the air," Sokolov reported. "Targets making only minimal evasive maneuvers and little jamming. Solid lock-on."
"The last act of overconfidence," Kundrin said. "They have no room to maneuver in any case. Too bad they're unmanned aircraft, eh, Captain?"
"Yes, sir. I'm concerned about those T-waves, or whatever they hit our fighter with."
"We'll see in a moment, won't we?"
"Missiles tracking perfectly...targets making slightly more aggressive maneuvers...channel-hop away from jamming, still locked on...three...two...one...now."
There were no other reports from the tactical action officer, which confused the regimental commander. "TAO, report!"
"Sir...sir, both missiles reporting ground contact!" Sokolov said in a low, confused voice. "Negative warhead detonation. Complete miss!"
"Release batteries and launch again!" Kundrin shouted. "Target range and bearing?"
"Second volley processing...missile three launched...missile four launched," Sokolov said. "Target range nine-zero, bearing steady at two-eight-zero."
"What of battery three? Status?"
"Battery three engagement..." And then his voice cut off with a sharp intake of breath.
Kundrin flew out of his seat and stared at the display. It was unbelievable..."They missed?" he exclaimed. "Another ground impact?"
"Battery three re-engaging...missile three launch...missile four..."
"Say range and bearing on battery three's target?"
"Range eight-zero, bearing steady at two-five-zero."
"That...that doesn't make sense," Kundrin said. "Both target bearings did not change even though they fell under attack? Something's not—"
"Sir, batteries two and three second-engagement missiles show ground impact as well!" Sokolov said. "All engagements missed! Battery two re-engaging. Battery three—"
"Negative! All batteries tight!" Kundrin shouted. "Inhibit auto engage!"
"Repeat that last, sir?"
"I said, all batteries tight, inhibit auto engagement!" Kundrin shouted. "We're being meaconed!"
"Meaconed? You mean, jammed, sir?"
"They're broadcasting false targets on our displays and making us fire at ghosts," Kundrin said.
"But we have full countermeasures and anti-jam algorithms in place, sir," Sokolov said. "Our systems are in perfect working order."
"We're not being jammed, dammit," Kundrin said. "Something's inside our system. Our computers believe they are processing actual targets."
The command network phone rang; only the regimental commander could answer it. "Tsentr."
"This is Rayetka." It was General Andrei Darzov himself, calling from Moscow. "We copied your notification of an attack response, but now we see you have canceled all engagements. Why?"
"Sir, I think we're being meaconed—we're reacting to false targets generated by our own sensors," Kundrin said. "I've inhibited automatic responses until..."
"Sir, battery two S-300 and Tor units receiving automatic engagement commands and are preparing to launch!" Sokolov shouted.
"I gave no such orders!" Kundrin shouted. "Countermand those orders! All batteries tight!"
"Tsentr, are you positive those are false targets?" Darzov asked.
"Every missile launched so far has hit the ground," Kundrin said. "Not one of our units has reported visual, optronic, or noise contact even though the targets are at very low altitude."
"S-300 battery two launching against new multiple inbound high-speed targets!" Sokolov reported. He ran over and pushed the communications officer out of the way, slapping on his headset. "Syeveer and Zapat batteries, this is Tsentr TAO, batteries tight, repeat, batteries tight! Ignore the computer's indications!" He hurriedly made out a date-time code for authentication—but as he did so, he watched as still more S-300 and Tor-M1 units launched missiles. "All units, this is Tsentr TAO, stop launch! Repeat, stop launch!"
"Stop those damned units from launching, Captain, now!" Kundrin shouted. There were now more targets appearing on the display—flying in exactly the same tracks, speed, altitude, and bearing as the first sets of targets! Soon battery one, the S-300 company at Reza International Airport, was beginning to launch missiles. "Rayetka, this is Tsentr, we're picking up more inbound hostile targets, but they're flying the exact same speed, altitude, and track as the first hostiles! Recommend we stop all responses and go to standby on all sensors. We're being spoofed, I'm positive."
There was a long pause, with the command net crackling and popping from the shifting encryption decoding routines; then: "Tsentr, this is Rayetka, deploy Fanar. Repeat, deploy Fanar. Stand by for engagement authentication."
"Repeat that last, Rayetka?" Kundrin asked. For God's sake, the regimental commander cried to himself, I just recommended to the guy that we shut everything down—now Darzo wants to roll out the biggest gun and the biggest sensor they had! "Repeat, Rayetka?"
"I said, deploy Fanar and stand by for engagement authentication," the order came back. It was followed by an authentication code.
"I copy, Rayetka, moving Fanar to firing position, standing by for engagement authentication." Darzov must be getting desperate, Kundrin thought. Fanar, the anti-spacecraft laser, was probably their last chance. The anti-aircraft artillery units scattered around Mashhad had no chance against fast, low-flying bombers. He picked up his regiment's command network phone: "Security, this is Tsentr, move Fanar to firing position and notify the crew to prepare to engage enemy aircraft." He gave the security commander an authentication code to move the trucks.
"Sir, we managed to get all units to respond to a weapons-tight order," Sokolov said. "We're down to twenty percent primary rounds available."
"Twenty percent!" Shit, they wasted eighty percent of their missiles on ghosts! "They had better be reloading, dammit!"
"We're in the process of reloading now, sir," Sokolov went on. "The Tor-M1 units will be done within fifteen minutes, and the S-300 units will be done before the hour."
"Get on it. The real attack may be happening at any moment. And make sure they do not respond to any more targets unless they have optronic verification!" Kundrin rushed to the exit, down the corridor, out the emergency exit, and up to the roof of the administration building. From there, using night-vision binoculars, he could see the progress of the security units.
The four Fanar trucks were just emerging from their hiding places. They had been hidden in a tunnel that ran under the runways which allowed vehicles to go from one side of the airport to another without going all the way around the runways. They were headed for a firefighting training pad on the north side of the runways, which had some old fuel tanks arranged to look like an airliner which could be filled with waste jet fuel and ignited to simulate a crashed airliner. The command vehicle was just now unfolding the huge electronically scanned radar antenna and datalink mast, which would allow the radar to tie into the S-300 fire control network.
Kundrin's secure portable radio crackled to life: "Tsentr, this is Rayetka," Darzov spoke. "Status."
"Fanar deployment under way, sir," Kundrin replied.
"Tsentr, this is the TAO," Sokolov radioed.
"Stand by, TAO," Kundrin said. "I'm talking to Rayetka."
"They are setting up on the southeast pad as directed?" Darzov asked.
Southeast pad? There was a fighter alert pad on the southeast side, but it was still in use by Revolutionary Guards Corps tactical attack helicopters and also as secure parking for the Russian transports. They had never briefed using it to employ the anti-spacecraft laser. "Negative, sir, we're using the north firefighting training pad, as briefed."
"Acknowledged," Darzov said. "Proceed."
Moments later, the TAO burst through the door to the roof observation post. "Stop, sir!" he shouted.
"What in hell is going on, Sokolov? What are you doing up here?"
"The authentication from Rayetka—it was not valid!" Sokolov said. "The order to deploy Fanar was not valid!"
"What?" A dull chill ran through Kundrin's head. He had assumed that because the person on the radio used the proper code name and was on the proper encrypted frequency that he was who he said he was and gave a valid order—he didn't wait to see if the authentication code checked...
...and he realized that he had just told whoever it was on the other end of that channel exactly where Fanar was located!"
He frantically raised his radio to his lips: "Security, this is Tsentr, cancel deployment, get those trucks back in hiding!" he shouted. "Repeat, get them into—!"
But at that exact moment there was a flash of light, and milliseconds later an impossibly thunderous explosion, followed by several more in quick succession. Kundrin and Sokolov were blown off their feet by the first concussion, and they frantically crawled away as crashing waves of raw heat roiled over them. They could do nothing but curl up into protective balls and cover their ears as the explosions continued one after the other.
It seemed to last an entire hour, but it was actually over in less than twenty seconds. Kundrin and Sokolov, their ears ringing from the deafening noise, crawled over to the shattered front of the administration building and peered out across the runways. The entire area north of the runways was on fire, centered on the firefighting training pad. The fire on the pad itself—obviously the burning chemicals used by the laser—seemed so hot and intense that it was radioactive. The alert aircraft parking area to the southeast had been hit too—every helicopter and transport was on fire.
Then they heard them, and in the brilliant reflection of the fires they soon saw them too, as plainly as if in daytime: a pair of American B-1 bombers, flying right down the runway. They obviously knew that all of the air defense units had been ordered to shut down their systems and not open fire. The first one wagged its wings as it passed by the administration building, and the second actually did an aileron roll, flying less than two hundred feet aboveground. When they finished their little airshow spectacle, they ignited afterburners, sped off into the night sky, and were soon out of sight.
LAS VEGAS, NEVADA
THAT SAME TIME
Stacy Anne Barbeau loved casinos, and she spent quite a bit of time in them on the Mississippi River in Louisiana and on the Gulf Coast in neighboring Mississippi. But this was the first time in many years that she had been in a big Las Vegas casino, and she was impressed. They were much more than gambling halls now—they were spectacular destinations, a sensory bombardment not only of lights, colors, and sounds, but of scenery, landscaping, architecture, and art that was truly amazing. The last time she was here, the decorations seemed cheesy and campy, almost Disneyesque. Not anymore. It was definitely Las Vegas elegant—bright, a little gaudy, loud, and extravagant, but it was elegant nonetheless.
"You know what I love the most about these places, darlin'—you can be completely anonymous so easily, even dressed like this," Barbeau said to her assistant Colleen Morna as they strode from the hotel elevators through the wide, sweeping hallway and across the rich red carpeting of a very large Italian-themed casino on the Strip in Las Vegas. She was wearing a silvery cocktail dress, diamond earrings and necklace, and carrying a mink stole, but except for the frequent and appreciative glances, she felt as if she was just another part of the scenery. "So where is 'Playgirl'?"
"Private poker room in the back," Morna said. She produced what appeared to be a thick ruby-encrusted brooch and pinned it to Barbeau's dress. "This is all you need to get in."
"It's ugly. Do I have to wear it?"
"Yes. It's an identification and tracking transponder—an RFID, or radio-frequency identification tag," Morna said. "They've been tracking us ever since I picked it up a half hour ago while you were getting dressed. They track all your movements; it sends information to all the cashiers, croupiers, maître d's, security, hotel staff, and even to the slot machines about who you are, what you play or do, and—more importantly to them, I'm sure—how much is left in your account. The security staff watches you with their cameras and automatically compares your description to their database to keep an eye on you while you're on the property. I think if you took more than one or two wrong turns anywhere around this place, they'd send a couple hospitality guys after you to steer you in the right direction."
"I like the sound of that—'hospitality guys,'" Barbeau cooed. "I don't much like the idea of being tagged like a brown bear in the woods, though."
"Well, keep it with you, because it's your room key, access to your line of credit, your charge card, and your admission pass to all the shows and VIP rooms—again, you don't need to know a thing because these guys will escort you everywhere you want to go. Anywhere."
"But they don't know who I am, do they?"
"I would assume they know exactly who you are, Senator," Morna said, "but this is Vegas—here, you are whoever you want to be. Tonight you're Robin Gilliam from Montgomery, telecommunications and oil money, married but here alone."
"Oh, do I have to be from Alabama?" she deadpanned. Morna rolled her eyes. "Never mind. So how did I get into this private poker room if I'm not who I say I am?"
"A fifty thousand dollar line of credit is the best way to start," Morna said.
"You used the billing codes from the White House for this trip for a line of credit in the casino? Smart girl."
"It's just to get us in the door, Senator—don't actually use any of it, or the sergeant at arms will crucify you," Morna said.
"Oh, pish on him—he's an old fuddy-duddy," Barbeau said.
Morna rolled her eyes, silently hoping she was kidding. Washington careers were ended by a lot less. "Everything is all set. The management is as attentive as they are discreet. They'll take good care of you. I'll be in the room next door to yours if you need me, and I've got a casino employee bought and paid for that will tell me exactly where you are at all times."
"Thanks, but I don't think I'll need a wingman tonight, darlin'," Barbeau said in her best man-slaying voice. "Captain Hunter 'Boomer' Noble will go down as easy as catching catfish in a barrel."
"What do you plan to do, Senator?"
"I plan to show Captain Noble the best way to get ahead in the United States Air Force, which is very simple: Don't cross a United States senator," she said confidently. She stuck out her chest and moved the mink aside. "I'll show him a couple advantages of pleasing me instead of opposing me. You're sure he's here?"
"He checked in last night and has been playing poker all day long," Morna said. "He's doing pretty well too—he's up a little."
"Oh, I'll make sure he's up, all right," Barbeau said. "Trust me."
"I know where his suite is—it's right down the hall from ours—and if he takes you there my guy will tell me," Morna went on.
"Any other ladies with him?"
"Just a few that have stopped by briefly at the table—he hasn't invited any of them to his room."
"We'll see about that, won't we?" Barbeau said. "Don't wait up, sugar."
Exactly as Colleen said, the casino staff knew she was coming without a word being spoken. As Barbeau left the main casino floor and began walking toward the ornate gold entryway of the private poker room, a man in a tuxedo with a communications earpiece in one ear smiled, nodded, and said, "Welcome, Miss Gilliam," as she passed by.
As she approached the doors she was met by a tall, good-looking man in a tuxedo and a woman in a tuxedo suit and skirt, carrying a beverage tray. "Welcome, Miss Gilliam," the man said. "My name is Martin, and this is Jesse, who will be your attendant for the rest of the evening."
"Why thank you, Martin," Barbeau said in her best Southern accent. "I'm quite taken by this extraordinary level of attention."
"Our goal is to assist you in any way possible to have the best evening while a guest at the hotel," Martin said. "Our motto is 'Anything at All,' and I will be here to be sure all your desires are met tonight." The waitress handed her the glass. "Southern Comfort and lime, I believe?"
"Exactly right, Martin. Thank you, Jesse."
"My job is to make you comfortable, get any dinner or show reservations you may like, get you a seat at any gaming table you'd prefer, and make any introductions while you're in the private hall. If there's anything at all you'd like—anything at all—please do not hesitate to tell Jesse or myself."
"Thank you, Martin," Barbeau said, "but I think I'd like to just...you know, prowl around a little bit to get comfortable. That's all right, isn't it?"
"Of course. Whenever you need anything, just motion to us. You don't have to look for us—we'll be looking out for you."
It was a very secure feeling, Barbeau thought, to know that she was being watched every second. She took her drink and began to stroll around the room. It was plush and ornate without being too ostentatious; there was just a hint of cigar smoke, not too bad, almost pleasant and reassuring. A room in the back had several sports games on huge wide-screen flat-panel monitors, with women who definitely didn't look like spouses hanging onto the shoulders of the spectators—male and female alike.
What happens in this place, Stacy thought as she took a sip of her drink, definitely stays in this place.
After a short hunt she finally found him, at a card table in the back: Hunter Noble, dressed in a T-shirt and jeans, with a single thick-link gold chain around his neck, an old-style metal POW bracelet on one wrist, and a black nylon Velcro watchband on the other wrist with its protective watch flap closed. He had an impressive stack of chips in front of him, and only two players and the dealer at the table with him—and the other players definitely looked perturbed, their chip stacks much lower than his, as if they were frustrated at being beat by this young punk. One of the other players had a cigarette in an ashtray beside him; Noble had an ashtray beside him too, but it was clean and empty.
Now that she saw him in his "native habitat," she liked what she saw. He was the perfect cross between lean and muscular—a naturally toned body without having to do a lot of weight lifting, not like McLanahan's chunky muscularity. His hair was short and naturally teased, without having to mousse it, which had to be the most unmanly thing Stacy had ever seen in her life. His movements were slow and easy, although she noticed his quick eyes when cards and chips started flying across the table in front of him. He certainly didn't miss much...
...and at that moment his eyes rested on her...and he didn't miss anything there, either. He smiled that mischievous naughty-boy smile, and his quick eyes danced, and she instantly felt herself being visually undressed once more—then, just as quickly, his attention was back to his game.
It was not too long afterward that Barbeau saw Martin supervising the dealer counting up Noble's winnings. He saw him ask Martin a question, the host responded, and soon he sauntered over to her table with a drink and a cigarette in his hand. "Pardon me, Miss Gilliam," he said, speaking very formally but with that same mischievous smile, "but I took the liberty of asking Martin who you were, and I thought I'd introduce myself. My name is Hunter Noble. I hope I'm not intruding."
Barbeau sipped her drink but eyed him over the rim of the glass, making him wait while she surveyed him. He simply stood before her patiently with that playful boyish smile on his face, standing casually but provocatively as well, as if he had no doubt that she would invite him to sit down. Well, shit, she thought, the guy flies hypersonic spaceplanes for a living—a mere woman isn't going to rattle him. "Of course not, Mr. Noble. Would you care to sit down?" Barbeau responded just as formally, enjoying playing the game of being strangers.
"Thank you, I would." He took a chair beside her, set his drink down, then leaned toward her. "Senator Barbeau? Is that you?"
"Captain Hunter 'Boomer' Noble," she said in response. "Fancy meeting you here, sir."
"Fancy nothing, Senator. Did you track me down here?"
"I don't know whatever you mean, Captain," Barbeau said. "The assistant hotel manager here happens to be a friend of mine, and he invited me to this wonderful VIP room when I came to town." She looked him over once again. "Where's your RFID tag, Captain?"
"I don't wear those things—I like tipping in cash and I can unlock my own room door without Big Brother doing it for me."
"I think it's fun, being surveilled all the time. Makes me feel very secure."
"You'll get tired of it," he said moodily. "You're here to shut down Dreamland, aren't you, Senator?"
"I'm here to talk with the SEALs who tried to assault the place, speak with General Luger about his actions, and report to the President," she replied.
"Then why are you here? Are you spying on me?"
"Why, Captain Noble, you sound like a man with something to hide," Barbeau said. "But I am surprised, quite frankly, to find a young Air Force captain who makes less than seventy thousand dollars a year before taxes here in a VIP gambling room, where the price of admission is usually a fifty-thousand-dollar line of credit with the casino, with such a large stack of chips in front of him."
"Playing poker for money is not against Air Force regulations, Senator. Neither is spending a good deal of my bachelor take-home pay on playing cards. Do you investigate guys who spend that much on cars or cameras?"
"I don't know of anyone who's been blackmailed by bookies or loan sharks because they buy camera gear," Barbeau said. "Being a habitual gambler certainly does look...how shall I say it, unseemly? For someone in such a highly critical job as yours, being such a gambling devotee—or perhaps even a gambling addict?—might look very suspicious to some."
"I'm not addicted to gambling," Boomer said defensively. The senator's eyes twinkled—she knew she had hit a nerve. "But why this charade, Senator? Why this campaign to destroy the program? You're opposed to the Black Stallion and the space station—fine. Why take the political opposition so personally?"
"I'm not an opponent of the XR-A9 project, Captain," Barbeau said, sipping her drink. "I think it's a remarkable piece of technology. But the space station has many very powerful opponents."
"Like Gardner."
"Many opponents," Barbeau repeated. "But some of the technology you use is of great interest to me, including the Black Stallion."
"Not to mention scoring some points with folks in the White House and dozens of defense contractors, too."
"Don't try to play politics with me, Captain—my family invented the game, and I learned from the best," Barbeau said.
"I see that. You're more than willing to destroy military careers for your own political gain."
"You mean General McLanahan? Perfect example of a smart, dedicated guy wading into political waters that were way over his head," she said dimissively, taking another sip. She was finally starting to feel relaxed, immersed in an atmosphere in which she was very comfortable...no, not just comfortable: one in which she was in control. McLanahan had destroyed himself, and because Hunter Noble cared about him, he was going to go down next.
Captain Hunter Noble was pretty, and obviously smart and talented, but this was business, and he would become just another one of her victims...after she had a little fun with him!
"He'll come out okay—as long as he backs off and lets me tell the White House what is best for the Air Force," Barbeau went on casually. "McLanahan's a war hero, for God's sake—everybody knows that. Very few people know what happened in Dreamland and Turkey." She snapped her fingers with a wave of her wrist. "It can be swept under the rug like that. With my help and with his maximum cooperation, he'll get off with a general court-martial and loss of his pension. But then he can get on with his life."
"Otherwise, you'll let him rot in prison."
Stacy Anne Barbeau leaned forward, giving him a good look at her bosom underneath her silvery low-cut neckline. "I'm not here to make anyone miserable, Captain—least of all you," she said. "The truth is, I would like your help."
"My help?"
"Next to McLanahan, you're the most influential person attached to the space project," she said. "The general is done for if what he's done in Dreamland and in Turkey gets leaked out. I don't think he'll cooperate with me. That leaves you."
"What is this, a threat? You're going to try to destroy me too?"
"I don't want to attack you, Captain," she said in a low voice. She looked him straight in the eye. "To be honest, I'm quite taken by you." She saw the look of surprise in his face and knew she had him by the balls. "I've been attracted to you since I first saw you in the Oval Office, and when I saw you here, looking at me like you were—"
"I wasn't looking at you," he said defensively, not too convincingly.
"Oh yes you were, Hunter. I felt it. You did too." He swallowed but said nothing. "What I'm trying to say, Hunter, is that I can take your career in a whole new direction if you'd let me. All you need to do is let me show you what I can do for you."
"My career is just fine."
"In the Air Force? That's fine for eggheads and Neanderthals, but not for you. You're smart, but you're savvy and in control. Those are special qualities. They will get suppressed in the military under layers upon layers of old-school bullshit and endless, faceless bureaucracy—not to mention the possibility of dying in combat or up in space, flying a jet built by the lowest bidder.
"I'm offering you a step out of that hellish cattle-call existence, Hunter," Barbeau went on in a low voice, pumping as much sincerity into it as she could. "How do you think other men and women rise above corporate Pentagon mediocrity and advance their futures?"
"The general did it by being dedicated to the mission and his fellow crewmembers."
"McLanahan did it by being Kevin Martindale's whipping boy," Barbeau said firmly. "If he died in any of those missions he sent him on, Martindale would have just found another mindless robot to activate. Is that what you want? Do you just want to be McLanahan's sacrificial lamb?" Again, Boomer didn't reply—she could see the wheels of doubt churning in his head. "So who's looking out for you, Hunter? McLanahan won't be in a position to do it. Even if he doesn't go to prison, he'll have a federal conviction and a less-than-honorable discharge on his record. You'll wither away too out there if you blindly follow idealistic men like McLanahan."
He didn't say it, but she knew what he was asking himself: How do I get out of this? He was putty in her hands, ready for the next step. "Come with me, Hunter," she said. "I'll show you how to rise above the swamp that McLanahan has stuck you in. I'll show you the real world, the one outside of spaceplanes and shadowy missions. With my help, you can dominate the real world. Just let me show you the way."
"And what do I need to do?"
She looked deeply in his eyes, took a deep breath, then gently placed a hand on his left thigh. "Just trust me," she said. "Place yourself in my hands. Do what I tell you, and I'll take you to places, introduce you to the most influential people who really want to hear what you have to say, and take you through the real corridors of power. That's what you want, isn't it?" She could feel those rock-hard thighs jump at her touch, and couldn't wait for those long legs to straddle her. He was practically gasping for air like a marathoner at the end of a race. "Let's go."
He stood, and she smiled and took his hand as he helped her to her feet. He's mine, she thought...mine.
She felt a little dizzy as she got to her feet—one glass of whiskey, after a half day of skipping meals preparing for this trip, was doing her in. After she dealt with Hunter Noble, she vowed to treat herself and Colleen to a late-night supper in the suite and toast her success. First Gardner, then McLanahan, and now this studly hard-body military astronaut.
"May I help you in any way, Miss Gilliam?" her waitress Jesse asked, appearing as if out of nowhere. She reached out as if to help steady her.
"No thank you, Jesse, I'm fine," Barbeau said. She watched as Martin came over and looked as if he was going to physically restrain Noble, who was discreetly following her, but she raised a hand. "Mr. Noble and I are going to take a walk together," she said. "Thank you, Martin."
"If you need anything, Miss Gilliam, just pick up a phone or give a signal—we'll be right there," Martin said.
"Thank you so much. I'm having a wonderful time," Barbeau said gaily. She tipped him fifty dollars, then headed for the door. Hunter opened the door for her; Martin took the door from him, and she noticed him giving Noble a stern warning glare...and he didn't tip him either. Well, she thought, maybe "Playgirl's" reputation was wearing a bit thin in here. That would be another weakness of his to explore if he didn't cooperate.
They walked together without talking until reaching the elevator, and then she took him by his slender waist, pulled him closer, and kissed him deeply. "I've wanted to do this ever since I first saw you," she said, pressing herself tightly against him. He whispered something in return, but the music in the elevator seemed a little loud, and she couldn't hear him.
At their floor they were met by a floor attendant. "Welcome, Mr. Noble, Miss Gilliam," she said brightly, obviously notified by the ever-present hotel security system of their arrival. "Is there anything I can do for you tonight? Anything at all?"
"No, I've got this one all taken care of myself," Barbeau heard herself say, reaching down between his legs and stroking him. "But if you'd care to join us a little later, sugar, that'd be fine, just fine." And then she heard herself giggle. Did she just giggle? That Southern Comfort was hitting her harder than she thought. Never party on an empty stomach, she reminded herself.
As she passed Colleen's room she pretended like she stumbled a bit and banged into her door just to give her a warning that she was back, and then they were at the door to the suite. "You just relax and let me do the drivin' for now, big boy," she said, starting to untuck his shirt from his pants even before he had the door open. "I'll show you how we like to party down on the bayou."
PRESIDENT'S PRIVATE RETREAT, BOLTINO, RUSSIA
SEVERAL HOURS LATER
"Why haven't you answered my calls, Gardner?" President Leonid Zevitin thundered. "I've been trying for hours."
"I've got my own problems, Leonid," President Joseph Gardner said. "As if you hadn't noticed, I've got to deal with a little mutiny over here."
"Gardner, McLanahan has bombed Mashhad, Iran!" Zevitin cried. "He's destroyed several Russian transports and killed hundreds of men and women! You said he would be forced under control! Why haven't you dealt with him yet?"
"I've been briefed about the attack," Gardner said. "I've also been briefed about the target—an anti-spacecraft laser that was supposedly used to shoot down one of our spaceplanes. Wouldn't happen to know anything about that, would you, Leonid? What were all those Russian personnel and transports doing in Mashhad?"
"Don't change the subject!" Zevitin shouted. "The Duma is going to meet soon, and they're going to recommend a permanent change in military posture, including a call-up of ready reserves, mobilization of the army and strategic air forces, and dispersal of mobile ballistic missiles and submarine forces. Was this your plan all along, Gardner—have McLanahan act crazy, attacking targets all over the planet, and forcing us to respond as if we are going to fight a world war? Because this is exactly what it looks like!"
"You think I'm conspiring with McLanahan? The guy is nuts! He's completely out of control! He's attacked American military forces, taken over a top secret military base, and stolen several highly classified aircraft and weapons. No one has any contact with him for almost half a day—we think he might have committed suicide on the space station."
Well, Zevitin thought, that was the best news he's heard in a long time. "No one will believe any of this," he told Gardner. "You have got to give me something to tell my Cabinet and the leaders in the Duma, Joe, or this thing could spin out of control. How did he do that attack on Mashhad, Joe?"
"It's a thing they call 'netrusion,' Leonid," Gardner said. Zevitin's eyes widened in surprise—the American President was actually going to tell him! "Some of McLanahan's aircraft and spacecraft have a system where they can not only jam radar and communications, but actually insert bogus code and signals into an enemy system. They can reprogram, crash, or control computers, invade networks, inject viruses, all that egghead shit."
"This is astounding!" Zevitin exclaimed. Yes—astounding that you're telling me all this! "That's how the bombers made it over Mashhad?"
"They made the air defenses around the city react to false targets," Gardner said. "The air defense guys apparently shut down their missile systems so they wouldn't shoot at stuff that wasn't there, and that let the bombers slip in. McLanahan also hacked into their encrypted radio transmissions and gave them false orders, which allowed the bombers to locate and attack the laser site."
"If all this is true, Joe, then we must put a deal in place to share this technology," Zevitin said, "or at least pledge not to use it except in time of declared war. Can you imagine if this technology got into the wrong hands? It could devastate our economies! We could be thrown back into the Stone Age in a flash!"
"It's all McLanahan's geeks at Dreamland coming up with this stuff," Gardner said. "I'm going to shut Dreamland down and have that bastard McLanahan shot. I think he's left the space station and is back at Dreamland. He's ignored my orders and done what he pleases for too long. I've got a friend, a powerful senator, who's going to try to bring McLanahan out in the open, and when she does I'll nail his ass to my wall."
"Who is the senator, Joe?"
"I'm not ready to divulge the name."
"It will lend credibility to my arguments before the Duma, Joe."
There was a bit of a pause; then: "Senator Stacy Anne Barbeau, the majority leader. She went to Dreamland to try to meet with McLanahan or Luger to try to defuse this situation."
He's got the Senate majority leader spying for him? This couldn't be better. Zevitin's mind was racing ahead. Dare he suggest it...? "You don't want to do that, Joe," he said carefully. "You don't want to expose yourself or Barbeau any further. McLanahan is a very popular man in your country, is he not?"
"Yes, unfortunately he is."
"Then let me propose this idea, Joe: as over the Black Sea and over Iran, let us do the deed for you."
"What?"
"You told us where those bombers would be and when, and we took care of them for you; you told us about the spaceplane and put them in a position where we could strike—"
"What? You did what with the spaceplane...?"
"Bring McLanahan out into the open," Zevitin went on, almost breathlessly. "Have Senator Barbeau tell us where he is. I'll send a team in to sanction him."
"You mean, a Russian hit team?"
"You don't want McLanahan's blood on your hands, Joe," Zevitin said. "You want him out of the way because he's much more than an embarrassment to you—he's a danger to the entire world. He's got to be stopped. If you have a person on the inside, have him or her contact us. Tell us where he is. We'll do the rest, and you don't have to know anything about it."
"I don't know if I can do that..."
"If you were seriously considering dispatching him yourself, then you are serious about the danger he poses not just to world peace, but to the safety and very existence of the United States of America. The man is a menace, pure and simple. He is a wild dog that needs to be put down."
"That's exactly what I said, Leonid!" Gardner said. "McLanahan has not just crossed the line, but I think he's become completely unhinged! He's brainwashed his men to attack American troops...or maybe he's used that 'netrusion' shit to brainwash them. He's got to be stopped before he takes down the entire country!"
"Then we are of one mind, Joe," Zevitin said. "I'll give you a number to call, a safe and secure blind drop, or you can code a message through the 'hot line.' You need not do anything except tell us where he is. You need not know a thing. This will be completely deniable."
There was a long pause on the line; then: "All right, Leonid. Convince your people that America doesn't want war and has no designs on Russia, and we'll work together to stop McLanahan." And he hung up.
This was too good to be true! Zevitin exclaimed to himself. Two of the top politicians in the United States were going to help him assassinate Patrick McLanahan! But who to trust with this project? Not his own intelligence bureau—there were too many shaky alliances, too many unknowns for this type of job. The only person he could trust was Alexandra Hedrov. Her ministry certainly had agents who could do this job.
He went into his bedroom adjacent to his executive office. Alexandra was sitting alone in bed in the darkness. The speakerphone was on; he had hoped she would listen in and be ready to give him advice. She was a valuable adviser and the person he trusted more than anyone in the entire Kremlin. "So, my love," Zevitin said, "what do you think? Gardner and Barbeau are going to tell us where McLanahan is! I need you to assemble a team, get them into Nevada, and be ready to strike." She was silent. Her knees were drawn up to her chest, her head down, touching her knees, her arms wrapped around her legs. "I know, love, this is ugly business. But this is an opportunity we can't miss! Don't you agree?" She remained still. "Darling...?" Zevitin flipped on the light switch...and saw that she was unconscious! "Alexandra! What's happened? Are you all right?"
"I can help you there, Mr. President." Zevitin turned...and standing in his closet, concealed by the darkness, was a figure in a dark gray uniform, a combination of a flight suit and body armor...a Tin Man battle armor system, he realized. He carried a large weapon, a combination sniper rifle and cannon, in his arms. "Raise your hands."
He did as he was told. "Who are you?" Zevitin asked. He took a step backward...toward the light switch, which if he could flip it off and back on quickly would send an emergency signal to his security team. "You're one of McLanahan's Tin Men, aren't you?"
"Yes," the man said in an electronically synthesized voice.
"McLanahan sent you to kill me?"
"No," Zevitin heard a voice say. He turned...and there, wearing another Tin Man battle armor suit but with the helmet removed, was Patrick McLanahan himself. "I thought I'd do that myself, Mr. President."
Zevitin whirled, pushed McLanahan, lunged for the light switch, and managed to flip it off, then on again. McLanahan impassively watched as Zevitin furiously moved the switch up and down. "Very impressive feat, sneaking past my guards, into my private residence, and into my bedroom," Zevitin said. "But now you'll have to fight your way past a hundred trained commandos. You'll never make it."
McLanahan's armored left hand snapped out, closed around Zevitin's wrist, and squeezed. Zevitin thought his hand had popped completely off his arm, and he sunk to his knees in pain, screaming in agony. "It was about sixty-two guards, and we took care of them all on the way in," McLanahan said. "We also bypassed your security system's link to the army base at Zagorsk—they'll think everything is normal."
"'Netrusion,' I believe you call it?"
"Yes."
"Ingenious. The whole world will know about it by tomorrow, and soon we'll unleash it on the rest of the world when we reverse-engineer the technology."
McLanahan's right hand whipped out and closed around Zevitin's neck. His face was purely impassive, emotionless. "I don't think so, Mr. President," he said.
"So. You've become an assassin now? The great air general Patrick Shane McLanahan has become a common killer. Betraying your oath and disobeying your commander-in-chief weren't enough for you, eh? Now you're going to commit the ultimate mortal sin and destroy a life for no other reason than a personal vendetta?"
McLanahan just stood there, no expression on his face, looking directly into Zevitin's sneering face; then he nodded and replied simply: "Yes, Mr. President," and he effortlessly squeezed his fingers together and clenched them until the body in his grasp went completely limp and lifeless. The two Americans stood there for a minute, watching the blood pour onto the polished wood floor and the body make a few twitches, until finally McLanahan let the body fall from his grasp.
"Didn't think you'd do it for a second there, boss," Major Wayne Macomber said in his electronic voice.
Patrick went into the closet and retrieved his helmet and electromagnetic rail gun. "I've been thinking about nothing else for a long time, Whack," he said. He put on his helmet and hefted his rail gun. "Let's go home."
MAIN LODGE, NAVAL SUPPORT FACILITY THURMONT (CAMP DAVID), MARYLAND
THAT SAME TIME
This is all going to shit, President Joseph Gardner said to himself. But it's not my damned fault. McLanahan needs to be gone, soonest. If he had to make a deal with the devil to do it, so be it.
He went from his private office back into the bedroom suite of the Camp David presidential retreat, where he found his houseguest—the staff sergeant he'd had aboard Air Force One—standing at the wet bar on the far side of the room, wearing nothing but an almost transparent negligee, open all the way down, her hands enticingly behind her. Damn, he thought, that was one hot future Air Force officer! "Hey, honey, sorry to take so long, but it couldn't wait. Fix us a drink, will you?"
"Fix it yourself, you fucking sleazeball," he heard, "then go shove it up your ass." Gardner whirled around...
...and found none other than Senator Stacy Anne Barbeau standing before him! "Stacy!" he blurted. "How in hell did you get in here?"
"Compliments of General McLanahan," he heard. He turned the other way and saw a figure in some sort of futuristic body armor and helmet standing by the wall. He heard a sound behind him and saw yet another figure in head-to-toe body armor and helmet, carrying a huge rifle, step into the suite.
"Who are you?" the President exclaimed. "How did you get in here?" He finally recognized who they were. "You're McLanahan's Tin Men! He sent you to kill me?"
"Never mind them, Joe!" Barbeau cried. "What was all that about? You made a deal with Zevitin to have McLanahan assassinated by Russian agents?"
"It's starting to look like a damn good idea, Stacy, don't you think?" Gardner asked. "This is exactly what I was afraid of—McLanahan is going to assassinate all his enemies and take over the government!"
"So to plan a strategy to deal with the crisis you bring a bimbo to Camp David, screw around with her awhile, then make a deal with the president of Russia to have an American general assassinated?"
Gardner whirled around. "Help! Help me!" he screamed. "I'm in the suite and there are armed men in here! Get in here! Help!"
One of the armored figures strode over to Gardner, put a hand behind his neck, and squeezed. Gardner's vision exploded into a cloud of stars from the sudden intense pain. All of his strength immediately left his body, and he collapsed to his knees. "They're all out for now, Mr. President," the armored figure said. "No one can hear you."
"Get away from me!" Gardner sobbed. "Don't kill me!"
"I should kill you myself, you piece of shit!" Barbeau shouted. "I wanted McLanahan out of the way, maybe embarrass or disgrace him if he didn't cooperate, but I wasn't going to kill him, you stupid idiot! And I certainly wasn't going to make a deal with the Russians to do it!"
"It's McLanahan's fault," Gardner said. "He's crazy. I had to do it."
The figure grasping Gardner's neck released him. Gardner collapsed to the floor, and the armored figure stood over him. "Listen to me carefully, Mr. President," the figure said in a weird computerized voice. "We've got you on tape admitting to conspiring with the Russians to shoot down American bombers and the Black Stallion spaceplane, and conspiring with the president of Russia to have Russian agents enter the country to assassinate an American general."
"You can't kill me!" Gardner cried. "I am the President of the United States!"
The figure slammed an armored fist right beside the President's head, then two inches down through the resawn maple floor and concrete foundation in the bedroom suite. Gardner screamed again and tried to scurry away, but the figure grasped him by the throat, putting his helmeted face right up to the President's. "I can kill you easily, Mr. President," the figure said. "We stopped the Navy SEALs, we stopped the Secret Service, and we stopped the Russian air force—we can certainly stop you. But we're not going to kill you."
"What do you want then?"
"Amnesty," the figure said. "Full and complete freedom from prosecution or investigation for everyone involved in actions against the United States or its allies from Dreamland, Battle Mountain, Batman, Tehran, and Constanţa. Full retirements and honorable discharges for everyone who doesn't want to serve under you as their commander-in-chief."
"What else?"
"That's all," the other figure said. "But to ensure that you'll do as we say, the Tin Men and CID units will disappear. If you cross us, or if anything happens to any of us, we'll come back and finish the job."
"You can't stop us," the first Tin Man said. "We'll find you no matter where you try to hide. You won't be able to track or detect us, because we can manipulate your sensors, computer networks, and communications any way we choose. We'll monitor all your conversations, your e-mails, your movements. If you betray us, we'll find you, and you'll simply disappear. Do you understand, Mr. President?" He looked at the two women in the room. "That goes for you two as well. We don't exist—but we'll be watching you. All of you."
## EPILOGUE
He that falls by himself never cries.
—TURKISH PROVERB
LAKE MOJAVE, NEVADA
SEVERAL WEEKS LATER
The young boy cast a fishing line into Lake Mojave from his spot at the tip of a rocky point beside the long, wide boat-launching ramp. Lake Mojave was not really a lake, just a wide spot of the Colorado River south of Las Vegas. It was a popular winter venue for seasonal residents, but they could begin to feel the onset of summer heat even now in early spring, and you could sense the stirring in the place that people were itching to leave. Not far behind the boy was his father, in shorts, sunglasses, nylon running sandals, and Tommy Bahama embroidered shirt, typing on a laptop computer in the shade of a covered picnic area. Behind him in the RV park, the "snowbirds" were packing up their campground and preparing to take their trailers, campers, and RVs to gentler climes. Soon only the most die-hard desert-lovers would stay to brave southern Nevada's brutally hot summer.
Amidst the bustle of the campground the man heard the sound of a heavier-than-normal car. Without turning or appearing to notice, he escaped out of his current program and called up another. With a push of a key, a remote wireless network camera on a telephone pole activated and began automatically tracking the newcomer. The camera zeroed in on the vehicle's license plate, and in a few seconds it had captured the letters and numbers and identified the vehicle's owner. At the same instant, a wireless RFID sensor co-located with the camera read a coded identification beacon broadcast from the vehicle, confirming its identity.
The vehicle, a dark H3 Hummer with tinted windows all the way around except for the windshield, parked in the white gravel parking lot between the marina restaurant and the launching ramp, and three men alighted. All wore jeans, sunglasses, and boots. One man in a safari-style tan vest stayed by the vehicle and started scanning the area. The second man wore an untucked white business shirt with the collar open and the sleeves rolled up, while the third also wore an open safari-style tan vest.
The man at the picnic table received a tiny beepbeepbeep in his Bluetooth wireless headset, telling him that a tiny millimeter-wave sensor set up in the park had detected that one of the men was carrying a large metallic object—and it wasn't a tackle box, either. The second man in the vest stopped about a dozen paces from the picnic area beside the ramp to the boat-launching ramp next to a garbage can and began scanning the area like the first. The third man walked up to the man at the picnic table. "Hot enough out here for you?" he asked.
"This is nothing," the man at the picnic table said. He set his laptop down, got to his feet, turned to the newcomer, and removed his sunglasses. "They say it'll get above a hundred by May and stay above a hundred and ten for all of June, July, and August."
"Swell," the newcomer said. "Cuts down on visitors, eh?" He looked past the man and to the boy fishing beside the boat ramp. "Cripes, can't believe how tall Bradley's getting."
"He'll be taller than the old man any day now."
"No doubt." The newcomer extended a hand. "How the hell are you, Patrick?"
"Just fine, Mr. President," Patrick McLanahan said. "You?"
"Fine. Bored. No, bored out of my skull," former President of the United States Kevin Martindale replied. He looked around. "Kind of a bleak place you got here, Muck. It's not San Diego. It's not even Vegas."
"The desert grows on you, especially if you come here in late winter and experience the gradual change in the temperature," Patrick said.
"You planning on staying?"
"I don't know, sir," Patrick said. "I bought a homesite and a hangar at the airpark in Searchlight. Don't know if I'm ready to build yet. The place is growing. I'm homeschooling Bradley now, but the schools here are getting better, they say, as more and more folks move to the area."
"And Jon Masters is just a little ways up Highway 95."
"Yeah, and he bugs me just about every day to come work for him, but I'm not sure," Patrick admitted.
"That hotshot astronaut Hunter Noble signed up with him. I heard he's a vice president already. But I'm sure they'll make a place for you if you want it."
"Been there, done that."
"There's another thing that we've both done before, Patrick," Martindale said.
"I figured you'd be showing up sooner or later about that."
"You have the Tin Men and the CIDs, don't you?"
"The what?"
"You're a horrible liar," Martindale said with a laugh.
"Is there any use trying to lie? I'm sure your intelligence network is good..."
"As good as the one you've reportedly built? I doubt it. I doubt it very much," the former President said. "Listen, my friend, you're still needed. The country needs you. I need you. Besides, the stuff you have stashed away is government property. You can't keep it." Patrick gave him a direct glance—just a fleeting one, but the meaning was loud and clear. "Okay, you probably can keep it, but you shouldn't just squirrel it away. You can do an awful lot of good with it." Patrick said nothing. Martindale took off his sunglasses and wiped them with a shirttail. "Heard the latest about Persia?"
"About the new president being assassinated?"
"When that hits the news the entire Middle East will go bonkers again, and Mohtaz will re-emerge from whatever rock he crawled under when the Russians left and claim the presidency again. The people want Queen Azar to take control of the government until new elections can be held, but she insists the prime minister, Noshahr, take charge."
"She's right."
"Noshahr's a bureaucrat, a bean counter. He can't run the country. Azar or Buzhazi should take charge under emergency authority until elections are held."
"He'll be fine, sir. If he's not, Azar will go to Parliament and recommend someone else. Buzhazi flat out won't do it."
"You think she'll ask Saqqez, the deputy prime minister?"
"I hope not. He's taken too many trips to Moscow to suit me."
Martindale nodded knowingly. "I knew you were keeping tabs on this stuff," he said. "Speaking of Moscow—what do you think about that replacement for Zevitin, Igor Truznyev, the former FSB chief?"
"He's a bloodthirsty goon," Patrick said. "He's doing a quiet little purge out there. The word is Hedrov will be next to be 'reassigned' to Siberia."
Martindale smiled and nodded. "Even I haven't heard that one yet, Patrick!" he said excitedly. "Thanks for the tip. I owe you one."
"Don't mention it, sir."
"Too bad about Zevitin, huh?" Martindale commented. "Unfortunate skiing accident, they said. That tree jumped out from nowhere and nearly took his head off, I hear. Poor bastard. Have you heard anything else about that?" Patrick had no comment. "Funny about that happening right around the same time Buzhazi attacks Mashhad and you come back from Armstrong all of a sudden. I guess strange things do happen in threes, huh?"
"Yes, sir."
"Yeah. Sure they do." Martindale put an arm around Patrick's shoulders. "You see, my friend, you can't leave the biz behind," he said. "It's in your blood. I can name a couple hundred hot spots in the world and you'll tell me something interesting about each one."
"Sir, I'm not interested in—"
"Mongolia," Martindale interjected. He smiled when he saw Patrick's eyes light up. "Aha, you know something. What is it?"
"I heard General Dorjiyn will be replaced as chief of staff because he's too chummy with the United States," Patrick said.
"So now he can run for president, right?"
"No, because he was born in Inner Mongolia—China—and proclaimed his allegiance to Beijing as a young officer," Patrick said. "But his son will run."
Martindale slapped his hands together. "Damn, I forgot about Myren Dorjiyn...!"
"Muren."
"Muren. Right. He graduated from Berkeley two years ago with a master's degree, right?"
"Double doctorate. Economics and government."
Martindale nodded, pleased that Patrick passed the two little tests he had given him. "See? I knew you were keeping up on this stuff!" Martindale exclaimed happily. "Come on back, Patrick. Let's join forces again. We'll set this world on fire."
Patrick smiled, then looked out at his son fishing and said, "I'll see you around, Mr. President," and walked out to join his son in the warm spring morning.
## ACKNOWLEDGMENTS
Thanks to fellow author Debbie Macomber and her husband, Wayne, for their generosity.
## AUTHOR'S NOTE
Your comments are welcome! E-mail me at readermail@airbattleforce.com or visit www.AirBattleForce.com to read my essays and commentary and get the latest updates on new projects, tour schedules, and more!
## About the Author
DALE BROWN is the author of numerous New York Times bestsellers, starting with Flight of the Old Dog in 1987. A former U.S. Air Force captain, he can often be found flying his own plane over the skies of Nevada.
www.dalebrown.info
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
Books by Dale Brown
_Iron Wolf_
_Starfire_
_Tiger's Claw_
_A Time for Patriots_
_Whiplash: A Dreamland Thriller_ (with Jim DeFelice)
_Drone Strike: A Dreamland Thriller_ (with Jim DeFelice)
_Collateral Damage: A Dreamland Thriller_ (with Jim DeFelice)
_Raven Strike: A Dreamland Thriller_ (with Jim DeFelice)
_Black Wolf: A Dreamland Thriller_ (with Jim DeFelice)
_Executive Intent_
_Dale Brown's Dreamland: End Game_ (with Jim DeFelice)
_Dale Brown's Dreamland: Retribution_ (with Jim DeFelice)
_Dale Brown's Dreamland: Revolution_ (with Jim DeFelice)
_Dale Brown's Dreamland: Satan's Tail_ (with Jim DeFelice)
_Dale Brown's Dreamland: Strike Zone_ (with Jim DeFelice)
_Dale Brown's Dreamland: Razor's Edge_ (with Jim DeFelice)
_Rogue Forces_
_Shadow Command_
_Strike Force_
_Edge of Battle_
_Act of War_
_Plan of Attack_
_Air Battle Force_
_Wings of Fire_
_Warrior Class_
_Battle Born_
_The Tin Man_
_Fatal Terrain_
_Shadow of Steel_
_Storming Heaven_
_Chains of Command_
_Night of the Hawk_
_Sky Masters_
_Hammerheads_
_Day of the Cheetah_
_Silver Tower_
_Flight of the Old Dog_
## Credits
Jacket design by Richard L. Aquan
Jacket photographs by Robert Llewellyn/ Workbook Stock/Jupiterimages
## Copyright
This book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.
SHADOW COMMAND. Copyright © 2008 by Air Battle Force, Inc. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
ePub edition April 2008 ISBN 9780061751592
10 9 8 7 6 5 4 3 2 1
## About the Publisher
Australia
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http://www.harpercollinsebooks.com
|
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Rome's Miraculous Crucifix "at Risk of Bursting" after Francis exposes it to Rain in St. Peter's Square
The 16th-century wooden crucifix was not shielded from the rain in St. Peter's Square
[UPDATE 01-APR-20: "Damage to 'miraculous crucifix' not as serious as reported, rector says"] God is making things very obvious if only we are willing to see. This past Friday, Mar. 27, 2020, the false pope in Rome, Jorge Bergoglio, held an extraordinary prayer event in the Vatican to petition God for an end to the Coronavirus pandemic. To that end, the blaspheming apostate had had the 16th-century Miraculous Plague Crucifix brought from the Roman parish church of San Marcello al Corso to St. Peter's Square. But Almighty God, whose divine providence governs all things, decided to send rain for the event (cf. Job 38:28). Francis, who does not believe in God's punishment, had not ordered the wooden crucifix to be shielded from the water, while for himself a large canopy had been arranged. The result of the Miraculous Crucifix being exposed to rain for almost two hours, was reported in the Italian newspapers Il Messaggero and Il Sussidiario. They state that the crucifix was damaged considerably by the rain, and this was discovered as soon as it was returned to San Marcello's:
The almost two hours of rain have inflated the centuries-old wood. In several places the ancient plaster has swelled up. Then in some sections the tempera paint used by the anonymous artist to draw the blood gushing from the side [of Christ] has loosened, the wooden surface of the hair has rippled, and the arms have also sustained damage. (From the Il Messaggero article, quoted by Il Sussidiario article; our translation with help from Deepl and via German.)
The Il Messaggero article has pictures of some of the damage (but is behind a paywall). At this point the Vatican's press office has -- understandably -- not cared to comment yet. Vatican News has released a video clip with the highlights, as well as video of the full ceremony:
[embedyt] https://www.youtube.com/watch?v=gYwOQb2D7Jg[/embedyt]
[embedyt] https://www.youtube.com/watch?v=JcUqLrbi9Cg[/embedyt]
As is quite evident from the video, the Miraculous Crucifix could easily have been moved to a dry spot, or a canopy could have been provided for it. Apparently it was not considered important enough.
The Miraculous Plague Crucifix up close
With regard to the history of the Miraculous Crucifix, a report by Catholic News Agency gives some background:
The crucifix was venerated as miraculous by Romans after it was the only religious image to survive unscathed from a fire that completely gutted the church on May 23, 1519. Less than three years later, Rome was devastated by the "black plague." Upon the request of Rome's Catholics, the crucifix was taken in procession from the convent of the Servants of Mary in Via del Corso to St. Peter's Square, stopping in each quarter of Rome. The procession continued 16 days, from August 4th to the 20th, 1522. When the crucifix was returned to St. Marcellus, the plague had disappeared from Rome. ("Miraculous crucifix moved to St. Peter's Square for Pope Francis' 'Urbi et Orbi' blessing", Catholic News Agency, Mar. 25, 2020; underlining added.)
Back in the 16th century, Rome of course was filled with real Catholics, people with deep Faith and great love of God. Today, the Vatican is filled with faithless blasphemers, heretics, and sodomites. No wonder that the crucifix that once miraculously survived a fire succumbs to the rain in the presence of Bergoglio. It makes sense that it would. In his special Urbi et Orbi meditation, Francis announced that the virtue of Faith "is not so much believing that you [God] exist, but coming to you and trusting in you." Lest anyone fear that maybe the Holy Trinity was punishing the world, he assured his hapless online viewers that this is not so. Addressing God, he brazenly declared: "It is not the time of your judgement, but of our judgement: a time to choose what matters and what passes away, a time to separate what is necessary from what is not. It is a time to get our lives back on track with regard to you, Lord, and to others." A few days earlier, Francis had opined that Coronavirus was nature's punishment for man's ecological sins. So at least he believes in some kind of retribution. But the real Catholic Church prays in her Prayer in Times of Epidemics:
We beseech thee, O Lord, grant us a hearing as we devoutly raise our petitions to thee, and graciously turn away the epidemic of plague which afflicts us; so that mortal hearts may recognize that these scourges proceed from thine indignation and cease only when thou art moved to mercy. (Source: Pope Pius XI, Rituale Romanum, Titulus IX, Caput X; translation taken from here; underlining added.)
God scourging a faithless people? What a concept! "The Lord is with you, because you have been with him. If you seek him, you shall find: but if you forsake him, he will forsake you" (2 Para [2 Chr] 15:2; cf. 2 Tim 2:12). On Mar. 15, Francis had already visited this Plague Crucifix in San Marcello's to pray before it. While this is obviously a pious and traditional act, in the case of "Pope" Francis it must be considered a mockery, since the man routinely commits blasphemies and heresies and even sanctions idolatry. In fact, for the Benediction part of his rain-drenched prayer service on Mar. 27, the false pope again refused to kneel, despite his kneel-ability demonstrated on other occasions. Why would God allow Francis to damage, perhaps forever ruin, a miraculous crucifix that survived fire and put an end to the plague in Rome? Let's answer the question this way: If saints work miracles as divine confirmation of their preaching and their works, what does a kind of anti-miracle testify to? Image source: youtube.com (screenshots) Licenses: fair use
Rome's Miraculous Crucifix "at Risk of Bursting" after Francis exposes it to Rain in St. Peter's Square
Bergoglio makes it possible…
The 16th-century wooden crucifix was not shielded from the rain in St. Peter's Square
[UPDATE 01-APR-20: "Damage to 'miraculous crucifix' not as serious as reported, rector says"]
God is making things very obvious if only we are willing to see.
This past Friday, Mar. 27, 2020, the false pope in Rome, Jorge Bergoglio, held an extraordinary prayer event in the Vatican to petition God for an end to the Coronavirus pandemic. To that end, the blaspheming apostate had had the 16th-century Miraculous Plague Crucifix brought from the Roman parish church of San Marcello al Corso to St. Peter's Square.
But Almighty God, whose divine providence governs all things, decided to send rain for the event (cf. Job 38:28). Francis, who does not believe in God's punishment, had not ordered the wooden crucifix to be shielded from the water, while for himself a large canopy had been arranged.
The result of the Miraculous Crucifix being exposed to rain for almost two hours, was reported in the Italian newspapers Il Messaggero and Il Sussidiario. They state that the crucifix was damaged considerably by the rain, and this was discovered as soon as it was returned to San Marcello's:
The almost two hours of rain have inflated the centuries-old wood. In several places the ancient plaster has swelled up. Then in some sections the tempera paint used by the anonymous artist to draw the blood gushing from the side [of Christ] has loosened, the wooden surface of the hair has rippled, and the arms have also sustained damage.
(From the Il Messaggero article, quoted by Il Sussidiario article; our translation with help from Deepl and via German.)
The Il Messaggero article has pictures of some of the damage (but is behind a paywall). At this point the Vatican's press office has — understandably — not cared to comment yet.
Vatican News has released a video clip with the highlights, as well as video of the full ceremony:
As is quite evident from the video, the Miraculous Crucifix could easily have been moved to a dry spot, or a canopy could have been provided for it.
Apparently it was not considered important enough.
The crucifix was venerated as miraculous by Romans after it was the only religious image to survive unscathed from a fire that completely gutted the church on May 23, 1519.
Less than three years later, Rome was devastated by the "black plague."
Upon the request of Rome's Catholics, the crucifix was taken in procession from the convent of the Servants of Mary in Via del Corso to St. Peter's Square, stopping in each quarter of Rome.
The procession continued 16 days, from August 4th to the 20th, 1522. When the crucifix was returned to St. Marcellus, the plague had disappeared from Rome.
("Miraculous crucifix moved to St. Peter's Square for Pope Francis' 'Urbi et Orbi' blessing", Catholic News Agency, Mar. 25, 2020; underlining added.)
Back in the 16th century, Rome of course was filled with real Catholics, people with deep Faith and great love of God. Today, the Vatican is filled with faithless blasphemers, heretics, and sodomites. No wonder that the crucifix that once miraculously survived a fire succumbs to the rain in the presence of Bergoglio.
It makes sense that it would. In his special Urbi et Orbi meditation, Francis announced that the virtue of Faith "is not so much believing that you [God] exist, but coming to you and trusting in you." Lest anyone fear that maybe the Holy Trinity was punishing the world, he assured his hapless online viewers that this is not so. Addressing God, he brazenly declared: "It is not the time of your judgement, but of our judgement: a time to choose what matters and what passes away, a time to separate what is necessary from what is not. It is a time to get our lives back on track with regard to you, Lord, and to others." A few days earlier, Francis had opined that Coronavirus was nature's punishment for man's ecological sins. So at least he believes in some kind of retribution.
But the real Catholic Church prays in her Prayer in Times of Epidemics:
We beseech thee, O Lord, grant us a hearing as we devoutly raise our petitions to thee, and graciously turn away the epidemic of plague which afflicts us; so that mortal hearts may recognize that these scourges proceed from thine indignation and cease only when thou art moved to mercy.
(Source: Pope Pius XI, Rituale Romanum, Titulus IX, Caput X; translation taken from here; underlining added.)
God scourging a faithless people? What a concept! "The Lord is with you, because you have been with him. If you seek him, you shall find: but if you forsake him, he will forsake you" (2 Para [2 Chr] 15:2; cf. 2 Tim 2:12).
On Mar. 15, Francis had already visited this Plague Crucifix in San Marcello's to pray before it. While this is obviously a pious and traditional act, in the case of "Pope" Francis it must be considered a mockery, since the man routinely commits blasphemies and heresies and even sanctions idolatry. In fact, for the Benediction part of his rain-drenched prayer service on Mar. 27, the false pope again refused to kneel, despite his kneel-ability demonstrated on other occasions.
Why would God allow Francis to damage, perhaps forever ruin, a miraculous crucifix that survived fire and put an end to the plague in Rome?
Let's answer the question this way: If saints work miracles as divine confirmation of their preaching and their works, what does a kind of anti-miracle testify to?
Image source: youtube.com (screenshots)
Licenses: fair use
in Novus Ordo Wire Coronavirus, Francis, Sacrilege, Vatican 0
Lent in the Vatican II...
Before and After Vatican II:...
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{"url":"http:\/\/www2.macaulay2.com\/Macaulay2\/doc\/Macaulay2-1.19\/share\/doc\/Macaulay2\/VectorGraphics\/html\/_show__Graph.html","text":"# showGraph -- Improved drawing of graphs\n\n## Description\n\nReturns a GraphicsObject that describes the input graph. Requires the package Graphs. Automatically used by the html output.\n\n i1 : needsPackage \"Graphs\"; i2 : R=QQ[x,y]; b=flatten entries basis(0,3,R) 2 3 2 2 2 3 o3 = {1, x, x , x , x y, x*y, x*y , y, y , y } o3 : List i4 : digraph select(b**b,a->a#1 % a#0 == 0 and first degree a#1 == first degree a#0 +1) o4 = Digraph{1 => {x, y} } 2 x => {x , x*y} 2 2 x*y => {x y, x*y } 2 y => {x*y, y } 2 x*y => {} 2 3 2 x => {x , x y} 2 x y => {} 2 2 3 y => {x*y , y } 3 x => {} 3 y => {} o4 : Digraph\n\n## For the programmer\n\nThe object showGraph is .","date":"2023-01-29 05:07:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7340513467788696, \"perplexity\": 6392.859891835218}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499700.67\/warc\/CC-MAIN-20230129044527-20230129074527-00195.warc.gz\"}"}
| null | null |
Click on play button to listen high quality(320 Kbps) mp3 of "Newsboys - We Believe (Official Music Video)". This audio song duration is 3:42 minutes. This song is sing by Newsboys. You can download Newsboys - We Believe (Official Music Video) high quality audio by clicking on "Download Mp3" button.
|
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Q: JoinColumn name not used in sql I have a problem with mapping many-to-one relationship without exact foreign key constraint set in database. I use OpenJPA implementation with MySql database, but the problem is with generated sql scripts for insert and select statements.
I have LegalEntity table which contains RootId column (among others). I also have Address table which has LegalEntityId column which is not nullable, and which should contain values referencing LegalEntity's "RootId" column, but without any database constraint (foreign key) set.
Address entity is mapped:
@Entity
@Table(name="address")
public class Address implements Serializable {
...
@ManyToOne(fetch=FetchType.LAZY, optional=false)
@JoinColumn(referencedColumnName="RootId", name="LegalEntityId", nullable=false, insertable=true, updatable=true, table="LegalEntity")
public LegalEntity getLegalEntity() {
return this.legalEntity;
}
}
SELECT statement (when fetching LegalEntity's addresses) and INSERT statment are generated:
SELECT t0.Id, .., t0.LEGALENTITY_ID FROM address t0 WHERE t0.LEGALENTITY_ID = ? ORDER BY t0.Id DESC [params=(int) 2]
INSERT INTO address (..., LEGALENTITY_ID) VALUES (..., ?) [params=..., (int) 2]
If I omit table attribute from mentioned statements are generated:
SELECT t0.Id, ... FROM address t0 INNER JOIN legalentity t1 ON t0.LegalEntityId = t1.RootId WHERE t1.Id = ? ORDER BY t0.Id DESC [params=(int) 2]
INSERT INTO address (...) VALUES (...) [params=...]
So, LegalEntityId is not included in any of the statements.
Is it possible to have relationship based on such referencing (to column other than primary key, without foreign key in database)? Is there something else missing?
Thanks in advance.
A: Try changing your FetchType to eager. OpenJPA should not query for the LegalEntity until it's requested by calling getLegalEntity.
Also, the absence of constraints shouldn't matter (somebody please correct me if I'm wrong).
|
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\section{Introduction}
In quantum metrology, the Mach--Zehnder interferometer (MZI) has been
exploited as a generic tool to realize precise measurement of phase. The
sensitivity of interferometer measurements is restricted by the shot-noise
limit, or the standard quantum limit (SQL) with respect to classic
resources. Therefore, how to improve the sensitivity of interferometers has
been received a lot of attention in recent years \cit
{Helstrom67,Holevo82,Caves81,Braunstein94,Braunstein96,Lee02,Giovannetti06,Zwierz10,Giovannetti04,Giovannetti11
, in which many quantum techniques have been utilized to improve measurement
precision than purely classical approaches.
Yurke et al. \cite{Yurke86} theoretically introduced the SU(1,1)
interferometer using an active nonlinear beam splitter (NBS) in place of a
passive linear beam splitter (LBS) for wave splitting and recombination. It
is called the SU(1,1) interferometer because it is described by the SU(1,1)
group, as opposed to SU(2) group for BSs. The NBS can be provided by the
optical parameter amplifier process or the four-wave mixing process. Due to
the quantum destructive interference in the SU(1,1) interferometer, the
noise accompanied by the amplification of the signal can revert to the level
of input. Benefiting from that, the signal-to-noise ratio improves. Because
it can be used to improve measurement sensitivity, this type of
interferometers has received extensive attention both experimentally \cit
{Jing11,Hudelist14,Chen15,Qiu16,Linnemann16,Lemieux16,Manceau17,Anderson17,Gupta17,Du18,Frascella19,Prajapati19,Du22}
and theoretically \cit
{Plick10,Ou12,Marino12,Li14,Gabbrielli15,Ma15,Chen16,Sparaciari16,Li16,Hu16,Gong17,Giese17,Li18,Guo18,Hu18,Ma18,Michael19,Gao20,Ou20,Gao22, Liang22
.
In the presence of environment noise, the measurement precision of
interferometers will be reduced because there are inevitable interactions
with the surrounding environment, which have been studied by many
researchers \cite{Dem09,Escher12,Dem12,Yue14,Berry13, Chaves, Dur,
Kessler,Brivio,Alipour,Escher11,Genoni11,Genoni12,Feng14}. For
interferometers, the photon losses is a typical decoherence process which
should be taken into account. The general frame for estimating the ultimate
precision limite in the presence of photo loss has been analyzed \cit
{Dem09,Escher11,Dem12,Yue14}, where this decoherence process can be
described by a set of Kraus operators, and the corresponding lower bounds in
quantum metrology is given by the quantum Cra\'{m}er-Rao bound (QCRB) usage
of quantum Fisher information (QFI) \cite{Braunstein94,Braunstein96}. It
establishes the best precision that can be attained with a given quantum
probe \cite{Toth, PezzBook, Demkowicz,Wang,Monras, Pinel, Liu,
Gao,Jiang,Safranek,Sparaciari15,Gao21,Chang22}.
For the SU(2) and SU(1,1) interferometers without losses, the QFIs of
unknown phase shifts only in the single arm case or in the two-arm case have
been investigated, where the results of QFI-only calculations will be
overestimated because many measurement processes do not include phase
reference (or phase-averaging) \cite{Jarzyna,Gong17CH,You19,Takeoka}. For
the unknown phase shifts in the two-arm case, the phase estimation is indeed
a two-parameter estimation problem. Although, the results of phase-averaging
method and multiparameter estimation are the same for some particular inputs
\cite{Takeoka,You19}. In general, the calculation of the quantum Fisher
information matrix (QFIM) is necessary when the unknown phase shifts are
applied in both arms \cite{Takeoka,You19,Lang1, Lang2, Ataman, Zhong2}. In
the presence of losses, the QFIM of SU(2) and SU(1,1) interferometers and
corresponding QCRBs are worth studying because there are inevitable
interactions with the surrounding environment in the measurement process.
In this work, we theoretically study the ultimate precision limit of SU(2)
and SU(1,1) interferometers in the presence of losses. In order to avoid
overestimation of QFI-only calculation, we give the phase sensitivity of
SU(2) and SU(1,1) interferometers with the QFIM approach for the unknown
phase shifts in the two-arm case. When inputs are arbitrary pure states, the
ultimate precision limit for the SU(2) and SU(1,1) interferometers with
optical losses in single arm and two arms are given. Taking the coherent
states $\otimes $ squeezed vacuum input as a specific example, we
numerically calculate and compare the sensitivities between the
single-parameter estimation and two-parameter estimation.
The paper is organized as follows. In Sec.~II, we firstly review the
QFIM of the SU(2) and SU(1,1) interferometers in the lossless case, and
derive the QFIM of them with noise with the method proposed by Escher et al.
\cite{Escher11}. In Sec. III and IV, we respectively investigate the QFIMs
in the case of optical losses in single arm and two arms. In Sec. V, we
numerically calculate and compare the QCRBs between the single-parameter
estimation and two-parameter estimation. Finally, we conclude with a summary
of our results.
\begin{figure}[t]
\centerline{\includegraphics[width=0.4\textwidth,angle=0]{fig1.eps}}
\caption{Lossy interferometer model. The losses in the interferometer are
modeled by adding fictitious beam splitters (a) before and (b) after the
phase shifts $\protect\phi _{a}$ and $\protect\phi _{b}$. The beam splitting
is the passive linear beam splitters (LBSs) or the active nonlinear beam
splitters (NBSs), corresponding the SU(2) interferometers or SU(1,1)
interferometers, respectively. $\hat{v}_{a}$ ($\hat{v}_{b}$) is the vacuum,
and $\protect\eta_{a}$ ($\protect\eta_{b}$) is transmission rate. In
realistic systems the photon losses are distributed throughout the arms of
the interferometer, not an event concentrated either before or after the
phase, which is handled with a distribution parameter.}
\label{fig1}
\end{figure}
\section{ QFIM of interferometers}
\label{sec:QOI}
\subsection{Ideal QFIM}
\label{sec:IQ} We consider a general interferometer, as shown in Fig.~\re
{fig1}, where the beam splitter can be either linear or nonlinear,
corresponding SU(2) and SU(1,1) interferometers, respectively. Consider the
initial input state $|\psi _{in}\rangle $. After the first LBS or NBS, the
state is labeled by $|\psi \rangle $. The two beams sustain phase shifts,
i.e., mode $a$ undergoes a phase shift of $\phi _{a}$\ and mode $b$
undergoes a phase shift of $\phi _{b}$, and the state is $\left\vert \psi
_{\phi }\right\rangle $. Then the phase transformation is written as
\begin{equation}
\hat{U}_{\phi }^{ab}=\exp [i\phi _{a}\hat{n}_{a}+i\phi _{b}\hat{n}_{b}]=\exp
(i\phi _{+}\hat{g}_{+})\exp (i\phi _{-}\hat{g}_{-}),
\end{equation
where $\phi _{\pm }=\phi _{a}\pm \phi _{b}$ and $g_{\pm }=(\hat{n}_{a}\pm
\hat{n}_{b})/2$ with $\hat{n}_{a}=\hat{a}^{\dag }\hat{a}$ and $\hat{n}_{b}
\hat{b}^{\dag }\hat{b}$.
The QFI is the intrinsic information in the quantum state and is not related
to the actual measurement procedure, which gives an upper limit to the
precision of quantum parameter estimation. However, the QFI-only calculation
was overestimated. In general, the phase estimation as a two-parameter
estimation problem such as $\phi _{a}$ and $\phi _{b}$ (or $\phi _{+}$ and
\phi _{-}$), and the quantum Fisher information matrix (QFIM) is necessary.
In the SU(2) and SU(1,1) interferometers, the estimated phases are $\phi
_{-} $ and $\phi _{+}$, respectively \cite{Kok2010}. For the estimation of
\phi _{+}$ and $\phi _{-}$ we can use the method of QFIM, which is given by
a two-by-two matrix
\begin{equation}
\mathcal{F}_{QFIM}=\left[
\begin{array}{cc}
F_{++} & F_{+-} \\
F_{-+} & F_{--
\end{array
\right] ,
\end{equation
where $F_{ij}=4(\langle \hat{g}_{i}\hat{g}_{j}\rangle -\langle \hat{g
_{i}\rangle \langle \hat{g}_{j}\rangle )$ ($i$, $j=+,-$), $\langle \cdot
\rangle $ denotes the average value $\left\langle \psi _{\phi }\right\vert
\cdot |\psi _{\phi }\rangle $ as shown in Fig.~\ref{fig1}. The estimation of
$\phi _{-}$ and $\phi _{+}$ are respectively given b
\begin{equation}
\Delta ^{2}\phi _{-}\geq \frac{1}{\mathcal{F}_{QFIM_{-}}}\text{, \ }\Delta
^{2}\phi _{+}\geq \frac{1}{\mathcal{F}_{QFIM_{+}}},
\end{equation
wher
\begin{eqnarray}
\mathcal{F}_{QFIM_{-}} &=&F_{--}-\Delta F_{-}\text{, }\mathcal{F
_{QFIM_{+}}=F_{++}-\Delta F_{+}, \notag \\
\Delta \mathcal{F}_{-} &=&\frac{F_{+-}F_{-+}}{F_{++}}\text{, }\Delta
\mathcal{F}_{+}=\frac{F_{+-}F_{-+}}{F_{--}}.
\end{eqnarray
When $\Delta \mathcal{F}_{\pm }\neq 0$, $\Delta \mathcal{F}_{\pm }$\ are the
overestimated Fisher information from QFI-only calculation. If one ignores
the nondiagonal terms, i.e., implicity assuming that $\phi _{+}$ ($\phi _{-}
$)\ is known a \textit{priori}, one could use the diagonal terms $F_{--} $ (
F_{++})$ to indicate\ the QFI of the single-parameter $\phi _{-}$ ($\phi
_{+} $) estimation, where the misleading overestimates the precision limit
\cite{Gong17,Takeoka,You19}.
For a two-input-port interferometer, the matrix elements $F_{ij}$ are given
by
\begin{eqnarray}
F_{++} &=&\langle \Delta ^{2}\hat{n}_{a}\rangle +\langle \Delta ^{2}\hat{n
_{b}\rangle +2Cov[n_{a},n_{b}], \notag \\
F_{--} &=&\langle \Delta ^{2}\hat{n}_{a}\rangle +\langle \Delta ^{2}\hat{n
_{b}\rangle -2Cov[n_{a},n_{b}], \notag \\
F_{+-} &=&F_{-+}=\langle \Delta ^{2}\hat{n}_{a}\rangle -\left\langle \Delta
^{2}\hat{n}_{b}\right\rangle ,
\end{eqnarray
where $\langle \Delta ^{2}\hat{n}_{i}\rangle =\langle \psi |\hat{n
_{i}^{2}|\psi \rangle -\langle \psi |\hat{n}_{i}|\psi \rangle ^{2}$ $(i=a$,
b)$, and $Cov[\hat{n}_{a},\hat{n}_{b}]=\langle \psi |\hat{n}_{a}\hat{n
_{b}|\psi \rangle -\langle \psi |\hat{n}_{a}|\psi \rangle \langle \psi |\hat
n}_{b}|\psi \rangle $. $Cov[\hat{n}_{a},\hat{n}_{b}]$ is the covariance of
two-mode field to describe the intermode correlation. After calculation, the
ultimate $\mathcal{F}_{QFIM_{-}}$\ and $\mathcal{F}_{QFIM_{+}}$ are\
rewritten as
\begin{eqnarray}
\mathcal{F}_{QFIM_{-}} &=&4\frac{\langle \Delta ^{2}\hat{n}_{a}\rangle
\langle \Delta ^{2}\hat{n}_{b}\rangle -Cov[n_{a},n_{b}]^{2}}{\langle \Delta
^{2}\hat{n}_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle
+2Cov[n_{a},n_{b}]}, \\
\mathcal{F}_{QFIM_{+}} &=&4\frac{\langle \Delta ^{2}\hat{n}_{a}\rangle
\langle \Delta ^{2}\hat{n}_{b}\rangle -Cov[n_{a},n_{b}]^{2}}{\langle \Delta
^{2}\hat{n}_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle
-2Cov[n_{a},n_{b}]}.
\end{eqnarray
and the overestimated Fisher information are cast into
\begin{eqnarray}
\Delta \mathcal{F}_{-} &=&\frac{\left( \langle \Delta ^{2}\hat{n}_{a}\rangle
-\langle \Delta ^{2}\hat{n}_{b}\rangle \right) ^{2}}{\langle \Delta ^{2}\hat
n}_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle +2Cov[n_{a},n_{b}]},
\label{DFM} \\
\Delta \mathcal{F}_{+} &=&\frac{\left( \langle \Delta ^{2}\hat{n}_{a}\rangle
-\langle \Delta ^{2}\hat{n}_{b}\rangle \right) ^{2}}{\langle \Delta ^{2}\hat
n}_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle -2Cov[n_{a},n_{b}]}.
\label{DFP}
\end{eqnarray
This is a general expression, which is dependent on the form of the input
state, and beam splitting transformation such as BS or NBS.
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth,angle=0]{fig2.eps}}
\caption{Phase sensitivity $\Delta\protect\phi$ of SU(2) interferometers
(top part) and SU(1,1) interferometers (bottom part) as a function of $N_
\protect\alpha}$ with $r=1.5$, $T=0.7$, and $G_{1}=1.2$. QCRB$^{k}_{p}$:
k\in$[+, SU(1,1) case; -, SU(2) case], $p\in$[s, QFI case; t, QFIM case].
The shaded parts represent the difference in phase sensitivities due to
overestimated QFI.}
\label{fig2}
\end{figure}
Using the linear correlation coefficient $J$ to describe the intermode
correlations \cite{Gerry05}, and the Mandel $Q$ parameter to describe the
intramode \cite{Mandel95}, which are given by
\begin{equation}
J=\frac{Cov[\hat{n}_{a},\hat{n}_{b}]}{\langle \Delta \hat{n}_{a}\rangle
\langle \Delta \hat{n}_{b}\rangle },\text{ }Q_{i}=\frac{\left\langle \Delta
^{2}\hat{n}_{i}\right\rangle -\langle \hat{n}_{i}\rangle }{\langle \hat{n
_{i}\rangle },
\end{equation
where $\langle \Delta \hat{n}_{i}\rangle =\sqrt{\langle \Delta ^{2}\hat{n
_{i}\rangle }$ $(i=a,b)$. Then, the ultimate $F_{QFIM_{-}}$\ and
F_{QFIM_{+}}$ are rewritten as
\begin{eqnarray}
\mathcal{F}_{QFIM_{-}} &=&\frac{4\sqrt{\left\langle \hat{n}_{a}\right\rangle
\left\langle \hat{n}_{b}\right\rangle (Q_{a}+1)(Q_{b}+1)}(1-J^{2})}{\sqrt
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}+\sqrt{\frac{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)
}+2J},\text{ } \text{ } \\
\mathcal{F}_{QFIM_{+}} &=&\frac{4\sqrt{\left\langle \hat{n}_{a}\right\rangle
\left\langle \hat{n}_{b}\right\rangle (Q_{a}+1)(Q_{b}+1)}(1-J^{2})}{\sqrt
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}+\sqrt{\frac{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)
}-2J}.\text{ } \text{ }
\end{eqnarray}
If $\langle \Delta ^{2}\hat{n}_{a}\rangle =\langle \Delta ^{2}\hat{n
_{b}\rangle $, the overestimated Fisher information $\Delta \mathcal{F}_{\pm
}$ will be $0$, and the results of phase estimation $\phi _{\pm }$ are from
the single-parameter QFI. Furthermore, considering $\left\langle \hat{n
_{a}\right\rangle =\left\langle \hat{n}_{b}\right\rangle =\bar{n}/2$, $\bar{
}$ is the average number of photons in the probe state, the results
\mathcal{F}_{QFIM_{\pm }}$ are reduced to
\begin{eqnarray}
\mathcal{F}_{QFIM_{-}} &=&\mathcal{F}_{QFI_{-}}=\bar{n}(Q+1)(1-J), \\
\mathcal{F}_{QFIM_{+}} &=&\mathcal{F}_{QFI_{+}}=\bar{n}(Q+1)(1+J).
\end{eqnarray
They are the same as that of the SU(2) \cite{Sahota15} and SU(1,1) \cit
{Gong17}, respectively. The intermode correlation $J$ ranges between $-1$
and $1$, which contribute at most a factor of $2$ improvement in the QFI.
The photon statistics within each arm of the interferometer are
super-Poissonian and sub-Poissonian for $Q>0$ and $-1<Q<0$, respectively.
Larger quantum enhancement originates from the photon variances within each
arm of the interferometers \cite{Sahota15,Gong17}.
In the case of $\langle \Delta ^{2}\hat{n}_{a}\rangle \neq \langle \Delta
^{2}\hat{n}_{b}\rangle $, for the SU(2) interferometers it exists when the
beam splitting ratio of the first LBS is not 50:50 without losses \cit
{Jarzyna}. However, in the presence of losses the optimal splitting ratio of
LBS for the SU(2) interferometers is no longer 50:50, leading to $\langle
\Delta ^{2}\hat{n}_{a}\rangle \neq \langle \Delta ^{2}\hat{n}_{b}\rangle $
\cite{Huang22,Cooper11}. For the SU(1, 1) interferometers, $\langle \Delta
^{2}\hat{n}_{a}\rangle \neq \langle \Delta ^{2}\hat{n}_{b}\rangle $ exists
as long as that the input states of the two inputs are not the same \cit
{Yu22}. Therefore, it is very necessary to estimate the phase sensitivity
with noise by using the method of QFIM.
\subsection{ Lossy QFIM}
In this section, the general model, developed by Escher \textit{et al}. \cit
{Escher11} for open quantum system metrology, is extended to the QFIM case.
We begin with a system $S$ and consider it along with the environment $E$.
In general, the enlarged state of system and environment ($S+E$) evolves a
\begin{equation}
|\Psi \rangle _{SE}=\hat{U}_{SE}(x_{+},x_{-})|\psi \rangle _{S}|0\rangle
_{E}=\sum_{l}\hat{\Pi}_{l}(x_{+},x_{-})|\psi \rangle _{S}|l\rangle _{E},
\end{equation
where $\hat{U}_{SE}(x_{+},x_{-})$ is the corresponding unitary operator, and
$|\psi \rangle _{S}$ is the initial state of the probe. $|0\rangle _{E}$\ is
the initial state of the environment, and $|l\rangle _{E}$\ are orthogonal
states of the environment $E$. $\hat{\Pi}_{l}(x_{+},x_{-})$ are $x_{+}$, and
$x_{-}$-dependent Kraus operators, which act on the system $S$. The
evolution of an open system can be described a
\begin{eqnarray}
\rho _{S} &=&Tr_{E}\left[ \hat{U}_{SE}\rho _{SE}(0)\hat{U}_{SE}^{\dag
\right] \notag \\
&=&\sum_{l}\hat{\Pi}_{l}(x_{+},x_{-})|\psi \rangle _{SS}\langle \psi |\hat
\Pi}_{l}^{\dag }(x_{+},x_{-}),
\end{eqnarray
where $\rho _{SE}(0)$ is the state of the systems and environment. For the
enlarged system-environment state, the QFI $\mathcal{C}_{QFIM}$\ is given by
\begin{equation}
\mathcal{C}_{QFIM}=\left[
\begin{array}{cc}
C_{++} & C_{+-} \\
C_{-+} & C_{--
\end{array
\right] ,
\end{equation
where
\begin{eqnarray}
C_{ij} &=&4[\left\langle \psi \right\vert _{S}\left\langle l\right\vert
_{E}\sum_{l}\frac{d\hat{\Pi}_{l}^{\dag }(x_{+},x_{-})}{d\phi _{i}}\frac{
\hat{\Pi}_{l}(x_{+},x_{-})}{d\phi _{j}}\left\vert \psi \right\rangle
_{S}\left\vert l\right\rangle _{E} \notag \\
&&-\left\langle \psi \right\vert _{S}\left\langle l\right\vert _{E}\sum_{l}
\frac{d\hat{\Pi}_{l}^{\dag }(x_{+},x_{-})}{d\phi _{i}}\hat{\Pi
_{l}(x_{+},x_{-})\left\vert \psi \right\rangle _{S}\left\vert l\right\rangle
_{E} \notag \\
&&\times \left\langle \psi \right\vert _{S}\left\langle l\right\vert
_{E}\sum_{l}-i\hat{\Pi}_{l}^{\dag }(x_{+},x_{-})\frac{d\hat{\Pi
_{l}(x_{+},x_{-})}{d\phi _{j}}\left\vert \psi \right\rangle _{S}\left\vert
l\right\rangle _{E}, \notag \\
i,j &\in &(+,-).
\end{eqnarray
The matrix elements $C_{ij}$ above can be expressed a
\begin{eqnarray}
C_{ij} &=&4[\left\langle \psi \right\vert _{S}\left\langle l\right\vert _{E
\hat{H}_{ij}(\phi _{+},\phi _{-})\left\vert \psi \right\rangle
_{S}\left\vert l\right\rangle _{E} \notag \\
&&-\left\langle \psi \right\vert _{S}\left\langle l\right\vert _{E}\hat{h
_{i}(\phi _{+},\phi _{-})\left\vert \psi \right\rangle _{S}\left\vert
l\right\rangle _{E} \notag \\
&&\times \left\langle \psi \right\vert _{S}\left\langle l\right\vert _{E
\hat{h}_{j}^{\prime }(\phi _{+},\phi _{-})\left\vert \psi \right\rangle
_{S}\left\vert l\right\rangle _{E}], \label{CQ}
\end{eqnarray
where
\begin{eqnarray}
\hat{H}_{ij}(\phi _{+},\phi _{-}) &=&\sum_{l}\frac{d\hat{\Pi}_{l}^{\dag
}(\phi _{+},\phi _{-})}{d\phi _{i}}\frac{d\hat{\Pi}_{l}(\phi _{+},\phi _{-}
}{d\phi _{j}}, \notag \\
\hat{h}_{i}(\phi _{+},\phi _{-}) &=&i\sum_{l}\frac{d\hat{\Pi}_{l}^{\dag
}(\phi _{+},\phi _{-})}{d\phi _{i}}\hat{\Pi}_{l}(\phi _{+},\phi _{-}),
\notag \\
\hat{h}_{j}^{\prime }(\phi _{+},\phi _{-}) &=&-i\sum_{l}\hat{\Pi}_{l}^{\dag
}(\phi _{+},\phi _{-})\frac{d\hat{\Pi}_{l}(\phi _{+},\phi _{-})}{d\phi _{j}}.
\end{eqnarray}
By using QFIM method, similar to lossless case, we obtain the estimation of
the phase difference $\phi _{-}$ and phase sum $\phi _{+}$ in the presence
of losses, which is bounded by the QFIM:
\begin{eqnarray}
\mathcal{C}_{QFIM_{-}} &=&C_{--}-\Delta C_{-}, \label{CQFIM} \\
\mathcal{C}_{QFIM_{+}} &=&C_{++}-\Delta C_{+}, \label{CQFIMP}
\end{eqnarray
where
\begin{equation}
\Delta C_{-}=\frac{C_{+-}C_{-+}}{C_{++}}\text{, }\Delta C_{+}=\frac
C_{+-}C_{-+}}{C_{--}}.
\end{equation
$\Delta C_{\pm }$\ are the overestimated Fisher information from QFI-only
calculation in the presence of losses. Because the additional freedom
supplied by the environment should increase the QFI. Therefore, $\mathcal{C
_{QFIM_{+}}$ ($\mathcal{C}_{QFIM_{-}}$) should be larger or equal to
\mathcal{F}_{QFIM_{+}}$\ ($\mathcal{F}_{QFIM_{-}}$) . The relation between
\mathcal{F}_{QFIM_{\pm }}$\ and $\mathcal{C}_{QFIM_{\pm }}$ is found to b
\begin{eqnarray}
\mathcal{F}_{QFIM_{i}} &=&\underset{\{\hat{\Pi}_{l}(x_{+},x_{-})\}}{\min
\mathcal{C}_{QFIM_{i}}\left[ |\psi \rangle ,\hat{\Pi}_{l}(x_{+},x_{-})\right]
, \notag \\
i &\in &(+,-).
\end{eqnarray}
Usually, losses in the interferometers can be modeled by adding the
fictitious beam splitters, and the lossy evolution of the field in two arms
is described by the Kraus operator $\hat{\Pi}_{l}(\phi _{+},\phi _{-})$ with
considering the phase shift. In realistic systems the photon losses are
distributed throughout the arms of the interferometer, which is described by
the parameter $\gamma $, instead of simply inserting the fictitious beam
splitters before or after the phase shift.
Therefore, in the presence of losses the matrix elements of $\mathcal{F
_{QFIM_{\pm }}$ are worked in three steps: (1) substituting the $\hat{\Pi
_{l}(\phi _{+},\phi _{-})$ into Eq.~(\ref{CQ}), we can obtain the $C_{ij}$,
which is a function of $\gamma $; (2) minimizing $\mathcal{C}_{QFIM_{\pm }}$
by the parameters $\gamma $, the optimal $\gamma _{opt}^{\pm }$ is obtained;
(3) substituting $\gamma _{opt}^{\pm }$ into $\mathcal{C}_{QFIM_{\pm }}$,
the minimum $\mathcal{C}_{QFIM_{\pm }}^{opt}$ is obtain, i.e., $\mathcal{F
_{QFIM_{\pm }}$ is achieved.
Next, we applied this model to analyze the phase estimation in the presence
of losses in the case of in one arm and in two arms.
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth,angle=0]{fig3a.eps}}
\centerline{\includegraphics[width=0.5\textwidth,angle=0]{fig3b.eps}}
\caption{Phase sensitivities of (a) SU(2) interferometers and (b) SU(1,1)
interferometers versus photon loss coefficient $\protect\eta$ with
single-parameter estimation and two-parameter estimation with $\left
\protect\alpha \right | =10$, $r=0.5$ and $R/T=0.25$ in (a), $G_{1}=1.2$ in
(b). QCRB$^{k}_{p}$: $k\in$[+, SU(1,1) case; -, SU(2) case], $p\in$[s, QFI
case; ls, QFI case with loss; t, QFIM case; lt, QFIM case with loss].}
\label{fig3}
\end{figure}
\section{Losses in one arm}
Photon losses, a very usual noise, may happen at any stage of the phase process and is modeled
by the fictitious LBS introduced in the interferometer arms. Firstly, we consider the photon losses in just one of the two arms, for example arm $a$. A possible set of Kraus operators describing
the process without considering the phase shift i
\begin{equation}
\hat{\pi}_{l_{a}}=\sqrt{\frac{(1-\eta _{a})^{l_{a}}}{l_{a}!}}\eta _{a}^{\hat
n}_{a}/2}\hat{a}^{l_{a}},
\end{equation
where $\eta _{a}$ quantifies the photon losses of arm $a$ ($\eta _{a}=1$,
lossless case; $\eta _{a}=0$, complete absorption).
When the photon losses before or after the phase shifts as shown in Fig. 1,
the Kraus operators $\hat{\pi}_{l_{a}}(\phi _{+},\phi _{-})$ including the
phase factor the general form ($\hbar =1$) is given by
\begin{eqnarray}
\hat{\Pi}_{la}(\varphi _{+},\varphi _{-}) &=&\sqrt{\frac{(1-\eta
_{a})^{l_{a}}}{l_{a}!}}e^{i\phi _{-}\frac{(\hat{a}^{\dagger }\hat{a}-\hat{b
^{\dagger }\hat{b}-\gamma l_{a})}{2}} \notag \\
&&\times e^{i\phi _{+}\frac{(\hat{a}^{\dagger }\hat{a}+\hat{b}^{\dagger
\hat{b}-\gamma l_{a})}{2}}\eta _{a}^{\hat{n}_{a}/2}\hat{a}^{^{l_{a}}},
\end{eqnarray
where $\gamma =0$ and $\gamma =-1$ describe\ the photons loss before and
after the phase shifts, respectively.
According to Eqs. (20) and (26). The matrix elements $C_{ij}$ of Eq. (19)
can be worked out:
\begin{align}
C_{++}& =\left[ \eta _{a}+\gamma \eta _{a}-\gamma \right] ^{2}\langle \Delta
^{2}\hat{n}_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle \notag \\
& +2\left[ \eta _{a}+\gamma \eta _{a}-\gamma \right] Cov[\hat{n}_{a},\hat{n
_{b}] \notag \\
& +\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta _{a}\langle \hat{n
_{a}\rangle , \\
C_{--}& =[\eta _{a}+\gamma \eta _{a}-\gamma ]^{2}\langle \Delta ^{2}\hat{n
_{a}\rangle +\langle \Delta ^{2}\hat{n}_{b}\rangle \notag \\
& -2\left[ \eta _{a}+\gamma \eta _{a}-\gamma \right] Cov[\hat{n}_{a},\hat{n
_{b}] \notag \\
& +\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta _{a}\left\langle \hat{n
_{a}\right\rangle , \\
C_{+-}& =C_{-+}=[\eta _{a}+\gamma \eta _{a}-\gamma ]^{2}\langle \Delta ^{2
\hat{n}_{a}\rangle \notag \\
& +\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta _{a}\left\langle \hat{n
_{a}\right\rangle -\langle \Delta ^{2}\hat{n}_{b}\rangle .
\end{align
Substituting the matrix elements $C_{ij}$ into Eq.~(\ref{CQFIM}) and Eq.~
\ref{CQFIMP}), one can obtain the $\mathcal{C}_{QFIM_{\pm }}$ for the case
of loss in one arm, where $\mathcal{C}_{QFIM_{-}}$ and $\mathcal{C
_{QFIM_{+}}$ are SU(2) and SU(1,1) interferometers, respectively. They are
given as follows:
\begin{eqnarray}
\mathcal{C}_{QFIM_{-}} &=&4\frac{\Upsilon _{0}}{\Upsilon _{1}+2J\Upsilon _{2
}, \label{OptC_} \\
\text{ \ }\mathcal{C}_{QFIM+} &=&4\frac{\Upsilon _{0}}{\Upsilon
_{1}-2J\Upsilon _{2}}, \label{OptC+}
\end{eqnarray
where
\begin{eqnarray}
\Upsilon _{0} &=&\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle (Q_{b}+1)[\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta
_{a} \notag \\
&&+(\eta _{a}+\gamma \eta _{a}-\gamma )^{2}(1-J^{2})(Q_{a}+1)], \notag \\
\Upsilon _{1} &=&\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)(\eta
_{a}+\gamma \eta _{a}-\gamma )^{2}+\left\langle \hat{n}_{b}\right\rangle
(Q_{b}+1) \notag \\
&&+\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta _{a}\left\langle \hat{n
_{a}\right\rangle , \notag \\
\Upsilon _{2} &=&\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)(\eta
_{a}+\gamma \eta _{a}-\gamma )\sqrt{\frac{\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)
}.\text{ \ }
\end{eqnarray
Next, we minimize $\mathcal{C}_{QFIM_{\pm }}$ by varying parameter $\gamma
, respectively.
\subsection{SU(2) interferometers}
In the case of SU(2) interferometers. To minimize the $\mathcal{C
_{QFIM_{-}} $, it requires to find the optimal $\gamma $, and we ge
\begin{equation}
\frac{d\mathcal{C}_{QFIM_{-}}}{d\gamma }=0.
\end{equation
After calculation, we obtain the optimal $\gamma _{opt}^{-}$, which is given
by
\begin{equation}
\gamma _{opt}^{-}=\frac{1}{(1-\eta _{a})+\frac{\eta _{a}}{(Q_{a}+1)(1-J^{2})
(1+J\sqrt{\frac{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}})}-1.
\end{equation
Substituting the optimal $\gamma _{opt}^{-}$ into $\mathcal{C}_{QFIM_{-}}$of
Eq. (\ref{OptC_}), the optimal bound $\mathcal{C}_{QFIM_{-}}^{opt}$ is
worked out:
\begin{eqnarray}
&&\mathcal{C}_{QFIM_{-}}^{opt}=\frac{1}{\Upsilon _{3}}4(1-J^{2})[\langle
\hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}(1-J^{2})\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1) \notag \\
&&+\langle \hat{n}_{a}\rangle ^{2}(\frac{\eta _{a}}{1-\eta _{a}})^{2}(\sqrt
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}+J)^{2}]. \label{CQ-opt}
\end{eqnarray
where
\begin{eqnarray}
\Upsilon _{3} &=&(\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
)^{2}\{[\frac{1+5J^{2}}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle ^{2}(Q_{a}+1)^{2}} \notag \\
&&+\frac{J^{2}}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}+\frac
2J(1+J^{2})}{\sqrt{\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle (Q_{a}+1)(Q_{b}+1)}} \notag \\
&&+\frac{4J}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}\sqrt{\frac
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}] \notag \\
&&+\langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}[(1-J^{2})[1+
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)} \notag \\
&&+J^{2}+4J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}]\} \notag \\
&&+(1-J^{2})^{2}\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1).
\end{eqnarray}
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth,angle=0]{fig4a.eps}}
\centerline{\includegraphics[width=0.5\textwidth,angle=0]{fig4b.eps}}
\caption{Phase sensitivity differences $\Delta \protect\phi _{s-t}$ between
the single-parameter estimation and the two-parameter estimation of (a)
SU(2) interferometer versus $R/T$ and (b) SU(1,1) interferometer versus $G$
under the condition of different photon loss coefficient $\protect\eta$,
where $|\protect\alpha|=2$, and $r=0.5$.}
\label{fig4}
\end{figure}
The bound $\mathcal{C}_{QFIM_{-}}^{opt}$ depends on the variance of the
initial state $\left\langle \Delta ^{2}\hat{n}_{a}\right\rangle $ and
\left\langle \Delta ^{2}\hat{n}_{b}\right\rangle $, the number of photons in
the initial state $\langle \hat{n}_{a}\rangle $, the intermode correlation
J $, and the losses $\eta _{a}$.
We check the optimal bound $\mathcal{C}_{QFIM_{-}}^{opt}$ in two limits. If
there is small dissipation, that is
\begin{equation}
\langle \Delta ^{2}\hat{n}_{b}\rangle ,\langle \Delta ^{2}\hat{n}_{a}\rangle
\ll \langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}.
\end{equation
Eq. (\ref{CQ-opt}) arrives a
\begin{eqnarray}
\mathcal{C}_{QFIM_{-}}^{opt} &=&\frac{1}{\Upsilon _{3L}}[4(1-J^{2})\lef
\langle \hat{n}_{a}\right\rangle (Q_{a}+1) \notag \\
&&\times (\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}+J)^{2}],
\end{eqnarray
where
\begin{eqnarray}
\Upsilon _{3L} &=&1+J^{2}\frac{\left\langle \hat{n}_{a}\right\rangle
(Q_{a}+1)}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}+\frac
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)} \notag \\
&&+2J(J^{2}+1)\sqrt{\frac{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}}+5J^{2} \notag \\
&&+4J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}.
\end{eqnarray}
When $\left\langle \hat{n}_{a}\right\rangle =\left\langle \hat{n
_{b}\right\rangle =\bar{n}/2$, and $\langle \Delta ^{2}\hat{n}_{a}\rangle
=\langle \Delta ^{2}\hat{n}_{b}\rangle $, the overestimated Fisher
information tends to $0$, then the result reduces to lossless case
F_{QFI_{-}}=\bar{n}(Q+1)(1-J)$ \cite{Sahota15}.
In the opposite, highly dissipative limit
\begin{equation}
\langle \Delta ^{2}\hat{n}_{b}\rangle ,\langle \Delta ^{2}\hat{n}_{a}\rangle
\gg \langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}},
\end{equation
Eq.~(\ref{CQ-opt}) is simplified as to
\begin{equation}
\mathcal{C}_{QFIM_{-}}^{opt}=\frac{\eta _{a}\langle \hat{n}_{a}\rangle }
1-\eta _{a}}(1-2J\frac{\langle \Delta \hat{n}_{b}\rangle }{\langle \Delta
\hat{n}_{a}\rangle })-\Delta \mathcal{C}_{QFIM_{-}}, \label{Eq41}
\end{equation
where the first term in $\mathcal{C}_{QFIM_{-}}$of Eq. (\ref{Eq41}) is the
bound from the single-parameter estimation, which coincides with the bound
by Escher \textit{et al.} \cite{Escher11} and Demkowicz-Dobrzanski \textit
et al. }\cite{Demkowicz}. The second term $\Delta \mathcal{C}_{QFIM_{-}}$ is
the overestimated Fisher information from QFI-only calculation in the
presence of losses in the SU(2) interferometers, which is given by
\begin{equation}
\Delta \mathcal{C}_{QFIM_{-}}=\frac{over_{U}^{-}}{over_{D}^{-}},
\end{equation
where
\begin{eqnarray}
over_{U}^{-} &=&\frac{\eta _{a}^{2}\langle \hat{n}_{a}\rangle ^{2}}{(1-\eta
_{a})^{2}}(1-2J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}) \notag \\
&&\times \lbrack 1-J^{2}+2(J+\sqrt{\frac{\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)
})^{2}] \notag \\
&&-\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
(1-J^{2})\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1) \notag \\
&&\times (2J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}+3), \notag \\
over_{D}^{-} &=&\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
[1-J^{2}+2(J+\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}})^{2}] \notag \\
&&+(1-J^{2})\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1).
\end{eqnarray
When $\left\langle \hat{n}_{a}\right\rangle =\left\langle \hat{n
_{b}\right\rangle =\bar{n}/2$ and $\langle \Delta ^{2}\hat{n}_{a}\rangle
=\langle \Delta ^{2}\hat{n}_{b}\rangle $, the overestimated Fisher
information $\Delta \mathcal{C}_{QFIM_{-}}\neq 0$ in the presence of losses,
it is different from lossless case.
\subsection{SU(1,1) interferometers}
In the case of SU(1, 1) interferometers. To minimize the $\mathcal{C
_{QFIM_{+}}$, it requires to find the optimal $\gamma $, and we obtai
\begin{equation}
\frac{d\mathcal{C}_{QFIM_{+}}}{d\gamma }=0. \label{Gama}
\end{equation
Based on Eq. (\ref{Gama}), the optimal $\gamma _{opt}^{+}$ is obtained
\begin{equation}
\gamma _{opt}^{+}=\frac{1}{(1-\eta _{a})+\frac{\eta _{a}}{(Q_{a}+1)(1-J^{2})
(1-J\sqrt{\frac{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}})}-1.
\end{equation
Substituting the optimal $\gamma _{opt}^{+}$ into $\mathcal{C}_{QFIM_{+}}$
of Eq. (\ref{OptC+}), the optimal bound $\mathcal{C}_{QFIM_{+}}^{opt}$ is
worked out:
\begin{eqnarray}
&&\mathcal{C}_{QFIM_{+}}^{opt}=\frac{4}{\Upsilon _{4}}[\langle \hat{n
_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}(1-J^{2})^{2}\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1) \notag \\
&&+\langle \hat{n}_{a}\rangle ^{2}(\frac{\eta _{a}}{1-\eta _{a}})^{2}(\sqrt
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}-J)^{2}(1-J^{2})],\text{ \ }
\end{eqnarray
where
\begin{eqnarray}
\Upsilon _{4} &=&(\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
)^{2}\{[\frac{1+5J^{2}}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle ^{2}(Q_{a}+1)^{2}} \notag \\
&&+\frac{J^{2}}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}-\frac
2J(1+J^{2})}{\sqrt{\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle (Q_{a}+1)(Q_{b}+1)}} \notag \\
&&-\frac{4J}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}\sqrt{\frac
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)}}] \notag \\
&&+\langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}(1-J^{2})[1+
\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)} \notag \\
&&+J^{2}-4J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}]\} \notag \\
&&+(1-J^{2})^{2}\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1).
\end{eqnarray}
Similar to the SU(2) interferometers, we can also derive two limits. For
small dissipation, that i
\begin{equation}
\langle \Delta ^{2}\hat{n}_{b}\rangle ,\langle \Delta ^{2}\hat{n}_{a}\rangle
\ll \langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}.
\end{equation
We get the reduced result
\begin{equation}
\mathcal{C}_{QFIM_{+}}^{opt}=4\frac{(\sqrt{\frac{\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)
}-J)^{2}(1-J^{2})\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}{\Upsilon
_{4L}},
\end{equation
where
\begin{eqnarray}
\Upsilon _{4L} &=&1+5J^{2}+J^{2}\frac{\left\langle \hat{n}_{a}\right\rangle
(Q_{a}+1)}{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}+\frac
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n
_{a}\right\rangle (Q_{a}+1)} \notag \\
&&-2J(1+J^{2})\sqrt{\frac{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}
\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}} \notag \\
&&-4J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}.
\end{eqnarray}
When $\left\langle \hat{n}_{a}\right\rangle =\left\langle \hat{n
_{b}\right\rangle =\bar{n}/2$, and $\langle \Delta ^{2}\hat{n}_{a}\rangle
=\langle \Delta ^{2}\hat{n}_{b}\rangle $, the overestimated Fisher
information tends to $0$, then the result reduces to lossless case
F_{QFI_{+}}=\bar{n}(Q+1)(1+J)$ \cite{Gong17}. Unlike the case of SU(2)
interferometers, here is $1+J$ not $1-J$.
The highly dissipative limit i
\begin{equation}
\langle \Delta ^{2}\hat{n}_{b}\rangle ,\langle \Delta ^{2}\hat{n}_{a}\rangle
\gg \langle \hat{n}_{a}\rangle \frac{\eta _{a}}{1-\eta _{a}}.
\end{equation
we obtai
\begin{equation}
\mathcal{C}_{QFIM_{+}}^{opt}=\frac{\eta _{a}\langle \hat{n}_{a}\rangle }
1-\eta _{a}}(1+2J\frac{\langle \Delta \hat{n}_{b}\rangle }{\langle \Delta
\hat{n}_{a}\rangle })-\Delta \mathcal{C}_{QFIM_{+}},
\end{equation
where
\begin{equation}
\Delta \mathcal{C}_{QFIM_{+}}=\frac{over_{U}^{+}}{over_{D}^{+}},
\end{equation
wit
\begin{eqnarray}
over_{U}^{+} &=&\frac{\eta _{a}^{2}\langle \hat{n}_{a}\rangle ^{2}}{(1-\eta
_{a})^{2}}(1+2J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}) \notag \\
&&\times \lbrack 1-J^{2}+2(J-\sqrt{\frac{\left\langle \hat{n
_{b}\right\rangle (Q_{b}+1)}{\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)
})^{2}] \notag \\
&&+\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
(1-J^{2})\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1) \notag \\
&&\times (2J\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}}-3), \notag \\
over_{D}^{+} &=&\frac{\eta _{a}\langle \hat{n}_{a}\rangle }{1-\eta _{a}
[1-J^{2}+2(J-\sqrt{\frac{\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1)}
\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)}})^{2}] \notag \\
&&+(1-J^{2})\left\langle \hat{n}_{b}\right\rangle (Q_{b}+1).
\end{eqnarray
Similar to SU(2) interferometer case, when $\left\langle \hat{n
_{a}\right\rangle =\left\langle \hat{n}_{b}\right\rangle =\bar{n}/2$ and
\langle \Delta ^{2}\hat{n}_{a}\rangle =\langle \Delta ^{2}\hat{n}_{b}\rangle
$, the overestimated Fisher information of $\Delta \mathcal{F}_{+}=0$ in
lossless case becomes $\Delta \mathcal{C}_{QFIM_{-}}\neq 0$ in the presence
of losses. Compare these two expressions $\Delta \mathcal{C}_{QFIM_{+}}$ and
$\Delta \mathcal{C}_{QFIM-}$, there is only difference in signs between
them. Since LBS and NBS are linear and nonlinear processes respectively, the
average photon number and photon fluctuation of the same input state after
LBS and NBS transformation will have significant differences, thus the phase
sensitivity will also have significant differences.
\section{Losses in two arms}
Interferometers with photon losses in both arms as shown in
Fig. \ref{fig1} can be treated in a similar way. A possible set of Kraus
operators describing the process is
\begin{align}
\hat{\Pi}_{l_{a},l_{b}}& =\sqrt{\frac{(1-\eta _{a})^{l_{a}}}{l_{a}!}}\sqrt
\frac{(1-\eta _{b})^{l_{b}}}{l_{b}!}} \notag \\
\times e^{i\phi _{-}\frac{\hat{n}_{a}-\hat{n}_{b}-\gamma _{a}l_{a}+\gamma
_{b}l_{b}}{2}}& e^{i\phi _{+}\frac{\hat{n}_{a}+\hat{n}_{b}-\gamma
_{a}l_{a}-\gamma _{b}l_{b}}{2}}\eta _{a}^{\frac{\hat{n}_{a}}{2}}\eta _{b}^
\frac{\hat{n}_{b}}{2}}\hat{a}^{l_{a}}\hat{b}^{l_{b}},
\end{align
where $\eta _{a}$ ($\eta _{b}$) quantifies the photon losses of arm $a$ ($b
). $\gamma _{a}=-1$ and $\gamma _{b}=-1$ ($\gamma _{a}=0$ and $\gamma _{b}=0
) describe\ the photons loss before (after) the phase shifts of arm $a$ and
arm $b$.
Substituting the $\hat{\Pi}_{l_{a},l_{b}}(\phi _{+},\phi _{-})$ into $C_{ij}
\ of Eq.~(\ref{CQ}), the matrix elements $C_{ij}$ are given as follows:
\begin{eqnarray}
&&C_{++}=[1-\Gamma _{a}(1-\eta _{a})]^{2}\left\langle \Delta \hat{n
_{a}^{2}\right\rangle +\Gamma _{a}^{2}(1-\eta _{a})\eta _{a}\left\langle
\hat{n}_{a}\right\rangle \notag \\
&&+[1-\Gamma _{b}(1-\eta _{b})]^{2}\left\langle \Delta \hat{n
_{b}^{2}\right\rangle +\Gamma _{b}^{2}(1-\eta _{b})\eta _{b}\left\langle
\hat{n}_{b}\right\rangle \notag \\
&&+2[1-\Gamma _{a}(1-\eta _{a})][1-\Gamma _{b}(1-\eta _{b})]Cov[\hat{n}_{a}
\hat{n}_{b}],
\end{eqnarray
\begin{eqnarray}
&&C_{--}=[1-\Gamma _{a}(1-\eta _{a})]^{2}\left\langle \Delta \hat{n
_{a}^{2}\right\rangle +\Gamma _{a}^{2}(1-\eta _{a})\eta _{a}\left\langle
\hat{n}_{a}\right\rangle \notag \\
&&+[1-\Gamma _{b}(1-\eta _{b})]^{2}\left\langle \Delta \hat{n
_{b}^{2}\right\rangle +\Gamma _{b}^{2}(1-\eta _{b})\eta _{b}\left\langle
\hat{n}_{b}\right\rangle \notag \\
&&-2[1-\Gamma _{a}(1-\eta _{a})][1-\Gamma _{b}(1-\eta _{b})]Cov[\hat{n}_{a}
\hat{n}_{b}],
\end{eqnarray
\begin{eqnarray}
&&C_{+-}=C_{-+}=[1-\Gamma _{a}(1-\eta _{a})]^{2}\left\langle \Delta \hat{n
_{a}^{2}\right\rangle +\Gamma _{a}^{2}(1-\eta _{a})\left\langle \hat{n
_{a}\right\rangle \notag \\
&&\times \eta _{a}-[1-\Gamma _{b}(1-\eta _{b})]^{2}\left\langle \Delta \hat{
}_{b}^{2}\right\rangle -\Gamma _{b}^{2}(1-\eta _{b})\eta _{b}\left\langle
\hat{n}_{b}\right\rangle ,
\end{eqnarray
where $\Gamma _{a}=\gamma _{a}+1$, $\Gamma _{b}=\gamma _{b}+1$.
Substituting the matrix elements $C_{ij}$ into $\mathcal{C}_{QFIM_{-}}$\ of
Eq. (\ref{CQFIM}) and into $\mathcal{C}_{QFIM_{+}}$\ of Eq. (\ref{CQFIMP}),
we can obtain the $\mathcal{C}_{QFIM_{-}}$ and $\mathcal{C}_{QFIM_{+}}$. For
convenience, considering $\gamma _{a}=\gamma _{b}=\gamma \equiv \Omega -1$,
\eta _{a}=\eta _{b}=\eta $, $\mathcal{C}_{QFIM_{-}}$ and $\mathcal{C
_{QFIM_{+}}$ are given by
\begin{equation}
\mathcal{C}_{QFIM_{-}}=4\frac{\Upsilon _{5}}{[1-\Omega (1-\eta
)]_{-}^{2}\chi _{-}+\Omega ^{2}(1-\eta )\eta \epsilon },
\end{equation
and
\begin{equation}
\mathcal{C}_{QFIM_{+}}=4\frac{\Upsilon _{5}}{[1-\Omega (1-\eta )]^{2}\chi
_{+}+\Omega ^{2}(1-\eta )\eta \epsilon },
\end{equation
where
\begin{eqnarray}
\Upsilon _{5} &=&[1-\Omega (1-\eta )]^{4}\zeta +\Omega ^{4}(1-\eta )^{2}\eta
^{2}\tau \notag \\
&&+[1-\Omega (1-\eta )]^{2}\Omega ^{2}(1-\eta )\eta \lambda ,
\end{eqnarray
an
\begin{eqnarray}
\chi _{\pm } &=&\left\langle \hat{n}_{a}\right\rangle (Q_{a}+1)+\left\langle
\hat{n}_{b}\right\rangle (Q_{b}+1) \notag \\
&&\mp 2J\sqrt{\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle (Q_{a}+1)(Q_{b}+1)}, \notag \\
\zeta &=&(1-J^{2})\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle (Q_{a}+1)(Q_{b}+1), \notag \\
\epsilon &=&\left\langle \hat{n}_{a}\right\rangle +\left\langle \hat{n
_{b}\right\rangle ,\tau =\left\langle \hat{n}_{a}\right\rangle \left\langle
\hat{n}_{b}\right\rangle , \notag \\
\lambda &=&\left\langle \hat{n}_{a}\right\rangle \left\langle \hat{n
_{b}\right\rangle \left[ (Q_{a}+1)+(Q_{b}+1)\right] .
\end{eqnarray}
Similar to single-arm loss case, we also minimize $\mathcal{C}_{QFIM_{-}}$
and $\mathcal{C}_{QFIM_{+}}$, corresponding to SU(2) and SU(1,1)
interferometers, respectively.
\subsection{SU(2) interferometers}
In the case of SU(2) interferometers. To minimize the $\mathcal{C
_{QFIM_{-}} $, it requires to find the optimal $\gamma $, and we ge
\begin{equation}
\frac{d\mathcal{C}_{QFIM_{-}}}{d\gamma }=0.
\end{equation
The equation above is more complex, and we cannot obtain an analytical
solution. Therefore, we check the $d\mathcal{C}_{QFIM_{-}}/d\gamma $
directly\ in two limits again. If there is small dissipation, we recover the
QFIM $\mathcal{F}_{QFIM_{-}}$ for the lossless case.
In the opposite, if there is highly dissipative limit, due to high intensity
we have $\left\langle \Delta \hat{n}_{b}^{2}\right\rangle _{0}\approx
\left\langle \Delta \hat{n}_{a}^{2}\right\rangle _{0}$, and consider a
special case $J=-1$ then we obtain the optimal $\gamma _{opt}^{H-}$, which
is given by
\begin{equation}
\gamma _{opt}^{H-}=\frac{\eta \left[ I-\tau \right] }{\left[ \eta \tau
+(1-\eta )\lambda \right] },
\end{equation
where the superscript $H$ indicates high loss. Substituting the optimal
\gamma _{opt}^{H-}$ into Eq. (\ref{CQFIMP}), the optimal bound $\mathcal{C
_{QFIM_{+}}^{H}$ is worked out:
\begin{eqnarray}
&&\mathcal{C}_{QFIM_{-}}^{H}=4\frac{1}{(\Lambda _{-}^{H})^{2}\chi
_{-}+(\Omega _{-}^{H})^{2}(1-\eta )\eta \epsilon }[(\Lambda
_{-}^{H})^{4}\zeta \notag \\
&&+(\Omega _{-}^{H})^{4}(1-\eta )^{2}\eta ^{2}\tau +(\Lambda
_{-}^{H})^{2}(\Omega _{-}^{H})^{2}(1-\eta )\eta \lambda ],
\end{eqnarray
wher
\begin{equation}
\Omega _{-}^{H}=\frac{\lambda }{\left[ \eta \tau +(1-\eta )\lambda \right] }
\text{ }\Lambda _{-}^{H}=\frac{\eta \tau }{\left[ \eta \tau +(1-\eta
)\lambda \right] }.
\end{equation}
\subsection{SU(1,1) interferometers}
Substituting the matrix elements $C_{ij}$ into Eq. (\ref{CQFIM}) and Eq.
\ref{CQFIMP}), ones can obtain the $\mathcal{C}_{QFIM_{-}}$ and $\mathcal{C
_{QFIM_{+}}$.
To minimize the $\mathcal{C}_{QFIM_{+}}$, it requires to find the optimal
\gamma $, and we obtain
\begin{equation}
\frac{d\mathcal{C}_{QFIM_{+}}}{d\gamma }=0.
\end{equation}
Similar to the case of SU(2) interferometers, due to the above equation is
more complex, we check the $d\mathcal{C}_{QFIM_{+}}/d\gamma $ directly\ in
two limits again. If there is small dissipation, we recover the QFIM
\mathcal{F}_{QFIM+}$ for the lossless case.
In the opposite, if there is highly dissipative limit, due to high intensity
we have $\left\langle \Delta \hat{n}_{b}^{2}\right\rangle _{0}\approx
\left\langle \Delta \hat{n}_{a}^{2}\right\rangle _{0}$, and we also consider
a special case $J=1$, we obtain the optimal $\gamma _{opt}^{H+}$, which is
given by
\begin{equation}
\gamma _{opt}^{H+}=\frac{\eta \left[ I-\tau \right] }{\left[ \eta \tau
+(1-\eta )\lambda \right] },
\end{equation
Substituting the optimal $\gamma _{opt}^{H+}$ into Eq. (\ref{CQFIMP}), the
optimal bound $\mathcal{C}_{QFIM_{+}}^{H}$ is cast into:
\begin{eqnarray}
&&\mathcal{C}_{QFIM_{+}}^{H}=4\frac{1}{(\Lambda _{+}^{H})^{2}\chi
_{+}+(\Omega _{+}^{H})^{2}(1-\eta )\eta \epsilon }[(\Lambda
_{+}^{H})^{4}\zeta \notag \\
&&+(\Lambda _{+}^{H}\Omega _{+}^{H})^{2}(1-\eta )\eta \lambda +(\Omega
_{+}^{H})^{4}(1-\eta )^{2}\eta ^{2}\tau ],
\end{eqnarray
where
\begin{equation}
\Omega _{+}^{H}=\frac{\lambda }{\left[ \eta \tau +(1-\eta )\lambda \right] }
\text{ }\Lambda _{+}^{H}=\frac{\eta \tau }{\left[ \eta \tau +(1-\eta
)\lambda \right] }.
\end{equation}
\section{QCRB and Numerical results}
The phase sensitivity of the interferometer can be obtained for a given
measurement scheme, such as homodyne measurement \cite{Li14}, parity
measurement \cite{Li16} or intensity measurement \cite{Plick10}, with usage
of error propagation formula. However, it is difficult to optimize over the
detection methods to obtain the optimal estimation schemes. Fortunately, the
QFI introduced by Braustein and Caves \cite{Braunstein94,Braunstein96} is
the intrinsic information in the quantum state and is not related to a
particular measurement scheme. Based on the QFI, the ultimate precision
bound of phase sensitivity is given by the QCRB
\begin{equation}
\Delta \phi _{QCRB}=\frac{1}{\sqrt{m\mathcal{F}}},
\end{equation
where $m$ is the number of independent repeats of the experience.
Now we present a typical example to apply our theory to describe the
difference between the QFI-only and QFIM. Here, We consider the case that
the two input modes of the interferometer. We input the $\left\vert \alpha
\right\rangle _{a}\otimes \left\vert \varsigma \right\rangle _{b}$, where
coherent state $\left\vert \alpha \right\rangle $ with $\alpha =\left\vert
\alpha \right\vert e^{i\theta _{\alpha }}$ with $\left\vert \alpha
\right\vert $ being a complex number and $\theta _{\alpha }$ being the
initial phase. And the squeezed vacuum state $\left\vert \varsigma
\right\rangle =\exp [1/2(\varsigma ^{\ast }\hat{a}^{2}-\varsigma (\hat{a
^{\dagger })^{2})$ with $\varsigma =re^{i\theta _{r}}$\ is squeezed
parameter where $r$ and $\theta _{r}$ are the squeezing amplitude and
squeezing angle respectively.
\subsection{SU(2) interferometers}
In the SU(2) interferometer, the average number of photons in the two arms
is $\langle \hat{n}_{a}\rangle ^{-}=T\left\vert \alpha \right\vert
^{2}+R\sinh ^{2}r$ and $\langle \hat{n}_{b}\rangle ^{-}=R\left\vert \alpha
\right\vert ^{2}+T\sinh ^{2}r$, when $2\theta _{\alpha }-\theta _{r}=0$, we
can get
\begin{eqnarray}
\langle \Delta ^{2}\hat{n}_{a}\rangle ^{-} &=&T^{2}\left\vert \alpha
\right\vert ^{2}+2R^{2}\sinh ^{2}r\cosh ^{2}r \notag \\
&&+TR(\left\vert \alpha \right\vert ^{2}e^{2r}+\sinh ^{2}r), \notag \\
\langle \Delta ^{2}\hat{n}_{b}\rangle ^{-} &=&R^{2}\left\vert \alpha
\right\vert ^{2}+2T^{2}\sinh ^{2}r\cosh ^{2}r \notag \\
&&+TR(\left\vert \alpha \right\vert ^{2}e^{2r}+\sinh ^{2}r),
\end{eqnarray
an
\begin{equation}
Cov[\hat{n}_{a},\hat{n}_{b}]^{-}=TR(\left\vert \alpha \right\vert
^{2}(1-e^{2r})+\sinh ^{2}r\cosh 2r).
\end{equation
where $T$ and $R$ are the reflectivity and transmissivity of the BS,
respectively. And using the above results one can ge
\begin{eqnarray*}
Q_{a}^{-} &=&\frac{R(T\left\vert \alpha \right\vert ^{2}(e^{2r}-1)+R\sinh
^{2}r\cosh 2r)}{T\left\vert \alpha \right\vert ^{2}+R\sinh ^{2}r}, \\
Q_{b}^{-} &=&\frac{T(R\left\vert \alpha \right\vert ^{2}(e^{2r}-1)+T\sinh
^{2}r\cosh 2r)}{R\left\vert \alpha \right\vert ^{2}+T\sinh ^{2}r}.
\end{eqnarray*
\begin{equation}
J^{-}=\frac{TR(\left\vert \alpha \right\vert ^{2}(1-e^{2r})+\sinh ^{2}r\cosh
2r)}{\sqrt
\begin{array}{c}
\left[
\begin{array}{c}
T\left\vert \alpha \right\vert ^{2}(T+Re^{2r}) \\
+R\sinh ^{2}r(T+2R\cosh ^{2}r
\end{array
\right] \\
\times \left[
\begin{array}{c}
R\left\vert \alpha \right\vert ^{2}(R+Te^{2r}) \\
+T\sinh ^{2}r(R+2T\cosh ^{2}r
\end{array
\right
\end{array
}}.
\end{equation}
\subsection{SU(1,1) interferometers}
Similar to the SU(2) interferometer case, we also consider the input
\left\vert \alpha \right\rangle _{a}\otimes \left\vert \varsigma
\right\rangle _{b}$ in the SU(1,1) interferometer. The average number of
photons in the two arms is $\langle \hat{n}_{a}\rangle ^{+}=G^{2}\left\vert
\alpha \right\vert ^{2}+g^{2}\cosh ^{2}r$ and $\langle \hat{n}_{b}\rangle
^{+}=G^{2}\sinh ^{2}r+g^{2}(\left\vert \alpha \right\vert ^{2}+1)$, when
2\theta _{g}-2\theta _{\alpha }-\theta _{r}=\pi $, $\theta _{g}$ is the
phase shift of the NBS for wave splitting and recombination. According to
the above conditions, we obtai
\begin{eqnarray}
\langle \Delta ^{2}n_{a}\rangle ^{+} &=&G^{4}\left\vert \alpha \right\vert
^{2}+2g^{4}\sinh ^{2}r\cosh ^{2}r \notag \\
&&+G^{2}g^{2}(\left\vert \alpha \right\vert ^{2}e^{2r}+\cosh ^{2}r), \notag
\\
\langle \Delta {}^{2}n_{b}\rangle ^{+} &=&g^{4}\left\vert \alpha \right\vert
^{2}+2G^{4}\sinh ^{2}r\cosh ^{2}r \notag \\
&&+G^{2}g^{2}(\left\vert \alpha \right\vert ^{2}e^{2r}+\cosh ^{2}r),
\end{eqnarray
an
\begin{equation}
Cov[\hat{n}_{a},\hat{n}_{b}]^{+}=G^{2}g^{2}(\left\vert \alpha \right\vert
^{2}(1+e^{2r})+\cosh ^{2}r\cosh 2r).
\end{equation}
Here, $G$ is the gain factors of NBS, for wave splitting and recombination
with $G^{2}-g^{2}=1$. The results are consistent with the results of Ref.
\cite{You19}. Using the above results we work ou
\begin{eqnarray}
Q_{a}^{+} &=&\frac{g^{2}(G^{2}\left\vert \alpha \right\vert
^{2}(1+e^{2r})+g^{2}\cosh ^{2}r\cosh 2r)}{G^{2}\left\vert \alpha \right\vert
^{2}+g^{2}\cosh ^{2}r}, \notag \\
Q_{b}^{+} &=&\frac{\left[
\begin{array}{c}
G^{2}\sinh ^{2}r\left[ 2G^{2}\cosh ^{2}r+g^{2}-1\right] \\
+g^{2}(g^{2}(\left\vert \alpha \right\vert ^{2}+1)+\left\vert \alpha
\right\vert ^{2}(G^{2}e^{2r}-1)
\end{array
\right] }{G^{2}\sinh ^{2}r+g^{2}(\left\vert \alpha \right\vert ^{2}+1)}.
\end{eqnarray
\begin{equation}
J^{+}=\frac{G^{2}g^{2}(\left\vert \alpha \right\vert ^{2}(1+e^{2r})+\cosh
^{2}r\cosh 2r)}{\sqrt
\begin{array}{c}
\left[
\begin{array}{c}
G^{2}\left\vert \alpha \right\vert ^{2}(G^{2}+g^{2}e^{2r}) \\
+g^{2}\cosh ^{2}r(G^{2}+2g^{2}\sinh ^{2}r
\end{array
\right] \\
\times \left[
\begin{array}{c}
g^{2}\left\vert \alpha \right\vert ^{2}(g^{2}+G^{2}e^{2r}) \\
+G^{2}\cosh ^{2}r(g^{2}+2G^{2}\sinh ^{2}r
\end{array
\right
\end{array
}}.
\end{equation}
Using the above results, we can obtain the numerical QCRBs in the case of
losses in one arm and in two arms with the coherent state and squeezed
vacuum state input for an example.
Firstly, we study the phase sensitivities of SU(1,1) interferometers or
SU(2) interferometers and compare the differences between the results given
by the QFI and QFIM phase estimation methods without loss. As shown in Fig.
\ref{fig2}, the phase sensitivities of the SU(1,1) interferometer or SU(2)
interferometer has difference between the single-parameter estimation and
the two-parameter estimation under certain conditions. The difference is due
to $\Delta F_{\pm }$ also known as overestimated Fisher Information. With
the increase of $N_{\alpha }$, the difference of phase sensitivity caused by
$\Delta F_{\pm }$ always exists, which corresponds to our conclusion in Sec.
\ref{sec:IQ}. Therefore, it is appropriate to use the two-parameter
estimation for the phase estimation of the interferometer.
Then, in the presence of loss of one arm, we study the phase sensitivities
of SU(1,1) interferometers or SU(2) interferometers in the method of QFIM,
as is depicted in Fig.~\ref{fig3}. The QCRBs of the single-parameter
estimation and the two-parameter estimation are very close, the sensitivity
of them is approximately equal at moderate loss. In order to investigate
this phenomenon, we study the difference of phase sensitivity $\Delta \phi
_{s-t}$ between single-parameter and two-parameter estimation by changing
\eta $. In the case of SU(2) interferometers, $\Delta \phi _{s-t}$ as a
function of $R/T$ under different photon loss coefficients $\eta $ is shown
in Fig.~\ref{fig4}(a). For a given $\eta $, there is a minimum value of
\Delta \phi _{s-t}$, which appears on the larger beam splitter ratio $R/T$
as $\eta $ increases. In the case of SU(1,1) ones, $\Delta \phi _{s-t}$ as a
function of $G$ under different photon loss coefficients $\eta $ is shown in
Fig.~\ref{fig4}(b). Similar to the SU(2) case, there is also a minimum value
of $\Delta \phi _{s-t}$, which appears on the larger $G$ as $\eta $
increases.
The overestimated Fisher Information in the ideal case gradually disappears
as the loss increases. With the further increase of the loss, the
overestimated Fisher Information reappears. The area of these minimum values
of $\Delta \phi _{s-t}$ corresponds to $\Delta C_{\pm }=0$, i.e.,
C_{+-}=C_{-+}=0$. Due to consider the single-arm loss (arm $a$), Eq.
(29) can be written as $C_{+-}=C_{-+}=[\eta _{a}+\gamma \eta _{a}-\gamma ]^{2}\langle
\Delta ^{2}\hat{n}_{a}\rangle +\left( \gamma +1\right) ^{2}(1-\eta _{a})\eta
_{a}\left\langle \hat{n}_{a}\right\rangle -\langle \Delta ^{2}\hat{n
_{b}\rangle \equiv \overline{\langle \Delta ^{2}\hat{n}_{a}\rangle }-\langle
\Delta ^{2}\hat{n}_{b}\rangle $, where $\overline{\langle \Delta ^{2}\hat{n
_{a}\rangle }$ can be seen as effective fluctuation due to loss. Because the
fluctuations $\overline{\langle \Delta ^{2}\hat{n}_{a}\rangle }$ and
\langle \Delta ^{2}\hat{n}_{b}\rangle $ are different, when the loss
increases gradually, $\overline{\langle \Delta ^{2}\hat{n}_{a}\rangle }$
decreases gradually. When it decreases to the same as the other arm, the
overestimated Fisher Information disappears. The loss continues to increase,
and the fluctuation $\overline{\langle \Delta ^{2}\hat{n}_{a}\rangle }$ in
turn continues to increase, leading to the re-emergence of overestimation. A
similar phenomenon also occurs in the loss of both arms.
\section{conclusion}
In conclusion, we theoretically extended the model developed by Escher
\textit{et al} \cite{Escher11} to the QFIM case with noise. Photon loss is a
very usual noise in optical systems, then we gave the QFIM expressions in the
case of single-arm photon loss and two-arm photon losses in the SU(2) and
SU(1,1) interferometers with arbitrary pure states input. The ultimate
precision limits of SU(2) and SU(1,1) interferometers with photon losses was
investigated and discussed by using coherent state $\otimes $ squeezed
vacuum state as an example. The overestimated Fisher Information existing in
the ideal case will occur disappear and revival phenomenon with loss
coefficient or splitter ratio changing. For a given loss coefficient, adjusting the splitter ratio is a method to optimize the sensitivity of quantum measurements in a lossy environment. This strategy is beneficial to quantum precision measurement in lossy environments.
\section{Acknowledgments}
This work is supported by the National Natural Science Foundation of China
Grants No.~11974111, No.~11874152, No.~12104423, No.~11654005, No. 11974116,
No.~91536114, Shanghai Municipal Science and Technology Major Project under
Grant No. 2019SHZDZX01; Innovation Program of Shanghai Municipal Education
Commission No. 202101070008E00099; the Fundamental Research Funds for the
Central Universities; the Shanghai talent program.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,143
|
Q: Equivalent characterization of compact operator Let $T: X \to Y$ be an operator between two normed vector spaces. Recall that $T$ is said to be a compact operator whenever $T$ maps the closed unit ball $U$ of $X$ onto a norm relatively compact subset of $Y$. In case $Y$ is Banach space, $T$ is compact if and only if $T(U)$ is a norm totally bounded subset of $Y$. In there, ı need help to prove that if $T(U)$ is a norm totally bounded subset of $Y$ then $T$ is compact. I've proved the other side. How can ı extend a totally bounded set to a compact set under which conditions?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,903
|
The court system in Chino Valley, Arizona is a government institution of Arizona to settle disagreements involving residents of, or events that occurred in, Chino Valley.
If you are immersed in a lawsuit, or any legal dispute, in Chino Valley, Arizona, it's extremely likely that you will have some contact with the court system.
Jury Duty: All adult U.S. citizens who live in Chino Valley are eligible to be called for jury duty in Chino Valley, Arizona. This is the only way that many people interact with the court system. If you are called to serve in jury duty (normally, every person is eligible to be called for jury duty once per year), when you get the letter from the court telling you to show up for jury duty (normally a few weeks after the letter arrives), you are legally obligated to do so on the appointed date. You are then placed in a large "juror pool," and called into a courtroom that has an upcoming trial. The lawyers for each side of the case will then ask each juror some questions to determine if they are biased. If the lawyer doesn't want a juror serving on the jury, they can dismiss the juror (they only have a limited number of dismissals, however). If you are dismissed, you're done. If you end up on the jury, you are now serving a very important role in the justice system, and it's important that you follow all of the judge's instructions.
Filing a Lawsuit: if you sue someone in Chino Valley, Arizona, you will most definitely interact with the local court system - a lot. Even if the case never goes to trial, you will have to deal with pre-trial motions, discovery requests, and case management conferences. You will definitely become more aware about the Chino Valley, Arizona court system than you ever wanted to be.
Being Sued: If you face the unenviable prospect of getting sued in Chino Valley, Arizona, you'll be spending a good deal of time dealing with the courts. You will have to file an answer to the complaint, discovery requests, motions, and many other documents with the court. Majority of these will happen whether or not the case even goes to trial.
Divorce: If you are divorcing your partner in Chino Valley, Arizona, you have to deal with the court system, particularly if you have disagreements over property or child custody which you cannot resolve on your own.
How Can A Chino Valley, Arizona Tort Lawyer Help?
If you find yourself in a situation in which you're likely to deal with the courts in Chino Valley, Arizona, chances are good that some difficult legal issues are involved.
You will probably need assistance navigating these legal issues. You should definitely consider contacting the services of a reliable litigation lawyer in Chino Valley, Arizona if you believe that you are going to have significant contact with the court system sometime soon.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,448
|
\section{Weighted Bergman spaces
}\label{weighted-bergman}
Let $\mathbb D$ denote the unit disc in the complex plane and let $H^2({\mathbb D})$ be the classical Hardy space, that is, the space of holomorphic functions on ${\mathbb D}$ such that
\begin{equation*}
\|f\|^2_{H^2({\mathbb D})}:=\sup_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2\, d\theta<+\infty.
\end{equation*}
Given a function $f\in H^2({\mathbb D})$ it is well--known (see, for instance, \cite{zhu-t}) that
\begin{equation}\label{hardy-limit}
\|f\|^2_{H^2({\mathbb D})}=\lim_{\nu\to-1^+}\|f\|^2_{A^2_{\nu}({\mathbb D})},
\end{equation}
where $A^2_\nu({\mathbb D})$ denotes the weighted Bergman space on ${\mathbb D}$ with norm
\begin{equation}\label{norm-disc}
\|f\|^2_{A^2_\nu({\mathbb D})}=(\nu+1)\int_{{\mathbb D}}|f(z)|^2(1-|z|^2)^{\nu}\, dz.
\end{equation}
Given $K_{{\mathbb D}}(z,w)=(1-\overline zw)^{-2}$, the reproducing kernel of the unweighted Bergman space $A^2(\bbH)$, notice that the weight in \eqref{norm-disc} satisfies
\begin{equation*}
(1-|z|^2)^{\nu}= K^{-\frac\nu2}_{{\mathbb D}}(z,z)\approx \delta^{\nu}(z),
\end{equation*}
where $\delta(z)$ is the distance of $z\in{\mathbb D}$ from the topological boundary $\partial{\mathbb D}$.
In light of the above, for $\nu>-1$ we define $A^2_\nu(\bbH)$ as the space of holomorphic functions on $\bbH$ such that
\begin{equation}\label{weighted-norm}
\|f\|^2_{A^2_\nu(\bbH)}:=C_\nu\int_{\bbH}|f(z)|^2 K^{-\frac\nu2}(z,z)\, dz<+\infty,
\end{equation}
where $z=(z_1,z_2)$, $K(z,w)$ is the reproducing kernel of the unweighted Bergman space $A^2(\bbH)$, $dz$ denotes the Lebesgue measure of ${\mathbb C}^2$ and $C_\nu$ is a normalization constant to be determined. In Section \ref{hardy-space} we will define the Hardy space $H^2(\bbH)$ as the limit of the $A^2_\nu(\bbH)$ spaces as $\nu\to-1$ in the sense of \eqref{hardy-limit}.
The unweighted Bergman space $A^2(\bbH)$ corresponds to the case $\nu=0$ but we drop the subscript in the notation $A^2_0(\bbH)$. In this case we set
\begin{equation*}
\|f\|^2_{A^2(\bbH)}=\frac{2}{\pi^2}\int_{\bbH}|f(z_1,z_2)|^2\, dz,
\end{equation*}
that is, we fix the constant $C_\nu=C_0$ in \eqref{weighted-norm} to be $2/\pi^2$. With such a normalization it is easily verified that $\{z_1^jz_2^k: j\geq0, j+k+1\geq0\}$ is an orthogonal basis for $A^2(\mathbb H)$ and that
\begin{equation*}
\|z_1^jz_2^k\|^2_{A^2}=\frac{2}{(j+1)(k+j+2)}.
\end{equation*}
We recall that if $\mathcal H$ denotes a reproducing kernel Hilbert space and $\{\varphi_n\}_{n}$ is an orthonormal basis for $\mathcal H$, then the reproducing kernel $K_{\mathcal H}$ is given by the sum $K_{\mathcal H}=\sum_{n}\varphi_n\overline\varphi_{n}$. Therefore, the Bergman kernel of $\bbH$ is given by
\begin{align*}
\begin{split}
K(z,w)&=\frac{1}{2}\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}(j+1)(k+j+2)(z_1\overline w_1)^j(z_2\overline w_2)^k=\frac{1}{2 z_2\overline w_2 (1-\frac{z_1\overline w_1}{z_2\overline w_2})^2(1-z_2\overline w_2)^2}.
\end{split}
\end{align*}
In the general case $\nu>-1$ we set
\begin{equation}\label{mu}
d\mu_\nu(z):= C_\nu K^{-\frac\nu 2}(z,z)\, dz= C_\nu 2^\frac\nu2 |z_2|^\nu(1-|z_1/z_2|^2)^\nu(1-|z_2|^2)^{\nu}\, dz
\end{equation}
where
\begin{equation}\label{Cnu}
C_\nu=\frac{(\nu+1)\Gamma(\frac32\nu+3)}{2^{\frac\nu2}\pi^{2}\Gamma(\nu+1)\Gamma(\frac\nu2+2)}.
\end{equation}
Notice that,
\begin{align*}
\int_\bbH K^{-\frac\nu 2}(z,z)\, dz&= 2^\frac\nu2\int_{\bbH}|z_2|^{\nu}(1-|z_1/z_2|^2)^{\nu}(1-|z_2|^2)^{\nu}\, dz\\
&=2^{\frac\nu2}4\pi^2\int_0^1\int_0^\rho r\rho^{\nu+1}(1-(r/\rho)^2)^\nu(1-\rho^2)^\nu\, drd\rho\\
&=2^{\frac\nu2}4\pi^2\int_0^1\int_0^1 r\rho^{\nu+3}(1-(r/\rho)^2)^\nu(1-\rho^2)^\nu\, drd\rho\\
&=2^\frac\nu2\pi^2\frac{\Gamma(\nu+1)\Gamma(\frac\nu2+2)}{(\nu+1)\Gamma(\frac32\nu+3)},
\end{align*}
where the last equality follows from from the identity
\begin{equation*}
\int_0^1 (1-x)^a x^b\,dx=\frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)}
\end{equation*}
whenever $\operatorname{Re}}%{\, \text{\rm Re\,} a,\operatorname{Re}}%{\, \text{\rm Re\,} b>-1$. Thus, we deduce from \eqref{mu} and \eqref{Cnu} that
\begin{equation*}
\int_{\mathbb H}\,d\mu_{\nu}(z)=1
\end{equation*}
for all $\nu>-1$.
In conclusion, the spaces $A^2_\nu(\bbH)$ are defined as
\begin{equation*}
A^2_\nu(\bbH)=\Big\{f\in \operatorname{Hol}(\bbH): \|f\|^2_{A^2_\nu}:=\int_{\bbH}|f(z_1,z_2)|^2\, d\mu_\nu(z)<+\infty\Big\}.
\end{equation*}
\begin{remark}\label{remark}\emph{
If $\nu\leq-1$ we have $\int_\bbH d\mu_\nu(z)=+\infty$ and the space $A^2_\nu(\bbH)$ is empty. In fact, $\|z_1^jz_2^k\|_{A^2_\nu}=+\infty$ for any non-negative integer $j$ and any integer $k$. Hence, exploiting the uniform convergence of the Laurent expansion $f=(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k=-\infty}^{+\infty}a_{jk}z_1^jz_2^k
$
of any function $f\in\operatorname{Hol}(\bbH)$, we conclude that $f$ does not belong to any $A^2_\nu(\bbH)$. Hence, $A^2_\nu(\bbH)=\emptyset$.}
\end{remark}
\medskip
We now recall the following proposition, proved in \cite{E-thesis}, about the diagonal behavior of the kernel $K$.
\begin{prop}[\cite{E-thesis}] The following facts hold true.
\begin{enumerate}
\item[$(i)$] Let $\delta(z)$ be the distance of $z$ to $\partial\bbH$, the topological boundary of $\bbH$. Then,
\begin{equation*}
K(z,z)\approx \delta(z)^{-2}
\end{equation*}
as $z$ tends to the origin.
\medskip
\item[$(ii)$] Let $p$ be any point in the distinguished boundary $d_b(\bbH)$. For any number $\beta\in (2,4]$ there exists a path $\gamma: [1/2,1]\to\overline\bbH$ such that $\gamma(1)=p$ and for all $u\in[1/2,1)$,
\begin{equation*}
K(\gamma(u),\gamma(u))\approx \delta(\gamma(u))^{-\beta}.
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
We refer the reader to Theorems 3.1.4 and 3.1.5 in \cite{E-thesis}.
\end{proof}
We want to prove that the spaces $A^2_\nu(\bbH)$ are reproducing kernel Hilbert spaces and study the regularity of the associated weighted Bergman projection. We need the following elementary lemma.
\begin{lem} \label{monomial-norm}
Let $\nu>-1$. Then, the monomial $z_1^jz_2^k$ belongs to $A^2_\nu$ if and only if
\begin{equation*}
j\geq0 \quad\text{and}\quad j+k+\frac\nu2+2>0.
\end{equation*}
\end{lem}
\begin{proof}
From \eqref{mu}, integrating in polar coordinates, we obtain
\begin{align*}
\|z_1^jz_2^k\|^2_{A^2_{\nu}}&=2^\frac\nu2 C_\nu\int_{\bbH}|z_1|^{2j}|z_2|^{2k} |z_2|^{\nu}(1-|z_1/z_2|^2)^\nu(1-|z_2|^2)^\nu\,dz\\
&=2^{\frac\nu2+2}\pi^{2}C_\nu \int_0^1\int_0^\rho r^{2j+1}\rho^{2k+\nu+1}(1-(r/\rho)^2)^\nu(1-\rho^2)^\nu\, drd\rho\\
&=2^{\frac\nu2+2}\pi^2 C_\nu\int_0^1\int_0^1 r^{2j+1}\rho^{2(j+k)+\nu+3}(1-r^2)^{\nu}(1-\rho^2)^{\nu}\, drd\rho\\
&=\frac{(\nu+1)\Gamma(\nu+1)\Gamma(\frac32\nu+3)}{\Gamma(\frac\nu2+2)}\frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)},
\end{align*}
where the last equality holds true if and only if
\begin{equation*}
j\geq 0 \quad\text{and}\quad j+k+\frac\nu2+2>0
\end{equation*}
as we wished to show.
\end{proof}
From the previous lemma we conclude that any function $f\in A^2_\nu(\bbH)$ is of the form
\begin{equation}\label{expansion-f}
f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}a_{jk}z_1^jz_2^k
\end{equation}
and
\begin{align}\label{expansion-norm}
\begin{split}
\|f\|^2_{A^2_\nu}=\frac{(\nu+1)\Gamma(\nu+1)\Gamma(\frac32\nu+3)}{\Gamma(\frac\nu2+2)}\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}\frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)}|a_{jk}|^2
\end{split}
\end{align}
Moreover, we have the following proposition.
\begin{prop}\label{Bergman-kernel} Let $\nu>-1$. The following properties hold.
\begin{enumerate}
\item[$(i)$] The space $A^2_\nu(\bbH)$ is a reproducing kernel Hilbert space.
\medskip
\item[$(ii)$] An orthonormal basis for $A^2_\nu(\bbH)$ is given by $
\{z_1^jz_2^k/\|z_1^jz_2^k\|_{A^2_\nu}\}_{(j,k)\in I_{\nu}},
$
where $ I_{\nu}=\big\{(j,k): j\geq0, j+k+\frac\nu2+2>0\big\}$.
\medskip
\item[$(iii)$] The reproducing kernel $K_\nu$ of $A^2_{\nu}(\bbH)$ is given by
\begin{equation*}
K_{\nu}(z,w)=\frac{\Gamma(\frac\nu2+2)}{\Gamma(\frac32\nu+3)}\frac{(z_2\overline w_2)^{-2}}{(1-\frac{z_1\overline w_1}{z_2\overline w_2})^{\nu+2}}\sum_{k>-\frac\nu2}^\infty\frac{\Gamma(k+\frac32\nu+1)}{\Gamma(k+\frac\nu2)}(z_2\overline w_2)^k.
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
Let $I_{\nu}=\big\{(j,k): j>0, j+k+\frac\nu2+2>0\big\}$ and set
\begin{equation*}
\nu_{jk}:=\frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)},
\end{equation*}
so that, from \eqref{expansion-norm}, we have
\begin{equation*}
\|f\|^2_{A^2_\nu}\approx\sum_{(j,k)\in I_\nu} \nu_{jk}|a_{jk}|^2.
\end{equation*}
Then, for any $z=(z_1,z_2)\in\bbH$,
\begin{align*}
|f(z_1,z_2)|&\leq \sum_{(j,k)\in I_\nu} |a_{jk}||z_1|^j|z_2|^k\leq \Big(\sum_{(j,k)\in I_\nu} |a_{jk}|^2 \nu_{jk}\Big)^{\frac12}\Big(\sum_{(j,k)\in I_\nu} |z_1|^{2j}|z_2|^{2k} \nu_{jk}^{-1}\Big)^{\frac12}.
\end{align*}
Since $(z_1,z_2)$ is a fixed point in $\bbH$ the latter factor in the above estimate is finite, whereas the first factor is comparable to the $A^2_\nu(\bbH)$ norm of the function $f$. Therefore, if $c_{\nu,z}$ denotes a positive constant depending on $\nu$ and $z=(z_1,z_2)\in\bbH$, we obtain
\begin{equation*}
|f(z_1,z_2)|\leq c_{\nu,z}\|f\|_{A^2_\nu},
\end{equation*}
that is, the point-evaluation functionals are bounded on $A^2_\nu(\bbH)$. Hence, the space $A^2_{\nu}(\bbH)$ is a reproducing kernel Hilbert space. This concludes the proof of $(i)$. The proof of $(ii)$ follows at once from Lemma \ref{monomial-norm} and \eqref{expansion-f}. Consequently, we obtain that the reproducing kernel $K_\nu$ of $A^2_\nu(\bbH)$ is given by
\begin{align}\label{kernel}
\begin{split}
K_\nu(z,w)&=K_\nu((z_1,z_2),(w_1,w_2))\\
&=\frac{\Gamma(\frac\nu2+2)}{(\nu+1)\Gamma(\nu+1)\Gamma(\frac32+3)}\sum_{(j,k)\in I_\nu}\frac{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)}{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}(z_1\overline w_1)^j(z_2\overline w_2)^k\\
&=\frac{\Gamma(\frac\nu2+2)}{\Gamma(\frac32\nu+3)}\frac{(z_2\overline w_2)^{-2}}{(1-\frac{z_1\overline w_1}{z_2\overline w_2})^{\nu+2}}\sum_{k>-\frac\nu2}^\infty\frac{\Gamma(k+\frac32\nu+1)}{\Gamma(k+\frac\nu2)}(z_2\overline w_2)^k
\end{split}
\end{align}
as we wished to show. The last equality follows from the identity
\begin{equation*}
\frac{1}{(1-z)^\lambda}=\frac{1}{\Gamma(\lambda)}\sum_{n=0}^\infty\frac{\Gamma(n+\lambda)}{n!}z^n,
\end{equation*}
which holds for $\operatorname{Re}}%{\, \text{\rm Re\,} \lambda>0$ and $|z|<1$. This concludes the proof of $(iii)$ and of the proposition.
\end{proof}
\begin{remark} \emph{
A closed formula for the kernel $K_\nu$ is immediately deduced. Given $\alpha,\beta$ and $\gamma$, real or complex parameters where $\gamma$ is not a non-positive integer, the Hypergeometric Function is defined as
\begin{equation*}
F(\alpha,\beta;\gamma;z)= \frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\beta)}\sum_{k=0}^{+\infty}\frac{\Gamma(k+\alpha)\Gamma(k+\beta)}{\Gamma(k+\gamma)k!}z^k.
\end{equation*}
It is easily seen that the radius of convergence of this series is $1$. For more properties of the Hypergeometric Function we refer the reader, for instance, to \cite{Lebedev}.
From \eqref{kernel} we get that
\begin{equation}\label{kernel-general}
K_{\nu}(z,w)=a_\nu \frac{(z_2\overline w_2)^{-1-\lceil\frac\nu2\rceil}}{(1-\frac{z_1\overline w_1}{z_2\overline w_2})^{\nu+2}}F\Big(\frac32\nu-\Big\lceil\frac\nu2\Big\rceil+2,1;\frac\nu2-\Big\lceil\frac\nu2\Big\rceil+1;z_2\overline w_2\Big)
\end{equation}
where $a_\nu=\frac{\Gamma(\frac\nu2+2)\Gamma(\frac32\nu-\lceil\frac\nu2\rceil+2)}{\Gamma(\frac32\nu+3)\Gamma(\frac\nu2-\lceil\frac\nu2\rceil+1)}$. Notice that when $\nu=2n,n\in\mathbb N_0$ it holds
\begin{equation*}
F\Big(\frac32\nu-\lceil\frac\nu2\rceil+2,1;\frac\nu2-\lceil\frac\nu2\rceil+1;z_2\overline w_2\Big)=(1-z_2\overline w_2)^{-2n-2}.
\end{equation*}
}
\end{remark}
\medskip
Given the weighted Bergman spaces $A^2_\nu(\bbH)$ it is a natural question to investigate the $L^p$ mapping properties of the associated weighted Bergman projection, that is, the operator
\begin{equation*}
P_\nu f(z_1,z_2):=\int_{\bbH} f(w_1,w_2) K_{\nu}((w_1,w_2),(z_1,z_2))\, d\mu_\nu(w),
\end{equation*}
where $d\mu_{\nu}$ is the measure defined in \eqref{mu}. We recall that $P_\nu$ is the orthogonal projection of $L^2_\nu(\bbH)$ onto $A^2_\nu(\bbH)$, where $L^2_\nu(\bbH)$ is the space of square-integrable functions with respect to the measure $d\mu_\nu$.
We now prove the necessary part of Theorem \ref{main-1}.
\proof[Necessary condition of Theorem \ref{main-1}]
Let us consider the function $f(z_1,z_2)=\overline z_2^{1+\lceil\frac\nu2\rceil}$ which belongs to $L^p_\nu$ for any $p\in(1,\infty)$. From the Laurent series of the kernel \eqref{kernel} we deduce
\begin{align*}
P_\nu f(z_1,z_2)= d_\nu z_2^{-1-\lceil\nu/2\rceil}
\end{align*}
for some positive constant $d_\nu$. Now,
\begin{align*}
\|P_\nu f\|^p_{L^p_\nu}&\approx \int_{\bbH}|z_2|^{\nu-(1+\lceil\nu/2\rceil)p}(1-|z_1/z_2|^{2})^{\nu}(1-|z_2|^2)^{\nu}\, dz\\
&\approx\int_0^1\int_0^1 r\rho^{\nu-(1+\lceil\nu/2\rceil)p+3}(1-r^2)^\nu(1-\rho^2)^\nu\, drd\rho,
\end{align*}
and this last integral diverges if $p\geq\frac{4+\nu}{1+\lceil\nu/2\rceil}=2+\frac{\nu-2\lceil\nu/2\rceil+2}{1+\lceil\nu/2\rceil}$. Therefore, $P_\nu$ cannot extend to a bounded operator if $p\geq 2+\frac{\nu-2\lceil\nu/2\rceil+2}{1+\lceil\nu/2\rceil}$. By a standard duality argument we also obtain that $P_\nu$ cannot be bounded if $1<p\leq2-\frac{\nu-2\lceil\nu/2\rceil+2}{\nu-\lceil\nu/2\rceil+3}$. Indeed, if $1/p+1/p'=1$, since $P_\nu$ is self-adjoint, we get
\begin{align*}
\|P_\nu f\|_{L^{p'}_\nu}&=\sup_{\|g\|_{L^p_\nu}=1}|\left< P_\nu f,g\right>|=\sup_{\|g\|_{L^p\nu}=1}|\left<f,P_\nu g\right>|\leq \sup_{\|g\|_{L^p_\nu}=1}\|P_\nu g\|_{L^p_\nu}\|f\|_{L^{p'}_\nu}.
\end{align*}
Therefore, the $L^p_\nu$-boundedness of $P_\nu$ would imply the $L^{p'}_{\nu}$-boundedness. Hence, $P_\nu$ cannot be bounded if $p\notin\Big(2-\frac{\nu-2\lceil\frac\nu2\rceil+2}{\nu-\lceil\frac\nu2\rceil+3}, 2+\frac{\nu-2\lceil\frac\nu2\rceil+2}{1+\lceil\frac\nu2\rceil}\Big)$. Using the identity $\lfloor\frac\nu2\rfloor=\lceil\frac\nu2\rceil-1$ it is easy to see that this condition coincides with the conditions \emph{(i),(ii)} and \emph{(iii)} as $\nu$ varies. The proof is concluded.
\qed
\medskip
The sufficient condition in Theorem \ref{main-1} will be proved by means of the classical Schur's lemma, which we now recall. For a proof of this result we refer the reader, for instance, to \cite[Appendix A]{Grafakos}.
\begin{lem}[Schur's lemma]\label{schur}
Let $(\mathcal X, d\mu_\mathcal X)$,$ (\mathcal Y, d\mu_\mathcal Y )$ be two $\sigma$-finite measure spaces. Let $T$ the integral operator given by
\begin{equation*}
Tf(x)=\int_{\mathcal Y}K(x,y)f(y)\, d\mu_\mathcal Y(y),
\end{equation*}
where $K$ is a measurable positive kernel on $\mathcal X\times\mathcal Y$. Let $1<p,p'<+\infty$ be such that $1/p+1/p'=1$. Suppose there exist positive functions $\psi:\mathcal Y\to (0,+\infty)$, $\varphi: \mathcal X\to (0,+\infty)$ such that
\begin{enumerate}[(i)]
\item$ \int_\mathcal Y K(x,y)\psi^{p'}\, d\mu_\mathcal Y(y)\leq C\varphi(x)^{p'}$;
\item $\int_{\mathcal X} K(x,y)\varphi^p(x)\, d\mu_{\mathcal X}(x)\leq C\psi(y)^p$.
\end{enumerate}
Then, $T: L^p(\mathcal Y)\to L^p(\mathcal X)$ is bounded.
\end{lem}
We also need the following lemmas. The first two lemmas are elementary estimates of which we omit the proofs.
\begin{lem}\label{cl-estimate}
Let $\tau\in\ (0,+\infty)$ and let $\rho\in (0,1)$. Then,
\begin{equation*}
\int_0^{2\pi}\frac{1}{|1-\rho e^{i\theta}|^{1+\tau}}\, d\theta\approx (1-\rho)^{-\tau}.
\end{equation*}
\end{lem}
\begin{lem}\label{cl-estimate2}
Let $\gamma>-1$, $\delta\in (0,+\infty)$ and $z\in \mathbb D$. Then,
\begin{equation*}
\int_{\mathbb D}\frac{(1-|w|^2)^\gamma}{|1-z\overline w|^{2+\gamma+\delta}}\,dw\lesssim (1-|z|^2)^{-\delta}.
\end{equation*}
\end{lem}
\begin{lem}\label{kernel-estimate2}
Let $\nu>-1$ and let $z\in\mathbb D$. Then,
\begin{equation*}
\Big|F\Big(\frac32\nu-\Big\lceil\frac\nu2\Big\rceil+2, 1;\frac\nu2-\Big\lceil\frac\nu2\Big\rceil+1;z\Big)\Big|\lesssim |1-z|^{-(\nu+2)}.
\end{equation*}
\end{lem}
\begin{proof}The proof follows from elementary properties of the Hypergeometric function. From \cite[Chapter 9.5]{Lebedev} we get
\begin{align*}
\Big|F\Big(\frac32\nu-\Big\lceil\frac\nu2\Big\rceil+2, 1;\frac\nu2-\Big\lceil\frac\nu2\Big\rceil+1;z\Big)\Big|&=|1-z|^{-(\nu+2)}\Big|F\Big(-\nu-1, \frac\nu2-\Big\lceil\frac\nu2\Big\rceil;\frac\nu2-\Big\lceil\frac\nu2\Big\rceil+1;z\Big)\Big|\\
&\lesssim |1-z|^{-(\nu+2)}
\end{align*}
where the last bound follows from estimates of $|F(\alpha, \beta; \gamma; z)|$ when $\operatorname{Re}}%{\, \text{\rm Re\,}(\gamma-\alpha-\beta)>0$ (\cite[Chapter 9.3]{Lebedev}).
\end{proof}
\begin{remark}\emph{From the previous lemma and equation \eqref{kernel-general} we deduce that the kernel $K_\nu$ satisfies the main estimate \eqref{kernel-estimate} for any $\nu>-1$.}
\end{remark}
We are now ready to prove the sufficient condition in Theorem \ref{main-1}.
\proof[Sufficient condition of Theorem \ref{main-1}]
We want to apply Schur's lemma to the positive kernel $|K_{\nu}|$ since the boundedness of the operator with such kernel would imply the boundedness of $P_\nu$. We choose
\begin{equation*}
f(z_1,z_2)= (1-|z_1/z_2|^2)^{-\alpha}(1-|z_2|^2)^{-\beta}|z_2|^{-\gamma}
\end{equation*}
as test function, where $\alpha,\beta,\gamma$ are positive real parameters to be chosen later. Let $(z,w)=((z_1,z_2),(w_1,w_2))$. Then, from Lemma \ref{kernel-estimate2} and the change of variables $(w_1/w_2,w_2)\mapsto (z_1,z_2)$ we get
\begin{align}\label{estimate-schur}
\begin{split}
\!\!\int_{\bbH}& |K_\nu(z,w)|f^p(w_1,w_2)\, d\mu_\nu(w)\!\approx\!\int_{\bbH}|K_\nu(z,w)||w_2|^{\nu-\gamma p}(1-|w_1/w_2|^2)^{\nu-\alpha p}(1-|w_2|^2)^{\nu-\beta p} dw\\
&\lesssim |z_2|^{-1-\lceil\frac\nu2\rceil}\int_{\bbH} |w_2|^{\nu-\lceil\frac\nu2\rceil-\gamma p-1}\frac{(1-|w_1/w_2|^2)^{\nu-\alpha p}(1-|w_2|^2)^{\nu-\beta p}}{|1-\frac{z_1\overline w_1}{z_2\overline w_2}|^{2+\nu}|1-z_2\overline w_2|^{2+\nu}}\, dw\\
&\lesssim |z_2|^{-1-\lceil\frac\nu2\rceil}\bigg(\int_{\mathbb D}\frac{(1-|w_1|^2)^{\nu-\alpha p}}{|1- \overline w_1\frac{z_1}{z_2}|^{2+\nu}}\, dw_1\bigg)\bigg(\int_{{\mathbb D}^*}\frac{|w_2|^{\nu-\lceil\frac\nu2\rceil-\gamma p+1}(1-|w_2|^2)^{\nu-\beta p}}{|1-z_2\overline w_2|^{2+\nu}}\, dw_2\bigg).
\end{split}
\end{align}
Now, we require $\nu-\alpha p>-1$ and we apply Lemma \ref{cl-estimate2} with $\gamma=\nu-\alpha p$ and $\delta=\alpha p$ to the first integral obtaining
\begin{align*}
\bigg(\int_{\mathbb D}\frac{(1-|w_1|^2)^{\nu-\alpha p}}{|1- \overline w_1\frac{z_1}{z_2}|^{2+\nu}}\, dw_1\bigg)\lesssim (1-|z_1/z_2|)^{-\alpha p}.
\end{align*}
For the second integral, applying Lemma \ref{cl-estimate} with $\tau=1+\nu$ and requiring the conditions $\nu-\beta p>-1$ and $\nu-\lceil\nu/2\rceil-\gamma p+2>-1$, we have
\begin{align*}
\int_{\mathbb D^*}|w_2|^{\nu-\lceil\frac\nu2\rceil-\gamma p+1}&\frac{(1-|w_2|^2)^{\nu-\beta p}}{|1-z_2\overline w_2|^{2+\nu}}\, dw_2\lesssim\int_0^1 \rho^{\nu-\lceil\frac\nu2\rceil-\gamma p+2}\frac{(1-\rho)^{\nu-\beta p}}{(1-|z_2|\rho)^{1+\nu}}\, d\rho\\
&=\sum_{k=0}^{+\infty}\frac{\Gamma(k+1+\nu)}{\Gamma(\nu+1)\Gamma(k+1)}|z_2|^{k}\int_0^1\rho^{k+\nu-\lceil\frac\nu2\rceil-\gamma p+2}(1-\rho)^{\nu-\beta p}\, d\rho\\
&=\sum_{k=0}^{+\infty}\frac{\Gamma(k+1+\nu)\Gamma(k+\nu-\lceil\frac\nu2\rceil-\gamma p+3)\Gamma(\nu-\beta p+1)}{\Gamma(\nu+1)\Gamma(k+1)\Gamma(k+2\nu-\lceil\frac\nu2\rceil-\gamma p-\beta p+4)}|z_2|^{k} \\
&\lesssim\sum_{k=0}^{+\infty}\frac{\Gamma(k+\beta p)}{\Gamma(k+1)}|z_2|^{k}\\
&\lesssim(1-|z_2|^2)^{-\beta p}.
\end{align*}
In conclusion, from \eqref{estimate-schur} we get
\begin{equation*}
\int_{\bbH} |K_\nu(z,w)|f^p(w_1,w_2)\, d\mu_\nu(w)\lesssim (1-|z_1/z_2|^2)^{-\alpha p}(1-|z_2|^2)^{-\beta p }|z_2|^{-\gamma p}
\end{equation*}
if the conditions
\begin{equation}\label{conditions}
\begin{cases}
\nu-\alpha p>-1\\ \nu-\beta p>-1\\ \nu-\lceil\frac\nu2\rceil-\gamma p+2>-1 \\ 1+\lceil\frac\nu2\rceil-\gamma p<0
\end{cases}
\end{equation}
are satisfied. We obtain the same estimate using the same test function $f$ and $p'$ instead of $p$. Therefore, we can apply Schur's lemma if the conditions \eqref{conditions} are satisfied simultaneously for $p$ and $p'$. This happens if
\begin{align*}
&\alpha\in (0,(\nu+1)/{p})\cap(0,(\nu+1)/{p'}),\qquad \beta\in (0,(\nu+1)/{p})\cap(0,\nu+1/{p'}),\\
&\gamma \in((1+\lceil\nu/2\rceil)/{p},(3+\nu-\lceil\nu/2\rceil)/{p})\cap((1+\lceil\nu/2\rceil)/{p'},(3+\nu-\lceil\nu/2\rceil)/{p'}).
\end{align*}
The parameters $\alpha$ and $\beta$ always vary in a non-trivial range, whereas, in order to have a non-trivial range for $\gamma$, we need that $p\in \Big(2-\frac{\nu-2\lceil\frac\nu2\rceil+2}{\nu-\lceil\frac\nu2\rceil+3}, 2+\frac{\nu-2\lceil\frac\nu2\rceil+2}{1+\lceil\frac\nu2\rceil}\Big)$ and the conclusion follows.
\qed
\medskip
\begin{remark}\emph{
Let $\Phi:{\mathbb D}\times{\mathbb D}^*\to\bbH$ be the biholomorphism $(w_1,w_2)\mapsto (w_1w_2,w_2)$.
We observe that the pull-back of $A^2_\nu(\bbH)$ via $\Phi$, which we denote by $A^2_{\nu}({\mathbb D}\times{\mathbb D}^*)$, is the space of holomorphic functions on ${\mathbb D}\times{\mathbb D}^*$ endowed with the norm
$$
\|f\|^2_{A^2_\nu({\mathbb D}\times{\mathbb D}^*)}:= c_\nu\int_{{\mathbb D}\times{\mathbb D}^*}|f(w_1,w_2)|^2|w_2|^{\nu} (1-|w_1|^2)^\nu(1-|w_2|^2)^{\nu}\, dw<+\infty,
$$
where $c_\nu=\frac{(\nu+1)\Gamma(\frac32\nu+3)}{\pi^2\Gamma(\nu+1)\Gamma(\frac\nu2+2)}$. Clearly, the map $f\mapsto \Phi'\cdot f\circ\Phi$, where $\Phi'=\operatorname{Jac}_{\mathbb C}(\Phi)$, is a surjective isometry from $A^2_\nu(\bbH)$ onto $A^2_\nu({\mathbb D}\times{\mathbb D}^*)$. We point out that for $\nu\in(-1,0]$ the functions in $A^2_\nu({\mathbb D}\times{\mathbb D}^*)$ are actually holomorphic on ${\mathbb D}\times{\mathbb D}$. \newline
\indent If $p\neq2$ we consider the weighted space $A^p_\nu({\mathbb D}\times{\mathbb D}^*,|w_2|^2)$, the space of holomorphic functions on ${\mathbb D}\times{\mathbb D}^*$ with norm
\begin{equation*}
\|f\|_{A^p_\nu({\mathbb D}\times{\mathbb D}^*,|w_2|^2)}=c_\nu\int_{{\mathbb D}\times{\mathbb D}^*}|f(w_1,w_2)|^p|w_2|^{\nu+2} (1-|w_1|^2)^\nu(1-|w_2|^2)^{\nu}\, dw<+\infty.
\end{equation*}
Then, $f\mapsto f\circ\Phi$ is a surjective isometry from $A^p_\nu(\bbH)$ onto $A^p_\nu({\mathbb D}\times{\mathbb D}^*,|w_2|^2)$.
}
\end{remark}
\section{A Hardy space on the Hartogs triangle}\label{hardy-space}
In this section we introduce a candidate Hardy space on the Hartogs triangle. If $\Omega=\{ z:\rho(z)<0\}$ is a
smoothly bounded domain in ${\mathbb C}^n$, the Hardy space $H^2(\Omega)$ is defined as
\begin{equation*}
H^2(\Omega) =\big\{ f\in\operatorname{Hol}(\Omega): \sup_{\varepsilon>0}
\int_{\partial\Omega_\varepsilon} |f|^2 d\sigma_\varepsilon <\infty
\big\} \,,
\end{equation*}
where $\Omega_\varepsilon=\{ z:\rho(z)<-\varepsilon\}$ and $d\sigma_\varepsilon$ is the
induced surface measure on $\partial\Omega_\varepsilon$, the topological boundary of $\Omega_\varepsilon$. Then, $H^2(\Omega)$ can be
identified with a closed subspace of $L^2(\partial\Omega,d\sigma)$ that we
denote by $H^2(\partial\Omega)$. The
Szeg\H o projection is the orthogonal projection
\begin{equation*}
S_\Omega : L^2(\partial\Omega,d\sigma) \to H^2(\partial\Omega) \,;
\end{equation*}
see \cite{Stein-holo}. Such a definition of Hardy space is not suitable for the Hartogs triangle. Having in mind \eqref{hardy-limit}, we introduce in this section a Hardy space $H^2(\bbH)$ as limit of the weighted Bergman spaces $A^2_\nu(\bbH)$ as follows. From \eqref{kernel-general} we get
\begin{align*}
\begin{split}
K_{-1}((z_1,z_2),(w_1,w_2)):= \lim_{\nu=-1} K_{\nu}((z_1,z_2),(w_1,w_2))=\frac{1}{(z_2\overline w_2-z_1\overline w_1)(1-z_2\overline w_2)}.
\end{split}
\end{align*}
We want $H^2(\bbH)$ to be the reproducing kernel Hilbert space associated to the kernel $K_{-1}$.
For $(s,t)\in (0,1)\times (0,1)$ we set
\begin{equation*}
\bbH_{st}=\Big\{(z_1,z_2)\in{\mathbb C}^2:|z_1|/s<|z_2|<t\Big\}\subsetneq \bbH
\end{equation*}
and
we define the Hardy space $H^2(\bbH)$ as
\begin{equation*}
H^2(\bbH)=\bigg\{f\in\operatorname{Hol}(\bbH):\sup_{(s,t)\in(0,1)\times(0,1)}\frac{1}{4\pi^2}\int_{d_b(\bbH_{st})}|f|^2\, d\sigma_{st}<+\infty\bigg\},
\end{equation*}
where $d\sigma_{st}$ denotes the induced surface measure on $d_b(\bbH_{st})$, the distinguished boundary of $\bbH_{st}$. We endow $H^2(\bbH)$ with the norm
\begin{equation*}
\|f\|^2_{H^2}:=\!\sup_{(s,t)\in(0,1)\times(0,1)}\frac{1}{4\pi^2}\int_{d_b(\bbH_{st})}\!|f|^2\, d\sigma_{st}=\!\sup_{(s,t)\in(0,1)\times(0,1)} \frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\!|f(st e^{i\theta}, t e^{i\gamma})|^2 st^2 d\theta d\gamma.
\end{equation*}
\begin{remark}\emph{
Unlike the classical setting, we point out that a boundary point, the origin, belongs to all the approximating domains $\bbH_{st}$. Also, our definition of $H^2(\bbH)$ is based on the approximating domain $\mathbb H_{st}$, which depends on two parameters $s,t$, but we could have also used the approximating domain $\bbH_t$ defined as
\begin{equation*}
\bbH_t=\Big\{(z_1,z_2)\in{\mathbb C}^2: |z_1|/t<|z_2|<t\Big\}.
\end{equation*}
The resulting space would be different from the one we considered. Hence, several different approaches are available to define a Hardy space on $\bbH$. A further investigation on this matter would be surely interesting. At the moment our goal is to characterize a reproducing kernel Hilbert space with a prescribed kernel that fits in a one-parameter family of reproducing kernels. As we now see, our space $H^2(\bbH)$ has this property.
}
\end{remark}
\begin{prop}
The Hardy space $H^2(\bbH)$ is a reproducing kernel Hilbert space with reproducing kernel
\begin{equation*}
K_{-1}((z_1,z_2),(w_1,w_2))=\frac{1}{(z_2\overline w_2-z_1\overline w_1)(1-z_2\overline w_2)}
\end{equation*}
\end{prop}
\begin{proof}
We first notice that $z_1^jz_2^k\in H^2(\bbH)$ if and only if $j\geq0$ and $j+k+1\geq0$. In fact,
\begin{align*}
\|z_1^jz_2^k\|^2_{H^2}&=\sup_{(s,t)\in(0,1)\times(0,1)}\frac{1}{
4\pi^2}\int_0^{2\pi}\int_0^{2\pi}|ste^{i\theta}|^{2j}|te^{i\gamma}|^{2k} st^2\, d\theta d\gamma\\
&=\sup_{(s,t)\in(0,1)\times(0,1)}|s^{2j+1}||t^{2(j+k+1)}|=1.
\end{align*}
Therefore, setting $I=\{(j,k): j\geq0 \wedge j+k+1\geq0\}$, $\{ z_1^jz_2^k\}_{(j,k)\in I}$ is an orthonormal basis for $H^2(\bbH)$ . Hence, any function $f\in H^2(\bbH)$ is of the form
\begin{equation}\label{hardy-power-series}
f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}a_{jk}z_1^jz_2^k
\end{equation}
and
\begin{equation*}
\|f\|^2_{H^2(\bbH)}=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}|a_{jk}|^2.
\end{equation*}
For any $z=(z_1,z_2)\in\bbH$ it holds
\begin{align*}
|f(z)|&=\Big|\sum_{j=0}^{+\infty}\sum_{j=-k-1}^{+\infty}\!a_{jk}z_1^jz_2^k\Big|\leq \bigg(\sum_{j=0}^{+\infty}\sum_{j=-k-1}^{+\infty}\!|a_{jk}|^2\bigg)^{\frac12}\bigg(\sum_{j=0}^{+\infty}\sum_{j=-k-1}^{+\infty} |z_1|^{2j}|z_2|^{2k}\bigg)^{\frac12}\leq c_z\|f\|_{H^2},
\end{align*}
where $c_z$ is a positive constant depending on the point $z$. It follows that point-evaluations are bounded functionals on $H^2(\bbH)$, that is, $H^2(\bbH)$ is a reproducing kernel Hilbert space with kernel given by
\begin{align*}
\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty} (z_1\overline w_1)^j(z_2\overline w_2)^k=\frac{1}{(z_2\overline w_2-z_1\overline w_1)(1-z_2\overline w_2)}=K_{-1}((z_1,z_2),(w_1,w_2))
\end{align*}
as we wished to show.
\end{proof}
\begin{prop}\label{Mf limit}
Let $f$ be in $H^2(\bbH)$. Then, $f\in A^2_{\nu}$ for all $\nu>-1$ and $\|f\|^2_{H^2} =\lim_{\nu\to-1}\|f\|^2_{A^2_\nu}$.
\end{prop}
\begin{proof}
For any $\nu>-1$ we have
\begin{align*}
\|f\|^2_{A^2_\nu}&=C_\nu2^{\frac\nu2}\int_{\bbH}|f(z_1,z_2)|^2|z_2|^\nu(1-|z_1/z_2|^2)^\nu(1-|z_2|^2)^\nu\,dz\\
&=C_\nu2^{\frac\nu2}\int_{{\mathbb D}\times{\mathbb D}^*}|f(z_1z_2,z_2)|^2|z_2|^{2+\nu}(1-|z_1|^2)^\nu(1-|z_2|^2)^\nu\, dz\\
&= C_\nu2^{\frac\nu2}\int_0^1\int_0^1\int_0^{2\pi}\int_{0}^{2\pi}|f(ste^{i\theta},t e^{i\gamma})|^2 (1-s^2)^\nu(1-t^2)^\nu st^{\nu+3}\, d\theta d\gamma ds dt\\
&\leq \pi^2 2^{\frac\nu2 }C_\nu\|f\|^2_{H^2} \frac{\Gamma(\frac 12)\Gamma(\nu+1)\Gamma(\frac \nu2+1)\Gamma(\nu+1)}{\Gamma(\nu+\frac 32)\Gamma(\frac 32\nu+2)}.
\end{align*}
Taking the limits on both sides and recalling \eqref{Cnu} we obtain that
\begin{equation}\label{upper-bound}
\|f\|^2_{H^2}\geq\lim_{\nu\to-1}\|f\|^2_{A^2_\nu}.
\end{equation}
Viceversa, given $\nu\in(-1,0)$ and $\varepsilon>0$ there exist $\delta_1,\delta_2>0$ such that
\begin{align*}
\|f\|^2_{A^2_\nu}&=C_\nu2^{\frac\nu2}\int_{\bbH}|f(z_1,z_2)|^2|z_2|^\nu(1-|z_1/z_2|^2)^\nu(1-|z_2|^2)^\nu\,dz\\
&=C_\nu2^{\frac\nu2}\int_{{\mathbb D}\times{\mathbb D}^*}|f(z_1z_2,z_2)|^2|z_2|^{2+\nu}(1-|z_1|^2)^\nu(1-|z_2|^2)^\nu\, dz\\
&\geq C_\nu2^{\frac\nu2}\int_{\delta_1}^1\int_{\delta_2}^1\int_0^{2\pi}\int_{0}^{2\pi}|f(ste^{i\theta},t e^{i\gamma})|^2 (1-s^2)^\nu(1-t^2)^\nu st^{\nu+3}\, d\theta d\gamma ds dt\\
&\geq \frac{\pi^2 2^{\frac\nu2} C_\nu}{(\nu+1)^2}(\|f\|^2_{H^2}-\varepsilon)\frac{(1-\delta_1^2)^{\nu+1}(1-\delta_2^2)^{\nu+1}}{(\nu+1)^2}.
\end{align*}
Taking the limit as $\nu\to-1$ we get
\begin{equation*}
\lim_{\nu\to-1}\|f\|^2_{A^2_\nu}\geq (\|f\|^2_{H^2}-\varepsilon).
\end{equation*}
Since $\varepsilon$ is arbitrary and \eqref{upper-bound} holds, we get that $\|f\|^2_{H^2}\lim_{\nu\to-1}\|f\|^2_{A^2_\nu}$ and the conclusion follows.
\end{proof}
Notice that to any function $f\in H^2(\bbH)$ we can associate a boundary value function $\widetilde f\in L^2(d_b(\bbH))$ defined as
\begin{equation}\label{boundary-value}
\widetilde f(e^{i\theta}, e^{i\gamma})=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}a_{jk}e^{ij\theta}e^{ik\gamma}.
\end{equation}
The function $\widetilde f$ is a boundary value function for $f$ since, setting
\begin{equation*}
f_{st}(e^{i\theta}, e^{i\gamma}):= f(st e^{i\theta}, t e^{i\gamma}),
\end{equation*}
we have
\begin{align*}
\lim_{(s,t)\to(1,1)}\|\widetilde f- f_{st}\|^2_{L^2(d_b(\bbH))}&=\lim_{(s,t)\to(1,1)}
\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}|a_{jk}|^2(1-s^j t^{j+k})^2=0
\end{align*}
by the dominated convergence theorem. Viceversa, any function $g\in L^2(d_b(\bbH)))$ of the form \eqref{boundary-value} automatically extends to a function in $H^2(\bbH)$. Therefore, we can identify the space $H^2(\bbH)$ with the closed space $H^2(d_b(\bbH))\subseteq L^2(d_b(\bbH))$ defined as
\begin{equation*}
H^2(d_b(\bbH)):=\bigg\{
f(e^{i\theta}, e^{i\gamma})=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}a_{jk}e^{ij\theta}e^{ik\gamma}: \|\{a_{jk}\}\|_{\ell^2}<+\infty\bigg\}.
\end{equation*}
From now on, we call Hardy space both the spaces $H^2(\bbH)$ and $H^2((d_b(\bbH)))$ and we denote by $f$ both the function in $H^2(\bbH)$ and its boundary value in $H^2(d_b(\bbH))$. This should cause no confusion and it will be clear from the context if we are working inside the domain $\bbH$ or on the distinguished boundary $d_b(\bbH)$. Wherever needed we will be more specific about notation and terminology.
We now consider the Szeg\H o projection associated to our Hardy space, that is, the Hilbert space projection operator $S: L^2(d_b(\bbH))\to H^2(d_b(\bbH))$ defined by
\begin{equation*}
f(e^{i\theta}, e^{i\gamma})=\sum_{j,k\in \mathbb Z} a_{jk}e^{ij\theta} e^{ik\gamma}\mapsto Sf(e^{i\theta}, e^{i\gamma}):=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}a_{jk}e^{ij\theta}e^{ik\gamma}.
\end{equation*}
If suitable interpreted, the Szeg\H o projection admits also the integral representation
\begin{align*}
\begin{split}
f\mapsto Sf(\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_1,\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_2)&=\frac{1}{4\pi^2}\int_{d_b(\bbH)} f(e^{i\theta},e^{i\gamma}) K_{-1}((\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_1,\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_2),(e^{i\theta},e^{i\gamma}))\, d\theta d\gamma
\end{split}
\end{align*}
where $(\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_1,\zeta} %%%%%%%%%%%%%%%%%%%%%%% NOTICE!!!! {\zeta_2)$ is any point in $d_b(\bbH)$. However, for our purposes it is enough to consider the Fourier series representation of $Sf$ and we no longer discuss its integral representation.
Since $d_b(\bbH)$ can be identified with the $2$-dimensional torus, from the classical theory of Fourier series on the torus the boundedness of $S$ will follow from the boundedness of the Fourier multiplier operator associated to the multiplier
\begin{equation}\label{szego-proj-2}
m(j,k)= \frac{1+\operatorname{sgn}(j)}{2}\cdot\frac{1+\operatorname{sgn}(j+k+1)}{2}.
\end{equation}
We recall that a Fourier multiplier operator on the $2$-dimensional torus is an operator of the form
\begin{equation*}
f\mapsto \sum_{(j,k)\in\bbZ^2} m(j,k)\widehat f(j,k) e^{ij\theta}e^{ik\theta},
\end{equation*}
where $\widehat f(j,k)$ denotes the $(j,k)$-Fourier coefficient of the function $f$ and $\{m(j,k)\}_{(j,k)\in\bbZ^2}$ is a bounded sequence.
We now prove Theorem \ref{main-2}.
\proof[Proof of Theorem \ref{main-2}]
The proof follows from classical results in harmonic analysis. Indeed, let us consider on $L^2({\mathbb R}^2)$ the Fourier multiplier operator associated to the multiplier
\begin{equation*}
\widetilde m(\xi,\eta)=\frac{1+\operatorname{sgn}(\xi)}{2}\cdot\frac{1+\operatorname{sgn}(\xi+\eta+1)}{2},
\end{equation*}
that is, the operator $Tf:= \mathcal F^{-1}(\widetilde m\mathcal F f)$, where $\mathcal F$ and $\mathcal F^{-1}$ denote the Fourier transform on $\mathbb R^2$ and its inverse respectively.
This operator is well-defined on the class of smooth compactly supported functions and extends to a bounded operator $T:L^p({\mathbb R}^2)\to L^p({\mathbb R}^2)$ for any $p\in(1,+\infty)$ by standard results on the Hilbert transform. By transference, see \cite{Grafakos-cl}, we obtain that the Fourier multiplier operator associated to $\widetilde m|_{\bbZ^2}$, that is, the Fourier multiplier operators associated to the multiplier defined in \eqref{szego-proj-2}, extends to a bounded operator $L^p(\partial{\mathbb D}\times\partial {\mathbb D})\to L^p(\partial {\mathbb D}\times\partial {\mathbb D})$ for any $p\in(1,+\infty)$. Therefore, the Szeg\H o projection extends to a bounded operator $S: L^p(d_b(\bbH))\to L^p(d_b(\bbH))$ for any $p\in(1,+\infty)$ as we wished to show.
\qed
\medskip
We conclude the section observing that there exists a surjective isometry from $H^2(\bbH)$ and $H^2({\mathbb D}\times{\mathbb D})$, the Hardy space on the bidisc. This latter space can be described as the space of functions $f\in\operatorname{Hol}({\mathbb D}\times{\mathbb D})$, $f(z_1,z_2)=\sum_{j,k\geq0} a_{jk}z_1^jz_2^k$ such that
\begin{equation*}
\|f\|^2_{H^2({\mathbb D}\times{\mathbb D})}:=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}|a_{jk}|^2<+\infty.
\end{equation*}
\begin{prop}\label{hardy-iso}
Let $\Phi:{\mathbb D}\times{\mathbb D}^*\to\bbH$ be the biholomorphic map $(w_1,w_2)\mapsto(w_1w_2,w_2)$ and set $\Phi'=\operatorname{Jac}_{{\mathbb C}}(\Phi)$. Then, the map
\begin{equation*}
f\mapsto \Phi'\cdot f\circ\Phi
\end{equation*}
is a surjective isometry from $H^2(\bbH)$ onto $H^2({\mathbb D}\times{\mathbb D})$.
\end{prop}
\begin{proof}
Given $f\in H^2(\bbH)$, from \eqref{hardy-power-series} we get
\begin{equation*}
\Phi'\cdot f\circ \Phi(w_1,w_2)=\sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}a_{jk}w_1^jw_2^{k+j+1}= \sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}\widetilde a_{jk}w_1^jw_2^{k},
\end{equation*}
where $\widetilde a_{jk}= a_{j(k-j-1)}$. Hence, $\Phi'\cdot f\circ \Phi\in\operatorname{Hol}({\mathbb D}\times{\mathbb D})$ and
\begin{equation*}
\|\Phi'\cdot f\circ \Phi\|^2_{H^2({\mathbb D}\times{\mathbb D})}=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}|\widetilde a_{jk}|^2= \sum_{j=0}^{+\infty}\sum_{k=-j-1}^{+\infty}|a_{jk}|^2=\|f\|^2_{H^2(\bbH)}.
\end{equation*}
Therefore, the map $
f\mapsto \Phi'\cdot f\circ\Phi$ is an isometry. The map is clearly surjective and the inverse is given by $(\Phi^{-1})'\cdot f\circ \Phi^{-1}$.
\end{proof}
\begin{remark}\emph{
We point out that the surjective isometry $f\mapsto \Phi'\cdot f\circ\Phi$ does not coincide with the expected result. In fact, let $\Omega_1,\Omega_2\subseteq{\mathbb C}$ be bounded domains with $\mathcal C^{\infty}$ boundaries and let $\varphi:\Omega_1\to\Omega_2$ be a biholomorphic mappings. Then, then map $f\mapsto\sqrt{\varphi'}\cdot f\circ\varphi$ is an isometric isomorphism between the Hardy spaces $H^2(\Omega_1)$ and $H^2(\Omega_2)$ (\cite[Chapter 12]{Bell-book}). In order to prove this result one has to prove that $\sqrt{\varphi'}$ is a well-defined holomorphic function on $\Omega_1$. Moreover, since $\Omega_1$ and $\Omega_2$ are smooth bounded domains, there is no ambiguity in defining the Hardy spaces on these domains. In our setting we would expect the isomorphism to be of the form $f\mapsto \sqrt{\Phi'}\cdot f\circ\Phi$ where $\Phi,\Phi'$ are defined as in Proposition \ref{hardy-iso}. However, this is not the case. The culprit may be the fact that there is no standard definition for the Hardy space on $\bbH$, whereas there is for the Hardy space on ${\mathbb D}\times{\mathbb D}$. With our definition of $H^2(\bbH)$ the expected isomorphism fails. Moreover, notice that $(\Phi^{-1})'(z_1,z_2)=z_2^{-1}$, thus the factor $\sqrt{(\Phi^{-1})'}$ in the expected isomorphism between $H^2(\bbH)$ and $H^2({\mathbb D}\times{\mathbb D})$ is not even well-defined.
}
\end{remark}
\section{Weighted Dirichlet and Dirichlet Spaces}\label{dirichlet-space}
In this section we enlarge our family of function spaces defining some weighted Dirichlet spaces $\mathcal D_\nu$, for $\nu\in(-2,-1)$ and a candidate Dirichlet space $\mathcal D$ for $\nu=-2$.
Given $f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k=-\infty}^{\infty} a_{jk}z_1^jz_2^k\in \operatorname{Hol}(\bbH)$, we define the functions
\begin{align*}
&f_1(z_1,z_2):=\sum_{j=1}^{+\infty}\sum_{k\neq-j}j(j+k)a_{jk}z_1^jz_2^{k-1};\\
&f_2(z_1,z_2):=\sum_{k\in\mathbb Z\backslash\{0\}}k a_{0k}z_2^{k-1};\\
&f_3(z_1,z_2):=\sum_{j=1}^{+\infty}j a_{j(-j)}z_1^j z_2^{-j-1}
\end{align*}
and define the operator
\begin{align}\label{operators-T}
Tf(z_1,z_2):=|z_2|(1-|z_1/z_2|^2)(1-|z_2|^2)f(z_1,z_2).
\end{align}
Then, we prove the following result. An analogous result in the setting of the polydisc was proved in \cite[Theorem B]{zhu-poly}.
\begin{thm}\label{bergman-derivative}
Let $\nu>-1$. A function $f$ belongs to $A^2_\nu(\bbH)$ if and only if $Tf_j, j=1,2,3$, belong to $L^2_\nu(\bbH)$.
\end{thm}
\begin{proof}
We first recall \eqref{expansion-f} and \eqref{expansion-norm}, that is, a function $f\in A^2_\nu(\bbH)$ if and only if
\begin{equation*}
f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}a_{jk}z_1^jz_2^k
\end{equation*}
and
\begin{equation*}
\|f\|^2_{A^2_\nu}=\frac{(\nu+1)\Gamma(\nu+1)\Gamma(\frac32\nu+3)}{\Gamma(\frac\nu2+2)}\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}\frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)}|a_{jk}|^2
\end{equation*}
Then,
\begin{align*}
&\|Tf_1\|^2_{L^2_\nu}= C_\nu 2^{\frac\nu2}\int_{\bbH}\Big|\sum_{j=1}^{+\infty}\sum_{k\neq-j}j(j+k)a_{jk}z_1^jz_2^{k-1}\Big|^2 |z_2|^{2+\nu}(1-|z_1/z_2|^2)^{2+\nu}(1-|z_2|^2)^{2+\nu}\, dz\\
&\,\,\,=4\pi^{2}2^{\frac\nu2} C_\nu\sum_{j=1}^{+\infty}\sum_{k\neq-j}j^2(j+k)^2|a_{jk}|^2\int_0^1\int_0^{\rho}r^{2j+1}\rho^{2k+\nu+1}(1-(r/\rho)^2)^{2+\nu}(1-\rho^2)^{2+\nu}\, drd\rho\\
&\,\,\,=\pi^{2}2^{\frac\nu2}C_\nu\sum_{j=1}^{+\infty}\sum_{k\neq-j}j^2(j+k)^2|a_{jk}|^2\int_0^1r^j(1-r)^{2+\nu}\,dr\int_0^1 \rho^{j+k+\frac\nu2+1}(1-\rho)^{2+\nu}\, d\rho\\
&<+\infty
\end{align*}
if and only if $\nu>-3$ and $j+k+\frac\nu2+1>-1$. In particular,
\begin{equation}\label{norm-T1}
\|Tf_1\|^2_{L^2_\nu}\approx\sum_{j=1}^{+\infty}\sum_{\substack{k\neq-j\\k>-j-\frac\nu2-2}}j^2(j+k)^2 \frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+4)\Gamma(j+k+\frac32\nu+5)}|a_{jk}|^2.
\end{equation}
Similarly, for $Tf_2$ we get
\begin{align*}
\|Tf_2\|^2_{L^2_\nu}&=4\pi^22^{\frac\nu2}C_\nu\sum_{k\in\bbZ\backslash\{0\}}k^2|a_{0k}|^2\int_0^1\int_0^{\rho}r\rho^{2k+\nu+1}(1-(r/\rho)^2)^{2+\nu}(1-\rho^2)^{2+\nu}\, drd\rho\\
&=\pi^{2}C_\nu\sum_{k\in\bbZ\backslash\{0\}}k^2|a_{0k}|^2\int_0^1(1-r)^{2+\nu}\,dr\int_0^1\rho^{k+\frac\nu2+1}(1-\rho)^{2+\nu}\, d\rho\\
&<+\infty
\end{align*}
if and only if $\nu>-3$ and $k+\frac\nu2+1>-1$.
In particular,
\begin{equation}\label{norm-T2}
\|Tf_2\|^2_{L^2_\nu}\approx\sum_{\substack{k\in\bbZ\backslash\{0\}\\k>-\frac\nu2-2}}k^2\frac{\Gamma(k+\frac\nu2+2)}{\Gamma(k+\frac32\nu+5)}|a_{0k}|^2.
\end{equation}
At last,
\begin{align*}
\|Tf_3\|^2_{L^2_\nu}&=4\pi^{2}2^{\frac\nu2}C_\nu\sum_{j=1}^{+\infty}j^2|a_{j(-j)}|^2\int_0^1\int_0^\rho r^{2j+1}\rho^{-2j+\nu+1}(1-(r/\rho)^2)^{2+\nu}(1-\rho^2)^{2+\nu}\, drd\rho\\
&=\pi^{2}2^{\frac\nu2}C_\nu\sum_{j=1}^{+\infty}j^2|a_{j(-j)}|^2\int_0^1r^j(1-r)^{2+\nu}\,dr\int_0^1\rho^{\frac\nu2+1}(1-\rho)^{2+\nu}\, d\rho\\
&<+\infty
\end{align*}
if and only if $\nu>-3$ and $\frac\nu2+2>-1$. Hence,
\begin{equation}\label{norm-T3}
\|Tf_3\|^2_{L^2_\nu}\approx\sum_{j=1}^{+\infty}j^2\frac{\Gamma(j+1)}{\Gamma(j+\nu+4)}|a_{j(-j)}|^2.
\end{equation}
The conclusion follows comparing \eqref{expansion-f} and \eqref{expansion-norm} with \eqref{norm-T1},\eqref{norm-T2} and \eqref{norm-T3}.
\end{proof}
From the previous proposition we deduce that we can we can endow $A^2_\nu$ with the equivalent norm
\begin{equation*}
\|f\|_{A^2_\nu}=|a_{00}|+\|Tf_1\|_{L^2_\nu}+\|Tf_2\|_{L^2_\nu}+\|Tf_3\|_{L^2_\nu}.
\end{equation*}
Moreover, we can use this norm to define some new spaces for $-2<\nu<-1$, namely, we define the weighted Dirichlet space $\mathcal D_\nu$ as
\begin{equation*}
\mathcal D_\nu =\big\{f\in\operatorname{Hol}(\bbH): \|f\|_{*}<+\infty\big\},
\end{equation*}
where
\begin{equation}\label{norm-Dnu}
\|f\|_{*}:=|a_{00}|+\|Tf_1\|_{L^2_\nu}+\|Tf_2\|_{L^2_\nu}+\|Tf_3\|_{L^2_\nu}.
\end{equation}
We have the following result.
\begin{thm}
There exists an inner product $\left<\cdot,\cdot\right>_{\mathcal D_\nu}$ on $\mathcal D_\nu$ such that $(\mathcal D_\nu,\|\cdot\|_{\mathcal D_\nu})$ is a reproducing kernel Hilbert space with kernel
\begin{equation*}
K_{\nu}(z,w)=c_\nu\frac{(z_2\overline w_2)^{-1}}{(1-\frac{z_1\overline w_1}{z_2\overline w_2})^{\nu+2}}F\Big(\frac32\nu+2,1;\frac\nu2+1,z_2\overline w_2\Big).
\end{equation*}
where $c_\nu= \frac{(\frac\nu2+1)}{(\frac32\nu+2)}$.
\end{thm}
\begin{proof}
From \eqref{operators-T} and \eqref{norm-Dnu} we get that any function $f\in \mathcal D_\nu$ is of the form
\begin{equation*}
f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}a_{jk}z_1^jz_2^k.
\end{equation*}
Moreover, notice that the formula for the $A^2_\nu(\bbH)$ norm in terms of the Laurent series coefficients, that is, formula \eqref{expansion-norm}, is meaningful for $-2<\nu<-1$ as well. Therefore, we endow $\mathcal D_\nu$ with the inner product
\begin{equation*}
\left<f,g\right>_{\mathcal D_\nu}:=\frac{(\nu+1)\Gamma(\nu+1)\Gamma(\frac32\nu+3)}{\Gamma(\frac\nu2+2)}\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}\frac{\Gamma(j+1)\Gamma(j+k+\frac\nu2+2)}{\Gamma(j+\nu+2)\Gamma(j+k+\frac32\nu+3)}a_{jk}\overline {b_{jk}}
\end{equation*}
where
\begin{equation*}
g(z_1,z_2):=\sum_{j=0}^{+\infty}\sum_{k>-j-\frac\nu2-2}b_{jk}z_1^jz_2^k.
\end{equation*}
From the properties of the Gamma function we deduce that the norm induced by this inner product is equivalent to the norm \eqref{norm-Dnu}. By means of the Cauchy-Schwartz inequality, similarly to the proof of Proposition \ref{Bergman-kernel}, we obtain that $\mathcal D_\nu$ is a reproducing kernel Hilbert space and the reproducing kernel $K_{\nu}$ is given by
\begin{align*}
K_{\nu}((z_1,z_2), (w_1,w_2)) &= \frac{(\frac\nu2+1)}{(\frac32\nu+2)}\frac{(z_2\overline w_2)^{-1}}{(1-\frac{z_1\overline w_1}{z_2\overline w_2})^{\nu+2}}F\Big(\frac32\nu+2,1;\frac\nu2+1,z_2\overline w_2\Big).
\end{align*}
as we wished to show.
\end{proof}
\medskip
Finally we define our candidate Dirichlet space $\mathcal D$, which corresponds to the case $\nu=-2$, as
\begin{equation*}
\mathcal D=\big\{f\in\operatorname{Hol}(\bbH): \|f\|_{\sharp}<+\infty\big\}
\end{equation*}
where
\begin{equation}\label{norm-Dirichlet}
\|f\|_{\sharp}:= |a_{00}|+\|Tf_1\|_{L^2_{-2}}+\|Tf_2\|_{L^2_{-2}}+\|Tf_3\|_{L^2_{-2}}.
\end{equation}
We recall that the space $L^2_{-2}(\bbH)$ is endowed with the norm \eqref{Lp-norm}, that is,
\begin{equation*}
\|f\|^2_{L^2_{-2}}\approx\int_{\bbH}|f(z_1,z_2)|^2|z_2|^{-2}(1-|z_1/z_2|^2)^{-2}(1-|z_2|^2)^{-2}\, dz,
\end{equation*}
and we observe that the measure
\begin{equation*}
(1-|z_1/z_2|^2)^{-2}(1-|z_2|^2)^{-2}|z_2|^{-2}\,dz= K(z,z)\, dz
\end{equation*}
is invariant for $\operatorname{Aut}(\bbH)$. The automorphisms of the Hartogs triangle are completely characterized and are described in the following proposition. We refer the reader to \cite{L,CX,K,K1}.
\begin{prop}
A function $\Psi\in \operatorname{Aut}(\bbH)$ if and only if $\Psi(z_1,z_2)=(\tilde z_1,\tilde z_2)$ where
\begin{align*}
&\tilde z_1=z_2\varphi\Big(\frac{z_1}{z_2} \Big),\qquad\quad\!\!\! \varphi\in\operatorname{Aut}({\mathbb D})\\
&\tilde z_2= c z_2,\qquad\quad\quad\quad c\in{\mathbb C},|c|=1.
\end{align*}
\end{prop}
The following proposition holds.
\begin{prop}\label{tau}
Let $d\tau$ be the measure
\begin{equation*}
d\tau:=K(z,z)\,dz=|z_2|^{-2}(1-|z_1/z_2|^2)^{-2}(1-|z_2|^2)^{-2}\,dz.
\end{equation*}
Then,
\begin{equation*}
\int_{\bbH}f\, d\tau=\int_{\bbH} (f\circ\Psi)\, d\tau
\end{equation*}
for all $\Psi\in\operatorname{Aut}(\bbH)$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{tau}]
It suffices to show that
\begin{equation*}
|K(\Psi(z),\Psi(z))||\Psi'(z)|^2\,dz=K(z,z)\, dz
\end{equation*}
for all $\Psi\in\operatorname{Aut}(\bbH)$. This equality holds if and only if
\begin{equation*}
\Big(1-\big|\varphi\big(z_1/z_2\big)\big|\Big)^{-2}\Big|\varphi'\big(z_1/z_2\big)\Big|^{2}\, dz=\Big(1-\big|z_1/z_2\big|\Big)^{-2}\, dz
\end{equation*}
for all $\varphi\in\operatorname{Aut}({\mathbb D})$. The conclusion now follows from the invariance of the measure $(1-|\eta|^2)^{-2}\, d\eta$ with respect to $\varphi$ on the unit disc.
\end{proof}
From \eqref{operators-T} and \eqref{norm-Dirichlet} we get that any function $f\in\mathcal D$ is of the form
\begin{equation*}
f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k=-j}a_{jk}z_1^jz_2^k.
\end{equation*}
We have the following result.
\begin{thm}\label{inner product-Dirichlet}
There exists an inner product $\left<\cdot,\cdot,\right>_{\mathcal D}$ on $\mathcal D$ such that $(\mathcal D,\|\cdot\|_{\mathcal D})$ is a a reproducing kernel Hilbert space with kernel
\begin{equation*}
K_{\mathcal D}((z_1,z_2),(w_1,w_2))=\frac{1}{z_1\overline w_1}\log\Big(\frac{1}{1-(z_1\overline w_1)/(z_2\overline w_2)}\Big)\log\Big(\frac{1}{1-z_2\overline w_2}\Big)
\end{equation*}
\end{thm}
\begin{proof}
We endow $\mathcal D$ with the inner product
\begin{equation*}
\left<f,g\right>_{\mathcal D}:=\sum_{j=0}^{+\infty}\sum_{k=-j}^{+\infty}(j+1)(j+k+1)a_{jk}\overline{b_{jk}}.
\end{equation*}
From the property of the Gamma function we get that the norm induced by this inner product is equivalent to the norm $\|\cdot\|_{\sharp}$. Similarly to the proof of Proposition \ref{Bergman-kernel}, we prove that $\mathcal D$ is a reproducing kernel Hilbert space by means of the Cauchy--Schwartz inequality and the reproducing kernel of $\mathcal D$ with respect to $\left<\cdot,\cdot\right>_{\mathcal D}$ is given by
\begin{align*}
K_{\mathcal D}((z_1,z_2),(w_1,w_2))&=\sum_{j=0}^{+\infty}\sum_{k=-j}^{+\infty}\frac{(z_1\overline w_1)^j(z_2\overline w_2)^k}{(j+1)(j+k+1)}\\
&=\frac{1}{z_1\overline w_1}\log\Big(\frac{1}{1-(z_1\overline w_1)/(z_2\overline w_2)}\Big)\log\Big(\frac{1}{1-z_2\overline w_2}\Big)
\end{align*}
as we wished to show.
\end{proof}
Finally, we prove that there exists a surjective isometry from $\mathcal D$ onto $\mathcal D({\mathbb D}^2)$, the Dirichlet space on the bidisc. Following \cite{AMPS2}, we recall that a holomorphic function $f(w_1,w_2)=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}a_{jk}w_1^jw_2^k$ is in $\mathcal D({\mathbb D}^2)$ if and only if
\begin{equation*}
\|f\|^2_{\mathcal D({\mathbb D}^2)}=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}|a_{jk}|^2(j+1)(k+1)<+\infty.
\end{equation*}
\proof[Proof of Theorem \ref{main-3}]
Let $f(z_1,z_2)=\sum_{j=0}^{+\infty}\sum_{k=-j}^{+\infty}a_{jk}z_1^jz_2^k\in \mathcal D$ and consider the map $f\mapsto\widetilde f:= f\circ\Phi$ where $\Phi(w_1,w_2)=(w_1w_2,w_2)$ is the biholomorphic map $\Phi:\mathbb D\times{\mathbb D}^*\to\bbH$, $(w_1,w_2)\mapsto(w_1w_2,w_2)$. Then,
\begin{align*}
\widetilde f(w_1,w_2)&=\sum_{j=0}^{+\infty}\sum_{k=-j}^{+\infty} a_{jk}(w_1w_2)^jw_2^k=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}a_{j(k-j)}w_1^j w_2^{k}=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}\widetilde a_{jk}w_1^j w_2^{k},
\end{align*}
where we set $\widetilde a_{jk}=a_{j(k-j-1)}$. Therefore, $\widetilde f\in \operatorname{Hol}({\mathbb D}^2)$ and
\begin{align*}
\|\widetilde f\|^2_{\mathcal D({\mathbb D}^2)}&=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}|\widetilde a_{jk}|^2(j+1)(k+1)\\
&=\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}|a_{j(k-j)}|^2(j+1)(k+1)\\
&=\sum_{j=0}^{+\infty}\sum_{k=-j}^{+\infty}|a_{jk}|^2(j+1)(j+k+1)\\
&=\|f\|^2_{\mathcal D}
\end{align*}
as we wished to show. The map $f\mapsto\widetilde f$ is clearly surjective and the inverse map is given by $g\mapsto g\circ\Phi^{-1}$ where $g\in \mathcal D({\mathbb D}^2)$.
\qed
\section{Concluding Remarks}\label{final-remarks}
We think that the family of holomorphic functions spaces we introduced is interesting and worth investigating. The results we prove in this paper are just a first step for this investigation and the techniques we use are classical and not excessively complicated. The major contribution of the paper is the very definition of the family of spaces itself. A few comments are needed.
Our proofs heavily rely on the Hilbert space setting and on the Laurent expansion of the functions involved. If we want to develop our spaces in a $L^p$ setting, $p\neq2$, we need a different approach, especially for the definition of the weighted Dirichlet and Dirichlet spaces. Let $\nu\in(-2,-1)$ and consider the weighted Dirichlet space on the disc $\mathcal D_\nu({\mathbb D})$, that is, the space of holomorphic functions on $\mathbb D$ endowed with the norm
\begin{equation*}
\|f\|^2_{\nu}=\Big\|\sum_{j\geq0} a_jz^j\Big\|^2_{\nu}:=\sum_{j\geq 0} \frac{\Gamma(j+1)\Gamma(\nu+2)}{\Gamma(j+\nu+2)}|a_j|^2.
\end{equation*}
Then, $\mathcal D_\nu({\mathbb D})$ is a reproducing kernel Hilbert space with kernel $K_\nu(z,w)=(1-z\overline w)^{-\nu-2}$. It is easily seen that $\|f\|_\nu<+\infty$ if and only if
\begin{equation*}
\int_{\mathbb D} |(1-|z|^2)\mathcal Rf(z)|^2(1-|z|^2)^\nu\, dz<+\infty,
\end{equation*}
where $\mathcal R$ is the radial derivative, that is, $\mathcal Rf(z)=\sum_{j=1}^{+\infty}ja_jz^j$. Therefore, we would like to define the spaces $\mathcal D_\nu(\bbH)$ in terms of a natural differential operator on $\bbH$ which plays the role of the radial derivative in the unit disc. This approach would be better suited for an investigation in a non-Hilbert setting.
Another point worth of further investigation is the definition of the Hardy space $H^2(\bbH)$. As we have seen, our Hardy space arise in a very natural way, but other approaches are available. Moreover, our space $H^2(\bbH)$ does not see the pathological geometry of the Hartogs triangle. It would be interesting to define a different, but still natural, Hardy space which keeps track of the origin and compare it with our space.
Finally, it would be interesting to use our approach to define some new function spaces on generalizations of the Hartogs triangle, such as, for instance, the fat or thin Hartogs triangles.
\bigskip
{\em Acknowledgments.} We warmly thank the anonymous referees for useful comments and remarks
that improved the presentation of this paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,004
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Q: Extension Methods for Indexers, would they be good? Extension Methods for Indexers, would they be good ?
I was playing around with some code that re-hydrates POCO's.
The code iterates around rows returned from a SqlDataReader and and uses reflection to assign properties from column values. Down my call stack I had a code a line like this :-
poco.Set("Surname", "Smith"); // uses extension method ...
The Set method was written as an extension method.
It would be great to have been able to write code like this
poco["Surname"] = "Smith"; // extension methods for indexers ?
ie I wanted to write an extension method for indexer
Are there good reason why .Net does not have extension methods for indexers?
Do other people have other good uses for extension method indexers ?
as an aside ...
If we could write extension methods for indexers then we could write code like this …
var poco = PocoFactory();
poco.Surname = "Smith"; // is this JavaScript ...
poco[Surname] = "Smith" ; // … or is this c# or both
Some snippets from my code
/////////////////////////////////////////////
// Client calling code
IDab dab = DabFactory.Create( "Northwind" );
string sql = @"select * from Customers ";
var persons = dab.ExecuteReader<NorthwindCustomer>(sql);
if (dab != null{
Assert.That(persons[0].CustomerID , Is.EqualTo("ALFKI"));}
/////////////////////////////////////////////
List<T> IDab.ExecuteReader<T>(string commandText)
{
List<T> pocos = new List<T>();
// setup connection
SqlDataReader reader = command.ExecuteReader(CommandBehavior.CloseConnection);
while (reader.Read())
{
Dictionary<string, int> colMappings = null ;
if (colMappings == null){
colMappings = reader.GetSqlDataReaderColumnMappings();}
T poco = new T();
poco.DbToMem<T>(reader, colMappings);
pocos.Add(poco);
}
}
// connection cleanup ...
return pocos ;
}
// the set extension method signature
public static void Set<T>(this T thisClientObject, string thisPropertyName, object newValue) where T : class
A: Indexers share a lot of commonality with properties (under the hood, an indexer is a property with an index), and extension properties don't exist. Agreed, there would be scenarios where they are handy.
Re the scenario presented - in some ways, this is quite like dynamic. Of course, if you declare an interface that has a string indexer, then your reader-code could use it directly - but it would be a lot of unnecessary work to implement this interface repeatedly!
Re the extension method; does this use regular reflection? You might want to look at tricks like HyperDescriptor, which might save a lot of CPU time if you are doing lots of this. Typical usage would then be:
PropertyDescriptorCollection props = TypeDescriptor.GetProperties(typeof(T));
while (reader.Read())
{
T poco = new T();
// abbreviated...
(per prop)
props[propName].SetValue(poco, cellValue);
}
You can optimise this further by looking at the returned columns first (once per grid, not per row), and only accessing matched columns...
Or alternatively, look at ORM tools; Expression can also be used to do data reading (I have a complete example of this somewhere on usenet, for DbLinq)
A: For re-hydrating Pocos, I would recommend taking a look at the AutoMapper Nuget package. It makes is really simple and greatly reduces the amount of code.
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Q: Android Wear 100% packet loss Was just writing my first wear app - everything works fine on the emulator - unfortunately on a real device ( Sony smartwatch 3 ) the app failed because it could not reach the internet. Even on adb shell I have problems
22|shell@tetra:/ $ ping google.com
PING google.com (173.194.112.33) 56(84) bytes of data.
^C
--- google.com ping statistics ---
6 packets transmitted, 0 received, 100% packet loss, time 5017ms
This is happening with wifi and without ( then from my understanding the network comes via bluetooth from the phone )
A reboot is not solving the problem and I am out of ideas
A: If your watch is connected to wifi and NOT connected to your phone (via BT), then you will have wifi access on your watch and ping should work (I just tested that). It is very important to make sure that your BT connectivity to your phone is disabled at that time otherwise, as long as there is a BT connectivity to your phone, WiFi is disabled by the framework.
I am not sure what application you are writing but you should not rely on the availability of direct wifi connectivity for your watch and should design your app to mostly rely on the companion phone, when one is paired and connected, and then other means (say, wifi) if watch is not connected but you still have wifi. Note that WiFi consumes a lot more battery so be mindful of that as well.
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Q: Not able to call generic method Here is my generic method signature:
public <T,R> R convertRFromT(T model, Class<R> response) {
R responseInstance = response.newInstance();
// perform some operations
return responseInstance;
}
Now when I try to call this method with
convertRFromT(exampleClassInstance, ABC.class);
Here exampleClassInstance is instance of some class and ABC is another class which can be created from example. But I'm getting error here that exampleClassInstance, ABC.class can't be applied to T, Class<R>.
Any idea what I'm doing wrong here?
Thanks in advance!
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The Full Spectrum Birthday Song captures the essence of feeling-friendliness in a narrative that springboards from the topic of birthdays, a dynamic time that stirs all manner of feeling. This musical learning tool is an invitation to celebrate all feelings every year, every day, every moment.
The Full Spectrum Birthday Song was first launched for partnerships between Emotion Literacy Advocates and twenty-three social service agencies in Oregon, Idaho and Washington. Through these agencies, ELA's song found integral application in multiple settings, for those working with children, parents and teachers. 8,000 CDs were requested and provided by generous ELA program sponsors included in the list shown here.
6th Annual Educational Resource Street Fair in Seattle where inquisitive children came to our table to reflect upon and talk about emotion and to receive ELA's birthday song as a gift.
Read more about this traveling innovation in community education-as-gift here.
Just as a point of information, I reviewed the CD for the family partnership program in my car on the way to an event, my nine-year old grandson (who I am raising) loves it! He can listen to it over and over again for, literally, hours…thanks so much for your generosity.
The Full Spectrum Birthday Song CDs were sent out to (325) families and community partners (with their quarterly newsletter). I wanted you to see the e-mail from Kelly DeLany, Oregon Post Adoption Resource Center Program Manager. The responses have been so positive. Thank you!
We just received the latest edition of Kidbits with the lovely enclosed CD. I'm playing it now over and over… I found The Full Spectrum Birthday Song laughter-inspiring and intriguing and it certainly covered the full range of birth feelings. I personally plan to play it at our next family birthday event and make it a 'full-spectrum celebration'. I have asked Jennifer Ricks to consider including it in her Positive Parenting training.
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Pamela Sackett's Full Spectrum Birthday Song is a wonderful song of celebration—not just of one's birthday, but of one's range of emotional expression, abilities & possibilities. We acknowledge and honor children's feelings and encourage them to express their feelings—that it's 'OK' to feel happy, sad, angry, etc. Pamela's Full Spectrum Birthday Song communicates these same values in a creative, musical and thoughtful way!
You did good…The Full Spectrum Birthday Song CDs work really well in several of our classes and we've shared them with other agencies that serve parents!
We hand out The Full Spectrum Birthday Song CDs at classes as part of our curriculum. Children are learning the song, moving to it and playing instruments with it.
The Full Spectrum Birthday Song is fresh and brings so much more to the meaning of celebrating birthdays. The song and music are a real work of art. Much soul and inspiration for celebrating all of life comes through even in just that two minute sample online.
Pamela's song reminds us of so many possibilities…let The Full Spectrum Birthday Song become the new tradition.
My three and a half year-old daughter loved the imagery in The Full Spectrum Birthday Song and requested we play 'that other birthday song' again…why settle for 'happy'? The Full Spectrum Birthday Song encourages exploration of all manner of birthday sentiments.
Birthdays are a time when we celebrate the passing of a year and the beginning of a new one. Each year has it's ups and downs and in the acceptance and appreciation of these fluctuations, we find the immensity and deep meaning and joy of life. Pamela's song asks why just a 'happy' birthday? Why not celebrate and love the whole reality of living? It's a simple song, humorous and one with a message as profound as the ages.
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Фьямметта, Фьяметта, устар. Фиаметта (, «огонек») (ок. 1310 — между 1350—1355 гг.) — псевдоним, которым итальянский писатель Джованни Боккаччо именует в ряде произведений, в том числе в одноимённой повести, свою возлюбленную. Как считается с XIX века, её настоящее имя Мария д'Аквино, тем не менее, до сих пор не доказано, была ли Фьямметта вымышленным персонажем, результатом литературной традиции (таким тропом её образ начали считать с 1930-40-х гг.), или все же реальным человеком.
Фьямметта в произведениях Боккаччо
История знакомства и отношений
Боккаччо в своих произведениях подробно рассказывает о своей любви к Фьямметте, хотя, разумеется, меру поэтического преувеличения установить нельзя. Он пишет, что первый раз образ возлюбленной явился к нему в видении, когда он, 15-летний, подъезжал к Неаполю (согласно «Амето», где Боккаччо себя выводит под псевдонимом Калеоне). Исследователи предполагают, что писатель облёк тут в поэтический образ действительную встречу с ней. К воспоминаниям о первой встрече с дамой он будет возвращаться неоднократно (в «Филоколо» (I, 4-6), «Филострато» (I, 17-30), «Фьямметта», «Амето» и «Любовном видении»). Затем на шесть лет (по другим указаниям, 7 лет и 4 месяца) он потерял её из виду и любил других женщин, но, наконец, состоялось знакомство, незадолго до которого Фьямметта опять явилась ему в видении.
Собственно встреча лицом к лицу произошла в церкви Св. Лаврентия в Страстную субботу, как считается, 12 апреля 1338 года, или немногим ранее. В церкви Фьямметта была одета в чёрное, а через несколько дней в церкви монастыря св. Михаила в Байи они встретились опять, и она была одета в зелёное. Он вступил с ней в разговор. Беседа шла о легенде о любви Флорио и Бьянкофьоре, и Фьямметта предложила Боккаччо написать историю этой любви. За то время, как Фьямметта уехала на лето из Неаполя в Саннио, он написал на эту тему свой роман «Филоколо». После этой встречи Боккаччо познакомился с мужем Фьяметты, подружился с ним и сделался частым гостем у них в доме. История встречи в церкви имеет прямые параллели с более ранними и равно знаменитыми историями знакомства Данте с Беатриче и Петрарки с Лаурой — то есть является типичной любовной историей эпохи dolce stil nuovo. Следует, кстати, отметить, что сам Боккаччо считал любовную историю Петрарки и Лауры выдуманной.
В отличие от Лауры и Беатриче, любовь Боккаччо к Фьямметте имела более земной и даже чувственный оттенок, но точно сказать о характере взаимоотношений между поэтом и музой нельзя. Сведения в его произведениях постоянно противоречат друг другу. Но судя по тому, что в своих сонетах Боккаччо жалуется на холодность своей возлюбленной, называет её мрамором, не согретым лучом любви, упрекает её в том, что ей гораздо дороже её честь, чем его любовь, и даже в пылу гнева высказывает желание видеть её состарившейся и подурневшей, следует предположить, что Фьямметта не позволяла ему переходить границ дозволенного. У пылкого Боккаччо, трое внебрачных детей которого от разных матерей известны по именам, такая ситуация действительно вызывала сильные эмоции. Исследователи предполагают, что если у них и была любовная связь, то только после написания «Филоколо», длиться же она могла около двух лет и закончиться летом 1338 или в 1339 году; вслед за этим Фьямметта оставила его ради другого возлюбленного. Начало плотских отношений, судя по намекам в тексте Боккаччо, могло иметь место, когда муж Фьямметты уехал из города, а безумный влюбленный прокрался к ней в дом, но установить, какова доля истины в этих намеках, не представляется возможным.
После разрыва — любовников ли, платонических возлюбленных ли, — который в любом случае был для него очень болезненным, Боккаччо пишет произведения, в которых пытается избыть свою страсть, и выводит в них пока ещё любимую женщину; затем же он успокаивается и продолжает использовать её образ как более мирное и сладкое воспоминание.
Дату смерти Фьямметты вычисляют по произведениям Боккаччо: когда он писал «Декамерон», то есть около 1350 года, она, вероятно, была ещё жива (некоторые предполагают, что она могла погибнуть во время чумы 1348 года, описанной в сборнике новелл «Декамерон», но тогда бы её образ в этой книге нес бы совсем другие интонации), а в 1355 году Боккаччо уже подпал под влияние другой неудачной любви, вызвавшей его сатиру «Корбаччо».
Мысли и любовь к усопшей он, тем не менее, сохранял до конца своих дней: так, незадолго до смерти, в 1374 году он написал по поводу известия о смерти Петрарки сонет (№CXXVI), в котором, обращаясь к Петрарке, говорит, что тот ушел туда, куда надеется вознестись всякая избранная Богом душа, туда, где находятся Лаура, Фьямметта, Данте, и просит Петрарку скорее призвать и его к себе — для того, чтобы он мог увидать снова ту, которая некогда воспламенила в нём любовь.
Личность Фьямметты
Судя по тем произведениям, которые посвящал ей Боккаччо, была прекрасно образована и интересовалась литературой. Согласно его словам, она на три года старше его, отличается необыкновенной красотой:
Список произведений
По её заказу написан «Филоколо» (ок. 1336—1338)
Ей посвящены «Тезеида» (ок. 1339—1341) и «Филострато» (ок. 1335-40). В «Тезеиде» его возлюбленная выведена под именем Хрисеида
Персонаж фигурирует в «Амето» (1341—1342): нимфы рассказами перевоспитывают пастуха, одна из них носит имя Фьяметты и персонифицирует Надежду.
«Любовное видение» (начало 1340-х) — поэт описывает своё блуждание-сон, в финале встречается со своей возлюбленной, названной Лучия. Поэма посвящена даме Марии.
Главный действующий персонаж повести «Фьямметта» / «Элегия мадонны Фьямметты» («L'amorosa Fiammetta», 1343)
«Фьезоланские нимфы» (1345). Намек на любовное увлечение поэта в первой октаве.
Сборник новелл «Декамерон» (1350-е) — одна из дам, рассказывающих новеллы; также она выведена поющей. (Ей принадлежат следующие новеллы: I, 5; II, 5; III, 6; IV, 1; V, 9; VI, 6; VII, 5; VIII, 6; IX, 5; X, 6).
«Канцоньере» — его стихи; исследователи предполагают, что она основная, или даже единственная их вдохновительница, хотя по имени она названа лишь в четырёх сонетах (№ XLV, XCVII, CII, CXXVI, ряд из них посмертные) и ещё одной пьесе.
Повесть «Фьямметта» и эволюция образа в творчестве писателя
«Фьямметта» — первый любовно-психологический роман в мировой литературе. Рассказчицей в повести выступает женщина, выведенная под именем Фьямметта. Она рассказывает о том, как была счастлива и замужем; вдруг ей снится страшный сон, будто она укушена змеей. На следующий день она влюбляется, в первый раз в жизни по-настоящему, в юношу по имени Панфило. Вскоре старик-отец просит Панфило приехать к нему во Флоренцию. Он уезжает, но обещает своей возлюбленной вернуться. Она его преданно ждет, мысленно подолгу с ним беседуя. Он не возвращается в положенный срок; Фьямметта узнает у краснеющей монахини, что Панфило женился. Фьямметта с разбитым сердцем, её здоровье ухудшается. Через год после отъезда Панфило из Флоренции возвращается слуга Фьяметты, который рассказывает, что женился вовсе не Панфило, а его отец. Панфило же влюбился в одну из флорентийских красавиц. Она хочет броситься с башни, но её останавливает кормилица. В конце Фьямметта сравнивает свои бедствия с таковыми же многих женщин древности и доказывает, что её были более тягостные.
Хотя рассказчиком данного романа является женщина, роман написан от первого лица, и автором его является мужчина: брошенный настоящей Фьямметтой Боккаччо «переворачивает» ситуацию и описывает душевные страдания покинутого влюбленного. А то, что повесть ведется от лица женщины, позволяет ему не скрывать непозволительную порой слабость. На русском языке издана в 1913 году в переводе Михаила Кузмина.
Исследователи творчества Боккаччо пишут о том, что свою возлюбленную покинутый Боккаччо никогда не забывал, «но с созданием "Элегии мадонны Фьямметты" для него она превращается в литературный персонаж, что, между прочим, несомненно помогло поэту залечить свою любовную рану. Известное замечание А. Н. Веселовского (Фьямметта — литературное переживание психологического момента, который перестал тревожить сердце, но продолжает занимать воображение) правильно лишь отчасти, — в исповеди Фьямметты ещё очень много неподдельной боли самого Боккаччо. Лишь позже — в "Нимфах" и "Декамероне" — эта боль стихает».
Мария д'Аквино
То, что настоящим именем Фьямметты является «Мария», угадывается по нескольким указаниям. В «Филоколо» (I,4) герой говорит о том, что его возлюбленная носит имя той, от которой пришло спасение мира. Другой псевдоним, которым награждает автор свою возлюбленную — Alleritam — является анаграммой имени Mariella, неаполитанской формы имени «Мария». В «Любовном видении» он различными экивоками указывает имя «Мария», а кроме того, пишет, что она принадлежит к роду святого Фомы Аквинского (который был сыном графа Ландульфа Аквинского); таким образом, если имя «Мария» устанавливается достаточно достоверно, принадлежность к роду графов Аквинских имеет меньше указаний-подсказок.
С XIX века традиционно предполагается, что под псевдонимом Фьямметты выведена женщина, чьим именем, таким образом, оказывается «Мария д'Аквино» (Maria d'Aquino; Maria dei Conti d' Aquino) — внебрачная дочь короля Неаполя Роберта Анжуйского (1277—1343) и, возможно, придворная дама при его дворе и дворе следующего монарха — внучки Роберта Джованны Неаполитанской. Как считается, её мать была супругой графа Аквинского, и эту же фамилию носила и родившаяся девочка.
Представители этой династии соответствующего хронологического периода могут являться её семьёй. Это графиня Аквинская, вероятная мать Марии — провансальская дама Сибилла Сабран (Sibilla Sabran), которая была замужем за графом Томмазо III Аквинским (Tommaso III di Aquino) (тем не менее, с тем же успехом к роду Аквинатов мог принадлежать не её отец-отчим, как считается общепринятым, а мать или даже муж).
Никакой дополнительной информации, кроме текстов Боккаччо, о биографии его музы нет. Обстоятельства рождения Марии интерполируются из событий жизни Фьямметты, о которых Боккаччо пишет в «Амето» (XXXV). Там повествуется о том, что вскоре после праздника, последовавшего за неким праздником, видимо коронацией Роберта (которого она называет именем царя из античной мифологии — «Мидасом»), монарх начал оказывать матери Фьямметты пристальное внимание, которое вскоре, видимо, увенчалось изнасилованием королём этой женщины: она «…против воли досталась ему в обладание. Утолив его вожделение, она получила просимое и, видя, что все осталось в тайне, умолчала о совершенном насилии. Если бы все это не привело к моему рождению, я бы, конечно, сказала, что она согрешила, не последовав примеру Лукреции». В другом фрагменте того же текста Фьямметта добавляет, что о её происхождении мать ей рассказала в детстве; что мать жила в доме мужа в тот момент, когда король её соблазнил, и она не уверена, кто именно является отцом ребёнка, муж или король. В «Филоколо» Боккаччо пишет, что возлюбленная старше его на три года, и, действительно, Роберт был коронован в 1309 году.
Мать Фьямметты рано умерла, её муж вскоре последовал за ней, отдав перед тем девочку в монастырь послушницей в Байи, где, как Фьямметта рассказывает в «Амето», у неё были родственники. Название обители не указано, но судя по всему, это был бенедиктинский монастырь святого Михаила. Там она обратила на себя внимание некоего дворянина, который при содействии короля Роберта добился её руки. Имя мужа Марии не сохранилось.
Но по сути, как подытоживают современные исследователи творчества писателя, соответствие боккачиевской Фьямметты реальному историческому лицу является легендой, которая, хотя и успела стать элементом классического восприятия Боккаччо, тем не менее, некоторыми критикуется: «Помимо полного отсутствия какого-либо упоминания или даже ссылки на столь необычного персонажа, как дочь короля, в генеалогическом древе и просто фамильных документах семьи Аквино, факт существования обольстительной фигуры Фьямметты находится под вопросом, и, скорее всего, она является абсолютно выдуманным и сконструированным образом, полностью основанном на литературных канонах этого времени, канонах, которые скрупулёзно диктовали детали развития сюжета и детали любовной истории, как для самого Боккаччо, так и для Фьямметты (которая, согласно этим традициям, должна оставаться вечно юной — так, в "Филоколо" она родилась в 1310 году, в "Комедии", судя по указанному возрасту — после 1313, в "Декамероне" — после 1321)».
Исторические личности
Возможная мать — Маргарита. В одной из итальянских генеалогий упоминается правнучка императора Священной Римской империи Фридриха II Гогенштауфен от незаконной дочери Маргариты Швабской (ок. 1230—1298) и Томмазо II Аквино, графа Ачерра (1273) — Маргерита Аквино (после 1328), была замужем три раза. Вышеназванный граф Томмазо III Аквинский был её старшим братом. Она считается любовницей короля Роберта I Сицилийского и, предположительно, могла быть матерью Марии Аквино.
Мария, дочь Елены? Историками литературы никак не упоминается известная из генеалогических таблиц некая незаконная дочь короля Роберта от неизвестной женщины, настоящим именем которой, предположительно, было Елена. Эта дочь короля упоминается как супруга некоего Andrea Thopia (1340), владетеля Matija (совр. область Мат, Албания), подконтрольной в то время Неаполю. У этой пары было трое детей: князь Албании Каролус Топиа, лорд Дураццо, Доминик Топиа, епископ Дураццо и архиепископ Зары, а также Георг.
Некая Мария Аквино была казнена в 1382 году по приказу Карла, герцога Дураццо, будущего короля Неаполя, по обвинению в соучастии в преступлениях Джованны I, внучки Роберта Анжуйского, и причастности к смерти Андрея, герцога Калабрийского и князя Венгерского в 1345 году.
В произведениях других авторов
Прерафаэлит Данте Габриэль Россетти написал картину «Видение Фьямметты», а также посвятил ей сонет «Fiametta. For a Picture».
Прерафаэлит Эмма Сэндис
Картина французского салонного живописца Лефевра «Фьямметта»
Картина британца William Clarke Wontner
Скульптурный бюст викторианского мастера Francis Derwent Wood
Оперетта «Боккаччо» Франца фон Зуппе — один из персонажей (см. дуэт Боккаччо и Фиаметты, mp3)
Библиография
D. Rastelli. Le fonti autobiografiche nella «Fiammetta». — «Humanitas», vol. III, 1948
А. В. Веселовский. Собр. соч., т. V. Пг., 1915, стр. 643—658
Janet Levarie Smarr. Boccaccio and Fiammetta: the narrator as lover, 1986
Примечания
Боккаччо
Женщины XIV века
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Kristen Kosmas and director Paul Willis in rehearsal for Kosmas's "There There." (Photo by Lia Aprile)
Feature | January 2013 | On the Scene January 1, 2013 Ben Gassman Leave a comment
Knocking Chekhov for a Loop
His themes and characters get upended by female artists with fresh insights into plays we thought we knew.
Maybe I'm making too much of a moment. Maybe it's Mac Wellman's fault. But hasn't Chekhov recently become something of a piñata in downtown New York theatre? The Russian master's canon, it seems, is being cracked open from all sides, particularly by female writers and theatremakers.
Take Annie Baker's critically acclaimed Uncle Vanya at Soho Repertory Theatre this past summer, which turned the company's Walkerspace into a parlor, replete with cushions and samovar, where audience and players alike could intimately revel and despair in the rhythmic brutalities of flawed people attacking and consoling each other using inadequate language. This Jan. 3–19, P.S. 122's COIL Festival features two inventive productions by artists not often associated with the naturalism or realism with which we label Chekhov (fairly or not): Half Straddle's Seagull (Thinking of You); and Kristen Kosmas's There There, a meditation on a minor character in Three Sisters, the misfit Vasily Vasilyevich Solyony.
The productions themselves are entirely coincidental, no doubt, but maybe there's something in the air—a cultural whisper that Chekhov's work, regularly celebrated in the commercial and not-for-profit repertory and always ripe for revival, is particularly ripe at this moment for stylized theatro-philosophical excursions by bold women—particularly ones who "don't traffic in realism" (as Kosmas puts it). Maybe 20 or 50 years on there will be a clear and convincing thesis (or several of them) on why all the Chekhovs at this cultural moment, but from here in the winds of Chekhov mania, the whys are hazy.
At a recent panel at the New Museum, where artistic director Tina Satter and her Half Straddle company are in residency, the artists tried to take some measure of this Chekhov wave. Organized by Satter and the museum's Travis Chamberlain, and moderated by Jessica Del Vecchio, "My Chekhov, Not Yours" brought together Satter, Kosmas and Baker; as well as Big Dance Theater's Annie-B Parson and Paul Lazar, who will present Man in a Case, an adaptation of two Chekhov short stories, this February at Connecticut's Hartford Stage, featuring Mikhail Baryshnikov.
As the conversation shuttled between these artists' variant aesthetic imperatives—Baker spoke of Vanya having been her favorite play for so long she's forgotten why; Satter of the connections she drew between The Seagull and her company; and Kosmas of her perplexed fascination and even reluctant identification with Solyony—a consistent point of return was playwright and teacher Mac Wellman, with whom Baker, Satter and Kosmas all studied at Brooklyn College. (Full disclosure: So did I—as does American Theatre associate editor Eliza Bent, who is a cast member of Half Straddle's Seagull.)
Baker, now a teacher herself at New York University, measures Chekhov's characters against characters her students find in the world—she sends them out on a Wellman-inspired quest to transcribe overheard dialogue, with all its attendant ums, ahs, stops, starts, non-responses and sudden cliffs. This is a bone Wellman has long been picking with mainstream American theatre writing: People don't actually respond to each other in conversation, he maintains, so why must they always do so in well-made American plays?
Chekhov's characters don't respond to one other—they struggle to say what they mean and aren't quite able to. Nor do they listen. They reach for each other or verbally push each other away. They trip over their words. They get stuck between themselves and the possibilities beyond themselves. The conversational veritas and communicative disintegration that Baker emphasizes with her students is essential to her own Vanya, and also galvanizes the current new works by Satter and Kosmas.
Satter's ladies are often stuck. They're at a remove. Whether on the tundra (as in 2008's The Knockout Blow), a difficult-to-swim-to island (2009's Family), or even behind the stadium bleachers (2012's Away Uniform), there are barriers confining them in an uncomfortable, perhaps even paralytic, place. Each time Half Straddle offers us a piece forged from a Satter script, a Chris Giarmo soundscape and the collaborative exhilaration of their collective, we expect the design feel of an American Apparel–outfitted Space Station Mir—there's a lot of love and some curve-ball catharsis, but nobody ever quite gets to their proverbial Moscow. So it seems natural for the company, in the most ambitious project of its five years of work, to take on Chekhov.
For Satter, the decision to engage with Chekhov "had everything to do with The Seagull," particularly the figure of Nina. "This beautifully flawed teenage actress," Satter reasons, "speaks in a slightly overwrought, subtextual way that feels so familiar to the writing I have done for young female characters." Satter connected aspects of Nina to Emily Davis (who plays a version of the character in the upcoming production), and went on to find Seagull parallels to all her company members (the venerated actress Arkadina, for instance, is refracted in veteran downtown performer Suzie Sokol). "On both personal and artistic levels," Satter notes, "the original play taps into dynamics my collaborators and I know from regularly working together, and the constant balancing out of personalities and egos, ideas about art-making, success and failure, and the inherent love and sense of family that underscores it all."
Satter's focus is on the feminine, the re-contextualization of desire, and family dynamics. In her company's modus operandi, gender is not fixed and sexuality is fluid, so she works between Chekhov's lines to find truths planted by the playwright, perhaps inadvertently, by way of his openness to the vagaries of the human spirit and the disconnect between thought and action. Satter describes her production as "aesthetically and textually like a snow globe of The Seagull being shook," wherein her methods for saying the unsaid merge with Chekhov's own.
In Act 4 of The Seagull, for example, Konstantin Treplev questions his own attempts to forge new forms in his writing: "The more I write, the more I think it's not a matter of old forms and new forms; what's important is to write without thinking about forms at all. Just write and pour out whatever's in your heart." Konstantin's realization speaks to Satter about her own work, and to Half Straddle's compulsion to present honest, sloppily complicated reflections of life in each stage moment.
Kosmas recently gave a talk at Whitman College in Walla Walla, Wash., where she teaches, that touched on how the unlikely circumstances of her play There There worked themselves into being. "I re-read Three Sisters a few years ago," she recalls, "and was struck by a character who actually has quite a small part in the play—in fact, most people, when I tell them I'm making a play inspired by Solyony, can't remember who he is or that he was even in the play. This has become one of the things about him that's the most interesting to me—his forgettable nature."
A refresher: The soldier Solyony is a morbidly digressive, possibly sociopathic hanger-on in the Prozorov home. There There is a dual-language, live-translated meditation on the character that, through a series of circumstantial mishaps, morphs into a "theatrical roller-coaster about being the completely wrong person in the totally wrong place at the exact wrong time doing all the wrong things," Kosmas proffers. In her script, Christopher Walken, on tour with a solo play inspired by Solyony, has mysteriously fallen off a ladder (offstage), and Karen, who once proofread the script, is asked to go on in his place, alongside Leo, a Russian translator. The play collages all of Solyony's language with some of Kosmas's own. "A part of me died in the duel," reflects Kosmas, appearing as Karen, standing in for Christopher Walken as Solyony. If that's a lot to wrap your head around, the language makes it easier—Kosmas has her own poetic language that mixes the lush and dreamlike with banal anxiety, curling cascading ideas into bursts of staccato clauses.
Language is both a propellant and vexing loop that controls Karen; she tries valiantly, in the service of her author and avatar Kosmas (and ostensibly in the service of the indisposed Walken) to take Solyony past where we leave him after the final act of Three Sisters, but we always end up back there, and it's uncertain whether we've been examining Solyony's emotional obstructions, or Karen's, or our own.
Because Kosmas is fearlessly intuitive as a writer, and lullingly defiant as a performer, we are never quite sure where Karen's mind will lead us. She says things we can't allow ourselves to say. Sokol's Arkadina from Seagull (Thinking of You) puts this sense of indirection and equivocation another way: "I just don't know what I actually want, or, I'm not going to admit it in a super real way." Which is the kind of double-speak that could use an irreverent translator. My attempt: "I think I might want this, and I'm trying as hard as I can to be clear about it." What's more Chekhovian than that?
Ben Gassman is a playwright from Queens, N.Y.
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\section*{Abstract}
\par
\textbf{The geometric phase (Berry phase) of an electronic wave function is the fundamental basis of the topological properties in solids. Modulating band structure provides a tuning knob for the Berry phase, and in the extreme case drives a topological phase transition. Despite the significant developments in topological materials study, it remains a challenge to tune between different topological phases while tracing the impact of the Berry phase on quantum charge transport, in the same material. Here we report both in a magnetotransport study of ZrTe$_5$. By tuning the band structure with uniaxial strain, we directly map a weak- to strong- topological phase transition through a gapless Dirac semimetal phase via quantum oscillations. Moreover, we demonstrate the impact of the strain-tunable spin-dependent Berry phase on the Zeeman effect through the amplitude of the quantum oscillations. We show that such a spin-dependent Berry phase, largely neglected in solid-state systems, is critical in modeling quantum oscillations in Dirac bands in topological materials.}
\section*{Main}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure1.png}
\caption{ZrTe$_5$ under strain. a. Application of uniaxial strain through substrate bending. b. Optical microscope image of ZrTe$_5$ microcrystal on Polyimide substrate with predefined electrodes. c. Atomic force microscope (AFM) image of the sample. The bottom panel show the cross section of the ZrTe$_5$ crystal, with a thickness of approximately 190 nm. d. Resistance versus temperature dependence under various external strains. e. Non-monotonic strain dependence of resistance at the fixed temperature of 20K, showing a resistance minimum at compressive strain $\approx -0.2\%$.}
\label{Fig1}
\end{figure*}
Tuning the band structure of a topological material causes a continuous evolution of Berry phase along the Fermi surface and can ultimately drive a topological phase transition by closing and reopening the bulk band gap\cite{KaneMele2005,KaneMele2005-2,BHZ2006,konig2007,Moore2007,Xu2012}. Both the evolution of the band gap and the Berry phase can be probed in quantizing magnetic fields, where Shubnikov de Haas (SdH) oscillations from Landau level (LL) quantization provides a tool to probe "Fermiology"\cite{shoenberg1984,Mikitik1999,Novoselov2005,Zhang2005}.
In addition to LL quantization, the Zeeman effect splits the spin-degenerate LLs and reduces the SU(2) Berry phase to the U(1) Berry phases, which play a crucial role in the SdH oscillations.
Interestingly, while the Zeeman effect has been extensively studied over the past decades\cite{luttingerperturbation,cohengfactor1960}, its connection to band topology and non-trivial Berry phase has emerged only recently. A Dirac band with a finite mass gap hosts spin-dependent Berry phase which can modify the SdH oscillations through the Zeeman effect, as recently demonstrated in the spin-zero effect\cite{Wang9145, PhysRevB.101.125118}. Such a spin-dependent Berry phase is expected to be fundamentally generic for Dirac-like bands, and is band parameter dependent. It is therefore desirable to conduct a comprehensive study of such effect in a system with tunable band parameters.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure2.png}
\caption{Magnetotransport measurements on ZrTe$_5$ under strain. a. Resistance versus magnetic field in various external strains. The curves are shifted in proportion to the strain for clarity, from the bottom to the top: -0.44, -0.34, -0.3, -0.24, -0.21, -0.19, -0.17, -0.15, -0.13, -0.11, -0.09, -0.07, -0.05, -0.03, 0, 0.1, 0.2, 0.3\%. b. SdH oscillations versus inverse magnetic field in various external strains. The curves ares shifted in proportion to the strain for clarity, from the bottom to the top: -0.44, -0.34, -0.3, -0.24, -0.17, -0.11, -0.05, 0, 0.1, 0.2, 0.3\%. Symbols are experimental data and solid lines are simulations using the two-component LK formula. c. Landau index, obtained from SdH oscillations, versus inverse magnetic field in large compressive and tensile strains. Here resistance peaks and valleys are assigned integer and half integer indices, respectively. The inset show the pockets of Fermi surface, with the Dirac band at the $\Gamma$ point and a parabolic side band between the R and E points dominating the SdH oscillations under strong compressive and tensile strains, respectively. d, e. SdH amplitude versus inverse magnetic field and temperature in strong compressive (-0.3\%) and tensile (0.3\%)strains. The symbols are experimental data and the solid lines are fittings using the LK formula. From the fittings, the effective cyclotron mass and Dingle temperature can be obtained. f. Under medium strains the SdH oscillations (open symbols) are directly simulated (solid lines) with the two-component LK formula at various temperatures: black: 3K; red: 8K; and blue: 12K.}
\label{Fig2}
\end{figure*}
Despite the significant developments in topological materials study in recent years, it remains a challenge to tune between different topological phases, and correspondingly the Berry phase, in the same material. The transition metal pentatelluride ZrTe$_5$ is a promising material for such study. ZrTe$_5$ is a van der Waals layered material with layer planes extending along the a- and c- lattice directions and stacking along the b direction. It hosts intriguing properties such as a resistance peak (Lifshitz transition) \cite{NatComm.15512, Chi2017}, chiral magnetic effect \cite{Li2016}, and 3D quantum Hall effect\cite{Tang2019}. In its 3D bulk form, ZrTe$_5$ has a Dirac-like low energy band structure, with sample-dependent mass gap rendering the material from Dirac semimetal to topological insulator (TI) \cite{Liu2016, Li2016, Chen816,PhysRevB.96.041101, PhysRevLett.121.187401,PhysRevB.95.195119}, suggesting extreme sensitivity to lattice deformations. Recently a strain-induced weak TI (WTI) to strong TI (STI) transition has been proposed \cite{PhysRevX.4.011002,SciRep.7.45667}, followed by experimental evidence in angle-resolved photoemission spectroscopy study \cite{Zhang2021}, and indirect charge transport evidences via the chiral anomaly effect \cite{Mutcheaav9771}. A charge transport study of the quantum oscillations which directly map the topological phase transition and reveal the spin-dependent Berry phase over tunable band parameters, however, is still lacking.
In this work, we study charge transport and SdH oscillations in ZrTe$_5$ under tunable uniaxial strain. In a magnetic field perpendicular to the a-c plane and the applied current, SdH oscillations and their evolution over strain allow direct mapping of the WTI-STI transition through the closing and reopening of the Dirac band mass gap. The dependence of the SdH oscillation amplitude on the Fermi energy and Dirac mass gap, tunable through the strain, is analyzed and compared with the quantum oscillation theory. Our results reveal that the spin-dependent Berry phase intrinsic to the Dirac bands, which has been largely neglected in previous studies, is critical in modeling the SdH oscillations in such topological materials.
The samples studied in this work are ZrTe$_5$ microcrystals mechanically exfoliated onto flexible Polyimide substrates (Fig. 1a-c), which allow application of external tensile and compressive strains along the a-axis through substrate bending, over a wide temperature range from room temperature down to below 4K (see Methods). Our ZrTe$_5$ exhibits a strain-tunable resistance peak at $T_p\approx 140K$(Fig.1d). At $T=20K$ a non-monotonic resistance versus strain dependence is observed (Fig.1e), consistent with the report in macroscopic ZrTe$_5$ crystals\cite{Mutcheaav9771}. A large strain gauge factor ranging $10^2-10^3$ generally presents throughout the temperature range up to room temperature, suggesting strain-tuning of the low energy electronic spectrum in ZrTe$_5$.
Next, we characterize the electrical resistance in magnetic field perpendicular to the a-c crystal plane. At the base temperature of $\approx4$K, SdH oscillations are clearly visible on the magnetoresistance background and evolve with changing strain (Fig.2a). Plotting the oscillatory part $\Delta R$ versus inverse magnetic field $1/B$, equally spaced resistance oscillations can be resolved (Fig.2b). In the limits of large compressive and tensile strains applied here, the oscillation amplitude monotonically decreases with increasing 1/B. Under mild compressive strains, however, the SdH oscillation amplitude appears non-monotonic, suggesting contributions from more than a single Fermi surface.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure3.png}
\caption{Strain tuning of band structure and spin-dependent Berry phase. a. Strain dependence of SdH oscillation period in inverse magnetic field, for the Dirac band (band 1) and the trivial band (band 2). The results are obtained by simulating the SdH oscillations. b, c. Strain tuning of mass gap size and Fermi energy. The mass gap and the Fermi energy are calculated from Eqs. (2) and (3), with $m_c$ and $E_F$ obtained from 2-component LK formula simulations of the the magnetic field and temperature dependence of the SdH oscillation curves. Tuning the external strain from compressive to tensile, the mass gap closes and reopens at around $-0.2\%$, consistent with the STI-WTI topological phase transition. d. Strain dependence of doping energy, the energy of the Fermi surface counted from the bottom of the conduction band. e. Theoretical intensity plot of SdH oscillation amplitude reduction factor from Zeeman effect, $R_S$, as functions of Fermi energy and mass gap. The dotted line shows the boundary between the STI phase (positive mass gap) and the WTI phase (negative mass gap). The symbols in the mass gap-Fermi energy parameter space correspond to the experimental data points at various external strains, evolving following the arrow from compressive to tensile. By convention we assign a negative sign to the mass gap on the WTI side with external strain larger than $\approx -0.2\%$. f. Theory(solid lines)/experiment(open symbols) comparison of $R_S$ as functions of mass gap and Fermi energy at various strains. The experimental values of $R_S$ are scaled by a common constant factor of 0.6 for comparison with the theory. The theoretical model qualitatively reproduces the experiment results by considering the spin-dependent Berry phase. By contrast, the inset shows the theoretical curve without considering the spin-dependent phase, which fails to explain the experimental data.}
\label{Fig3}
\end{figure*}
Our analysis of the SdH oscillations focuses on mapping out the strain-dependent mass gap, Fermi level, and the spin-dependent Berry phase. We model ZrTe$_5$ with the anisotropic Dirac Hamiltonian (see Methods). In quantizing magnetic fields, the extremal cross-sectional area of the Fermi surface is $S_F = \pi (\mu^2 - \Delta^2) / (\hbar^2 v^2)$. The corresponding cyclotron mass is $m_c = \frac{\hbar^2 \partial S_k}{2 \pi \partial
\mu} = \frac{\mu}{v^2}$. Here $\mu$ is the Fermi energy, $\Delta$ is the mass gap, and $v$ ($\approx 5\times 10^5m/s$\cite{Liu2016}) is the Dirac band velocity in the a-c plane. The SdH oscillations can be modeled with the Lifshitz–Kosevich (LK) formula: to the first order, each single band contributes to the SdH oscillations:
\begin{equation}
\Delta R/R_0 \propto R_D R_T \cos \left( \frac{\hbar S_{F }}{e B} + \phi_{B }
+ \pi + \delta\right)
\label{tdepeq}
\end{equation}
We note that the conventional (parabolic band) LK formula and its Dirac version \cite {PhysRevB.71.125124} have the same mathematic form with their corresponding band parameters (see Supplementary Information). In Eq.(1) $R_0$ is the zero magnetic field resistance. Defined for Dirac band, $R_D=\exp{(-2\pi^2k_B T_d |\mu|/\hbar e B v^2)}$ is the amplitude reduction factor from disorder scattering, where $T_d$ is the Dingle temperature characterizing the level of disorder,and $k_B$ is the Boltzmann constant. $R_T=\xi/\sinh{(\xi)}$ is the amplitude reduction factor from temperature, where $\xi=2\pi^2 k_B T \mu/\hbar e B v^2$. $\phi_B$ is the Berry phase, and $\delta=\pm\pi/4$ in 3D materials.
Now we consider the Zeeman effect, which splits a spin-degenerate band into two, each with a different Fermi surface extremal cross-sectional area: $S_{F \uparrow / \downarrow} = S_F \pm \alpha B$, and Berry phase: \ $\phi_{B \uparrow
/ \downarrow} = \phi_B \pm \phi_s$ \cite{PhysRevB.101.125118}. Here $\alpha$ describes the splitting of extremal cross-sectional area of the Fermi surface and $\phi_s$ is the spin-dependent part of the Berry phase. The overall quantum oscillation is a summation of the two oscillation terms from the two spin bands:
$\cos \left( \frac{\hbar S_{F \uparrow}}{e B} + \phi_{B \uparrow}
+ \pi +\delta \right) + \cos \left( \frac{\hbar S_{F \downarrow}}{e B} + \phi_{B
\downarrow} + \pi + \delta \right) \\= 2\cos \left( \frac{\hbar S_F}{e B} + \phi_B + \pi + \delta \right) \cos \left( \frac{\hbar \alpha}{e} +\phi_s \right)$
Interestingly $S_F$ and $\phi_B$, which are the common (spin-independent) part of the two spin bands before the
Zeeman splitting, determines the period and phase offset of the quantum oscillations; while the difference between the two spin bands determines the amplitude.
For systems with nontrivial Berry curvature, the Berry phase splitting $\phi_s$ along the Fermi surface will be nonzero. In particular, for a Dirac electron system, we find (see Methods)
\begin{eqnarray}
& \phi_s = \pi\frac {\Delta}{\mu} \\
& \phi_{B \uparrow} = \pi + \phi_s = \pi \left( 1 +
\frac{\Delta}{\mu} \right) & \nonumber\\
& \phi_{B \downarrow} = \pi - \phi_s = \pi \left( 1 -
\frac{\Delta}{\mu} \right) &
\end{eqnarray}
Defining the Zeeman effect-induced SdH amplitude reduction factor: $R_s=\cos \left( \frac{\hbar \alpha}{e } + \phi_s \right)$, we derive for the Dirac band:
\begin{equation}
\Delta R/R_0 =A R_D R_T R_s \cos \left( \frac{\pi (\mu^2 - \Delta^2)}{e B \hbar^{} v^2} +\delta \right) ,
\end{equation}
Further, the coefficient $\alpha$ is computed from the g-factor tensor obtained from first principle calculations at given values of $\Delta$ and $\mu_F$.
\begin{figure*}[tp]
\centering
\includegraphics[width=\textwidth]{Figure4.png}
\caption {Evolution of band structure and Zeeman effect over strain. a. Evolution of Dirac band and Fermi level through the STI to WTI phase transition with increasingly tensile strain. b. Schematic energy dispersion with $k_z=0$ and magnetic field in the z(b) direction. Green circles indicate the boundary of extremal cross-sections for the spin bands. The spin-dependent Berry phases are $\phi$ and $-\phi$ over the two circles. c. spin-dependent Berry phase $\phi_s$ as a function of strain, calculated with mass gap and Fermi energy obtained from the SdH oscillations. d. Strain tuning of effective g-factor.}
\label{Fig4}
\end{figure*}
Under large compressive and large tensile strains, we observed approximately single SdH oscillation frequencies and a monotonic decay of SdH oscillation amplitude over increasing $1/B$, suggesting that SdH oscillations are dominated by single bands there. The linear dependence of the LL indices on $1/B$ (Fig. 2c) extrapolates to the index values of close to 0 and 1/2 at $1/B = 0$, corresponding to a SdH oscillation phase change from approximately zero to $\pi$. Fitting the $1/B$ and $T$ dependence of the SdH oscillation amplitude to the LK formula (examples shown in Fig.2d-f) provides estimations to the cyclotron mass and the Dingle temperature at these particular strains.
More generally, the SdH oscillations show a complex $1/B$ dependence which can be modeled considering contributions from two bands: $\Delta R=\Delta R_1+\Delta R_2$. Here band 1 is a Dirac band with a common Berry phase of $\pi$ for its two spins bands and hence a overall SdH phase which is much smaller than $\pi$ (Fig.~2c). Band 2, a ``trivial'' parabolic band possess a overall SdH phase which is close to $\pi$. We note that evidence of charge transport contribution from a secondary band has been reported previously\cite{Chi2017}. The change of SdH phase shown in Fig.~2c is due to the strain-evolution of the relative contribution from the two bands, instead of band topology transition within the same band.
Fig. 2b compares the measured SdH oscillations with the two-component LK formula simulations at base temperatures. Here the Dirac band SdH oscillations are modeled with Eq.(4), while the SdH oscillations from the trivial band are modeled with the conventional LK formula, with parameters including $T_d$, Fermi energy and cyclotron mass $m_c$. The simulation reproduces the experimental observations, with small deviations which may be attributed to the choices of background curves when extracting the oscillatory part of the data. The more general comparisons including temperature dependence are shown in the Supplementary Information. Fig.~3a-c plots the key simulation parameters. Focusing on the the Dirac band, a closing and re-opening of mass gap with tuning external strain happens at a compressive strain of $\approx -0.2\%$. This, remarkably, is a direct transport evidence of the theoretically predicted WTI to STI transition. Associated with such transition, the Fermi energy (measured from the center of the mass gap) shows a non-monotonic strain dependence, reaching a minimum close to the transition strain where mass gap vanishes. From the mass gap and the Fermi energy, we calculate the electron doping: $E_F-\Delta$, which characterizes the energy of the Fermi level in relation to the bottom of the conduction band. The result indicates a maximum electron doping at the WTI-STI transition, which decreases when strain-tuned away from the transition. Generally the analysis of the band parameters suggests a band evolution with strain which is depicted in Fig.~4a.
Next we focus on the impact of Zeeman effect on SdH oscillation amplitude ($R_s$). Experimentally we obtain $R_s$ at every strain by quantitative simulation of the SdH oscillations. We then compare the results with the theoretical expectation: $R_s=\cos \left( \frac{\hbar \alpha}{e } + \phi_s \right)$. Here $\frac{\hbar\alpha}{e}$, which is associated with the splitting of extremal cross-sectional area of the Fermi surface, follows (see Methods):
\begin{equation}
\frac{\hbar\alpha}{e}
= - \pi \frac{\Delta}{\mu} - \pi\frac{\hbar^2}{m_e}\frac{\mu}{v^2}\left(\frac{g_{p,z}}{2}(1+\frac{\Delta}{\mu})-\frac{g_{s,z}}{2}(1-\frac{\Delta}{\mu})\right) \label{eq:halpha}
\end{equation}
We adopt the g-factors for the $s$ and $p$ orbitals $g_{p,z} = 9.66, g_{s,z} = -6.45$, as computed from first-principle calculations (DFT-mBJ) \cite{song2021first}. The spin dependent part of the Berry's phase $\phi_s=\pi \frac{\Delta}{\mu}$ is calculated from the mass gap and Fermi energy obtained from simulating the SdH oscillations. The theory shows good qualitative agreement with the data on the strain dependence of $R_s$. By contrast, the conventional modeling of Zeeman effect on SdH oscillations which only considers the Fermi surface splitting (Eq.~(5)) completely fails to match with the experimental observations. This comparison definitively highlights the importance of spin-dependent Berry phase in the Zeeman effect.
The significant strain-tunability of the spin-dependent Berry phase $\phi_s$ is shown in Fig.4c. Accompanied by the topological phase transition where the mass gap vanishes, $\phi_s$ passes through zero when the system is in the Dirac semimetal phase. We also compute the effective g-factor by comparing $R_s$ with the conventional Zeeman effect induced SdH amplitude reduction factor $\cos{(\pi g m_c/(2m_e))}$\cite{shoenberg1984,Liu2016}. This leads to $g=2m_e(\hbar \alpha/e + \phi_s)/(\pi m_c)$, whose strong strain tunability is shown Fig.4d.
In conclusion, we have carried out a magnetotransport study on the evolution of band topology and non-trvial Berry phase over strain-tunable band parameters in ZrTe$_5$. The strain-dependent SdH oscillations allow direct mapping of the closing and reopening of the Dirac band mass gap, which is consistent with a WTI-STI transition in ZrTe$_5$. Moreover we observed the non-trivial Berry phase and its dependence on band parameters over the transition. Such a spin-dependent Berry phase is generic and intrinsic to Dirac band structure, and is a critical factor in modeling Zeeman effect in SdH oscillations in topological materials.
\section*{Methods}
\subsection*{Sample fabrication and characterization}
The samples studied in this work are ZrTe$_5$ microcrystals mechanically exfoliated onto flexible Polyimide substrates (Fig. 1a,b), which allow application of tensile and compressive strains over a wide temperature range from room temperature down to ~4K. ZrTe$_5$ single crystals were synthesized by chemical vapor transport (CVT) method, with iodine as transport agency.
Stoichiometry amounts of Zr(4N) and Te(5N) powder, together with 5mg/mL I$_2$, were
loaded into a quartz tube under argon atmosphere. The quartz tube was flame sealed
and then placed in a two-zone furnace, a temperature gradient from 480°C to 400°C was
applied. After 4 weeks reaction, golden, ribbon-shaped single crystals were obtained, of typical size about $0.6\times0.6\times5mm$. The X-ray diffraction characterization of the crystals is shown in the Supplementary Information.
To avoid degradation of the material from ambient exposure, the crystals are exfoliated and press-contacted on predefined gold electrodes on 120$\mu$m-thick Polyimide substrates in Ar environment, and are encapsulated with poly(methyl methacrylate) (PMMA) which both protects the crystals from degradation and facilitates the application of strain. A detailed description on the sample fabrication and on the contact/crystal interface can be found in reference\cite{Mills2019}. All measurements were done with electric current applied along the a-axis. Typical contact resistance over a ~10 $\mu m^2$ contact area is between few hundred ohms to a few kilo-ohms, and does not change significantly over the application of strain through substrate bending. Uniaxial strain is applied to the samples using a 4-point bending setup, with motorized precision control to the substrate curvature\cite{Guan107.193102,Guan2017}. Because the substrate is much thicker than the microcrystals, significant uniaxial strain can be achieved at relatively mild substrate curvatures under which the strain on the microcrystals is predominantly tensile/compressive due to the elongation/compression of the top substrate surface where the microcrystals are PMMA-pinned. Besides external strain applied through substrate bending, both the crystals, the substrates, and the encapsulating PMMA layer go through thermal expansion/contraction upon temperature change. Because of such complication it is difficult to precisely characterize the absolute strain on the crystals. Here we focus on the dependence of the transport characteristics on the change of strain induced by substrate bending, which is described in the discussion below as "external strain". In magnetotransport measurements, we limit the temperature range between 3-20K, where the change of thermal expansion is small in comparison to the range of the external strain. All charge transport measurements were performed in a Oxford Variable Temperature Insert (VTI) with a superconducting magnet, using standard lock-in technique.
\subsection*{Computing the \texorpdfstring{$g$}{lg}-factor}
To compute the effect of Zeeman coupling on quantum oscillations, we study the $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian near $\Gamma$, which is an anisotropic gapped Dirac Hamiltonian \cite{song2021first}:
\begin{equation}
H(\mathbf{k} ) =
\begin{pmatrix}
\Delta \sigma_0 & \sum_i v_i k_i \sigma_i \\
\sum_i v_i k_i \sigma_i & -\Delta \sigma_0
\end{pmatrix},
\label{eq:H0}
\end{equation}
in the basis $|p,\frac{1}{2} \rangle$, $|p, -\frac{1}{2}\rangle$, $|s, \frac{1}{2}\rangle$, $|s, -\frac{1}{2} \rangle$, where $\pm \frac{1}{2}$ refers to the $z$-component of spin.
In Eq.~(\ref{eq:H0}), $\Delta$ is half the mass gap; $v_{x,y,z}$ are the anisotropic Dirac cone velocities; $\sigma_{x,y,z}$ are the Pauli matrices; and $\sigma_0$ is the $2\times 2$ identity matrix.
The eigenvalues of $H(\mathbf{k})$ are
\begin{equation}
E_\pm = \pm \sqrt{ \Delta^2 + \sum_i v_i^2 k_i^2},
\label{eq:E0}
\end{equation}
where all bands are doubly-degenerate due to the combination of time-reversal and inversion symmetry.
The eigenvectors are given by
\begin{equation}
| \psi^\pm_{m,\mathbf{k}} \rangle = \sqrt{ \frac{ E_\pm + \Delta}{2E_\pm } } \begin{pmatrix}
u_m \\ \frac{v}{E_\pm + \Delta} \bm{p \cdot \sigma} u_m \end{pmatrix},
\label{eq:eigenstates}
\end{equation}
where $u_1= (1,0)^T$, $u_2 = (0,1)^T$
and we have defined the rescaled coordinates $p_i = v_i k_i /v$.
The Zeeman coupling for the four-band model is:
\begin{equation}
H^Z_0 =
\mu_B \begin{pmatrix}
\sum_i g_i^p B_i & 0 \\
0 & \sum_i g_i^s B_i
\end{pmatrix},
\label{eq:HZ0}
\end{equation}
where $g_{x,y,z}^{s,p}$ are the matrix-valued $g$-factors for the $s$ and $p$ orbitals, which are anisotropic due to the crystal symmetry and which differ from their bare value due to coupling with all the other bands not in the four-band model.
To compute the quantum oscillations, we must downfold this four-band model into the two bands at the Fermi level.
We seek an \textit{effective} $g$-factor for the two degenerate bands that make up the Fermi surface, such that when projected onto those bands, the Zeeman Hamiltonian takes the form:
\begin{equation}
H^Z_{mm'} = \mu_B \mathbf{g}_{mm'} \cdot \mathbf{B},
\end{equation}
where $\mu_B = \frac{e\hbar}{2m_e}$ is the Bohr magneton and $m,m'$ index the two bands at the Fermi surface.
The effective $g$-factor has two contributions:
\begin{align}
\bm{g} = \bm{g}_0 + \bm{g}_D,
\label{eq:gdecompose}
\end{align}
where $\bm{g}_0$ is the projection of $g^s$ and $g^p$ (defined by $H^Z_0$) onto the two conduction bands at the Fermi level and $\bm{g}_D$ is an extra orbital contribution that we will describe below.
For a magnetic field in the $z$ direction, using the expressions for the conduction band eigenstates from Eq.~(\ref{eq:eigenstates}),
\begin{align}
\bm{g}_{0,z}
&= \left( \frac{E_+ + \Delta}{2E_+} \right)\left( g_z^p + g_z^s \frac{v^2(p_z^2 - p_x^2 - p_y^2)}{(E_+ + \Delta)^2} \right) \sigma_z,
\end{align}
where the Pauli matrix $\sigma_z$ acts in the basis of the two bands at the Fermi level.
Ultimately, we will only need the $g$-factor at the extremum of the Fermi surface, where $p_z=0$ and $v^2(p_x^2 + p_y^2) = \mu^2 - \Delta^2$. In this case,
\begin{equation}
\bm{g}_{0,z}^\text{ext} = \frac{1}{2}\left(g_z^p \left(1+\frac{1}{\gamma}\right)-g_z^s \left(1-\frac{1}{\gamma}\right)\right) \sigma_z
\label{eq:gelse}
\end{equation}
where $\gamma = \frac{\mu}{\Delta}$ is a dimensionless constant.
Returning to the second term in Eq.~(\ref{eq:gdecompose}),
$\bm{g}_D$ is the orbital contribution to the $g$-factor from the Dirac cone~\cite{Xiao2010Berry}:
\begin{equation}
\mathbf{g}_{D,mm'}(\mathbf{k}) = \frac{im_e}{\hbar^2}\sum_{ijk}\varepsilon_{ijk}\hat{\mathbf{e}}_k \langle \partial_i \psi_{m,\bm{k}}| \left( H(\bm{k}) - E_{m,\bm{k}} \right) |\partial_j \psi_{m',\bm{k}}\rangle ,
\label{eq:gfactor2}
\end{equation}
where $\psi_{m,\mathbf{k}} $ are the eigenvectors of Eq.~(\ref{eq:H0}).
The first step to evaluate $\bm{g}_D$ is to find the derivatives of the eigenstates:
\begin{align}
\frac{\partial |\psi_{m,\bm{k}}\rangle}{\partial p_i}
&= \sqrt{\frac{E+\Delta}{2E}}\left( \begin{array}{c}
-\frac{\Delta v^2 p_i}{2E^2(E+\Delta)}u_m \\
-\frac{(\Delta+2E)v^3 p_i}{(E+\Delta)^22E^2}\bm{p \cdot \sigma}u_m+\frac{v}{E+\Delta}\sigma_iu_m
\end{array}\right),
\label{eq:deigenstates}
\end{align}
where we have dropped the subscript/superscript $\pm$ on $E$ and $\psi$ to reduce clutter.
After some algebra, it follows that:
\begin{widetext}
\begin{align}
\left(H-E_{m,\bm{k}}\right)\frac{\partial |\psi_{m,\bm{k}}\rangle}{\partial p_i}
&=\sqrt{\frac{E+\Delta}{2E}} \frac{1}{E(E+\Delta)}\left(\begin{array}{c}
\Delta v^2p_i u_m + iEv^2\sum_{jk}\varepsilon_{jik}p_j\sigma_ku_m\\
v^3p_i \bm{p\cdot\sigma}u_m - E(E+\Delta)v\sigma_iu_m
\end{array}\right),
\end{align}
which yields the matrix element
\begin{align}
\langle \frac{\partial \psi_{m,\bm{k}}}{\partial p_i}|H(\bm{k}) - E_{m,\bm{k}}|\frac{\partial \psi_{m',\bm{k}}}{\partial p_j}\rangle
&= \frac{v^2}{2E^2} u_m^\dagger \left[ \frac{-i v^2\varepsilon_{jkl}p_ip_k \sigma_l + iv^2 \varepsilon_{ikl} p_jp_k \sigma_l}{E+\Delta} - i E \varepsilon_{ijk} \sigma_k + \frac{ v^2 p_i p_j }{E} - E \delta_{ij}\right]u_{m'}
\end{align}
\end{widetext}
The terms symmetric under the exchange of the $i$ and $j$ indices (i.e., the last two terms) will cancel in the sum in Eq.~(\ref{eq:gfactor2}). Thus, applying Eq.~(\ref{eq:gfactor2}), the extra orbital contribution to the $g$-factor is given by:
\begin{widetext}
\begin{align}
\bm{g}_{D}
&=\frac{g^L}{2\tilde{\gamma}^2}\sum_{ijk} \left[\tilde{\gamma} \varepsilon_{ijk}\varepsilon_{ijk}\sigma_k-\frac{2v^2/\Delta^2}{1+\tilde{\gamma}}\sum_{qr}\varepsilon_{iqr}\varepsilon_{ijk}p_qp_j\sigma_r\right]\frac{v_iv_j}{v^2}\hat{\bm{e}}_k
\label{eq:gfactorresult}
\end{align}
\end{widetext}
where $g^{L} \equiv \frac{m_ev^2}{\hbar^2\Delta}$ and $\tilde{\gamma} \equiv \frac{E}{\Delta}$ are dimensionless constants.
We now simplify this result by taking $B$ in the $z$-direction and restricting to the boundary of the extremal cross-section, where $p_z = 0$ and $v^2(p_x^2+p_y^2)=\mu^2-\Delta^2$. Plugging this into Eq.~(\ref{eq:gfactorresult}), we obtain the $z$-component of $g$:
\begin{equation}
\bm{g}_{D,z}^\text{ext} = \frac{g^L}{\tilde{\gamma}^2}\frac{v_xv_y}{v^2}\sigma_z
\label{eq:gdirac}
\end{equation}
The total effective $g$-factor at the extremal cross-section of the Fermi surface is found by combining Eqs.~(\ref{eq:gelse}) and (\ref{eq:gdirac}):
\begin{equation}
\bm{g}_{z}^\text{ext} = \left[\frac{m_e\Delta v_xv_y}{\hbar^2\mu^2}+\frac{g_z^p}{2}\left(1+\frac{\Delta}{\mu}\right)-\frac{g_z^s}{2}\left(1-\frac{\Delta}{\mu}\right) \right]\sigma_z
\label{eq:gextfinal}
\end{equation}
This expression shows that the \textit{g}-factor in the $\hat{z}$-direction is independent of momentum on the boundary of the extremal cross-section and is diagonal, i.e., $|\psi_{m,\bm{k}}\rangle$ are eigenstates of the Zeeman coupling.
\subsection*{Quantum oscillations}
We now discuss the quantum oscillations of the anisotropic Dirac semi-metal.
As discussed in the main text, the quantum oscillations for a doubly degenerate electronlike Fermi surface are proportional to
\begin{align}
\Delta R/R_0 &\propto \cos \left( \frac{\hbar S_F}{e B} + \phi_B + \pi + \delta \right) \cos \left( \frac{\hbar \alpha}{e} +\phi_s \right)
\label{eq:LK}
\end{align}
where $\phi_{B,\uparrow / \downarrow} = \phi_B \pm \phi_s$ are the Berry phases around the extremal Fermi surface for each of the two spins and $S_{F,\uparrow / \downarrow} = S_F \pm \alpha B$ are the extremal areas of the Fermi surface for each spin.
The Berry connection is defined by \cite{Xiao2010Berry}: $\bm{A}_{mm'}(\bm{k}) = \sum_k i\langle \psi_{m,\bm{k}}|\frac{\partial}{\partial k_k}|\psi_{m',\bm{k}}\rangle \bm{e}_k $.
By plugging in the expressions for the eigenstates and derivatives of eigenstates in Eqs.~(\ref{eq:eigenstates}) and (\ref{eq:deigenstates}), we find the explicit expression:
\begin{multline}
\bm{A}(\bm{k}) =\frac{v}{\Delta^2}\frac{1}{2\tilde{\gamma}(\tilde{\gamma}+1)}\left[(p_y\sigma_z-p_z\sigma_y)v_x\bm{e}_x+ \right.\\
\left. (p_z\sigma_x-p_x\sigma_z)v_y\bm{e}_y+(p_x\sigma_y-p_y\sigma_x)v_z\bm{e}_z \right]
\end{multline}
To find the Berry phase, we ultimately need $\bm{A}(\bm{k})\bm{\cdot}\mathrm{d}\bm{k}$
on the extremal cross-section, where $p_z=0$ and $\mathrm{d}\bm{k}$ is in the $x-y$ plane, which yields:
\begin{align}
\bm{A}(\bm{k})\bm{\cdot}\mathrm{d}\bm{k}
&=\frac{v_xv_y}{\Delta^2}\frac{\sigma_z}{2\tilde{\gamma}(\tilde{\gamma}+1)}\left(k_y\bm{e}_x-k_x\bm{e}_y\right)\bm{\cdot} \mathrm{d}\bm{k},
\end{align}
where $\sigma_z$ is acting in the space of the two conduction bands at the Fermi level.
We find the Berry phase around each extremal ring of the Fermi surface by integrating over the the boundary of the extremal cross-section:
\begin{equation}
\oint \bm{A}(\bm{k})\bm{\cdot}\mathrm{d}\bm{k} \!=\! \frac{v_xv_y}{2 \Delta^2\tilde{\gamma}(\tilde{\gamma}+1)}\oint \left(k_y\bm{e}_x-k_x\bm{e}_y\right)\bm{\cdot} \mathrm{d}\bm{k}\label{eq:phi}
\end{equation}
The loop integral can be turned into an area integral using Stokes theorem:
\begin{align}
\iint \nabla_{\bm{k}}\times \left(k_y\bm{e}_x-k_x\bm{e}_y\right) \cdot \mathrm{d}\bm{S}_{\bm{k}} &= -2 \iint d S_\mathbf{k} \\&= - 2\pi \frac{\mu^2 - \Delta^2}{v_y v_x},
\end{align}
which yields:
\begin{equation}
\phi_{B,\uparrow/\downarrow} = \pi\left(1 \pm \frac{\Delta}{\mu}\right), \label{eq:berryphase}
\end{equation}
where we have added $2\pi$ in order to ensure the Berry phase in the range of $0$ to $2\pi$.
The last term that we need to evaluate the Lifschitz-Kosevich formula is the area of the extremal Fermi surfaces.
Since the effective $g$-factor at the extremal Fermi surfaces (Eq.~(\ref{eq:gextfinal})) is diagonal in the basis of the two conduction bands, the two extremal Fermi surfaces satisfy the equation:
\begin{equation}
\mu = \sqrt{ \Delta^2 + v_x^2 k_x^2 + v_y^2 k_y^2 } \pm \mu_B \bm{g}_z^\text{ext} B,
\end{equation}
accounting for the fact that the magnetic field is in the $z$ direction and the extremal Fermi surfaces have $p_z = 0$.
Thus, each extremal cross-section forms an ellipse whose area is given by:
\begin{equation}
S = \frac{\pi}{v_x v_y} \left[ \left( \mu \mp \mu_B \bm{g}_z^\text{ext} B \right)^2 - \Delta^2 \right]
\end{equation}
Thus, the area-splitting term $\alpha$ in Eq.~(\ref{eq:LK}) is given by:
\begin{equation}
\alpha = - \frac{\pi e\hbar \mu \bm{g}_z^\text{ext} }{m_ev_x v_y},
\end{equation}
where we have used $\mu_B = e\hbar/2m_e$. Plugging in the result for $\bm{g}_z^\text{ext}$ from Eq.~(\ref{eq:gextfinal}),
\begin{equation}
\frac{\hbar\alpha}{e} = - \frac{\pi \Delta }{\mu} - \frac{\pi \hbar^2 \mu }{m_ev_x v_y}\left[ \frac{g_z^p}{2}\left(1+\frac{\Delta}{\mu}\right)-\frac{g_z^s}{2}\left(1-\frac{\Delta}{\mu}\right) \right] \label{eq:alpha}
\end{equation}
Plugging in the calculation of the Berry phase (Eq.~(\ref{eq:berryphase})) and the area difference (Eq.~(\ref{eq:alpha})) to the Lifschitz-Kosevich formula (Eq.~(\ref{eq:LK})), we derive the expression for quantum oscillations:
\begin{widetext}
\begin{align}
\Delta \rho
&\propto \cos\left( \frac{\pi \hbar^2 \mu }{m_ev_x v_y}\left[ \frac{g_z^p}{2}\left(1+\frac{\Delta}{\mu}\right)-\frac{g_z^s}{2}\left(1-\frac{\Delta}{\mu}\right) \right] \right)\cos\left(\frac{\hbar S_F}{eB}\pm\frac{\pi}{4}\right)
\end{align}
\end{widetext}
\section*{References}
\bibliographystyle{apsrev4-1}
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BT may come early with IPOs
Miles Gregory-Costello
Wednesday August 23, 2000 5:00 pm
British Telecommunications may have to speed up plans to dispose of subsidiaries such as BT Wireless after being forced to put back a much-needed fundraising via a bond issue.
BT has had to delay its roadshow for the sale of $10bn (€ 11.1bn) of bonds planned for September following Standard & Poor's decision to place the UK's leading telecoms company on review for possible downgrade.
Analysts...
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,887
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{"url":"https:\/\/sysive.buzz\/article\/6-3-some-details-of-glycolysis","text":"# 6.3: Some details of glycolysis (2023)\n\n1. Last update\n2. Save as PDF\n\u2022 ID of the page\n16445\n\n## A. Glycolysis, Stage 1\n\nReaction 1:In the first reaction of glycolysis, the enzymeHexoquinaserapidly phosphorylates glucose entering the cell and formsGlucose-6-phosphat(G-6-P). As shown below is the overall responseexergonic; \u00d6free energy exchangefor the reaction is -4 Kcal per mole of G-6-P synthesized.\n\nThat is acoupled reaction, in whichphosphorylationGlucose is coupled to ATP hydrolysis. Free energy from ATP hydrolysis (an energetically favorable reaction) drives glucose phosphorylation (an energetically favorable reaction).efavorable reaction). The reaction is toobiologically irreversible, as shown by the single vertical arrow. Excess dietary glucose can be stored in most cells (especially liver and kidney cells) as a highly branched polymer of so-called glucose monomersGlycogen. In green algae and plants, the glucose produced by photosynthesis is stored as a starch polymer. When glucose is needed for energy, hydrolysis of glycogen and starch forms glucose-1-phosphate (G-1-P), which is then convertedG-6-P.\n\nLet's consider the energetics (free energy flux) of the reaction catalyzed by hexokinase. This reaction can be viewed as the sum of two reactions shown below.\n\nRemember that ATP hydrolysis is aexergonic reaction, releasing ~7 Kcal\/mol (rounded off!) in a closed system under standard conditions. The condensation reaction of glucose phosphorylation occurs with a DGo of +3 Kcal\/mol. That is aendergonicoReaction under standard conditions. By summing the free energy changes of the two reactions, we can calculate the total DGo of -4 Kcal\/mol for the coupled reaction under standard conditions in a closed system.\n\nThe reactions above are written as if they were reversible. However, we have said that the overall coupled response isbiologically irreversible. Where's the contradiction? To explain, we say that an enzyme-catalyzed reaction is biologically irreversible if the products have a relatively low affinity for the enzyme's active site, making the catalysis (acceleration) of the reverse reaction very inefficient. Enzymes that catalyze biologically irreversible reactions must not be based on reactants but are often allosterically regulated. Such is the case with hexokinase. Imagine a cell reducing its G-6-P consumption because its energy needs are being met. What happens when the levels of G-6-P in the cells increase? You can expect the hexokinase reaction to slow down so the cell isn't wasting a valuable nutritious energy source. OAllosterische Regulationof hexokinase is shown below.\n\nWhen the G-6-P concentration in the cell increases,excessG-6-P binds to an allosteric site on hexokinase. The enzyme's conformational change is then transferred to the active site, thereby inhibiting the reaction.\n\n152 Glycolysis Stage 1, Reaction 1\n\nReaction 2:In this slightly endergonic and reversible reactionIsomerasecatalyzes the isomerization ofG-6-PforFructose-6-P(F-6-P), as shown below.\n\nReaction 3:In this biologically irreversible reaction6-Phosphofructokinase(6-P-Fructokinase) catalyzes the phosphorylation of F-6-P to makeFructose-1,6-diphosphat(F1,6 diP). This is also acoupled reaction, where ATP supplies the second phosphate. The total reaction is written as the sum of two reactions as shown below.\n\nLike the hexokinase reaction, the6-P-FrutoquinaseThe reaction is a coupled, exergonic and allosterically regulated reaction. Severalallosteric effectors, including ATP, ADP and AMP, and long-chain fatty acids regulate this enzyme.\n\nReactions 4 and 5:These are the final reactions of the first stage of glycolysis. InReaction 4, F1,6 diP (a 6-C sugar) is reversibly cleavedDihydroxyacetonphosphat(DHAP) EGlycerinaldehyd-3-phosphat(G-3-P). EmReaction 5(also reversible) the DHAP is converted into another G-3-P. Here are the reactions:\n\nThe net result is the formation of two molecules G-3-P in the final reactions ofstage 1of glycolysis. the enzymesF-diP-AldolaseeTriose-P-Isomeraseboth catalyze freely reversible reactions. Furthermore, both reactions occur with, and therefore are, a positive free energy changeendergonico. The sum of the free energy changes for the splitting of F1.6 diP into two G-3-Ps is a whopping +7.5 Kcal per mole, an energetically very unfavorable process.\n\nIn short, in the endstage 1from glycolysis we consume two molecules of ATP and split a 6-C carbohydrate into two 3-C carbohydrates. We also saw two biologically irreversible and allosterically regulated enzymes.\n\n153 glycolysis level 1; reactions 2-5\n\n## B. Glycolysis, Stage 2\n\nWe will only follow one of the two G-3-P molecules that are generated at the end of thestage 1of glycolysis, but remember that both will go throughLevel 2of glycolysis.\n\nReaction 6:This is a redox reaction. G-3-P becomes too oxidized1,3, diphosphoglyceric acid(1.3, in PG) and NAD+ is reduced to NADH. The by gLyceraldehyd-3-phosphatDehydrogenaseis shown below.\n\nDarinfreely reversible endergonicreaction, a hydrogen molecule (H2) is removed from the G-3-P leaving the phosphoglyceric acid. This short-lived oxidation intermediate is phosphorylated to produce1,3-diphosphoglyceric acid(1,3diPG). At the same time, the hydrogen molecule is converted into a hydride ion (H-) and a proton (H+). Reduce H ionsSHE+to NADH, leaving the protons in solution. Remember that this is all happening at the active site of the same enzyme!\n\nAlthough it catalyzes a reversible reaction,G-3-P-Desidrogenaseis allosterically regulated. In contrast to the regulation of hexokinase, however, that of G-3-P dehydrogenase is more complicated! The regulator is NAD+ and the mechanism of allosteric regulation ofG-3-P-Desidrogenasecalled by NAD+negative willingness to cooperate. It turns out that the higher the [NAD+] in the cell, the lower the enzyme's affinity for more NAD+ and the faster the reaction in the cell! The mechanism is explained at the link below.\n\n154 glycolysis level 2; reaction 6\n\nReaction 7:The reaction shown below is catalyzed byPhosphoglyceratkinase. It is freely reversible andexergonic, production of ATP and3-phosphoglyceric acid(3PG).\n\nCatalysis is the transfer of phosphate groups between molecules by kinasesPhosphorylation at the substrate level, usually the phosphorylation of ADP to produce ATP. In thiscoupled reactionthe free energy released by the hydrolysis of a 1,3-DiPG phosphate is used to produce ATP. Remember that this reaction occurs twice per starting glucose. Up to this point, two ATPs have been synthesized in glycolysis. We call 1.3 diPGvery high-energy phosphate compound.\n\nReaction 8:This freely reversible endergonic reaction moves phosphate from carbon number 3 of 3PG to carbon number 2 as shown below.\n\nI had movedifPhosphoglycerato-Mutasecatalyze the transfer of functional groups within a molecule.\n\nReaction 9:In this reaction (shown below)Enolasecatalyzes the conversion of 2PG toPhosphoenolpyruvat(PEP).\n\nReaction 10:This reaction leads to the formation ofpyruvic acid(Pyruvate), as shown below. remember again,Two pyruvates are produced per starting glucose molecule.\n\nthe enzymePyruvatkinasecouples orbiologically irreversible,exergonic hydrolysis of a phosphate from PEP and transfer of the phosphate to ADP in acoupled reaction. The product of the reaction, PEP, is anothervery high energyphosphate compound.\n\n155 glycolysis level 2; reactions 7-10\n\nPyruvate kinase is allosterically regulated by ATP, citric acid, long chain fatty acids, F1,6 diP and one of its own substrates, PEP.\n\nEmincomplete glycolysis (aerobic), pyruvate is oxidized in the mitochondria during respiration (cfAlternative targets for pyruvateabove).fermentationsyour name iscomplete glycolysisbecause pyruvate is reduced to one end product or another. Remember that muscle fatigue occurs when skeletal muscles use anaerobic digestion for energy during intense exercise. If pyruvate is too reducedlactic acid(lactate), the accumulation of lactic acid leads to muscle fatigue. the enzymeLactate dehydrogenase(LDH) that catalyzes this reaction is regulated but not allosterically. Instead, different muscle tissues regulate LDH by producing different versions of the enzyme! Click on the link below for an explanation.\n\n156 Fermentation: regulation of pyruvate reduction IS NOT allosteric!\n\n## C. A chemical and energy balance for glycolysis\n\nCompare the balances of complete glycolysis (fermentation) with lactic acid andincompleteGlycolysis (aerobic) showing chemical products and energy transfers.\n\nThere are two reactionsLevel 2glycolysis, each producing one molecule of ATP. Since each of these reactions occurs twice per starting molecule of glucose, Stage 2 of glycolysis produces four molecules of ATP. Sincestage 1For example, if two ATPs are consumed, the net yield of chemical energy as ATP at the end of glycolysis is two ATPs, whether complete for lactate or incomplete for pyruvate! Because they can't use oxygen, anaerobes have to make do with the meager 15 Kcal of ATP they make from fermentation. Since there are potentially 687 kcal available to fully burn one mole of glucose, there is much more free energy to capture during the rest of the breath.\n\n157 Equilibrium of Glycolysis\n\nAlso remember that the only redox reaction in aerobic glycolysis occurs in stage 2. This is the oxidation of G-3-P, a 3C glycolysis intermediate. Now look at the redox reaction and a fermentation pathway. When pyruvate, also a 3C intermediate, was reduced, there wasno net glucose oxidation(i.e. glycolytic intermediates) in complete glycolysis.\n\nYou will have recognized by now that glycolysis is an energetically favorable network (Downhill,spontaneous) reaction pathway in a closed system with globally negative \u0394Go. Glycolysis is also normally spontaneous in most of our cells, driven by a constant need for energy to perform cellular work. Thus, the actual free energy of glycolysis, or \u0394G', is also negative. In fact, glycolysis occurs in actively breathing cells releasing more free energy than in a closed system. In other words, the \u0394G' of glycolysis in active cells is more negative than the \u0394Go of glycolysis!\n\nNow let's look at gluconeogenesis, the Atkins diet, and some not-so-normal circumstances for a moment when glycolysis is essentially reversed, at least in some cell types. Under these conditions, glycolysis is energetically unfavorable and it is these reverse reactions that are associated with a \u0394G'! Negative!\n\nTop Articles\nLatest Posts\nArticle information\n\nAuthor: Dean Jakubowski Ret\n\nLast Updated: 03\/17\/2023\n\nViews: 5915\n\nRating: 5 \/ 5 (70 voted)\n\nAuthor information\n\nName: Dean Jakubowski Ret\n\nBirthday: 1996-05-10\n\nAddress: Apt. 425 4346 Santiago Islands, Shariside, AK 38830-1874\n\nPhone: +96313309894162\n\nJob: Legacy Sales Designer\n\nHobby: Baseball, Wood carving, Candle making, Jigsaw puzzles, Lacemaking, Parkour, Drawing\n\nIntroduction: My name is Dean Jakubowski Ret, I am a enthusiastic, friendly, homely, handsome, zealous, brainy, elegant person who loves writing and wants to share my knowledge and understanding with you.","date":"2023-03-23 04:49:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5046593546867371, \"perplexity\": 9633.022725233981}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296944996.49\/warc\/CC-MAIN-20230323034459-20230323064459-00277.warc.gz\"}"}
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\section{Introduction}
The \defn{Lyusternik-Schnirelmann category} or \defn{LS-category} of a topological space $M$ is the numerical homotopy invariant $\mbox{Cat}(M)$ defined to be the least number of open subsets $U\subset M$, whose inclusion is nullhomotopic, that are required to cover $M$. Although it is now the subject of a full theory in connection with algebraic topology, it was originally introduced by Lyusternik and Schnirelmann in a course on the global calculus of variations, when $M$ is a smooth closed manifold without boundary~\cite{LS}. In this case they show that any smooth real-valued function $f$ defined on $M$ has at least $\mbox{Cat}(M)$ critical points. The difference with Morse theory is that $f$ is allowed to have degenerate critical points. Rewiews on the Lyusternik-Schnirelmann theory are for instance~\cite{James,MR1990857,bartsch}.
If $M$ is a topological space, acted on continuously and properly by a topological group $G$, then the LS-category has an equivariant analogue $\mbox{Cat}_G(M)$. It has first been introduced by\cit{Fadell}~\cite{fadell} and\cit{Marzantowicz}~\cite{marzantowicz} for compact groups, and by\cit{Colman}~\cite{colman} for finite groups. A substantial part of the theory has been extended for proper group actions by\cit{Ayala, Lasheras and Quintero}~\cite{ayala}.
A \defn{$G$-categorical} open subset of $M$ is a $G$-invariant open subset $U\subset M$ admitting a $G$-deformation retract onto a $G$-orbit (cf. Definition \ref{def: G-categorical open subset}). The numerical homotopy invariant $\mbox{Cat}_G(M)$ is the least number (possibly infinite) of $G$-categorical open subsets that are required to cover $M$. If $M$ is a smooth manifold and $G$ is a Lie group, a class of $G$-categorical open subsets consists of \defn{$G$-tubular} open subsets, which are essentially tubular neighbourhoods of group orbits (cf. Definition \ref{def:G tubular open subset}). This fact is a direct application of the Tube Theorem \ref{Tube theorem}.
This invariant is in general difficult to compute and we are usually only able to know an estimation of it, in term of the cup length of $M$. We obtain a new formula to reduce the calculation of $\mbox{Cat}_G\left(M\right)$ to the calculation of the equivariant LS-category of the minimal orbit-type strata of $M$. In general, any topological space $M$ can be written as a disjoint union of smaller subsets $M_{\beta}$, called \defn{strata}, indexed on some strictly partially ordered set $(\mathcal{B},\prec)$. Those strata are required to fit in a specific way and form themselves a strictly partially ordered set (cf. Section \ref{isotropytype}). A stratum is minimal if it is minimal with respect to the strict partial order defined on them. If $M$ is a proper $G$-manifold, the strata $M_{\beta}$ are generally the connected components of the orbit-type submanifolds. We use a modified definition of orbit-type stratum (cf. Definition \ref{def: orbit type stratum}). We say that an \defn{orbit-type stratum} is a $G$-orbit of a connected component of the subset of $M$ of all the points having the same stabilizer.
On a large class of proper $G$-manifolds $M$, including symplectic toric manifolds, we observe that $M$ can be entirely covered by a subcover of its minimal orbit-type strata, made of $G$-tubular open subsets. Besides this cover is the smallest cover, made of $G$-categorical open subsets, that we can take. Such covers are called \defn{minimal $G$-tubular covers} and are discussed in Section \ref{s:G-categorical tubular covers}. However those covers do not exist in general. A non-example is given, when $M$ is a non-Hamiltonian compact $S^1$-manifold (cf. Example \ref{non-example}). By using the natural stratification of the moment polytope, we show in Section \ref{s: tubular toric} that every symplectic toric manifold admits a minimal $G$-tubular cover, where $G$ in this case is a torus having half the dimension of $M$ and acting effectively on it (cf. Theorem \ref{thm toric}). When $M$ admits a minimal $G$-tubular cover, we show that the calculation of $\mbox{Cat}_G(M)$ is intrinsically reduced to those of the minimal orbit-type strata of $M$. Explicitly, we obtain the localization formula
\begin{equation*}
\mbox{Cat}_G(M)=\sum\mbox{Cat}_{G}\left(M_{\beta}\right)
\end{equation*}
where the summation is taken over the minimal orbit-type strata $M_{\beta}$ (cf. Section \ref{s: loc formula}, Theorem \ref{localization thm first} and Corollary \ref{localization thm}). The result of\cit{Bayeh and Sarkar} (cf. \cite{MR3422738} Theorem $5.1$), which states that the equivariant Lyusternik-Schnirelmann category of a quasitoric manifold is precisely the number of fixed points of the torus action, is a consequence of Theorem \ref{thm toric} and of our localization formula.
\paragraph{Acknowledgements.} We would like to thank Yael Karshon and Eckhard Meinrenken for useful discussions and suggestions.
\vspace{1cm}
After completing this work, which forms part of the thesis \cite{moi}, we discoverd the paper of \cit{Hurder and T\"oben} \cite{MR2492297} which contains one of our main theorems (Theorem \ref{localization thm first}) and uses a similar approach.
\section{Terminologies}
We work with smooth manifolds and, except stated otherwise, the term \defn{submanifold} refers to an embedded submanifold. A smooth \defn{action} of a Lie group $G$ on $M$ is a group homomorphism $G\to \mbox{Diff}(M)$. We denote the action map by $(g,m)\in G\times M\mapsto g\cdot m\in M$. A \defn{$G$-manifold} is a pair $(M,G)$ where $M$ is a smooth manifold acted on by a Lie group $G$. If the action is proper then we refer to $(M,G)$ as a \defn{proper $G$-manifold}. A smooth map $f:M\to N$ between two $G$-manifolds is \defn{$G$-equivariant} if $f(g\cdot m)=g\cdot f(m)$ for all $g\in G$ and $m\in M$. The \defn{stabilizer} of $m\in M$ is the subgroup $G_m=\lbrace g\in G\mid g\cdot m=m\rbrace$. We say that the action of $G$ on $M$ is \defn{free} if all the stabilizers $G_m$ are equal to the trivial group $\mathbbm{1}$. The \defn{group orbit} or \defn{$G$-orbit} of a point $m\in M$ is the set $G\cdot m=\lbrace g\cdot m\mid g \in G\rbrace$. We use the notation $\mathfrak{g}\cdot m$ to mean the tangent space to $G\cdot m$ at $m$. The space $\mathfrak{g}$ stands for the Lie algebra of $G$ with Lie bracket $[\cdot,\cdot ]$, obtained by identifying $\mathfrak{g}$ with the left invariant vector fields on $G$.
\subsection*{Local models of G-orbits}
Let $(M,G)$ be a proper $G$-manifold. In this case, the group orbits are embedded submanifolds and all the stabilizers $G_m$ are compact. Let $N\subset T_mM$ be a $G_m$-invariant complement to $\mathfrak{g}\cdot m$ in $T_mM$. Given a subgroup $K$ of $G$, there is a (left) $K$-action on the product $G\times N$ given by
\begin{equation}
k\cdot (g,\nu)=(gk^{-1},k\cdot \nu).
\end{equation}
This action is free and proper by freeness and properness of the action on the first factor. The orbit space $G\times_KN$ is thus a smooth manifold whose points are equivalence classes of the form $[(g,\nu)]$, and the orbit map $\rho:G\times N\to G\times_KN$ is a smooth surjective submersion. Moreover the group $G$ acts smoothly and properly on $G\times_KN$, by left multiplication on the first factor. The theorem below states that, given a $G_m$-invariant neighbourhood of zero $N_0\subset N$, the associated bundle $G\times_KN_0$ defines a \defn{local model} for some $G$-invariant open subset $U\subset M$.
\begin{theorem}[Tube Theorem (cf. \cite{MR2021152} Theorem $2.3.28$)]\label{Tube theorem}
Let $(M,G)$ be a proper $G$-manifold and set $K=G_m$ for some $m\in M$. Let $N_0\subset N$ be an open $K$-invariant neighbourhood of $0$. Then, there exists a $G$-invariant neighbourhood $U\subset M$ of $m$ and a $G$-equivariant diffeomorphism
\begin{equation}\label{tau tube}
\varphi:G\times_K N_0\to U
\end{equation}
sending $[(e,0)]$ on $m$.
\end{theorem}
\subsection*{G-categorical open subsets}
Although the $LS$-category is defined for topological spaces, our interest is mainly about proper $G$-manifolds $(M,G)$. The terminologies are thus defined in this setting. Given $(M,G)$, a homotopy $H:M\times [0,1]\to M$ which satisfies $H(g\cdot m,t)=g\cdot H(m,t)$ for every $g\in G$, $m\in M$ and $t\in [0,1]$ is called a \defn{$G$-homotopy}. We write $H_t(m)=H(m,t)$. Let $B\subset A$ be two $G$-invariant subsets of $M$. A \defn{$G$-deformation retract of $A$ onto $B$} is a $G$-homotopy $H:A\times [0,1]\to A$ such that $H_0(a)=a$ and $H_1(a)\in B$ for every $a\in A$, and $H(b,1)=b$ for every $b\in B$.
\begin{definition}\label{def: G-categorical open subset}
\normalfont A $G$-invariant subset $U\subset M$ is called \defn{$G$-categorical} if there exists a $G$-deformation retract of $U$ onto the orbit $G\cdot x$ of some $x\in U$.
\end{definition}
\begin{definition}
\normalfont The \defn{equivariant LS-category of $M$}, denoted $\mbox{Cat}_G(M)$, is the least number of $G$-categorical open subsets $U\subset M$ that are required to cover $M$. We set $\mbox{Cat}_G(M)=\infty$ if such a cover does not exist. The non-equivariant LS-category $\mbox{Cat}(M)$ is obtained by setting $G=\mathbbm{1}$.
\end{definition}
\begin{example}\label{ex:TETRAHEDRON}
\normalfont The equivariant version of the LS-category is in general different from its non-equivariant analogue, as shown in the examples below.
\begin{enumerate}[label=(\roman*)]
\item Let $M=S^1\times \mathbb{R}$ with cylindrical coordinates $(\theta,z)$. Define an $S^1$-action on $M$ by $\phi\cdot (\theta,z)=(\theta+\phi,z)$. The cylinder itself is an $S^1$-categorical open subset with $S^1$-deformation retract $H:M\times [0,1]\to M$ given by $H((\theta,z),t)=(\theta,(1-t)z)$. Therefore, $\mbox{Cat}_{S^1}(M)=1$. However we require two contractible open subsets to cover $M$, which yields $$1=\mbox{Cat}_{S^1}(M)<\mbox{Cat}(M)=2.$$
\item Consider the complex projective space $M=\mathbb{C} P^2$ endowed with the $S^1$-action $\theta\cdot [z_0:z_1:z_2]=[e^{i\theta}z_0:z_1:z_2].$ For $i=0,1,2$, the open subsets $U_i=\lbrace [z_0:z_1:z_2]\mid z_i\neq 0\rbrace$ are $S^1$-invariant. On $U_0$, an $S^1$-deformation retract onto an orbit is given by
$$H([z_0:z_1:z_2],t)=[z_0:(1-t)z_1:(1-t)z_2].$$ The image $H_1(U_0)$ is the single point $[1:0:0]$ which is a fixed point of the action, hence an $S^1$-orbit. Similar homotopies can be found on $U_1$ and $U_2$, respectively. Therefore $\mbox{Cat}_{S^1}(M)$ is at most three. The fact that we have an equality follows from Proposition \ref{atmost} below. We conclude that $$\mbox{Cat}_{S^1}(M)=\mbox{Cat}(M)=3.$$
\item
The rotations of a tetrahedron form a group $\mathbb{T}$ of order $12$, which is a zero-dimensional Lie subgroup of $SO(3)$. This group acts on $M=S^2$. We construct a cover of $M$ by three $\mathbb{T}$-categorical open subsets as follows:
Pick a point $x_1\in M$ and its opposite point $y_1\in M$. The $\mathbb{T}$-orbit of $x_1$ forms a spherical tetrahedron with vertices $x_1,x_2,x_3,x_4$. Similarly the $\mathbb{T}$-orbit of $y_1$ forms another spherical tetrahedron with vertices $y_1,y_2,y_3,y_4$. For each $i<j$ denote by $p_{ij}$ the middle point of the geodesic arc joining $x_i$ and $x_j$ (cf. Figure \ref{spherical tetrahedrons}).
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{9cm}
\centering
\includegraphics[height=5cm]{sphericalTligne.pdf}
\caption{\small Spherical tetrahedrons on the sphere.}
\label{spherical tetrahedrons}
\end{minipage}
\end{figure}
For each $i$, let $D_i\subset M$ be an open disk centered at $x_i$ such that
$$\overline{D_i}\cap\overline{D_j}=\lbrace p_{ij}\rbrace\quad \forall i<j. $$
In the same way, let for each $i$, an open disk $E_i\subset M$ centered at $y_i$ with the property
$$\forall i<j\quad \overline{E_i}\cap\overline{E_j}=\lbrace p_{kl}\rbrace\quad \mbox{where} \quad k,l\notin \lbrace i,j\rbrace\quad k<l $$ as shown in Figure \ref{spherical}.
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{9cm}
\centering
\includegraphics[height=4cm]{spherical.pdf}
\caption{\small Disk $E_4$ centered at $y_4$.}
\label{spherical}
\end{minipage}
\end{figure}
Finally we define for each $i<j$, an open subset $B_{ij}\subset M$ containing $p_{ij}$ such that
\begin{eqnarray*}
x_k\in\overline{B_{ij}}\setminus B_{ij} && \forall k=i,j\\
y_k\in\overline{B_{ij}}\setminus B_{ij} && \forall k\neq i,j
\end{eqnarray*}
We obtain the following $\mathbb{T}$-categorical open subsets:
$$D=\bigcup\limits_{i=1}^4 D_i$$ which retracts in a $\mathbb{T}$-equivariant way onto the orbit $\mathbb{T}\cdot x_1,$
$$E=\bigcup\limits_{i=1}^4 E_i$$ which retracts in a $\mathbb{T}$-equivariant way onto the orbit $\mathbb{T}\cdot y_1,$ and
$$B=\bigcup\limits_{i<j} B_{ij}$$ which retracts in a $\mathbb{T}$-equivariant way onto the orbit $\mathbb{T}\cdot p_{12}.$ Those three subsets form a cover of $M$. This cover is in fact the smallest that we can take, by Proposition \ref{atmost} below. Hence $$3=\mbox{Cat}_{\mathbb{T}}(M)>\mbox{Cat}(M)=2.$$ \label{tetrahedron}
\end{enumerate}
\end{example}
\section{Stratifications and orbit-type strata} \label{isotropytype}
A \defn{partition} of a topological space $M$ is a cover of $M$ by pairwise disjoint subsets. Clearly every topological space admits a partition into its connected components. If our topological space is endowed with a group action, we can choose a partition which also encodes the information about the group action. For example, a proper $G$-manifold $(M,G)$ can be partitioned into locally closed (connected) submanifolds called the orbit-type strata, each of them being a union of group orbits with the same orbit-type.
\subsection*{Stratifications}\label{s: stratification}
There are several ways to define stratifications. The one we present here is the definition used by\cit{Kirwan} in her thesis \cite{Kirwan}. It is more flexible than the standard definition of\cit{Duistermaat and Kolk} \cite{kolk} (Definition $2.7.3$), especially for applications to algebraic geometry. A \defn{strict partial order} $\prec$ on $\mathcal{B}$ is a binary relation which is irreflexive and transitive. Note that in this case if $\alpha,\beta\in\mathcal{B}$ are such that $\alpha\prec\beta$, then $\beta\not\prec\alpha$.
For example the set of conjugacy classes of subgroups of $G$ admits the strict partial order $\prec_{conj}$, where we say that $(K)\prec_{conj} (H)$ if and only if $H$ is conjugate to a proper subgroup of $K$.
\begin{example}
\normalfont The group $\mathbb{T}$ has four conjugacy classes, namely $(\mathbb{T}),(\mathbb{Z}_3),(\mathbb{Z}_2)$ and $(\mathbbm{1})$. There are partially ordered with respect to $\prec_{conj}$ as shown in Figure \ref{T}.
\vspace{0.5cm}
\centering
\begin{figure}[h!]
\centering
\begin{minipage}[t]{9cm}
\[
\begin{xy}
(0,0)*+{\mathbb{T}}="T";%
(-10,-10)*+{\mathbb{Z}_3}="z";(10,-10)*+{\mathbb{Z}_2}="zz";%
(0,-20)*+{\mathbbm{1}}="i";%
{\ar@{-} "T";"zz"};{\ar@{-} "T";"z"};%
{\ar@{-} "zz";"i"};{\ar@{-} "z";"i"};%
\end{xy}
\]\caption{\small Conjugacy classes of subgroups of $\mathbb{T}$ where the order goes up to down i.e. $(\mathbb{T})$ is minimal with respect to $\prec_{conj}$.}
\label{T}
\end{minipage}
\end{figure}
\end{example}
\begin{definition}
\normalfont A collection $\left\lbrace M_{\beta}\mid \beta\in \mathcal{B}\right\rbrace$ of subsets of a topological space $M$ is \defn{locally finite} if each compact subset of $M$ meets only finitely many $M_{\beta}$. A locally finite collection $\left\lbrace M_{\beta}\mid \beta\in \mathcal{B}\right\rbrace$ of locally closed (non-empty) topological subspaces of $M$ form a \defn{$\mathcal{B}$-decomposition} or \defn{stratification} of $M$ if $M$ is the disjoint union of the \defn{strata} $M_{\beta}$, and there is a strict partial order $\prec$ on the indexing set $\mathcal{B}$ such that
\begin{equation}\label{closure condition}
\overline{M}_{\beta}\subset\bigcup_{\alpha\preceq\beta}M_{\alpha}
\end{equation}
for every $\beta\in\mathcal{B}$. We say that the stratification is \defn{smooth} if $M$ is a smooth manifold and every $M_{\beta}$ is a locally closed submanifold.
\end{definition}
Given a stratification of $M$, a strict partial order can be defined on the strata as follows
\begin{equation}\label{partial order strata}
M_{\alpha}< M_{\beta} \quad \Longleftrightarrow\quad \alpha\prec\beta.
\end{equation}
We say that a stratum $M_{\beta}$ is \defn{minimal} with respect to \eqref{partial order strata} if there is no $\alpha\in\mathcal{B}$ such that $M_{\alpha}< M_{\beta}$. Of course minimal strata are not unique because we just have a partial ordering. If $M_{\beta}$ is a minimal stratum, \eqref{closure condition} implies that $\overline{M}_{\beta}\subset M_{\beta}$. In particular $\overline{M}_{\beta}=M_{\beta}$ i.e. $M_{\beta}$ is closed in $M$.
\subsection*{Orbit-type strata}
Let $G$ be a Lie group and $H\subset G$ be a closed subgroup. The \defn{conjugacy class} of $H$ is the set $(H)=\lbrace L\subset G\mid L=gHg^{-1}\mbox{ for some }g\in G\rbrace.$
Given a $G$-manifold $(M,G)$, we define the set $$M_{(H)}:=\left\lbrace m\in M\mid G_m\in (H)\right\rbrace$$ which is the union of all the $G$-orbits in $M$ with orbit-type $(H)$. Using the definitions and the $G$-invariance of $M_{(H)}$, it is shown in \cite{MR2021152} (Proposition $2.4.4$) that $M_{(H)}=G\cdot M_H$ where $$M_H=\lbrace m\in M\mid G_m=H\rbrace.$$ Note that the biggest subgroup of $G$ which leaves $M_H$ invariant is the normalizer $N_G(H)=\left\lbrace g\in G\mid gHg^{-1}=H\right\rbrace.$ Furthermore this action induces a well-defined free action of the quotient group $N_G(H)/H$ on $M_H$. Write
$$M_H=\coprod_{b\in\mathcal{B}_H} M_{H,b}$$ as the disjoint union of its connected components, indexed on some set $\mathcal{B}_H$. Given $b\in \mathcal{B}_H$, we define the equivalence class $(b)$ to be the set of indices $a\in \mathcal{B}_H$ such that $G\cdot M_{H,a}=G\cdot M_{H,b}$. Let $\mathcal{B}$ be the set of pairs $\beta=((H),(b))$ where $(H)$ is the conjugacy class of some closed subgroup of $G$ and $b\in \mathcal{B}_H$.
\begin{definition}\label{def: orbit type stratum}
\normalfont For $\beta=((H),(b))\in \mathcal{B}$, we define an \defn{orbit-type stratum} $M_{\beta}$ to be the $G$-orbit of the connected component $M_{H,b}$ of $M_H$.
\end{definition}
We use here a modified definition of the standard definition which states that an orbit-type stratum is a connected component of $M_{(H)}$. If the $G$-action on $M$ is proper, the connected components $M_{H,b}$ are locally closed embedded submanifolds of $M$ and so are their $G$-orbits (cf. \cite{MR2021152} Proposition $2.4.7$).
The example below illustrates the difference between the standard definition of orbit-type strata and ours. With our definition, the orbit-type strata might not be connected.
\begin{example}
\normalfont Think of $\mathbb{R}^*=\mathbb{R}\setminus\lbrace 0\rbrace$ as a multiplicative group and let it act on $M=\mathbb{R}^2$ by $t\cdot (x,y)=(x,ty)$. The stabilizers of points of $M$ are either equal to the trivial group $\mathbbm{1}$, or equal to $\mathbb{R}^*$. Then $M_{(\mathbb{R}^*)}=M_{\mathbb{R}^*}$ is the $x$-axis, and $M_{(\mathbbm{1})}=M_{\mathbbm{1}}=H_+\cup H_-$ where $H_{\pm}=\lbrace (x,y)\in M\mid \pm y > 0\rbrace$. According to the standard definition of orbit-type strata, there are two strata with orbit-type $(\mathbbm{1})$, namely the connected components $H_+$ and $H_-$; and one stratum with orbit-type $(\mathbb{R}^*)$, the $x$-axis.
With our definition, there is one stratum with orbit-type $(\mathbb{R}^*)$ which is the $x$-axis; but there is only one stratum with orbit-type $(\mathbbm{1})$ which is $H_+\cup H_-$. Indeed, $M_{\mathbbm{1}}$ has two connected components, $H_+$ and $H_-$. The $\mathbb{R}^*$-orbits of each of them coincide. There is thus only one stratum with orbit-type $(\mathbbm{1})$ and it is not connected.
\end{example}
We define a strict partial order $\prec$ on $\mathcal{B}$ as follows: for $\alpha=((K),(a))$ and $\beta=((H),(b))$,
\begin{equation}\label{partial order orbit}
\alpha\prec \beta\quad\Longleftrightarrow\quad \alpha\neq\beta\quad\mbox{and}\quad M_{\alpha}\cap\overline{M}_{\beta}\neq \varnothing.
\end{equation}
By $\alpha\neq\beta$ we mean that the associated orbit-type strata $M_{\alpha}$ and $M_{\beta}$ are distinct.
\begin{proposition}[\cit{Sjamaar and Lerman} \cite{MR1127479}]\label{stratification by orbit type}
Let $(M,G)$ be a proper $G$-manifold and let $(\mathcal{B},\prec)$ as above with partial order \eqref{partial order orbit}. Then the orbit-type strata $\lbrace M_{\beta}\mid \beta\in\mathcal{B}\rbrace$ form a smooth stratification of $M$.
\end{proposition}
\begin{remark} In \cite{MR1127479}, \cit{Sjamaar and Lerman} use the standard definition of orbit-type strata. In particular those are connected submanifolds of $M$. In \cite{moi} we give a proof using the modified definition of orbit-type strata (cf. \cite{moi} Proposition $2.2.9$).
\end{remark}
\begin{example}\label{ex: stratum}\normalfont
Given an equivalence class $(H)$, the corresponding orbit-type strata might not all have the same dimension, as shown in the following example, appearing in\cit{Delzant} \cite{delzant} and\cit{Sjamaar and Lerman}~\cite{MR1127479}.
\begin{enumerate}[label=(\roman*)]
\item Let $M=\mathbb{C} P^2$ endowed with the $S^1$-action $$\theta\cdot [z_0:z_1:z_2]=[e^{i\theta}z_0:z_1:z_2].$$ The set $M_{S^1}$ has two connected components namely, the point $[1:0:0]$ and a copy of $\mathbb{C} P^1$, which consists of the points of the form $[0:z_1:z_2]$. Since $S^1$ acts trivially on each of these components, they form themselves two orbit-type strata, which are closed submanifolds of $M$. Since the action is free anywhere else, the last orbit-type stratum is $M\setminus \left(\lbrace [1:0:0]\rbrace \cup \mathbb{C} P^1\right)$. It has orbit-type $(\mathbbm{1})$ and is an open dense submanifold of $M$.
\item Let $M=S^2$ be the $2$-sphere embedded in $\mathbb{R}^3$, equipped with the $S^1$-action which rotates the sphere about the $z$-axis. There are three orbit-type strata namely, $M_{(\mathbbm{1})}$ which is diffeomorphic to $S^1\times (-1,1)$ and the two closed connected components of $M_{S^1}$ that are the North and South pole. \label{ex: stratum sphere}
\item \label{ex: stratum T} The group $\mathbb{T}$ acts on $M=S^2$. This group contains a copy of the cyclic group of order three $C_3\simeq \mathbb{Z}_3$ for each vertex, one copy of $\mathbb{Z}_2$ for each axis joining the middle point of an edge and the middle point of the opposite edge, and the identity (cf. Figure ~\ref{fig:TT}).
\vspace{0.3cm}
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{9.5cm}
\centering
\includegraphics[height=4cm]{tetrahedrons.pdf}
\caption{\small On the left hand side we fix a vertex $v$ and permute the three other vertices. As a subgroup it is isomorphic to $C_3$. On the right hand side we permute $v_1,v_2$ and $v_3,v_4$. This subgroup is isomorphic to $\mathbb{Z}_2$.}
\label{fig:TT}
\end{minipage}
\end{figure}
There are two minimal strata with orbit-type $(\mathbb{Z}_3)$, one minimal stratum with orbit-type $(\mathbb{Z}_2)$, and one open dense stratum with orbit type $(\mathbbm{1})$ (cf. Figure \ref{ex: stratum tetrahedron}). Indeed, when $H=\mathbb{Z}_3$, the eight points forming $M_{(H)}$ are a union of two $\mathbb{T}$-orbits. There are thus two strata with orbit-type $(\mathbb{Z}_3)$. For $H=\mathbb{Z}_2$, the six points forming $M_{(H)}$ are a single $\mathbb{T}$-orbit and form a single stratum.
\vspace{0.3cm}
\begin{figure}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$H$ & $M_H$ & $M_{(H)}$ \\
\hline
$\mathbb{T}$ & $\varnothing$ & $\varnothing$ \\
\hline
&&\\
\raisebox{25pt}{$\mathbb{Z}_3$} & \includegraphics[height=2cm]{T1.pdf} & \includegraphics[height=2cm]{T2.pdf}\\
&&\\
\hline
&&\\
\raisebox{30pt}{$\mathbb{Z}_2$}& \includegraphics[height=2cm]{T3.pdf} & \includegraphics[height=2.5cm]{T4.pdf}\\
&&\\
\hline
$\mathbbm{1}$& $M\setminus\lbrace 14\mbox{ points }\rbrace$& $M\setminus\lbrace 14\mbox{ points }\rbrace$\\
\hline
\end{tabular}
\caption{\small Orbit-type strata of $(M,\mathbbm{T})$ where $M=S^2$.}\label{ex: stratum tetrahedron}
\end{center}
\end{figure}
\end{enumerate}
\end{example}
In general, using the strict partial order $\prec_{conj}$ on the conjugacy classes of subgroups of $G$ is not enough to guarantee that we have a good stratification. For instance, in Example \ref{ex: stratum} \ref{ex: stratum T}, we have $(\mathbb{T})\prec_{conj}(\mathbb{Z}_2)$ but there are no strata with orbit-type $(\mathbb{T})$. However we have the following lemma:
\begin{lemma}\label{lemma conj}
If $\alpha=((K),(a))$ and $\beta=((H),(b))$ then $$\alpha\prec\beta\quad\Longrightarrow\quad (K)\prec_{conj} (H).$$
\end{lemma}
\begin{Proof}
By definition $\alpha\prec\beta$ implies that there exists some $x\in M_{\alpha}\cap\overline{M}_{\beta}$. In particular $x\in M_{\alpha}$ and then $G_x\in (K)$. By the Tube Theorem \ref{Tube theorem}, there is a $G$-invariant open neighbourhood $U\subset M$ of $x$, locally modelled by a fixed associated bundle $G\times_{G_x}N_0$, in which $x$ reads $[(e,0)]$.
By definition of the adherence, there is a sequence $(x_n)_{n\in\mathbb{N}}\subset M_{\beta}$ converging to $x$ in $M$, with stabilizers $G_{x_n}\in (H)$. For $n$ big enough, $x_n\in U$ and it can thus be identified with some point $[(g_n,\nu_n)]\in G\times_{G_x}N_0$. The stabilizer of $[(g_n,\nu_n)]$ is $$G_{[(g_n,\nu_n)]}=g_n(G_x)_{{\nu}_n}g_n^{-1}$$ and is thus conjugate to a proper subgroup of $G_x$, because by assumption $M_{\alpha}$ and $M_{\beta}$ are disjoint. Since $G_x\in (K)$ and $G_{x_n}\in (H)$, it follows that $(K)\prec_{conj}(H)$.
\end{Proof}
\subsection*{Stratification of a convex polytope by open faces}\label{stratification polytope}
There is a natural stratification of a convex polytope into vertices, edges and higher dimensional faces. Let $\Delta\subset (\mathbb{R}^n)^*$ be a $n$-dimensional convex polytope. Let $X_1,\dots,X_d$ in $\mathbb{R}^n$ be the outward-pointing normal vectors to the facets. Then there exists real numbers $\lambda_1,\dots,\lambda_d$ such that $\Delta$ reads
\begin{equation*}
\Delta=\bigcap_{i=1}^d\left\lbrace \mu\in (\mathbb{R}^n)^*\mid \langle \mu,X_i\rangle\leq\lambda_i\right\rbrace.
\end{equation*}
Let $\mathcal{B}$ be the set of subsets (possibly empty) $\beta\subset\lbrace 1,\dots,d\rbrace.$ For each $\beta\in\mathcal{B}$ we consider the intersection
\begin{equation*}
F_{\beta}=\bigcap_{i\in \beta}\left\lbrace \mu\in\Delta\mid \langle \mu,X_i\rangle=\lambda_i\right\rbrace.
\end{equation*}
If $F_{\beta}\neq\varnothing$, its relative interior $\overset{\circ}{F}_{\beta}$ is called a $l$-dimensional open face of $\Delta$ where $l$ is equal to $n$ minus the cardinality of $\beta$. We equip $\mathcal{B}$ with the strict partial order
\begin{equation}\label{partial order polytope}
\alpha\prec\beta\quad\Longleftrightarrow\quad \alpha\neq\beta\quad\mbox{and}\quad \overset{\circ}{F}_{\alpha}\cap \overline{\overset{\circ}{F}_{\beta}}\neq\varnothing.
\end{equation}
With this strict partial order, the collection $\lbrace \overset{\circ}{F}_{\beta}\mid \beta\in\mathcal{B}\rbrace$ forms a $\mathcal{B}$-stratification of $\Delta$. A strict partial order is defined on the set of faces by
\begin{equation*}
\overset{\circ}{F}_{\alpha}< \overset{\circ}{F}_{\beta}\quad \Longleftrightarrow\quad\alpha\prec\beta.
\end{equation*}
Finally note that, if $\alpha\prec \beta$ then $\beta\subset \alpha$.
\section{G-tubular covers}\label{s:G-categorical tubular covers}
If $(M,G)$ is a proper $G$-manifold, the Tube Theorem \ref{Tube theorem} allows us to produce $G$-categorical open subsets in the following way: any $m\in M$ admits a $G$-invariant neighbourhood $U\subset M$ such that the map $\varphi:Y_0\to U$ defined in \eqref{tau tube} is a $G$-equivariant diffeomorphism. Here $$Y_0=G\times_{G_m}N_0$$ where $N_0$ is a fixed neighbourhood of zero in some subspace $N\subset T_mM$, complementary to $\mathfrak{g}\cdot m$ in $T_mM$, on which $G_m$ acts linearly. The proper $G$-manifold $Y_0$ is a local model for $U$, in which $m$ reads $\varphi^{-1}(m)=[e,0]$. The $G$-homotopy $F:Y_0\times [0,1]\to Y_0$ defined by
\begin{equation*}
F([(g,\nu)],t)=[(g,(1-t)\nu)].
\end{equation*}
is a $G$-deformation retract of $Y_0$ onto the orbit $G\cdot [e,0]$. By using the fact that $\varphi$ is a $G$-equivariant diffeomorphism, the open subset $U=\varphi(Y_0)$ is $G$-categorical since the $G$-homotopy $H:U\times [0,1]\to U$ given by
\begin{equation}\label{Gtubular homotopy}
H(p,t)=\varphi\left(F(\varphi^{-1}(p),t)\right)
\end{equation}
is a $G$-deformation retract of $U$ onto $G\cdot m$.
\begin{definition}\label{def:G tubular open subset}
\normalfont A $G$-categorical open subset $U\subset M$ as above, with associated $G$-deformation retract as in \eqref{Gtubular homotopy}, is called a \defn{$G$-tubular open subset} of $M$. A cover of $M$ made of $G$-tubular open subsets is called a \defn{$G$-tubular cover} of $M$.
\end{definition}
\subsection*{Minimal G-tubular covers}
Clearly, every $m\in M$ admits a neighbourhood which is a $G$-tubular open subset of $M$. Consequently, $G$-tubular covers of $M$ always exist. The question is whether they can be refined. Let $\mathcal{U}$ be any $G$-tubular cover of $M$. We know that $M$ can be decomposed into the disjoint union of its orbit-type strata $\lbrace M_{\beta}\mid \beta\in\mathcal{B}\rbrace$, which form themselves a smooth stratification of $M$. Let $\mathcal{B}'\subset\mathcal{B}$ be the biggest subset of indices $\beta\in\mathcal{B}$ such that $M_{\beta}$ is minimal with respect to \eqref{partial order strata}. Consider the disjoint union $M'$ of all the strata $M_{\beta}$ with $\beta\in\mathcal{B}'$. From $\mathcal{U}$ we extract a subcover $\mathcal{U}'$, chosen as small as possible such that $\mathcal{U}'$ covers $M'$. In particular $\mathcal{U}'$ is a refinement of $\mathcal{U}$. We ask the following:
\vspace{0.5cm}
\hypertarget{label}{(Q)} \textit{Does it exist a subset $\mathcal{U}'\subset \mathcal{U}$, obtained as above, which covers $M$?}
\vspace{0.5cm}
The answer is in general negative (cf. Example \ref{non-example}). However it is positive for all the proper $G$-manifolds listed in Example \ref{ex:TETRAHEDRON}, where $\mathcal{U}'$ is constructed explicitly.
\begin{definition}\label{minimal tubular cover}
\normalfont Let $(M,G)$ be a proper $G$-manifold. $\mathcal{U}'$ obtained as above is called a \defn{minimal $G$-tubular cover} if the following holds:
\begin{enumerate}[label=(\roman*)]
\item $\mathcal{U}'$ is a cover of $M$.\label{minimal i}
\item For each minimal orbit-type stratum $M_{\beta}$, the set $$\mathcal{V}'_{\beta}=\lbrace V_{\beta}=U\cap M_{\beta}\mid U\in\mathcal{U}'\rbrace$$ is the smallest cover by $G$-categorical open subsets of $M_{\beta}$, where the topology of $M_{\beta}$ is the subset topology.\label{minimal ii}
\end{enumerate}
\end{definition}
We discuss the simplest example where such a cover exists. Let $S^2\subset\mathbb{R}^3$, on which $S^1$ acts by rotations about the $z$-axis. This action has two minimal orbit-type strata, namely the North and South pole. Two small disks centered at those points are $S^1$-tubular open subsets and can be taken sufficiently big so that they form a minimal $S^1$-tubular cover of $S^2$. In this example, a disk centered at the North pole can be extended until its closure meets the South pole. The impossibility to extend it further relies on the fact that such neighbourhoods are constructed by mean of the Riemannian exponential map. This map is no longer injective if the disk contains two opposite points on the sphere. The next proposition gives another answer to this fact by using the properties of $G$-tubular open subsets.
\begin{proposition}\label{atmost}
Let $(M,G)$ be a proper $G$-manifold. If $U\subset M$ is a $G$-tubular open subset which intersects a minimal orbit-type stratum $M_{\beta}$, then $U$ retracts onto the orbit $G\cdot x$ of some $x\in M_{\beta}$. In particular $G$-tubular open subsets intersect at most one minimal orbit-type stratum.
\end{proposition}
\begin{Proof}
Let $\beta=((H),(b))\in\mathcal{B}$ such that $M_{\beta}$ is a minimal orbit-type stratum. Let $U\subset M$ be a $G$-tubular open subset of $M$ such that $U\cap M_{\beta}\neq\varnothing$, and let $H:U\times [0,1]\to U$ be a $G$-deformation retract of $U$ onto $G\cdot x$ for some $x\in M$. By contradiction, assume that $x\in M_{\alpha}$ for some $\alpha=((G_x),(a))\neq\beta$.
Each point $y\in U\cap M_{\beta}$ has stabilizer $G_y\in (H)$. By $G$-equivariance of the homotopy, $G_y$ is a subgroup of $G_{H_1(y)}$ which is itself conjugate to $G_x$, as $H_1(y)$ and $x$ lie on the same orbit. In particular $(G_x)\prec_{conj}(H)$. Two cases occur:
\begin{enumerate}[label=(\roman*)]
\item If $M_{\beta}\cap\overline{M}_{\alpha}\neq\varnothing$, then $\beta\prec\alpha$ since $\beta\neq\alpha$. By Lemma \ref{lemma conj} we get $(H)\prec_{conj}(G_x)$ which is a contradiction.
\item If $M_{\beta}\cap\overline{M}_{\alpha}=\varnothing$ we must use the assumption that $U$ is $G$-tubular. Let $G\times_{G_x}N_0$ be the local model for $U$. Given $y\in M_{\beta}$ we define the $G$-equivariant path $y(t)=H_t(y)$, where $t\in [0,1]$. In the local model, $y$ reads $[g,\nu]$ and $y(t)$ reads $[g,\nu_t]$ where $\nu_t=(1-t)\nu$. We can assume without lost of generality that $(G_x)_{\nu}=H$. Observe that, by linearity of the $G_x$-action on $N_0$, we have $(G_x)_{\nu_t}=(G_x)_{\nu}=H$ for all $t\neq 1$. Hence $G_{[g,\nu_t]}=g(G_x)_{\nu_t}g^{-1}=gHg^{-1}$ for every $t\neq 1.$ In particular, $y(t)\in M_{\beta}$ for all $t\neq 1$.
Since the path $y(t)$ starts at $y\in M_{\beta}$ and ends on $G\cdot x\subset M_{\alpha}$, there is some $t_0\in [0,1]$ such that $y(t_0)\in \overline{M_{\alpha}}$. The parameter $t_0$ is chosen the smallest such that this occurs. If $t_0\neq 1$, the previous argument shows that $y(t_0)\in M_{\beta}\cap\overline{M_{\alpha}}$, which is a contradiction. Otherwise, since $y(t)\in M_{\beta}$ for all $t<t_0$, there is a sequence $(y_n)_{n\in\mathbb{N}}\subset M_{\beta}$ which converges to $y(t_0)$. By closedness of $M_{\beta}$, this yields $y(t_0)\in M_{\beta}\cap \overline{M_{\alpha}}$, which is again a contradiction. We conclude that $x\in M_{\beta}$.
\end{enumerate}
\end{Proof}
The answer to question \hyperlink{label}{(Q)} is in general negative. In the example below, $(M,G)$ is a proper $G$-manifold with $M$ compact, and the action admits only one minimal orbit-type stratum $M_{\beta}$ which is a single $G$-orbit.
\begin{example}\label{non-example}
\normalfont
Think of $M=S^3$ as the set of unit vectors $(z_1,z_2)\in \mathbb{C}^2$ equipped with the $S^1$-action
\begin{equation*}
\theta\cdot (z_1,z_2)=(e^{i\theta}z_1,e^{2i\theta}z_2).
\end{equation*}
This action has only one minimal orbit-type stratum $M_{\beta}$ with $\beta=((\mathbb{Z}_2),(b))$ for some index $b\in \mathcal{B}_{\mathbb{Z}_2}$. Explicitly
\begin{equation*}
M_{\beta}=\left\lbrace (0,z_2)\in \mathbb{C}^2\mid |z_2|^2=1\right\rbrace
\end{equation*}
which is diffeomorphic to a circle. In particular $M_{\beta}$ is the $S^1$-orbit of the point $(0,1)\in S^3$. The $S^1$-invariant open subset $$U=\left\lbrace (z_1,z_2)\in S^3\mid |z_1|^2<\frac{2}{3}\right\rbrace$$
is an $S^1$-invariant tubular neighbourhood of the minimal orbit-type stratum, and is diffeomorphic to a solid torus (cf. Figure \ref{hopf}). Since $M_{\beta}$ is an $S^1$-orbit, $U$ is an $S^1$-tubular open subset. We may choose $\mathcal{U'}=\lbrace U\rbrace$. This cover satisfies \ref{minimal ii} of Definition \ref{minimal tubular cover} but it does not satisfy \ref{minimal i}, since it does not cover $M$. To cover $M$ we require the additional open subset
$$V=\left\lbrace (z_1,z_2)\in S^3\mid |z_1|^2>\frac{1}{3}\right\rbrace$$ which is also a solid torus, understood as an $S^1$-invariant tubular neighbourhood of the $S^1$-orbit of $(1,0)$ (cf. Figure \ref{hopf}). It is therefore $S^1$-categorical and then $\mbox{Cat}_{S^1}\left(M\right)\leq 2$. There is in fact equality because otherwise it would mean that $S^3$ is contractible onto a circle, which is untrue.
\vspace{0.5cm}
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{9cm}
\centering
\includegraphics[height=6cm]{torushopf.pdf}
\caption{\small Representation in $\mathbb{R}^3$ of the sphere $S^3$ with a point removed. The stratum $M_{\beta}$ is a circle closing at infinity and the tori around it form a solid torus, which is a tubular neighbourhood.}\label{hopf}
\end{minipage}
\end{figure}
\end{example}
\section{Localization Formula}\label{s: loc formula}
\iffalse
\begin{definition}
\normalfont Let $(X,G)$ be a $G$-space and let $A,B\subset X$ be $G$-invariant. A $G$-deformation retract $H:A\times [0,1]\to X$ of $A$ onto $B$ has the \defn{$G$-homotopy extension property} if it can be extended into a $G$-deformation retract $\widetilde{H}:X\times [0,1]\to X$ of $X$ onto $B$ such that $\restr{\widetilde{H}_t}{A}=H_t$ for all $t\in [0,1]$.
\end{definition}
\fi
In this section we obtain a localization formula (cf. Corollary \ref{localization thm}) for proper $G$-manifolds which admit a minimal $G$-tubular cover. This formula says in particular that the equivariant LS-category of a proper $G$-manifold is intrinsic to the equivariant LS-category of its minimal orbit-type strata. The theorem below holds in general, without any assumption on the proper $G$-manifold.
\begin{theorem}\label{localization thm first}
Let $(M,G)$ be a proper $G$-manifold and write $M$ as the disjoint union of its orbit-type strata $\lbrace M_{\beta}\mid \beta\in\mathcal{B}\rbrace$. Let $\mathcal{B}'$ be the biggest subset of $\mathcal{B}$ such that $M_{\beta}$ is minimal for every $\beta\in\mathcal{B}'$. Then $$\mbox{Cat}_G(M)\geq\sum_{\beta\in\mathcal{B}'} \mbox{Cat}_{G}(M_{\beta}).$$
\end{theorem}
\begin{Proof}
Let $\mathcal{U}$ be a $G$-tubular cover of $M$. Choose $U\in\mathcal{U}$ such that $U\cap M_{\beta}\neq \varnothing$ for some $\beta\in \mathcal{B}'$, say $\beta=((H),(b))$. By Proposition \ref{atmost}, $U$ does not intersect any other minimal stratum and the $G$-deformation retract $H:U\times [0,1]\to U$ retracts onto an orbit $G\cdot x$ of some $x\in M_{\beta}$. The set $V_{\beta}=U\cap M_{\beta}$ is open in $M_{\beta}$ for the subset topology, and it is $G$-invariant because so are $U$ and $M_{\beta}$.
Let $G\times_{G_x}N_0$ be the local model for $U$. Given $y\in V_{\beta}$ we define the $G$-equivariant path $y(t)=H_t(y)$, where $t\in [0,1]$. In the local model, $y$ reads $[g,\nu]$, and $y(t)$ reads $[g,\nu_t]$ where $\nu_t=(1-t)\nu$. Since $(G_x)_{\nu}\in (H)$, we use the linearity of the $G_x$-action on $N_0$, to obtain $(G_x)_{\nu_t}=(G_x)_{\nu}=(H)$ for all $t\in [0,1]$. Hence $$G_{[g,\nu_t]}=g(G_x)_{\nu_t}g^{-1}\in (H)\quad\mbox{for all}\quad t\in [0,1].$$ In particular, $y(t)\in M_{\beta}$ for all $t\in [0,1]$. Because $y\in V_{\beta}$ is arbitrary and $[0,1]$ is compact, the map $F:V_{\beta}\times [0,1]\to V_{\beta}$ given by $F_t(y)=y(t)$ is a homotopy. It is clearly $G$-equivariant by construction and defines a $G$-deformation retract of $V_{\beta}$ onto $G\cdot x$. It follows that $V_{\beta}$ is $G$-categorical.
Let $\mathcal{U}_{\beta}\subset\mathcal{U}$ be the subset of all $U\in\mathcal{U}$ such that $U\cap M_{\beta}\neq\varnothing$. Then $$\mathcal{V}_{\beta}=\lbrace V_{\beta}=U\cap M_{\beta}\mid U\in\mathcal{U}_{\beta}\rbrace$$ is a cover of $M_{\beta}$ by $G$-categorical open subsets, which is not necessarily a minimal cover. This procedure associates to each $\beta\in\mathcal{B}'$ a cover $\mathcal{V}_{\beta}$ of $M_{\beta}$.
Proposition \ref{atmost} says that, if $\alpha,\beta\in\mathcal{B}'$ are distinct, then $\mathcal{U}_{\alpha}\cap\mathcal{U}_{\beta}=\varnothing$. In particular, each $V_{\beta}\in\mathcal{V}_{\beta}$ is determined by a unique $U\in\mathcal{U}_{\beta}$.
Therefore $$\mbox{Cat}_G(M)\geq \sum_{\beta\in\mathcal{B}'} \mbox{Cat}_{G}(M_{\beta}).$$
\end{Proof}
Theorem \ref{localization thm first} had already been obtained by\cit{Hurder and T\"oben} (cf. \cite{MR2492297} Theorem $3.7$). However if $(M,G)$ admits a minimal $G$-tubular cover, the following occurs:
\begin{cor}[Localization Formula]\label{localization thm}
Let $(M,G)$ be a proper $G$-manifold which admits a minimal $G$-tubular cover. Decompose $M$ into its orbit-type strata $\lbrace M_{\beta}\mid\beta\in\mathcal{B}\rbrace$. Let $\mathcal{B}'$ be the biggest subset of $\mathcal{B}$ such that $M_{\beta}$ is minimal for every $\beta\in\mathcal{B}'$. Then $$\mbox{Cat}_G(M)=\sum_{\beta\in\mathcal{B}'} \mbox{Cat}_{G}(M_{\beta}).$$
\end{cor}
\begin{Proof}
By Theorem \ref{localization thm first}, $\mbox{Cat}_G(M)\geq \sum_{\beta\in\mathcal{B}'} \mbox{Cat}_{G}(M_{\beta}).$ The other inequality is a direct consequence of the properties of a minimal $G$-tubular cover (cf. Definition \ref{minimal tubular cover}).
\end{Proof}
\begin{proposition}\label{cat norm}
Let $(M,G)$ be a proper $G$-manifold which admits a minimal $G$-tubular cover. Assume $M_{\beta}$ is a minimal orbit-type stratum with $\beta=((H),(b))$. Then $$\mbox{Cat}_G\left(M_{\beta}\right)=\mbox{Cat}_{N_G(H)}\left(M_{H,b}\right).$$
\end{proposition}
\begin{Proof}
Let $\mathcal{U}'$ be a minimal $G$-tubular cover of $M$ and let $M_{\beta}$ be a minimal orbit-type stratum. By definition of $\mathcal{U}'$, the set $\mathcal{V}_{\beta}=\lbrace V_{\beta}=U\cap M_{\beta}\mid U\in\mathcal{U}'\rbrace$ is the smallest cover by $G$-categorical open subsets of $M_{\beta}$, where the topology of $M_{\beta}$ is the subset topology.
For every $V_{\beta}\in\mathcal{V}_{\beta}$, let $\widetilde{V}_{\beta}=V_{\beta}\cap M_{H,b}$. Then $\widetilde{V}_{\beta}$ is an $N_G(H)$-invariant open subset of $M_{H,b}$, for the subset topology. Let $H:V_{\beta}\times [0,1]\to V_{\beta}$ be a $G$-deformation retract of $V_{\beta}$ onto some orbit $G\cdot x$ of $x\in M_{\beta}$. Then the $N_G(H)$-homotopy $F:\widetilde{V}_{\beta}\times [0,1]\to \widetilde{V}_{\beta}$ defined by
$$F_t=\restr{H_t}{\widetilde{V}_{\beta}}\quad\mbox{for each}\quad t\in [0,1]$$
is an $N_G(H)$-deformation retract of $\widetilde{V}_{\beta}$ onto the orbit $N_G(H)\cdot x$. Therefore the set $\widetilde{\mathcal{V}}_{\beta}=\lbrace \widetilde{V}_{\beta}=V_{\beta}\cap M_{H,b}\mid V_{\beta}\in\mathcal{V}_{\beta}\rbrace$ is a cover of $M_{H,b}$ made of $N_G(H)$-categorical open subsets. This cover is minimal by assumption and because $M_{\beta}=G\cdot M_{H,b}$. We thus get $$\mbox{Cat}_G\left(M_{\beta}\right)=\mbox{Cat}_{N_G(H)}\left(M_{H,b}\right).$$
\end{Proof}
The reader is invited to compare the above result with \cite{marzantowicz} (Proposition $2.1$).
\begin{example}
\normalfont
\begin{enumerate}[label=(\roman*)]
We verify Theorem \ref{localization thm} on the examples discussed above.
\item Let $M=S^2\subset \mathbb{R}^3$ on which $S^1$ acts by rotations about the $z$-axis. The minimal strata have orbit-type $(H)$ where $H=S^1$, namely
\begin{equation*}
M_{\beta_b}=\left\lbrace (0,0,(-1)^{b-1})\right\rbrace,
\quad \beta_b=((S^1),(b))\quad\mbox{and}\quad b=1,2.
\end{equation*}
Then
\begin{equation*}
\mbox{Cat}_{S^1}\left(M_{\beta_1}\right)+\mbox{Cat}_{S^1}\left(M_{\beta_2}\right)=1+1=\mbox{Cat}_{S^1}\left(M\right).
\end{equation*}
\item Let $M=\mathbb{C} P^2$ equipped with the action of $S^1$
$$\theta\cdot [z_0:z_1:z_2]=[e^{i\theta}z_0:z_1:z_2].$$ The minimal strata $M_{\beta_1}$ and $M_{\beta_2}$ have orbit-type $(H)=(S^1)$. There is a $\mathbb{C} P^1$ and the single point $[1:0:0]$, respectively. Therefore
$$\mbox{Cat}_{S^1}\left(M_{\beta_1}\right)+\mbox{Cat}_{S^1}\left(M_{\beta_2}\right)=2+1=\mbox{Cat}_{S^1}\left(M\right).$$
\item Let $M=S^2$ acted on by the group $\mathbb{T}$ as in Example \ref{ex:TETRAHEDRON} \ref{tetrahedron}. There are three minimal orbit-type strata. Two of them, $M_{\beta_1}$ and $M_{\beta_2}$, have orbit-type $(\mathbb{Z}_3)$. The last minimal stratum $M_{\alpha}$ has orbit-type $(\mathbb{Z}_2)$. We find
\begin{equation*}
\mbox{Cat}_{\mathbb{Z}_3}\left(M_{\beta_1}\right)+\mbox{Cat}_{\mathbb{Z}_3}\left(M_{\beta_2}\right)+\mbox{Cat}_{\mathbb{Z}_2}\left(M_{\alpha}\right)=1+1+1=\mbox{Cat}_{\mathbb{T}}\left(M\right)
\end{equation*}
\end{enumerate}
\end{example}
\section{Tubular covers of symplectic toric manifolds}\label{s: tubular toric}
In this section we show that symplectic toric manifolds admit a minimal tubular cover. Such a cover is constructed explicitly in Theorem \ref{thm toric}. Let $\mbox{T}$ be torus with Lie algebra $\mathfrak{t}$ and dual Lie algebra $\mathfrak{t}^*$. The smooth action of $\mbox{T}$ on a symplectic manifold $(M,\omega)$ is \defn{Hamiltonian} if there exists a momentum map $\Phi_{\mbox{\tiny T}}:M\to \mathfrak{t}^*$. The quadruple $(M,\omega,\mbox{T}, \Phi_{\mbox{\tiny T}})$ is called a \defn{Hamiltonian $\mbox{T}$-manifold}.
\begin{definition}
\normalfont Let $\mbox{T}$ be an $n$-dimensional torus. A Hamiltonian $\mbox{T}$-manifold $(M,\omega,\mbox{T},\Phi_{\mbox{\tiny T}})$ is called a \defn{symplectic toric manifold} if $(M,\omega)$ is a $2n$-dimensional compact connected symplectic manifold and the Hamiltonian action of $\mbox{T}$ on $M$ is effective.
\end{definition}
For symplectic toric manifolds, the image $\Phi_{\mbox{\tiny T}}(M)$ of the momentum map is a \defn{Delzant polytope} i.e. a convex polytope $\Delta\subset (\mathbb{R}^n)^*$ which is \defn{simple} i.e. each vertex $x$ meets exactly $n$ edges, \defn{rational} i.e. the edges meeting at a vertex $x$ are of the form $x+t\alpha_{x,i}$ where $\alpha_{x,i}\in (\mathbb{Z}^n)^*$, \defn{smooth} i.e. for each vertex $x$ the isotropy weights $\alpha_{x,1},\dots,\alpha_{x,n}$ form a $\mathbb{Z}$-basis of $(\mathbb{Z}^n)^*$.
This observation is due to \cit{Delzant} (cf. \cite{delzant} Lemmas $2.2$ and $2.4$). Delzant also proved that $\Delta$ determines entirely the symplectic toric manifold $(M,\omega,\mbox{T},\Phi_{\mbox{\tiny T}})$, up to $\mbox{T}$-equivariant symplectomorphisms (cf. \cite{delzant} Theorem $2.1$). His proof relies on a well-known result of convexity obtained independently by\cit{Atiyah} \cite{MR642416} and\cit{Guillemin and Sternberg} \cite{MR664117}, which states that the image of a momentum map for the action of a torus (not necessarily effective) on a compact connected symplectic manifold is a convex polytope.
We recall some standard facts about Morse theory applied to a symplectic toric manifold $(M,\omega,\mbox{T},\Phi_{\mbox{\tiny T}})$. The reader is referred to the book of\cit{Guillemin and Sjamaar} (cf. \cite{MR2252111} Section $3.6$) for details. Let $M_{\mbox{\tiny T}}$ be the fixed point set of $\mbox{T}$. For every $m\in M_{\mbox{\tiny T}}$, the torus acts on the tangent space at $m$. There is a $\mbox{T}$-invariant complex structure on $M$ such that $T_mM$ is a complex $\mbox{T}$-representation
with weight space decomposition $$\mathbb{C}_{\alpha_{m,1}}\oplus \dots\oplus \mathbb{C}_{\alpha_{m,n}}$$ where $\alpha_{m,1},\dots,\alpha_{m,n}\in\mathfrak{t}^*$ are the weights of the representation. A \defn{generic component} of the momentum map $\Phi_{\mbox{\tiny T}}:M\to\mathfrak{t}^*$ is a component $\phi^{\xi}=\langle \Phi_{\mbox{\tiny T}}(\cdot),\xi\rangle$ where $\xi\in\mathfrak{g}$ is generic i.e. $\alpha_{m,i}(\xi)\neq 0$ for every $m\in M_{\mbox{\tiny T}}$ and $i=1,\dots,n$. In this case, the critical points of $\phi^{\xi}$ are isolated and $\phi^{\xi}$ is a Morse function whose critical set is precisely $M_{\mbox{\tiny T}}$. Moreover every critical point of $\phi^{\xi}$ has even index. Therefore symplectic toric manifolds possess an extra structure given by the properties of the $\mbox{T}$-action. This structure is used to construct a minimal $\mbox{T}$-tubular cover of $M$.
\begin{theorem}\label{thm toric}
Let $(M,\omega,T,\Phi_{\mbox{\tiny T}})$ be a symplectic toric manifold. Then $M$ admits a minimal $\mbox{T}$-tubular cover.
\end{theorem}
\begin{Proof}
Let $\lbrace M_{\beta}\mid \beta\in\mathcal{B}\rbrace$ be the $\mathcal{B}$-stratification of $M$ into orbit-type strata, with strict partial order \eqref{partial order orbit}. Since $\mbox{T}$ is compact, there are only finitely many minimal orbit-type strata $M_{\beta_1},\dots,M_{\beta_{\ell}}$. By assumption on the $\mbox{T}$-action, each $M_{\beta_i}$ is an isolated fixed point $m_i\in M_{\mbox{\tiny T}}$. Then there is $\xi_i\in\mathfrak{t}$ such that $\phi^{\xi_i}$ is a generic component of the momentum map, which takes its minimum at $m_i$. Let $-\nabla\phi^{\xi_i}$ be the gradient vector field associated to this component, with corresponding flow $\varphi_t$. Since the image of the momentum map $\Phi_{\mbox{\tiny T}}(M)$ is a Delzant polytope $\Delta$, the $\mathcal{B'}$-stratification $\lbrace \overset{\circ}{F}_{\beta'}\mid \beta'\in\mathcal{B}'\rbrace$ of $\Delta$ by open faces (cf. Section \ref{stratification polytope}) coincides with the $\mathcal{B}$-stratification by orbit-type of $M$. In other words, for every $i=1,\dots,\ell$, we can associate to $\beta_i\in\mathcal{B}$ a unique index $\beta'_i\in\mathcal{B}'$ such that $\Phi_{\mbox{\tiny T}}(M_{\beta_i})$ is precisely the zero-dimensional face $\overset{\circ}{F}_{\beta'_i}$. For each other index $\alpha\in\mathcal{B}$ there is a unique $\alpha'\in\mathcal{B}'$ such that $\Phi_{\mbox{\tiny T}}(M_{\alpha})=\overset{\circ}{F}_{\alpha'}$. Define an open subset $V_{\beta'_i}\subset \mathfrak{t}^*$ by
\begin{equation*}
V_{\beta'_i}=\bigcup_{\beta'_i\preceq\alpha'}\overset{\circ}{F}_{\alpha'}.
\end{equation*}
By continuity and $\mbox{T}$-invariance of $\Phi_{\mbox{\tiny T}}$, the subset $U_{\beta_i}=\Phi_{\mbox{\tiny T}}^{-1}(V_{\beta'_i})$ is a $\mbox{T}$-invariant open neighbourhood of $m_i$ in $M$. It reads
\begin{equation*}
U_{\beta_i}=\bigcup_{\beta_i\preceq\alpha}M_{\alpha}.
\end{equation*}
For every $m\in U_{\beta_i}\setminus \lbrace m_i\rbrace$, the flow line $\varphi_t(m)$ is defined for every $t\in\mathbb{R}$, by compacity of $M$. By construction of $U_{\beta_i}$, the point $m$ belongs to some orbit-type stratum $M_{\alpha}$ with $\beta_i\prec\alpha$. Since $\varphi_t$ is stratum-preserving, $\varphi_t(m)\in U_{\beta_i}$ for every $t\in\mathbb{R}$. Moreover the only critical point of $\phi^{\xi_i}$ in $U_{\beta_i}$ is $m_i$, and it is a minimum. Hence $\varphi_t(m)$ tends to $m_i$ as $t$ tends to infinity. Therefore the continuous map
\begin{eqnarray*}
f_m:[0,1[ & \longrightarrow & U_{\beta_i}\\
t & \longmapsto & \varphi_{\frac{t}{1-t}}(m)
\end{eqnarray*}
extends by continuity into a map $\widetilde{f}_m:[0,1]\to U_{\beta_i}$ with $\widetilde{f}_m(1)=m_i$. Then the map
\begin{eqnarray*}
H:U_{\beta_i}\times [0,1] & \longrightarrow & U_{\beta_i}\\
(m,t) & \longmapsto & \widetilde{f}_m(t)
\end{eqnarray*}
is a $\mbox{T}$-deformation retract of $U_{\beta_i}$ onto the orbit $\mbox{T}\cdot m_i=m_i$. In particular $U_{\beta_i}$ is $\mbox{T}$-categorical for every $i=1,\dots,\ell$. It is clear that $\mathcal{U}=\lbrace U_{\beta_i}\rbrace_{i=1}^{\ell}$ is a cover of $M$ made of $\mbox{T}$-tubular open subsets, which are themselves tubular neighbourhoods of the closed strata. By Proposition \ref{atmost}, this cover is the smallest that we can take and then $\mathcal{U}$ is minimal.
\end{Proof}
The result below, due to \cit{Bayeh and Sarkar} (cf. \cite{MR3422738} Theorem $5.1$), is then a direct consequence of the Localization Formula.
\begin{cor}[\cite{MR3422738} Theorem $5.1$]\label{corollary toric}
Let $(M,\omega,T,\Phi_{\mbox{\tiny T}})$ be a symplectic toric manifold. Then $\mbox{Cat}_{\mbox{\tiny T}}(M)$ coincides with the cardinality of $M_{\mbox{\tiny T}}$.
\end{cor}
Our choice to consider symplectic toric manifolds makes the proof of Theorem \ref{thm toric} relatively straightforward for two reasons. The first reason is that the fixed points of the $\mbox{T}$-action are isolated, and the second reason is that the stratification by orbit-type strata of $M$ coincides with the stratification by open faces of the polytope.
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Is it noticeably faster to filter by an integer compared to a string in ElasticSearch? I have 3 item archetypes in my app, Weapon, Armour, Misc.
A user can only search in one category at a time for now. So, I was wondering if I should index the archetype as an integer (0, 1, 2), or as a string "weapon", "armour", "misc"?
I'm using a boolean query. Will it be faster to filter by an integer, a string, or is there almost no difference?
A: No difference.
Ensure it's actually a filter you use. Then it'll be cached as a bitmap, and just as fast on later uses.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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The Pride caucus is open to all union members who identify as LGBTQIA+. This includes, but is not limited to, people who identify as lesbian, gay, bisexual, pansexual, transgender, queer, genderqueer, gender nonconforming, agender, genderfluid, intersex, Two Spirit, asexual, aromantic, and demiromantic. We seek to pursue changes in institutional policy that will safeguard the rights and affirm the identities of people who identify as part of the LGBTQIA+ community at Oregon State University, regardless of whether they are graduate students, employees, or union members. In addition to general members, the Pride Caucus will consist of at least one officer from the Executive Council to serve as a general note-taker and point-person for budgetary considerations. All union members are encouraged to attend and observe meetings; however, we aim to amplify the
voices of those with lived experiences pertaining to our policy goals.
Contact the Pride Caucus by emailing pride@cge6069.org
The Pride Caucus will meet twice a term in Weeks 3 & 7 at Westminster House (101 NW 23rd St.) Check the CGE Calendar for specific dates/times.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}\label{sec:Introduction}
The Standard Model (SM) of particle physics has, so far, been recognized as the most successful model in theoretical physics, proving its validity over a wide range of subatomic phenomena and energy scales \cite{ATLAS:2012yve,D0:1995jca,CDF:1995wbb,GargamelleNeutrino:1973jyy,UA1:1983crd,UA1:1983mne,UA2:1983mlz}. However, despite its immense success, there is an experimental evidence that indicates certain missing features for which the SM can not provide a satisfactory explanation. Among these are the lack of a dynamical mechanism behind neutrino mass generation and the absence of a plausible dark matter (DM) candidate, as well as the emergence of slowly but steadily growing anomalies such as those of the magnetic moment of the muon \cite{Bennett_2006,PhysRevLett.126.141801}, the $\mathrm{R_{D,D^*}}$ \cite{LHCb:2017smo,PhysRevLett.118.211801} and the $\mathrm{R_{K,K^*}}$ flavour observables \cite{BELLE:2019xld,LHCb:2021trn}. Regarding the muon $g-2$ and B-physics anomalies, one should stress that the corresponding deviations for the SM predictions do not pass the $5\sigma$ threshold for discovery, and thus are only considered as an indication for possible New Physics (NP) effects.
To address these shortcomings one typically extends the SM framework by introducing new beyond-the-SM (BSM) fields and new interactions. There are three potential avenues for one to follow in this case. One possibility relies in a phenomenology driven approach, and either simply add new BSM fields that address a given problem \cite{PhysRevLett.127.061802,Dorsner:2019itg}, or work under the effective field theory (EFT) formalism attempting to establish a bound on the NP energy scale by confronting the model predictions with the data \cite{Capdevila:2017bsm,Buttazzo:2017ixm}. One can also adopt a more fundamental avenue and build a high-energy scale framework from which the SM gauge group, its Yukawa sector and particle content are emergent at low energy-scale. In the latter approach, one typically resorts to the so-called Grand Unified Theories (GUTs), potentially providing a first-principles understanding of the origins of the specific SM structure. For such a class of models, one attempts to encapsulate the electroweak (EW) symmetry group, $\mathrm{SU(3)_C\times SU(2)_L \times U(1)_Y}$ into a larger gauge group, with the SM and NP fields being embedded into its representations. This strategy may result in various types of unification in the sectors of matter fields and forces such as gauge and/or Yukawa couplings unification, Higgs-lepton unification, quark-lepton unification etc, in some cases even yielding relatively light NP fields in the low-energy SM-like EFT spectrum. The latter may be observable at current or near-future collider experiments, offering a potential of probing the GUT-scale physics through either direct observation of those new states or studies of their impact on precision SM observables, such as the muon $g-2$ and flavour changing neutral current (FCNC) processes. In this context, some of the authors have previously constructed a novel class of GUT models based on the trinification gauge group supplemented with a horizontal family symmetry in both supersymmetric (SUSY) and non-SUSY versions \cite{Camargo-Molina:2016bwm,Camargo-Molina:2016yqm,Camargo-Molina:2017kxd,Morais:2020odg,Morais:2020ypd,CarcamoHernandez:2020owa}. In particular, it has been demonstrated that successful unification of the gauge couplings can be achieved in a SUSY formulation in consistency with proton decay constraints \cite{Morais:2020odg,Morais:2020ypd}. For this work, we shall focus on the latter scenario, which shares many common features with other GUT implementations and thus represents a good playground for collider searches.
The SHUT (\textbf{S}upersymmetric \textbf{H}iggs-\textbf{U}nified \textbf{T}rinification) model \cite{Morais:2020odg,Morais:2020ypd} predicts various types of NP at low energies. The extended fermion sectors include heavy left-handed and sterile right-handed neutrinos as well as vector-like fermions (VLFs), such as EW-singlet vector-like quarks (VLQs) and EW-doublet vector-like leptons (VLLs). The latter can potentially be at the reach of Run-III and the high-luminosity (HL) phases of the Large Hadron Collider (LHC) \cite{Freitas:2020ttd}. An extended scalar sector features a minimum of three Higgs doublets at low scales, required for consistency with the measured Cabibbo–Kobayashi–Mas\-kawa (CKM) quark mixing matrix.
In this article we continue the phenomenological exploration of SHUT inspired models, focusing on the collider signatures of a down-type VLQs, expected to emerge at a TeV energy scale. Three generations of VLQs are predicted within such a GUT framework with the first two generations being light enough to be potentially accessible at the LHC, with masses of $\mathcal{O}(1)~\mathrm{TeV}$, whereas the 3rd generation VLQ is expected to hold a mass of $\mathcal{O}(100)~\mathrm{TeV}$ i.e. it is too heavy to be probed in collider measurements in the near future. As a particular promising discovery channel, we consider multi-jet signatures with one charged lepton and missing energy, probing the couplings of the VLQ with light quarks. For this, we apply Deep Learning (DL) techniques to help separating signal from background classes. To this end, we rely on two types of data in the training phase, namely, we use the abstract jet images \cite{Alves:2019ppy} and the numerical data containing kinematic information from the final states. To train the Neural Network (NN) on the data, we construct a specific Hybrid Net, which is designed to make use of both data formats. Optimization of the NN hyper-parameters is performed by using an evolution algorithm, which optimises the Asimov metric. For a thorough description of our methods and techniques we refer the reader to our previous work \cite{Freitas:2020ttd}.
The paper is organized as follows. In Sec.~\ref{sec:SHUT} we provide a brief introduction into the SHUT model, providing particular details relevant for our phenomenological analysis. In Sec.~\ref{sec:Search} we present the methodology implemented in the search for VLQs at the LHC, including a full description of the chosen topology as well as a detailed description of the DL algorithms and datasets used for solving the considered classification problem. This section is finalised with the presentation and subsequent discussion of the numerical results. Finally, concluding remarks are given in Sec.~\ref{sec:conclusions}. Additional information is presented in appendix, including the VLQ Feynman rules \ref{app:Feynman-Rules}, the NN architecture \ref{app:NN_Asimov} and kinematic and angular distributions \ref{app:Kinematics}.
\section{VLQ sector of the SHUT model}
\label{sec:SHUT}
In this section, we begin by briefly introducing the SHUT model, with the focus on the main properties that arise from the GUT description, without delving deep into the mathematical inner workings of this framework. A more detailed description of the considered GUT framework can be found in earlier works e.g.~in Refs.~\cite{Camargo-Molina:2016yqm,Camargo-Molina:2017kxd,Camargo-Molina:2016bwm,Morais:2020odg,Morais:2020ypd}, while a comprehensive discussion of the low-energy limit, including the physical Lagrangian that is used in this work, can be found in Ref.~\cite{Freitas:2020ttd}.
The SHUT model is a particularly promising flavoured GUT (F-GUT) scenario whose group-theoretical structure is inspired by the $\mathrm{E_6}$ exceptional gauge group extended with a local $\mathrm{SU(2)_F\times U(1)_F}$ family symmetry, where all matter content, including the Higgs sector, is unified in fundamental (\textbf{27,2}) and (\textbf{27,1}) representations. It then proposes a first-principles' explanation for some of the well-known problems within the SM, such as a common origin of all fundamental gauge forces (with exception of gravity) as well as the observed flavour hierarchies in the SM fermionic sectors. In practical terms, the SHUT model starts off with the trinification gauge symmetry, supplemented with the $\mathrm{SU(2)_F\times U(1)_F}$ family symmetry, whose representations have a common origin from the referred (\textbf{27,2}) and (\textbf{27,1}) multiplets of the $\mathrm{E_6} \times \mathrm{SU(2)_F\times U(1)_F}$ gauge group.
Due to SUSY, the Higgs sector is unified with the charged lepton and neutrinos sectors, thus, sharing common flavor properties with fermions at high energies. The superpotential of the SHUT model reads \cite{Morais:2020ypd}
\begin{equation}
\label{eq:Superpotential}
\begin{aligned}
W_3 = \varepsilon_{IJ}\qty(\mathcal{Y}_1\bm{L}^I\bm{Q}^3_\mathrm{L}\bm{Q}_R^J - \mathcal{Y}_2\bm{L}^I\bm{Q}^J_\mathrm{L}\bm{Q}_R^J + \mathcal{Y}_2\bm{L}^3\bm{Q}^I_\mathrm{L}\bm{Q}_R^J) \,, \end{aligned}
\end{equation}
where $\varepsilon_{IJ}$ is the two-dimensional Levi-Civita symbol, $\mathcal{Y}_1$ and $\mathcal{Y}_2$ are the two unified Yukawa couplings, while subscripts $\mathrm{L}/\mathrm{R}$ denote $\mathrm{SU(3)_{L/R}}$ superfields. The indices $I,J=1,2$ represent $\mathrm{SU(2)_F}$ doublets, while flavour singlets are denoted with the ``3'' in superscript. The left-handed and right-handed components of the SM-like leptons are embedded in the $\bm{L}$ superfield. Right-handed neutrinos also reside in this representation, which helps address the neutrino mass problem via a radiative seesaw mechanism \cite{Camargo-Molina:2017kxd,Morais:2020ypd}. The SM quarks are then found in $\bm{Q}_{\text{L,R}}$.
At low energies, the SM gauge group is fully realized, with additional exotic physics, including an extended scalar sector and vector-like fermions. Full unification of the field content results in a reduced freedom in the Yukawa sector, with the presence of only two $\mathcal{Y}_1$ and $\mathcal{Y}_2$ couplings at high scales, which provide the leading contributions to the generation of quark masses at tree level and the necessary means to radiatively induce strong hierarchies as observed in Nature. Indeed, as soon as the scalar components of $\bm{L}$ develop vacuum expectation values (VEVs) that break the SHUT symmetry down to that of the SM, the down-type VLQs, second- and third-generation SM-like quarks gain tree-level masses, while the charged leptons and first-generation quarks develop their masses at quantum level via loop corrections. This fact offers the necessary ingredients to potentially explain the observed splitting in the fermion spectra implying the relative lightness of the charged leptons, neutrinos and the quarks of first generation.
In Ref.~\cite{Morais:2020ypd}, various benchmark scenarios assuming three light Higgs doublets yielding a consistent CKM mixing matrix, were explored in detail. The further presence of new VLQs results in an extended $3\times 6$ quark mixing matrix, defined as
\begin{equation}\label{eq:VLQs_CKM}
V_\text{CKM} = U_L^u \dotproduct P \dotproduct (U_L^d)^\dagger = \begin{pmatrix}
V_{\mathrm{CKM}}^{\mathrm{SM}} & V_{\mathrm{CKM}}^{\mathrm{VLQs}}
\end{pmatrix} \,,
\end{equation}
where $P$ is a projection operator, defined as $P = \begin{pmatrix}\mathbb{1}_{3\times 3} \hspace{0.5em} | \hspace{0.5em} 0_{3\times 3}\end{pmatrix}$. Here, a $3\times 3$ matrix $V_{\mathrm{CKM}}^{\mathrm{SM}}$ represents the CKM-like quark mixing of the SM while a $3\times 3$ matrix $V_{\mathrm{CKM}}^{\mathrm{VLQs}}$ contains a mixing between the down-type VLQs with up-type chiral quarks of the SM. Two of them are relatively light, with a certain hierarchy between them, and are expected to emerge at a TeV energy scale. A third heavy VLQ appears beyond the reach of current or near-future colliders.
In the present work we explore the LHC discovery potential of the lightest VLQ denoted as $D$, ignoring its heavier counterparts for simplicity. Its mass,
\begin{equation}
m_\text{D}\sim \omega\mathcal{Y}_2\,,
\label{eq:mVLQ}
\end{equation}
with $0.01 \sim \mathcal{Y}_2 \ll \mathcal{Y}_1 \sim 1$ fixed by the charm and top quark masses respectively, and with $\omega \sim 100$ TeV being an intermediate Left-Right symmetry breaking scale, is expected to emerge not far above the TeV scale. As a particular benchmark scenario inspired by the phenomenologically relevant ranges of the SHUT model parameters found in Ref.~\cite{Morais:2020ypd}, we adopt the following values of the vector-like $D$-quark mixing elements in the extended CKM matrix:
\begin{equation}
V_{\rm uD} \simeq 5.1 \times 10^{-6} \,, \qquad
V_{\rm cD} \simeq 2.6 \times 10^{-5} \,, \qquad
V_{\rm tD} \simeq 0.016 \,.
\label{eq:CKM_matrix}
\end{equation}
In our numerical analysis below, we consider only light quark jets in the final states of VLQ decays $D\to W+$jets, so only $V_{\rm uD}$ and $V_{\rm cD}$ will be relevant in what follows. These will be fixed as above, whereas a numerical scan will be performed over the VLQ mass $m_{\rm D}$ within a phenomenologically acceptable range (see below).
\section{VLQ search at the LHC}
\label{sec:Search}
The generation of events in proton-proton collisions at the LHC follows the standard methods employed by the particle physics community. In this work, we use a specially designed chain of computational tools developed in the previous work of Ref.~\cite{Freitas:2020ttd}, and here we only briefly elaborate on the corresponding machinery. First, the Lagrangian density is constructed utilizing the \verb|SARAH| package \cite{Staub:2013tta}, which can be used to derive interaction vertices between the fields, their spectra and mixing. The latter are then outputted into \verb|UFO| \cite{Degrande:2011ua} files that can be used in Monte-Carlo event generators. In particular, here we employ \verb|MadGraph| \cite{Alwall:2014hca} for hard-scattering events and \verb|Pythia8| \cite{Sjostrand:2014zea} for hadronization/showering processes. Fast-simulation of the detector is implemented using \verb|Delphes| \cite{deFavereau:2013fsa} (with the ATLAS default card), while the kinematic data extraction is done in \verb|ROOT| \cite{Brun:1997pa}. We generate 250k events, for both VLQ production signal and backgrounds, at a centre-of-mass energy of $\sqrt{s}=14~\mathrm{TeV}$.
First, it is instructive to analyse the current constraints from direct searches of down-type VLQs. From the experimental side, among the well-known NP states a VLQ is one of the most searched for at the LHC. Since it is a coloured particle, its production rate is expected to be relatively high. In this regard, typical lower bounds on the VLQ mass range from 1.4 TeV up to 2.0 TeV (see the latest exclusion bounds, as of July 2021, in the summary plots of Ref.~\cite{ATLAS_twiki_VLQs}). However, one must carefully analyse the assumptions made in these analyses. First, the vast majority of current searches focus on dominant couplings to the third generation of chiral quarks (see e.g. Refs.~\cite{ATLAS:2018cjd,ATLAS:2018alq,ATLAS:2018uky,ATLAS:2017nap,ATLAS:2016scx,ATLAS:2015ktd}). Such an assumption is not the most general, as there is no reason for the VLQ to not couple to lighter quarks. Indeed, there have also been the searches assuming dominant light-quark couplings -- see, for instance, Refs.~\cite{ATLAS:2015lpr,ATLAS:2011tvb}. In Ref.~\cite{ATLAS:2015lpr}, in particular, a search for pair production of heavy VLQs has been performed, with their subsequent decays into light jets, that can go via either neutral ($D\rightarrow Zq$) or charged ($D\rightarrow Wq$) currents. This work excludes the VLQs with masses below 690 GeV at 95\% confidence level. Such a constraint is quite relevant since the same topology will be used in our analysis below. In the second work \cite{ATLAS:2011tvb}, the same VLQ decay modes into light jets have been considered, but in single VLQ production channels. For the neutral $D\rightarrow Zq$ decay channel, the VLQ mass has been constrained to be below 760 GeV, whereas a tighter lower bound of 900 GeV is obtained in the charged $D\rightarrow Wq$ decay channel. These constraints, however, are not relevant for the particular benchmark point that we have shown in the previous section. For example, we have calculated the production cross-section in \verb|MadGraph| for single production of the VLQ (using the benchmark point shown in Eq.~\eqref{eq:CKM_matrix}), for which we obtained $\sigma = 1.678 \times 10^{-4}$ pb for a mass of 800 GeV, which is still consistent with the cross-section limits suggested by Ref.~\cite{ATLAS:2011tvb}. For the sake of generality, in this work we consider the VLQ mass to be greater than 800 GeV.
As mentioned before, the chosen topology of the signal is equivalent to that of Ref.~\cite{ATLAS:2015lpr}, with VLQ pair-production undergoing via gluon fusion. An example for the leading-order Feynman diagram, with the contribution from the gluon triple vertex, is shown in Fig.~\ref{fig:VLQ-pair}. This and other relevant contributions (see Fig.~1 of \cite{ATLAS:2015lpr}) are automatically included at the level of event generation in \texttt{MadGraph}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.40\textwidth]{figures/VLQ_double_production.pdf}
\caption{One of the leading-order Feynman diagrams for the pair-production of a $D$-type VLQ. Here, $g$ corresponds to gluons from initial colliding protons. The $D$-quark decays into a light jet $j = u, c, \bar{u},\bar{c}$ and a $W$ boson, with one of the $W$'s further decaying via a leptonic channel, with one charged lepton ($\ell^- = e^-, \mu^-$) and the corresponding neutrino ($\bar{\nu}_e, \bar{\nu}_\mu$), and the other decaying via a hadronic channel into light jets. In the hadronic $W$ decay, both the up- and down-type SM quarks are considered.}
\label{fig:VLQ-pair}
\end{figure}
The main irreducible backgrounds for this topology include:
\begin{itemize}
\item Pair-production of top quarks, $t\bar{t}$, with one of the $W$ bosons decaying fully leptonically, whereas the other one decays into two light jets;
\item Diboson production plus two light jets. In this case, one of the vector bosons decays into two light jets whereas the other one ($W$ boson) decays into a charged lepton and a neutrino.
\end{itemize}
While the NN models will do most of the heavy lifting, we will also consider simple selection criteria to help discriminate the signal events from the background. In particular, we consider that
\begin{equation}\nonumber
\begin{aligned}
&p_T(\ell^\pm) > 25 \text{ GeV} \,, \\
&\rm{MET} > 20 \text{ GeV} \,, \\
&\abs{\eta(\ell^\pm)} \leq 2.5 \,,\\
\end{aligned}
\end{equation}
with $\eta = -\ln(\tan(\theta/2))$, $\theta$ is the angle with respect to the beam axis, and MET is defined as a missing transverse energy. With this, we extract all relevant kinematic and angular information about the final states' configurations, such as the transverse momentum, pseudo-rapidity and mass distributions of pairs of produced particles. A comprehensive list of the observables that we utilise in this work is provided in Table~\ref{tab:vars_VLQ}.
\begin{table}[H]
\centering
\captionsetup{justification=raggedright,singlelinecheck=false}
\resizebox{1.0\textwidth}{!}{\begin{tabular}{|c|c|c|c|}
\toprule \toprule
\hline
& Dimension-full & \multicolumn{2}{|c|}{Dimensionless} \\
\hline
\hline
\midrule
\makecell{VLQ production} &
\makecell{$p_T(\ell^-)$, $E(\ell^-)$, $p_T(j_1)$, \\
$E(j_1)$, $p_T(j_2)$, $E(j_2)$, \\
$p_T(j_3)$, $E(j_3)$, $p_T(j_4)$, \\
$E(j_4)$, MET ,$M_T(W)$, \\
$M(W_{jj})$, $M_D (\mathrm{MET},\ell^-,j_1)$,\\
$M_D (j_2,j_3,j_4)$, $M_D(\mathrm{MET},\ell^-,j_2)$,\\
$M_D (j_1, j_3, j_4)$, $M(j_1,j_2)$, $M(j_1,j_3)$,\\
$M(j_1,j_4)$, $M(j_2,j_3)$, $M(j_2,j_4)$,
$M(j_3,j_4)$} &
\makecell{$\eta(\ell^-)$, $\eta(j_1)$, $\eta(j_2)$, \\
$\eta(j_3)$, $\eta(j_4)$, $\phi(\ell^-)$,\\
$\phi(j_1)$, $\phi(j_2)$, $\phi(j_3)$, \\
$\phi(j_4)$, $\cos(\theta_{\ell^-,\bar{\nu}_\ell})$,\\
$\cos(\theta_{\ell^-,j_1})$,$\cos(\theta_{\ell^-,j_2})$,\\
$\cos(\theta_{\ell^-,j_3})$,$\cos(\theta_{\ell^-,j_4})$,\\
$\cos(\theta_{j_1,j_2})$,$\cos(\theta_{j_1,j_3})$,\\
$\cos(\theta_{j_1,j_4})$,$\cos(\theta_{j_2,j_3})$,\\
$\cos(\theta_{j_2,j_4})$,$\cos(\theta_{j_3,j_4})$} &
\makecell{$\Delta R (j_1,\ell^-)$, $\Delta R (j_2,\ell^-)$,\\
$\Delta R (j_3,\ell^-)$, $\Delta R (j_4,\ell^-)$,\\
$\Delta R (j_1,j_2)$, $\Delta R (j_1,j_3)$,\\
$\Delta R (j_1,j_4)$, $\Delta R (j_2,j_3)$,\\
$\Delta R (j_2,j_4)$, $\Delta R (j_3,j_4)$,\\
$\Delta \Phi (j_1,\ell^-)$, $\Delta \Phi (j_2,\ell^-)$,\\
$\Delta \Phi (j_3,\ell^-)$, $\Delta \Phi (j_4,\ell^-)$,\\
$\Delta \Phi (j_1,j_2)$, $\Delta \Phi (j_1,j_3)$,\\
$\Delta \Phi (j_1,j_4)$, $\Delta \Phi (j_2,j_3)$,\\
$\Delta \Phi (j_2,j_4)$, $\Delta \Phi (j_3,j_4)$}\\
\hline
\hline
\end{tabular}}
\caption{The final-state kinematic and angular distributions selected for the DL analysis for double production of the $D$-VLQ. All observables are calculated in the laboratory frame. We define $j_1$ as the leading jet with the greatest transverse momentum and $j_4$ as a sub-leading light jet having the lowest transverse momentum, $p_T$ is the transverse momentum, $E$ is the energy, MET is the missing transverse energy, $\phi$ is the azimuthal angle, $\theta$ is the polar angle between two final states, $\Delta R$ is defined as the Euclidean distance in the $(\eta,\phi)$ plane, i.e. $\Delta R(i,j)= \sqrt{\Delta\phi^2_{ij} + \Delta\eta^2_{ij}}$ with $\Delta\phi_{ij} = \phi_j - \phi_i$ and $\Delta \eta_{ij} = \eta_j - \eta_i$. Here, $W$ is reconstructed from one charged lepton plus missing energy, whereas $W_{jj}$ is reconstructed from two light jets.}
\label{tab:vars_VLQ}
\end{table}
\subsection{LHC analysis: Hybrid Net approach}
\label{subsubsec:DL}
In this section, we present the DL methodology used in our analysis, which involves a combination of both image and tabular datasets. The network, which we name as Hybrid Net in what follows, will take into account both data types in the classification of background and signal classes. The image datasets will be trained in a convolutional model, whereas kinematic data are trained through a linear sequential model, after which we combine the predictions of both.
Jet images, first introduced in Refs.~\cite{Cogan:2014oua,deOliveira:2015xxd}, are constructed by mapping particle hits in the detector into a grid in the $(\eta,\phi)$ plane, where the intensity of the pixel is directly proportional to the energy deposited by the particles. These methods, however, can be quite inefficient since in a typical jet-image there can exist plenty of blank pixels, making it computationally inefficient when passing through convolutional layers in the training phase. While various approaches have been proposed in the literature (such as particle clouds \cite{Qu:2019gqs} or through the use of graph networks \cite{Ju:2020xty,Farrell:2018cjr}), in this work we consider abstract jet-images as first discussed by one of the authors and collaborators in Ref.~\cite{Alves:2019ppy}. In this approach, besides associating each pixel with the energy of a particle, we also map characteristics of the particle into various geometrical shapes. For example, if it is a charged particle in a detector, we map it to a circle. In this scenario, the radius of the polygons is directly proportional to the particle energy. With this, we then provide additional features for the neural model to learn. Examples of these types of images can be seen in Fig.~\ref{fig:jet_images}.
\begin{figure}[htb!]
\centering
\subfloat[Signal event]{\includegraphics[width=0.30\textwidth]{figures/D4_exemplo.png}}
\hspace{1.65em}
\subfloat[$t\bar{t}$ event]{\includegraphics[width=0.30\textwidth]{figures/ttbar_exemplo.png}}
\hspace{1.65em}
\subfloat[$WW+$ jets event]{\includegraphics[width=0.30\textwidth]{figures/WW_plus_jet_exemplo1.png}}
\caption{Example of an abstract jet image in the $(\eta,\phi)$ plane. Green polygons are representative of photons, red circles are charged particles while blue hexagons indicate neutral hadrons. The radius of the polygons is proportional to the energy left by the particle in the calorimeter tower. In this image we have an event for a proton-proton collision at $\sqrt{s}=14$ TeV, with subsequent VLQ double production (to the right) and the corresponding background events ($t\bar{t}$ in the centre and $WW+$ jets in the right).}
\label{fig:jet_images}
\end{figure}
In order to construct such images, we impose the following conditions:
\begin{itemize}
\item Charged particles hitting the detector are mapped into blue circles, whose radius is given as $r = \log{p_T}$, where $p_T$ is the transverse momentum of the particle;
\item Neutral hadrons are mapped into hexagons, with the radius given as $r = \log(E_T)$, where $E_T$ is the transverse energy;
\item Photons are mapped into squares, with the radius given by the same formula as the neutral hadrons.
\end{itemize}
The resulting images are then stored in the \verb|.png| format, with the pixel size of $224\times224$, requiring that both hadrons and photons are restricted to be in the kinematic ranges of $\abs{\eta} \leq 5.0$ and $\abs{\phi} \leq \pi$.
We use both the image and tabular information and feed it into a NN. The images pass through a convolutional network, whereas the corresponding kinematic information is passed through a linear model. We then combine the results by summing together the predictions of both networks leading to the final result. The hyper-parameter optimization designed to maximize the statistical significance is implemented via evolutionary algorithms as described in previous works \cite{Morais:2021ead,Bonilla:2021ize,Freitas:2020ttd}. Training of the NNs based on the statistical significance follows the analysis first performed by A. Elwood and D. Kr{\"u}cker \cite{Elwood:2018qsr}. In this work, only linear networks are part of the optimization procedure, whereas the convolutional network for the images is fixed to a ResNet-34 network \cite{he2015deep}. Furthermore, we consider the following list of hyper-parameters:
\begin{itemize}
\item Number of neurons of the sequential model: 256, 512, 1024 and 2048;
\item Number of layers of the sequential model: 1, 2, 3, 4 and 5;
\item Bottleneck architecture\footnote{We define a bottleneck architecture as a neural model with an ever decreasing number of neurons at each layer.}: True or False;
\item Initialisers in the sequential model: Normal, He normal and He uniform;
\item Optimisers in the sequential model: Adam, SGD, Adamax and nadam;
\item Vision models: ResNet-34.
\end{itemize}
The evolutionary algorithm builds sets of Hybrid Net models. Both the linear and convolution parts of the algorithms are constructed using \verb|PyTorch| \cite{paszke2019pytorch}. In the convolution part, we also utilized \verb|FastAI| \cite{2020arXiv200204688H}, which employs \verb|PyTorch| as backend. Given this methodology, let us now turn to a discussion of numerical results.
\subsection{Numerical results}
\label{subsubsec:results}
We first begin by presenting the results for a particular (fixed) VLQ mass. Focusing on the lightest $D$-VLQ with a mass of $800~\mathrm{GeV}$ and applying the selection criteria described above, we obtained the following production cross-sections:
\begin{equation}\label{eq:cross_sections}
\begin{aligned}
& \sigma (\bar{D}D) = 1.06~\mathrm{fb}\, ,\\
& \sigma (\bar{t}t) = 14970.50~\mathrm{fb}\, ,\\
& \sigma (WW\mathrm{+jets}) = 1094.94~\mathrm{fb}\,.
\end{aligned}
\end{equation}
The variation of the cross-section with the VLQ mass is shown in Fig.~\ref{fig:xsec_points}. In this figure, we also plot dashed lines to indicate the allowed sensitivity expected for future runs of the LHC. In particular, we show that for the benchmark values of the VLQ-light quark mixing elements indicated in Eq.~\eqref{eq:CKM_matrix}, the run-III phase of the LHC can potentially probe $D$-type VLQs up to 1.9 TeV in this channel, while the HL phase has a sufficient sensitivity to exclude VLQs up to 2.37 TeV.
In order to evaluate the performance of the network, we show in Fig.~\ref{fig:ROC_and_PCS} the receiver operating characteristic (ROC) plot. Such a plot can serve as a measurement of how well the network performs on the data. In particular, these results are shown for the validation dataset, i.e.~for the dataset that the network was not trained on. Within the labeling of the curves, we also indicate the area under the curve (AUC), which is related to the accuracy. As one can readily observe, we obtain a very good accuracy indicating an efficient separation between the signal and background classes. Further confirmation of this fact can be given by the predicted confidence scores for the samples plotted on the left-hand-side of Fig.~\ref{fig:ROC_and_PCS}. Here, we can clearly see that the model successfully separates the signal classes (mainly populating the convolutional NN scores of around one) from the background (which mainly populates a score of around zero). Of course, such separation is not perfect, with some contamination still present, in a sense that some background and signal overlap in some regions of the convolutional NN parameter domain.
\begin{figure}[htb!]
\centering
\includegraphics[width=\textwidth]{figures/DF_ROC_resnet18_BCE.pdf}
\caption{Performance results for the Hybrid network. On the left, we plot the ROC and, on the right, -- the predicted confidence score for our samples. The signal is indicated with a dashed blue line, while the $t\bar{t}$ background -- in red, and $WW$+jets -- in green.}
\label{fig:ROC_and_PCS}
\end{figure}
The histograms with the kinematic and angular observables that are utilized by the NN are plotted in appendix, see Figs.~\ref{fig:Dimensionfull-vars} and \ref{fig:Dimensionless-vars}. One can observe that the angular distributions for the signal tends to be overshadowed by the backgrounds, with the signal following the same structure as the background, with the only exception being the pseudo-rapidity distributions. Note that the signal has a dominant peak at $\eta\sim 0$, whereas the backgrounds spread over the entirety of the allowed range $\abs{\eta}\leq 2.5$. The kinematics, on the other hand, offers the most significant discriminating power, with distributions being peaked in kinematic regions where the SM backgrounds are less dominant. This is true for all transverse momentum, energy and MET distributions. Similarly, mass distributions also peak in higher invariant mass domains compared to those of the SM backgrounds.
\begin{figure}[htb!]
\centering
\includegraphics[width=\textwidth]{figures/Cross_section.pdf}
\caption{The production cross-section (in femtobarn) as a function of the VLQ mass (in GeV). Two horizontal dashed lines are shown, one for the run-III target luminosity and the other for the HL phase of the LHC, which represent the minimal cross section for which a single event can be produced, i.e. $N_{\mathrm{exp}}=\sigma\mathcal{L} = 1$ for the dashed lines. The area filled in gray represents the region where neither the HL-LHC or run-III LHC have the required sensitivity to detect the $D$-VLQ for the chosen benchmark~\eqref{eq:CKM_matrix}. The $y$-axis is shown in logarithmic scale.}
\label{fig:xsec_points}
\end{figure}
Each individual event corresponds to a particular value of the physical observables transported over into the tabular data. Each event will have an associated jet image. A combination of both is used in the classification procedure, and then the statistical significance is computed using the validation dataset. For completeness, we present the results utilising distinct estimates, ones being more conservative than the others. For this purpose, in this work we consider the following three distinct metrics:
\begin{itemize}
\item The Asimov estimate, defined as
\begin{equation}\label{eq:Asimov_sig}
\mathcal{Z}_A = \Bigg[2\Bigg((s + b)\ln\Bigg(\frac{(s+b)(b+\sigma_b^2)}{b^2 + (s+b)\sigma_b^2}\Bigg) -\frac{b^2}{\sigma_b^2}\ln\Bigg(1+\frac{\sigma_b^2 s}{b(b+\sigma_b)}\Bigg)\Bigg)\Bigg]^{1/2} \,,
\end{equation}
where $s$ is the number of signal events, $b$ is the number of background events and $\sigma_b$ is the uncertainty of the background. For this metric, we assume a 1\% systematic uncertainty. This is the most conservative metric and, in fact, the most realistic one.
\item Another calculation of the Asimov metric is also considered, with $\mathcal{Z}(<1\%)$, where the background is known within an error of $10^{-3}$. This is the least conservative metric, since it assumes that most of the contributions to the background are very well understood, requiring all physics backgrounds to be well known. This metric should be then understood as the ultimate best case scenario.
\item Somewhat between the other two metrics, we also consider the standard metric, $s/\sqrt{s+b}$, that is typically employed in the experimental context.
\end{itemize}
\begin{figure}[hb!]
\centering
\includegraphics[width=\textwidth]{figures/sig_plot.pdf}
\caption{The statistical significance as a function of the integrated luminosity for pair-production of 800 GeV $D$-VLQs. The red curve corresponds to the $s/\sqrt{s+b}$ metric, the blue curve -- to the Asimov estimate $\mathcal{Z}_A$ and the green line -- to the adapted Asimov significance $\mathcal{Z}(<1\%)$. There are two vertical dashed lines marking the run-III luminosity ($300~\mathrm{fb^{-1}}$) and the HL-LHC luminosity ($3000~\mathrm{fb^{-1}}$). Both the $y$-axis and $x$-axis are shown in logarithmic scale.}
\label{fig:lum_vs_sig}
\end{figure}
\begin{figure*}[t!]
\captionsetup{justification=raggedright,singlelinecheck=false}
\subfloat{{\includegraphics[width=0.33\textwidth]{figures/sig_sqrt_bkgd-5p2911.pdf} }}
\subfloat{{\includegraphics[width=0.33\textwidth]{figures/Z_resnet50_sys-5p29110p01.pdf} }}
\subfloat{{\includegraphics[width=0.33\textwidth]{figures/Z_1perc-5p2911.pdf} }}\\
\caption{The statistical significance for a hypothetical discovery of VLQs as a function of the NN score for the different metrics considered in this work. The VLQ mass is 800 GeV, and the collider luminosity is $\mathcal{L}=3000~\mathrm{fb}$. The significance is computed following the implementation of an evolution algorithm whose architecture is detailed in Appendix~\ref{app:NN_Asimov}. From left to right, we plot the $s/\sqrt{s+b}$ metric, the lenient Asimov metric $\mathcal{Z}(<1\%)$ and the Asimov metric $\mathcal{Z}_A$ with 1\% systematics.
\label{fig:sig_plots}}
\end{figure*}
Considering the HL phase of the LHC, which is expected to reach an integrated luminosity of $\mathcal{L} = 3000~\mathrm{fb^{-1}}$, we can determine the median significance for the considered metrics as a function of the NN score, as shown in Fig.~\ref{fig:sig_plots}. The maximal significances for each metric are then determined to be
\begin{itemize}
\item $s\sqrt{s+b} = 48.59\sigma$;
\item $\mathcal{Z}(<1\%) = 79.20\sigma$;
\item $\mathcal{Z}_A = 56.94\sigma$,
\end{itemize}
i.e.~we consistently obtain significances greater than $5\sigma$ for all considered metrics. Such a strong signature of VLQs at the HL run of the LHC can be expected, since VLQs are coloured particles, and their production is enhanced at hadron colliders.
It is also relevant to analyse the impact for other luminosities, in particular, for the run-III of the LHC, which is set to deliver $\mathcal{L}=300~\mathrm{fb^{-1}}$ of data. In Fig.~\ref{fig:lum_vs_sig} we plot the statistical significance of all metrics as a function of the collider luminosity. Specifically, the maximal median significances obtained for the run-III luminosity read as follows:
\begin{itemize}
\item $s\sqrt{s+b} = 15.37\sigma$;
\item $\mathcal{Z}(<1\%) = 25.04\sigma$;
\item $\mathcal{Z}_A = 19.16\sigma$,
\end{itemize}
i.e. the significance is above $5\sigma$ for all considered metrics indicating that the VLQ production can also be probed at run-III.
\begin{table}[htb!]
\begin{center}
\captionsetup{justification=raggedright,singlelinecheck=true}
\begin{tabular}{c|c|l|l|cll|c|l|l|c|l|l}
Benchmarks & \multicolumn{3}{c|}{$800~\mathrm{GeV}$} & \multicolumn{3}{c|}{$1200~\mathrm{GeV}$} & \multicolumn{3}{c|}{$1700~\mathrm{GeV}$} & \multicolumn{3}{c|}{$2200~\mathrm{GeV}$} \\[2mm] \hline
\multirow{3}{*}{\makecell{$\hphantom{.}$ \quad \hspace{2.6em}$s/\sqrt{s+b}$ \\ \hspace{-3.6em} $300~\mathrm{fb^{-1}}$ \hspace{0.3em}$\mathcal{Z}_A$ \\ \hspace{3.9em} $\mathcal{Z}(<1\%)$}} & \multicolumn{3}{c|}{\redBU $15.37\sigma$\Tstrut} & \multicolumn{3}{c|}{$3.51\sigma$} & \multicolumn{3}{c|}{$0.71\sigma$} & \multicolumn{3}{c|}{$-$} \\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $19.16\sigma$} & \multicolumn{3}{c|}{$1.29\sigma$} & \multicolumn{3}{c|}{$0.05\sigma$} & \multicolumn{3}{c|}{$-$}\\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $25.04\sigma$\Bstrut} & \multicolumn{3}{c|}{\redBU $5.58\sigma$} & \multicolumn{3}{c|}{$1.15\sigma$} & \multicolumn{3}{c|}{$-$} \\ \hhline{=============}
\multirow{3}{*}{\makecell{$\hphantom{.}$ \quad \hspace{2.6em}$s/\sqrt{s+b}$ \\ \hspace{-3.6em} $1000~\mathrm{fb^{-1}}$ \hspace{0.3em}$\mathcal{Z}_A$ \\ \hspace{3.9em} $\mathcal{Z}(<1\%)$}} & \multicolumn{3}{c|}{\redBU $28.05\sigma$\Tstrut} & \multicolumn{3}{c|}{\redBU $6.41\sigma$} & \multicolumn{3}{c|}{$1.29\sigma$} & \multicolumn{3}{c|}{$-$} \\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $34.39\sigma$} & \multicolumn{3}{c|}{$2.33\sigma$} & \multicolumn{3}{c|}{$0.10\sigma$} & \multicolumn{3}{c|}{$-$} \\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $45.72\sigma$\Bstrut} & \multicolumn{3}{c|}{\redBU $10.19\sigma$} & \multicolumn{3}{c|}{$2.09\sigma$} & \multicolumn{3}{c|}{$-$} \\ \hhline{=============}
\multirow{3}{*}{\makecell{$\hphantom{.}$ \quad \hspace{2.6em}$s/\sqrt{s+b}$ \\ \hspace{-3.6em} $3000~\mathrm{fb^{-1}}$ \hspace{0.3em}$\mathcal{Z}_A$ \\ \hspace{3.9em} $\mathcal{Z}(<1\%)$}} & \multicolumn{3}{c|}{\redBU $48.59\sigma$\Tstrut} & \multicolumn{3}{c|}{\redBU $11.10\sigma$} & \multicolumn{3}{c|}{$2.24\sigma$} & \multicolumn{3}{c|}{$0.53\sigma$} \\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $56.94\sigma$} & \multicolumn{3}{c|}{$3.93\sigma$} & \multicolumn{3}{c|}{$0.16\sigma$} & \multicolumn{3}{c|}{$0.0090\sigma$} \\ \cline{2-13}
& \multicolumn{3}{c|}{\redBU $79.20\sigma$\Bstrut} & \multicolumn{3}{c|}{\redBU $17.66\sigma$} & \multicolumn{3}{c|}{$3.60\sigma$} & \multicolumn{3}{c|}{$0.84\sigma$} \\ \hhline{=============}
\end{tabular}
\caption{The statistical significance for the metrics considered in this work, for various masses of the VLQ and different collider luminosities. Scenarios which pass the $5\sigma$ requirement for potential exclusion (or hypothetical discovery) are displayed in {\redBU red}. Points marked with ``--'' indicate that the cross-section is not enough to produce one event at the corresponding luminosity.}
\label{tab:VLQ_significance_table}
\end{center}
\end{table}
In order to complement these results further, we implement the same machinery in a numerical scan over the VLQ mass, up to the detectability limit of the HL phase, which corresponds up to the VLQ mass of around 2.2 TeV. These results are summarised in Table~\ref{tab:VLQ_significance_table}. As the main result, a 5$\sigma$ significance is obtained for VLQ masses of up to 1.2 TeV. In particular, at this mass point, we note that we can exclude the VLQ at $\mathcal{L}=3000~\mathrm{fb^{-1}}$ with the significance of $17.66\sigma$ and $11.10\sigma$ for lenient metrics. The conservative Asimov estimate does not get above the discovery threshold, however, with a significance of $\mathcal{Z}_A = 3.93\sigma$. In a similar manner, we determine an exclusion bound over the VLQ mass at $\mathcal{L}=300~\mathrm{fb^{-1}}$. However, this result is obtained for the most lenient metric considered in this work, with $\mathcal{Z}(<1\%) = 5.58\sigma$, whereas for the stringiest metric we find $\mathcal{Z}_A = 1.29\sigma$. Above 1.2 TeV, all metrics provide the statistical significance bellow 5$\sigma$ for all considered collider luminosities, and therefore, no exclusion (or hypothetical discovery) can be placed there. We notice, however, that $\mathcal{Z}(<1\%)=3.60\sigma$ is found for the VLQ mass of 1.7 TeV. Despite being below 5$\sigma$, it may still appear as a signal anomaly in the data. A combination of this channel with other possible ones (such as single VLQ production and new topologies via 3rd generation decays) might lead to a combined significance above $5\sigma$ for such a values of the mass.
\section{Conclusions}
\label{sec:conclusions}
In this paper, we continue the DL phenomenological analysis, as first started in our previous work \cite{Freitas:2020ttd}, of exotic vector-like fermions that naturally arise in the low-energy limit of a F-GUT model based on the trinification gauge group. In particular, we have focused on possible collider signatures of the weak-isospin-singlet $D$-type VLQ at the LHC, via pair-production topologies, with subsequent VLQ decay into lighter chiral quarks. The latter topology results in a single charged lepton, light jets and a neutrino in the final state particularly convenient for experimental analysis at the LHC. In order to accomplish this task, we have performed Monte-Carlo simulations and utilized sophisticated DL methods for efficient separation of SM backgrounds from the VLQ signal. This enables us to compute the statistical significance for a hypothetical discovery of the considered VLQ. In the context of the VLQ detection prospects at the LHC, we have utilised an evolutionary algorithm that optimises the construction of a NN based on the maximisation of the statistical significance. For this purpose, we have exploited both the kinematic tabular data as well as the detector jet images as inputs in the classification procedure through the use of a Hybrid Net.
With this technique, we have found that the down-type VLQs with a mass of up to 1.2 TeV can be excluded both at the run-III and the HL phase of the LHC. In particular, we have determined that at $\mathcal{L}=3000~\mathrm{fb^{-1}}$ one rules out VLQ at this mass scale with statistical significance of $\mathcal{Z}(<1\%)=17.66\sigma$ (and $\mathcal{Z}(<1\%)=5.58\sigma$ at $\mathcal{L}=300~\mathrm{fb^{-1}}$). For the mass of 1.7 TeV, we have found that $\mathcal{Z}(<1\%)=3.60\sigma$, which, in combination with other channels, may be useful to probe VLQ at this mass and even to set an exclusion bound.
\section*{Acknowledgments}
The work developed in this article is supported by the projects PTDC/FIS-PAR/31000/2017 and CERN/FIS-PAR/0014/2019. J.G., F.F.F., and A.P.M. are supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020.
%
A.P.M.~is also supported by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19.
%
J.G. is also directly funded by FCT through a doctoral program grant with the reference 2021.04527.BD.
%
R.P.~is supported in part by the Swedish Research Council grant, contract number 2016-05996, as well as by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 668679).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}\label{sec:1}
Low-density parity-check (LDPC) codes, invented by Gallager \cite{Gallager1963} in the $1960$s, constitute a hot research topic since they are a main competitor to turbo codes \cite{Mackay, Chung2001, Lentmaier2010, Kudekar2010}. Recently, a connection between LDPC codes and codes based on graphs was shown (see, for example, \cite{Schmidt2003, Kim2007, Sullivan2006, Barg2006, WovenHypergraph}), which opens new perspectives in searching for powerful LDPC codes. Moreover, coding theory methods can be applied in describing and searching for graphs better than previously known. For example, in \cite{ISIT2009, PPI} compact representations based on convolutional LDPC codes for famous bipartite graphs such as Heawood's, Tutte's, and Balaban's graphs\cite{Bondy1976} are presented.
Typically, LDPC codes have a minimum distance which is less than that of the best known linear codes, but due to their structure they are suitable for low-complexity iterative decoding, like for example the believe-propagation algorithm. An important parameter determining the efficiency of iterative decoding algorithms for LDPC codes is the \textit{girth}, which determines the number of independent iterations\cite{Gallager1963} and is a parameter of the underlying graph. The minimum distance seems not to play an important role within iterative decoding algorithms, since the error-correcting capabilities of such a suboptimal procedure are often less than those guaranteed by the minimum distance. In fact, it was shown in \cite{Dolecek2009} that the performance of LDPC codes in the high signal-to-noise (SNR) region is predominantly dictated by the structure of the smallest absorbing sets. However, as the size of these absorbing sets is upper-bounded by the minimum distance, LDPC codes with large minimum distance are of particular interest.
LDPC codes can be characterized as either random/pseudo-random or nonrandom, where nonrandom codes can be subdivided into regular or irregular \cite{Kou2001, Johnson2001, Johnson2001-2, Fossorier2004, Kim2007, Sullivan2006, TannerClass, Tanner2002, Tanner2004, Zhang2010, Milenkovic2006, Esmaeili2010, Smarandache2004, Wang2008, Fan2000, ISIT2009, ISIT2010}, while random/pseudo-random codes are always irregular \cite{Richardson2001, Hu2005}. A $(J,K)$-regular (nonrandom) LDPC code is determined by a parity-check matrix with exactly $J$ ones in each column and exactly $K$ ones in each row.
The class of quasi-cyclic (QC) $(J, K)$-regular LDPC codes is a subclass of regular LDPC codes with low encoding complexity. Such codes are most suitable for algebraic design and are commonly constructed based on combinatorial approaches using either finite geometries \cite{Kou2001} or Steiner Triple Systems \cite{Johnson2001, Johnson2001-2}, having girth $g \geq 6$. Amongst other algebraic constructions leading to QC LDPC codes with larger girth we would like to mention \cite{TannerClass}, where a class of QC LDPC codes of rate $R=2/5$ with girth up to $12$ based on subgroups of the multiplicative group of the finite field $\mathbb{F}_{p}$ was obtained. The same method was used for convolutional codes in \cite{Tanner2004}.
Although QC LDPC codes are not asymptotically optimal they can outperform random or pseudorandom LDPC codes (from asymptotically optimal ensembles) for short or moderate block lengths \cite{Fossorier2004}. This motivates searching for good short QC LDPC codes.
The problem of finding QC LDPC codes with large girth and large minimum distance for a wide range of code rates was considered in several papers, for example, \cite{Fossorier2004, Kim2007, Tanner2004, Sullivan2006, Esmaeili2010, Milenkovic2006}. Codes with girth at most $12$ are constructed in \cite{TannerClass, Fossorier2004, Tanner2004, Milenkovic2006}, while \cite{Sullivan2006} gives examples of rather short codes with girth $14$. Codes with girth up to $18$ with $J \geq 3$ are presented in \cite{Esmaeili2010} and it is shown that QC LDPC codes with girth $\geq 14$ and block length between $34,\!000$ and $92,\!000$ outperform random codes of the same block length and rate.
Most of the papers devoted to constructing nonrandom LDPC codes with large girth combine some algebraic techniques and computer search. Commonly these procedures start by choosing a proper base matrix (also called weight or degree matrix) or the corresponding base graph (also called seed graph \cite{Tanner2002} or protograph\cite{Thorpe2004}). The references \cite{Tanner2002, Fossorier2004} are focused on all-ones base matrices, while in \cite{Kim2007, Esmaeili2010} base matrices are constructed from Steiner Triple Systems and integer lattices. In both cases, a system of inequalities with integer coefficients describing all cycles of a given length is obtained and suitable labels or degrees are derived. For example, if we replace all nonzero entries in the base matrix by permutation matrices \cite{Thorpe2004, Fan2000}, circulant matrices \cite{Fossorier2004, TannerClass, Sullivan2006, Wang2008, Esmaeili2010}, or sums of circulant matrices \cite{Smarandache2004}, we obtain the corresponding QC LDPC block codes. On the other hand, if we replace all nonzero entries in the base matrix by either monomials or binomials, we obtain the corresponding (parent) LDPC convolutional codes\cite{Tanner2004, ISIT2010, Smarandache2004}.
Notice that both constructing the inequalities and the labeling require significant computational efforts. Some methods directed towards reducing the computational complexity of these steps can be found in \cite{Milenkovic2006, Wang2008}.
Parameters of the so far shortest QC LDPC block codes with $J=3$ and girth $6,8$, and $10$ found via computer search are presented in \cite{Wang2008}, improving previous results from \cite{Fossorier2004}.
As mentioned earlier, it is important that the constructed QC LDPC block codes have large minimum distance for achieving a suitable upper-bound on their error-correcting performance at high SNR. It is proved in \cite{Mackay} that the minimum distance of QC LDPC codes whose base parity-check matrices are $J \times K$ all-one matrices is upper-bounded by $(J+1)!$. However, considering base matrices with zeros leads to QC LDPC codes with larger minimum distance. For example, in \cite{Smarandache2004} it is shown that replacing all nonzero entries in the base matrix by sums of circulants and all zero entries by all-zero matrices, increases the minimum distance of the resulting code while preserving its regularity. For LDPC convolutional codes this approach implies that a parity-check matrix contains binomials instead of monomials. The corresponding upper-bound on the minimum distance of such LDPC codes is presented in \cite{Smarandache2004}. A particular case of this upper-bound, valid only for codes with zeros and monomials is derived in \cite{ISIT2009}.
In \secref{2}, we introduce notations for generator and parity-check matrices of convolutional codes and for their corresponding tailbiting block codes. \secref{3} focuses on bipartite graphs, biadjacency matrices, and their relation to parity-check matrices of LDPC block codes. Our constructions of base and voltage matrices, used when searching for LDPC block codes with large girth, are introduced in \secref{4}. Bounds on the girth and the minimum distance for QC $(J, K)$-regular LDPC block codes are discussed in \secref{5}. New search algorithms for QC LDPC block codes constructed from all-one matrices, Steiner Triple Systems, and QC regular matrices are presented in \secref{6}. Moreover, depending on the desired girth, algorithms of different complexity for constructing the set of inequalities and searching for suitable labelings are described. A new algorithm for computing the minimum distance of QC $(J, K)$-regular LDPC codes is described in \secref{7} and used to compute the minimum distance of our newly found codes. Moreover, we determined the hitherto unknown minimum distance for some of the shortest known LDPC codes given in \cite{Esmaeili2010}. In \secref{8}, we present new examples of $(J, K)$-regular QC LDPC codes in the form of tailbiting LDPC codes with girth between $10$ and $24$. This representation is compact and it is possible to apply low-complexity encoding, searching, and decoding procedures well developed for convolutional and tailbiting block codes \cite{RolfKam, beast}. In particular, the presented codes with girth $10$ and $12$ are shorter than the codes presented in \cite{Wang2008} and \cite{Sullivan2006, Zhang2010}, respectively. Moreover, our codes with girth $14$ to $18$ are shorter than the corresponding codes presented in \cite{Esmaeili2010}. \secref{9} concludes the paper with some final remarks.
\section{Generator and Parity-Check Matrices}\label{sec: 2}
Consider a rate $R=b/c$ binary convolutional code $\mathcal{C}$ with the semi-infinite generator matrix
\begin{IEEEeqnarray}{rCl}
\label{eq: semi-infinite-generator-matrix}
G & = & \left(\begin{array}{ccccccc}
G_0 & G_1 & \ldots & & G_{m_g} \\
& G_0 & G_1 & \ldots & & G_{m_g} \\
& & \ddots & \ddots & & & \ddots
\end{array}\right)
\end{IEEEeqnarray}
of memory $m_g$ where $G_i$, $i=0,1,\ldots,m_g$, are $b \times c$ binary matrices. Its semi-infinite syndrome former
\begin{IEEEeqnarray}{rCl}
H^T & = & \left(\begin{array}{ccccccc}
H^T_0 & H^T_1 & \ldots & & H^T_m \\
& H^T_0 & H^T_1 & \ldots & & H^T_m \\
& & \ddots & \ddots & & & \ddots
\end{array}\right)
\end{IEEEeqnarray}
of memory $m$, where in general $m \neq m_g$, $H_j$, $j=0,1,\ldots,m$, are $(c-b) \times c$ binary matrices, and $T$ denotes transpose. Clearly $G$ and $H$ satisfy
\begin{IEEEeqnarray}{rCl}
\label{eq: GH-equal-zero}
G H^T & = & \bs{0}
\end{IEEEeqnarray}
and
\begin{IEEEeqnarray}{rCl}
\bs{v} H^T & = & \bs{0}
\end{IEEEeqnarray}
where
\begin{IEEEeqnarray}{rCl}
\bs{v} & = & \bs{u} G
\end{IEEEeqnarray}
is the code sequence and $\bs{u}$ is the information sequence.
Next we tailbite the semi-infinite generator matrix \eqref{eq: semi-infinite-generator-matrix} to length $M$ $c$-tuples, where $M> \max \{m, m_g\}$. Then we obtain the $Mb \times Mc$ generator matrix of the quasi-cyclic (QC) block code $\mathcal{B}$ as
{\setlength{\arraycolsep}{2pt}
\begin{IEEEeqnarray}{rCl}
\label{eq: tb-generator-matrix}
G_{\text{TB}} & = & \left(\begin{array}{ccccccccc}
G_0 & G_1 & \ldots & & G_{m_g} \\
& G_0 & G_1 & \ldots & & G_{m_g} \\
& & \ddots & \ddots & & & \ddots \\
& & & G_0 & G_1 & \ldots & & G_{m_g} \\
G_{m_g} & & & & G_0 & G_1 & \ldots & G_{m_g - 1} \\
G_{m_g - 1} & G_{m_g} & & & & \ddots & \ddots & \vdots \\
\vdots & & \ddots & & & & \ddots & G_1 \\
G_1 & G_2 & \ldots & G_{m_g} & & & & G_0
\end{array}\right)\IEEEeqnarraynumspace
\end{IEEEeqnarray}}
Every cyclic shift of a codeword of $\mathcal{B}$ by $c$ places modulo $Mc$ is a codeword.
The corresponding tailbiting parity-check matrix is the $M(c-b) \times Mc$ matrix
{\setlength{\arraycolsep}{2pt}
\begin{IEEEeqnarray}{rCl}
\label{eq: tb-parity-check-matrix}
H_{\text{TB}} & = & \left(\begin{array}{ccccccccc}
H_0 & & & & H_m & H_{m-1} & \ldots & H_1 \\
H_1 & H_0 & & & & H_m & & H_2 \\
\vdots & H_1 & \ddots & & & & \ddots & \vdots \\
& \vdots & \ddots & H_0 & & & & H_m \\
H_m & & & H_1 & H_0 & & & \\
& H_m & & \vdots & H_1 & \ddots & & \\
& & \ddots & & \vdots & \ddots & \ddots & \\
& & & H_m & H_{m-1} & \ldots & H_1 & H_0
\end{array}\right)\IEEEeqnarraynumspace
\end{IEEEeqnarray}}
It is easily shown that $G_{\text{TB}}$ and $H_{\text{TB}}$ satisfy
\begin{IEEEeqnarray}{rCl}
G_{\text{TB}} H_{\text{TB}}^T & = & \bs{0}
\end{IEEEeqnarray}
given that \eqref{eq: GH-equal-zero} is fulfilled.
The parity-check matrix for the convolutional code $\mathcal{C}$ can also be written as the $(c-b) \times c$ polynomial matrix
\begin{IEEEeqnarray}{rCl}
H(D) & = & H_0 + H_1 D + H_2 D^2 + \cdots + H_m D^m
\end{IEEEeqnarray}
or, equivalently, as
{\setlength{\arraycolsep}{3pt}
\begin{IEEEeqnarray}{rCl}
\label{eq: convolutional-parity-check-matrix}
H(D) & = & \left(\begin{array}{cccc}
h_{11}(D) & h_{12}(D) & \ldots & h_{1c}(D) \\
h_{21}(D) & h_{22}(D) & \ldots & h_{2c}(D) \\
\vdots & \vdots & \ddots \\
h_{(c-b)1}(D) & h_{(c-b)2}(D) & \ldots & h_{(c-b)c}(D)
\end{array}\right)\IEEEeqnarraynumspace
\end{IEEEeqnarray}}
In the sequel we mostly consider parity-check matrices with only monomial entries $h_{ij}(D) = D^{w_{ij}}$ of degree $w_{ij}$, where $w_{ij}$ are nonnegative integers. Clearly, such a parity-check matrix $H(D)$ can be represented by its \textit{degree matrix} $W = \left( w_{ij} \right)$, $i=1,2,\ldots,c-b$ and $j=1,2,\ldots,c$. Note that starting from \secref{8} we will relax the restriction to only monomial entries and also include zero entries.
\begin{example}\label{ex: 1}
Consider the rate $R=1/4$ convolutional code $\mathcal{C}$ with parity-check matrix
\begin{IEEEeqnarray}{c}
\label{eq: example_conv_parity_check}
H(D) = \left( \begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & D & D \\
1 & D & 1 & D
\end{array}\right)
\end{IEEEeqnarray}
whose degree matrix is
\begin{IEEEeqnarray*}{c}
W = \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1
\end{array}\right)
\end{IEEEeqnarray*}
Tailbiting \eqref{eq: example_conv_parity_check} to length $M=2$, we obtain the tailbitten $6 \times 8$ parity-check matrix of a QC LDPC block code
\begin{IEEEeqnarray}{rCl}
\label{eq: example_tb_parity_check}
H_{\rm TB} & = & \borderarray{(}{)}{2em}{1.5ex}{ccccc|cccc}{
~~~~ & \sm{1}&\sm{2}&\sm{3}&\multicolumn{1}{c}{\sm{4}}&\sm{5}&\sm{6}&\sm{7}&\sm{8}\vspace{1mm}\\
\sm{1}& 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
\sm{2}& 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\
\sm{3}& 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ \cline{2-9}
\sm{4}& 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
\sm{5}& 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
\sm{6}& 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\
}
\end{IEEEeqnarray}
In particular, every cyclic shift of a codeword by $c=4$ places modulo $Mc=8$ is a codeword.
\end{example}
Due to the restriction to monomial entries in $H(D)$, $H_{\text{TB}}$ is $(J,K)$\textit{-regular}, that is, it has exactly $J$ and $K$ ones in each column and row, respectively. Moreover, to fulfill the \textit{low} density criterion, $M$ has to be much larger than $J$ and $K$, and thus the matrix $H_{\text{TB}}$ is sparse.
Note that the first $c$ columns of $H_{\text{TB}}$ are repeated throughout the whole matrix in a cyclicly shifted manner. By reordering the columns as $1,c+1,2c+1,\ldots,(M-1)c+1$, $2,c+2,2c+2,\ldots,(M-1)c+2$, \textit{etc.~} and the rows as $1,(c-b)+1,2(c-b)+1,\ldots,(M-1)(c-b)+1$, $2,(c-b)+2,2(c-b)+2,\ldots,(M-1)(c-b)+2$, \textit{etc.~} we obtain a parity-check matrix of an equivalent $(J, K)$-regular LDPC block code constructed from circulant matrices
\begin{IEEEeqnarray}{rCl}
\label{eq: circulant-ldpc_parity-check-matrix}
H_{\text{C}}
& = &
\left( \begin{array}{llll}
I_{w_{11}} & I_{w_{12}} & \cdots & I_{w_{1c}} \\
I_{w_{21}} & I_{w_{22}} & \cdots & I_{w_{2c}} \\
\cdots & \cdots & \cdots & \cdots \\
I_{w_{(c-b)1}} & I_{w_{(c-b)2}} & \cdots & I_{w_{(c-b)c}} \\
\end{array}\right)\IEEEeqnarraynumspace
\end{IEEEeqnarray}
where $w_{ij}$ are the entries of the degree matrix $W$ and $I_{w_{ij}}$ denotes an $M \times M$ \textit{circulant matrix}, that is, an identity matrix with its rows shifted cyclically to the left by $w_{ij}$ positions. Note, that the $(J, K)$-regular LDPC block code determined by $H_{\text{C}}$ is not quasi-cyclic, although equivalent to the QC block code determined by $H_{\text{TB}}$.
\begin{examplecontd}{1}
We return to \eqref{eq: example_tb_parity_check} in \exref{1} and reorder the columns as $1,5,2,6,3,7,4,8$ and the rows as $1,4,2,5,3,6$. Then we obtain the equivalent rate $R=1-6/8$ $(3,4)$-regular LDPC block code with parity-check matrix
\begin{IEEEeqnarray}{rl}
\label{eq: example_tb_qc_parity_check}
{\setlength{\arraycolsep}{-3pt}
\begin{array}{c}
\\ H_{\text{C}} =
\end{array}} &
\borderarray{(}{)}{2em}{1.5ex}{ccc|cc|cc|cc}{
~~~~ & \sm{1}&\multicolumn{1}{c}{\sm{5}}&\sm{2}&\multicolumn{1}{c}{\sm{6}}&\sm{3}&\multicolumn{1}{c}{\sm{7}}&\sm{4}&\sm{8}\vspace{1mm}\\
\sm{1}& 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\
\sm{4}& 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\cline{2-9}
\sm{2}& 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\
\sm{5}& 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\\cline{2-9}
\sm{3}& 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\
\sm{6}& 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\
}
\end{IEEEeqnarray}
\end{examplecontd}
\section{Graphs and Biadjacency Matrices}\label{sec: 3}
A graph $\mathcal{G}$ is determined by a set of \textit{vertices} $\mathcal{V} = \{v_i\}$ and a set of \textit{edges} $\mathcal{E} = \{e_i\}$, where each edge connects exactly two vertices. The \textit{degree of a vertex} denotes the number of edges that are connected to it. If all vertices have the same degree $c$, the \textit{degree of the graph} is $c$, or, in other words, the graph is $c$-\textit{regular}.
Consider the set of vertices $\mathcal{V}$ of a graph partitioned into $t$ disjoint subsets $\mathcal{V}_k$, $k=0,1,\ldots,t-1$. Such a graph is said to be $t$-\textit{partite}, if no edge connects two vertices from the same set $\mathcal{V}_k$, $k=0,1,\ldots,t-1$.
A \textit{path} of length $L$ in a graph is an alternating sequence of $L+1$ vertices $v_i$, $i=1,2,\ldots,L+1$, and $L$ edges $e_i$, $i=1,2,\ldots,L$, with $e_i \neq e_{i+1}$. If the first and the final vertex coincide, that is, if $v_1 = v_{L+1}$, then we obtain a \textit{cycle}. A cycle is called \textit{simple} if all its vertices and edges are distinct, except for the first and final vertex which coincide. The length of the shortest simple cycle is denoted the \textit{girth} of the graph. In \cite{Fossorier2004} it was proven that the girth of a graph coincide with the minimum distance of the block code, whose parity-check matrix corresponds to the \textit{incidence matrix} of the graph. Moreover, the girth determines the number of independent iterations in belief-propagation decoding \cite{Gallager1963}.
Every parity-check matrix $H$ of a rate $R=k/n$ LDPC block code can be interpreted as the \textit{biadjacency matrix}\cite{Asratian1998} of a bipartite graph, the so-called \textit{Tanner graph} \cite{Tanner1981}, having two disjoint subsets $\mathcal{V}_0$ and $\mathcal{V}_1$ containing $n$ and $n-k$ vertices, respectively. The $n$ vertices in $\mathcal{V}_0$ are called \textit{symbol nodes}, while the $n-k$ vertices in $\mathcal{V}_1$ are called \textit{constraint nodes}. If the underlying LDPC block code is $(J, K)$-regular, the symbol and constraint nodes have degree $J$ and $K$, respectively.
Consider the Tanner graph with the biadjacency matrix $H_{\text{TB}}$, corresponding to a QC $(J, K)$-regular LDPC code, obtained from the parity-check matrix of a tailbiting LDPC block code. Clearly, by letting the tailbiting length $M$ tend to infinity, we obtain a convolutional parity-check matrix $H(D)$ \eqref{eq: convolutional-parity-check-matrix} of the parent convolutional code $\mathcal{C}$. In terms of Tanner graph representations, this procedure corresponds to unwrapping the underlying graph and extending it in the time domain towards infinity. Hereinafter, we will call the girth of this infinite Tanner graph the \textit{free girth} and denote it $g_{\text{free}}$.
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.1]
\foreach \x in {1,2,3,...,8}
\path coordinate (S\x) at (1.1*\x, 3);
\foreach \x in {1,2,3,...,6}
\path coordinate (C\x) at (1.3*\x+0.6, 0);
\draw (C1) -- (S1); \draw (C1) -- (S2); \draw (C1) -- (S3); \draw (C1) -- (S4);
\draw (C2) -- (S1); \draw (C2) -- (S2); \draw (C2) -- (S7); \draw (C2) -- (S8);
\draw (C3) -- (S1); \draw (C3) -- (S3); \draw (C3) -- (S6); \draw (C3) -- (S8);
\draw (C4) -- (S5); \draw (C4) -- (S6); \draw (C4) -- (S7); \draw (C4) -- (S8);
\draw (C5) -- (S3); \draw (C5) -- (S4); \draw (C5) -- (S5); \draw (C5) -- (S6);
\draw (C6) -- (S2); \draw (C6) -- (S4); \draw (C6) -- (S5); \draw (C6) -- (S7);
\foreach \x in {1,2,3,...,8}
{
\draw[fill=black] (S\x) circle (3pt);
\node at ($(S\x) + (0,0.3)$) {\makebox(0,0)[b]{\small $s_\x$}};
}
\foreach \x in {1,2,3,...,6}
{
\draw[fill=white] (C\x) circle (3pt);
\node at ($(C\x) - (0,0.3)$) {\makebox(0,0)[t]{\small $c_\x$}};
}
\end{tikzpicture}
\caption{\label{fig: example_graph}Tanner graph with $8$ symbol nodes ($s_i$, $i=1,2,\ldots,8$) and $6$ constraint nodes ($c_i$, $i=1,2,\ldots,6$).}
\vspace{1em}
\end{figure}
\begin{examplecontd}{1}
Interpreting \eqref{eq: example_tb_qc_parity_check} as a biadjacency matrix, we obtain the corresponding Tanner graph $\mathcal{G}$ as illustrated in \figref{example_graph} with $8$ symbol nodes and $6$ constraint nodes, having girth $g=4$.
\end{examplecontd}
\section{Base Matrices, Voltages, and their Graphs}\label{sec: 4}
A binary matrix $B$ is called \textit{base matrix} for a tailbiting LDPC block code if its parent convolutional code with parity-check matrix $H(D)$ has only monomial or zero entries and satisfies
\begin{IEEEeqnarray}{c}
B = H(D)\big|_{D=1}
\end{IEEEeqnarray}
that is, all nonzero entries in $H(D)$ are replaced by $D^0 = 1$. Different tailbiting LDPC block codes can have the same base matrix $B$.
The \textit{base graph} $\mathcal{G}_{\text{B}}$ follows as the bipartite graph, whose biadjacency matrix is given by the base matrix $B$. Denote the girth of such a base graph $g_{\text{B}}$. The terminology ``base graph'' originates from graph theory and is used, for example, in \cite{Kelley2008}. It differs from the terminology used in \cite{Thorpe2004, Sullivan2006}, where \textit{protograph} or \textit{seed graph} are used.
Consider the additive group $(\Gamma, +)$, where $\Gamma=\{\gamma\}$. From the base graph $\mathcal{G}_{\text{B}} = \{\mathcal{E}_{\text{B}},\mathcal{V}_{\text{B}}\}$ we obtain the \emph{voltage graph} \cite{Gross1974,Exoo2008} $\mathcal{G}_{\text{V}} = \{\mathcal{E}_{\text{B}},\mathcal{V}_{\text{B}}, \Gamma\}$ by assigning a voltage value $\gamma(e,v,v^\prime)$ to the edge $e$ connecting the vertices $v$ and $v^\prime$, satisfying the property $\gamma(e,v,v^\prime)=-\gamma(e,v^\prime,v)$. Although the graph is not directed, the voltage of the edge depends on the direction in which the edge is passed. Finally, define the \emph{voltage of the path} to be the sum of the voltages of its edges.
Let $\mathcal{G} = \{\mathcal{E},\mathcal{V}\}$ be a \emph{lifted graph} obtained from a voltage graph $\mathcal{G}_{\text{V}}$, where $\mathcal{E}=\mathcal{E}_{\text{B}}\times \Gamma$ and $\mathcal{V}=\mathcal{V}_{\text{B}}\times \Gamma$. Two vertices $(v,\gamma)$ and $(v^\prime,\gamma^\prime)$ are connected in the lifted graph by an edge if and only if $v$ and $v^\prime$ are connected in the voltage graph $\mathcal{G}_{\text{V}}$ with the voltage value of the corresponding edge given by $\gamma(e,v,v^\prime) = \gamma - \gamma^\prime$.
It is easy to see that cycles in the lifted graph correspond to cycles in the voltage graph with zero voltage. Consequently, the girth $g_{\text{V}}$ of a voltage graph follows as the length of its shortest cycle with voltage zero, which is equal to the free girth $g_{\text{free}}$ \cite{Sullivan2006, Milenkovic2006}. A voltage assignment corresponds directly to selecting the degrees of the parity-check monomials in $H(D)$.
In the following we start from a base graph $\mathcal{G}_{\text{B}}$ and use a voltage assignment based on the monomial degrees $w_{ij}$ of the degree matrix $W$ to determine the corresponding voltage graph $\mathcal{G}_{\text{V}}$. The edge voltage from the constraint node $c_i$ to the symbol node $s_j$ is denoted by $\mu_{ij}$ while the opposite direction from symbol node $s_j$ to constraint node $c_i$ follows as $\bar{\mu}_{ji}$, $i=1,2,\ldots,(c-b)$ and $j=1,2,\ldots,c$, where
\begin{IEEEeqnarray}{C}
\label{eq: voltage_labeling}
\left\{ \,
\begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode\IEEEeqnarraystrutsizeadd{2pt}{2pt}][c]{rCl}
\mu_{i j} & = & w_{ij} \\
\bar{\mu}_{j i} & = & -w_{ij}
\end{IEEEeqnarraybox} \right.
\end{IEEEeqnarray}
When searching for LDPC convolutional codes with given free girth $g_{\text{free}}$, we use integer edge voltages, that is, we deal with an infinite additive group. However, when searching for QC LDPC block codes with given girth $g$, obtained by tailbiting a parent convolutional code to length $M$, we use a group of modulo $M$ residues, that is, \eqref{eq: voltage_labeling} is replaced by
\begin{IEEEeqnarray}{C}
\label{eq: voltage_labeling_mod_M}
\left\{ \,
\begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode\IEEEeqnarraystrutsizeadd{2pt}{2pt}][c]{rCll}
\mu_{i j} & = & w_{ij} & \mod M\\
\bar{\mu}_{j i} & = & -w_{ij} & \mod M
\end{IEEEeqnarraybox} \right.
\end{IEEEeqnarray}
The definitions of path and cycle in a voltage graph coincide with those in a regular graph, except for the additional restriction that two neighboring edges may not connect the same nodes in reversed order. The \textit{voltage} of a path or cycle within a voltage graph, follows as the sum of all edge voltages involved.
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.1]
\foreach \x in {1,2,3,4}
\path coordinate (S\x) at (2.5*\x, 2.7);
\foreach \x in {1,2,3}
\path coordinate (C\x) at (2.5*\x+1.25, 0);
\draw[->, shorten >=5pt] (C1) -- (S1) [xshift=-5pt] node[pos=0.3]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C1) -- (S2) [xshift=-5pt] node[pos=0.9]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C1) -- (S3) [xshift=-8pt] node[pos=0.9]{\scriptsize $0$};
\draw[->, shorten >=6pt] (C1) -- ($(S4) + (0,0.1)$) [xshift=-10pt] node[pos=0.95]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C2) -- (S1) [xshift=-6pt] node[pos=0.6]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C2) -- (S2) [xshift=-4pt] node[pos=0.7]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C2) -- (S3) [xshift=+4pt] node[pos=0.7]{\scriptsize $1$};
\draw[->, shorten >=5pt] (C2) -- (S4) [xshift=+6pt] node[pos=0.6]{\scriptsize $1$};
\draw[->, shorten >=6pt] (C3) -- ($(S1) + (0,0.1)$) [xshift=+10pt] node[pos=0.95]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C3) -- (S2) [xshift=+8pt] node[pos=0.9]{\scriptsize $1$};
\draw[->, shorten >=5pt] (C3) -- (S3) [xshift=+5pt] node[pos=0.9]{\scriptsize $0$};
\draw[->, shorten >=5pt] (C3) -- (S4) [xshift=+5pt] node[pos=0.3]{\scriptsize $1$};
\foreach \x in {1,2,3,4}
{
\draw[fill=black] (S\x) circle (5pt);
\node at ($(S\x) + (0,0.3)$) {\makebox(0,0)[b]{\small $s_\x$}};
}
\foreach \x in {1,2,3}
{
\draw[fill=white] (C\x) circle (5pt);
\node at ($(C\x) - (0,0.3)$) {\makebox(0,0)[t]{\small $c_\x$}};
}
\end{tikzpicture}
\vspace{1em}
\caption{\label{fig: voltage_graph} Bipartite graph with $4$ symbol nodes ($s_i$, $i=1,2,3,4$) and $3$ constraint nodes ($c_i$, $i=1,2,3$). Since the edges are labeled according to \eqref{eq: voltage_labeling}, this corresponds to a voltage graph.}
\end{figure}
\begin{examplecontd}{1}
The bipartite graph whose biadjacency matrix is given by the base matrix $B$ of the rate $R=1/4$ $(3, 4)$-regular LDPC convolutional code $\mathcal{C}$ is illustrated in \figref{voltage_graph}. As the edges are labeled according to \eqref{eq: voltage_labeling}, \figref{voltage_graph} corresponds to a voltage graph with girth $g_{\text{V}} = 4$ (for example, $s_1$ $\to$ $c_1$ $\to$ $s_2$ $\to$ $c_2$ $\to$ $s_1$). The edge from, for example, constraint node $c_2$ to symbol node $s_3$ is labeled according to
\begin{IEEEeqnarray*}{rClClCl}
\mu_{23} & = & -\bar{\mu}_{32} & = & w_{23} & = & 1
\end{IEEEeqnarray*}
As the free girth of the infinite Tanner graph, corresponding to the parent convolutional code $\mathcal{C}$, determined by the convolutional parity-check matrix $H(D)$ is equal to the girth of the voltage graph, we can conclude that $g_{\text{free}} = g_{\text{V}} = 4$.
If we neglect all edge labels, we would obtain the corresponding base graph.
\end{examplecontd}
\section{Bounds on the girth and the minimum distance of $(J \geq 3, K)$ QC LDPC block codes}\label{sec: 5}
There are a number of approaches which can be applied to construct and search for QC $(J=2,K)$-regular LDPC block and convolutional codes \cite{PPI} or the bipartite graphs constructed by their incidence matrices. Since every LDPC convolutional code can be represented by a bipartite Tanner graph using the biadjacency matrix, these techniques can be applied to $(J\ge 3,K)$ QC LDPC codes. Moreover, bounds on the girth and the minimum distance of $(J=2,K)$ QC LDPC codes \cite{PPI} can be generalized to an arbitrary $J$.
\begin{theorem}\label{th: one}
The minimum distance $d_{\text{min}}$ and the girth $g$ of an $(n,k,d_{\text{min}})$ QC LDPC block code $\mathcal{B}$ obtained from a rate $R=b/c$ convolutional code $\mathcal{C}$ with free distance $d_{\text{free}}$ and girth $g_{\rm free}$ by tailbiting to length $M$ are upper-bounded by the inequalities
\begin{IEEEeqnarray*}{rCl}
d_{\rm min} &\leq & d_{\rm free} \\
g &\leq & g_{\rm free}
\end{IEEEeqnarray*}
\end{theorem}
\begin{IEEEproof}
The first statement follows directly from the fact that any codeword $\bs{v}(D)$ of the tailbiting block code $\mathcal{B}$, obtained from the parity-check matrix $H(D)$ of the parent convolutional code $\mathcal{C}$, satisfies
\begin{IEEEeqnarray}{c}
\label{eq: proof_theorem_1}
\bs{v}(D) H^T(D) = \bs{0} \mbox{ mod } (D^M-1)
\end{IEEEeqnarray}
Since the parent convolutional code $\mathcal{C}$ satisfies \eqref{eq: proof_theorem_1} without reduction modulo $(D^M-1)$ and reducing modulo $(D^M-1)$ does not increase the weight of a polynomial, the first statement follows directly.
For the second statement we consider the voltage graph $\mathcal{G}_{\text{V}}$ representation of the parent convolutional code $\mathcal{C}$ with girth $g_{\text{V}} = g_{\text{free}}$ together with the Tanner graph representation of the QC LDPC block code $\mathcal{B}$ with girth $g$. Similar to the relations between the free distance $d_{\text{free}}$ and the minimum distance $d_{\text{min}}$, there exists a relation between each cycle within the voltage graph $\mathcal{G}_{\text{V}}$ of the parent convolutional code and the Tanner graph $\mathcal{G}$ of the corresponding block code obtained by tailbiting to length $M$. The edge voltages for every cycle in $\mathcal{G}_{\text{V}}$ have to sum up to zero. Similarly, every cycle in $\mathcal{G}$ corresponds to a cycle in $\mathcal{G}_{\text{V}}$ such that its edge voltages have to sum up to zero modulo $M$. With the same argument as before it follows directly that
\begin{IEEEeqnarray*}{rClCl}
g & \leq & g_{\text{V}} & = & g_{\text{free}}
\end{IEEEeqnarray*}
\end{IEEEproof}
In \cite{PPI} a lower bound on the girth of a voltage graph $g_{\text{V}}$ was found via the girth of the corresponding base graph $g_{\text{B}}$ for ordinary graphs. It is straightforward to generalize this bound:
Consider a base graph of a QC $(J \geq 3, K)$-regular LDPC convolutional code with girth $g_{\text{B}}$ and let $d_s$ denote the $s$th generalized minimum Hamming distance of the linear $M((J-2)c+b) \times JMc$ block code determined by the parity-check matrix which corresponds to the incidence matrix of the Tanner graph. In order words, $d_s$ corresponds to the number of nontrivial (not identically zero) positions of an $s$-dimensional linear subcode.
\begin{theorem}\label{th: two}
There exist a tailbiting length $M$ and a voltage assignment, such that the girth $g$ of the Tanner graph for the corresponding TB block code of length $N=Mc$ satisfies the inequality
\begin{IEEEeqnarray}{rClCl}
\label{eq: girth_lower_bound}
g & \geq & 2 \max \left\{ g_{\text{B}} + \left\lceil g_{\text{B}}/2 \right\rceil, d_2 \right\} & \geq & 3g_{\text{B}}
\end{IEEEeqnarray}
where $d_2$ is the second generalized minimum Hamming distance, that is, the minimum support of a subcode of dimension two. We have equality in \eqref{eq: girth_lower_bound}, if the underlying base graph consists of two connected cycles, having at least one common vertex.
\end{theorem}
\begin{IEEEproof}
According to \thref{one}, any cycle in the Tanner graph of a QC LDPC block code corresponds to a cycle of the same length in the voltage graph. As the labels of the voltage Tanner graph can be freely chosen, it is enough to prove that there is no zero cycle shorter than $2(g_{\text{B}} + \lceil g_{\text{B}}/2 \rceil)$, that is, no such cycle whose voltage is zero regardless of the labeling of the base graph. In particular, such a cycle is also known as an \textit{inevitable cycle} \cite{Kim2007} or \textit{balanced cycle} \cite{Sullivan2006}. The number of times each edge in such a cycle of the voltage graph is passed in different directions has to be even. This cycle cannot be simple, since in a simple cycle each edge is passed in one direction only. Hence, the cycle passes through the vertices of a subgraph which contains at least two different cycles, corresponding to two different nonzero codewords. The minimum distance of the base graph is $g_{\text{B}}$, and, according to the Griesmer bound, the smallest length of a linear code with two nonzero codewords of minimum distance $d$ is $d+ \lceil d/2 \rceil$, and, hence, the first lower bound of inequality \eqref{eq: girth_lower_bound} follows.
Consider the second lower bound. The definition of the second generalized minimum Hamming distance implies that the smallest subgraph with two cycles has to have at least $d_2$ edges. Thus, the second of the two lower bounds gives the precise value of the girth of a subgraph containing two connected cycles, having at least one common vertex. Otherwise, $d_2$ is a lower bound.
\end{IEEEproof}
The bounds are tighter than the $3g_{\text{B}}$ bound \cite{Kim2007, Kelley2008} but not tight if the shortest non-simple cycle consists of two simple cycles connected by a path.
Finally, we want to recall an upper bound on the achievable girth and minimum distance. We start by reformulating the theorem on the achievable girth by Fossorier \cite{Fossorier2004} and thereby generalize it to include base matrices with zero elements.
\begin{theorem}\label{th: three-a}
Consider the parity-check matrix $H(D)$ of a rate $R=b/c$ convolutional code with base matrix $B$. Denote the corresponding base graph $\mathcal{G}_{\text{B}}$ and let $B^\prime$ be the $2 \times 3$ submatrix
\begin{equation}
B^\prime = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right)
\end{equation}
If the base matrix $B$, after possibly reordering its rows and columns, contains the submatrix $B^\prime$, then the girth $g_{\text{V}}$ of the corresponding voltage graph $\mathcal{G}_{\text{V}}$ is upper-bounded by
\begin{equation}
g_{\text{V}} \leq 12 \label{eq: upper_bound_girth}
\end{equation}
regardless of the voltage assignment.
\end{theorem}
\begin{IEEEproof}
The subgraph determined by the $2 \times 3$ submatrix $B^\prime$ contains $3$ symbol nodes, $2$ constraint nodes, and $6$ edges. Moreover, there exist $3$ shortest cycles of length $4$. Thus, the base graph $\mathcal{G}_{\text{B}}$ has girth $g_{\text{B}}=4$ and its second generalized Hamming distance is $d_2 = 6$. Applying \thref{two}, we obtain the precise value of the achievable girth as $2d_2 = 12$, which completes the proof.
\end{IEEEproof}
For parity-check matrices with only nonzero monomial entries, the inequality \eqref{eq: upper_bound_girth} was proven in \cite{Fossorier2004}.
Moreover, let $H(D)$ be the parity-check matrix of a rate $R=b/c$ $(J, K)$-regular LDPC convolutional code with free distance $d_{\text{free}}$. By tailbiting to length $M$ we obtain a QC LDPC block code of block length $Mc$ and minimum distance $d_{\text{min}}$. As proven in \cite{Mackay} for parity-check matrices without zero elements and reformulated in \cite{ISIT2009} for parity-check matrices with zero elements, the corresponding minimum distance $d_{\text{min}}$ can be upper-bounded by
\begin{equation}
d_{\text{min}} \leq d_{\text{free}} \leq (c-b+1)! \label{eq: upper_bound_minimum_distance}
\end{equation}
For parity-check matrices with only nonzero monomials, the inequality simplifies to $(J+1)!$.
\section{Searching for QC LDPC block codes with large girth}\label{sec: 6}
When searching for QC LDPC block codes with large girth, we start from a base graph of a rate $R=b/c$ $(J, K)$-regular LDPC convolutional code. Using the following algorithm, we determine a suitable voltage assignment based on the group of nonnegative integers, such that the girth of this voltage graph is greater than or equal to a given girth $g$. Afterwards we replace all edge labels by their corresponding modulo $M$ residuals, where we try to minimize $M$ while preserving the girth $g$. Using the concept of biadjacency matrices leads to the corresponding degree matrix $W$ and we obtain the parity-check matrix of a convolutional code whose bipartite graph has girth $g = g_{\text{free}}$. Tailbiting to lengths $M$, yields the rate $R=Mb/Mc$ QC LDPC block code whose parity-check matrix is equal to the biadjacency matrix of a bipartite graph with girth $g$.
The algorithm for determining a suitable voltage assignment for a base graph consists of the following two main steps:
\begin{enumerate}
\item Construct a list containing all inequalities describing cycles of length smaller than $g$ within the base graph.
\item Search for such a voltage assignment of the base graph that all inequalities are satisfied.
\end{enumerate}
The efficiency of the second step, searching for a suitable voltage assignment, depends on the chosen representation for the list of inequalities determined during the first step. In general, when searching for all cycles of length $g$ roughly $(J-1)^g$ different paths have been considered. However, by using a similar idea as in \cite{beast} when searching for a path within a trellis, we create a tree of maximum depth $g/2$ and search only for identical nodes within the tree and thereby reduce the complexity to roughly $(J-1)^{g/2}$.
\subsection*{Creating a tree structure}
Utilizing the base graph of a rate $R=b/c$ $(J, K)$-regular LDPC convolutional code, with $c$ symbol and $c-b$ constraint nodes, we construct a separate subtree starting with each of the $c$ symbol nodes.
Before describing the algorithm, we have to introduce some notations. A node in the tree will be denoted by $\xi$ and has a unique parent node $\xi^{\text{p}}$. The underlying base graph is bipartite, that is, every node $\xi$ in the tree with $\xi \in \mathcal{V}_i$ is only connected to nodes $\xi^\prime \in \mathcal{V}_{j}$ with $i,j \in \{0,1\}$, $i \neq j$, where $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ are the sets of symbol and constraint nodes, respectively. In other words, a symbol node is only connected to constraint nodes and vice versa. Moreover, every node $\xi$ is characterized by its depth $\ell(\xi)$ and its number $n(\xi)$, where $n(\xi) = i$ follows directly from $\xi = s_i$ or $\xi = c_i$ depending on whether its depth $\ell(\xi)$ is even or odd. In particular, $\xi \in \mathcal{V}_{\ell(\xi) \mod 2}$.
Having introduced those basic notations, we can grow $c$ separate subtrees, with the root node $\xi_{i,\text{root}}$ of the $i$th subtree being initialized with $\xi \in \mathcal{V}_0$ and depth $\ell(\xi) = 0$. Extend every node $\xi \in \mathcal{V}_i$ at depth $\ell(\xi) = n$ with $i = n \mod 2$ by connecting it with the nodes $\xi^\prime \in \mathcal{V}_{i+1 \mod 2}$ at depth $n + 1$ according to the underlying base graph, except $\xi^\text{p}$ which is already connected to $\xi$ at depth $n - 1$.
Finally we label the edges according to \eqref{eq: voltage_labeling} and obtain the voltage for node $\xi$ in the $i$th subtree as the sum of the edge voltages of the path $\xi_{i,\text{root}} \to \xi$.
\begin{figure}
\centering
\begin{tikzpicture}
\foreach \x in {1,2,3,4}
\path coordinate (S\x) at (2.5*\x, 2.7);
\foreach \x in {1,2,3}
\path coordinate (C\x) at (2.5*\x+1.25, 0);
\draw[->, shorten >=5pt] (C1) -- (S1) [xshift=-10pt] node[pos=0.3]{\scriptsize $\mu_{1 1}$};
\draw[->, shorten >=5pt] (C1) -- (S2) [xshift=-10pt] node[pos=0.9]{\scriptsize $\mu_{1 2}$};
\draw[->, shorten >=5pt] (C1) -- (S3) [xshift=-13pt] node[pos=0.9]{\scriptsize $\mu_{1 3}$};
\draw[->, shorten >=6pt] (C1) -- ($(S4) + (0,0.1)$) [xshift=-15pt] node[pos=0.95]{\scriptsize $\mu_{1 4}$};
\draw[->, shorten >=5pt] (C2) -- (S1) [xshift=-10pt] node[pos=0.6]{\scriptsize $\mu_{2 1}$};
\draw[->, shorten >=5pt] (C2) -- (S2) [xshift=-9pt] node[pos=0.7]{\scriptsize $\mu_{2 2}$};
\draw[->, shorten >=5pt] (C2) -- (S3) [xshift=+11pt] node[pos=0.7]{\scriptsize $\mu_{2 3}$};
\draw[->, shorten >=5pt] (C2) -- (S4) [xshift=+13pt] node[pos=0.6]{\scriptsize $\mu_{2 4}$};
\draw[->, shorten >=6pt] (C3) -- ($(S1) + (0,0.1)$) [xshift=+14pt] node[pos=0.95]{\scriptsize $\mu_{3 1}$};
\draw[->, shorten >=5pt] (C3) -- (S2) [xshift=+12pt] node[pos=0.9]{\scriptsize $\mu_{3 2}$};
\draw[->, shorten >=5pt] (C3) -- (S3) [xshift=+11pt] node[pos=0.9]{\scriptsize $\mu_{3 3}$};
\draw[->, shorten >=5pt] (C3) -- (S4) [xshift=+11pt] node[pos=0.3]{\scriptsize $\mu_{3 4}$};
\foreach \x in {1,2,3,4}
{
\draw[fill=black] (S\x) circle (5pt);
\node at ($(S\x) + (0,0.3)$) {\makebox(0,0)[b]{\small $s_\x$}};
}
\foreach \x in {1,2,3}
{
\draw[fill=white] (C\x) circle (5pt);
\node at ($(C\x) - (0,0.3)$) {\makebox(0,0)[t]{\small $c_\x$}};
}
\end{tikzpicture}
\caption{\label{fig: voltage_graph_variables} A bipartite voltage graph with $4$ symbol nodes ($s_i$, $i=1,2,3,8$) and $3$ constraint nodes ($c_i$, $i=1,2,3$) with its edges labeled according to \eqref{eq: voltage_labeling}.}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.2, transform shape]
\path coordinate (start) at (0, 5.6);
\draw[fill=black] (start) circle (5pt);
\node[below] at ($(start) - (0,0.15)$) {\small $s_1$};
\foreach \x in {1,2,3}
{
\path coordinate (C\x) at (2.8, 2.8*\x);
\foreach \y in {2,3,4}
\path coordinate (S\x\y) at (5, 2.8*\x-2.4+0.8*\y);
}
\draw[->, shorten >=6pt] (start) -- (C1) [xshift=+14pt] node[pos=0.4]{\scriptsize $-\mu_{1 1}$};
\draw[->, shorten >=6pt] (start) -- (C2) [yshift=+5pt, xshift=+14pt] node[pos=0.4]{\scriptsize $-\mu_{2 1}$};
\draw[->, shorten >=6pt] (start) -- (C3) [xshift=+14pt] node[pos=0.4]{\scriptsize $-\mu_{3 1}$};
\draw[->, shorten >=6pt] (C1) -- (S12) [yshift=-5pt] node[pos=0.4]{\scriptsize $\mu_{1 2}$};
\draw[->, shorten >=6pt] (C1) -- (S13) [yshift=+5pt] node[pos=0.65]{\scriptsize $\mu_{1 3}$};
\draw[->, shorten >=6pt] (C1) -- (S14) [yshift=+9pt] node[pos=0.4]{\scriptsize $\mu_{1 4}$};
\draw[->, shorten >=6pt] (C2) -- (S22) [yshift=-5pt] node[pos=0.4]{\scriptsize $\mu_{2 2}$};
\draw[->, shorten >=6pt] (C2) -- (S23) [yshift=+5pt] node[pos=0.65]{\scriptsize $\mu_{2 3}$};
\draw[->, shorten >=6pt] (C2) -- (S24) [yshift=+9pt] node[pos=0.4]{\scriptsize $\mu_{2 4}$};
\draw[->, shorten >=6pt] (C3) -- (S32) [yshift=-5pt] node[pos=0.4]{\scriptsize $\mu_{3 2}$};
\draw[->, shorten >=6pt] (C3) -- (S33) [yshift=+5pt] node[pos=0.65]{\scriptsize $\mu_{3 3}$};
\draw[->, shorten >=6pt] (C3) -- (S34) [yshift=+9pt] node[pos=0.4]{\scriptsize $\mu_{3 4}$};
\foreach \x in {1,2,3}
{
\draw[fill=white] (C\x) circle (5pt);
\node[below] at ($(C\x) - (0,0.15)$) {\small $c_\x$};
\foreach \y in {2,3,4}
{
\draw[fill=black] (S\x\y) circle (5pt);
\node[below] at ($(S\x\y) - (0,0.15)$) {\small $s_\y$};
}
}
\end{tikzpicture}
\caption{\label{fig: tree} A tree representation with maximum depth two, starting with symbol node $s_1$.}
\end{figure}
Clearly, all subtrees together contain all paths of a given length in the voltage graph. Moreover, taking into account that the girth $g$ of bipartite graphs is always even, we can conclude that in order to check all possible cycles of length at most $g-2$ in the voltage graph, it is sufficient to grow the corresponding $c$ subtrees up to depth $(g - 2)/2$ and to construct voltage inequalities for all node pairs $(\xi, \xi^\prime)$ in the same subtree $i$ with the same number $n(\xi) = n(\xi^\prime)$ and depth $\ell(\xi) = \ell(\xi^\prime)$ but different parent nodes $\xi^{\text{p}} \neq \xi^{\prime\text{p}}$.
The corresponding voltage inequality for the node pair $(\xi, \xi^\prime)$ follows directly as the difference between the voltages for the paths from $\xi_{i,\text{root}}$ to $\xi$ and $\xi^\prime$, respectively, that is, $(\xi_{i,\text{root}} \to \xi) - (\xi_{i,\text{root}} \to \xi^\prime)$.
If there exists a cycle $g^\prime < g$, then at depth $g^\prime/2$ there exists at least one pair of nodes $(\xi, \xi^\prime)$, whose corresponding voltage inequality is not satisfied, that is, is equal to zero. If there is no cycle shorter than $g$ in the voltage graph, then there are no such pairs, and all voltage inequalities are satisfied.
\begin{example}\label{ex: 2}
Consider the rate $R=1/4$ $(3, 4)$-regular LDPC convolutional code given by \eqref{eq: example_conv_parity_check}. The corresponding base graph, with four symbol nodes $s_i \in \mathcal{V}_0$, $i=1,2,3,4$, and three constraint nodes $c_i \in \mathcal{V}_1$, $i=1,2,3$, is illustrated in \figref{voltage_graph_variables}.
In the following, we shall search for a set of edge labels, that is, monomial degrees in $W$, such that the corresponding voltage graph has at least a given girth $g$. Thus, we label the branches by the general edge voltages according to \eqref{eq: voltage_labeling} and obtain a bipartite voltage graph.
In order to find a suitable labeling for the edge voltages from the $i$th constraint node $c_i$ to the $j$th symbol node $s_i$, that is $\mu_{ij}$, $i=1,2,3$, $j=1,2,3,4$, whose underlying voltage graph has at least girth $g = 6$, we have to grow $4$ subtrees up to length $(g-2)/2 = 2$, with their root nodes being initialized by $s_i$, $i=1,2,3,4$.
The subtree with root node $s_1$ is illustrated in \figref{tree}. Clearly, at depth $\ell(\xi)=1$ there are no identical nodes, while at depth $\ell(\xi)=2$ there are $3 \times {3 \choose 2} = 9$ pairs of identical nodes ($n(\xi) = n(\xi^\prime)$), but with different parents. Taking into account that a similar subtree is constructed using the remaining three symbol nodes $s_2$, $s_3$ and $s_4$ as root nodes, we obtain in total $36=4 \times 9$ node pairs, which all correspond to a voltage inequality.
For example, the voltage inequality obtained by checking all node pairs $(\xi, \xi^\prime)$ with $\xi = s_2$, that is, $n(\xi) = 2$, at depth $\ell(\xi)=2$ in the subtree starting with symbol node $s_1$, are
\begin{IEEEeqnarray*}{rCl}
-\mu_{11} + \mu_{12} - \mu_{22} + \mu_{21} & \neq & 0 \\
-\mu_{11} + \mu_{12} - \mu_{32} + \mu_{31} & \neq & 0 \\
-\mu_{21} + \mu_{22} - \mu_{32} + \mu_{31} & \neq & 0
\end{IEEEeqnarray*}
Note that amongst all $36$ voltage inequalities, there are only $18$ unique ones.
\end{example}
\begin{algorithm}[~TR]{Constructing a tree representation}
\begin{enumerate}
\item Grow $c$ separate subtrees according to the underlying base graph up to depth $g/2 -1$, with the root node $\xi_{i,\text{root}}$ of the $i$th subtree being initialized with $\xi \in \mathcal{V}_0$ and depth $\ell(\xi) = 0$.
\item Extend every node $\xi \in \mathcal{V}_i$ at depth $\ell(\xi) = n$ with $i = n \mod 2$ by connecting it with the nodes $\xi^\prime \in \mathcal{V}_{i+1 \mod 2}$ at depth $n + 1$ according to the underlying base graph, except $\xi^\text{p}$ which is already connected to $\xi$ at depth $n - 1$. Denote the set of all nodes within the $i$th subtree by $\mathcal{T}_i$.
\end{enumerate}
\end{algorithm}
\subsection*{Searching for a suitable voltage assigment}
Using the obtained subtrees $\mathcal{T}_i$, $i=1,2,\ldots,c$, with maximum depth $g/2 - 1$, we have found all cycles of length smaller than or equal to $g-2$ as well as their corresponding voltage inequalities.
The same cycle might be found several times within the $c$ subtrees. Moreover, two different cycles can correspond to the same voltage inequality.
We continue by creating a reduced list $\mathcal{L}$ of node pairs $(\xi, \xi^\prime)$ of all $c$ subtrees $\mathcal{T}_i$, $i=1,2,\ldots,c$, containing all unique voltage inequalities. Thereby, we remove all duplicate cycles, as well as different cycles corresponding to the same voltage inequality. Using the reduced list $\mathcal{L}$ we can reduce the obtained $c$ subtrees $\mathcal{T}_i$, $i=1,2,\ldots,c$, in a similar way by removing all nodes, not participating in any of the cycles in $\mathcal{L}$, and denote the reduced subtree by $\mathcal{T}_{i,\text{min}}$. In other words, we remove all nodes in $\mathcal{T}_i$, $i=1,2,\ldots,c$, which only participate in already known cycles or new cycles with already known voltage inequalities.
In the following we present two different approaches for finding suitable edge labels (edge voltages), which we shall denote as Algorithm A and Algorithm B.
In Algorithm A, we label the edges of the reduced subtrees $\mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$, with a set of randomly chosen voltages. For every node pair $(\xi, \xi^\prime)$ in the list $\mathcal{L}$, we calculate the voltage of the corresponding cycle as the difference of the path voltages $\xi_{i,\text{root}} \to \xi$ and $\xi_{i,\text{root}} \to \xi^\prime$. If none of these cycle voltages is equal to zero, the girth of the underlying base graph with such a voltage assignment is greater than or equal to $g$.
In Algorithm B, we discard the list $\mathcal{L}$ and only focus on the $c$ reduced subtrees $\mathcal{T}_{i,\text{min}}$. After labeling their edges with a set of randomly chosen voltages, we sort the nodes $\xi$ of each subtree according to their path voltage $\xi_{i,\text{root}} \to \xi$. If there exists no pair of nodes $(\xi, \xi^\prime)$ with the same path voltage, number $n(\xi)=n(\xi^\prime)$, and depth $\ell(\xi)=\ell(\xi^\prime)$, but different parent nodes $\xi^\text{p} \neq \xi^{\prime\text{p}}$, the girth of the underlying base graph with such a voltage assignment is greater than or equal to $g$.
A formal description of those two algorithms is given below:
\begin{algorithm}[~A]{Constructing a system of voltage inequalities and searching for an optimum voltage assignment using a list}
\begin{enumerate}
\item Create a reduced list $\mathcal{L}$ of node pairs $(\xi, \xi^\prime)$ for all $c$ subtrees $\mathcal{T}_i$, $i=1,2,\ldots,c$, containing all node pairs $(\xi, \xi^\prime)$ with a unique voltage inequality, having the same number $n(\xi) = n(\xi^\prime)$, depth $\ell(\xi) = \ell(\xi^\prime)$, but different parent nodes $\xi^{\text{p}} \neq \xi^{\prime\text{p}}$.
\item Reduce each of the $c$ subtrees $\mathcal{T}_i$ by removing all nodes, which do not participate in any of the found cycles corresponding to the voltage inequalities in $\mathcal{L}$, and denote the reduced subtree structure by $\mathcal{T}_{i,\text{min}}$.
\item Assign randomly chosen voltages to the edges of all trees and perform the following steps:
\begin{enumerate}
\item Find the voltages for all paths leading from the root node $\xi_{i,\text{root}}$ of the $i$th reduced subtree $\mathcal{T}_{i,\text{min}}$ to all nodes $\xi \in \mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$.
\item Determine the voltage inequality for all cycles $(\xi, \xi^\prime) \in \mathcal{L}$, as the difference of the corresponding path voltages in $\mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$, computed previously.
\item If all voltage inequalities are satisfied, the girth of the underlying base graph with such a voltage assignment is greater than or equal to $g$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
\begin{algorithm}[~B]{Constructing a system of voltage inequalities and searching for an optimum voltage assignment using a tree}
\begin{enumerate}
\item Construct the reduced list $\mathcal{L}$ and the reduced subtrees $\mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$, as in Algorithm A without storing the corresponding list $\mathcal{L}$.
\item Assign randomly chosen voltages to the edges of all trees and perform the following steps:
\begin{enumerate}
\item Find the voltages for all paths from the root node $\xi_{i,\text{root}}$ to all nodes within $\mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$, and sort all elements within $\mathcal{T}_{i,\text{min}}$ according to their voltages.
\item Search for a pair of nodes $(\xi, \xi^\prime)$ in the sorted list with the same path voltage, number $n(\xi)=n(\xi^\prime)$, and depth $\ell(\xi)=\ell(\xi^\prime)$, but different parent nodes $\xi^\text{p} \neq \xi^{\prime\text{p}}$.
\item If no such pair exists, then the girth of the corresponding voltage graph with such a voltage assignment is greater than or equal to $g$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
\subsection*{Complexity}
Denote the sum of all nodes in the reduced tree $\mathcal{T}_{i,\text{min}}$, $i=1,2,\ldots,c$, and the number of unique inequalities in the list $\mathcal{L}$ by $N_{\text{T}}$ and $N_{\text{L}}$, respectively, that is,
\begin{IEEEeqnarray*}{rCltRCl}
N_{\text{T}} & = & \sum_{i=1}^c \left| \mathcal{T}_{i,\text{min}}\right| & \quad and \quad &
N_{\text{L}} & = & \left| \mathcal{L} \right|
\end{IEEEeqnarray*}
where $\left| \mathcal{X} \right|$ denotes the number of entries in the set $\mathcal{X}$.
Algorithm A requires $N_{\text{T}}$ summations for computing the path voltages and $N_{\text{L}}$ comparisons for finding cycles, leading to the complexity estimate $N_{\text{T}} + N_{\text{L}}$. Algorithm B requires the same number of $N_{\text{T}}$ summations for computing the path voltages, roughly $N_{\text{T}} \log_2 N_{\text{T}}$ operations for sorting the set, and $N_{\text{T}}$ comparisons, leading to a total complexity estimate of $N_{\text{T}} \log_2 N_{\text{T}}$.
\begin{table}
\centering
\caption{\label{tab: complexity}Complexity of a search for suitable voltage assignment for QC LDPC block codes with girth $g \leq 12$}
\renewcommand\arraystretch{1.5}
\begin{tabular}[t]{c!{\shvline[1pt]}c|c!{\shvline[1pt]}c|c!{\shvline[1pt]}c|c}
\multirow{2}{*}{$K$} & \multicolumn{2}{c!{\shvline[1pt]}}{$g=8$} & \multicolumn{2}{c!{\shvline[1pt]}}{$g=10$} & \multicolumn{2}{c}{$g=12$} \\
& $N_{\text{T}}$ & $N_{\text{L}}$ & $N_{\text{T}}$ & $N_{\text{L}}$ & $N_{\text{T}}$ & $N_{\text{L}}$ \\ \shhline[1pt]
$4$ & $53$ & $42$ & $150$ & $231$ & $269$ & $519$ \\ \shhline
$5$ & $93$ & $90$ & $286$ & $645$ & $581$ & $1905$ \\ \shhline
$6$ & $142$ & $165$ & $485$ & $1470$ & $1060$ & $5430$ \\ \shhline
$7$ & $200$ & $273$ & $759$ & $2919$ & $1742$ & $12999$ \\ \shhline
$8$ & $267$ & $420$ & $1120$ & $5250$ & $2663$ & $27426$ \\ \shhline
$9$ & $343$ & $612$ & $1580$ & $8766$ & $3859$ & $52614$ \\ \shhline
$10$ & $428$ & $855$ & $2151$ & $13815$ & $5358$ & $93735$ \\ \shhline
$11$ & $522$ & $1155$ & $2845$ & $20790$ & $7210$ & $157410$ \\ \shhline
$12$ & $625$ & $1518$ & $3674$ & $30129$ & $9446$ & $251889$ \\ \shhline[1pt]
\end{tabular}
\end{table}
In \tabref{complexity} the values of $N_{\text{T}}$ and $N_{\text{L}}$ are given when searching for suitable voltage assignment for a $(J, K)$-regular rate $R=1-J/K$ QC LDPC convolutional codes with $J=3$ and arbitrary $K \geq 4$ and girth $g$ constructed from all-ones base matrices. In this case, up to $g = 10$, Algorithm A is preferable, while when searching for voltage assignment with girth $g \geq 12$, Algorithm B should be used.
In the general case we have to consider all node pairs, and as $N_{\text{L}}$ is roughly $N^2_{\text{T}}$ we conclude that Algorithm B performs asymptotically better (when $N_{\text{T}} \to \infty$).
\section{Minimum Distance of QC LDPC Codes}\label{sec: 7}
Usually the girth of the Tanner graph of an QC LDPC block code is considered to be the most important parameter that affects the performance of belief-propagation decoding, as it determines the number of independent iterations \cite{Gallager1963}. Therefore, most research is focused on finding QC LDPC block codes with large girth, while their corresponding minimum distance is mostly unknown. In \cite{Dolecek2009} it was shown, that the performance of belief-propagation decoding algorithms at high SNRs depends on the structure and the size of the smallest absorbing sets, which however can be upper-bounded by the minimum distance. This is the rationale for computing the minimum distance of the shortest known QC LDPC block codes as well as our search for QC LDPC codes with both large girth and large minimum distance.
Our method of calculating the minimum distance is based on the well-known fact that the minimum distance of a linear block code $\mathcal{B}$ with parity-check matrix $H$ is equal to the minimum number of columns of $H$ which sum up to zero.
\begin{table}[t]
\centering
\caption{\label{tab: girth6}Degree matrices for QC LDPC codes with girth $g=6$}
\renewcommand\arraystretch{1.2}
\setlength\tabcolsep{0.5em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}l}
$K$ & $(n, k)$ & $d_{\text{min}}$ & $M$ & \multicolumn{1}{c}{$W^\prime$} \\ \shhline[1pt]
\multicolumn{5}{c}{short codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$(20, 7)$} & \multirow{2}{*}{$6$} & \multirow{2}{*}{$5$} & $1,2,4$ \\
& & & & $3,1,2$ \\ \shhline
\multirow{2}{*}{$5$} & \multirow{2}{*}{$(25, 12)$} & \multirow{2}{*}{$6$} & \multirow{2}{*}{$5$} & $1,2,3,4$ \\
& & & & $3,1,4,2$ \\ \shhline
\multirow{2}{*}{$6$} & \multirow{2}{*}{$(42, 23)$} & \multirow{2}{*}{$4$} & \multirow{2}{*}{$7$} & $1,2,3,4,6$ \\
& & & & $3,5,2,1,4$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(49, 30)$} & \multirow{2}{*}{$4$} & \multirow{2}{*}{$7$} & $1,2,3,4,5,6$ \\
& & & & $3,5,2,1,6,4$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(72, 47)$} & \multirow{2}{*}{$4$} & \multirow{2}{*}{$9$} & $1,2,3,4,5,7,8$ \\
& & & & $3,6,2,1,8,5,4$ \\ \shhline
\multirow{2}{*}{$9$} & \multirow{2}{*}{$(81, 56)$} & \multirow{2}{*}{$4$} & \multirow{2}{*}{$9$} & $1,2,3,4,5,6,7,8$ \\
& & & & $3,6,2,1,8,7,5,4$ \\ \shhline
\multirow{2}{*}{$10$} & \multirow{2}{*}{$(110, 79)$} & \multirow{2}{*}{$6$} & \multirow{2}{*}{$11$} & $1,2,3,4,5 ,6,8,9,10$ \\
& & & & $3,1,7,2,10,9,4,6,5 $ \\ \shhline
\multirow{2}{*}{$11$} & \multirow{2}{*}{$(121, 90)$} & \multirow{2}{*}{$4$} & \multirow{2}{*}{$11$} & $1,2,3,4,5 ,6,7,8,9,10$ \\
& & & & $3,1,7,2,10,9,8,4,6,5$ \\ \shhline
\multirow{2}{*}{$12$} & \multirow{2}{*}{$(156, 119)$} & \multirow{2}{*}{$6$} & \multirow{2}{*}{$13$} & $1,2,3,4,5,6 ,7,8 ,10,11,12$ \\
& & & & $3,1,8,2,9,12,4,11,5, 7, 6, $ \\ \shhline[1pt]
\multicolumn{5}{c}{large distance codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$(92, 25)$} & \multirow{2}{*}{$22$} & \multirow{2}{*}{$23$} & $1,2,4$ \\
& & & & $5,3,12$ \\ \shhline
\multirow{2}{*}{$5$} & \multirow{2}{*}{$(245, 100)$} & \multirow{2}{*}{$22$} & \multirow{2}{*}{$49$} & $1,3,10,14$ \\
& & & & $40,31,33,30$ \\ \shhline
\multirow{2}{*}{$6$} & \multirow{2}{*}{$(414, 209)$} & \multirow{2}{*}{$22$} & \multirow{2}{*}{$69$} & $3,4,21,26,67$ \\
& & & & $34,15,64,33,44$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(763, 438)$} & \multirow{2}{*}{$22$} & \multirow{2}{*}{$109$} & $1, 3,11,15,45,93$ \\
& & & & $101, 34,18,9,1,4$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(1224, 767)$} & \multirow{2}{*}{$22$} & \multirow{2}{*}{$153$} & $2,10,26,57,89,4,49$ \\
& & & & $22,19,5,23,61,90,123$ \\ \shhline[1pt]
\end{tabular}
\end{table}
Consider the $M(c-b) \times Mc$ parity-check matrix $H_{\text{TB}}$ of the $(J,K)$-regular rate $R=Mb/Mc$ tailbiting block code $\mathcal{B}$ with block-length $N=Mc$ \eqref{eq: tb-parity-check-matrix}. Starting with each of the first $c$ columns of $H_{\text{TB}}$ as a root, we will construct $c$ separate trees, where each node $\xi$ is characterized by its depth $\ell(\xi)$ and partial syndrome state column-vector $\bs{\sigma}(\xi)$.
Initialize the partial syndrome state of the root $\xi_{i,\text{root}}$ of the $i$th tree with column $i$ of the corresponding parity-check matrix, that is, $\bs{\sigma}(\xi_{i,\text{root}}) = \bs{h}_i$, $i=1,2,\ldots,c$. Then grow each tree in such a way, that every branch between any two nodes $\xi$ and $\xi^\prime$ is labeled by a column vector $\bs{h}_j$, $j \neq i$, such that $\bs{\sigma}(\xi^\prime) = \bs{\sigma}(\xi) + \bs{h}_j$, where every branch label on the path $\xi_{i,\text{root}} \to \xi^\prime$ does not occur more than once.
Consider now a certain node $\xi$ with nonzero state $\bs{\sigma}(\xi) = (\bs{\sigma}_1(\xi)\,\bs{\sigma}_2(\xi)\ldots\bs{\sigma}_{(c-b)}(\xi))^T$, where $\bs{\sigma}_j(\xi)$, $j=1,2,\ldots,c-b$ is a $M \times 1$ column vector. If we assume that the $k$th position of the column-vector $\bs{\sigma}(\xi)$ is nonzero, then there are at most $K-1$ columns which can cancel this nonzero position and have not been considered previously. Therefore, every node $\xi$ has at most $K-1$ children nodes per nonzero position.
Such a tree would grow until all possible linear combinations have been found. Therefore, we assume that the minimum distance is restricted by $d_{\text{min}} < t$, that is, the maximum depth of the tree is $t-1$. Consequently, a node $\xi$ at depth $\ell(\xi)$ will not be extended, if the number of nonzero positions in its partial syndrome state column-vector $\bs{\sigma}(\xi)$ exceeds $J(t-\ell(\xi)-1)$, since at most $J$ ones can be canceled by each branch.
By initially reordering the columns of the parity-check matrix $H_{\text{TB}}$ in such a way that each of the $c-b$ nonoverlapping blocks of $M$ rows contains not more than a single one per column, we can strengthen the stopping criterion as follows: A node $\xi$ at depth $\ell(\xi)$ will not be extended, if the number of nonzero positions in each of its partial syndrome state column-vectors $\bs{\sigma}_j(\xi)$, $j=1,2,\ldots,c-b$ exceeds $(t-\ell(\xi)-1)$, since at most $1$ one in each block can be canceled by each branch. In particular, such a reordering of the parity-check matrix $H_{\text{TB}}$ corresponds directly to the parity-check matrix $H_{\text{C}}$ \eqref{eq: circulant-ldpc_parity-check-matrix} of the equivalent $(J,K)$-regular LDPC block code constructed from circulant matrices.
\begin{algorithm}[~MD]{Determine the minimum distance of a rate $R=b/c$ $(J, K)$-regular LDPC block code}
\begin{enumerate}
\item Assume a suitable restriction $t$ on the minimum distance $d_{\text{min}} < t$.
\item Grow $c$ separate trees as follows:
\begin{enumerate}
\item Initialize the root node of the $i$th tree by $\bs{\sigma}(\xi_{\text{root},i}) = \bs{h_i}$ with depth $\ell(\xi) = 0$.
\item Extend all nodes $\xi$ as long as the Hamming weights of their partial syndrome states $w_{\text{H}}(\bs{\sigma}(\xi)) \leq J(t-\ell(\xi)-1)$ (Note, for codes with blocks of $M$ rows containing only a single one, this criterion can be strengthen to $w_{\text{H}}(\bs{\sigma}_j(\xi)) \leq (t-\ell(\xi)-1)$, $j=1,2,\ldots,c-b$).
\item The minimum distance $d_{\text{min}}$ follows directly as
\begin{IEEEeqnarray*}{rCl}
d_{\text{min}} & = & \min_{\xi} \left\{ \ell(\xi) \;\vert\; \bs{\sigma}(\xi) = \bs{0} \right\}
\end{IEEEeqnarray*}
If there is no node $\xi$ whose partial syndrome state $\bs{\sigma}(\xi) = \bs{0}$, then the minimum distance is lower-bounded by $d_{\text{min}} > t$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
\begin{table}[t]
\centering
\caption{\label{tab: girth8}Degree matrices for QC LDPC codes with girth $g=8$}
\renewcommand\arraystretch{1.2}
\setlength\tabcolsep{0.5em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}l}
$K$ & $(n, k)$ & $d_{\text{min}}$ & $M$ & \multicolumn{1}{c}{$W^\prime$} \\ \shhline[1pt]
\multicolumn{5}{c}{short codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$(36, 11)$} & \multirow{2}{*}{$6$} & \multirow{2}{*}{$9$} & $1,4,6$ \\
& & & & $5,2,3$ \\ \shhline
\multirow{2}{*}{$5$} & $(65, 28)$ & \multirow{2}{*}{$10$} & \multirow{2}{*}{$13$} & $1,3,7,11$ \\
& \scriptsize \big($(75, 32)$\cite{Esmaeili2010}\big) & & & $10,4,5,6$ \\ \shhline
\multirow{2}{*}{$6$} & $(108, 56)$ & \multirow{2}{*}{$10$} & \multirow{2}{*}{$18$} & $2,3,5,7,9$ \\
& \scriptsize \big($(156, 80)$\cite{Esmaeili2010}\big) & & & $4,6,13,1,16$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(147, 86)$} & \multirow{2}{*}{$10$} & \multirow{2}{*}{$21$} & $2,3,8,15,17,20$ \\
& & & & $4,6,7,9,12,13$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(200, 127)$} & \multirow{2}{*}{$8$} & \multirow{2}{*}{$25$} & $1,3,4,10,14,15,19$ \\
& & & & $5,6,11,24,2,9,12$ \\ \shhline
\multirow{2}{*}{$9$} & \multirow{2}{*}{$(270, 182)$} & \multirow{2}{*}{$8$} & \multirow{2}{*}{$30$} & $1,3,10,16,23,25,26,28$ \\
& & & & $2,6,5,9,8,12,14,22$ \\ \shhline
\multirow{2}{*}{$10$} & \multirow{2}{*}{$(350, 247)$} & \multirow{2}{*}{$8$} & \multirow{2}{*}{$35$} & $2,6,7,18,19,26,29,31,34$ \\
& & & & $4,5,3,13,10,16,12,11,23$ \\ \shhline
\multirow{2}{*}{$11$} & \multirow{2}{*}{$(451, 330)$} & \multirow{2}{*}{$8$} & \multirow{2}{*}{$41$} & $1,4,8,20,27,28,29,33,39,40$ \\
& & & & $5,7,6,9,10,19,13,21,14,35$ \\ \shhline
\multirow{2}{*}{$12$} & \multirow{2}{*}{$(564, 425)$} & \multirow{2}{*}{$8$} & \multirow{2}{*}{$47$} & $3,7,8,22,24,27,29,35,40,41,43$ \\
& & & & $6,2,4,5,14,16,1,21,28,9,34$ \\ \shhline[1pt]
\multicolumn{5}{c}{large distance codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$(116, 31)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$29$} & $3,14,21$ \\
& & & & $7,1,17$ \\ \shhline
\multirow{2}{*}{$5$} & \multirow{2}{*}{$(225, 92)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$45$} & $1,3,10,14$ \\
& & & & $40,31,33,30$ \\ \shhline
\multirow{2}{*}{$6$} & \multirow{2}{*}{$(431, 218)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$72$} & $3,4,21,26,67$ \\
& & & & $34,15,64,33,44$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(777, 446)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$111$} & $3,11,15,45,93,110$ \\
& & & & $34,18,9,1,4,101$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(1280, 802)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$160$} & $2,4,10,26,49,57,89$ \\
& & & & $22,90,19,5,123,23,61$ \\ \shhline
\multirow{2}{*}{$9$} & \multirow{2}{*}{$(1386, 926)$} & \multirow{2}{*}{$20$} & \multirow{2}{*}{$154$} & $6,9,26,65,79,99,124,153$ \\
& & & & $24,16,14,1,46,62,137,84$ \\ \shhline[1pt]
\end{tabular}
\end{table}
\section{Search results}\label{sec: 8}
When presenting our search results for QC ($J=3, K$)-regular LDPC block codes with different girth we will distinguish two cases.
We started by searching for QC ($J=3, K$)-regular LDPC block codes using an all-one base matrix $B$, applied the algorithms as described above and obtained QC ($3, K$)-regular LDPC block codes with girth $g=6,8,10$, and $12$ as given in \tabsref{girth6}{girth12}. These codes correspond to a parity-check matrix $H(D)$ of a convolutional code with only monomial entries as given, for example, in \eqref{eq: convolutional-parity-check-matrix}.
However, according to \thref{three-a} the achievable girth $g$ of a QC $(J,K)$-regular LDPC code, constructed in such a way, is limited by $g \leq 12$. Thus, in order to find QC $(J=3, K)$-regular LDPC block codes with girth $g > 12$ as presented in \tabsandref{girthgreater}{girth20}, we have to allow zero entries in our base matrix $B$. This is a straight-forward generalization of the restriction to only monomial entries in the parity-check matrix $H(D)$ of the underlying convolutional code.
\begin{table}
\centering
\caption{\label{tab: girth10}Degree matrices for QC LDPC codes with girth $g=10$}
\renewcommand\arraystretch{1.2}
\setlength\tabcolsep{0.4em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}l}
$K$ & $(n, k)$ & $d_{\text{min}}$ & $M$ & \multicolumn{1}{c}{$W^\prime$} \\ \shhline[1pt]
\multicolumn{5}{c}{short codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & $(148, 39)$ & \multirow{2}{*}{$14$} & $37$ & $1,14,17$ \\
& \scriptsize \big($(144, 38)$\cite{Esmaeili2010}\big) & & \scriptsize \big($39$\cite{Wang2008}\big) & $11,6,2$ \\ \shhline
\multirow{2}{*}{$5$} & $(305, 124)$ & \multirow{2}{*}{$24$} & $61$ & $2,20,54,60$ \\
& \scriptsize \big($(550, 222)$\cite{Esmaeili2010}\big) & & \scriptsize \big($61$\cite{TannerClass}\big) & $26,16,31,48$ \\ \shhline
\multirow{2}{*}{$6$} & $(606, 305)$ & \multirow{2}{*}{$24$} & $101$ & $2,24,25,54,85$ \\
& \scriptsize \big($(780, 392)$\cite{Esmaeili2010}\big) & & \scriptsize \big($103$\cite{Wang2008}\big) & $21,15,11,8,59$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(1113, 638)$} & \multirow{2}{*}{$24$} & $159$ & $2,14,27,67,97,130$ \\
& & & \scriptsize \big($160$\cite{Wang2008}\big) & $21,24,1,6,75,58$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(1752, 1097)$} & \multirow{2}{*}{$24$} & $219$ & $3,14,26,63,96,128,183$ \\
& & & \scriptsize \big($233$\cite{Wang2008}\big) & $24,6,19,46,4,77,107$ \\ \shhline
\multirow{2}{*}{$9$} & \multirow{2}{*}{$(2871, 1916)$} & \multirow{2}{*}{$24$} & $319$ & $6,9,26,65,99,153,233,278$ \\
& & & \scriptsize \big($329$\cite{Wang2008}\big) & $24,16,14,1,62,84,200,137$ \\ \shhline
\multirow{4}{*}{$10$} & \multirow{4}{*}{$(4300, 2912)$} & \multirow{4}{*}{$24$} & & $9,11,26,67,101,161,\ldots$ \\
& & & $430$ & $\quad 233,302,395$ \\
& & & \scriptsize \big($439$\cite{Wang2008}\big) & $23,5,1,54,33,96,\ldots$ \\
& & & & $\quad 120,104,244$ \\ \shhline
\multirow{4}{*}{$11$} & \multirow{4}{*}{$(6160, 4482)$} & \multirow{4}{*}{$24$} & & $2,11,25,62,101,162,225,\ldots$ \\
& & & $560$ & $\quad 268,421,492$ \\
& & & \scriptsize \big($577$\cite{Wang2008}\big) & $24,21,5,55,6,59,178,\ldots$ \\
& & & & $\quad 132,204,311$ \\ \shhline
\multirow{4}{*}{$12$} & \multirow{4}{*}{$(8844, 6635)$} & & & $2,22,23,63,101,147,219,\ldots$ \\
& & & $737$ & $\quad 322,412,569,601$ \\
& & & \scriptsize \big($758$\cite{Wang2008}\big) & $16,9,6,58,34,91,126,\ldots$ \\
& & & & $\quad 155,185,298,232$ \\ \shhline[1pt]
\multicolumn{5}{c}{large distance codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$(176, 46)$} & \multirow{2}{*}{$24$} & \multirow{2}{*}{$44$} & $1,14,17$ \\
& & & & $11,6,2$ \\ \shhline[1pt]
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{\label{tab: girth12}Degree matrices for QC LDPC codes with girth $g=12$}
\renewcommand\arraystretch{1.2}
\setlength\tabcolsep{0.3em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}l}
$K$ & $(n, k)$ & $d_{\text{min}}$ & $M$ & \multicolumn{1}{c}{$W^\prime$} \\ \shhline[1pt]
\multicolumn{5}{c}{short codes} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & $(292, 75)$ & \multirow{2}{*}{$24$} & $73$ & $2,25,33$ \\
& \scriptsize \big($( 444, 113)$\cite{Esmaeili2010}\big) & & \scriptsize \big($97$\cite{Sullivan2006}\big) & $18,6,5$ \\ \shhline
\multirow{2}{*}{$5$} & $(815, 328)$ & \multirow{2}{*}{$24$} & $163$ & $5,33,42,117$ \\
& \scriptsize \big($(1700, 682)$\cite{Esmaeili2010}\big) & & \scriptsize \big($181$\cite{TannerClass}\big) & $36,35,25,57$ \\ \shhline
\multirow{2}{*}{$6$} & $(1860, 932)$ & \multirow{2}{*}{$24$} & $310$ & $1,24,38,145,246$ \\
& \scriptsize \big($(4680, 2342)$\cite{Esmaeili2010}\big) & & \scriptsize \big($393$\cite{Zhang2010}\big) & $16,36,5,82,110$ \\ \shhline
\multirow{2}{*}{$6$} & \multirow{2}{*}{$(1836, 920)$} & \multirow{2}{*}{$24$} & $306$ & $9,36,38,154,204$ \\
& & & \scriptsize \big($393$\cite{Zhang2010}\big) & $33,1,13,54,123$ \\ \shhline
\multirow{2}{*}{$7$} & \multirow{2}{*}{$(3962, 2266)$} & & $566$ & $3,10,33,147,297,442$ \\
& & & \scriptsize \big($881$\cite{Sullivan2006}\big) & $31,22,4,93,133,219$ \\ \shhline
\multirow{2}{*}{$8$} & \multirow{2}{*}{$(6784, 4242)$} & & $848$ & $4,24,31,143,303,498,652$ \\
& & & \scriptsize \big($1493$\cite{Sullivan2006}\big) & $32,9,6,70,130,193,222$ \\ \shhline
\multirow{4}{*}{$9$} & \multirow{4}{*}{$(12384, 8258)$} & & & $4,20,32,160,284,\ldots$ \\
& & & $1376$ & $\quad 569,794,1133$ \\
& & & \scriptsize \big($2087$\cite{Sullivan2006}\big) & $30,7,1,92,169,\ldots$ \\
& & & & $\quad 350,437,645$ \\ \shhline
\multirow{4}{*}{$10$} & \multirow{4}{*}{$(21030, 14723)$} & & \multirow{4}{*}{$2103$} & $6,13,28,150,291,565,\ldots$ \\
& & & & $\quad 678,1258,1600$ \\
& & & & $30,16,5,64,225,207,\ldots$ \\
& & & & $\quad 491,838,746$ \\ \shhline
\multirow{4}{*}{$11$} & \multirow{4}{*}{$(34507, 25098)$} & & \multirow{4}{*}{$3137$} & $9,11,24,150,306,508,\ldots$ \\
& & & & $\quad 666,1279,1765,1958$ \\
& & & & $31,28,1,83,131,160,\ldots$ \\
& & & & $\quad 429,550,956,1391$ \\ \shhline
\multirow{4}{*}{$12$} & \multirow{4}{*}{$(56760, 42572)$} & & \multirow{4}{*}{$4730$} & $3,15,22,140,286,537,\ldots$ \\
& & & & $\quad 811,1113,1878,2524,3349$ \\
& & & & $31,26,1,66,95,210,373,\ldots$ \\
& & & & $\quad 729,878,1365,1644$ \\ \shhline[1pt]
\end{tabular}
\end{table}
\subsection*{Case I: monomial entries}
In \tabsref{girth6}{girth12}, parity-check matrices of short known QC $(J=3, K)$-regular LDPC block codes with girth $g=6,8,10$, and $12$ together with those of large minimum distance are presented. When searching for such codes, we applied the following restrictions to reduce the number of possible voltage assignments:
\begin{itemize}
\item As the girth of a voltage graph is defined as the shortest cycle with voltage zero, and the sign of the voltage depends on the direction of the edge, we can add the same arbitrary offset to the voltage of all edges being connected to the same node. Thus, without loss of generality, we set the voltage of all edges connected to one specific symbol node as well as all edges connected to one specific constraint node to voltage zero. (For consistency with codes constructed from Steiner Triple System, that will be introduced later, we choose the first symbol node and the last constraint node. This corresponds directly to a degree matrix with zeros in its first column and last row.)
For example, the degree matrix of the $(J=3, K=4)$ QC LDPC block code with girth $g=8$ from \tabref{girth8} follows directly as
\begin{IEEEeqnarray*}{rCl}
W & = &
\left(\begin{array}{cccc}
0 & 1 & 4 & 6 \\
0 & 5 & 2 & 3 \\
0 & 0 & 0 & 0
\end{array}\right)
\end{IEEEeqnarray*}
\item Furthermore, to reduce the number of only permuted degree matrices, we assume that
\begin{itemize}
\item The first row is sorted in ascending order.
\item When sorting the first and the second row in ascending order, the second row is lexicographically less than the first row.
\item The maximum degree is less than the tailbiting length $M$ for which there exists a QC $(J=3, K)$-regular LDPC block code with given girth $g$.
\end{itemize}
\item QC $(J=3, K=4)$-regular LDPC block codes were found by exhaustive search over the previously defined set of restricted edge voltages.
\item QC $(J=3, K=N)$-regular LDPC block codes with $N > 4$ were obtained by adding one additionally, randomly chosen column to the best degree matrices of codes with $K=N-1$ having the same girth $g$. The maximum degree in this additional column is limited by twice the maximum degree of the previous code.
\end{itemize}
Using these restrictions, the obtained QC $(J=3, K)$-regular LDPC block codes with girth $g=6,8,10$, and $12$ are presented in \tabsref{girth6}{girth12}.
The first column $K$ denotes the number of nonzero elements per row, which corresponds to the number of columns in $H(D)$ and $W$, due to the all-ones base matrix $B$. As all entries in the first column and the last row of the degree matrix $W$ are zero, they are omitted in the submatrix $W^\prime$ which is given in the corresponding last column.
Consider now the parity-check matrix $H(D)$ of the rate $R=1-J/K$ convolutional code $\mathcal{C}$, with only monomial entries corresponding to the degree matrix $W$. By tailbiting the semi-infinite parity-check matrix $H$ to length $M$ (given in the forth column), we obtain the parity-check matrix $H_{\text{TB}}$ of an $(n, k)$ block code $\mathcal{B}$ with minimum distance $d_{\text{min}}$, where $(n, k)$ and $d_{\text{min}}$ follow from the second and third column, respectively. Note that due to linear dependent rows in $H_{\text{TB}}$ the rank of $\mathcal{B}$ might be less than $M(c-b)$.
The codes presented in \tabsandref{girth6}{girth8} coincide with the QC LDPC block codes found by the ``hill-climbing'' algorithm \cite{Wang2008}, but we determined their minimum distance with our algorithm described in \secref{7}. \tabsandref{girth10}{girth12} contain new QC $(J=3, K)$-regular LDPC block codes, which, to the best of our knowledge, are shorter than the previously known codes obtained from an all-ones base matrix \cite{Wang2008, TannerClass, Sullivan2006, Zhang2010}. In particular, these codes are significantly shorter than those presented in \cite{Esmaeili2010}, which are obtained from base matrices with zeros. However, due to the zeros in their base matrix, the minimum distance of the LDPC block codes in \cite{Esmaeili2010} can exceed $(J+1)!$. For example, we determined the minimum distance of the $(444, 113)$ QC $(3, 4)$-regular LDPC block code with girth $g=12$ in \cite{Esmaeili2010} to be $d_{\text{min}} = 28$, while the corresponding code in \tabref{girth12}, that is, the $(292, 75)$ QC $(3, 4)$-regular LDPC block code, has only minimum distance $d_{\text{min}}=24$, but shorter block length. Using the BEAST \cite{beast}, we calculated the free distance of the corresponding parent convolutional code for the code in \cite{Esmaeili2010} to be $d_{\text{free}} = 46$. Therefore, using our approach and a larger tailbiting length it would be possible to construct corresponding QC $(3, 4)$-regular LDPC block codes with minimum distance up to $46$.
\subsection*{Case 2: monomial or zero entries}
In order to find QC $(J=3, K)$-regular LDPC block codes with girth $g \geq 14$, we have to allow zero entries in our base matrix $B$; that is, relax the restriction from only monomial entries in $H(D)$ to include also zero entries. According to \thref{two}, a code with girth $g$ exists if the corresponding base graph has girth $g_{\text{B}}$ satisfying \eqref{eq: girth_lower_bound}. Additionally, as we are searching for codes with short block length, we consider the shortest possible base matrices $B$.
\begin{table*}[t]
\centering
\caption{\label{tab: girthgreater}Degree matrices for QC LDPC codes with girth $g = 14$ to $18$}
\renewcommand\arraystretch{1.2}
\setlength\tabcolsep{0.3em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}l}
$K$ & $g$ & $(n, k)$ & $M$ & Base graph & \multicolumn{1}{c}{$W^{\prime}$} \\ \shhline[1pt]
\multirow{2}{*}{$4$} & \multirow{2}{*}{$14$} & $(1812, 453)$ & $151$ & $\text{STS}(9)$ & $0, 123, 36, 3, 2, 79, 4, 7, 52, 4, 1$ \\
& & \scriptsize \big($(2208, 732)$\cite{Esmaeili2010}\big) & \scriptsize \big($184$\cite{Esmaeili2010}\big) & $(9 \times 12)$ & $0, 96, 23, 11, 1, 37, 12, 2, 61, 1, 4$ \\ \shhline
\multirow{2}{*}{$5$} & \multirow{2}{*}{$14$} & $(9720, 3888)$ & \multirow{2}{*}{$486$} & $\text{S-STS}(13)$ & $423, 0, 437, 5, 237, 235, 170, 333, 260, 109, 241, 2, 114, 5, 2, 428, 92, 228, 299$ \\
& & \scriptsize \big($(11525, 4612)$\cite{Esmaeili2010}\big) & & $(12 \times 20)$ & $0, 0, 0, 445, 465, 51, 440, 22, 111, 307, 433, 4, 285, 2, 1, 4, 113, 282, 5$\\ \shhline
\multirow{4}{*}{$6$} & \multirow{4}{*}{$14$} & & & & $1037, 0, 1051, 1105, 933, 1027, 962, 1000, 665, 805, 646, 2, \ldots$ \\
& & $(29978, 14989)$ & $1153$ & $\text{STS}(13)$ & $\quad 906, 5, 2, 1095, 788, 633, 913, 264, 51, 772, 672, 686, 737$ \\
& & \scriptsize \big($(37154, 18579)$\cite{Esmaeili2010}\big) & \scriptsize \big($1429$\cite{Esmaeili2010}\big) & $(13 \times 26)$ & $0, 0, 0, 1112, 1132, 51, 1107, 22, 807, 921, 1100, 4, 952, 2, \ldots$ \\
& & & & & $\quad 1, 4, 905, 949, 5, 0, 1111, 922, 620, 351, 140$ \\ \shhline
$7$ & $14$ & $n=80000000$ & $800000$ & $\text{STS}(25)$ & \quad \textit{available at \cite{online}}\\ \shhline
\multirow{2}{*}{$4$} & \multirow{2}{*}{$16$} & \multirow{2}{*}{$(7980, 1995)$} & \multirow{2}{*}{$665$} & $\text{STS}(9)$ & $0, 468, 99, 3, 2, 305, 43, 9, 251, 3, 2$ \\
& & & & $(9 \times 12)$ & $0, 351, 41, 6, 8, 215, 18, 1, 79, 1, 8$ \\ \shhline
\multirow{4}{*}{$5$} & \multirow{4}{*}{$16$} & & \multirow{4}{*}{$2562$} & & $937, 0, 1551, 1264, 1670, 2119, 1973, 1960, 1848, 1223, 1806, \ldots$ \\
& & $(51240, 20496)$ & & $\text{S-STS}(13)$ & $\quad 15, 1761, 1, 2, 2175, 1169, 1768, 548$ \\
& & \scriptsize \big($(62500, 25002)$\cite{Esmaeili2010}\big) & & $(12 \times 20)$ & $0, 0, 0, 2367, 2491, 126, 2296, 66, 1197, 582, 2200, 9, \ldots$ \\
& & & & & $\quad 1836, 2, 1, 0, 1757, 1833, 4$ \\ \shhline
\multirow{4}{*}{$6$} & \multirow{4}{*}{$16$} & & & & $8328, 0, 8393, 8106, 7840, 8289, 8143, 8130, 6821, 7393, 6779, 15, 7931, \ldots$ \\
& & $(227032, 113516)$ & $8732$ & $\text{STS}(13)$ & $\quad 1, 2, 8345, 7339, 6741, 7390, 1557, 498, 6357, 5666, 5001, 1684$ \\
& & \scriptsize \big($(229476, 114740)$\cite{Esmaeili2010}\big) & \scriptsize \big($8826$\cite{Esmaeili2010}\big) & $(13 \times 26)$ & $0, 0, 0, 8537, 8661, 126, 8466, 66, 7367, 7424, 8370, 9, 8006, 2, 1, 0, \ldots$\\
& & & & & $\quad 7927, 8003, 4, 0, 8412, 5799, 4553, 2142, 6293$ \\ \shhline
\multirow{2}{*}{$4$} & \multirow{2}{*}{$18$} & \multirow{2}{*}{$(32676, 8169)$} & $2723$ & $\text{STS}(9)$ & $0, 853, 217, 6, 2, 1108, 75, 20, 586, 1, 5$ \\
& & & \scriptsize \big($2855$\cite{Esmaeili2010}\big) & $(9 \times 12)$ & $0, 1797, 97, 3, 4, 485, 33, 37, 246, 1, 5$ \\ \shhline
\multirow{4}{*}{$5$} & \multirow{4}{*}{$18$} & & \multirow{4}{*}{$13588$} & & $10484, 0, 12275, 10611, 9703, 10786, 10227, 11122, 3263, 7933, \ldots$ \\
& & $(271760, 108704)$ & & $\text{S-STS}(13)$ & $\quad 3129, 21, 9554, 1, 2, 12183, 7837, 3084, 8297$ \\
& & \scriptsize \big($(371100, 92777)$\cite{Esmaeili2010}\big) & & $(12 \times 20)$ & $0, 0, 0, 12012, 13041, 498, 12534, 223, 7947, 8356, \ldots$ \\
& & & & & $\quad 12213, 13, 10701, 2, 1, 0, 9550, 10698, 4$ \\ \shhline[1pt]
\end{tabular}
\end{table*}
\subsection*{Case 2-I: Steiner Triple Systems}
When searching for QC $(J=3, K)$-regular LDPC block codes with girth $g=14,16$, and $18$, we started with a (shortened) base graph constructed by using Steiner triple systems of order $n$, that is, $\text{STS}(n)$ \cite{Johnson2001, Johnson2001-2, Thorpe2004}.
For all $n$, where $n \mod 6$ is equal to $1$ or $3$, there exists a Steiner triple system of order $n$. Then we construct a $(J, K)$-regular, $(c-b) \times c$ base matrix $B$ with entries $b_{ij}$, $i=1,2,\ldots,c-b$ and $j=1,2,\ldots,c$, in such a way that the positions of the nonzero entries in each column correspond to a Steiner triple system of order $(c-b)$. Denote such a $(c-b) \times c$ base matrix $B_{\text{STS}(c-b)}$.
Using the obtained $(J, K)$-regular $(c-b) \times c$ base matrix $B$, we search for a set of edge labels, such that the corresponding voltage graph has at least girth $g$.
In general, it is possible, without loss of generality, to label a certain subset of edges of the voltage graph simultaneously with zero voltage and thereby decreasing the number of possible labelings. The following algorithm constructs a $(c-b) \times c$ base matrix $B$ based on $\text{STS}(c-b)$ and reorders the matrix to maximize the number of zero entries in its lower left corner. Using such a base matrix, it is always possible to label the last nonzero entry in each column with degree zero. Moreover, in each of the remaining rows at the top of the base matrix, we can label at least one nonzero entry with degree zero.
(Hereinafter we will always choose at least the first element in the remaining rows to be labeled with zero voltage).
\begin{algorithm}[~STS]{Construction of a $(J, K)$-regular $(c-b) \times c$ base graph $B$ obtained from $\text{STS}(c-b)$}
\begin{enumerate}
\item Initialize a counter $u$ to zero and denote the current row and column by $s$ and $t$, respectively, starting from the right-most entry in the last row, that is, $s=c-b$ and $t=c$.
\item Set the $K-u$ elements in row $s$ and column $t,t-1,\ldots,t-K+u+1$ to one, that is, $b_{ij} = 1$ for $i=s$ and $j=t,t-1,\ldots,t-K+u+1$.
\item Choose the remaining $J-1$ nonzero positions in each of those $K-u$ columns to fulfill the properties of a Steiner Triple System. If possible, choose the positions $b_{ij}$ to minimize $i$. In other words, try to avoid using the lowest rows $s-1, s-2,\ldots$, if possible, despite of the restrictions imposed by the Steiner Triple System.
\item Finally, decrease $t$ by $K-u$, set $s$ to $s-1$, denote the number of nonzero elements in the new row $s$ by $K-u$ and continue with Step $2$ until all $c$ columns are used, that is, $t=0$.
\end{enumerate}
\end{algorithm}
By removing the last row and last $K$ columns of the $(J, K)$-regular $(c-b) \times c$ base matrix $B$ constructed using $\text{STS}(c-b)$, we obtain a shortened $(c-b-1) \times (c-K)$ ($J, K-1$)-regular base matrix $B^\prime$, which we denote $B_{\text{S-STS}(c-b)}$. By deleting columns and rows, it is also possible to obtain intermediate codes, which are, however, irregular.
\begin{example}\label{ex: basests}
In the following we shall construct the $(J=3, K$) base matrices $B$ of dimension $9 \times 12$ $(K=4)$, dimension $13 \times 26$ $(K=6)$ and dimension $25 \times 100$ $(K=7)$. Using Algorithm STS, we obtain the following Steiner Triple Systems of order $9$ ($\text{STS}(9)$), $13$ ($\text{STS}(13)$) and $25$ ($\text{STS}(25)$).
\begin{IEEEeqnarray*}{rLLLL}
\IEEEeqnarraymulticol{5}{l}{\text{\textbf{STS}}(9) =} \\
\Big\{ & \{2, 3, 5\}, & \{1, 4, 6\}, & \{1, 3, 7\}, & \{2, 6, 7\},\\[.1em]
& \{4, 5, 7\}, & \{1, 2, 8\}, & \{5, 6, 8\}, & \{3, 4, 8\},\\[.1em]
& \{1, 5, 9\}, & \{2, 4, 9\}, & \{3, 6, 9\}, & \{7, 8, 9\} \Big\} \\[1.2em]
\IEEEeqnarraymulticol{5}{l}{\text{\textbf{STS}}(13) =} \\
\Big\{ & \{0, 3, 6\}, & \{0, 2, 7\}, & \{1, 5, 7\}, & \{3, 4, 7\}, \\[.1em]
& \{3, 5, 8\}, & \{1, 4, 8\}, & \{2, 6, 8\}, & \{2, 4, 9\}, \\[.1em]
& \{5, 6, 9\}, & \{0, 1, 9\}, & \{1, 3, 10\}, & \{0, 4, 10\}, \\[.1em]
& \{6, 7, 10\}, & \{2, 5, 10\}, & \{8, 9, 10\}, & \{7, 8, 11\}, \\[.1em]
& \{4, 6, 11\}, & \{1, 2, 11\}, & \{0, 5, 11\}, & \{3, 9, 11\},
\end{IEEEeqnarray*}
\begin{IEEEeqnarray*}{rLLLL}
& \{10, 11, 12\}, & \{7, 9, 12\}, & \{0, 8, 12\}, & \{1, 6, 12\}, \\[.1em]
& \{4, 5, 12\}, & \{2, 3, 12\} \Big\} \\[1.2em]
\IEEEeqnarraymulticol{5}{l}{\text{\textbf{STS}}(25) =} \\
\Big\{ & \{ 4, 5,10\}, & \{ 1, 9,10\}, & \{ 7, 8,11\}, & \{ 1, 6,11\}, \\[.1em]
& \{ 2, 3,12\}, & \{ 0, 9,12\}, & \{ 6, 8,12\}, & \{ 8, 9,13\}, \\[.1em]
& \{ 6, 7,13\}, & \{ 0, 5,13\}, & \{ 2,10,13\}, & \{ 3, 4,14\}, \\[.1em]
& \{ 1,12,14\}, & \{ 0, 2,14\}, & \{ 7, 9,14\}, & \{ 5,11,14\}, \\[.1em]
& \{ 5, 6,15\}, & \{ 3,10,15\}, & \{ 4,12,15\}, & \{ 1, 7,15\}, \\[.1em]
& \{ 0, 8,15\}, & \{11,13,16\}, & \{ 5, 7,16\}, & \{ 6,10,16\}, \\[.1em]
& \{ 2, 8,16\}, & \{ 3, 9,16\}, & \{ 0, 4,16\}, & \{ 9,11,17\}, \\[.1em]
& \{12,13,17\}, & \{ 1, 3,17\}, & \{ 4, 7,17\}, & \{ 0, 6,17\}, \\[.1em]
& \{ 2, 5,17\}, & \{ 8,17,18\}, & \{ 3,11,18\}, & \{ 2, 4,18\}, \\[.1em]
& \{13,15,18\}, & \{ 0,10,18\}, & \{ 1,16,18\}, & \{ 6,14,18\}, \\[.1em]
& \{ 9,18,19\}, & \{ 4, 8,19\}, & \{14,15,19\}, & \{10,11,19\}, \\[.1em]
& \{ 0, 3,19\}, & \{ 2, 7,19\}, & \{12,16,19\}, & \{ 1, 5,19\}, \\[.1em]
& \{17,19,20\}, & \{ 9,15,20\}, & \{10,12,20\}, & \{ 0,11,20\}, \\[.1em]
& \{ 5, 8,20\}, & \{ 1, 4,20\}, & \{13,14,20\}, & \{ 3, 7,20\}, \\[.1em]
& \{ 2, 6,20\}, & \{ 5,18,21\}, & \{ 4, 6,21\}, & \{ 1,13,21\}, \\[.1em]
& \{16,17,21\}, & \{10,14,21\}, & \{ 2, 9,21\}, & \{ 3, 8,21\}, \\[.1em]
& \{11,15,21\}, & \{ 7,12,21\}, & \{19,21,22\}, & \{18,20,22\}, \\[.1em]
& \{ 0, 7,22\}, & \{10,17,22\}, & \{ 3, 5,22\}, & \{ 6, 9,22\},
\end{IEEEeqnarray*}
\begin{IEEEeqnarray*}{rLLLL}
& \{ 2,15,22\}, & \{ 1, 8,22\}, & \{11,12,22\}, & \{ 4,13,22\}, \\[.1em]
& \{14,16,22\}, & \{20,21,23\}, & \{ 0, 1,23\}, & \{ 6,19,23\}, \\[.1em]
& \{15,16,23\}, & \{ 2,11,23\}, & \{ 7,18,23\}, & \{ 5,12,23\}, \\[.1em]
& \{14,17,23\}, & \{ 4, 9,23\}, & \{ 8,10,23\}, & \{ 3,13,23\}, \\[.1em]
& \{ 0,21,24\}, & \{22,23,24\}, & \{ 1, 2,24\}, & \{16,20,24\}, \\[.1em]
& \{ 7,10,24\}, & \{ 8,14,24\}, & \{13,19,24\}, & \{ 3, 6,24\}, \\
& \{12,18,24\}, & \{15,17,24\}, & \{ 5, 9,24\}, & \{ 4,11,24\}\Big\}
\end{IEEEeqnarray*}
Each number $1,2,\ldots,J$ occurs $K$ times within the set of Steiner triples. However, the chosen Steiner triples are not uniquely determined.
The corresponding base matrices of dimension $9 \times 12$ $\text{STS}(9)$, dimension $13 \times 26$ $\text{STS}(13)$, and dimension $25 \times 100$ $\text{STS}(25)$ are sparse matrices with nonzero elements only in column $i$ and row $j$, where the $i$th Steiner Triple contains the value $j$. The $9 \times 12$ $(3, 4)$-regular base matrix constructed from $\text{STS}(9)$ denoted by $B_{\text{STS}(9)}$ is given, for example, by
{\setlength{\arraycolsep}{3.5pt}
\begin{IEEEeqnarray}{rCl}
\label{eq: base_sts9}
B_{\text{STS(9)}} & = & \borderarray{(}{)}{2em}{1.5ex}{ccccccccccccc}{
~~~~ & \sm{1}&\sm{2}&\sm{3}&\multicolumn{1}{c}{\sm{4}}&\sm{5}&\sm{6}&\sm{7}&\sm{8}&\sm{9}&\sm{10}&\sm{11}&\sm{12}\vspace{1mm}\\
\sm{1}& 0 & \bs{1} & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
\sm{2}& \bs{1} & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
\sm{3}& \bs{1} & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
\sm{4}& 0 & \bs{1} & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
\sm{5}& \bs{1} & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\
\sm{6}& 0 & \bs{1} & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
\sm{7}& 0 & 0 & \bs{1} & \bs{1} & \bs{1} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\sm{8}& 0 & 0 & 0 & 0 & 0 & \bs{1} & \bs{1} & \bs{1} & 0 & 0 & 0 & 1 \\
\sm{9}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bs{1} & \bs{1} & \bs{1} & \bs{1}
}\IEEEeqnarraynumspace
\end{IEEEeqnarray}}
Entries that correspond to edges in the base graph that can be, according to Algorithm STS, labeled with zero voltage are marked in bold.
By removing the last row and the last $K=4$ columns, the corresponding shortened $8 \times 8$ $(3, 3)$-regular base matrix $B_{\text{S-STS}(9)}$ follows directly as
{\setlength{\arraycolsep}{3.5pt}
\begin{IEEEeqnarray}{rCl}
\label{eq: base_sts9_shortened}
B_{\text{S-STS(9)}} & = & \borderarray{(}{)}{2em}{1.5ex}{ccccccccc}{
~~~~ & \sm{1}&\sm{2}&\sm{3}&\multicolumn{1}{c}{\sm{4}}&\sm{5}&\sm{6}&\sm{7}&\sm{8}\vspace{1mm}\\
\sm{1}& 0 & \bs{1} & 1 & 0 & 0 & 1 & 0 & 0 \\
\sm{2}& \bs{1} & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
\sm{3}& \bs{1} & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\
\sm{4}& 0 & \bs{1} & 0 & 0 & 1 & 0 & 0 & 1 \\
\sm{5}& \bs{1} & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
\sm{6}& 0 & \bs{1} & 0 & 1 & 0 & 0 & 1 & 0 \\
\sm{7}& 0 & 0 & \bs{1} & \bs{1} & \bs{1} & 0 & 0 & 0 \\
\sm{8}& 0 & 0 & 0 & 0 & 0 & \bs{1} & \bs{1} & \bs{1}
}
\end{IEEEeqnarray}}
This corresponds to removing the four Steiner Triples of $\text{STS}(9)$ containing the number of the last row. Shortening the $9 \times 12$ base matrix $B_{\text{STS}(9)}$ constructed from $\text{STS}(9)$ to obtain a shortened $8 \times 8$ base matrix $B_{\text{S-STS}(9)}$ is unpractical as its code rate is $R=1-8/8=0$. However, by shortening the $13 \times 25$ base matrix $B_{\text{STS}(13)}$ in the same way we obtain a $12 \times 20$ base matrix $B_{\text{S-STS}(13)}$ with the feasible code rate $R=8/20$.
\end{example}
In \tabref{girthgreater} the obtained QC ($J=3, K$)-regular LDPC block codes with girth $g=14,16$, and $18$ constructed from Steiner Triple Systems are presented. While the number of nonzero elements in each column is fixed to $J=3$, the number of nonzero elements in each row $K$ is specified in the first column. The second column corresponds to the obtained girth $g$, while in the third and forth columns we give the dimensions of the block code $(n, k)$ after tailbiting to length $M$. And the fifth column contains which Steiner Triple System ($\text{STS}(n)$) is used.
Finally, in the last column $W^{\prime}$ we give the degrees of the corresponding degree matrix $W$ in a compact way. As we have constructed the base matrices in such a way that the last nonzero entry in each column and the first entry in all other rows of the base matrix is labeled with a zero voltage, these entries are omitted. An entry of $W^{\prime}$ in column $j$ and row $i$ corresponds to the voltage degree of the $(j+1)$th nonzero entry in the $i$th row of the corresponding base matrix.
\begin{table}[t]
\centering
\caption{\label{tab: girth20}Properties of QC LDPC codes with girth $g \geq 20$}
\renewcommand\arraystretch{2.2}
\setlength\tabcolsep{0.4em}
\begin{tabular}[t]{c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c!{\shvline[1pt]}c}
$K$ & $g$ & $(n,k)$ & $M$ & Base graph (\tabref{girth8})\\ \shhline[1pt]
$4$ & $20$ & $(1296000, 324002)$ & $36000$ & $(27 \times 36)$, $g=8$ \\ \shhline
$5$ & $20$ & $(31200000, 12480002)$ & $480000$ & $(39 \times 65)$, $g=8$ \\ \shhline
$6$ & $20$ & $(518400000, 259200002)$ & $4800000$ & $(54 \times 108)$, $g=8$ \\ \shhline
$4$ & $22$ & $(7200000, 1800002)$ & $200000$ & $(27 \times 36)$, $g=8$ \cite{Esmaeili2010} \\ \shhline
$5$ & $22$ & $(325000000, 130000002)$ & $5000000$ & $(39 \times 65)$, $g=8$ \\ \shhline
$4$ & $24$ & $(39600000, 9900002)$ & $1100000$ & $(27 \times 36)$, $g=8$ \\ \shhline[1pt]
\end{tabular}
\end{table}
\subsection*{Case 2-II: $(J, K)$-regular LDPC block codes}
When searching for QC $(J=3, K)$-regular LDPC block codes with girth $g=20, 22$ and $24$, we started with previously obtained QC $(J=3, K)$-regular LDPC block codes of smaller block size and smaller girth, and (re-)applied our algorithms.
The obtained results for QC $(J=3, K)$-regular LDPC block codes with girth $g=20, 22$ and $24$ are presented in \tabref{girth20}. They are all but one based on previously obtained $(J=3, K)$-regular LDPC block codes with girth $g=8$ (\textit{cf.~} \tabref{girth8}), as specified in their last column in \tabref{girth20}. As before, the first column $K$ denotes the number on nonzero elements in each column; then we give the obtained girth $g$ and the dimensions of the block code $(n, k)$ after tailbiting to length $M$. The corresponding degree matrices are too large and are omitted in \tabref{girth20}, but are available at \cite{online}.
These codes are (probably) unpractical due to their huge block length. However, the table illustrates that by interpreting QC $(J, K)$-regular LDPC block codes as base matrices and re-applying our algorithms we can find QC $(J, K)$-regular LDPC block codes of ``any'' girth $g$.
\section{Conclusions}\label{sec: 9}
Using the relation between the parity-check matrix of QC LDPC block codes and the biadjacency matrix of bipartite graphs, new searching techniques have been presented. Starting from a base graph, a set of edge voltages is used to construct the corresponding voltage graph with a given girth.
By representing bipartite graphs in different ways, lower and upper bounds on the girth as well as on the minimum distance of the corresponding tailbiting block code have been discussed.
New algorithms for searching iteratively for bipartite graphs with large girth and for determining the minimum distance of the corresponding QC LDPC block code have been presented. Depending on the given girth, the search algorithms are either based on all-ones matrices, Steiner Triple Systems, or QC block codes. Amongst others, new QC regular LDPC block codes with girth between $10$ and $24$ have been presented including their minimum distance if possible. In particular, the previously unknown minimum distance, for some known codes with girth $6$ and $8$, has been determined.
\section*{Acknowledgements}
This research was supported in part by the Swedish Research Council under Grant 621-2007-6281.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Gödel-encoding beer orders 30 June 2017
Every few months I go to the Wellington for beer and food (bringing the food with me; they don't have a food licence, but do provide plates and cutlery).
Tour de Force, Christianna Brand 29 June 2017
1955 detective fiction; sixth and last of Brand's novels of Inspector Cockrill. On a Conducted Tour (an early package holiday) in Italy, one of the group is murdered. Nobody has a strong motive, and everyone seems to have an alibi. But the local Grand Duke is determined to execute someone…
Pyramid 104: Dungeon Fantasy Roleplaying Game 28 June 2017
Pyramid, edited by Steven Marsh, is the monthly GURPS supplement containing short articles with a loose linking theme. This time it's one of the three issues promised during the Dungeon Fantasy RPG kickstarter.
Penric and the Shaman, Lois McMaster Bujold 27 June 2017
2016 fantasy novella in the World of the Five Gods (formerly known as Chalion), second in the Penric sub-series. Now a trained sorcerer, Penric is lent to a detective chasing the murderer of a young nobleman; of course it's more complicated than that.
Lethal Weapon season 1 26 June 2017 - 1 comment
2016-2017, 18 episodes: Murtaugh is a straight-arrow cop, Riggs is his ker-razy partner.
Airs Above the Ground, Mary Stewart 25 June 2017
1965 mystery/thriller or romantic suspense. Vanessa March, married two years, had a fight with her husband Lewis just before he left on a business trip to Stockholm. But then he shows up on a newsreel about a circus fire in Austria… with his arm round a blonde.
Thirsty Meeples June 2017 24 June 2017
High Island Blues, Ann Cleeves 23 June 2017
1996 detective fiction; eighth and apparently last of Cleeves's novels of amateur private detectives George and Molly Palmer-Jones. George is commissioned to look into a possible charity fraud, then whipped off to Texas to assist an old birdwatching friend who's being accused of murder.
The Tripartite Problem Report 22 June 2017
I think I have come up with a new and potentially useful paradigm for computer-related problem reports.
The Great Weevil Infestation 21 June 2017
For a few weeks, we had weevils.
The Beckoning Lady, Margery Allingham 20 June 2017
1955 classic English detective fiction; fifteenth of Allingham's novels of Albert Campion. Campion's friends and country neighbours the Cassands are having their annual party; but three corpses will go some way towards spoiling the fun.
Westworld season 1 19 June 2017
2016 science fiction, 10 episodes. At the Westworld holiday resort, the android "hosts" exist to be shot, slept with, and generally taken advantage of. But some of them are starting to remember.
Femme Fatale, Carole Nelson Douglas 18 June 2017
2003 historical mystery, seventh of Douglas's novels about Irene Adler. Goaded by the scandal-mongering reporter Nellie Bly, Irene travels back to New York to look into a past of which she has little memory… but someone seems to be killing people who figured in it.
May-June 2017 Trailers 17 June 2017
Some trailers I've seen recently, and my thoughts on them. (Links are to youtube. Opinions are thoroughly personal.)
The Dispossessed, Ursula K. Le Guin 16 June 2017
1974 Hugo- and Nebula-award-winning science fiction. Shevek is a physicist working on a new theory of time, but he finds political obstruction even in an anarchist utopia. He travels to the mother world to continue his work.
High point value campaigns in GURPS 15 June 2017 - 1 comment
Thomas Thornberry posted, and Douglas Cole forwarded, a plea for information about high point value campaigns in GURPS: how can they be made to work, when skills become so high that failure is highly unlikely?
Marlow Tabletop and Board Games 5 June 2017 14 June 2017
Murder of Crows, Annie Bellet 13 June 2017
2014 modern fantasy, second in the series. Jade Crow is training to fight against the big nasty magician, but her father calls for help. You really shouldn't go home again.
Pyracantha in summer 2017 12 June 2017 - 3 comments
In 2013 we planted a pyracantha at the front of the house. Last year it didn't do much. Images follow: cc-by-sa on everything.
Too Like the Lightning, Ada Palmer 11 June 2017
2016 Hugo-nominated science fiction, first book of Terra Ignota. Even in a techno-utopia, there are people who are unhappy with the rules.
Van Helsing season 1 10 June 2017 - 2 comments
2016 post-apocalyptic horror, 13 episodes. Vampires have always walked the earth, and after a Yellowstone-eruption-induced loss of sunlight they've taken over. Three years later, a descendant of Abraham van Helsing helps to fight against them.
Spinsters in Jeopardy, Ngaio Marsh 09 June 2017
1954 classic English detective fiction; seventeenth of Marsh's novels of Inspector Roderick Alleyn. Alleyn is combining work for the Sûreté with a family holiday in the South of France, but sees a possible murder from the train, and then things get even more complicated. US vt The Bride of Death.
Election, Terrorism, Revolutionary Communism 08 June 2017 - 5 comments
I can't help noticing an obvious historical parallel which I haven't heard people talking about.
The Obelisk Gate, N K Jemisin 07 June 2017
2016 Hugo-nominated science-fantasy. As the world ends, two orogenes try to protect their little patches of it. Definitely don't try to start the series here.
UK Games Expo 2017 06 June 2017 - 3 comments
UK Games Expo continues to expand; the trade hall was still in NEC Hall 1, but all the tournaments which had shared space with it were pushed into one of the other halls at the back. With images; cc-by-sa on everything.
The Shape of Dread, Marcia Muller 05 June 2017
1990 mystery; ninth in Muller's series about Sharon McCone, private investigator in San Francisco. Tracy Kostakos, rising comedian, disappeared two years ago; Bobby Foster confessed to kidnapping and killing her. But there's no body, he's recanted the confession, and All Souls Legal Cooperative is handling his appeal.
GURPS Hot Spots: The Silk Road, Matt Riggsby 04 June 2017
This historical GURPS supplement looks at the Silk Road during the peak years of its importance, between roughly the second and tenth centuries AD.
Death on the Air and Other Stories, Ngaio Marsh 03 June 2017
1989 collection of short mystery stories, some featuring Roderick Alleyn.
Marlow Tabletop and Board Games 17 May 2017 02 June 2017
This Meetup-based boardgames group continues to meet at the Marlow Donkey – and, unusually, I was available on a Wednesday for the mid-month meeting.
London Particular, Christianna Brand 01 June 2017
1952 detective fiction; fifth of Brand's novels of Inspector Cockrill. Raoul Vernet, fading roué, had his head bashed in on a foggy night – in a nearly empty house. But everyone's got something to hide. US vt Fog of Doubt.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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package log
// Writer is an io.Writer which logs to Logger at the given Level.
type Writer struct {
Level Level
Logger *Logger
}
// NewWriter creates a new Writer.
func NewWriter(level Level, logger *Logger) *Writer {
return &Writer{level, logger}
}
// Write msg to the logger.
func (w *Writer) Write(msg []byte) (int, error) {
w.Logger.Printf(w.Level, "%s", msg)
return len(msg), nil
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,480
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Q: Osmnx basic stats from gdf I am trying to calculate the basic stats of a graph created from a gdf using the following code:
ox.utiles_graph.graph_from_gdf(n,e)
Nodes and edges have been saved previously and resulted from:
G = graph.graph_from_place(some_place_name)
n,e = ox.graph_to_gdfs(G)
n.save_to_file(filname,driver="ESRI Shapefile")
e.save_to_file(filname,driver="ESRI Shapefile")
I can plot the graph and also calculate travel times from it but I am unable to use the basic stats function.
I get the following error:
---> 56 ox.basic_stats(G)
~/anaconda3/envs/ox/lib/python3.9/site-packages/osmnx/stats.py in basic_stats(G, area, clean_int_tol, clean_intersects, tolerance, circuity_dist)
346 stats["edge_length_avg"] = stats["edge_length_total"] / stats["m"]
347 stats["streets_per_node_avg"] = streets_per_node_avg(G)
--> 348 stats["streets_per_node_counts"] = streets_per_node_counts(G)
349 stats["streets_per_node_proportions"] = streets_per_node_proportions(G)
350 stats["intersection_count"] = intersection_count(G)
~/anaconda3/envs/ox/lib/python3.9/site-packages/osmnx/stats.py in streets_per_node_counts(G)
80 """
81 spn_vals = list(streets_per_node(G).values())
---> 82 return {i: spn_vals.count(i) for i in range(int(max(spn_vals)) + 1)}
83
84
ValueError: max() arg is an empty sequence
Any clue how I can soleve this?
A: If you want to save your graph to disk and then work with it again later, you should save it as a GraphML file. If you must save it as node/edge shapefiles for some reason, you'll need to manually calculate the street per node counts if you want to calculate basic stats:
import osmnx as ox
import networkx as nx
G = ox.graph_from_gdfs(some_nodes, some_edges)
spn = ox.utils_graph.count_streets_per_node(G)
nx.set_node_attributes(G, values=spn, name="street_count")
stats = ox.basic_stats(G)
Note that if you do it manually this way, your street per node counts will probably be somewhat inaccurate due to periphery effects. You won't have this problem if you just download the graph the usual way then save it as a GraphML file.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Das Tholosgrab von Berbati () oder Grab des Lehrers () ist ein mykenisches Kuppelgrab im Tal von Berbati in der Argolis in Griechenland. Es liegt etwa 500 m nördlich des Mastos-Hügels, auf dem sich eine antike Siedlung befand, und etwa 3 km westlich des Ortes Prosymna.
Erforschung
Das Grab wurde 1935 entdeckt und im selben Jahr von Axel W. Persson ausgegraben. Die Veröffentlichung durch den Ausgräber wurde jedoch zunächst durch den Zweiten Weltkrieg und dann durch Perssons frühen Tod verhindert. 1970 wurden die unvollständigen Aufzeichnungen aufgefunden und erst 1984 durch Barbro Santillo Frizell publiziert. Während des Berbati-Limnes-Surveys erhielt das Grab die Bezeichnung FS 515 (engl. find spot = Fundstelle). 1994 untersuchte man FS 516 oberhalb des Grabs, mit dem dieser im Zusammenhang stand.
Beschreibung
Der Zuweg hatte eine Länge von 8 m und war 2,25 m breit. Er wurde seitlich von Mauern aus flachen, länglichen Natursteinen in pseudoisodomem Mauerwerk flankiert. Die Fassade wurde aus akkurat gearbeiteten Blöcken errichtet. Der Eingang war 1,60 m breit und 3,27 m hoch. Durch einen 3,86 m langen Korridor (Stomion) gelangte man in die runde Grabkammer. Der Korridor war ursprünglich mit großen Decksteinen bedeckt. Über diesen Decksteinen gab es ein sogenanntes Entlastungsdreieck in Kraggewölbe, das die Last der darüberliegenden Steine auf die Seitenwände ableitete und so die Decksteine entlastete. Die Tholos mit einem Durchmesser von 8 m wurde aus grob bearbeiteten Steinen in Kyklopischem Mauerwerk errichtet. In die Zwischenräume zwischen den Steinen wurden zur Stabilisierung kleine Steine gesteckt. Nach oben wurden die Steine immer weiter nach innen versetzt und bildeten eine Kuppel aus Kraggewölbe. Die Kuppel, die heute eingestürzt ist, war ursprünglich etwa 8 m hoch. Die Mauern sind noch zwischen 2,50 m und 4,50 m Höhe erhalten. Aufgrund der ähnlichen Bauweise zu dem Kyklopengrab von Mykene datierte Alan Wace das Grab in SH II A (etwa 1500–1450 v. Chr.). Es hat auch Ähnlichkeit mit dem später entdeckten Tholosgrab von Kazarma, das auch in die frühmykenische Zeit datiert wird.
Funde
Auf dem ursprünglichen Boden der Grabkammer fand man die Überreste von mindestens 18 Beisetzungen. Das Grab war in der Antike ausgeraubt worden und so fand man nur bescheidene Grabbeigaben. Hierunter befanden sich verschiedene Keramikgefäße und konische Tassen. Außerdem fand man meterlangen Bleidraht, der vermutlich dem Beschweren und Versteifen der Kleidung der Toten diente. Einige Gefäße datierte man zunächst in SH II A, was auch gut mit der Bauweise korrelierte. Bei einer Redatierung bevorzugte man jedoch eine Zuordnung zu SH III A1 (etwa 1400–1375 v. Chr.) und verlegte auch den Bau des Grabes in diese Zeit.
In einer höheren Schicht fand man ein einzelnes männliches Begräbnis aus SH III B (etwa 1300–1200 v. Chr.). Der Tote scheint einen höheren sozialen Status besessen zu haben. So wurde eine Steinbank über die gesamte Breite des Grabes errichtet. Unter den reicheren Beigaben befanden sich verzinnte Kylikes und Schalen sowie Tonfiguren. Eine Figur stellte eine gebärende Frau auf einem Stuhl dar. Wenige geometrische Tonscherben fand man in einer etwas höheren Schicht. An dem Hang oberhalb des Grabes fand man eine Keramikansammlung gleicher Zeitstellung. Sie bestand aus Trink- und Miniaturgefäßen und anderen Opfergaben. Sie waren vermischt mit Erde, Asche und verbrannten Knochen. Man vermutet, dass man sich hier vom späten 8. Jahrhundert v. Chr. bis in die Archaische Zeit am Tholosgrab traf und kultische Handlungen zur Ahnen- oder Heldenverehrung vollzog.
In römischer Zeit diente das Grab einer profanen Nutzung. So fand man 1–1,15 m über der untersten Schicht bei einer Kochstelle mit Ascheschicht Teile eines großen Kochgeschirrs, 4 Öllampen und zahlreiche Tierknochen. Der Zugang erfolgte damals über eine Treppe aus großen Blöcken rechts des Korridors.
Die Funde befinden sich in den Archäologischen Museen von Mykene und Nafplio.
Literatur
Weblinks
Σουηδικό Ινστιτούτο Αθηνών, «Η ζωή γύρω από μια ανασκαφή». Ασίνη, Μιδέα, Δενδρά, Μπερμάντι, (Αργολίδα, 1922–1959)
The Classical Review: Erik J. Holmberg: A Mycenaean Chamber Tomb near Berbati in Argolis
L'Antiquité Classique, Comptes Rendus: Erik J. Holmberg, A Mycenaean Chamber Tomb near Berbati in Argolis
Mykenisches Tholosgrab
Geographie (Argos-Mykene)
Archäologischer Fundplatz auf dem Peloponnes
Grabbau in Argos-Mykene
Grabbau in Europa
Archäologischer Fundplatz in Europa
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,945
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Apart from the details the result seems clear
Categories: Labour UK state
Demoralisation or fightback? That is the main question after June 8, writes Eddie Ford
With Theresa May robotically saying "strong and stable" at every single opportunity, this general election campaign so far has proved to be less than thrilling. But, leaving that aside for now, there have been some very worrying regional polls for the Labour Party - all of which strongly indicate, as we have long predicted, that the party is heading towards a crushing defeat on June 8.
Despite a lot of the left believing the myth that Scotland is an eternal Tory-free zone, an ORB International poll shows that the Tories are significantly ahead of Labour there with 27%, as opposed to Labour's 16%.1 At the moment, as our readers will know, the Scottish National Party has a near monopoly in Scotland when it comes to the House of Commons, with the three major all-Britain parties having only one seat each. But that was a bit of a freak result, thanks to the marvels of the first-past-the-post electoral system - it was never going to be repeated or bettered, except in the dreams of nationalists.
Obviously things could change, but, when you factor in other polls, it is more than reasonable to expect that the Tories will emerge after June 8 as the second party in Scotland, picking up between five to seven seats (perhaps even up to 10, if they have a very good night). Meanwhile, it seems like the Liberal Democrats will gain two seats and the SNP will lose about nine or so. With regards to Labour, you can do the maths. Just as ominously, if not more so, it looks almost certain that Glasgow will fall to the SNP on May 4 in the local elections - the first loss of control since 1980. Interestingly, since the 1930s onwards Labour control of Glasgow has been punctuated by periods of rule from the Progressive Party (or 'Progressives') - essentially a lash-up between the Unionist Party (ie, Scottish Tories) and Liberals to keep Labour out, something that happened all over Scotland until the PP was dissolved in the late 1970s.2
Anyhow, it looks like there will be no revival of Labour in Scotland, but rather of the Tories instead - not that this should not surprise us, even if it may come as a slight shock to dewy-eyed leftwingers in England who believe everything they read in Socialist Worker. Quite inevitably, Nicola Sturgeon's reaction to the Brexit vote was to immediately raise the prospect of another referendum on independence, given that Scotland voted to stay in, and to come out in clear opposition to Brexit - unlike Labour, which trots out the indistinct and rather pathetic formulation, 'We respect the vote, but …' Meaning that the SNP can pick up those anti-Brexit votes, while the Tories are now well positioned to pick up those who are anti-European Union and Labour could be squeezed out - neither fish nor fowl. What is it trying to say about Europe? Your guess is as good as mine.
Ditto in Wales, of course. According to another ORB poll, the Tories are on a fairly incredible 56% - making the party now stronger in Wales than in the 'Tory heartlands' such as the south-east, where they have 50% support.3 If true, this will be the first time the Conservatives have had a majority in Wales since 1859, when the franchise was massively restricted - only one in seven men had the vote and, of course, no women. This obviously represents a huge political change. Then again, Wales voted 'leave' by 52.5%, so Theresa May is trying to capitalise on this fact. By contrast, Labour is only on 32%. Similarly, in London, generally regarded as cosmopolitan and more leftwing - but here we have the same phenomenon. The Tories are on 43% and Labour has 32%. Again, quite a remarkable figure.
Needless to say, Labour looks set to take a drubbing on May 4, which will see 2,370 seats contested in 27 county councils, six English unitary authorities, one metropolitan authority and two mayoralties - there will also be votes for six new "combined authority" mayors in Greater Manchester, Liverpool City Region, Tees Valley, West Midlands, Cambridgeshire and Peterborough, and the West of England. As previously mentioned in this paper, Labour could lose around 125 council seats - which would be the worst local election results for an opposition party since 1985, excluding the years when there was also a general election. On the other hand, the Tories seem on course to gain more than 100 seats - chiefly at the expense of the UK Independence Party, which is on the road to nowhere (it would be a minor miracle if Ukip ends up with any MPs after June 8). The Lib Dems too could see a net gain of about 100 seats - possibly retaking control of Somerset and Cornwall from the Tories, the party clawing back support in the region despite a majority Brexit vote.
Labour will be particularly judged by its performance in Nottinghamshire, Lancashire and Derbyshire - it currently controls the latter and is the largest party in the other two counties. Of the councils which are up for grabs, Durham looks like the only place where it is certain to maintain control - but watch this space.According to some pollsters, Labour might be able to mitigate the embarrassment with victories in the elections for the new 'metro mayors' in Manchester and Liverpool - but that is not much consolation, if it happens.
When it comes to the wider, national, picture, the polls are consistently grim for Labour. ICM's weekly poll for The Guardian still has the Tories ahead by 19-points - getting 47% to Labour's 28% (with both the Lib Dems and Ukip on 8%).4 The same poll also had some questions on the general campaign so far. Questioned about whether the leaders were running a good or bad campaign, Theresa May was the only one to get a positive rating (41% good, 22% bad) - whilst Jeremy Corbyn's ratings were almost a mirror image (21% good, 40% bad), with 'Dr' Paul Nuttall getting the very worse results (8% good, 31% bad). As for the Financial Times's 'poll of polls' tracker, which calculates a time-weighted average of seven pollsters' most recent polls, it currently gives the Conservatives an 18-point lead over Labour (47% to 28%).5
Perhaps the most interesting findings - or depressing, depending on how you look at it - come in a recent YouGov poll, which shows that 48% of those who voted for Ed Miliband in the last election will be taking their votes elsewhere, whilst the Tories will retain 77% of theirs.6The Lib Dems and Ukip will only hang on to 47% and 43% respectively. It almost goes without saying that the unusually high rate of 'churn' between political parties is directly linked to the Brexit vote.Breaking down the results by the vote in last year's referendum, it can be seen that many 'leave' voters are coming over to the Tories from Ukip - and to a lesser extent Labour - while many 'remain' voters are drifting toward the Lib Dems from Labour. There is also a small, but significant number of 'remainers' who have switched to the Lib Dems from the Greens. Chris Curtis, YouGov's political researcher, pointed out that if these voters were to drift back to the party they voted for last time, then this would "disproportionately help" Labour - but if they start to move towards the Conservatives, just like other voters have, "we could be heading for an even bigger Tory victory than many currently anticipate". Showing the Zeitgeist, Coral's latest odds for the election have the Tories on 1/20 and Labour 12/1.7
Christ-like
Now, there is still quite a long way to go, but it is for Theresa May to lose this election, as opposed to Jeremy Corbyn winning it - something that looks extraordinary unlikely at the moment. The fact of the matter is that, from the moment Corbyn looked like he was going to win the Labour leadership contest in the summer of 2015, there has been a civil war raging in the party - with one rebellion after another, orchestrated resignations, massive leaking, knives in the back, attempts to give Corbyn a nervous breakdown. You name it, the Labour right have tried it - with the full backing of the media, of course.
Corbyn's response to this has been almost Christ-like - by which I mean the Jesus Christ as portrayed in the Bible: the one who always turns the other cheek, lets people rob him, urges his compatriots to pay their taxes to Caesar, passively accepts brutal Roman occupation, loves his enemies, etc. Needless to say, this writer does not believe a word of it - the genuine Jesus of Galilee was an apocalyptic revolutionary communist who wanted to totally overturn the established order, not the creepy other-worldly figure depicted in the holy book.
But this is how Corbyn seems to be behaving - they attack him and he offers up his cheek for another slapping. He appeases the right, having them in his shadow cabinet, he even appeases the right when it comes to the 'anti-Zionism equals anti-Semitism' campaign - that despite there being not a shadow of doubt about the involvement of the Israeli embassy, the US Christian right and the active connivance of Labour Friends of Israel and the Jewish Labour Movement (which, despite its name, is a Zionist organisation). Absurdly, all of this has been conclusively proved - not by Corbyn instigating his own investigation - but by an outside party, Al Jazeera. His reaction to the fabricated smears about 'anti-Semitism' was to give a speech to the Labour Friends of Israel fringe meeting at the 2016 Labour conference - where, instead of denouncing Israel as a colonial-settler project and declaring his solidarity with the oppressed Palestinians, he described Tel Aviv as having a "happy atmosphere". And a few days before told a reporter from the Jewish News that he admires Israel's "verve and spirit".8 Talk about love thy enemy. Labour Zionism is nothing to do with international socialism and the world solidarity of the working class - it promotes sectarian division and chauvinism. Unlike the Jeremy Corbyn of today, the Jeremy Corbyn of 1984 was well aware of this - hence his support for a motion calling for severing the ties between Poale Zion (ie, Jewish Labour Movement) and the Labour Party.9 Can we have this Jeremy Corbyn back, please?
Meanwhile, a number of Labour parliamentary candidates who have said that, when it comes to their local publicity, Jeremy Corbyn will not feature. Or, when questioned as to whether Corbyn would make a good prime minister, have refused to give a straight reply. You would have thought that they would actually fight a bit more, as it must surely matter to them in terms of results - being evasive on the doorstep is hardly going to increase their chances of getting elected. But you will not see such reticence from the Tories, who will play the Corbyn card for all it is worth - saying, like Boris Johnson, that the Labour leader is a "mutton-headed old mugwump", a threat to national security and a friend of terrorists.10 And the Labour right are treacherously joining in these Tory-led attacks, whether it be the likes of Tony Blair or Lord Roy Hattersley.
But for us in the CPGB, at least, the main issue is not so much the final details of the result, as we have a pretty good idea of them already. No, the real question is what happens after June 8? Sadly, like visitors from another planet, a lot of the left appear to have genuinely believed that Corbyn would sail to victory if he adopted a few leftish economic policies like the £10-per-hour minimum wage, renationalisation of the railways after the franchises have run out, promises of more money for the NHS, introduction of four new bank holidays, banning unpaid internships, etc - whilst gibbering on about fairness and niceness that could come from the lips of any Tory.11 Predictably, Corbyn's programme of managing capitalism - far less 'radical' than anything we saw in 1945 - has not proven to be popular with the post-Brexit electorate, as reflected in the dismal poll ratings. Regardless, the CPGB calls for a Labour vote, with active backing for proven leftwing candidates.
So what is the left going to do under these highly unfavourable circumstances? The very real danger is that the hundreds of thousands who joined because of Corbyn are going to become demoralised and eventually drop out. Or alternatively, as we urge, they will put the blame squarely on the right - not the Labour leader. It is not as though we do not have criticisms of Corbyn - when we called for a vote for him in the leadership election, it was despite his politics, not because of it. His election opened up an opportunity for the left to democratise and transform the Labour Party. But, at the end of the day, the right has been sabotaging Corbyn from day one - and should therefore be held responsible for the election defeat. Marxists in the party must redouble their efforts to win the argument and drive out the right.
eddie.ford@weeklyworker.co.uk
1. www.telegraph.co.uk/news/2017/04/29/exclusive-telegraph-orb-poll-reveals-conservatives-take-lead.
2. https://en.wikipedia.org/wiki/Progressives_(Scotland).
3. www.telegraph.co.uk/news/2017/04/29/five-key-charts-show-people-will-vote-general-election-according.
4. http://ukpollingreport.co.uk/blog/archives/9865.
5. https://ig.ft.com/elections/uk/2017/polls.
6. www.independent.co.uk/news/uk/politics/general-election-2017-corbyn-jeremy-labour-theresa-may-conservatives-poll-yougov-abandon-party-a7708956.html.
7. http://sports.coral.co.uk/political-specials/uk/uk-politics/next-general-election-6127461.html?market_set=ALL#market-144855019.
8. http://jewishnews.timesofisrael.com/jeremy-corbyn-praises-israel-at-jewish-news-debate.
9. http://jewishnews.timesofisrael.com/corbyn-backed-motion-to-severe-ties-with-jewish-labour-group.
10. www.thesun.co.uk/news/3422855/dont-feel-sorry-for-helpless-jeremy-corbyn-he-poses-an-enormous-threat-to-our-country.
11. http://press.labour.org.uk/post/160128951404/the-next-labour-government-will-transform-the.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 2,754
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Q: Handle self-references when flattening dictionary Given some arbitrary dictionary
mydict = {
'first': {
'second': {
'third': {
'fourth': 'the end'
}
}
}
}
I've written a small routine to flatten it in the process of writing an answer to another question.
def recursive_flatten(mydict):
d = {}
for k, v in mydict.items():
if isinstance(v, dict):
for k2, v2 in recursive_flatten(v).items():
d[k + '.' + k2] = v2
else:
d[k] = v
return d
It works, giving me what I want:
new_dict = recursive_flatten(mydict)
print(new_dict)
{'first.second.third.fourth': 'the end'}
And should work for just about any arbitrarily structured dictionary. Unfortunately, it does not:
mydict['new_key'] = mydict
Now recursive_flatten(mydict) will run until I run out of stack space. I'm trying to figure out how to gracefully handle self-references (basically, ignore or remove them). To complicate matters, self-references may occur for any sub-dictionary... not just the top level. How would I handle self-references elegantly? I can think of a mutable default argument, but there should be a better way... right?
Pointers appreciated, thanks for reading. I welcome any other suggestions/improvements to recursive_flatten if you have them.
A: One way you can do it using set and id. Note this solution also uses generators which means we can start using our flattened dict before the entire result is computed
def recursive_flatten (mydict):
def loop (seen, path, value):
# if we've seen this value, skip it
if id(value) in seen:
return
# if we haven't seen this value, now we have
else:
seen.add(id(value))
# if this value is a dict...
if isinstance (value, dict):
for (k, v) in value.items ():
yield from loop(seen, path + [k], v)
# base case
else:
yield (".".join(path), value)
# init the loop
yield from loop (set(), [], mydict)
Program demo
mydict = {
'first': {
'second': {
'third': {
'fourth': 'the end'
}
}
}
}
for (k,v) in recursive_flatten (mydict):
print (k, v)
# first.second.third.fourth the end
mydict['new_key'] = mydict
for (k,v) in recursive_flatten (mydict):
print (k, v)
# first.second.third.fourth the end
We can make a slight modification if you would like to see output for self-referential values
# if we've seen this value, skip it
if (id(value) in seen):
# this is the new line
yield (".".join(path), "*self-reference* %d" % id(value))
return
Now the output of the program will be
first.second.third.fourth the end
first.second.third.fourth the end
new_key *self-reference* 139700111853032
A: I'm not sure what your definition of "graceful" is, but this can be done with some bookkeeping of what has been seen before in a set of object ids:
class RecursiveFlatten:
def __init__(self):
self.seen = set()
def __call__(self, mydict):
self.seen.add(id(mydict))
d = {}
for k, v in mydict.items():
if isinstance(v, dict):
if id(v) not in self.seen:
self.seen.add(id(v))
for k2, v2 in self(v).items():
d[k + '.' + k2] = v2
else:
d[k] = v
return d
def recursive_flatten(mydict):
return RecursiveFlatten()(mydict)
Testing it out gives me what I expect
mydict = {
'first': {
'second': {
'third': {
'fourth': 'the end'
}
},
'second2': {
'third2': 'the end2'
}
}
}
mydict['first']['second']['new_key'] = mydict
mydict['new_key'] = mydict
print(recursive_flatten(mydict))
Out:
{'first.second2.third2': 'the end2', 'first.second.third.fourth': 'the end'}
|
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| 1,087
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{"url":"http:\/\/www.reddit.com\/r\/math\/comments\/1z01md\/everything_about_category_theory\/?sort=old","text":"[\u2013] 1 point2 points \u00a0(28 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI wanted to learn Haskell. To do it properly, I should first learn about category theory, I thought. The first thing would be to look up the definition of class. Then, I realized, it's not even well-defined in ZF or ZFC!\n\nSo, I apparently have to choose a better foundation. I'm told there are \"material set theories\" and \"structural set theories\", and that there's also type theory, which is supposed to have the best of those two worlds. That sounds great to me, but which type theory should I use? Which one is the latest and greatest? If you don't set me straight on this, I might just go with Martin-L\u00f6f due to geographical proximity.\n\nI won't even bother to install a Haskell compiler just yet.\n\n[\u2013] 3 points4 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHomotopy type theory\n\n[\u2013] 10 points11 points \u00a0(11 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAre you joking? That's like saying that before I learn to cook I should learn all about heat and organic chemistry, so I need a PhD in Physics and another one in Chemistry. So forget about learning to cook for another eight or ten years.\n\n[\u2013] -1 points0 points \u00a0(10 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHaskell is strongly motivated by and designed based on category theoretical concepts. Drop into the IRC and just listen to some of the discussions going on there. It's all category theory.\n\nSome would argue that leaning the theoretical foundation of a language is the Right Way to learn it, especially in this case. I'm among them.\n\n[\u2013] 5 points6 points \u00a0(9 children)\n\nsorry, this has been archived and can no longer be voted on\n\nMissing the point. OP didn't say he was going to learn Category theory. He said he WASN'T going to learn Category theory till he found a suitable underlying set theory. That's absurd. It's like saying you can't learn basic arithmetic because you have doubts about the foundations of logic.\n\n[\u2013] 1 point2 points \u00a0(7 children)\n\nsorry, this has been archived and can no longer be voted on\n\nSeems like just another way to do it. I don't see the issue.\n\n[\u2013] 0 points1 point \u00a0(6 children)\n\nsorry, this has been archived and can no longer be voted on\n\nOne's vastly more inefficient than the other.\n\n[\u2013] 0 points1 point \u00a0(5 children)\n\nsorry, this has been archived and can no longer be voted on\n\nInefficient at what, exactly?\n\n[\u2013] 0 points1 point \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nBecoming competent at a given skill, whether it's writing Haskell or learning to cook or whatever.\n\nIt would take decades to start from first principles when you could hit the ground running and be making solid progress by year's end.\n\n[\u2013] 1 point2 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAssuming that your goal is to write and deploy a program for the sake of it and have no other goal.\n\n[\u2013] -2 points-1 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nFor what other reason would you wish to learn a programming language but to use it? And why would you needlessly delay learning it by several orders of magnitude of time for no practical benefit to your goal?\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI'm sure I can do without any knowledge about the underlying things, but I want to learn about them. The reason I study mathematics and program is that I enjoy it.\n\n[\u2013]Algebra 16 points17 points \u00a0(6 children)\n\nsorry, this has been archived and can no longer be voted on\n\nwanted to learn Haskell. To do it properly, I should first learn about category theory,\n\nDespite the stupid popular opinion, this is very much not the case.\n\nCategory theory was originally a tool for algebra and topology to study homology in the 1930s. In the 60s, Grothendieck reformulated algebraic geometry with the notion of schemes and Lawvere showed that category could be applied to logic (notably, quantifiers can be realized as adjunctions).\n\nThat is, categories theory is a mathematical tool. A programmer should be more concerned that their program is bug-free and efficient, rather than worrying about presheaves or the yoneda lemma.\n\nDon't be lured in by technical terms. If you want to study semantics of programming languages, then yes, learn the tools. But you won't be studying Haskell. You will want to look at untyped and simply-typed lambda calculi, System F, and Martin-Lof type theory.\n\nAnd if you just want to learn Haskell, then just go do it. Ignore the words you don't know, or ask in #haskell. Some terms, like monad and functor, are used -- but their meanings are warped. There is a kind of \"pseudo-\" category theory that came about when programmers decided that the theoretical tools they knew almost worked... and so they banged them with a hammer until they fit. You don't need to know anything about a categorical monad to use Haskell monads.\n\nStill, some tools are useful to know about. Monoids and semigroups are invaluable if just for their ability to communicate common patterns. (A semigroup is something you can map-reduce over. A monoid is a semi-group where there's a \"do nothing\" action).\n\nSo, I apparently have to choose a better foundation\n\nWhere did you get this idea?\n\nNo one thinks in terms of foundations. A foundation is a nice-to-have. Something that helps logicians sleep at night.\n\nYou can treat types as sets, and ignore the technical inconsistencies.\n\nNo one with a job has ever learned ZF. No one has ever written a nontrivial proof in it either.\n\nMaterial set theories are how everyone learns in school. Structural set theory is something invented by Lawvere. You don't need to formally learn it, but rather just understand the basic idea: sets alone are boring, what we care about is structure (topologies, group multiplication, ordering, actions). But when you take an element without mentioning what set it came from, you don't know how it is intended to be used. Is 2 a prime number? Well, in the integers, yes, it is. But in Z[\u221a2], it's not (...but \u221a2 is). Structural set theories are also much more natural if you want to use category theory.\n\nThe \"latest and greatest\" foundations (although that's a fucking terrible measure to choose anything by) is homotopy type theory. Why is it so great? Probably because there's a fields medalist who said it was important. Why is it actually pretty cool? Because it solves a lot of people's problems simultaneously. It's structural (in the loose sense I mentioned above). It allows you to (in theory) do homotopy theory synthetically. It gives a novel interpretation of equality. It solves the type theorists' problem of functional extensionality. It's a constructive logic, which makes a small niche very happy. And it formalizes the often appealed to notion that you can transport structures from one type to another. (That is, if the unit complex numbers form a group, and the unit complex numbers are topologically a circle, then the circle can also be made into a group).\n\nBut tl;dr, Haskell doesn't require category theory.\n\n[\u2013] 0 points1 point \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAnd if you just want to learn Haskell, then just go do it. Ignore the words you don't know, or ask in #haskell. Some terms, like monad and functor, are used -- but their meanings are warped.\n\nI pretty much expect them to be warped. Still I would like to know what a monoid really is when I go to learn what it means in Haskell.\n\nI want to learn category theory anyway. This just seems like a good order to do things in.\n\n[\u2013]Algebra 2 points3 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nA monoid is an algebraic structure and a generalization of a group.\n\nIn particular, a monoid is:\n\n\u2022 A set M\n\u2022 An associative binary operation * : MxM \u2192 M\n\u2022 An element 1 in M called the identity such that for all m in M, 1m = m1 = m.\n\nIt's commutative if nm = mn for all n and m. It's called a group if every n has a unique element n{-1} such that nn{-1} = n{-1}n = 1.\n\nA monoid homomorphism is a structure-preserving map between two monoids. If \u03c6 : M \u2192 N is a monoid homomorphism, then \u03c6(mn) = \u03c6(m)\u03c6(n) and \u03c6(1) = 1 (note that 1 means the identity from M on the LHS and the identity from N on the RHS).\n\nImportant examples:\n\n\u2022 The natural numbers. (Note that * is addition and 0 is the identity).\n\u2022 The natural numbers again (with * being multiplication and 1 as the identity)\n\u2022 Strings (* is concatenation, 1 is the empty string)\n\u2022 Lists of Integers (* is concatenation, 1 is the empty list).\n\u2022 The integers (because they are a group).\n\u2022 Square matrices (with * being matrix multiplication and 1 is the identity matrix).\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nExcellent! Clear and to the point!\n\nI must confess an embarrassing mistake. I actually meant monad when I mistakenly wrote monoid. I suspect that would take longer to define and build upon many more prerequisites.\n\n[\u2013] 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nWell, you're right about it involving more prerequisites. It's not so many (if you get as far as understanding the definition of a category, functor, and natural transformation, then it's easy to pick up the definition at least), but the generalisation will be relatively empty unless you actually have some other categories to care about than the category of Haskell types and Haskell-definable functions between them, and it would be a shame to not be able to show the relationship to adjunctions and such, which will require some more prerequisites.\n\nIf your goal is just to write programs, this is a remarkable waste of time, and will probably be easier anyway if you already understand how things work out in Haskell.\n\nFrom the perspective of programming, a monad is just a particular \"shape\" that shows up in the API of many different libraries, and in Haskell, we choose to abstract over it, so we can have a bunch of code which works with all those libraries. You don't even really have to know what a monad is in Haskell to be able to use those libraries (but it kind of helps to know how the type class works, and be able to recognise that the operations in Control.Monad will work together with various libraries that you're using).\n\nIn particular, it's just a parameterised type constructor T with specified operations of these types:\n\nreturn :: a -> T a\n(>>=) :: T a -> (a -> T b) -> T b\n\n\nsatisfying a few laws which correspond to some ways in which we'd expect to be able to refactor code. (For the mathematicians in the audience, we're using the equivalent definition of a monad as a Kleisli triple).\n\nThe usual idea is that T a is some type of \"computations\" producing \"results\" of type a. (e.g. parsers, memory transactions, I\/O actions, nondeterministic machines of some sort, etc. etc.), and then for any value v, the computation return v will be the one which \"does nothing\" (whatever that means in context) apart from producing v as its result. The type of (>>=) (read \"bind\") is a bit more of a mouthful, but the idea is that it takes some initial computation whose result has type a, and a function from values of type a to further computations to be performed, and glues those together in the \"obvious\" way to get a computation whose result is the result of the second computation. So it captures the idea of doing first one thing and then another, but where we're allowed to make the second thing to do depend on the result of the first.\n\nAnd as far as it goes in Haskell, that's all you need to provide in order to define a monad. Once you have that, there's a library of other stuff which comes for free, as well as many others defined elsewhere where various things are abstracted over the choice of a monad.\n\nliftM :: (Monad m) => (a -> b) -> (m a -> m b)\n\n\nwhich for any monad m, lets you take an arbitrary function f of type (a -> b), and turn it into a function (m a -> m b), which basically works by acting on the result(s) of the given computation using the given function f.\n\nThe monad m itself (which acts on types to produce other types), along with this liftM, together define an endofunctor on the category of Haskell types. (We also have a type class called Functor for such endofunctors in Haskell, but for weird historical reasons, it's not a superclass of Monad. That should be changing soonish.)\n\nAnother little one is:\n\njoin :: (Monad m) => m (m a) -> m a\n\n\nThe description of monads you'll find in category theory textbooks will generally define them as endofunctors T on some category together with natural transformations eta: 1 -> T and mu: T2 -> T, satisfying some laws. The eta corresponds to return, and mu corresponds to join.\n\nKnowing that connection won't likely be all that useful to you if you're just getting to work writing a program, but if you're interested in the mathematical side of things, there are many examples of monads all over the place. The double dual construction in linear algebra is a functor. The embedding eta of each vector space V into its double dual (V**) provide the components of a natural transformation. What they don't typically bother with in first linear algebra courses is that there's also a natural transformation mu consisting of maps (V**)** -> V**. This (**, eta, mu) form a monad on the category of vector spaces.\n\nAnother example comes from analysis. The completion of a metric space gives a functor on the category of metric spaces (and contraction maps). There's an embedding of each metric space into its completion which gives a natural transformation eta from the identity functor to the completion functor, and moreover there's a natural isomorphism mu between the completion of the completion and the completion. That defines a monad on the category of metric spaces.\n\nFrom topology, we can consider the category of all subsets of a topological space with the arrows being inclusion maps between them. The map which sends each subset to its closure defines a functor on this category. There is an inclusion of each subset into its closure, and there is an inclusion of the closure of the closure into the closure (in fact, they're equal). As you can imagine, this gets us a monad.\n\nAnother wide class of examples comes from considering the functors on Set which send each set X to the set of all algebraic expressions of some sort in the elements of X (e.g. all K-linear combinations for K some field, or all polynomials, or all group words, etc.). There tends to be a natural map from X to the given set of expressions which maps each indeterminate to the expression with just that indeterminate, as well as another natural map which takes expressions in (expressions in X) and collapses them down to expressions in X, by distributing out coefficients, or whatever else is necessary in the given example. The algebras for these monads tend to actually be the algebraic structures of the sort that the expressions were for. e.g. The algebras of the monad of formal K-linear combinations will actually be (all!) vector spaces over the field K.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\n[\u2013]Algebra 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nA minor (very minor point).... The first paper on category theory was \"On the General Theory of Natural Equivalences\" This was published in 1942 not the 1930s. Their idea was to establish a framework in which they could frame questions about natural transformations. Natural transformations had been around since the 30s but there was really no firm foundation.\n\nBy the way, If you click on the above link, you will go to a link that asks you for 34 bucks for a copy of the paper. If you just enter the title of the paper into Google, you can probably get it free.\n\n[\u2013]Logic 3 points4 points \u00a0(6 children)\n\nsorry, this has been archived and can no longer be voted on\n\nOn top of what everyone else has said (really, you're making a mountain out of a molehill here), just use \"small\" categories and Grothendieck universes.\n\nMike Shulman's post that you linked to is about homotopy type theory, which is a slight modification of (intensional) Martin-L\u00f6f type theory. (Or rather, a collection of related variations on Martin-L\u00f6f type theory).\n\n[\u2013] 1 point2 points \u00a0(5 children)\n\nsorry, this has been archived and can no longer be voted on\n\nOn top of what everyone else has said (really, you're making a mountain out of a molehill here), just use \"small\" categories and Grothendieck universes.\n\nI want to learn category theory for its own sake too, not just for programming. It just seems like a good order to do it in. Is your recommendation above intended to cover only that which would be applicable in Haskell?\n\nThank you for your description of the type theory!\n\n[\u2013]Logic 1 point2 points \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nNo, that was just a method for providing category theory with a rigorous foundation. It really has no applicability to Haskell, since the \"category of Haskell programs\" (which isn't actually a category), is countable, so size issues don't come into play.\n\nAs for learning category theory: don't worry about making things sit on top of a rigorous foundation. If you're not at a point where you know how to hack around with foundations to make things like \"class-sized structure\" rigorous, you probably won't need to think seriously about size issues for a while. ;)\n\nAnyway, category theory is incredibly abstract, and like all abstract things, it's often better to learn a bit of their applications, and then turn around and use intuition from there to understand the abstract thing (and then turn back around and apply those abstractions to learn more about the applications). In other words, it's probably better to learn some Haskell, and use intuition from there to learn about category theory.\n\nYou don't need to know category theory to be a Haskell programmer; it can (and will) help, but it will only really help once you have a deep enough understanding of Haskell to apply categorical reasoning to what you're doing.\n\n[\u2013] 1 point2 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAs for learning category theory: don't worry about making things sit on top of a rigorous foundation.\n\nIt just feels so unsatisfactory not to even be aware of the proper foundation! In any case, I of course want to learn something about type theory anyway.\n\nYou don't need to know category theory to be a Haskell programmer\n\nI don't doubt that at all. Knowing about the actual category theoretical concepts as you encounter their namesakes in Haskell just seems like the most fun way to do it. I will want to know about category theory anyway.\n\n[\u2013] 1 point2 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nYou can always worry about putting any foundation on yet another foundation. Even once you get to set theory, well, what are sets really? This tends to be the place where most people stop, but there are ways to continue. Pretty much any logical system will have models in something else, if not many other somethings.\n\nAn important property of how we've collectively structured mathematics is that it's built in layers where you might switch out the underlying foundations for another without changing too much apart from perhaps some finer details of the layer in question. To the extent that those underlying details are important, you'll learn about them as they actually get applied.\n\nSo don't worry too much about starting at a higher level, and then fleshing out some ways in which it could be put in terms of another foundational system later. If the foundations are what interests you, by all means go for it, but just realise that it can be a bit of a distraction in some ways.\n\n[\u2013] 0 points1 point \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nEven once you get to set theory, well, what are sets really? This tends to be the place where most people stop, but there are ways to continue.\n\nWhat are those ways?\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nWell, one thing you might do is to consider sets to be homotopy classes of spaces A such that if x and y are any two points in A, then the type of paths from x to y is either contractible or empty. (i.e. there's a unique path up to homotopy, or no path.)\n\nThis is an impractical approach unless you have an abstract theory of homotopy types, like the one provided by homotopy type theory. HoTT identifies sets as being particular homotopy types along the lines I described. They can be made to satisfy Lawvere's properties for an elementary theory of the category of sets, if an appropriate form of the axiom of choice for types is taken to hold.\n\nIf you had an abstract foundational theory of categories, you might define sets to be those categories for which the only arrows are the identity arrows. Such a formalism is a bit hard to construct, so at present, there are only a few preliminary approaches to doing that.\n\nOf course, you can do the boring thing and construct models of sets in other set theories too. For instance, you can consider models of ZF set theory, NBG and Lawvere's ETCS in each other.\n\nMany set theorists spend much of their time considering the relationships between variations of ZF with various large cardinal axioms by examining the models they have in each other.\n\nThere's a lot of interest lately in finding other approaches to foundational logical theories based on something with \"more structure\" than sets have. First order theories of sets tend to be intolerable for most mathematicians to use directly on a day to day basis. Even though we have quite trustworthy computer programs which could help to check that proofs are correct if given in this form, almost nobody can actually be bothered to do it! Everyone simply writes things in such a way that everyone can be convinced that they could write everything out formally if forced to, and descending to that formalism is used as a tool for clearing up occasional misunderstandings.\n\nOne of the things which is very awkward about set theory is that the formalism doesn't know anything about the notions of isomorphism that we build on top of it, so when formalising any proof, you constantly find yourself explicitly transporting results across isomorphisms that we gloss over at a higher level of discussion, and while this is mostly straightforward, it's quite tedious and there are always differences in how it must be done.\n\nSo there's some hope that other approaches to foundations might be more usable, while at the same time still giving us much of the freedom to build the structures we're interested in that set theory has. (Perhaps we can even hope that some of the constructions that have been awkward in set theory become easier.)\n\nHoTT is exciting in that regard because its notion of equality (which is geometrically identified with the paths between points in a space, and thus behaves differently in different types) is sufficiently general to capture most if not all of the notions of equivalence or isomorphism that we would ever need, and it can formally transport constructions and theorems across that notion of equality. This makes it potentially far more usable to directly formalise our work in, and have it be machine checkable.\n\n[\u2013] 2 points3 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThe point of foundations is not to worry about them.\n\nUse Grothendieck universes if you really must, but seriously, except at very high\/technical levels of category theory, no one cares.\n\n[\u2013] 7 points8 points \u00a0(21 children)\n\nsorry, this has been archived and can no longer be voted on\n\nWow, timely. I've been struggling to understand two basic things about category theory:\n\nFirst, while I can see the use of category theory as a convenient language for discussing structures in various settings, I don't grok what it's applications are in terms of proving power. This is vague -- some examples:\n\nIn set theory, I can prove that |X| < |P(X)|, which has immediate implications, e.g., that there exist undecidable languages. In group theory, I can prove Lagrange's theorem, which also has immediate implications, e.g., the number of achievable positions on a Rubik's Cube divides the number achievable if I disassemble and reassemble it.\n\nAre there any parallels from category theory?\n\nSecond, I've read statements like this: \"Category theory is an alternative to set theory as a foundation for mathematics.\" But I haven't seen a good exposition of this -- any pointers?\n\n[\u2013] 2 points3 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI think its power comes from the ability to generalize back into category theory, prove a stronger result (where there is a little less imposed structure), and then project back into your specific category and tidy up the details.\n\nI took a course in higher module theory\/homological algebras, and I can't even imagine trying to prove some of these results without relying on categorical results. One particular example that comes to mind is the notion of a splitting short exact sequence (of R-modules, where R is commutative). The two most common definitions I've seen are as follows: for a short sequence [; 0 \\rightarrow A \\stackrel{f}{\\rightarrow} B \\stackrel{g}{\\rightarrow} C \\rightarrow 0 ;], we say it splits if i)[; B \\cong A \\oplus C ;] or ii) there exists an R-linear map [; h: C \\rightarrow B ;] such that [; g \\circ h = 1_{C} ;]. As it turns out, in an abelian category (which [; \\operatorname{{}_R Mod} ;] is, at least when R is commutative), there is a third equivalent characterization: iii) there exists an R-linear map [; i: B \\rightarrow A ;] such that [; i \\circ f = 1_{A} ;]. This result follows quickly from the Splitting Lemma, but is probably much more difficult to prove via module theory alone. EDIT: Nevermind. Bad example.\n\nI can't comment on the quote about category theory, though, as my knowledge in Category Theory is still too basic.\n\n[\u2013] 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nCould you elaborate a little on how your example shows that category theory has power in proving things? I learned that in the category of R-modules, left-split <=> right-split <=> the middle term is a direct sum, long before I knew what a category was. Do you think you could give the proof you have in mind (i.e. a proof that looks much easier when stated in terms of abelian categories than just for R-modules)?\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nTruth be told, I'd never seen a proof involving only module theory - if I recall correctly, neither Dummit & Foote nor Hungerford prove or even state the left-splitting characterization.* Looking back at the proof for the Splitting Lemma, however, I see no reason why the same, rather straightforward proof, couldn't apply to both. Damn. I really thought I'd had a good example too.\n\n*EDIT: Skimmed D&F. Seems they do talk about it. Double-fail today.\n\n[\u2013]Low-Dimensional Topology 10 points11 points \u00a0(11 children)\n\nsorry, this has been archived and can no longer be voted on\n\n\"Category theory is an alternative to set theory as a foundation for mathematics.\"\n\nThis is probably referring to Lawvere's elementary theory of the category of sets (ETCS): you can assert that a \"category of sets\" exists which satisfies certain axioms, and sets are defined as the objects of this category. Then a particular collection of those axioms turns out to be equivalent to ZFC.\n\nFor example, one axiom states that the category has a terminal object T, meaning that every object S has a unique morphism S->T; but the equivalent notion to this in classical set theory is that T is a 1-element set, since there is exactly one function from any set to the 1-element set. Now how do you define \"elements\" of sets, when sets are themselves just objects of some category? In classical set theory, an element of a set is equivalent to a function from the 1-element set to your set, because you can identify the element with the image of that function. So in ETCS, an element of a set is just a morphism T->S, and so Hom(T,S) stands in for the collection of elements of S. Another example is the ZF axiom that there is an \"empty set\": now in ETCS this is just an object E such that there are no morphisms T->E.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks for the reference -- I've only been able to give it a cursory glance, but it looks right on target.\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIsn't the existence of an empty set a theorem in ZF, rather than an axiom?\n\n[\u2013] 0 points1 point \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nI think its an axiom or at the very least near one. Similar to the initial element that must be exhibited in Peano arithmetic, the empty set is often used to provide the construction of many (all?) elementary sets through algorithms involving operations on sets. This allows us to be economical axiomatically, in the sense that you only directly assume the existence of a single element and alude to the rest with construction rules.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThe axiom of infinity gives us the existence of a set, then we can use the axiom of subsets to say that for P(y) false for all y in an inductive set z that exists by the axiom of infinity,\n[;\\exists x : y \\in x \\Leftrightarrow (y \\in z \\ \\land P(y));]\ni.e. there exists a set that contains no elements.\n\n[\u2013] 2 points3 points \u00a0(6 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIsn't the empty set usually introduced as an initial object?\n\n[\u2013]Low-Dimensional Topology 1 point2 points \u00a0(5 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI don't know how it's usually done, but suppose we define it as above: it's an object E with no morphisms 1->E, where I'm now using 1 to denote the terminal object. If there are two morphisms f,g:E->S for some set S, then it is vacuously true that f(x)=g(x) for all x in E (i.e. given any x:1->E, the compositions fx and gx are equal) because there are no x in E; thus axiom #4 in the linked PDF says that f=g. In other words, there is at most one morphism E->S for any S; now we just need to show that such a morphism exists.\n\nThe product axiom (#5 in the linked PDF) implies that for any set S, the sets E and S have a product ExS, so there are morphisms ExS->E and ExS->S. Now if ExS has an element, i.e. a morphism 1->ExS, then by composition there's an element 1->ExS->E of E, which is impossible, so ExS is an empty set E' and thus we have morphisms E'->E and E'->S which are both injections since E' has no elements. The subobject classifier axiom (#8) implies that we have a commutative diagram\n\nE' -> 1\n| |\nV V\nE -> 2\n\n\nin which the morphism E'->E is an inverse image of the map 1->2 under the map E->2. But now the identity E->E is an inverse image as well (recall that the map E->2 is unique), so there is a unique isomorphism E->E' and then the composition of this with the above morphism E'->S is our desired morphism E->S. It follows that E is an initial object.\n\n[\u2013] 1 point2 points \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nInteresting. Anyway, it still seems weird to me to assert the non-existence of a morphism.\n\n[\u2013]Low-Dimensional Topology 1 point2 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThat's true, but I guess the two perspectives are equivalent: suppose you declare as an axiom that there's an initial object E. Then there's a unique morphism E->1, and if there were also a morphism 1->E then the compositions E->1->E and 1->E->1 would have to be the identity morphisms, because there are unique morphisms E->E and 1->1 since they're initial and terminal respectively. This means that E is isomorphic to 1, so 1 is also an initial object. But this means that every set has exactly one element, which contradicts the existence of the two-element set 2 or the natural number system; we conclude that there are no morphisms 1->E after all.\n\n[\u2013] 1 point2 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nStrictly speaking, that depends on how you define 2 and N. The terminal category is (locally?) BiCartesian BiClosed with a natural numbers object after all.\n\n[\u2013]Low-Dimensional Topology 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nI agree, but I'm using the axioms in the ETCS article I originally linked to. It follows from their axioms that 2 and N have multiple elements, so my point was that if you assume all of the ETCS axioms except for \"there is a set with no elements\", then the statement \"there is a set with no elements\" is equivalent to \"there is an initial object.\"\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAh, I'm more used to the original axioms from Lawvere.\n\n[\u2013]Number Theory 10 points11 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nMaybe I can manage to give you a flavour of the power of category theory by telling you about the Yoneda lemma. In essence, the Yoneda lemma tells you that you can recover an object from knowing just the maps into the object. I like the analogy with particle physics: you can figure out properties of a particle by throwing stuff at it.\n\nThis means that in essence you can identify an object A of a category with the functor Hom(-,A). This is called the Yoneda embedding.\nSo you replace objects of your category with these representable functors: [; A \\mapsto \\mathrm{Hom}(-,A) ;]\nand then you can still keep track of the morphisms between objects\n[; \\mathrm{Hom}(A,B) \\leftrightsquigarrow \\mathrm{Nat}\\left (\\mathrm{Hom}(-,A), \\mathrm{Hom}(-,B) \\right ) ;]\nas being the same as natural transformations between the associated functors.\n\nIt might seem a bit trite, but it's a really powerful way to think about things. For instance in algebraic geometry, you are often led to consider so called moduli spaces, which are spaces which parametrise families of objects. For instance you can think of the moduli space of lines in the plane through the origin, and that forms a circle (in this case it's called the projective line).\nThe insight the Yoneda lemma brings to the table is that you can consider the functor which is the moduli problem itself. For instance, for the problem of parametrising lines through the origin, you are looking at the functor\n[; F(X) = \\left \\{ \\text{lines through the origin on } \\mathbb{A}^2_X \\right \\} ;]\nwhere [; \\mathbb{A}^2_X ;] is the plane over X.\nThe question of whether such a moduli space exists is then the same as whether this functor is representable, i.e. of the form [; \\mathrm{Hom}(-,A) ;] for some object A of your category. This is why representability is so important in algebraic geometry. And whenever you do have a representable functor, you automatically for free get a universal object family over A, corresponding to [; \\text{id} \\in \\mathrm{Hom}(A,A) ;]. In the case of the projective line (over the real numers), you get the M\u00f6bius strip. This is a universal varying family of lines: for instance, any family of lines over the circle is obtained by specialising the M\u00f6bius strip (specifically, pulling back along a map to the M\u00f6bius strip). This is the same as saying that you can obtain any family of lines over the circle by taking a piece of paper and repeatedly twisting it. So it's a universal family of lines, you can obtain any other family of lines by it (this leads to classifying spaces for the general linear group, Grassmannians and Stiefel manifolds).\n\nThis is a very powerful principle. I'll repeat it, in different words. The Yoneda lemma guarantees that whenever you have an object that classifies families (projective spaces classify families of lines, classifying spaces classify principal bundles, etc), you automatically have a universal family over your object, such that any other family is obtained by specialising this universal family.\nIf you have a \"moduli space of widgets\" that classifies all widgets, then there is a universal family of widgets defined over this moduli space such that any family of widgets over any space is a specialisation of it.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks for the reply. I'm still trying to parse... :)\n\n[\u2013]Number Theory 5 points6 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAnother perspective is that category theory is a useful organising principle. You put groups into the category of groups, sets into the category of sets, etc. But you forget everything except how maps compose, you don't even remember what the maps are. It's surprisingly powerful. The Elementary Theory of the Category of Sets that mathman1 mentions is just that: it describes what the category of sets is, and all it's doing is talking about dots and arrows, there's no real extra structure. Amazingly, you find that pinning down the structure of the category alone is enough to provide a foundation of mathematics which is very close to equivalent to ZFC (you need to weaken the axiom of replacement slightly, but that's it).\n\nI recently talked about how you can't recover a group from its category of representations. You only have dots and arrows, and it's a pretty hard exercise. I think it's a fun thing to seriously think about: pick a finite group you like, and for each of its representations write down a dot on a piece of paper (give them labels corresponding to the representation). Then write down arrows between representations when there's an intertwiner between the representations. This gives an object which is called a quiver (the difference with a category is that you're not remembering how to compose arrows, but don't mind that). It looks really really tricky to be able to figure out what the group is from simply this kind of data; you only have dots and arrows (and the data of how arrows compose)...\n\nIn that answer I explained that indeed you can't recover the group from this category of representations, but you can if you remember the tensor product structure of representations (Tannaka reconstruction). Once you internalise how little data you actually have on the level of the category, it's really neat!\n\n[\u2013]Algebra 3 points4 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nRethinking Set Theory explains the basic idea behind \"categories for foundations\".\n\nAn important thing to emphasize (that threw me off when I was first learning) is that category theory isn't inherently about the foundations of mathematics. You should think of a category as being the same thing (in spirit) as a group or a module. Another way to think of it as a hyper-powered partial- (or rather pre-) order.\n\nBill Lawvere in the 60s was a proponnet for category theory as an alternative foundations. It led to some interesting new ways to think about foundations (notably, structural set theory, where objects do not have an identity outside of their set, and for example, you talk not about subsets but of embeddings of one set into another via a map). Probably most importantly (from what little I know), Lawvere showed that logical quantification arises from adjunctions.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nBill Lawvere in the 60s was a proponnet for category theory as an alternative foundations.\n\nIf you use his ETCS, isn't that the foundation and category theory then built upon it?\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIf you want to start out simply you should look up things like the definition of a \"product\", this immediately shows you that a product of two sets\/groups\/topological spaces is (in a certain sense) unique, and it is possible to define what a \"product\" is in any category, although they may not always exist.\n\n[\u2013] 2 points3 points \u00a0(22 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI've been taking a course on Category Theory and there's something really basic I do not understand.\n\nOpposite categories.\n\nSay for example my category is [; \\text{Set} ;] , how exactly do I interpret the arrows in [; \\text{Set}^{\\text{op}} ;] ? If I have a function [; f:A\\rightarrow B ;] in [; \\text{Set} ;], do I interpret its opposite arrow as the relation [; \\{(f(x),x)\\in B\\times A\\mid x\\in A\\} ;] or what?\n\n[\u2013] 3 points4 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nArrows are not functions. They are just arrows.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI know that in general they're not functions, but in the case of [; \\text{Set} ;] they definitely are functions. I take your point that in general I shouldn't think of them as functions, or their equivalent between classes\/collections, though.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI believe that, in the case of [; \\operatorname{Set} ;], all that happens is that your arrows (maps) are reversed.\n\n[\u2013]Number Theory 7 points8 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nGood question. The thing to really grok is that a category is nothing but a bunch of dots and arrows, with a way of composing the arrows. The arrows don't really have an intrinsic meaning or representation.\n\nSo when someone talks about, for instance, the opposite category of the category of sets, well, the arrows are just that, arrows, which are going in the opposite direction.\n\nNow there's such a thing as a concrete category: this is a category together with a faithful functor to the category of sets. This gives you an interpretation of arrows as maps between sets. It's important that there exist categories which are not concretisable (you can't find such a functor).\n\nIn the case you asked, you're in luck, the opposite category of the category of sets is concretisable. To see this, you just take the power set functor. This is a contravariant functor on the category of sets (it's equal to the functor [; \\mathrm{Hom}(-,2) ;] where 2 is a 2-element set; 2 is a subobject classifier), it sends a set to its power set, and a function between sets to the \"preimage function\" between power sets. It's a faithful (covariant) functor from the opposite category of the category of sets to the category of sets.\nFrom this, it follows that a category is concretisable if and only if its opposite is (the opposite would have a faithful functor to [; \\text{Set}^{\\text{op}} ;], and just compose that with the functor we just constructed).\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nSo my intuition should in general be to not worry too much about what the arrows represent. That is actually very helpful. I already suspected that was kind of the case but it's nice to have it confirmed\/spelled out. Thank you a lot. Edit: Now that I think about it, that makes a lot of sense when considering isomorphisms between categories.\n\nI'll have to come to this comment tomorrow when I'm a little more sober to really understand your explanation of Setop though.\n\n[\u2013]Algebra 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nSo my intuition should in general be to not worry too much about what the arrows represent.\n\nYup.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nOf course, an arrow A -> B in Setop is just a function B -> A.\n\n[\u2013]Mathematical Physics 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nIf you don't mind my asking, what book are you using for the course?\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI'm not actually using a book, and I don't mind you knowing which country I'm in so I'll link you to the notes.\n\nCategory Theory: http:\/\/www.staff.science.uu.nl\/~ooste110\/syllabi\/catsmoeder.pdf\n\nThe second half of the course, Topos Theory: http:\/\/www.staff.science.uu.nl\/~ooste110\/syllabi\/toposmoeder.pdf\n\n[\u2013]Arithmetic Geometry 8 points9 points \u00a0(9 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAn arrow X \u2192 Y in the category Setop is a function from Y to X. Similarly, an arrow G \u2192 H in the category Grpop is a group homomorphism from H to G.\n\n[\u2013] 0 points1 point \u00a0(5 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAn arrow X \u2192 Y in the category Setop is a function from Y to X.\n\nBut suppose you have non-injective f \\in YX, how is fop \\in XY when it's not a function? I'm pretty sure that's what they're asking. I suspect you're using 'function' more generally than MediocreAtMaths.\n\n[\u2013]Arithmetic Geometry 3 points4 points \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nYou're using the same notation XY to denote Hom-sets in two different categories. To avoid ambiguity, let's denote by C(X, Y) the set of morphisms from X to Y in the category C.\n\nWhat I'm saying is that, by construction, Cop has the same objects as C, and Cop(X, Y) = C(Y, X). So, if C = Set, a morphism f \u2208 Setop(X, Y) = Set(Y, X) is a morphism in Set from Y to X, that is, a set-theoretic function (in the ordinary sense) with domain Y and codomain X.\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nTake f:{1,2}->{1,2} where f(1) = 1 = f(2). What's the set-theoretic opposite of f? I'm now confused by that and suspect that's what confused the original question asker.\n\n[\u2013]Arithmetic Geometry 3 points4 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nThe corresponding morphism in Setop is the exact same set-theoretic function. It's easier to see if you replace one of the sets with an isomorphic (but non-equal) copy: take the set-theoretic function f: {1, 2} \u2192 {3, 4} defined by f(1) = 3 = f(2). Note that f \u2208 Set({1, 2}, {3, 4}). The corresponding morphism fop in the opposite category is the exact same set-theoretic function, but fopSetop({3, 4}, {1, 2}). In other words, it's the same function set-theoretically, but the objects that are category-theoretically labelled \"domain\" and \"codomain\" of fop are reversed in Setop.\n\nIn other words:\n\n\u2022 A morphism from X to Y in Set is a function with (set-theoretic) domain X and (set-theoretic) codomain Y.\n\u2022 A morphism from X to Y in Setop is a function with (set-theoretic) codomain X and (set-theoretic) domain Y.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI think the problem I had with that, and many others do too, is that we expect the opposite map to be the inverse. That made it clearer though, thanks for bearing with me :).\n\n[\u2013]Algebra 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nYou shouldn't think of Setop like that. Functions from X to Y do not necessarily correspond to functions from Y to X. For example, there are 2 functions from {a} to {b, c}. But there is only one function from {b, c} to {a}.\n\nIt's better to just think of the op category as a place where you formally turn the arrows around. That's it. It's a good way to think about contravariant functors.\n\n[\u2013]Arithmetic Geometry 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nFunctions from X to Y do correspond to arrows in Setop from Y to X, though. That's the only point I'm making.\n\n[\u2013]Algebra 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAh... OK.\n\n[\u2013]Algebra 4 points5 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nThink of them as purely formal constructions.\n\nYou can attempt to reason about them more literally... and sometimes that's useful. But not always.\n\nFor instance, in a pre-order category, the opposite cat is the dual pre-order. (You just reverse the order of things, simple).\n\nA frame is what you get when you take the open set lattice of a topological space and forget what the points were. A frame morphism maps open sets to open sets. However, we all know from topology that a continuous map doesn't (necessarily) preserve open sets.... rather, open sets pull back to open sets. In other words, the opposite of a frame morphism is a \"continuous\" function (albeit, you have forgotten the points). The dual category of frames is called the category of locales.\n\nA classic example in algebraic topology. The category of affine varieties has affine sets as objects (subsets of Cn defined by the vanishing-set of a collection of complex polynomials). A regular morphism between two affine varieties is a map which gives each coordinate in terms of a polynomial.\n\nIt turns out this category is equivalent to the opposite category of reduced C-algebras of finite type. (Roughly, every object is a \"nice\" polynomial ring).\n\nHow does this equivalence come about? Well, the definition of a regular morphism says that for every variable in the codomain, you need a polynomial (in the variables of the domain). You might equivalently say that you have a mapping from variables in the codomain... a function... but one going the wrong way. It turns out this (combined with the fact the variables of a C-algebra form a \"basis\") is enough to specify a C-algebra morphism.\n\nThat wasn't very clear. Tl;dr, good luck!\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThankfully I'm also taking a course on Algebraic Geometry, so that was a lot clearer than you might think. Thank you a lot :)\n\n[\u2013] 2 points3 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIf you want a concrete interpretation of Setop, it\u2019s (equivalent to) the category of complete atomic boolean algebras.\n\nBut most of the time (at least for me) it\u2019s easier to think of it as the category whose objects are just sets, and where an arrow A\u2192B is a function from B to A.\n\n[\u2013]Logic 6 points7 points \u00a0(13 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHey! Timely. Yoneda lemma: what does it mean?\n\nI can state it; I can prove it, I can even come up with some contrived examples. But I really have no idea what it means, and when it comes up in a proof, it always seems to come out of nowhere. Any intuition, tips, insights?\n\n[\u2013]Algebraic Geometry 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nMaybe there's a deeper meaning that I ignore, but what it says to me is that everything you want to know about an object in some category, you can know it by knowing the morphisms that involve such object.\n\nOr in a more abstract and blunt way, you shouldn't care about the meaning of the objects on your category, the whole deal relies in the relations an object has with all others\n\n[\u2013]Number Theory 3 points4 points \u00a0(3 children)\n\nsorry, this has been archived and can no longer be voted on\n\nFor me, the essence of it is that to understand an object A of a category C, it's equivalent to understand the Hom functor\n[; \\mathrm{Hom}(-,A) \\colon \\mathcal{C}^{\\text{op}} \\to \\text{Set} ;]\nThere's nothing more to it than that, but it's surprisingly powerful as a mode of thought if you take it seriously.\n\nThis mathoverflow thread contains many insightful perspectives on the topic.\n\n[\u2013]Logic 1 point2 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks! I think your reply to DoctorZook is very helpful. I'll have to mull over it a bit, but it seems to provide the sort of intuition I'm looking for. Urs Schreiber's reply in the linked MO thread also seems very helpful, and feels a bit more applicable to things I know. Thanks for the link!\n\n[\u2013]Number Theory 4 points5 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nIt also allows you to talk about internal objects in a category. For instance, you could give the usual definition of a group object A in a category (it's an object A, together with multiplication and inversion which are morphisms in the category, with additional axioms expressed as commutative diagrams), but instead you can say it's simply a group-valued representable functor!\nThat is, consider first the usual representable functor\n[; \\mathrm{Hom}(-,A) \\colon \\mathcal{C}^{\\text{op}} \\to \\text{Set}.;]\nThen, if you have a functorial way of lifting this to a functor\n[; \\mathrm{Hom}(-,A) \\colon \\mathcal{C}^{\\text{op}} \\to \\text{Grp},;]\nit means A is actually a group object in your category!\n\nThis is the kind of parallel the Yoneda lemma allows, it's useful everywhere (for instance, the description of affine group schemes as spectra of Hopf algebras is an immediate application).\n\n[\u2013]Logic 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\n!!!! Woah! That's very cool. I'm gonna play around with this! This is an application that I can sink my teeth into to really understand what's going on. thanks again.\n\n[\u2013]Algebra 4 points5 points \u00a0(4 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThe one secret I learned from Awodey's video lectures on YouTube was that the Yoneda lemma is really only important because it tells you that the Yoneda embedding is actually an embedding.\n\nThe Yoneda embedding is a functor that takes any object A in C to the functor Hom(-, A) in Presheaves(C) (= Set{C{op}}). The fact its an embedding says that it is full and faithful. That is, the Hom-sets between A and A' are the same (up to isomorphism) as the natural transformations between Hom(-, A) and Hom(-, A').\n\nIn particular, that means that (up to isomorphism), we have taken the category C and added extra objects and morphisms to it. In effect, we are taking a kind of closure of the category!\n\nIt's very similar to compactification of a topological space, or algebraic closure of a field, or completing a metric space, etc.\n\nThe category of presheaves on C is a very nice category. It has all finite limits and colimits. It's cartesian closed. It's a topos.\n\n[\u2013]Logic 1 point2 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIn effect, we are taking a kind of closure of the category!\n\nYeah, Urs Schreiber's comment in the MO thread \/u\/presheaf link to says something very similar. I think this is a good perspective for me to take, at least until I can internalize (heh) what's going on.\n\n[\u2013]Algebra 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\n(Just to be clear, I don't know wtf I'm talking about either. I had had this question for a long time, and I haven't head my head deep enough in to really tell how useful the damn thing is :)\n\n[\u2013]Logic 2 points3 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHeh. If I had a dime for every time I explained something I don't understand, I'd... well, I could buy a coffee or two.\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHere's a precise way to say what \"closure\" means here. Let C be a category and Psh(C) the category of presheaves of sets on C. One might say that Psh(C) is obtained by freely adjoining colimits to C because of the following result. If D is a category which is cocomplete, i.e. has all colimits, then the functor Fun(Psh(C),D) --> Fun(C,D) given by composition with the Yoneda embedding C --> Psh(C) is an equivalence. In particular, any functor from C into a cocomplete category extends to a functor on Psh(C), which is unique up to equivalence. This is an example of a Kan extension.\n\n[\u2013] 3 points4 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThere are several perspectives on this.\n\nOne perspective is that a presheaf on a category [; C ;] is a right [; C ;]-module (a covariant functor [; C \\to \\mathbf{Set} ;] is a left [; C ;]-module). In this context, the Yoneda lemma is a generalization of [; \\operatorname{Hom}_R(R,M) = M ;] (for this to be literally true, you need to use an enriched-category-theory version of the Yoneda lemma). The co-Yoneda lemma, which states that every presheaf is canonically a colimit of representable presheaves, then generalizes [; R \\otimes_R M = M ;]. For more on this see Mac Lane-Moerdijk, Sheaves in Geometry in Logic, Chapter VII (Geometric Morphisms), or these nLab pages: Yoneda reduction, co-Yoneda lemma.\n\nAnother is to show that the Yoneda embedding (map x to Hom(-,x)) is fully faithful; this formalizes the principle that we can and should \"understand things by how they relate to others\".\n\nIt can be used to express values of functors as hom-sets, which may be more manipulable; for example, to define the mapping complex [X,Y] of simplicial sets, we compute\n\n[; [X,Y]_n = \\operatorname{Hom}(\\Delta^n, [X,Y]) = \\operatorname{Hom}(X \\times \\Delta^n, Y) ;]\n\nSimilarly, it may be used to construct morphisms between objects defined solely by universal properties. As an example of this, let [; \\Omega = \\{\\text{true},\\text{false}\\} ;]. A proposition on a set [; X ;] is a map [; X \\to \\Omega ;] (note that we can identify propositions on [; X ;] with subsets of [; X ;] via [; P \\mapsto P^{-1}(\\text{true}) ;] and [; S \\mapsto -\\in S ;]). We can define an AND operator on propositions by [; (P\\land Q)(x) \\iff P(x) \\text{ and }Q(x) ;]. Since this is defined \"in the same way\" for all pairs of propositions, it specifies a natural transformation [; \\operatorname{Hom}(-,\\Omega)\\times\\operatorname{Hom}(-,\\Omega) \\to \\operatorname{Hom}(-,\\Omega) ;]. But [; \\operatorname{Hom}(-,\\Omega)\\times\\operatorname{Hom}(-,\\Omega) = \\operatorname{Hom}(-,\\Omega\\times\\Omega) ;], so by the Yoneda lemma this gives a map [; \\Omega\\times\\Omega \\to \\Omega ;]; thus the Yoneda lemma allows us to construct the AND operator on truth values from the AND operator on propositions.\n\nIn topos theory, [; \\Omega ;] is a more general subobject classifier, and the above is what lets us construct the logical operators on [; \\Omega ;] from operations on subobjects.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI\u2019ve come to think that talking about the Yoneda \u201clemma\u201d is underselling it: I like to think of it as a very useful algorithm for translating between internal and external arrows and equations in a category. By internal, I mean expressed in terms of the arrows of the category itself, and by external I mean expressed in terms of presheaves on the category.\n\nOf course this is a personal view, and I\u2019m fairly unusual as someone who works (sometimes) in category theory but prefers solving problems to building every-more-abstract theories. There are other ways one can think about it.\n\nAt least in computer science, people sometimes seem to be too wary of proofs that use Yoneda, preferring purely internal arguments. This is irrational, since the Yoneda procedure allows the external proof to be internalised in a systematic way, so you can think of the external proof as being just a concise description of an internal one.\n\nIt often turns out that proofs can be made much simpler and concise by working externally. I had an interesting experience about a year and a half ago, where someone wrote to me wanting an internal version of a proof that I\u2019d written about the external version of. What externally was a simple observation turned into three pages of massive commutative diagrams, when we extracted an internal version of the proof. The internal proof is also very non-obvious, and I don\u2019t believe anyone would ever have discovered it working purely internally.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI like to think of the Yoneda lemma as being a completion theorem for categories.\n\nIt says that given an arbitrary, potentially \"poor\" small category C, we can embed it into a category which is both complete and cocomplete, has all exponentials, is not only Cartesian closed (meaning that you can interpret the lambda calculus in it), but locally cartesian closed as well (you can interpret dependently typed languages), has a subobject classifier making it into a topos, and many other nice properties.\n\nIt accomplishes this fully and faithfully, so not only is this completion very rich with practically all the widgets we could ever want, there's a pristine copy of our original category sitting inside it. If we can use all these extra widgets to prove the existence of an arrow yA -> yB, then that arrow must have come from exactly one arrow A -> B in our original category.\n\n[\u2013]Differential Geometry 1 point2 points \u00a0(6 children)\n\nsorry, this has been archived and can no longer be voted on\n\nCan someone suggest a good text for learning category theory? Preferably a text which uses modern terminology and notation.\n\n[\u2013]Logic 2 points3 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nFor a first encounter?\n\nBorceuox's 3 volume tome on categorical algebra seems to be the most complete, and is very readable. The first volume covers \"category theory\", and the other two delve into its use in algebra. One problem with this book is it doesn't cover monads until rather late (Chapter 4 of Volume 2). You could probably read this chapter without reading anything else in Vol 2 (perhaps Chapter 3 would be a good warmup).\n\nUnless you're coming in with a strong background in algebra, it might be a bit high level. In this case, I'd start with Awodey's book, which requires less background, but doesn't cover nearly as much. It's also definitely emphasizes applications to logic.\n\nIf you're looking for something that is more \"light reading\", Goldblatt's Topoi is (IMO) incredibly well-written, but isn't a deep enough text to get anything more than the flavor of category theory. It also, even more than Awodey's book, focuses on logic--it really is a book on categorical logic. It might be better as a companion to one of the other texts to look at when you get lost.\n\n[\u2013]Differential Geometry -1 points0 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks for the recommendations! I'm weak in algebra, since my undergrad degree was in physics and I mostly took analysis courses on the side. I'm learning differential geometry from John and Jeffrey Lee's texts now and I see a lot of category-theoretic terminology so I wanted to be more familiar with it.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nI'll second Borceux. I've been reading the first volume off and on the past few months and while there are some occasional errors, overall it's really good. I like it much better than Mac Lane.\n\n[\u2013]Algebra 4 points5 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nAwodey is the standard intro text for logicians. It's pretty readable.\n\nMacLane's Categories for the Working Mathematician is not hard to read, but is geared towards mathematicians and isn't a great introduction to the subject.\n\nPierce has a small book for computer scientists. It's pretty good up until the natural transformations chapter, and then it gets kinda hard.\n\nLawvere has a book Conceptual Mathematics (aka, the \"baby book\"). It starts off as if categories are something a 10-year-old could grok, but difficulty escalates at some point.\n\nThere are a number of free books (Triples, Toposes, and Theories, Categories for Computing Science, Categories, Types, and Structures) worth looking at.\n\nAluffi's Algebra Chapter 0 is a book on abstract algebra that promotes (very basic) categorical language early on.\n\nHowever, overall, most introductions are pretty terrible.\n\n[\u2013] 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nI'm going to second the recommendation of Awodey's text, but I'm going to go a little further and make that recommendation regardless of whether your background is in general mathematics, logic, computer science, or elsewhere.\n\nIt's reasonably self-contained, though it does assume just a bit of ability to read things written in a mathematically mature style, it develops the more important examples internally. It's pretty modern in terms of the way in which it covers the material that it does.\n\nThe explanations of concepts are quite good. Though I'd gone over the proof before, I don't think I really understood Yoneda's lemma properly until I read Awodey's coverage of it.\n\nIf it has a major downside, it's that it stops just as the story is really getting good. It does serve as a very good starting point though, in that once you've gone through it, you should have a solid foundation to start picking up additional material from a wide variety of places.\n\n[\u2013]Differential Geometry -1 points0 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks! I took a look at it and it seems like a great text.\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nWhat prerequisites are necessary to begin studying category theory? I've considered taking an independent study course and learning it but I wasn't sure if I'd be prepared enough as a 3rd year undergraduate.\n\n[\u2013]Algebraic Geometry 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nI began studying around that time and it was hard.\n\nYou could study this since beggining your undergraduate as it doesnt require you to know that many things by itself. However, if you want to truly understand the definitions and the deepness of the theorems you'll want to have a solid background of mathematical maturity, a good experience with algebraic topology, some algebra for other motivating examples and maybe learn it along some algebraic geometry. Knowledge in logic would be great too, but I dont think you NEED to.\n\nI guess it depends on your program and yourself, but specially the algebraic topology background is hard to get early in your studies, not impossible tho\n\n[\u2013]Logic 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nyou'll want to have a solid background of mathematical maturity, a good experience with algebraic topology, [...]\n\nGood experience with logic can replace algebraic topology here, in my experience.\n\n[\u2013] 2 points3 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIs there much use for category theory in functional analysis\/operator theory?\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIs there a good intuitive explanation behind the notion of Kan extension and its importance? Just looking at the technical definition, there is no way I can \"get\" this quote by MacLane:\n\nThe notion of Kan extensions subsumes all the other fundamental concepts of category theory.\n\n[\u2013]Arithmetic Geometry 1 point2 points \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nIt's a very natural concept. I have a functor F: C -> D, and another functor i: C -> C', and I want to get a \"best approximation\" F' : C' -> D which is somehow the best approximation of F on the category C'.\n\nUltimately, all you can hope for a Ran F or Lan F from C' to D, where the set of natural transformations NatTrans(Lan F, G) is isomorphic to NatTrans(F, G o i) for any functor G: C' -> D or Ran F which instead satisfies NatTrans(G, Ran F) is isomorphic to NatTrans(G o i, F).\n\nYou'll notice that this is an adjointness relationship(and I believe adjointness in general is a special case of Kan extensions!). In fact limits and colimits are special cases of Kan extensions too.\n\nIf you're willing to sit and spend an hour reading:\n\nhttp:\/\/ncatlab.org\/nlab\/show\/Kan+extension#idea\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nThanks! This is a much better read than the wikipedia page (although it will take me much longer than an hour!). I liked this bit:\n\nTo a fair extent, category theory is all about Kan extensions and the other universal constructions: limits, adjoint functors, representable functors, which are all special cases of Kan extensions \u2013 and Kan extensions are special cases of these.\n\n[\u2013]Logic 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nErrr, so this is really specific, but in section III.7 of categories for the working mathematician, as he's going through colimits of representable functors, he says that a given functor K is the colimit of a diagram given by a functor M:JD -> SetsD . Am I correct in thinking this should instead be M: J -> SetsD .\n\nI know it's not exactly what this thread is for, but I couldn't think of a better place to ask :(\n\n[\u2013]Arithmetic Geometry 3 points4 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nFun fact: Let n be a positive integer. There exists an abelian category An where An has n distinct isomorphism classes of objects.\n\nIf n=1, then take the zero category which is trivially abelian. Otherwise n is at least 2. Let's construct such an An. Consider Vn the category of Q-vector spaces of dimension at most aleph (n-2). This admits a Serre subcategory Vfin of finite dimensional vector spaces. We can then consider the quotient category(really a localization), Vn \/Vfin =: V. The morphisms in V are fairly complicated, but the only isomorphism classes in V are the following:\n\nFinite dimensional vector spaces, which are identified with zero.\n\nCountable dimensional vector spaces,\n\nAleph 1 dimensional vector spaces,\n\n.\n\n.\n\n.\n\nAleph n-2 dimensional vector spaces,\n\nThus we get n distinct isomorphism classes of objects in V.\n\nThis is pretty neat! If you think about it, it's strange that an abelian category can have finitely many objects(up to isomorphism), but what happens here is that the hom sets become very complicated, whereas Obj V is relatively simple.\n\n[\u2013] 2 points3 points \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHey r\/math. There are two category-theoretic constructions I don't understand very well. Both have pretty detailed Wikipedia pages and are documented in a lot of books, but every source that I've found is relatively opaque and I've been having difficulty internalizing all of the objects and morphisms and functors that are flying around. These two objects are monads and (co)limits.\n\nFor each, I'm wondering if someone could help explain to me (a) why these constructions are important and (b) some concrete examples in mathematics where such things arise. I know that the Seifert van Kampen theorem can be phrased in terms of limits, but the exposition in May's book is difficult for me to read.\n\nSome further questions I have: is there a sense in which these limits are a generalization to limits in metric spaces? And is there a sense in which these monads are related to the monads in functional programming languages?\n\nThanks!\n\n[\u2013]Algebraic Geometry 2 points3 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nOkay, so first you need to know the definition: you have a diagram in some category, which just means a collection of objects and morphisms, like A -> B or X -> Y <- Z or maybe ...->C2 -> C1 -> C0 as it doesn't have to be finite. Then the limit of the diagram is an object in your category which has maps to everything in your diagram (so in the last one, there are maps D -> Ci for each i) such that everything commutes, and it has the universal property that if you have another such object with morphisms, it factors through the limit.\n\nFirst thing to note: limits don't have to exist in your category. And they are dependant on the category you take them in, if you take the limit in the category of abelian groups that will be different than the limit in the category of groups, for instance.\n\nExample 1: If your diagram is A->B then A maps to both A and B by the obvious choices, and everything commutes, and anything which maps to both A and B in a way that commutes has those maps factor through A. Thus A is the limit of that diagram.\n\nExample 2: The limit of a diagram like X Z (no maps between them) is called a product. In the category of sets its the cartesian product, in the category of abelian groups its the direct product, in the category of topological spaces its the product space, etc. A limit encompasses all these constructions at once. If you add a twist like X->Y<-Z now you have the fibered product. In say Top, this is just the subset of X x Z where the elements of the ordered pair map to the same point in Y. If you are in the category of abelian groups and Z = 0, then this is the kernel of the morphism X -> Y.\n\nExample 3: In the category of abelian groups, ...->C1->C0 gives the inverse limit construction.\n\nThe power of this is that while you may have been originally introduced to all these constructions independently, product spaces, inverse limits, kernels, etc. but really they are the same idea: there is some diagram and the construction is the universal object mapping to that diagram (I studied mathematics for way too long before someone told me what a universal object actually is: it's just another word for terminal object, so in this sense you take the category of objects with morphisms to the diagram, and it is terminal there, everything maps to it).\n\nA colimit is everything here but backwards: you have a diagram and the object is universal in that everything in the diagram maps to it, and any other object mapped to by the diagram has those maps factor through the colimit. Examples: disjoint union in Top, direct sum, cokernel, and direct limit in Ab, tensor product in the category of rings.\n\nThe colimit of a diagram like X<-Y->Z is called the pushout, and in the category of groups specifically, it is the free product with amalgamation. This is where the weirdness in SvK comes in: if you're like me SvK is the first place you ever see the free product, let alone free product with amalagamation, so it seems very weird and out of nowhere. But in terms of colimits, you just say that the fundamental group of the union is the pushout of the fundamental groups of the open sets and their intersection. This doesn't give you (right away) the actual construction, but it may give you everything you care about: the universal property.\n\nI think any connections with limits in the analytic sense are extremely tenuous. One connection is the p-adics. Basically you give the rational numbers a strange metric and Qp is the completion with respect to that metric: adding in things to make cauchy sequences converge. But you can also construct it by taking the inverse limits of the groups Z\/(pn), then taking the field of fractions. I don't know any other examples, but would be interested to hear them.\n\n[\u2013]Algebraic Topology 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nYes! Monads are definitely related to the monads in Haskell. In fact, I'm fairly new to category theory, and most of the stuff I do understand is either from Haskell or (co)homology theory.\n\nIn Haskell, all monads operate on the category Hask of haskell types, with morphisms given by (computable) functions. So here, the endofunctor is an instance of the Functor typeclass -- it maps the type a to m a, and any functions (morphisms) of type a -> b to m a -> m b. The unit 1_C -> T (i.e. Hask -> T(Hask)) is defined by a polymorphic function return of type a -> m a, and fmap which is the raising (a -> b) -> (m a -> m b). Finally, join is the natural transformation T2 -> T, which takes m (m a) -> m a.\n\nThe important thing here is that you can raise all computation in Hask up to T(Hask), and stay only \"one level up\" from Hask, and that's the idea of a monad, as far as I know. All relationships in a category C can be raised to relationships in T(C), and furthermore relations between C and T(C) stay in T(C) without having to move up to T2 (C). People more familiar with category theory at large can feel free to correct me on this point.\n\nAlso, all monads arise as the composition of two adjoint functors. For example, the list monad [] arises from a free functor F from Hask to the category of monoids, and the forgetful functor G back. F takes a type a to the free monoid on the elements of a -- the most general associative operation on a collection of symbols (here, elements of a) is just string concatenation. The forgetful functor takes a sequence of elements back to the list of those elements. By composing these functors you get the return function for this monad -- x maps to \"x\", which maps to [x]. Furthermore, if we apply this again to [x], we get \"[x]\" and then [[x]], which can be flattened to [x] by list flattening.\n\n[\u2013] 0 points1 point \u00a0(2 children)\n\nsorry, this has been archived and can no longer be voted on\n\nIt seems people here are more interested in Categories with very little structure. I prefer ones enriched in cochain complexes over C.\n\nAlso while we're at it Braided Monoidal Categories are very fun to play with. You can get away with messing with ribbons.\n\nIn fact why don't we enrich in more. We can talk about enriching in Cat or even just go all the way up to \\infty.\n\n[\u2013]Algebra 0 points1 point \u00a0(1 child)\n\nsorry, this has been archived and can no longer be voted on\n\nI like braided categories because I like Hopf algebras :)\n\n[\u2013] 1 point2 points \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nRemembering how you forgot is really useful. I could say it the real way, but this sounds sillier.\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nFor me, I am less fascinated with \"Category Theory\" as a mathematical framework than the extent to which many objects of mathematical study are encoded in the shapes of certain categories and functors. Category theory offers a different perspective for approaching questions about these objects - and one of the things which leads to proof is being able to consider the same thing in different ways.\n\nAn example I am particularly fond if is the construction of the classifying space of a finite p-group as the nerve of the category of elementary abelian subgroups, with morphisms given by inclusion (see the book Homotopy theoretic methods in group cohomology by Dwyer and Henn for details).\n\n[\u2013] 0 points1 point \u00a0(0 children)\n\nsorry, this has been archived and can no longer be voted on\n\nHas category theory been applied to dynamical systems or control theory ? I am not looking for that one paper that appeared in 1970ish but anything more recent.","date":"2014-10-02 17:38:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7827349305152893, \"perplexity\": 705.2121294031597}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-41\/segments\/1412037663762.45\/warc\/CC-MAIN-20140930004103-00145-ip-10-234-18-248.ec2.internal.warc.gz\"}"}
| null | null |
Q: How can I change CSS with JavaScript without the use of an ID I have a custom scroll wheel and I want to make a button to switch between two classes for that scroll wheel, for example let's say I have these CSS lines to style the scroll wheel:
body::-webkit-scrollbar {
width: 1em;
}
body::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px rgba(180, 88, 88, 0.473);
background-color: rgb(131, 69, 69);
}
body::-webkit-scrollbar-thumb {
background-color: rgb(185, 115, 115);
outline: 1px solid slategrey;
}
But when I click the button I want it to change to this CSS:
body::-webkit-scrollbar {
width: 1em;
}
body::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px DIFFERENT COLOR;
background-color: DIFFERENT COLOR;
}
body::-webkit-scrollbar-thumb {
background-color: DIFFERENT COLOR;
outline: 1px solid slategrey;
}
Everything I have read uses the ID to change the CSS for that element but I can't do that with a scroll wheel. I hope this explained it well enough please ask me if you want me to add more code, examples etc.
A: You can just toggle a class on the body which changes the styling.
Below is an extra snippet that makes use of CSS variables.
const changeBtn = document.querySelector('.change-btn');
changeBtn.addEventListener('click', () => {
document.body.classList.toggle('changed');
});
/* For demo purposes */
body {
height: 500vw;
}
/* Default styling */
body::-webkit-scrollbar {
width: 1em;
}
body::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px rgba(180, 88, 88, 0.473);
background-color: rgb(131, 69, 69);
}
body::-webkit-scrollbar-thumb {
background-color: rgb(185, 115, 115);
outline: 1px solid slategrey;
}
/* Changed styling */
body.changed::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px orange;
background-color: teal;
}
body.changed::-webkit-scrollbar-thumb {
background-color: lime;
}
<button type='button' class='change-btn'>Change</button>
Bonus with CSS variables
const changeBtn = document.querySelector('.change-btn');
changeBtn.addEventListener('click', () => {
document.body.classList.toggle('changed');
});
body {
height: 500vw; /* For demo purposes */
--track-color: rgb(131, 69, 69);
--thumb-color: rgb(185, 115, 115);
--shadow-color: rgba(180, 88, 88, 0.473);
}
/* Default styling */
body::-webkit-scrollbar {
width: 1em;
}
body::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px var(--shadow-color);
background-color: var(--track-color);
}
body::-webkit-scrollbar-thumb {
background-color: var(--thumb-color);
outline: 1px solid slategrey;
}
/* Changed styling */
body.changed {
--track-color: teal;
--thumb-color: lime;
--shadow-color: orange;
}
<button type='button' class='change-btn'>Change</button>
A: You can select elements on jquery without it's ID by using classNames, data-attributes, etc.
$('button').on('click', function(){
$('body::-webkit-scrollbar').css({
'width' : '1em',
...
})
$('body::-webkit-scrollbar-track').css({
'background-color' : 'DIFFERENT COLOR',
'another-property' : 'another-value'
...
})
$('.class-name').css({
'width' : '1em',
...
})
$('[data-foo="bar"]').css({
'width' : '1em',
...
})
$('.class-name.with-another-classname').css({
'width' : '1em',
...
})
$('.class-name .with-children').css({
'width' : '1em',
...
})
})
A: JavaScript does not require ID's. The alternative to getElementByIdis the usage of querySelector. querySelector can be used to target classes, Id's or tags.
In your case the best solution however would be to target the body tag and to add a class to the body that applies your changes you predefine in CSS like the sample below:
function addClass() {
document.querySelector('body').classList.add('red');
}
html {
color: white;
}
body {
background-color: blue;
}
body.red {
background-color: red;
}
<button onclick="addClass()">Turn background red</button>
A: I think your best option is to do it like this:
function toggleBodyActive(){
document.body.classList.toggle('active');
}
/* set bodyheight for scrolling */
body{ height:200vh; }
body::-webkit-scrollbar {
width: 1em;
}
body::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px rgba(180, 88, 88, 0.473);
background-color: rgb(131, 69, 69);
}
body::-webkit-scrollbar-thumb {
background-color: rgb(185, 115, 115);
outline: 1px solid slategrey;
}
/* same but when body has .active class */
body.active::-webkit-scrollbar {
width: 1em;
}
body.active::-webkit-scrollbar-track {
-webkit-box-shadow: inset 0 0 6px rgba(0, 188, 88, 0.473);
background-color: rgb(31, 169, 69);
}
body.active::-webkit-scrollbar-thumb {
background-color: blue;
outline: 1px solid slategrey;
}
<div onclick="toggleBodyActive();">toggle 'active'</div>
A: You can use querySelector for that
document.querySelector('body')
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,689
|
\section{Motivation and Overview}
An intriguing 2.4 standard deviation discrepancy \cite{atomsNewMeasurement2019,atomsTheoryReview2019,HarvardMagneticMoment2008,MullerAlpha2018} recently arose between the Standard Model's most precise prediction and the measured value (Fig.~\ref{fig:MeasuredAndPredicted}). The best measurement \cite{HarvardMagneticMoment2008,HarvardMagneticMoment2011} determines the electron magnetic moment in Bohr magnetons ($\mu$/$\mu_B$) to $3$ parts in $10^{13}$ -- the most precisely determined property of an elementary particle. The SM prediction requires Dirac theory, quantum electrodynamics, hadronic and weak interaction contributions \cite{atomsTheoryReview2019}.
The part in $10^{12}$ agreement between SM prediction and measurement that stood for years gave way as a result of a more precise measurement of the latter. The discrepancy triggered new theoretical investigations into possible physics beyond the SM \cite{gardner2019light,ALightComplexScalarForTheElectronAndMuonAnomalousMagneticMoments,PhysRevD.98.075011,PhysRevD.98.113002,PhysRevD.99.095034}. As this work was being reported, a second new $\alpha$ measurement \cite{RbAlpha2020Nature} contradicted the first, giving a SM prediction that disagrees with electron's measurement by 1.6 standard deviations, but in the other direction.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/gComparison.pdf}
\caption{Comparison of the measured electron magnetic moment \cite{HarvardMagneticMoment2008} with the standard model predictions \cite{atomsTheoryReview2019,MullerAlpha2018,RbAlpha2020Nature}} \label{fig:MeasuredAndPredicted}
\end{figure}
A one-particle, quantum cyclotron is at the heart of past and future measurements \cite{HarvardMagneticMoment2008,atomsNewMeasurement2019}. A single electron, suspended indefinitely in a Penning trap, is cooled enough that it initially occupies only one of the two stable cyclotron ground states, one with spin down and one with spin up (Fig.~\ref{fig:quantumstates}). Transitions are driven between these states and a third, the first excited cyclotron state with spin down. The state of the quantum cyclotron is detected after the drives are turned off using quantum jump spectroscopy. The angular cyclotron and anomaly drive frequencies, $\omega_c$ and $\omega_a$, that produce one-quantum transitions, determine the magnetic moment in Bohr magnetons,
\begin{equation}
\pm\frac{\mu}{\mu_B} = 1 + \frac{\omega_a}{\omega_c} = \frac{g_\pm}{2}.
\end{equation}
The plus and minus signs are for the positron and electron, and the g-values $g_\pm$, divided by 2, are other names for the ratio of moments. The frequency $\omega_c$ is the electron cyclotron frequency. The anomaly frequency $\omega_a = \omega_s - \omega_c$ is the difference between the electron spin precession frequency $\omega_s$ and its cyclotron frequency. The anomaly frequency $\omega_a$ is directly measured instead of the spin frequency $\omega_s$ because the uncertainty in $\mu_/\mu_B$ is thereby reduced by about a factor of $\omega_c/\omega_a\approx10^3$ \cite{Review}. The resonance line shapes from which these frequencies are extracted have intrinsically different shapes.
The use of quantum nondemolition (QND) detection methods completely evades detection backaction for determining the quantum state of the cyclotron and spin motion. Nonetheless, detection backaction still prevented better measurements of the cyclotron and spin transition frequencies to better determine the magnetic moments. This backaction produced a very wide and asymmetric quantum jump spectroscopy line shape when cyclotron transitions were driven to determine the cyclotron frequency. Even though resonant frequencies can be extracted from broad and asymmetric lines in principle, in practice this causes a susceptibility to systematic uncertainties. Significant progress in precision frequency measurements typically takes place only when narrower and more symmetric line shapes are produced. We recently proposed a very promising method for circumventing this detector backaction for the frequency measurements \cite{Fan2020BackActionPRL}. The cyclotron line shape would be much more symmetric, and orders of magnitude narrower, than for previous measurements.
In this work we describe the quantum calculation that is carried out to predict the narrow quantum-jump line shapes \cite{Fan2020BackActionPRL} much more completely. A master equation is solved for a driven quantum cyclotron with a QND coupling to a detection oscillator, the latter being coupled to a thermal reservoir. The predicted quantum jump line shapes are very different than was predicted for the case when the detection oscillator was more strongly coupled to its environment \cite{BrownLineshapePRL,BrownLineshape}. We also present for the first time quantum calculations for (1) driven anomaly transitions, (2) directly driven spin flips, and (3) spin flips produced by simultaneous cyclotron and anomaly drives. The additional calculations make it possible to evaluate and contrast possibilities for making new measurements of the electron and positron magnetic moments.
Key to these calculations, and the possibility to measure the electron and positron magnetic moments much more accurately, is decoupling the detection oscillator from its thermal environment by a factor of 100 during the time in the measurement when one-quantum transitions are being driven. The parameters used in the calculation are those realized in a very recent experimental demonstration of one way that this could be done \cite{FanRFSwitch2020}, while also allowing the necessary coupling to be restored for quantum state readout.
The outcome of the calculation is that it now seems feasible to carry out new electron and positron magnetic measurements that are an order of magnitude more accurate than was previously possible. This would make it possible to investigate the discrepancies between the most precise prediction of the standard model of particle physics, and the most accurate measurement of a property of an elementary particle \cite{atomsNewMeasurement2019,atomsTheoryReview2019}.
Details of the quantum system are given in Sec.~\ref{sec:QuantumCyclotron}. The Hamiltonian and master equation of the system are presented in Sec.~\ref{sec:HamiltonianForTheQuantumSystem}. Calculations of single photon excitations of cyclotron and anomaly transitions are given in Sections~\ref{sec:OneDriveExcitations} and \ref{sec:AnomalyLineshape}, respectively.
Sec.~\ref{sec:DirectSpinFlips} does the same for directly driven spin flips. Sec.~\ref{sec:twophotoncalculation} predicts the quantum-jump line shape for simultaneously applied cyclotron and anomaly drives. Sec.~\ref{sec:Comparison} contrasts the relative advantages of the different methods, and Sec.~\ref{sec:summary} provides a summary.
\section{Quantum Cyclotron}
\label{sec:QuantumCyclotron}
A one-electron quantum cyclotron is at the heart of the approach being investigated here. An electron or positron in a Penning trap is confined within a spatially uniform magnetic field $B\hat{z}$, along with an electrostatic quadrupole potential \cite{Review}. The possibility to use only the ground and first excited cyclotron states of a single isolated electron has already been demonstrated and used for measurement \cite{HarvardMagneticMoment2008}. The two lowest levels of the quantum cyclotron are separated by an energy $\hbar \omega_c$, where $\omega_c$ is the angular cyclotron frequency introduced above. The spin up (quantum number $m_s=1/2$) and spin down ($m_s=-1/2$) states are separated in energy by $\hbar \omega_s$, where $\omega_s$ is the spin precession frequency discussed above. This one-particle quantum cyclotron has a Hamiltonian
\begin{equation}
H=\hbar \omega_s \left(a_s^\dagger a_s
- \tfrac{1}{2}
\right) +\hbar \omega_c (a_c^\dagger a_c + \tfrac{1}{2}).
\label{eq:HQuantumCyclotron}
\end{equation}
The spin raising and lowering operators are
\begin{equation}
\begin{split}
a_s^\dagger\left|\downarrow\right\rangle&=\left|\uparrow\right\rangle \\ a_s\left|\uparrow\right\rangle&=\left|\downarrow\right\rangle,
\label{eq:spinoperator}
\end{split}
\end{equation}
and $a_c^\dagger$ and $a_c$ are harmonic raising and lowering operators for the cyclotron motion \cite{Review}.
An electrostatic quadrupole potential added to the magnetic field makes a Penning trap that can suspend a single charged particle indefinitely within an extremely high vacuum \cite{PbarMass}. The electron (of charge $-e$ and mass $m$) oscillates along the magnetic field direction in a harmonic oscillator potential energy,
\begin{equation}
W(z)=\tfrac{1}{2} m \omega_z^2 z^2
\end{equation}
and $\omega_z$ is the angular axial oscillation frequency. The electrostatic quadrupole shifts the cyclotron frequency slightly in a well understood way \cite{InvarianceTheorem,Review} that can be neglected for the purposes of this calculation.
This axial motion is used to make quantum nondemolition (QND) measurements of one-quantum spin and cyclotron transitions \cite{QuantumCyclotron,QNDScience1980,QNDReview1980, QNDreview1996,1996MarkQND}. A small magnetic bottle gradient, $B_2 z^2$, is added to the spatially uniform magnetic field, $B_0$, of the Penning trap,
The addition modifies the axial trapping potential and shifts the frequency of the axial oscillation. A QND detection of a one-quantum cyclotron excitation is possible because it shifts
the axial frequency from $\omega_z$ to $\omega_z+\delta_c$, with
\begin{equation}
\delta_c = \frac{e B_2}{m} \frac{\hbar}{m \omega_z} \approx 2\pi~ \times(3~\rm{Hz})
\end{equation}\cite{Review}, without changing the cyclotron state. (A two-quantum cyclotron excitation would be $2 \delta_c$ and so on, as will be discussed later and quantified in Eq.~(\ref{eq:axialfrequencyshift}). The one-quantum shift is just large enough to be detectable. The relative shift is $\delta_c/\omega_z = 1.5 \times 10^{-8}$ for demonstrated experimental values \cite{HarvardMagneticMoment2011} ($B_2 = 1500$ T/m$^2$ and $\omega_z/(2 \pi ) = 200$ MHz.).
This bottle shift can be decreased in two ways -- by decreasing the magnetic gradient $B_2$ or by increasing the axial frequency, $\omega_z$.
Since a next generation experiment\cite{atomsNewMeasurement2019} uses $B_2=660$ T/m$^2$, we choose the intermediate value $B_2=1200$ T/m$^2$ for the illustrations in this paper.
The magnetic gradient is unfortunately also responsible for a backaction that broadens the range of frequencies over which a driven cyclotron excitation or spin flip can occur. The cyclotron and spin frequencies in Eq.~(\ref{eq:HQuantumCyclotron}) both acquire a small $z^2$ dependence,
\begin{eqnarray}
&&\omega_{c}(z)=\omega_{c} + \tfrac{eB_2}{m}z^2\\ &&\omega_{s}(z)=\omega_{s} + \tfrac{g}{2}\tfrac{eB_2}{m}z^2,
\label{eq:spindependentfreq}
\end{eqnarray}
where the g-value is $g_+$ for a positron and $g_-$ for an electron. A one-quantum axial excitation within the magnetic bottle gradient shifts the cyclotron frequency by the same $\delta_c$. A thermal distribution over $\bar{n}_z$ axial states (Eq.~(\ref{eq:Averagenz})) thus makes the cyclotron frequency fluctuate over a spread of frequencies that is of order $\bar{n}_z \delta_c$.
Two relativistic shifts must be mentioned, both arising from the ``relativistic mass increase." The largest is the increase of the effective mass due to the energy of cyclotron motion \cite{Review},
\begin{equation}
\delta_{r} = -\frac{\hbar\omega_c}{mc^2} \omega_c \approx - 2\pi\times (180~\rm{Hz}).
\end{equation}
It is only a $1$ part in $10^9$ shift of the cyclotron frequency per cyclotron quantum, but it is a large shift compared to the experimental precision that can being attained. The cyclotron transition frequency between quantum number $n_c$ and $n_c+1$ is shifted by $(n_c+\tfrac{1}{2})\,\delta_r$. The cyclotron frequency between the ground and first excited cyclotron states with spin down shift by half of $\delta_r$. The shift is thus extremely important in that a cyclotron drive that excites the first spin-down excited state, will not excite a cyclotron excitation of the spin-up ground state.
However, for the purposes of this calculation it can simply be absorbed into $\omega_c$.
The second relativistic shift,
\begin{equation}
\delta_{cr} = -\frac{\hbar\omega_c}{2mc^2} \omega_z \approx - 2\pi\times (0.12~\rm{Hz}), \end{equation}
is about 1000 times smaller. It
comes from the increase of the effective mass due to the zero-point energy of the axial oscillation. This coupling has much the same effect in coupling the motions to allow QND detection as does a magnetic bottle \cite{Review}. It also produces a corresponding backaction. This relativistic coupling is neglected here because it is 25 times smaller than the coupling caused by the magnetic bottle gradient considered above.
A spin flip shifts the angular axial frequency by $\delta_s = (g/2) \,\delta_c$. This is nearly the same size as the corresponding cyclotron frequency shift because $\frac{g}{2}$ differs from 1 by only a part in 1000, and experiments are not able to resolve these two shifts from each other. The frequency difference $\omega_a=\omega_s-\omega_c$ is measured rather than $\omega_s$ \cite{HarvardMagneticMoment2008}, and the thousand times smaller shift, $\delta_a = \delta_s-\delta_c$, is thus also important.
Table~\ref{table:Frequencies} gives the typical trapped electron frequencies, damping rates, and quantum numbers used in this calculation. The spin and cyclotron frequencies are for an electron in a $B=5.3$ T magnetic field, and $\gamma_c$ is the rate at which the first excited cyclotron state radiates spontaneous emission to return to its ground state. This radiation rate is substantially inhibited by a surrounding cylindrical trap cavity \cite{CylindricalPenningTrapDemonstrated, InhibitionLetter,HarvardMagneticMoment2011}. The spin-up cyclotron ground state radiates with a time constant so long that we treat it as stable.
The axial frequency depends upon the trap size and the applied trapping potential \cite{CylindricalPenningTrap,Gabrielse84h}. Its damping rate $\gamma_z$ depends upon the quality factor and inductive reactance of the damping and detection circuit to which it is coupled \cite{ElectronCalorimeter}. The maximal damping rate in Tab.~\ref{table:Frequencies} applies during particle detection. For this calculation, we assume that this rate is electronically reduced by a factor of 100 during the time that spin and cyclotron transitions are driven, a number that has been experimentally demonstrated \cite{FanRFSwitch2020}. The average quantum number is for thermal equilibrium with a circuit kept at 0.1 K, the ambient temperature that has been maintained for measurements using a dilution refrigerator \cite{HarvardMagneticMoment2008}.
The magnetron orbit of a trapped particle is important experimentally but not for this calculation. It is a motion at a much lower frequency. The average quantum number in the table pertains for the sideband cooling limit \cite{Review}, and its radiation damping rate is completely negligible. The broadening due to magnetron motion is smaller than that due to axial motion by a factor of $\omega_m/\omega_z\approx1/1000$, and we drop the magnetron motion term to simplify the calculation. If necessary, the Hamiltonian and master equation in Sec.~\ref{sec:HamiltonianForTheQuantumSystem} and Sec.~\ref{sec:OneDriveExcitations} can be naturally generalized to include it.
\begin{table}[htbp!]
\begin{tabular}{c|l|l|l}
& frequency & \begin{tabular}{@{}c@{}}damping\\time\end{tabular} & \begin{tabular}{@{}c@{}}quantum\\number\end{tabular} \\
\hline\hline
spin & $\omega_s/2\pi\approx148.5$ GHz& $\gamma_s^{-1}\approx 10^8$ s & $m_s=\pm\tfrac{1}{2}$ \\
cyclotron & $\omega_c/2\pi\approx148.3$ GHz& $\gamma_c^{-1}\approx 5$ s & $\bar{n}_c=0$ \\
axial & $\omega_z/2\pi\approx 200$ MHz& $\gamma_z^{-1}\approx 0.2$ s & $\bar{n}_z=10$ \\
magnetron & $\omega_m/2\pi\approx 133$ kHz& $\gamma_m^{-1}\approx10^{17}$ s & $\bar{n}_m=10$ \\
anomaly & $\omega_a/2\pi\approx 170$ MHz& --- & ---
\end{tabular}
\caption{The frequencies, damping rates, and quantum numbers used for this calculation are typical for an electron in a Penning trap \cite{HarvardMagneticMoment2008}.}
\label{table:Frequencies}
\end{table}
Tables ~\ref{table:Frequencies} and \ref{table:FrequenciesCompared} list the parameters used for this calculation. They are mostly what has been realized experimentally. Table
\ref{table:Frequencies} gives frequencies, damping times and quantum number for the spin, cyclotron, axial an magnetron motion of an electron or positron in a Penning trap. Table~ \ref{table:FrequenciesCompared} compares the important frequency offsets and corresponding time constants.
\begin{table}[htbp!]
\begin{tabular}{c |c |c}
ang.\ frequency or rate & frequency (Hz) & time constant (s) \\
\hline
\hline
$\delta_a$ & $0.003$ & $60$\\
$\gamma_z$ & $0.003$ & $60$\\
$\bar{n}_z\delta_a$ & $0.03$ & $6$\\
$\gamma_c$ & $0.03$ & 6\\
$\bar{n}_z\gamma_z$ & $0.03$ & $6$\\
$\delta_c$ & $3$ & $0.06$\\
$\bar{n}_z\delta_c$ & $30$ & $0.006$
\label{tab:twodrivefrequencyscale}
\end{tabular}
\caption{Hierarchy of angular frequencies and rates that are in reach for a new generations of measurements. The numerical values are frequencies in Hz and times in seconds, with $\delta_a/2\pi = 0.003 $ Hz and $\delta_a^{-1} = 60$ s, for example.}
\label{table:FrequenciesCompared}
\end{table}
One motivation for this calculation is evaluating the possibilities that open if a greatly reduced axial damping rate pertains while cyclotron and anomaly transitions are driven. The rate can be electronically switched \cite{FanRFSwitch2020} to the low value in the table just before drives are applied, to make one-quantum anomaly and cyclotron transitions with an electron largely uncoupled from the bath. After the drives are turned off, the damping rate can be electronically switched to a much larger values, as needed to detect the particle state and to damp the axial motion.
\section{Hamiltonian}
\label{sec:HamiltonianForTheQuantumSystem}
\newcommand{Schr\"{o}dinger~}{Schr\"{o}dinger~}
The basic Hamiltonian for the quantum cyclotron,
\begin{equation}
H_0 =\hbar\omega_s\left(a_s^{\dagger}a_s-\tfrac{1}{2}\right)
+ \hbar\omega_c
\left(a_c^{\dagger}a_c+\tfrac{1}{2}\right) +\hbar \omega_z \left(a_z^{\dagger}a_z+\tfrac{1}{2}\right),
\label{eq:H0}
\end{equation}
is the sum of independent spin, cyclotron and axial terms.
The raising and lowering operators for the spin ($a_s^\dagger$ and $a_s$), cyclotron ($a_c^\dagger$ and $a_c$) and axial ($a_z^\dagger$ and $a_z$) motions are introduced in Ref.~\cite{Review}, along with relationships to the position and momentum operators. The eigenstates for $H_0$ are direct products of independent spin, cyclotron, and axial eigenstates $\left|m_s,n_c,n_z\right\rangle$, with
\begin{equation}
E_0(m_s,n_c,n_z)
=\hbar\omega_sm_s+\hbar\omega_c\left(n_c+\tfrac{1}{2}\right)+\hbar\omega_z\left(n_z+\tfrac{1}{2}\right)
\label{eq:E0}
\end{equation}
as the resulting energy eigenvalues, with $m_s = \pm 1/2$, $n_c=0, 1, ...$ and $n_z=0,1,...\,$.
The magnetron motion of a particle in a Penning trap is neglected because it introduces no significant complications, and because it can be cooled to a small radius that does not change during a measurement.
The addition of a magnetic bottle gradient adds a coupling term to make the Hamiltonian, $H = H_0 + V$, with
\begin{equation}
V = \tfrac{\hbar}{2}\left[\delta_s \left(a_s^{\dagger}a_s-\tfrac{1}{2}\right)+\delta_c \left(a_c^{\dagger}a_c+\tfrac{1}{2}\right) \right] (a_z^\dagger + a_z)^2, \label{eq:V}
\end{equation}
when contributions smaller by order $\omega_z/\omega_c$ are neglected.
This is a QND coupling because $[H_0,V]=0$. The result is that the energy eigenstates of $H=H_0+V$ are the same uncoupled states $\left|n_c, m_s, n_z\right\rangle$ that are the energy eigenstates of $H_0$. The magnetic bottle shifts the energy eigenvalues to
\begin{equation}
\begin{split}
E(m_s,&n_c,n_z) = E_0(m_s,n_c,n_z) \\
&+\hbar\delta_c\left(n_c+\tfrac{1}{2}\right)\left(n_z+\tfrac{1}{2}\right) + \hbar\delta_sm_s\left(n_z+\tfrac{1}{2}\right).
\label{eq:energystate}
\end{split}
\end{equation}
That this coupling makes it possible to detect that quantum spin and cyclotron states can be seen by rewriting the energy eigenvalues as
\begin{equation}
E(m_s,n_c,n_z)
=\hbar\omega_sm_s+\hbar\omega_c\left(n_c+\tfrac{1}{2}\right)+\hbar\widetilde{\omega}_z\left(n_z+\tfrac{1}{2}\right).
\label{eq:0}
\end{equation}
Monitoring the effective axial oscillation frequency
\begin{equation}
\widetilde{\omega}_z=\omega_z+m_s \delta_s+(n_c+\tfrac{1}{2})\delta_c,
\label{eq:axialfrequencyshift}
\end{equation}
thus reveals the spin and cyclotron states via their quantum numbers. A feature of the QND detection is that the axial detection backaction upon these quantum states is completely evaded. Repeated measurements, made to see if something else is changing these states, do not in themselves change the quantum state.
Critical to this work is that the QND coupling $V$ that completely evades detection backaction in the determination of the quantum spin and cyclotron states, does not do so for a measurement of either $\omega_s$ or $\omega_c$. This can be seen by writing the energy eigenvalues in the alternate form,
\begin{equation}
E(m_s,n_c,n_z) = \hbar \widetilde{\omega}_s m_s+
\hbar \widetilde{\omega}_c (n_c+\tfrac{1}{2}) + \hbar \omega_z (n_z+\tfrac{1}{2}).
\end{equation}
Despite the QND coupling, the effective spin, cyclotron and anomaly frequencies all have shifts that go as the axial quantum number
\begin{subequations}\begin{align}
\widetilde{\omega}_s &= \omega_s + \delta_s (n_z+\tfrac{1}{2}),\\
\widetilde{\omega}_c &= \omega_c + \delta_c (n_z+\tfrac{1}{2}),\\
\widetilde{\omega}_a &= \omega_a + \delta_a (n_z+\tfrac{1}{2}).
\label{eq:CyclotronShift}
\end{align}
\label{eq:Shifts}%
\end{subequations}
These detection backaction shifts cannot be completely evaded because a shift due to axial zero point motion remains even if the axial detection motion would be cooled to its $n_z=0$ ground state. Because the shifts in this limit are orders of magnitude smaller than what has been attained, we focus upon how these zero-point limits can be attained. We call this ``circumventing'' detection backaction because of the possibility to achieve these limits while axial detection states well above $n_z=0$ are populated \cite{Fan2020BackActionPRL}.
Electron and positron magnetic moment measurements require the determination of the cyclotron and the anomaly frequencies, $\omega_c$ and $\omega_a$. These frequencies can be determined observing the rate of quantum jumps between the lowest cyclotron and spin states as a function of the frequency of external driving forces introduced to make these transitions.
Because the axial detection motion is coupled to a thermal reservoir there is a thermal distribution of axial states. This spreads out the range of spin, cyclotron and anomaly frequencies at which a spin, cyclotron or anomaly drive causes one-quantum transitions. The detection backaction thus significantly broadens the observed spin, cyclotron and anomaly resonance line shapes from which the needed frequencies must be detected.
Switching from the Schr\"{o}dinger~ picture to the interaction picture transforms away the
well-understood spin, cyclotron and axial motions in the absence of a magnetic bottle. Terms that go as $a_za_z$ and $a_z^\dagger a_z^\dagger$ oscillate rapidly and hence average to zero in the interaction picture. The resulting interaction Hamiltonian $\widetilde{V} = e^{i H_0 t/\hbar} V e^{-i H_0 t/\hbar}$ is
\begin{equation}
\begin{split}
\widetilde{V}=&\left[
\hbar\delta_s\left(a_s^{\dagger}a_s-\tfrac{1}{2}\right)
+\hbar\delta_c\left(a_c^{\dagger}a_c+\tfrac{1}{2}\right)
\right] \left(a_z^{\dagger}a_z+\tfrac{1}{2}\right).
\end{split}
\end{equation}
We continue using the time-independent raising and lowering operators from the Schr\"{o}dinger~ picture (rather than transforming these to the interaction picture). The interaction picture Hamiltonian has an energy scale set by the tiny bottle shifts, $\delta_c$ and $\delta_s$, rather than by the much larger frequencies $\omega_c$, $\omega_s$ and $\omega_z$.
\begin{figure}
\centering
\includegraphics[width=\the\columnwidth]{Figures/quantumstatesAll_ms.pdf}
\caption{Quantum states of a particle in a Penning Trap. Cyclotron transition and anomaly transition of $n_z=0$ state are also shown with dotted lines. Each cyclotron state has infinite number of axial substates.} \label{fig:quantumstates}
\end{figure}
Figure \ref{fig:quantumstates} represents the lowest of these quantum energy levels, with spin down states ($m_s=-1/2$) on the left and spin up states ($m_s=1/2$) on the right. The lowest of the infinite ladder of cyclotron states are shown ($n_c = 0, 1$), as are the lowest three of the infinite ladder of axial states ($n_z=0, 1, 2$). For the driving forces we will consider, the electron will essentially occupy only the three cyclotron and spin state combinations
\begin{equation}
\begin{split}
\left|1,n_z\right\rangle&\equiv\left|n_c=0, m_s=-\tfrac{1}{2},n_z\right\rangle,\\ \left|2,n_z\right\rangle&\equiv\left|n_c=1, m_s=-\tfrac{1}{2},n_z\right\rangle,\\ \left|3,n_z\right\rangle&\equiv\left|n_c=0, m_s=+\tfrac{1}{2},n_z\right\rangle,
\end{split}
\label{eq:ThreeStates}
\end{equation}
with $n_z = 0, 1, ...$.
These are the basis of time-independent states used for this calculation. The basis would shrink to only three states if the axial motion would be cooled to its quantum ground state.
The electromagnetic drives that oscillate at angular frequencies $\omega_s + \epsilon_s$, $\omega_c + \epsilon_c$ and $\omega_a + \epsilon_a$ to drive spin, cyclotron and anomaly transitions are described by the Hamiltonians
\begin{eqnarray}
V_s(t) &=& \tfrac{1}{2} \hbar \Omega_s \left[a_s^\dagger e^{-i (\omega_s + \epsilon_s) t} + a_s e^{i (\omega_s + \epsilon_s) t} \right]
\label{eq:Drives}\\
V_c(t) &=& \tfrac{1}{2} \hbar \Omega_c \left[a_c^\dagger e^{-i (\omega_c + \epsilon_c) t} + a_c e^{i (\omega_c + \epsilon_c) t} \right]
\label{eq:Drivec}\\
V_a(t) &=& \tfrac{1}{2} \hbar \Omega_a \left[a_a^\dagger e^{-i (\omega_a + \epsilon_a) t} + a_a e^{i (\omega_a + \epsilon_a) t} \right].
\label{eq:Drivea}
\end{eqnarray}
The positive Rabi frequencies $\Omega_s$, $\Omega_c$ and $\Omega_a$ quantify the drive strengths, and $\epsilon_s$, $\epsilon_c$ and $\epsilon_a$ are detunings of the drives from resonance.
In the interaction picture the Hamiltonian drive terms are
\begin{eqnarray}
\widetilde{V}_s(t)&=&\tfrac{1}{2}\hbar\Omega_s \left[ a^\dagger_s e^{-i\epsilon_s t} + a_s e^{i\epsilon_s t} \right]\\
\widetilde{V}_c(t)&=&\tfrac{1}{2}\hbar\Omega_c \left[ a^\dagger_c e^{-i\epsilon_c t} + a_c e^{i\epsilon_c t} \right]\\
\widetilde{V}_a(t)&=&\tfrac{1}{2}\hbar\Omega_a \left[ a^\dagger_a e^{-i\epsilon_a t} + a_a e^{i\epsilon_a t} \right].
\label{eq:driveeq}
\end{eqnarray}
An anomaly transition is a simultaneous cyclotron and spin transition. The raising operator for an anomaly transition from $\left|2,n_z\right\rangle$ to $\left|3,n_z\right\rangle$, for example, requires
$a_a^\dagger=a_s^\dagger a_c$, a lowering of the cyclotron state followed by a raising of the spin state. A transition from the spin down ground state to the spin up ground state is accomplished by $a_a^\dagger a_c^\dagger$.
The axial and cyclotron motions are both coupled to a thermal bath, with damping rates of $\gamma_z$ and $\gamma_c$, respectively.
An ambient bath temperature of 0.1 K is assumed because it has been demonstrated in experiments \cite{HarvardMagneticMoment2011}. The energy for a one-quantum axial excitation, $\hbar\omega_z/k_B=0.01$ K in temperature units, is instead much smaller than 0.1 K. The axial state is thus a Boltzmann distribution with an average quantum number
\begin{equation}
\bar{n}_z=\left[\exp\left({\frac{\hbar\omega_z}{k_BT}}\right)-1\right]^{-1} \approx \frac{k_B T}{\hbar \omega_z}\approx 10.
\label{eq:Averagenz}
\end{equation}
It may be possible to cool this motion further using cavity sideband cooling \cite{atomsNewMeasurement2019}, but this is not assumed here. A cyclotron excitation requires an energy of $\hbar\omega_c/k_B=7.1$ K that is much larger than the $0.1$ K bath temperature. The result is that
\begin{equation}
\bar{n}_c=\left[\exp\left({\frac{\hbar\omega_c}{k_BT}}\right)-1\right]^{-1} = 1.2\times10^{-32} \approx 0.
\end{equation}
The cyclotron motion essentially remains in its $n_c=0$ ground state \cite{QuantumCyclotron} unless an excitation drive is applied.
For an electron or positron coupled to a thermal bath, a density operator must be used. The density operator in the Schr\"{o}dinger~ picture, $\rho$, and the interaction picture, $\tilde{\rho}$ are related by
\begin{equation}
\widetilde{\rho} = e^{i H_0 t /\hbar} \rho e^{-i H_0 t/\hbar}.
\end{equation}
Both $\rho$ and $\widetilde{\rho}$ can be expanded in the infinite base of time-independent states in Eq.~(\ref{eq:ThreeStates}). The diagonal elements are the probabilities to be in each basis state. These are invariant under a change between the Schr\"{o}dinger~ and interaction pictures. Also invariant are the traces,
\begin{eqnarray}
P_l = \sum_{n_z=0}^{\infty} \langle l,n_z |\rho \left|l,n_z\right\rangle = \sum_{n_z=0}^{\infty} \langle l,n_z |\tilde{\rho} \left|l,n_z\right\rangle,
\label{eq:Pl}
\end{eqnarray}
that are the total probabilities to be in each of the 3 spin and cyclotron states. Here, $l$ denotes the label 1, 2, or 3 introduced in Fig.~\ref{fig:quantumstates} and Eq.~(\ref{eq:ThreeStates}).
The Schr\"{o}dinger~ picture density operator, $\rho$, evolves in time as described by a Lindblad master equation \cite{Lindblad1,Lindblad2,jacobs_2014},
\begin{equation}
\begin{split}
\frac{d\rho}{dt}&=-\frac{i}{\hbar}\left[{H_0} + V + {V}_s + {V}_c + {V}_a,\rho\right] \\
&-\frac{\gamma_c}{2}\left(a_c^\dagger a_c{\rho}-2a_c {\rho} a_c^\dagger+{\rho} a_c^\dagger a_c\right)\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_za_z^\dagger{\rho}-2a_z^\dagger{\rho} a_z+{\rho} a_za_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z{\rho}-2a_z {\rho} a_z^\dagger+{\rho} a_z^\dagger a_z\right).
\end{split}
\label{eq:MasterEquationSchroedinger}
\end{equation}
The coherent time evolution is described by the commutator term. The incoherent spontaneous emission from the cyclotron motion (from the first excited cyclotron state to its ground state) is described by the nonlinear terms in line two. (As noted earlier, the heating of the cyclotron motion by the thermal black-body radiation for low temperature surroundings can be neglected.) The coupling of the axial motion and the thermal bath is described by the last two lines. The bath temperatures comes in via the average axial quantum number $\bar{n}_z$ of Eq.~(\ref{eq:Averagenz}).
The interaction picture density operator, $\tilde{\rho}$, evolves as
\begin{equation}
\begin{split}
\frac{d\tilde{\rho}}{dt}&=-\frac{i}{\hbar}\left[\widetilde{V}+ \widetilde{V}_s+ \widetilde{V}_c + \widetilde{V}_a,\tilde\rho\right] \\
&-\frac{\gamma_c}{2}\left(a_c^\dagger a_c\tilde{\rho}-2a_c \tilde{\rho} a_c^\dagger+\tilde{\rho} a_c^\dagger a_c\right)\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_za_z^\dagger\tilde{\rho}-2a_z^\dagger\tilde{\rho} a_z+\tilde{\rho} a_za_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z\tilde{\rho}-2a_z \tilde{\rho} a_z^\dagger+\tilde{\rho} a_z^\dagger a_z\right).
\end{split}
\label{eq:MasterEquationI}
\end{equation}
As for the Hamiltonian, we use the time-independent, raising and lowering operators from the Schr\"{o}dinger~ picture. The damping terms transform to have the same form in both pictures. Explicit calculation are done using the interaction picture because it is simpler. $H_0$ is removed,
and $\widetilde{V}_s+\widetilde{V}_c + \widetilde{V}_a$ varies much less rapidly in time than does $V_s+V_c + V_a$.
\section{Driven Cyclotron Excitations}
\label{sec:OneDriveExcitations}
\newcommand{\ArgBoth}{\left(
\tfrac{\epsilon}{\delta},
\tfrac{\gamma_c}{\delta},
\tfrac{\gamma_z}{\delta},
\bar{n}_z
\right)}
\newcommand{\ArgC}{\left(
\tfrac{\epsilon_c}{\delta_c},
\tfrac{\gamma_c}{\delta_c},
\tfrac{\gamma_z}{\delta_c},
\bar{n}_z
\right)}
\newcommand{\ArgA}{\left(
\tfrac{\epsilon_a}{\delta_a},
\tfrac{\gamma_c}{\delta_a},
\tfrac{\gamma_z}{\delta_a},
\bar{n}_z
\right)}
\newcommand{\Arg}{\left(
\tfrac{\epsilon}{\delta},
\tfrac{\gamma_c}{\delta},
\tfrac{\gamma_z}{\delta},
\bar{n}_z}
\subsection{Cyclotron Master Equation}
\label{sec:CyclotronLineshape}
\newcommand{\ket}[2]{\left|#1,#2\right\rangle}
\newcommand{\bra}[2]{\left\langle#1,#2\right|}
\newcommand{\me}[4]{\left\langle#1,#2\right|\tilde{\rho}\left|#3,#4\right\rangle}
A weak cyclotron drive, $V_c$, excites cyclotron states $\left|2,n_z\right\rangle$ from an initial state that is a thermal distribution of spin down, cyclotron ground states, $\left|1,n_z\right\rangle$. The drive provides no mechanism to flip the spin, so the states $\left|3,n_z\right\rangle$ are not populated. For a weak drive, $\Omega_c \ll \gamma_c$, the probability of a cyclotron excitation is very small. We neglect the possibility of a second cyclotron excitation that follows the first, from the excited state $\left|2,n_z \right\rangle$ to a higher state, because this is much smaller still. The Hermitian density operator for cyclotron excitation,
\begin{eqnarray}
\tilde{\rho} &= \tilde{\rho}_{11} + \tilde{\rho}_{12} + \tilde{\rho}_{21} +\tilde{\rho}_{22}
= \begin{pmatrix} \tilde{\rho}_{11} & \tilde{\rho}_{12} \\ \tilde{\rho}_{21} & \tilde{\rho}_{22} \end{pmatrix}
\end{eqnarray}
is the sum of four operators, each defined by \begin{equation}
\tilde{\rho}_{jk} \equiv \sum_{n_z,n^\prime_z} \ket{j}{n_z} \me{j}{n_z}{k}{n^\prime_z} \bra{k}{n^\prime_z}.
\label{eq:rhojk}
\end{equation}
Since $\tilde{\rho}$ is Hermitian, $\tilde{\rho}_{21}=\tilde{\rho}_{12}^\dagger.$
The initial density operator at time $t=0$ is diagonal with respect to the axial quantum numbers,
\begin{equation}
\begin{split}
&\left\langle 1,n_z|\tilde{\rho}|1,n_z\right\rangle = p_{n_z}(T) =\\
&=\left[1-\exp\left({-\frac{\hbar\left(\omega_z-\frac{1}{2}\delta_a\right)}{k_BT}}\right)\right]\exp\left({-\frac{n_z\hbar\left(\omega_z-\frac{1}{2}\delta_a\right)}{k_BT}}\right)\\
&\approx \left[1-\exp\left({-\frac{\hbar\omega_z}{k_BT}}\right)\right]\exp\left({-\frac{n_z\hbar\omega_z}{k_BT}}\right)
\end{split}
\label{eq:11distribution}
\end{equation}
with Boltzmann factors as its nonzero elements. The approximation is nearly exact because $\delta_a \ll \omega_z$. In the weak drive limit, we would expect this distribution of initial states to remain essentially unchanged.
The probability $P_2$ from Eq.~(\ref{eq:Pl}), that the system is excited by one quantum from its spin-down, cyclotron ground state,
\begin{equation}
P_2 = \sum_{n_z} \me{2}{n_z}{2}{n_z} = \textrm{Tr}\left[\tilde{\rho}_{22}\right].
\label{eq:Pc}
\end{equation}
is the sum of the probabilities for excitation to any of the states $\ket{2}{n_z}$.
Either the Schrodinger or interaction picture density operator can be used since their diagonal elements are identical.
Determining $\tilde{\rho}_{22}$ requires solving the master equation
\begin{equation}
\begin{split}
\frac{d}{dt} &\begin{pmatrix} \tilde{\rho}_{11} & \tilde{\rho}_{12} \\ \tilde{\rho}_{21} & \tilde{\rho}_{22} \end{pmatrix} = -{i}\left(a_z^{\dagger}a_z+\tfrac{1}{2}\right)
\begin{pmatrix} 0 & -\delta_c\tilde{\rho}_{12} \\ \delta_c\tilde{\rho}_{21} & 0 \end{pmatrix}\\
&-i\frac{\Omega_c}{2}
\begin{pmatrix}
i2\mathrm{Im}[\tilde{\rho}_{21}e^{i\epsilon_c t}]
& e^{i\epsilon_c t}\left(\tilde{\rho}_{22}-\tilde{\rho}_{11}\right) \\
e^{-i\epsilon_c t}\left(\tilde{\rho}_{11}-\tilde{\rho}_{22}\right)
& i2\mathrm{Im}[\tilde{\rho}_{12}e^{-i\epsilon_c t}]
\end{pmatrix}\\
&-\frac{\gamma_c}{2}
\begin{pmatrix} -2\tilde{\rho}_{22} & \tilde{\rho}_{12} \\ \tilde{\rho}_{21} & 2\tilde{\rho}_{22} \end{pmatrix}\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_za_z^\dagger\tilde{\rho}-2a_z^\dagger\tilde{\rho} a_z+\tilde{\rho} a_za_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z\tilde{\rho}-2a_z \tilde{\rho} a_z^\dagger+\tilde{\rho} a_z^\dagger a_z\right).
\end{split}
\label{eq:CyclotronMasterEquation}
\end{equation}
The first line describes time evolution of the density matrix by $\widetilde{V}$. The diagonal terms are 0 because $\ket{1}{n_z}$ and $\ket{2}{n_z}$ are eigenstates of $\widetilde{V}$ for the QND measurement. The non-diagonal terms represents the differing bottle shift for $\ket{1}{n_z}$ and $\ket{2}{n_z}$. The second line describes the electromagnetic cyclotron drive. The third term describes synchrotron radiation from the excited cyclotron state at a rate $\gamma_c$. The fourth and fifth terms arise from the axial damping and reservoir excitation. They do not change $P_c$ because they do not change either the cyclotron or spin state.
The axial damping terms in the master equation (Eq.~(\ref{eq:CyclotronMasterEquation})) generate no coherence between axial states. Only axially diagonal terms, $\langle i,n_z|\tilde{\rho}|j,n_z\rangle$ are nonzero, where $i$ and $j$ are the labels for the states we consider (Eq.~(\ref{eq:ThreeStates})). The transformation
\begin{equation}
p_{ij;n_z}(t)=\langle i,n_z|\tilde{\rho}(t)|j,n_z\rangle \, e^{i(i-j) \epsilon_c t}.
\label{eq:pjkn}
\end{equation}
makes these coefficients carry all the time dependence. Notice that the probability to be in each of the cyclotron and spin states of Eq.~(\ref{eq:Pl}) is also the trace
\begin{eqnarray}
P_l = \sum_{n_z=0}^{\infty} \langle l,n_z | p \left|l,n_z\right\rangle,
\label{eq:Plp}
\end{eqnarray}
where $p$ has components $p_{jk}$, and $l$ denotes 1, 2, or 3 in Fig.~\ref{fig:quantumstates} and Eq.~(\ref{eq:ThreeStates}). This is because
the diagonal matrix elements with $i=j$ are equal to the those for the density operator in the Schrodinger picture and the interaction picture. For the cyclotron excitation being considered in this section, $P_3=0$ because the states $|3,n_z\rangle$ are never populated.
The differential equations after the transformation are
\begin{subequations}
\begin{align}
\frac{d}{dt}&p_{11;n_z}(t)\nonumber\\
&=\left[-\gamma_z\left(2\bar{n}_z+1\right)n_z-\gamma_z\bar{n}_z\right]p_{11;n_z}(t)\nonumber\\
&+\gamma_cp_{22;n_z}(t)-\Omega_c\textrm{Im}\left[p_{12;n_z}\right]\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{11;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{11;n_z+1}(t)\label{eq:masterdifferentialA}\\
\frac{d}{dt}&p_{12;n_z}(t)\nonumber\\
&=\big[i\left(-\epsilon_c+\delta_c\left(n_z+\tfrac{1}{2}\right)\right)\nonumber\\
&-\tfrac{1}{2}\gamma_c-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\big]p_{12;n_z}(t)\nonumber\\
&-i\frac{\Omega_c}{2}(p_{22;n_z}(t)-p_{11;n_z}(t))\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{12;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{12;n_z+1}(t)\label{eq:masterdifferentialB}
\\
\frac{d}{dt}&p_{22;n_z}(t)\nonumber\\
&=\left[-\gamma_c-\gamma_z\left(2\bar{n}_z+1\right)n_z-\gamma_z\bar{n}_z\right]p_{22;n_z}(t)\nonumber\\
&+\Omega_c\textrm{Im}\left[p_{12;n_z}\right]\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{22;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{22;n_z+1}(t).
\label{eq:masterdifferentialC}
\end{align}
\end{subequations}
These equations are to be solved for the initial conditions $p_{12;n_z}(0)=p_{22;n_z}(0)=0$ and $p_{11,n_z}(0) = p_{n_z}(T)$. Because the states $|3,n_z\rangle$ are never populated, the $p_{j,k} = 0$ when either $j=3$ or $k=3$.
Equations~(\ref{eq:masterdifferentialA}-\ref{eq:masterdifferentialC}) is a matrix equation for the vectors $\vec{p}_{ij}(t)$ with components $p_{ij;n_z}$,
\begin{subequations}
\begin{align}
\frac{d}{dt}\vec{p}_{11}(t)
&=\mathbf{R}(0,0,0)\, \vec{p}_{11}(t)-\Omega_c\textrm{Im}\left[\vec{p}_{12}(t)\right]+\gamma_c\vec{p}_{22}(t)\\
\frac{d}{dt}\vec{p}_{12}(t)
&=\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)\, \vec{p}_{12}(t)-i\frac{\Omega_c}{2}(\vec{p}_{22}(t)-\vec{p}_{11}(t))\\
\frac{d}{dt}\vec{p}_{22}(t)
&=\mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22}(t)+\Omega_c\textrm{Im}\left[\vec{p}_{12}(t)\right].
\end{align}
\label{eq:MatrixEquation}%
\end{subequations}
The states $|3,n_z\rangle$ are never populated so all
The non-zero elements of the time-independent matrix are
\begin{subequations}
\begin{align}
\mathbf{R}(\epsilon,\delta,\gamma_c)_{n_z,n_z-1} = & \gamma_z\bar{n}_z n_z\label{eq:Ra}\\
\mathbf{R}(\epsilon,\delta,\gamma_c)_{n_z,n_z} = &
i\left[-\epsilon+(n_z+\tfrac{1}{2})\delta \right]-\tfrac{1}{2}\gamma_c\nonumber\\
&-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\label{eq:Rb}\\
\mathbf{R}(\epsilon,\delta,\gamma_c)_{n_z,n_z+1} = & \gamma_z (\bar{n}_z+1) (n_z+1).\label{eq:Rc}
\end{align}
\label{eq:R}%
\end{subequations}
\noindent The initial conditions for the vector differential equations above are $\vec{p}_{11}(0)=\vec{p}(T)$ and $\vec{p}_{12}(0)=\vec{p}_{22}(0)=0.$
\subsection{Steady-State Cyclotron Line Shape}
After transients have died out in a time
\begin{equation}
t\gg\gamma_c^{-1},
\end{equation}
a weak drive with $\Omega_c\ll \gamma_c$ produces a steady state in which driven cyclotron excitation balances the incoherent spontaneous emission of synchrotron radiation. Clearly,
\begin{eqnarray}
&P_1 = \textrm{Tr}[p_{11}] = &\sum_{n_z} {p}_{11;n_z}(t)\approx \sum_{n_z}p_{n_z}(T) = 1 \\
&P_2 = \textrm{Tr}[p_{22}] = &\sum_{n_z} p_{22;{n_z}}(t) \ll 1
\label{eq:p11}
\end{eqnarray}
and terms involving $\vec{p}_{22}$ are negligibly small compared to those involving $\vec{p}_{11}$.
The resulting steady state, from Eq.~(\ref{eq:MatrixEquation}) with the time derivatives set to zero and the mentioned approximation is described by
\begin{eqnarray}
&\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)\, \vec{p}_{12}+i\frac{\Omega_c}{2}\vec{p}(T)=0\label{eq:SteadyStatep12}
\\
&\mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22}+\Omega_c\textrm{Im}\left[\vec{p}_{12}\right]=0.
\label{eq:SteadyStatep22}
\end{eqnarray}
The latter can be simplified because
\begin{equation}
\sum_{{n_z}=0}^{\infty}\left( \mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22} \right)_{n_z}
=-\gamma_c\textrm{Tr}\left[{p}_{22}\right],
\end{equation}
because axial damping does not change the total population in states $|2,{n_z}\rangle$, and because $\mathbf{R}(0,0,2\gamma_c)$ has a simple structure.
The result is a steady
state probability for weak drive cyclotron excitation, $P_2$ as defined in Eq.~(\ref{eq:Pl}), given by
\begin{eqnarray}
& ~~~~~~~P_2& =P(\Omega_c,\epsilon_c,\delta_c)\\
&P(\Omega,\epsilon,\delta)\,& \equiv -\frac{\Omega^2}{2\gamma_c} \textrm{Im}\left[
\sum_{{n_z}=0}^{\infty}\left(
i \mathbf{R}(\epsilon,\delta,\gamma_c)^{-1}\vec{p}(T)
\right)_{n_z}\right].
\label{eq:P}
\end{eqnarray}
We use arguments without subscripts in $P(\Omega,\epsilon,\delta)$ because this function with other arguments will also describes other steady-state line shapes in what follows.
For the limiting case of a $T=0$ bath, $\bar{n}_z = 0$ and $\vec{p}(T)$ collapses to a single element ${p}_0(T)=1$. Only the reciprocal of $\mathbf{R}(\epsilon,\delta,\gamma_c)_{0,0} =-i\epsilon +i\delta/2- \frac{1}{2}\gamma_c$ contributes to Eq.~(\ref{eq:P}). The steady state line shape for a weak drive, $P(\Omega,\epsilon,\delta)$, thus becomes a Lorentzian, \begin{equation}
P_0(\Omega,\epsilon,\delta)
= \left( \frac{\Omega}{\gamma_c} \right)^2
\frac{ \left(\frac{1}{2}\gamma_c\right)^2 }
{\left(\epsilon-\frac{1}{2}\delta\right)^2 + \left(\frac{1}{2}\gamma_c\right)^2}
\label{eq:Lorentian}
\end{equation}
in the $T=0$ limit.
The full width at half maximum of this line shape is $\gamma_c$. The line shape maximum is shifted to $\epsilon=\delta/2$. That this shift is due to the coupling of zero-point fluctuations of the axial motion can be seen by setting $n_z=0$ for the appropriate frequency in Eq.~(\ref{eq:Shifts}). The steady state probability for being excited with a resonant weak drive is $( \Omega/\gamma_c)^2$. This is a very small fraction for a weak drive with $\Omega \ll \gamma_c$.
The symmetric and narrow Lorentzian cyclotron line shape that would pertain for $T=0$ would be ideal experimentally in some respects. Cavity sideband cooling with a extremely small $\gamma_z$ has been proposed \cite{Review} as way to attain this limit. This calculation, however, is an investigation of what can be done for a temperature of 0.1 K, an achieved temperature that is close to but not at this limit.
\subsection{Classical Brownian Motion Line Shape Limit}
\label{sec:ClassicalLimit}
Before the quantum treatment of the coupled spin, cyclotron and axial system presented above, the calculated line shape that was compared to experiment \cite{BrownLineshape,BrownLineshapePRL,Review} assumed the axial detector motion was a classical harmonic oscillation driven by thermal noise. The Brownian motion line shape that resulted from a weak drive is given in terms of a line shape function,
\begin{equation}
\begin{split}
&\chi (\epsilon,\gamma_z,\bar{n}_z) \\
& = \frac{4}{\pi} \textrm{Re}\Bigg[\frac{\gamma^{\prime}\gamma_z}{\left(\gamma^{\prime}+\gamma_z\right)^2}\sum_{k=0}^{\infty}\frac{\left(\gamma^{\prime}-\gamma_z\right)^{2k}\left(\gamma^{\prime}+\gamma_z\right)^{-2k}}{\left(k+\frac{1}{2}\right)\gamma^\prime+\frac{1}{2}\left(\gamma_c-\gamma_z\right)-i\epsilon}\Bigg],
\end{split}
\label{eq:classicallineshape1}
\end{equation}
in our notation. (The argument $\epsilon$ for $\chi (\epsilon,\gamma_z,\bar{n}_z)$, like for the function $P$ of Eq.~(\ref{eq:P}), will be equal to one of $\epsilon_c$, $\epsilon_a$ or $\epsilon_s$, as will be specified in context.) The bath temperature $T$ enters via
\begin{equation}
\gamma^\prime=\sqrt{\gamma_z^2+4i\gamma_z\bar{n}_z\delta},
\label{eq:classicallineshape2}
\end{equation}
since this bath temperature determines $\bar{n}_z$, while $\delta$ and $\epsilon$ are equal to $\delta_c$ ($\delta_a$) and $\epsilon_c$ ($\epsilon_a$) respectively for cyclotron (anomaly) transition.
The steady state pertains when the transition rate $(\pi/2) \Omega^2 \chi$ (Eq.~(5.19) of \cite{Review}) equals the decay rate $\gamma_c\times P(\Omega,\epsilon,\delta)$. Thus
\begin{equation}
P(\Omega,\epsilon,\delta) = \frac{\pi\Omega^2}{2\gamma_c} \chi (\epsilon,\gamma_z,\bar{n}_z)
\end{equation}
is the classical, Brownian motion line shape.
\subsection{Discussion of the Quantum Cyclotron Line Shape}
The quantum steady-state lineshape
(solid in Fig,~\ref{fig:ClassicalAndQuantumCyc}) is very close to the Brownian motion steady-state line shape (dashed in Fig,~\ref{fig:ClassicalAndQuantumCyc})
when $\bar{n}_z\gamma_z \gg \delta_c$. This was true for the 2008 measurement
for which $\bar{n}_z\gamma_z\approx6\delta_c$
(using parameters from Table.~\ref{table:FrequenciesComparedOld}). For weaker axial damping the two line shapes predict every different results, however.
\begin{figure}[]
\centering
\includegraphics[width=\the\columnwidth]{Figures/ClassicalLimitCyc.pdf}
\caption{Comparison of quantum calculation (solid) and classical calculation (dashed) with the different $\gamma_z's$ for weak drive ($\Omega_c=0.1\gamma_c$) in cyclotron transition. The two calculations agree when $\bar{n}_z\gamma_z>\delta_c$ as illustrated for damping rates $\gamma_z$ for (a-c) that are 1000, 100, 10 times the value in Table. \ref{table:FrequenciesCompared}. The lineshape for the $\gamma_z$ in the table is presented later in Fig.~\ref{fig:CyclotronLineshape}. }
\label{fig:ClassicalAndQuantumCyc}
\end{figure}
The master equation for driven cyclotron excitation can be solved numerically to reveal the time evolution of the probabilities. It can also be integrated directly to examine the effect of power broadening when the weak drive condition ($\Omega_c \ll \gamma_c$) is not satisfied. Both will be illustrated.
Figure~\ref{fig:CyclotronTimeEvolution} illustrates the time evolution for a cyclotron drive that is weak ($\Omega_c = 0.1 \gamma_c$, resonant ($\epsilon_c = \delta_c/2$) for the realistic experimental conditions in Table~\ref{table:FrequenciesCompared}. The probability to be in the $\left|2,n_z\right\rangle$ states increases from zero to reach a steady state for $t\gg 1/\gamma_c$. The cyclotron damping time $1/\gamma_c$ sets the scale for the transients to die out. The much larger probability to be in the initial $\left|1,n_z\right\rangle$ states stays close to unit probability. The black curve in the figure shows the small decrease from unit probability needed to conserve probability.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/CyclotronTimeEvolution_gammaC_2.eps}
\caption{Time evolution in response to a weak and resonant cyclotron drive applied for 10 cyclotron damping times, indicated by vertical gray lines. The probability to be in the $\left|2,n_z\right\rangle$ states (blue) reaches a steady state after transients die out on a time scale give by the cyclotron damping time, $1/\gamma_c$. The probability to be in the $\left|1,n_z\right\rangle$ states is shown in black with unit probability subtracted out.}
\label{fig:CyclotronTimeEvolution}
\end{figure}
The resonance line shape for cyclotron excitation is obtained by numerically integrating the master equation from the stated boundary conditions at time $t=0$ to time $t$ for various values of the drive detuning, $\epsilon_c$, as illustrated in Fig.~\ref{fig:CyclotronLineshape}. The probability to be in the states $\left|2,n_z\right\rangle$ at time $t=10\gamma_c^{-1}$ is shown for a cyclotron drive that is weak ($\Omega_c = 0.1 \gamma_c$), for the realistic experimental conditions in Table~\ref{table:FrequenciesCompared}.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/CyclotronLineshape_2.pdf}
\caption{Quantum cyclotron line shape (solid) with clearly resolved axial quantum states (for a weak cyclotron drive with the larges peak normalized to 1) for the quantum calculation (solid), but not for the classical Brownian motion line shape (dashed). The quantum line shape is a huge improvement on the line shape used for the best measurement (dotted).}
\label{fig:CyclotronLineshape}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/OnePhotonCycPowerBroadening.pdf}
\caption{Cyclotron line shape for the resolved $n_z=0$ axial state and a weak drive (solid curve, $\Omega_c=0.1~\gamma_c$) has a full-width at half maximum of about $3 \gamma_c$. The master equation integrated for 10 cyclotron damping times and the steady-state line shape (solid) coincide. A 10 times stronger drive (dashed, $\Omega_c=\gamma_c)$, only slightly increases the linewidth.}
\label{fig:CyclotronFirstPeak}
\end{figure}
The first narrow peak to the left in the figure shows the probability versus drive frequency for making a cyclotron excitation from the cyclotron ground state while the axial motion is in its ground state with $n_z=0$. The series of narrow cyclotron resonances, the first evidence of axial quantization, are for successively higher values of $n_z$ going right. Resolving these narrow peaks becomes possible only for the small axial damping rate that is now possible experimentally \cite{FanRFSwitch2020}. Each of the peak corresponds to one quantum excitation of cyclotron motion for different $n_z$. This quantum line shape is very different than was observed previously, and it is completely inconsistent with the classical cyclotron line shape, of course. The narrow peaks correspond to resolved quantum states of the axial motion which could not previously be observed. The left peak is for $n_z=0$, the next for $n_z=1$, and so on. There are many peaks because the average axial quantum number is $\bar{n}_z = 10$ for the experimental conditions in Table ~\ref{table:FrequenciesCompared}. The individual peaks are resolved because two conditions are met. First, $\bar{n}_z\gamma_z \ll \delta_c$, i.e.\ the width of each axial state, $\bar{n}_z\gamma_z$, is much smaller than the magnetic bottle shift per axial quantum, $\delta_c$. Second, $\gamma_c \ll \delta_c$, i.e.\ the cyclotron damping width is much smaller than the magnetic bottle shift per axial quantum, $\delta_c$.
The good news from this calculation for potential measurements is how much narrower the $n_z=0$ resonance peak is compared to the cyclotron line shape used for the last electron magnetic moment measurement (dotted in Fig.~\ref{fig:CyclotronLineshape} with experimental parameters in Table~\ref{table:FrequenciesComparedOld}). In fact, the linewidth of the $n_z=0$ peak is only a factor of 3 larger than the cyclotron linewidth, $\gamma_c$ (Fig \ref{fig:CyclotronFirstPeak}). This is consistent with the indication from Eq.~(\ref{eq:masterdifferentialC}) that the linewidth is of order $\gamma_c + 2\bar{n}_z \gamma_z$. Cavity-inhibition of spontaneous emission makes $\gamma_c$ very small \cite{InhibitionLetter}. A low temperatures makes $\bar{n}_z$ small, and the previously mentioned new method makes $\gamma_z$ small \cite{FanRFSwitch2020}.
\begin{table}[htbp!]
\begin{tabular}{c |c |c}
ang.\ frequency or rate & frequency (Hz) & time constant (s) \\
\hline
\hline
$\delta_a$ & $0.004$ & $40$\\
$\gamma_z$ & $1$& $0.16$\\
$\bar{n}_z\delta_a$ & $0.09$ & $1.7$\\
$\gamma_c$ & $0.03
$ & 6\\
$\bar{n}_z\gamma_z$ & $23$& $0.007$\\
$\delta_c$ & $4$ & $0.04$\\
$\bar{n}_z\delta_c$ & $92$ & $0.0017$
\label{tab:twodrivefrequencyscale}
\end{tabular}
\caption{Hierarchy of angular frequencies and rates used on the best completed experiments \cite{HarvardMagneticMoment2008,HarvardMagneticMoment2011}, to be compared with the previous table. The axial temperature was also as low as $\bar{n}_z=23$. The numerical values are frequencies in Hz and times in seconds.}
\label{table:FrequenciesComparedOld}
\end{table}
More good news for possible measurements is that the $n_z=0$ peak is quite symmetric about its center frequency. This is generally a big help in precisely identifying the center frequency of a resonance. The dotted line in Fig.~\ref{fig:CyclotronLineshape} illustrates the big contrast to the highly asymmetric classical line shape used for previous measurements.
The small probability, $3.1 \times 10^{-4}$, that a weak cyclotron drive ($\Omega_c=0.1\gamma_c$) will make an excitation within 10 cyclotron damping times (53 seconds) is of some concern.
However, increasing the cyclotron drive strength to $\Omega_c = \gamma_c$ increases the probability for an excitation to $2.2\times 10^{-2}$ while increasing the full linewidth from 3 to only 3.6 cyclotron decay widths (solid and dashed curves in Fig.~\ref{fig:CyclotronFirstPeak}). This cyclotron linewidth is narrow enough to make possible magnetic moment measurements that are orders of magnitude more accurate than the current limit (assuming the anomaly frequency is determined with a similar accuracy). Because the power broadening is so small, even stronger drives could be used to track a slowly drifting magnetic field \cite{HarvardMagneticMoment2008}.
The offset of the $n_z=0$ resonance from $\epsilon_c=0$ to $\epsilon_c = \delta_c/2$ is due to the zero point motion of the quantum axial oscillator. Measuring this peak and its neighbor would determine this offset more accurately than is needed for dramatically improved magnetic moment measurements, since these two peaks are spaced by twice the offset. This could be an important new option for precisely measuring the offset.
In summary, this quantum calculation demonstrates the exciting possibility to fully resolve the axial quantum structure in the cyclotron line shape. With the achievable reductions in axial damping in Table~\ref{table:FrequenciesCompared}, a cyclotron resonance for a particle in its axial ground state can be fully resolved. This will make it possible to determine the cyclotron frequency (one of two frequencies needed for a magnetic moment measurement) orders of magnitude more precisely. The broad cyclotron linewidth (larger than $\bar{n}_z \delta_c$) that limited past measurements is essentially removed.
\section{Calculating the Anomaly Line Shape}
\label{sec:AnomalyLineshape}
\subsection{Anomaly Master Equation}
An anomaly drive $V_a$ will transfer population from a thermal distribution of stable, spin-up, cyclotron ground states, $|3,n_z\rangle$ to the unstable states, $|2,n_z\rangle$. These states will then decay via the spontaneous emission of synchrotron radiation to the stable spin-down ground states $|1,n_z\rangle$. The attractive feature for measurement is that there is no need to detect an unstable state population before it decays.
The density operator needed to describe anomaly transitions, \begin{equation}
\tilde{\rho} \equiv \begin{pmatrix}
\tilde{\rho}_{22} & \tilde{\rho}_{23} \\
\tilde{\rho}_{32} & \tilde{\rho}_{33}
\end{pmatrix},
\end{equation}
does not need to include the stable lower states, $|1,n_z\rangle$, though it must include decay to these states.
It has the upper and lower energy states in the same relative matrix locations as in the previous section.
What must be calculated is the loss of probability from the initial state during the time that the drive is applied, since this is the probability that a spin-flip transition takes place.
The master equation in the interaction representation is then a lot like Eq.~(\ref{eq:CyclotronMasterEquation}), with the indices $1\rightarrow 2$ and $2\rightarrow 3$,
\begin{equation}
\begin{split}
\frac{d}{dt}& \begin{pmatrix}
\tilde{\rho}_{22} & \tilde{\rho}_{23} \\
\tilde{\rho}_{32} & \tilde{\rho}_{33}
\end{pmatrix}\\
=&-{i}\left[a_z^{\dagger}a_z+\frac{1}{2}\right]
\begin{pmatrix}
0 & -\delta_a\tilde{\rho}_{23} \\
\delta_a\tilde{\rho}_{32} & 0
\end{pmatrix}\\
&-i\frac{\Omega_a}{2}\begin{pmatrix}
i2\mathrm{Im}[\tilde{\rho}_{32}e^{i\epsilon_a t}] & e^{i\epsilon_at}\left(\tilde{\rho}_{33}-\tilde{\rho}_{22}\right) \\
e^{-i\epsilon_a t}\left(\tilde{\rho}_{22}-\tilde{\rho}_{33}\right) & i2\mathrm{Im}[\tilde{\rho}_{23}e^{-i\epsilon_a t}]
\end{pmatrix}\\
&-\frac{\gamma_c}{2}\begin{pmatrix}
2\tilde{\rho}_{22} & \tilde{\rho}_{23} \\
\tilde{\rho}_{32} & 0
\end{pmatrix}\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_za_z^\dagger\tilde{\rho}-2a_z^\dagger\tilde{\rho} a_z+\tilde{\rho} a_za_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z\tilde{\rho}-2a_z \tilde{\rho} a_z^\dagger+\tilde{\rho} a_z^\dagger a_z\right).
\end{split}
\label{eq:eqanom}
\end{equation}
The term that is different is the cyclotron damping term that is proportional to $\gamma_c$. This is because the lower rather than the upper of the two sets of states is unstable. The vanishing element in the matrix comes because the states $|3,n_z\rangle$ do not decay.
The discussion follows essentially the same steps discussed in the previous section. The differential equations are
\begin{subequations}
\begin{align}
\frac{d}{dt}\vec{p}_{22}(t)
&=\mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22}(t)-\Omega_a\textrm{Im}\left[\vec{p}_{23}(t)\right]\\
\frac{d}{dt}\vec{p}_{23}(t)
&=\mathbf{R}(\epsilon_a,\delta_a,\gamma_c)\, \vec{p}_{23}(t)-i\frac{\Omega_a}{2}(\vec{p}_{33}(t)-\vec{p}_{22}(t))\\
\frac{d}{dt}\vec{p}_{33}(t)
&=\mathbf{R}(0,0,0)\, \vec{p}_{33}(t)+\Omega_a\textrm{Im}\left[\vec{p}_{23}(t)\right]
,
\end{align}
\label{eq:MatrixEquationAnom}%
\end{subequations}
These equations are to be solved for the initial conditions $\vec{p}_{33}(0)=\vec{p}(T)$ and $\vec{p}_{23}(0)=\vec{p}_{22}(0)=0.$
\subsection{Quasi Steady State Solution}
Coherent, driven anomaly transitions can balance the incoherent spontaneous emission of synchrotron radiation to produce a quasi steady state. For a weak drive ($\Omega_a\ll \gamma_c$), the system remains mostly in its initial state, so
\begin{eqnarray}
&P_3 = \textrm{Tr}[p_{33}] = &\sum_{n_z} {p}_{33;n_z}(t)\approx \sum_{n_z}p_{n_z}(T) = 1 \\
&P_2 = \textrm{Tr}[p_{22}] = &\sum_{n_z} p_{22;{n_z}}(t) \ll 1.
\label{eq:LargeRho33}
\end{eqnarray}
The quasi steady state pertains in the time range
\begin{equation}
\gamma_c^{-1}\ll t \ll \gamma_c^{-1} \left(\frac{\gamma_c}{\Omega_a}\right)^2.\label{eq:TimeRange}
\end{equation}
The time must be long enough for transients to die out. It must be short enough that Eq.~(\ref{eq:LargeRho33}) remains valid, with the upper time limit justified presently. What is detected is the probability $P_1$ to end up in the spin-down cyclotron ground state. This probability increases as
\begin{equation}
\frac{dP_1}{dt}=\gamma_cP_2
\label{eq:dP1dt}
\end{equation}
via synchrotron emission from $P_2$ at rate $\gamma_c$.
For the quasi steady state, the time derivatives of $\vec{p}_{22}$ and $\vec{p}_{23}$ are set to zero in Eq.~(\ref{eq:LargeRho33}), though that of $\vec{p}_{33}$ is not, so that
\begin{eqnarray}
&\mathbf{R}(\epsilon_a,\delta_a,\gamma_c)\, \vec{p}_{23}-i\frac{\Omega_a}{2}\vec{p}(T)=0\label{eq:SteadyStatep12}
\\
&\mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22}-\Omega_a\textrm{Im}\left[\vec{p}_{23}\right]=0.
\label{eq:SteadyStatep22}
\end{eqnarray}
Because $\mathbf{R}(0,0,2\gamma_c)$ has a simple structure,
\begin{equation}
P_2=\textrm{Tr}\left[{p}_{22}\right]
= -\frac{1}{\gamma_c}
\sum_{{n_z}=0}^{\infty}\left(\mathbf{R}(0,0,2\gamma_c)\, \vec{p}_{22}\right)_{n_z}.
\label{eq:simplified22}
\end{equation}
Eqs.~(\ref{eq:dP1dt}-\ref{eq:simplified22}) together give a quasi steady state rate
\begin{equation}
\frac{dP_1}{dt}= \gamma_cP(\Omega_a,\epsilon_a,\delta_a)
\label{eq:dP1dtSteady}
\end{equation}
that is the same function that described the steady state for cyclotron excitation Eq.~(\ref{eq:P}) multiplied by $\gamma_c$. With anomaly arguments rather than cyclotron arguments, however, the function takes an entirely different shape.
When the drive is applied for time $t_d$ and then turned off, the probability $P_1$ eventually becomes the integral of $dP_1/dt$ at time $t_d$ plus $P_2(t_d)$, because the latter probability is transferred to the spin-down ground state by spontaneous emission from the cyclotron excited state.
Approximating $dP_1/dt$ with the quasi steady state value in Eq.~(\ref{eq:dP1dtSteady}) gives
\begin{equation}
P_{1;\mathrm{total}} \approx \left(t_d\gamma_c+1\right)P(
\Omega_a,\epsilon_a,\delta_a).
\end{equation}
This slightly overstates the transition probability because $dP_1/dt$ increases before the quasi steady state is established, but the line shape is approximately right.
The $T=0$ limit of the quasi-steady-state anomaly line shape for a weak drive becomes a Lorentzian, $\left(t_d\gamma_c+1\right)P_0(\Omega_a,\epsilon_a,\delta_a)$, similar to what was discussed for cyclotron resonance. On resonance, the quasi steady state probability to be in state $\ket{2}{0}$ at $T=0$ is $( \Omega_a/\gamma_c)^2$. This is extremely small for a weak anomaly drive with $\Omega_a \ll \gamma_c$. For the cases we consider, with temperatures not far from 0, we expect that the rate to transfer population from the initial $\ket{3}{n_z}$ states to the final $\ket{1}{n_z}$ states goes as this small probability times the rate $\gamma_c$ to decay form $\ket{2}{n_z}$ to $\ket{1}{n_z}$. The population transfer will be small (as needed to have a quasi steady state) as long as the time is short compared to the inverse of this rate, which gives the upper time limit in Eq.~(\ref{eq:TimeRange}).
\subsection{Discussion of the Anomaly Line Shape}
Figure~\ref{fig:AnomalyTimeEvolution} is a numerical solution to the master equation for an anomaly drive that is weak ($\Omega_a = 0.1 \gamma_c$) and resonant (at a drive detuning $\epsilon_a = 5 \delta_a$) for the realistic experimental conditions in Table~\ref{table:FrequenciesCompared}.
The probability $P_2$ increases from zero to reach a quasi steady state in several cyclotron damping times, whereupon the probability $P_1$ increases linearly. The probability to be in the initial spin-up ground state, $P_3$, decreases only slightly from unity to conserve probability.
\begin{figure}[htbp!]
\includegraphics[width=\the\columnwidth]{Figures/AnomalyTimeEvolution_gammaC.eps} \hfill
\caption{Time evolution in response to a weak ($\Omega_a = 0.1 \gamma_c$) and resonant anomaly drive (at a drive detuning $\epsilon_a = 5 \delta_a$) with a cyclotron damping times indicated by vertical grid lines.} \label{fig:AnomalyTimeEvolution}
\end{figure}
The resonance line shapes for driven anomaly transitions in Fig.~\ref{fig:AnomalyLineshape} are for a weak drive ($\Omega_a = 0.1 \gamma_c$) and the realistic experimental conditions in Table~\ref{table:FrequenciesCompared}.
The probability $P_{1;\mathrm{total}} = P_1(t_d) + P_2(t_d)$ is plotted versus the detuning $\epsilon_a$ of the drive from $\omega_a$. The sold curve is obtained by numerically integrating the master equation for ten cyclotron damping times, $t=10\gamma_c^{-1}$. The quasi steady state solution (dashed) overestimates the probability because it takes some time to increase the transition rate to the steady state. However, the normalized line shapes in Fig.~\ref{fig:AnomalyLineshape}b shows that the quasi steady state line shape correctly predicts the shape.
In Fig.~\ref{fig:AnomalyLineshape}, the classical Brownian motion line shape (dotted) is remarkably close to the solution to the master equation obtained by direct integration (solid), quite unlike the case for the cyclotron line shape. Figure \ref{fig:ClassicalAndQuantumAnom} compares quantum and classical calculations with three realizable values of $\gamma_z$. For the best measurement \cite{HarvardMagneticMoment2008}, with $\bar{n}_z\gamma_z=6\times10^3\delta_a$ (Table.~\ref{table:FrequenciesComparedOld}), the two calculations predicts same line shape.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/AnomalyLineshape_2.eps}
\includegraphics[width=\the\columnwidth]{Figures/AnomalyLineshapeNormalized_2.eps}
\caption{(a) Anomaly line shape for spin flip transition induced by a weak anomaly drive.
The integrated solution of the master equation for time $10/\gamma_c$ (solid) is compared to the quasi steady state solution (dashed) and the classical Brownian motion line shape (dotted). (b) The integrated and steady-state solutions coincide when normalized to their peak probability. These line shape is much narrower than the $\pm 300$ ppt uncertainty of the best measurement (represented by the "error bar").
}
\label{fig:AnomalyLineshape}
\end{figure}
What is so different from the case of the cyclotron lineshape is that the circumvention of detection backaction that was possible in the cyclotron case is not possible for the anomaly lineshape. The axial quantum states are not resolved within the anomaly lineshape for the realistic parameters of Table \ref{table:FrequenciesCompared}. The reason is that the anomaly frequency shift per axial quantum of excitation is about 10 times smaller than both the cyclotron damping width $\gamma_c$ and the axial decoherence width $\bar{n}_z\gamma_z$. The anomaly frequency must be extracted from a resonance line with a calculated linewidth that is about $2.2~\gamma_c$. The shape is slightly asymmetric with a tail toward higher frequencies because more populated axial states have $n_z > \bar{n}_z$.
The good news that the calculation nonetheless brings for measurements is that the predicted linewidth (for the realistic conditions of Table ~\ref{table:FrequenciesCompared}) is much narrower than previously realized. The ``error bar'' in the figure corresponds to the $\pm 300$ ppt uncertainty (ppt = 1 part in $10^{12}$) of the most accurate measurement to date \cite{HarvardMagneticMoment2008,HarvardMagneticMoment2011}.
The full halfwidth of the predicted lineshape is 60\% of the error bar, so a modest linesplitting of only a factor of 6 would suffice for a ten times more accurate measurement of the electron magnetic moment.
\begin{figure}[]
\centering
\includegraphics[width=\the\columnwidth]{Figures/ClassicalLimitAnom.pdf}
\caption{Comparison of quantum calculation (solid) and classical calculation (dashed) with the different $\gamma_z's$ for weak drive ($\Omega_c=0.1\gamma_a$) and 10$\gamma_c^{-1}$ drive time in anomaly transition. The damping rates $\gamma_z$ for (a-c) are 1000, 100, 10 times the value in Table. \ref{table:FrequenciesCompared}, respectively.}
\label{fig:ClassicalAndQuantumAnom}
\end{figure}
\subsection{Temperature and Damping Dependence}
Once the detection backaction is circumvented \cite{Fan2020BackActionPRL}, determining $\omega_c$ from the cyclotron lineshape should no longer be the leading impediment to measuring the electron and positron magnetic moments orders of magnitude more precisely than has been possible. Since a similar method is not available for measuring the anomaly frequency $\omega_a$, this promises to be the central challenge for future measurements. The lineshape prediction discussed in the previous section suggests the possibility for a ten-fold improvement if the experimental parameters that currently seem feasible (Tab. \ref{table:FrequenciesCompared}) are realized. The purpose of this section is to search for possible reductions in anomaly linewidth that may be possible with reductions in axial temperature, cyclotron damping rate, and axial damping rate beyond the values in the table.
Fig.~\ref{fig:FutureAnomalyPossibilities} shows anomaly line shapes for a weak drive ($\Omega_a=\gamma_c/10$) for temperatures of 100 mK (black solid), 50 mK (black dashed) and 25 mK (black dotted). The other parameters used are from Table \ref{table:FrequenciesCompared}). The most accurate measurement was done at an ambient temperature of 100 mK \cite{HarvardMagneticMoment2008}, with a demonstrated electron cyclotron temperature of 100 mK and a demonstrated axial temperature as low as 230 mK. The temperature in the table assumes that with better detectors under development, that the latter temperature can be reduced to the ambient. However,
dilution refrigerators can reach lower temperatures if the heat load can be made low enough. Also, cavity-sideband cooling is a possible method to reduce the axial temperature below the ambient apparatus temperature \cite{atomsNewMeasurement2019}. The anomaly lineshapes clearly reduce and the lines become more symmetric for lower axial temperatures.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/AnomalyLoweringTemperatureAndGammac.eps}
\caption{Probability of a spin-flips caused by a driven anomaly transitions for cyclotron and axial reservoir temperatures of 100 mK (solid), 50 mK (dashed) and 25 mK (dotted). The black curves are the experimentally accessible parameters in Table \ref{table:FrequenciesCompared}. The blue curves are for a ten-fold reduction in the cyclotron damping rate below the value in the table.
}
\label{fig:FutureAnomalyPossibilities}
\end{figure}
The blue curves in Fig.~\ref{fig:FutureAnomalyPossibilities} show the large anomaly lineshape reduction that comes from lowering the cyclotron radiation rate by a factor of ten. The most accurate experiment achieved the low damping rate in the table by using a microwave cavity to suppress the spontaneous emission of synchrotron radiation \cite{InhibitionLetter} by a factor of about 200. An lower loss microwave cavity could further reduce the cyclotron damping rate to produce the narrower lineshapes. This would slow the measurement because it takes several cyclotron damping times for the population excited to states $\ket{2}{n_z}$ to decay to the ground state, but the damping rate could varied by tuning $\omega_c$ closer or further from cavity microwave resonances \cite{HarvardMagneticMoment2011}.
Reducing the axial temperature without reducing $\gamma_c$ reduces the linewidth somewhat. A bigger consequence is that the doing so reduces the asymmetry of the line shape, which should make it possible to identify the resonance frequency more reliably. The effects of the axial damping rate have also been investigated. Further reductions in the axial damping rate do not noticeably change any of the curves in Fig. \ref{fig:FutureAnomalyPossibilities}.
The possibly to use cavity sideband cooling of the axial motion has been mentioned as a possible route to narrower resonance linewidths \cite{atomsNewMeasurement2019}. Once the cooling is stopped, the axial motion would then reequilibrate at the bath temperature at a rate $\gamma_z$. This is not a steady state, of course, but we can investigate the possibility by directly integrating the master equation.
Fig.~\ref{fig:AnomalyCavitySidebandCooling} shows the probability of a spin-flip caused by a weak anomaly drive ($\Omega_c=0.1 \gamma_c$) applied for a 100 mK temperature bath (solid). For this illustration, the axial motion is initially assumed to be cooled to the $T=0$ limit so that only the lowest axial quantum state is initially populated. This causes the linewidth to narrow from $\pm 190$ ppt to $\pm 130$ ppt (dashed). The line shape also is more symmetric about its center, and the offset frequency is smaller.
The drive is applied for time $10/\gamma_c$ in this illustration, which is one axial damping time $1/\gamma_z$. For the parameters we are using for this illustration (Table~\ref{table:FrequenciesCompared}), the linewidth gets broader for shorter driving times because of the limited drive duration, so narrower resonances would come for a smaller $\gamma_z$.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/AnomalyCavitySidebandCoolingNormalized.eps}
\caption{Probability of a spin-flips caused by a driven anomaly transitions for a 100 mK temperature bath (solid). If the axial motion is initially cooled to the $T=0$ limit so that only $n_z=0$ is initially populated, then the linewidth narrows from $\pm 190$ ppt to $\pm 130$ ppt, becomes more symmetric and has a slightly smaller offset frequency.
}
\label{fig:AnomalyCavitySidebandCooling}
\end{figure}
Achieving detection circumvention by resolving the axial states in the anomaly line shape, just as for the cyclotron line shape, would require increasing the bottle shift $\delta_a$ per axial quantum by a factor of 100 or more. This is to make the bottle shift much larger than both the axial decoherence width ($\bar{n}_z\gamma_z$) and the cyclotron damping width ($\gamma_c$). The solid curve in Fig. \ref{fig:Anomaly:LargeBottle} shows the anomaly line shape for the parameters in Table \ref{table:FrequenciesCompared}. The anomaly lineshape broadens for a 10 times larger bottle gradient. For a 100 times larger bottle the line begins to separate into peaks that correspond to individual axial quantum states. Magnetic bottle gradients of the size needed have been produced, but only for Penning traps that are smaller than is otherwise desirable for electron and positron measurements \cite{PbarMagneticMoment2013,ProtonMagneticMoment,OneProtonSpinFlipHarvard,MainzSpinFlips,BASEQOverM2015,BasePbarMagneticMoment,BaseProtonMagneticMomemt}. However, the figure illustrates that resolving the axial quantum states is not an advantage in that the linewidth of the lowest resolved peak is a bit bigger than the anomaly linewidth already considered. As mentioned above, the linewidth from both the cyclotron damping and the axial decoherence broadening do not decrease with bottle gradient size.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/OnePhotonAnom2008NormLargeB2.pdf}
\caption{Probability of a spin-flips caused by a driven anomaly transitions for the typical magnetic bottle in Table \ref{table:FrequenciesCompared} (solid), 10 times larger bottle (dashed) and 100 times larger bottle(dotted).
}
\label{fig:Anomaly:LargeBottle}
\end{figure}
This cursory survey of anomaly line shapes reveals no obvious way to make a single large additional reduction in the anomaly linewidth beyond the order of magnitude that has been discussed.
\section{Directly Driven Spin Flips}
\label{sec:DirectSpinFlips}
A spin drive $V_s$ transfers population between the spin-down and spin up cyclotron ground states, $|1,n_z\rangle$ and $|3,n_z\rangle$, both of which are stable. In this section, we apply a spin-flip drive with a Rabi frequency $\Omega_s$ to an initial population in $|1\rangle$, with no cyclotron or anomaly drives (i.e.\ $\Omega_c=0$ and $\Omega_a=0$). If only one axial detection state was populated this would be the prototypical ``Rabi flopping'' of the two states of a spin qubit. A distribution of axial detection states has a backaction that makes a superposition of spin frequencies, the effect of which is calculated and discussed here.
The master equation for the density operator describing this case, in the interaction representation, is
\begin{equation}
\begin{split}
\frac{d}{dt} &\begin{pmatrix} \tilde{\rho}_{11} & \tilde{\rho}_{13} \\ \tilde{\rho}_{31} & \tilde{\rho}_{33} \end{pmatrix} = -{i}\left(a_z^{\dagger}a_z+\tfrac{1}{2}\right)
\begin{pmatrix} 0 & -\delta_s\tilde{\rho}_{13} \\ \delta_s\tilde{\rho}_{31} & 0 \end{pmatrix}\\
&-i\frac{\Omega_s}{2}
\begin{pmatrix}
i2\mathrm{Im}[\tilde{\rho}_{31}e^{i\epsilon_s t}]
& e^{i\epsilon_s t}\left(\tilde{\rho}_{33}-\tilde{\rho}_{11}\right) \\
e^{-i\epsilon_s t}\left(\tilde{\rho}_{11}-\tilde{\rho}_{33}\right)
& i2\mathrm{Im}[\tilde{\rho}_{13}e^{-i\epsilon_s t}]
\end{pmatrix}\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_za_z^\dagger\tilde{\rho}-2a_z^\dagger\tilde{\rho} a_z+\tilde{\rho} a_za_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z\tilde{\rho}-2a_z \tilde{\rho} a_z^\dagger+\tilde{\rho} a_z^\dagger a_z\right).
\end{split}
\label{eq:SpinMasterEquationInitial}
\end{equation}
This master equation is the same as for driven cyclotron transitions (Eq.~(\ref{eq:CyclotronMasterEquation})) except that the state $\vert2\rangle$ is replaced by $\vert3\rangle$ and the damping term $\gamma_c$ is replaced by $\gamma_s\approx0$ (see Tab.~\ref{table:Frequencies}).
The master equation can be solved exactly in the same way as the cyclotron transition. We assume the initial population is distributed in the state $\vert1,n_z\rangle$ with the Boltzmann distribution as Eq.~(\ref{eq:11distribution}). In vector form,
Eq.~(\ref{eq:SpinMasterEquationInitial}) is
\begin{subequations}
\begin{align}
\frac{d}{dt}\vec{p}_{11}(t)
&=\mathbf{R}(0,0,0)\, \vec{p}_{11}(t)-\Omega_s\textrm{Im}\left[\vec{p}_{13}(t)\right]\label{eq:MatrixEquationSpinA}
\\
\frac{d}{dt}\vec{p}_{13}(t)
&=\mathbf{R}(\epsilon_s,\delta_s,0)\, \vec{p}_{13}(t)-i\frac{\Omega_s}{2}(\vec{p}_{33}(t)-\vec{p}_{11}(t))\label{eq:MatrixEquationSpinB}
\\
\frac{d}{dt}\vec{p}_{33}(t)
&=\mathbf{R}(0,0,0)\, \vec{p}_{33}(t)+\Omega_s\textrm{Im}\left[\vec{p}_{13}(t)\right].
\label{eq:MatrixEquationSpinC}
\end{align}
\label{eq:MatrixEquationSpin}%
\end{subequations}
In general, these equations will be solved for an initial values of the density operators at $t=0$.
Since the damping between the spin states $\vert1\rangle$ and $\vert3\rangle$ is essentially zero, the steady state is not as obvious as in the case for cyclotron transitions. The axial decoherence term $\bar{n}_z\gamma_z$ in Eq.~(\ref{eq:MatrixEquationSpin}) does not induce transition between $\vert1\rangle$ and $\vert3\rangle$, but there is still a useful quasi steady state solutions for the ``weak" drive limit, $\Omega_s \ll \bar{n}_z\gamma_z$.
\subsection{Steady State}
If the spin-flip drive is applied for a long time, $t\gg \bar{n}_z \gamma_z/\Omega_s^2,$ and $\bar{n}_z\gamma_z \ne 0$, there is a steady state described by setting the time derivatives in Eqs.~(\ref{eq:MatrixEquationSpin}) to zero,
\begin{subequations}
\begin{align}
&\mathbf{R}(0,0,0)\, \vec{p}_{11}(t)-\Omega_s\textrm{Im}\left[\vec{p}_{13}(t)\right]=0\label{eq:MatrixEquationSpinSteadyA}
\\
&\mathbf{R}(\epsilon_s,\delta_s,0)\, \vec{p}_{13}(t)-i\frac{\Omega_s}{2}(\vec{p}_{33}(t)-\vec{p}_{11}(t))=0\label{eq:MatrixEquationSpinSteadyB}
\\
&\mathbf{R}(0,0,0)\, \vec{p}_{33}(t)+\Omega_s\textrm{Im}\left[\vec{p}_{13}(t)\right]=0.
\label{eq:MatrixEquationSpinSteadyC}
\end{align}
\label{eq:MatrixEquationSpinSteady}%
\end{subequations}
Summing Eqs.~(\ref{eq:MatrixEquationSpinSteadyA}) and (\ref{eq:MatrixEquationSpinSteadyC}) over $n_z$, and using
\begin{equation}
\sum_{{n_z}=0}^{\infty}\left( \mathbf{R}(0,0,0)\, \vec{p}_{11} \right)_{n_z}
=\sum_{{n_z}=0}^{\infty}\left( \mathbf{R}(0,0,0)\, \vec{p}_{33} \right)_{n_z}
=0,
\end{equation}
gives $\vec{p}_{13}=0$ and equal populations
\begin{equation}
\sum_{{n_z}=0}^{\infty}\left( \vec{p}_{11} \right)_{n_z}
=\sum_{{n_z}=0}^{\infty}\left( \vec{p}_{33} \right)_{n_z}
=\frac{1}{2}.
\label{eq:SteadyStatePopulationSpin}
\end{equation}
of spin up and spin down states. The interaction of the axial motion with its thermal reservoir produces a spread of Rabi Flopping frequencies, averages out the net Rabi flopping between the two spin states.
\subsection{Quasi Steady State}
A steady state with equal spin up and spin down populations is not useful for determining the spin frequency $\omega_s$. There is a quasi steady state, however.
For a ``weak drive" with $\Omega_s\ll\bar{n}_z\gamma_z$ that does not appreciably change the initial Boltzmann distribution $\vec{p}_{11} \approx \vec{p}(T)$ of axial states, there is an excitation rate
\begin{equation}
\frac{dP_3}{dt} =\frac{d}{dt}\left[\sum_{{n_z}=0}^{\infty}\vec{p}_{33;n_z}(t)\right].
\end{equation}
The drive must be applied for a time in the range
\begin{equation}
\left(\bar{n}_z\gamma_z\right)^{-1}\ll t \ll\left(\frac{\Omega_s^2}{\bar{n}_z\gamma_z}\right)^{-1},
\end{equation}
long compared to the dephasing time $\bar{n}_z \gamma_z$ but short compared to the time for approaching the steady-state with equal spin-up and spin-down populations. Fig.~\ref{Fig:SpinFlipLineShape} shows an example of the time evolution.
\begin{figure}
\centering
\includegraphics[width=\the\columnwidth]{Figures/SpinFlipTimeEvolutionForPaper.pdf}
\includegraphics[width=\the\columnwidth]{Figures/SpinFlipTimeEvolutionDerivativeForPaper.pdf}
\caption{The probabilities for spin up (red) and and spin down minus 1 (black) in (a), and for the derivatives of these probabilities (b) as a function of time for a spin flip drive tuned to the $n_z=0$ cyclotron resonance, with Rabi frequency $\Omega_s/(2\pi) = 0.01$ Hz and other parameters in Tab~\ref{table:FrequenciesCompared}. The horizontal grids are spaced by $(\bar{n}_z \gamma_z)^{-1}$ The vertical grids show quasi steady state values.
}
\label{Fig:SpinFlipLineShape}
\end{figure}
Eq.~(\ref{eq:MatrixEquationSpinC}) gives
\begin{equation}
\frac{d}{dt}\vec{p}_{33}(t)=\Omega_s\textrm{Im}\left[\vec{p}_{13}(t)\right],
\end{equation}
when a small excitation $\vec{p}_{33} \approx 0$ is assumed. The steady-state $\vec{p}_{13}(t)$ comes from solving Eq.~(\ref{eq:MatrixEquationSpinB}),
\begin{equation}
\begin{split}
\vec{p}_{13}(t)
&=\mathbf{R}^{-1}(\epsilon_s,\delta_s,0)\left[-i\frac{\Omega_s}{2}(\vec{p}_{33}(t)-\vec{p}_{11}(t))\right]\\
&=i\frac{\Omega_s}{2}\mathbf{R}^{-1}(\epsilon_s,\delta_s,0)\, \vec{p}(T),
\end{split}
\end{equation}
for $\vec{p}_{11} \approx \vec{p}(T)$ and $\vert\vec{p}_{33}\vert \ll 1$. The transition rate is then
\begin{equation}
\frac{dP_3}{dt} =\frac{\Omega_s^2}{2}\mathrm{Im}\left[\sum_{{n_z}=0}^{\infty}\left(i\mathbf{R}^{-1}(\epsilon_s,\delta_s,0) \vec{p}(T)\right)_{n_z}\right].
\label{eq:TransitionRateSpin}
\end{equation}
This rate is essentially the line shape defined in Eq.~(\ref{eq:P}) except for the cyclotron damping rate $\gamma_c$ and the parameters for spin flip transition $\Omega_s$, $\epsilon_s$ and $\delta_s$.
\begin{figure}
\centering
\includegraphics[width=\the\columnwidth]{Figures/OnePhotonSpinNorm2.pdf}
\caption{(a) Quantum spin-flip line shape (solid) for a weak drive with the largest peak normalized to 1 and for the classical Brownian motion line shape(dashed). The line shape used for the best measurement (dotted) is also shown. (b) Normalized probability of a spin-flip transition (solid) compared to the cyclotron transition (dashed). The horizontal scale is $\epsilon_s/\delta_s$ for the spin transition and $\epsilon_c/\delta_c$ for the cyclotron transition.
}
\label{Fig:SpinFlipLineShape}
\end{figure}
The directly driven spin flip transition rate in Fig.~(\ref{Fig:SpinFlipLineShape}a) is very similar to the cyclotron line shape of Fig.~(\ref{fig:CyclotronLineshape}a) for same experimental conditions in Table~\ref{table:FrequenciesCompared}. A spin-flip resonance for every axial quantum state is clearly resolved. The line shapes for classical calculation (Eq.~(\ref{eq:classicallineshape1})) and for the best measurement parameters (Tab.~\ref{table:FrequenciesComparedOld}) are also shown for comparison. Figure~\ref{Fig:SpinFlipLineShape}(b) compares the line shape for spin flip (solid line) compared to the one for the cyclotron transition (dashed line) with the parameters in Tab.~\ref{table:FrequenciesCompared}. Since the magnetic bottle parameters are related by $\delta_s=g/2\times\delta_c\approx1.001\delta_c$, the transition rate line shape for spin flip (Eq.~(\ref{eq:TransitionRateSpin})) is much the same as the cyclotron line shape. The only difference from the small damping rate $\gamma_s\approx0$ appears when focusing on the $n_z=0$ peak (fig.~\ref{Fig:SpinFlipLineShape}(b)). Because of the negligible spin-flip damping rate, the full-width at the half maximum of the peak $2\bar{n}_z\gamma_z$ is slightly narrower than the cyclotron's linewidth $\gamma_c+2\bar{n}_z\gamma_z$. The spin transition line shape peaks can be made even narrower by reducing $\gamma_z$ further.
The possibility to use the spin-flip transition probability is discussed and compared to alternatives in Sec.~\ref{sec:Comparison}.
\section{Two-drive Spin Flips}
\label{sec:twophotoncalculation}
\subsection{Master Equation}
Spin flips (from the spin down ground state $\left|1\right\rangle$ to the spin up ground state $\left|3\right\rangle$) can also be driven using simultaneously applied cyclotron and anomaly drives ($\Omega_c > 0$ and $\Omega_a > 0$ with $\Omega_s=0$) instead of the direct spin flip drive discussed in Sec.~\ref{sec:DirectSpinFlips}. A practical advantage is that the stable final state would remain unchanged as long as is needed to detect it. This was true for the anomaly transitions considered above, but not for cyclotron transitions that must be detected before cyclotron decay. A quantum calculation is needed to ascertain whether two photon transitions would be less sensitive to slow drifts of the magnetic field insofar as the spin and cyclotron motion will experience the same average magnetic field.
The three sets of states in Eq.~(\ref{eq:ThreeStates}) are involved in flipping the spin via the two drives. The density operator in the interaction picture can be written in terms of operators $p_{jk}$ proportional to the operators $\tilde{\rho}_{jk}$ of Eq.~(\ref{eq:rhojk}), such that
\begin{equation}
\begin{split}
\tilde{\rho} =&
\begin{pmatrix}
p_{11} & p_{12} e^{i\epsilon_c t} & p_{13} e^{i(\epsilon_c+\epsilon_a)t} \\
p_{21} e^{-i\epsilon_c t} & p_{22} & p_{23} e^{i \epsilon_a t} \\
p_{31} e^{-i(\epsilon_c+\epsilon_a)t} & p_{32} e^{-i \epsilon_a t} & p_{33}
\end{pmatrix}.
\end{split}
\end{equation}
The transformation puts the master equation in the form
\begin{equation}
\begin{split}
\frac{d}{dt} &
\begin{pmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23}\\
p_{31} & p_{32} & p_{33}
\end{pmatrix}
=\\
&-i
\begin{pmatrix}
0 & \epsilon_c p_{12} & (\epsilon_c+\epsilon_a)p_{13} \\
-\epsilon_c p_{21} & 0 &\epsilon_a p_{23}\\
-(\epsilon_c+\epsilon_a)p_{31} & -\epsilon_a p_{32} & 0
\end{pmatrix} \\
& -{i}\left(a_z^{\dagger}a_z+\tfrac{1}{2}\right)
\begin{pmatrix}
0 & -\delta_c p_{12} &-\delta_s p_{13}\\
\delta_c p_{21} & 0 & -\delta_a p_{23}\\
\delta_s p_{31} & \delta_a p_{32} & 0
\end{pmatrix}\\
&-\frac{i\Omega_c}{2}
\begin{pmatrix}
i2 \mathrm{Im}[ p_{21}] & p_{22}- p_{11} & p_{23}\\
p_{11}- p_{22} &i2 \mathrm{Im}[p_{12}] & p_{13}\\
-p_{32} &-p_{31} &0
\end{pmatrix}\\
& -\frac{i\Omega_a}{2}
\begin{pmatrix}
0 &-p_{13} &-p_{12}\\
p_{31} &i2 \mathrm{Im}[ p_{32}] & p_{33}- p_{22}\\
p_{21} &p_{22}- p_{33} &i2 \mathrm{Im}[p_{23}]
\end{pmatrix}\\
&-\frac{\gamma_c}{2}
\begin{pmatrix}
-2p_{22} & p_{12} & 0 \\
p_{21} & 2p_{22} & p_{23}\\
0 & p_{32} & 0
\end{pmatrix}\\
&-\frac{\gamma_z}{2}\bar{n}_z\left(a_z a_z^\dagger p -2a_z^\dagger p a_z+p a_z a_z^\dagger\right)\\
&-\frac{\gamma_z}{2}\left(\bar{n}_z+1\right)\left(a_z^\dagger a_z p-2a_z p a_z^\dagger+p a_z^\dagger a_z\right).
\end{split}
\label{eq:eqtwophotonp}
\end{equation}
All time dependence is now within the components $p_{ij}$.
For a thermal distribution of initial axial states, the operators $p_{ij}$ are axially diagonal, with diagonal components, $p_{ij;n_z}=\langle i,n_z|p_{ij}|j,n_z\rangle$. The master equation is then given by the differential equations,
\begin{subequations}
\begin{align}
\frac{d}{dt}&p_{11;n_z}(t)\nonumber\\
&=\left[-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\right]p_{11;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{11;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{11;n_z+1}(t)\nonumber\\
&-\Omega_c\textrm{Im}\left[p_{12;n_z}(t)\right]+\gamma_cp_{22;n_z}(t)\\
\frac{d}{dt}&p_{22;n_z}(t)\nonumber\\
&=\left[-\gamma_c-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\right]p_{22;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{22;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{22;n_z+1}(t)\nonumber\\
&+\Omega_c\textrm{Im}\left[p_{12;n_z}(t)\right]-\Omega_a\textrm{Im}\left[p_{23;n_z}(t)\right]\\
\frac{d}{dt}&p_{33;n_z}(t)\nonumber\\
&=\left[-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\right]p_{33;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{33;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{33;n_z+1}(t)\nonumber\\
&+\Omega_a\textrm{Im}\left[p_{23;n_z}(t)\right]\\
\frac{d}{dt}&p_{12;n_z}(t)\nonumber\\
&=\big[i\left(-\epsilon_c+\delta_c\left(n_z+\tfrac{1}{2}\right)\right)\nonumber\\
&-\tfrac{1}{2}\gamma_c-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\big]p_{12;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{12;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{12;n_z+1}(t)\nonumber\\
&-i\frac{\Omega_c}{2}(p_{22;n_z}(t)-p_{11;n_z}(t))+i\frac{\Omega_a}{2}p_{13;n_z}(t)\\
\frac{d}{dt}&p_{23;n_z}(t)\nonumber\\
&=\big[i\left(-\epsilon_a+\delta_a\left(n_z+\tfrac{1}{2}\right)\right)\nonumber\\
&-\tfrac{1}{2}\gamma_c-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\big]p_{23;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{23;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{23;n_z+1}(t)\nonumber\\
&-i\frac{\Omega_a}{2}(p_{33;n_z}(t)-p_{22;n_z}(t))-i\frac{\Omega_c}{2}p_{13;n_z}(t)\\
\frac{d}{dt}&p_{13;n_z}(t)\nonumber\\
&=\big[i\left(-\left(\epsilon_c+\epsilon_a\right)+\delta_s\left(n_z+\tfrac{1}{2}\right)\right)\nonumber\\
&-\gamma_z(2\bar{n}_z+1)n_z-\gamma_z\bar{n}_z\big]p_{13;n_z}(t)\nonumber\\
&+\gamma_z\bar{n}_zn_zp_{13;n_z-1}(t)+\gamma_z(\bar{n}_z+1)(n_z+1)p_{13;n_z+1}(t)\nonumber\\
&-i\frac{\Omega_c}{2}p_{23;n_z}(t)+i\frac{\Omega_a}{2}p_{12;n_z}(t).
\end{align}
\end{subequations}
The general time-dependent solution of these equations has initial conditions $p_{11,n_z}(0) = p_{n_z}(T)$, with $p_{ij,n_z}(0)=0$ all other $i$ and $j$.
In terms of $\mathbf{R}(\epsilon,\delta,\gamma_c)$ from Eq.~(\ref{eq:R}) and vectors $\vec{p}_{ij}$ with components $p_{ij;n_z}=\langle i,n_z|p_{ij}|j,n_z\rangle$, the vector equations of motion are
\begin{subequations}
\begin{align}
\frac{d}{dt}\vec{p}_{11}(t)
&=\mathbf{R}(0,0, 0)\vec{p}_{11}(t) \nonumber \\
&-\Omega_c\mathrm{Im}[\vec{p}_{12}(t)]+\gamma_c\vec{p}_{22}(t)\\
\frac{d}{dt}\vec{p}_{22}(t)&=\mathbf{R}(0,0, 2\gamma_c)\vec{p}_{22}(t)\nonumber \\
&+\Omega_c\mathrm{Im}[\vec{p}_{12}(t)]-\Omega_a\mathrm{Im}[\vec{p}_{23}(t)]\\
\frac{d}{dt}\vec{p}_{33}(t)&=\mathbf{R}(0,0,0)\vec{p}_{33}(t)\nonumber \\
&+\Omega_a\mathrm{Im}[\vec{p}_{23}(t)]\\
\frac{d}{dt}\vec{p}_{12}(t)&=\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)\vec{p}_{12}(t)\nonumber \\
&-i\tfrac{\Omega_c}{2}\left(\vec{p}_{22}(t)-\vec{p}_{11}(t)\right)+i\tfrac{\Omega_a}{2}\vec{p}_{13}(t)\\
\frac{d}{dt}\vec{p}_{23}(t)&=\mathbf{R}(\epsilon_a,\delta_a, \gamma_c)\vec{p}_{23}(t)\nonumber \\
&-i\tfrac{\Omega_a}{2}\left(\vec{p}_{33}(t)-\vec{p}_{22}(t)\right)-i\tfrac{\Omega_c}{2}\vec{p}_{13}(t)\\
\frac{d}{dt}\vec{p}_{13}(t)&=\mathbf{R}(\epsilon_c+\epsilon_a,\delta_s,0)\vec{p}_{13}(t)\nonumber \\
&-i\tfrac{\Omega_c}{2}\vec{p}_{23}(t)+i\tfrac{\Omega_a}{2}\vec{p}_{12}(t).
\end{align}
\label{eq:TwoPhotonVector}%
\end{subequations}
The initial conditions are $\vec{p}_{11}(0)=\vec{p}(T)$, with $\vec{p}_{ij}(0)=0$ for all other $i$ and $j$. Small, non-resonant excitations to more highly excited states are neglected. For the parameters being considered in this work, we found that simultaneously solving 900 differential equations determine the solution to the master equation numerically for cyclotron and anomaly drives applied at the same time.
\subsection{Quasi Steady State}
A quasi steady state is produced when weak cyclotron and anomaly drives, with
\begin{eqnarray}
&\Omega_c \ll \gamma_c\\
&\Omega_a \ll \gamma_c,
\end{eqnarray}
are applied for a time $t$ in the range
\begin{equation}
\gamma_c^{-1}\ll t \ll \gamma_c/\Omega_a^2.
\end{equation}
The time must be long compared to the cyclotron damping time to allow transients to dies out. It must be short compared to the time it takes to transfer an appreciable population to the spin up spin states.
Fig.~\ref{fig:SimultaneousTimeEvolution} illustrates the time evolution for weak drives ($\Omega_c=\Omega_a = 0.1 \gamma_c$ that are resonant, and for the realistic experimental conditions in Table~\ref{table:FrequenciesCompared}. The sum of the probabilities to be in the states $\left|l,n_z\right\rangle$
\begin{equation}
P_l = \sum_{n_z=0}^{\infty} p_{ll;n_z}
\end{equation}
(from Eqs.~(\ref{eq:Pl}) and (\ref{eq:Plp})) is plotted for $l=1,2,3$. The drives are turned on at time $t=0$ and the time evolution shown continues for ten cyclotron damping period, to $t=10/\gamma_c$. The probability $P_2$ to be driven into the $\left|2,n_z\right\rangle$ states (blue) increases from zero to reach a quasi steady state after the transients die out in several cyclotron damping times $1/\gamma_c$. The probability $P_1$ to remain in the initial $\left|1,n_z\right\rangle$ states, minus unit probability, is shown in black. It remains at essentially unit probability, decreasing only slightly to conserve probability.
The much smaller probability (solid red) to transition to the cyclotron ground states with spin up, $\left|3,n_z\right\rangle$, gradually increases at first, and then increases linearly for much of the $10/ \gamma_c$ time evolution. The solid red curve in Fig.~\ref{fig:SimultaneousD3} illustrates how it is the derivative $dP_3/dt$ that reaches a quasi steady state.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/SimultaneousTimeEvolution_3.pdf}
\caption{Time evolution driven by weak and resonant cyclotron and anomaly drives applied for 10 cyclotron damping times, the latter indicated by vertical gray lines. The probability to be in the $\left|2,n_z\right\rangle$ states (blue) reaches a steady state after transients die out on a time scale give by the cyclotron damping time, $1/\gamma_c$. The probability to be in the initial $\left|1,n_z\right\rangle$ states, with unit probability subtracted out, is shown in black. The probability to be be driven to the final $\left|3,n_z\right\rangle$ states is shown in red.} \label{fig:SimultaneousTimeEvolution}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/SimultaneousD3_Gray.pdf}
\caption{Rate $dP_3/dt$ for simultaneously applied weak cyclotron and anomaly drives ($\Omega_c = \Omega_a=\gamma_c/10$) that are resonant. The quasi steady state estimate (dashed) slightly overstates the transition rate.
}
\label{fig:SimultaneousD3}
\end{figure}
An approximate analytic expression for the quasi steady state rate \begin{equation}
\frac{dP_3}{dt}=\frac{d}{dt} \sum_{n_z=0}^\infty p_{33;n_z}(t)=
\Omega_a\mathrm{Im}\left[\sum^\infty_{n_z=0}{p}_{23;n_z}(t)\right],
\label{eq:E6}
\end{equation}
comes from summing Eq.~(\ref{eq:TwoPhotonVector}c) over all axial states and simplifying using $\sum^\infty_{n_z=0}\left(\mathbf{R}(0,0,0)\vec{p}_{33}\right)_{n_z}=0$. The quasi steady state is also described by
\begin{subequations}
\begin{align}
&\vec{p}_{11} = \vec{p}(T) \label{eq:E1}\\
&\mathbf{R}(0,0, 2\gamma_c)\vec{p}_{22}+\Omega_c\mathrm{Im}[\vec{p}_{12}]=0\label{eq:E2}\\
&\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)\vec{p}_{12}+i\tfrac{\Omega_c}{2}\vec{p}_{11}=0\label{eq:E3}\\
&\mathbf{R}(\epsilon_a,\delta_a, \gamma_c)\vec{p}_{23} -i\tfrac{\Omega_a}{2}(\vec{p}_{33}-\vec{p}_{22})-i\tfrac{\Omega_c}{2}\vec{p}_{13}=0\label{eq:E4}\\
&\mathbf{R}(\epsilon_c+\epsilon_a,\delta_s,0)\vec{p}_{13}+i\tfrac{\Omega_a}{2}\vec{p}_{12}=0.
\label{eq:E5}
\end{align}
\end{subequations}
The first of these equations states that $\vec{p}_{11}$ then remains at the initial thermal equilibrium value, $\vec{p}_{n_z}(T)$. The remaining equations assume $\vert{p}_{11;n_z}\vert\gg\vert{p}_{22;n_z}\vert$, $\vert{p}_{11;n_z}\vert\gg\vert{p}_{13;n_z}\vert$, and $\vert{p}_{12;n_z}\vert\gg\vert{p}_{32;n_z}\vert$, and the time derivatives of $\vec{p}_{12}$, $\vec{p}_{22}$ and $\vec{p}_{23}$ are neglected. The drives must also be applied for a time $t \gg (\bar{n}_z\gamma_z)^{-1}$, so that the
time derivative of $\vec{p}_{13}$ can be neglected because of the decoherence of the superposition of axial states at a rate $\bar{n}_z\gamma_z$. (This last condition does not hold when there in only one axial state in the $T \rightarrow 0$ limit.)
The solutions to these linear equations are
\begin{subequations}
\begin{align}
\vec{p}_{12}&=-i\tfrac{\Omega_c}{2}\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)^{-1}\vec{p}(T)\label{eq:eqtwophotonPLinear1}
\\
\vec{p}_{22}&=-\Omega_c\mathbf{R}(0,0,2\gamma_c)^{-1}\mathrm{Im}[\vec{p}_{12}]\label{eq:eqtwophotonPLinear2}
\\
\vec{p}_{13}&=-i\tfrac{\Omega_a}{2}\mathbf{R}(\epsilon_c+\epsilon_a,\delta_s,0)^{-1}\vec{p}_{12}\label{eq:eqtwophotonPLinear3}
\\
\vec{p}_{23}&=\tfrac{1}{2}\mathbf{R}(\epsilon_a,\delta_a, \gamma_c)^{-1}
\left[i\Omega_c\vec{p}_{13}-i\Omega_a\vec{p}_{22}+i\Omega_a\vec{p}_{33}\right].
\label{eq:eqtwophotonPLinear4}
\end{align}
\end{subequations}
The last of these equations does not yet describe a steady state because it depends upon the growing $\vec{p}_{33}$.
Eq.~(\ref{eq:eqtwophotonPLinear4}) can be substituted into Eq.~(\ref{eq:E6}) to allow an estimate of how rapidly $\vec{p}_{33}$ grows in time. The time dependent parts are
\begin{equation}
\begin{split}
\frac{d}{dt} \sum_{n_z=0}^{\infty} {p}_{33;n_z}
&=\frac{\Omega_a^2}{2}\mathrm{Im}\left[i\sum^\infty_{n_z}\left(\mathbf{R}(\epsilon_a,\delta_a, \gamma_c)^{-1} \vec{p}_{33}\right)_{n_z}\right]+\mathcal{C},
\end{split}
\label{eq:p33timeconstant}
\end{equation}
with $\mathcal{C}$ represents terms which do not depend upon time for $t\gg\gamma_c^{-1}$. Roughly speaking, $\vec{p}_{33}$ (i.e.\ the diagonal elements of $p_{33}$) approaches its steady state with a rate going as $\Omega_a^2|\mathbf{R}(\epsilon_a,\delta_a, \gamma_c)^{-1}|$. The magnitude of the transformation matrix roughly goes as its eigenvalues, $\bar{n}_z\gamma_z + \gamma_c>\gamma_c$. This means that the time constant is longer than $\Omega_a^2/\gamma_c$, more than a thousand seconds with realistic parameters in Table~\ref{table:FrequenciesCompared}. For a realistic drive time $t\ll(\Omega_a^2/\gamma_c)^{-1}$, the last term in Eq.~(\ref{eq:eqtwophotonPLinear4}) can be neglected, as needed to make a steady state equation.
The quasi steady state spin-flip rate as a function of detunings ${dP_3(\epsilon_c,\epsilon_a)}/{dt}$ is thus
\begin{subequations}
\begin{align}
\frac{dP_3(\epsilon_c,\epsilon_a)}{dt}
&=\frac{\Omega_a^2\Omega_c^2}{8}\textrm{Im}
\Bigg[
\sum^\infty_{n_z=0}
\bigg(i \mathbf{R}(\epsilon_a,\delta_a,\gamma_c)^{-1} \vec{W} \bigg)_{n_z} \Bigg],\label{eq:twophototransitionA}
\\
\vec{W} = &-2\mathbf{R}(0,0,2\gamma_c)^{-1}\mathrm{Im}[i\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)^{-1}\vec{p}(T)]\nonumber\\
&-\mathbf{R}(\epsilon_c+\epsilon_a,\delta_s,0)^{-1}\mathbf{R}(\epsilon_c,\delta_c,\gamma_c)^{-1}\vec{p}(T)
\label{eq:twophototransitionB}
\end{align}
\end{subequations}
using Eqs.(\ref{eq:E6}) and (\ref{eq:eqtwophotonPLinear1}-\ref{eq:eqtwophotonPLinear4}). The first term in $\vec{W}$ describes sequential one photon transitions (Fig.~\ref{fig:SimultaneousContour}(b)). The second term in $\vec{W}$, depending as it does upon $ \mathbf{R}(\epsilon_c+\epsilon_a,\delta_s,0)$, adds the effect of direct two photon transitions (Fig.~\ref{fig:SimultaneousContour}(c)).
Figure~\ref{fig:SimultaneousD3} compares this quasi steady state derivative (dashed) from the complete solution (solid). The derivative rises to almost the quasi steady state value and then begins to decrease.
\subsection{Line shapes for Simultaneous Cyclotron and Anomaly Drives}
\label{sec:DisplayedTwoDriveLineshapes}
Quasi steady state line shapes ${dP_3(\epsilon_c,\epsilon_a)}/{dt}$ are illustrated in
Figs.~\ref{fig:twophotoweakdrive} and
\ref{fig:SimultaneousContour} as a function of the anomaly and cyclotron drive frequencies. The vertical scale is the detuning $\epsilon_a$ of the anomaly drive from $\omega_a$, scaled by $\delta_a$. The horizontal scale is the detuning $\epsilon_c$ of the cyclotron drive from $\omega_c$, scaled by $\delta_c$. The contours are for probabilities of making a transition from the initial spin-down ground states to spin-up states.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/TwoPhotonlineshape.pdf}
\caption{The two-drive line shapes change dramatically as a function of the axial damping rate, $\gamma_z$. Contours of the quasi steady state ${dP_3(\epsilon_c,\epsilon_a)}/{dt}$ are shown as a function of the scaled detunings from $\omega_c$ and $\omega_a$ of the cyclotron and anomaly drives. Parameters other than $\gamma_z$ are from Tab.~\ref{table:FrequenciesCompared}.}
\label{fig:twophotoweakdrive}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/TwoPhotonlineshapeContour.pdf}
\includegraphics[width=\the\columnwidth]{Figures/TwoPhotonlineshapeZoomSeparate.pdf}
\caption{(a) The quasi steady state line shape ${dP_3(\epsilon_c,\epsilon_a)}/{dt}$ for the resolved peak corresponding to $n_z=0$ in Fig.~\ref{fig:twophotoweakdrive}f. The the contours shown are at 75\%, 50\% and 25\% amplitudes relative to the peak value. The anomaly and cyclotron drive frequencies are given in terms of the scaled detunings of these frequencies from $\omega_a$ and $\omega_c$. The interior black lines indicate the drive frequencies scanned in Figs.~\ref{fig:SimultaneousCyclotronLineshape} and \ref{fig:SimultaneousAnomalyLineshape}. (b) and (c) shows the contribution from the first and second terms in Eq.~(\ref{eq:twophototransitionB}) respectively. The colors indicate the amplitude relative to the peak value in (a). The dotted line, $\epsilon_c + \epsilon_a = \delta_s/2$, indicates when the cyclotron and anomaly drive frequencies sum to the resonance $\omega_s+\delta_s/2$. }
\label{fig:SimultaneousContour}
\end{figure}
The dependence of the line shapes upon the axial damping rate $\gamma_z$ is illustrated in Fig.~\ref{fig:twophotoweakdrive}. Except for this damping rate, the experimental parameters from Table~\ref{table:FrequenciesCompared} are used. As the axial damping rate is lowered, the contributions from individual axial quantum states become resolved as resolved ``islands'' in Fig.~\ref{fig:twophotoweakdrive}f, for the lowest axial damping realized in the laboratory so far \cite{FanRFSwitch2020} while yet allowing quantum jump spectroscopy.
The narrowest transition peak in Fig.~\ref{fig:twophotoweakdrive}(f), corresponding to $n_z=0$, are potentially the most useful for measuring an electron or positron magnetic moment. Fig.~\ref{fig:SimultaneousContour}(a) shows the contour of
$dP_3(\epsilon_c,\epsilon_a)/dt$. The contours shown are at 75\%, 50\% and 25\% of the peak amplitude. The anomaly and cyclotron drive frequencies are specified as scaled detunings of these frequencies from $\omega_a$ and $\omega_c$. Fig.~\ref{fig:SimultaneousContour}(b) and (c) shows the decomposed contributions from the first and second terms in Eq.~(\ref{eq:twophototransitionB}). The dotted line shows where $\epsilon_c + \epsilon_a = \delta_s/2$, which corresponds to the sum of two drive frequencies being equal to $\omega_s+\delta_s/2$. The sum of (b) and (c) gives the tilted contour in (a).
The drives for this example are weak, with $\Omega_c=\Omega_a=\gamma_c/10$, and the realistic experimental parameters of Table~\ref{table:FrequenciesCompared}) are used. Notice that the peak of the contour slightly deviates from $\epsilon_c + \epsilon_a = \delta_s/2$. The anomaly resonance does not resolve into separate peaks for various $n_z$, and the composite peak is thus shifted from $\epsilon_a=\delta_a/2$.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/SimultanteousCyclotronLineshape_TopAxisAdded.pdf}
\caption{Cyclotron resonance line shape that is a horizontal slice through the maximum spin-flip probability of the contour plot in Fig.~\ref{fig:SimultaneousContour}(a). The quasi steady state line shape (dashed) has the same shape as is numerically calculated (solid) but with a slightly different amplitude.
}
\label{fig:SimultaneousCyclotronLineshape}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\the\columnwidth]{Figures/SimultaneousAnomalyLineshape_TopAxisAdded.pdf}
\caption{Anomaly line shape that is a vertical slice through the maximum spin-flip probability of the contour plot in Fig.~\ref{fig:SimultaneousContour}(a). The quasi steady state line shape (dashed) has the same shape as is numerically calculated (solid) but with a slightly different amplitude.
}
\label{fig:SimultaneousAnomalyLineshape}
\end{figure}
A significant challenge to using two drives is the extremely small transition rates. The maximum rate is for a cyclotron drive detuning $\epsilon_c = \delta_c/2$ (Fig.~\ref{fig:SimultaneousCyclotronLineshape}) and a anomaly detuning of about $\epsilon_a = 0.2\gamma_c=2\delta_a$ (Fig.~\ref{fig:SimultaneousAnomalyLineshape}). Weak drives, with $\Omega_c = \Omega_a = \gamma_c/10$, applied for a time $t = 10 /\gamma_c \approx 53$ s, a time long enough for cyclotron transients to die out, avoids power broadening of the resonance lines. However, the transition probability in time $t$ is then approximately given by $\frac{d P_3}{dt} \times t$. The challenge is that the maximum transition probability is then about $2.5 \times 10^{-5}$ (Fig.~\ref{fig:SimultaneousD3}). This is a factor of $10^3$ times smaller than the peak cyclotron excitation rate for the one-drive case, and is likely too small to be useful. A careful study will thus be required to determine the drive strengths and durations that can be used to get an acceptable rate and broadening. This seems possible, and such transitions have been used in experiments to prepare the desired spin state for measurement \cite{HarvardMagneticMoment2008}. However, numerical solutions of the differential equations will be required since the weak drive limit will not apply. The optimisation, when it is well motivated, would take some time to carry out given the size of the parameter space, even though we have demonstrated that it is feasible with the codes available.
An intriguing possibility is that measurements made using simultaneous cyclotron and anomaly drives might be much less sensitive to magnetic field drifts because an cyclotron excitation and an anomaly transition would both take place before the field could drift much. (Considerable time passed between the measurements of these frequencies in past measurements.) A study of this possibility would also require using the numerical solutions of the differential equations along with a realistic model of linear or quadratic magnetic field drift. Again, although we have demonstrated that this should be possible, it would take considerable time to carry this out.
\section{Prospects for Electron and Positron Magnetic Moment Measurements with Significant Accuracy Improvements}
\label{sec:Comparison}
\subsection{First Possibility}
As discussed in Sec.~\ref{sec:QuantumCyclotron}, two extremely precise frequency measurements must be made to use the quantum cyclotron to determine the electron or positron much more precisely. For all measurements so far, the anomaly and cyclotron frequencies have been measured, and
\begin{equation}
\pm \mu/\mu_B = 1 + \omega_a/\omega_c
\end{equation}
has been used to deduce the magnetic moment in Bohr magnetons. A $3 \times 10^{-14}$ measurement, ten times more precise than existing measurements, requires that the frequency ratio be measured to $3 \times 10^{-11}$.
For a cyclotron frequency measurement at 150 GHz, the largest impediment
to an improved measurement is the detection backaction width $\bar{n}_z \delta_c$ that is of order 30 Hz and $2 \times 10^{-10}$. Fortunately, our proposal to circumvent detector back action \cite{Fan2020BackActionPRL} elaborated in this work provides a way to keep this backaction from contributing to the uncertainty of a new measurement. We showed that the remaining cyclotron line shape is very symmetric, with widths coming from cyclotron damping width $\gamma_c$ and axial decoherence $\bar{n}_z \gamma_z$. In Sec.~\ref{sec:OneDriveExcitations} we saw that these contribution together were about 3 time the cyclotron damping width, a width of 0.1 Hz and fractional width of $6 \times 10^{-13}$. These values are a factor of 50 smaller than is needed for the the contemplated measurement.
With the cyclotron detection backaction circumvented, the anomaly frequency uncertainty becomes the largest challenge. The three linewidth contributions are $\gamma_c$ from cyclotron decay, $\bar{n}_z \gamma_z$ from axial state decoherence, and $\bar{n_z}\delta_a$ from detection backaction, the latter contributing asymmetry to the line shape. These are all comparable in size at about 0.03 Hz which, because the cyclotron frequency is 1000 times smaller than the anomaly frequency, is a much larger fractional uncertainty of about $2\times 10^{-10}$. The desired measurement uncertainty thus seems attainable if the anomaly frequency can be extracted from the resonance line shape with an uncertainty ten times smaller than these contributions to the anomaly line shape. This work thus suggests that a ten times improved measurement should be possible.
\subsection{Second Possibility}
Because the uncertainty in the anomaly frequency now seems to be the largest challenge for an improved measurement, we consider the option of instead determining the magnetic moment from the ratio of the spin and cyclotron frequencies,
\begin{equation}
\pm \mu/\mu_B = \omega_s/\omega_c.
\end{equation}
A direct spin-flip drive (Sec.~\ref{sec:DirectSpinFlips}) or simultaneous cyclotron and anomaly drives (Sec.~\ref{sec:twophotoncalculation}) to determine $\omega_s$. The daunting challenge is that this frequency ratio must then be determined to the desired precision in the electron and positron magnetic moment of $3 \times 10^{14}$, a factor of 1000 better than for possibility one above.
For the cyclotron linewidth of of 0.1 Hz and fractional linewidth of $6 \times 10^{-13}$ noted above, the cyclotron frequency would need to be extracted to a precision that was at least 30 times narrower than the anticipated linewidth. This may not be an unreasonable linesplitting given that the line shape should be symmetric about the cyclotron frequency once the the detection backaction is circumvented.
Directly driving spin flips to determine the spin frequency to the same precision would also be required, the first time that this would be realized with a quantum cyclotron. A two photon cyclotron plus anomaly transition would be an alternative. As for the cyclotron line shape, detection backaction that would make the resonance line shape broad and asymmetric can be circumvented. Because the two spin states are effectively stable, there would be no contributions to the line width from decay of an unstable state though the axial decoherence width $\bar{n}_z \gamma_z$ would persist. This is an alternative route to a new measurement, in principle.
\subsection{Magnetic Field Instability}
The quantum calculations support the viability of both of the measurement possibilities outlined above. For the immediate future, however, measurements will almost certainly rely upon the first possibility -- measuring an electron or positron's anomaly and cyclotron frequencies. The reason is that the magnetic field produced by the best of superconducting solenoids drifts in time. The demonstrated drift rates (about 1 part in $10^{10}$ per hour \cite{Helium3NMR2019}) is slow enough to make it possible to alternate determinations of the anomaly and cyclotron frequencies rapidly enough to make a new measurement. This source of systematic uncertainty had to be carefully managed already in past measurements \cite{HarvardMagneticMoment2011}.
To obtain the same precision using the second measurement possibility, alternating instead measurements of the spin and cyclotron frequencies, requires measuring these frequencies 1000 times more rapidly or producing a much more stable magnetic field. The source of laboratory magnetic field instability and its reduction, whether by better solenoid design or shielding against changes in magnetic flux, is an interesting and important topic but it is beyond the scope of this calculation.
\section{Summary and Conclusions}
\label{sec:summary}
A quantum calculation is carried out for a driven one-electron quantum cyclotron with a quantum nondemolition (QND) coupling to a harmonic detection motion. The quantum spin and cyclotron motions have a QND coupling to a quantum axial detection motion, which in turn is coupled to a thermal reservoir. External drives are applied to produce one-quantum transitions between the lowest spin and cyclotron states.
A master equation is used to describe the driven motion of this open quantum system. Convenient steady state solutions and resonance lineshapes for weak drives are presented, illustrated and discussed. Numerical solutions reveal the time evolution and check the steady state line shapes. Calculations of driven cyclotron excitations and driven anomaly transitions are presented, along with calculations for directly driven spin flips and spin flips driven by simultaneous anomaly and cyclotron drives. For a next generation of measurements, the first two of these four drive options turn out to be the most promising. For weakly driven cyclotron and spin excitations, the predicted steady-state lineshapes for experimental parameters that have recently become accessible, are very different than the Brownian motion prediction used to interpret past measurements.
An exciting result is the emergence of extremely narrow quantum resonances that appear within the cyclotron resonance line, corresponding to resolved quantum states of the axial detection oscillator. These symmetric lines are about 100 time narrower than the broad and asymmetric cyclotron line shape that has been the biggest obstacle to a new generation of magnetic moment measurements. Resolving these narrow peaks circumvents the detection backaction that would otherwise cause broad and asymmetric cyclotron resonance lines, reducing it to what is caused by only the zero-point motion of the detection motion, even many more detection states are populated. The circumvention opens the way to the much more precise measurements of the cyclotron frequency that are needed to determine the electron and positron magnetic moments.
Given the new method to measure the cyclotron frequency extremely accuracy, measuring the anomaly frequency precisely will become the biggest challenge to more precise magnetic moment measurements. The anomaly line shape cannot be resolved into narrow symmetric peaks that correspond to individual quantum states of the axial detection motion. Nonetheless, the calculations suggest that an anomaly line shape can be produced that will make possible measurements that are perhaps an order of magnitude more precise. An initial survey of the effect of changing experimental parameters (e.g.\ cyclotron damping rate and lower ambient temperature) upon the anomaly lineshape identifies possible future upgrade paths, though none of these by itself is a large step.
The electron and positron magnetic moments are the most precise predictions of the Standard Model of Particle Physics -- the fundamental mathematical description of physical reality.
Whether the current discrepancy between the measured electron magnetic moment and the Standard Model prediction is a hint of physics beyond the Standard Model
is not yet known, but it warrants investigation. The calculation and methods in this work indicate how this may be possible.
\bigskip
\section{acknowledgements}
A preliminary version of some of this work is in a thesis \cite{ThesisDUrso}. This work was supported by the NSF, with X.\ Fan partially supported by the Masason Foundation. B.\ D'Urso, S.\ E.\ Fayer, T.\ G.\ Myers, B.\ A.\ D.\ Sukra and G.\ Nahal provided useful comments.
\bibliographystyle{prsty_gg}
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Latest & Upcoming Games Reviews | Gamesofficial.com
Nintendo to launch new handheld gaming device
Game manufacturing giant Nintendo is likely to launch a new portable gaming device in 2016. The information came from a statement from the IHS senior principal analyst Hiroshi Hayase.
The upcoming console is expected to be the first out of the two next generation devices planned by the company. Hayase at the HIS display forum held in Tokyo said, "We expect a small recovery in shipments of flat-panel displays for game devices because of Nintendo's new game hardware expected to be released in 2016".
However, the analyst could not confirm about the size of the console but has claimed that the product would be a standalone device and will be different from the Wii U GamePad that has a screen of its own but requires a Wii U to function. The company has refused to comment anything on the development but have assured that there will be more news about the NX later in 2016.
While Nintendo has not provided a timeline for the launch, the analysts have expected that the new device would be unveiled at E3 in June 2016. Nintendo now needs to launch something out of the box so as to keep up with the competitors like PS4 and the Xbox One.
Photo Credits: bgr
handheld Nintendo
Nintendo handheld game
Nintendo new handheld device
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{"url":"https:\/\/zbmath.org\/?q=an:0635.62035","text":"# zbMATH \u2014 the first resource for mathematics\n\nSome families of multivariate symmetric distributions related to exponential distribution. (English) Zbl\u00a00635.62035\nThis paper introduces a family of multivariate symmetric distributions, which includes the one with i.i.d. exponential components as its special member. This family, denoted by $$F_ n$$, is defined as scale mixtures of the uniform distribution on the surface of the $$l_ 1$$ unit sphere and studied from several aspects such as distribution functions, probability density functions, marginal and conditional distributions and components\u2019 independence.\nA more general family $$T_ n$$ in which the survival functions are functions in $$l_ 1$$ norm and an important subset $$D_{n,\\infty}$$ of scale mixtures of random vectors with i.i.d. exponential components are also discussed. The relationships among these three families and some applications are given.\n\n##### MSC:\n 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H10 Multivariate distribution of statistics\nFull Text:\n##### References:\n [1] Block, H.W, Multivariate exponential distribution, (), 55-59 [2] Cambanis, S; Huang, S; Simons, G, On the theory of elliptically contoured distributions, J. multivariate anal., 11, 368-385, (1981) \u00b7 Zbl\u00a00469.60019 [3] Cambanis, S; Keener, R; Simons, G, On \u03b1-symmetric multivariate distributions, J. multivariate anal., 13, 213-233, (1983) \u00b7 Zbl\u00a00513.60024 [4] Chemielewski, M.A, Elliptically symmetric distributions: review and bibliography, Internat. statis. rev., 49, 67-74, (1981) \u00b7 Zbl\u00a00467.62047 [5] Eaton, M.L, On the projections of isotropic distributions, Ann. statist., 9, 391-400, (1981) \u00b7 Zbl\u00a00463.62016 [6] Edwards, J, () [7] Feller, W, () [8] Gupta, R.D; Richards, D.St.P, Multivariate Liouville distributions, J. multivariate anal., 23, 233-256, (1987) \u00b7 Zbl\u00a00636.62038 [9] Jensen, D.R, Multivariate distributions, (), 43-55 [10] Johnson, N.L; Kotz, S, () [11] Marshall, A.W; Olkin, I, Multivariate exponential distributions, Marshall-Olkin, (), 59-62 [12] Ressel, P, De Finetti-type theorems: an analytical approach, Ann. probability, 13, 898-922, (1985) \u00b7 Zbl\u00a00579.60012 [13] Williamson, R.E, Multiply monotone functions and their Laplace transforms, Duke math. J., 23, 189-207, (1956) \u00b7 Zbl\u00a00070.28501\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-12-03 13:10:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7966848015785217, \"perplexity\": 5454.352861695522}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964362879.45\/warc\/CC-MAIN-20211203121459-20211203151459-00299.warc.gz\"}"}
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Home » Photogallery » Zero Box Office Collections: This is where Shah Rukh Khan's latest film fits in his all-time top 10 list
Zero Box Office Collections: This is where Shah Rukh Khan's latest film fits in his all-time top 10 list
Dec 24, 2018, 23:39 PM IST
Zero Box Office Collections: The expectation of the audience for a Shah Rukh Khan movie is quite high, as over the years, he has created an aura around himself with his acting prowess. Most believe that his movies can't fail. The success he has had has been massive indeed! However, Zero, which was released last week, has failed to pull the audience to theaters in the numbers that were initially expected, but even then, the movie has shot into Shah Rukh Khan's top 10 list.
Zero opening weekend collections has been outed and it came in at Rs 59.07 crore, far below the earnings of his other top films -- Happy New Year, Chennai Express, and Dilwale. This figure slots Zero collection into in 5th place in terms of opening weekend collections among Shah Rukh Khan's last 10 films. Check them out here:
1. Happy New Year box opening weekend collection: Shah Rukh Khan's musical heist action comedy film, released in 2014, earned Rs 104 crore by the opening weekend. The film directed by Farah Khan, had star cast include Deepika Padukone, Abhishek Bachchan, Sonu Sood, Boman Irani, Vivaan Shah and Jackie Shroff.
2. Chennai Express opening weekend collection: This action comedy film, released in 2013, earned Rs 93.66 crore by the opening weekend. The movie was directed by Rohit Shetty the film had Shah Rukh Khan and Deepika Padukone in the lead role.
3. Dilwale opening weekend collection: This Shah Rukh Khan romantic action film was released in 2015 and it earned Rs 64.09 crore in the opening weekend. The movie directed by Rohit Shetty had star cast include Shah Rukh Khan, Kajol, Varun Dhawan and Kriti Sanon in lead roles. Johnny Lever and Varun Sharma were in supporting roles.
4. Raees opening weekend collection: A Rahul Dholakia directorial movie, it was released in 2017 andearned Rs 59.83 crore in the opening weekend. The film had cast stars Shah Rukh Khan, Mahira Khan and Nawazuddin Siddiqui in the lead roles.
5. Zero box office collection: The film's opening weekend collections was at Rs 59.07 crore, which is not even close to the earnings of Happy New Year, Chennai Express, and Dilwale.
Ra.One
6. Ra.One opening weekend collection: An Anubhav Sinha-directed superhero film, released in 2011, it had collected Rs. 55 crore in the opening weekend. The film cast actors include Kareena Kapoor, Shah Rukh Khan, and Arjun Rampal in leading roles.
7. Fan opening weekend collection: This was a Maneesh Sharma directed action thriller film and it was released in 2016. It collected Rs. 52.35 crore in the opening weekend. In the film Shah Rukh Khan plays double roles of film star Aryan Khanna and obsessive fan Gaurav Chandna.
8. Jab Tak Hai Jaan opening weekend collection: A Yash Chopra directed romantic drama film,released in 2012, had earned Rs 49.22 crore in the opening weekend. The film starred actors Shah Rukh Khan, Katrina Kaif and Anushka Sharma in lead roles.
9. Don 2 opening weekend collection: This was a Farhan Akhtar directed film and it released in 2011. It earned Rs 48.39 crore in the opening weekend. The film stars Shah Rukh Khan, Priyanka Chopra, Lara Dutta, Om Puri, Boman Irani and Kunal Kapoor. The film is the sequel to the Don, which was released in 2006.
10. Jab Harry Met Sejal opening weekend collection: Another romantic comedy film, it was released in 2017, and it could manage to earn Rs. 45.75 crore in the opening weekend. The film, directed by Imtiaz Ali, featured Anushka Sharma and Shah Rukh Khan in the lead roles.
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Q: Soundness and Completeness for a single Model Only? Question modified to hopefully answer the questions (I'm a physicist to all might not be mathematically watertight)
In Enderton "A Mathematical Introduction to Logic", logical Implication is defined as:
Logical Implication $\models$ : Let $\Gamma$ be a set of wffs (well formed functions), $\varphi$ a wff. Then $\Gamma$ logically implies $\varphi$ iff for every structure X for the language and every satisfaction function s:V $\mapsto $ |X| such that X satisfies every member of $\Gamma$ with s, X also satisfies $\varphi$ with s.
In Enderton a model is defined as :
X is a model of $\varphi$ iff X satisfies $\varphi$ with every function s:V $\mapsto $ |X|. X is a model of a set $\Sigma$ of sentences iff it is a model of every member of $\Sigma$, written as $X \models \varphi$ (Enderton uses $\models_X \varphi$, to clarify that $\models_X$ isnt the Logical implication symbol $\models$).
As a result logical implication $\models$ includes all models and not just one.
Enderton then proves Soundness and Completeness between $\models$ and $\vdash$:
$$ \Gamma \models \varphi \iff \Gamma \vdash \varphi \tag{1}$$
Question : As the Forcing Truth Lemma (taken from Kunen) looks to be effectively a form of (1) but only applies to one model (G being like a satisfaction function):
$$ \forall G \; [ M[G] \models \varphi[G] \iff \exists p \in G (p \Vdash \varphi)^M] \tag{2}$$
are there general techniques for changing $\vdash$ / $\Vdash$ into a similar model specific deductive system (call it MSDS) to make Completeness and Soundness hold for a single model?
$$ \forall s (X,\Gamma[s] \models \varphi \iff \Gamma[s] \; MSDS \; \varphi) \tag{3}$$
or maybe start with (3) and see what properties MSDS needs (and maybe slightly modify the satisfaction functions used) ? If this could be done then MSDS would only deduce what is true in the specific Model and be incapable of deducing anything else.
A: I'm not totally sure I understand your question, but I think I'm able to make some helpful comments.
First, let's consider $\Gamma\models_X \varphi$ when $\Gamma$ and $\varphi$ have no free variables. I'll assume that the definition of this relation is: If $X\models \Gamma$, then $X\models \varphi$.
We can indeed modify $\vdash$ to a relation $\vdash_X$ so that $\Gamma\models_X \varphi$ iff $\Gamma\vdash_X \varphi$, when $\Gamma$ and $\varphi$ have no free variables. Just use the original proof system used to define $\vdash$, but add every sentence true in $X$ as an axiom!
This might not be so satisfactory, since we'd like to be able to recognize proofs, and there is no general algorithm for determining which sentences are true in an arbitrary structure $X$. If the complete theory of $X$ is computably (recursively) axiomatizable, then it suffices to add the sentences in the computable axiomatization of $X$ as axioms to our proof system, and the result is a computable proof system defining $\vdash_X$. On the other hand, there are structures, like $(\mathbb{N},+,\times)$, whose theories are not computably enumerable. For such structures, there is no hope to have a computable proof system defining $\vdash_X$, since we could computably enumerate proofs $\vdash_X \varphi$ in such a system, and thus enumerate $\mathrm{Th}(\mathbb{N},+,\times)$.
Ok, what about the case of free variables? When $\Gamma$ and $\varphi$ have free variables, I'll assume the definition of $\Gamma\models_X \varphi$ is: For all assignments $s$ from the free variables of $\Gamma$ and $\varphi$ to $X$, if $X\models \Gamma[s]$, then $X\models \varphi[s]$.
In this case, simply adding the sentences true in $X$ as axioms is not sufficient. This amounts to restricting attention to all models of $\mathrm{Th}(X)$, i.e., to all structures elementarily equivalent to $X$. This strategy works to understand entailment between sets of sentences, which will be the same for all structures elementarily equivalent to $X$. But different structures elementarily equivalent to $X$ might differ on entailments between sets of formulas with free variables.
As an example, suppose the language consists of infinitely many constant symbols $\{c_n\mid n\in \mathbb{N}\}$. Let $X$ have underlying set $\mathbb{N}$, with $c_n$ interpreted as $n$. Let $\Gamma = \{x\neq c_n\mid n>0\}$. Then $\Gamma\models_X (x = c_0)$, since $0$ is the only element of $X$ which is not equal to any $c_n$ with $n>0$. Of course, in an elementary extension $X\prec X'$, $X'$ has elements which are not equal to $c_n$ for any $n$, so $\Gamma\not\models_{X'}(x = c_0)$. So using the definition of $\vdash_X$ proposed above, we do not have $\Gamma\vdash_X (x = c_0)$.
This example also shows that $\models_X$ is not compact, and hence there is no finitary proof system giving $\Gamma\models_X\varphi$ iff $\Gamma\vdash_X \varphi$. Explicitly: there is no finite subset $\Delta\subset_{\mathrm{fin}} \Gamma$ such that $\Delta\models_X(x = c_0)$. In fact, there is no proper subset $\Delta\subsetneq \Gamma$ such that $\Delta\models_x(x = c_0)$. A proof of $(x = c_0)$ from $\Gamma$ must use every formula in $\Gamma$, and hence must be infinite.
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\section{Introduction}
Rare gases (RG) are important laser media in the vacuum ultraviolet (VUV)
spectral region \cite{Hutchinson, McCusker}, and this is one of the reasons
for the undying interest in these chemically inactive species. Normally, they
are almost atomic gases, because electronically unexcited RG dimers are only
weakly bound (except radioactive Rn$_{2}$ with dissociation energy of about
400 K \cite{Runeberg}). In contrast to this, the electronically excited RG
dimers (excimers) in their lowest states 1$_{u}$ and 0$_{u}^{+}$ are strongly
bound. These excimers can be created with the help of various excitation
sources (charged particle beams, synchrotron radiation, etc.), and their decay
produces photons in the VUV region, forming two intense continua observed long
ago for all RGs (except Rn) \cite{Ar-1972, He-1973, Ne-1973, Kr-1974, Xe-1976}
(see also \cite{Becker-2002, Ulrich-2004} for the recent Ne and He data).
A vast literature exists on the RG dimers in their ground electronic state
\cite{Runeberg, Aziz-90, Aziz-93, Woon-94, Aziz-97, Cybulski, Faas, Tang,
Slavicek}, because they are good prototypes for studying van der Waals
interactions. At the same time, there is still a lack of knowledge about the
RG excimer states, which are of crucial importance for the development of
efficient light sources in the VUV region \cite{Ulrich-2004, Wieser-97,
APL-98, PP-2000, APL-2001, PRA-2003}. Of special interest in this context are
the heavier excimers for which the laser emission has been reported
\cite{Hutchinson, McCusker}. The observed laser effect is related to the
so-called second continua, broad structureless emission bands with maxima at
126, 146 and 172 nm for Ar$_{2}^{\ast}$, Kr$_{2}^{\ast}$, and Xe$_{2}^{\ast}$,
respectively. They correspond to transitions from the vibrationally relaxed
1$_{u}$ and 0$_{u}^{+}$ states to the repulsive part of the ground electronic
state 0$_{g}^{+}$, in contrast to the so-called first continua originating
from the highest vibrational levels of the 0$_{u}^{+}$ excimer state, and
therefore being spectrally close to the atomic $^{3}$P$_{1}\rightarrow$ $^{1}%
$S$_{0}$ resonance lines. The first continuum is related to the excimers'
classical outer turning point region, while the weak oscillatory emission from
the region of their inner turning points has also been observed under
selective synchrotron radiation excitation \cite{Moeller}.
All these important spectroscopic features can be uniformly treated
\cite{Selg-2003, Selg-2007}, but to this end, one needs reliable potential
energy curves (PECs) and coordinate dependencies of the related transition
moments, as well as a reasonable model to describe vibrational relaxation of
the excimers. In this paper, we concentrate on the first two issues, while the
relaxation dynamics and the time-dependent emission spectra will be studied in
a separate paper \cite{Selg-Future}. Valuable information on the energetic
position, absolute numbering and radiative lifetimes of the higher vibrational
levels for all heavier RG excimers has been obtained in a series of supersonic
jet expansion experiments \cite{Stoicheff-Xe, Stoicheff-Kr, Stoicheff-Ar,
Madej}, using tunable VUV radiation to produce high-resolution fluorescence
excitation spectra. Unfortunately, these high-quality reference data concern
only a narrow range of vibrational levels near the dissociation limit, and one
cannot construct reliable PECs for RG excimers solely on this basis. Various
\textit{ab initio} and semi-empirical potentials are available
\cite{Castex-Xe, Castex-Ar, Gadea, Yates, Messing, Audouard, Jonin}, which
satisfactorily reproduce some of the characteristic spectroscopic features,
but not the whole variety of available experimental data. Apart from
imperfections of the PECs of the excimer states, the discrepancies between
theoretical and experimental spectra may be caused by inaccuracies of the
ground state potentials in the short distance range.
Thus, the spectroscopic properties of RG dimers certainly need further
investigation. This is the motivation for the present study, the main goal of
which is to construct accurate reference potentials of relatively simple
analytic form, by directly fitting their parameters to the observed data, and
taking account of the relevant theoretical considerations. In principle, the
approach that will be described below can be applied to any RG dimers, since
they have much in common. For several reasons, however, only Ar$_{2}$
molecules are under examination in this paper. First, compared with Xe and Kr,
the effects of spin-orbit coupling are much less pronounced in Ar, which makes
the theoretical analysis more simple and, hopefully, more adequate. Second, a
sufficient amount of good-quality experimental data are available for Ar gas,
which, again, is an advantage compared with Xe and Kr. Third, there is a
practical point of special interest in Ar$_{2}^{\ast}$ excimers, as they produce
the highest energy photons (9.8 eV) for the potential laser applications, which
is the field of great interest and permanent development \cite{Neeser, Tanaka}.
It therefore seems reasonable to test the method on Ar$_{2}$ molecules, leaving
more complex cases for the future analysis.
The reference potential approach to different spectral problems has been
described in detail elsewhere (see, e.g., \cite{Selg-2003, Selg-2006} and
references therein). In this paper, we are going to construct smooth
multi-component exactly solvable reference potentials for the lowest excimer
states 1$_{u}$ and 0$_{u}^{+}$, and for the ground electronic state 0$_{g}%
^{+}$ of the Ar$_{2}$ molecule. These approximate PECs should be realistic in
a wide range of nuclear separations related to the observed spectroscopic
features. To achieve this goal, fully quantum mechanical calculations of
bound-bound and bound-free Franck-Condon factors for 1$_{u}$ $\rightarrow$
0$_{g}^{+}$ and 0$_{u}^{+}$ $\rightarrow$ 0$_{g}^{+}$ transitions have been
performed. The procedure has to be repeated many times, varying the parameters
of the PECs and the transition moments. In this way, step by step, the
calculated energy levels and their radiative lifetimes, as well as the
intensity patterns of Franck-Condon factors, can be fitted to the available
experimental data.
The paper is organized as follows. In Section II, a short overview of the
theoretical background of the method is given, specific properties of the
reference potentials for the 0$_{g}^{+}$, 0$_{u}^{+}$ and 1$_{u}$ electronic
states of Ar$_{2}$ are described (in sub-sections A, B, and C, respectively),
and the details of the fitting procedure are explained. Section III is devoted
to demonstrating the calculated Franck-Condon factors, transition
probabilities and radiative lifetimes for the vibrational levels of the
0$_{u}^{+}$ and 1$_{u}$ states. Finally, a brief discussion of the obtained
results and future prospects of the research are given in Section IV, which
concludes the paper.
\section{Exactly solvable multi-component reference potentials}
At first sight, the reference potential approach we are going to adopt may
seem naive and unjustified. However, the solution to a problem always
depends on the context. Modern spectroscopic analysis often involves many
thousand items of high resolution input data, but unfortunately, such
detailed information is not yet available neither for the ground state of Ar$%
_{2}$ molecule nor for the excimer states 1$_{u}$ and 0$_{u}^{+}$. Indeed,
the scattering data for these species are almost lacking. Moreover, even the
total number of the bound states is not conclusively established, and only a
narrow range of discrete levels has been accurately ascertained for the excimer
states \cite{Stoicheff-Ar}, while their absolute numbering still needs to be
confirmed. Under these circumstances, there is no chance to deduce the "real"
PECs directly from the experimental data. One can only take some steps towards
this goal by combining the available experimental data with the relevant
theoretical results. In this context, the simple but mathematically rigorous
approach in question has proven to be quite useful.
To begin with the analysis, let us recall another important peculiarity of
the system under study. Namely, for all RGs, there is a big difference
(about 1.4 \AA\ in the case of Ar$_{2}$)\ between the minima of the tiny
potential well of the ground state and those for the excimer states. It
means that bound-free transitions to the repulsive wall of the ground state
PEC are responsible for the major part of the observed fluorescence spectra.
Of course, as we demonstrate, bound-bound transitions are also very
important to reveal the details of the first emission continuum.
Unfortunately, this fine structure, which should contain tens of spectral
lines in the range of less than 0.1 eV width (compared with more than 4.5 eV
width of the overall spectrum), has not yet been detected experimentally.
Moreover, the spectral resolution in fluorescence experiments on Ar and other
RGs is typically $\sim $10 meV. Therefore, at the present level of our
knowledge, one should not put too much effort into deducing the PECs that would
reproduce the observed (relatively few) level positions with the utmost
accuracy. Formally, one can always achieve this specific goal, but this does not
guarantee the reliability of the potentials in the major part of the domain.
Indeed, as is well known, even the full knowledge of the discrete energy
spectrum combined with the full scattering information is insufficient to
uniquely determine the potential \cite{Chadan}.
In view of the above, we set the following criteria to the quality of the
reference potentials for the Ar$_{2}^{\ast }$ excimer states: 1) these PECs
should reproduce the observed bound-bound transition energies \cite%
{Stoicheff-Ar} with $\sim $1 meV accuracy; 2) they should reproduce all
details of the fluorescence spectra in the range from about 7 to 11.63 eV
with $\sim $10 meV accuracy. To achieve these goals, we use a fitting method
which is based on repeated and accurate calculation of the Franck-Condon
factors. Naturally, this strictly quantum mechanical procedure presumes
highly efficient solution of the Schr\"{o}dinger equation, because this
elementary act has to be performed many million times within a reasonable
time scale. For this reason, we try to adopt an analytic approach of calculating
the energy eigenfunctions, which indirectly means that we have to construct
exactly solvable approximants (reference potentials) to the "real" PECs. If one
sets a "physical constraint", requiring continuity of the potential and its
first derivative in the whole physical domain, the number of suitable options
becomes quite limited. Certainly one of the best choices is to compose a
reference potential of several Morse-type \cite{Morse} pieces%
\begin{equation}
V(r)=V_{k}+D_{k}\left[ \exp(-\alpha_{k}(r-R_{k}))-1\right] ^{2},\qquad
r\in(0,\infty),
\end{equation}
where $V_{k},$ $D_{k},$ $\alpha_{k}$ and $R_{k}$ are real (not definitely
positive!) constants, and the subscript $k=0,1,2...$labels different
components smoothly joined at the boundary points $X_{k+1}$.
As is well known, the classical Morse potential belongs to the family of shape
invariant potentials \cite{SUSY}, and its energy eigenvalue problem can be
solved with the help of solely algebraic techniques. The shape invariance is
lost, if there are several analytically different components, as assumed in
Eq. (1). Consequently, the discrete energy levels of such multi-component
potentials cannot be given in an explicit analytic form. With some concession,
we can still preserve the term "exactly solvable" for this kind of piece-wise
potentials, because the two linearly independent solutions of the related
Schr\"{o}dinger equation can be always found analytically to any desired
accuracy. Indeed, introducing a dimensionless variable $y_{k}\equiv2a_{k}%
\exp(-\alpha_{k}(r-R_{k})),$ the Schr\"{o}dinger equation for a Morse-type PEC
can be converted into a confluent hypergeometric form \cite{Bateman}%
\begin{equation}
y_{k}\frac{d^{2}G(a_{k},\mu_{k};y_{k})}{dy_{k}^{2}}+(2\mu_{k}+1-y_{k}%
)\frac{dG(a_{k},\mu_{k};y_{k})}{dy_{k}}+(a_{k}-\mu_{k}-1/2)G(a_{k},\mu
_{k};y_{k})=0.
\end{equation}
Here $a_{k}\equiv\sqrt{D_{k}/C}/\alpha_{k},C\equiv\frac{\hbar^{2}}{2m}$ is a
characteristic constant ($C=0.10464$ meV$\cdot$\AA $^{2}$ for Ar$_{2}$),
$\mu_{k}\equiv\frac{\sqrt{(V_{k}+D_{k}-E)/C}}{\alpha_{k}},$ and the solutions
of the Schr\"{o}dinger equation read (up to normalization) $\Psi=\exp
(-y_{k}/2)y_{k}^{\mu_{k}}G(a_{k},\mu_{k};y_{k}).$ Note that the parameters
$a_{k},\mu_{k}$ and the coordinate $y_{k}$ may be imaginary, if $D_{k}<0$ or
the total energy $E>V_{k}+D_{k}.$
There are several possibilities to construct the fundamental system of
solutions for Eq. (2) (see \cite{Bateman}, Chapter 6, for a thorough
overview). For example, one can make use of the special solutions%
\begin{equation}
G_{1}=\Phi(-a_{k}+\mu_{k}+1/2,2\mu_{k}+1;y_{k}),\text{ }G_{2}=y_{k}^{-2\mu
_{k}}\Phi(-a_{k}-\mu_{k}+1/2,-2\mu_{k}+1;y_{k}),
\end{equation}
where the symbols%
\begin{equation}
\Phi(a,c;x)\equiv1+\frac{a\cdot x}{1!\cdot c}+\frac{a(a+1)\cdot x^{2}}{2!\cdot
c(c+1)}+...
\end{equation}
denote confluent hypergeometric functions. Correspondingly, the two linearly
independent solutions of the original Schr\"{o}dinger equation read%
\begin{equation}
\Psi_{k1}=y_{k}^{\mu_{k}}S(a_{k},\mu_{k};y_{k}),\text{ }\Psi_{k2}=y_{k}%
^{-\mu_{k}}S(a_{k},-\mu_{k};y_{k}),
\end{equation}
where we have introduced another very useful function%
\begin{equation}
S(a,c;x)\equiv\exp(-x/2)\Phi(-a+c+1/2,2c+1;x),
\end{equation}
which can be evaluated as follows \cite{Tricomi}:%
\begin{gather}
S(a,c;x)=\sum_{n=0}^{\infty}B_{n},\text{ }B_{0}=1,\text{ }B_{1}=-\frac
{ax}{2c+1},\\
B_{n}=\frac{x}{n(2c+n)}(-aB_{n-1}+\frac{x}{4}B_{n-2}),\text{ }%
n=2,3,...\nonumber
\end{gather}
From Eqs. (5) to (7) one can infer that $\Psi_{k1}$ and $\Psi_{k2}$\ are
always complex conjugates, if $E>V_{k}+D_{k}.$
\subsection{Four-component reference potential for the ground state of
Ar$_{2}$}
After the brief theoretical introduction, let us try to put the ideas into
practice. Our fitting procedure does not actually involve the parameters of
the ground state, but only those of the excimer states 1$_{u}$ and 0$_{u}%
^{+}.$ In other words, the PEC for the ground state is assumed to be fixed.
Nevertheless, in order to solve the whole quantum mechanical problem exactly, we
have to construct an exactly solvable (in the above-mentioned sense) reference
potential for the ground state as well. Of course, such a constraint is
technical rather than conceptual. The Schr\"{o}dinger equation can be always
solved numerically, but in the present context the main problem is how to fix
a sufficiently realistic PEC. The shape of the ground state potential for
Ar$_{2}$\ is accurately known in the intermediate and long-distance range, but
the knowledge about its repulsive short-distance part is rather ambiguous. As
mentioned, this region is very important to interpret the observed
spectroscopic features, thus our aim is to describe the repulsive wall as
adequately as possible.
The resulting reference PEC for Ar$_{2}$\ is shown in Fig. 1, and its
parameters are given in Table 1. All four components have the simple analytic
form of Eq. (1), and their parameters have been ascertained from the least
squares fit to the \textit{ab initio} CCSD(T) daug-cc-pV5Z-33211 potential by
Fern\'{a}ndez and Koch \cite{Fernandez}. This particular PEC has been
preferred, because this is the only work to date where both the ground state
and the excimer state (0$_{u}^{+}$) potentials have been calculated on the
same theoretical basis. These PECs nicely reproduce the observed spectra
related to the excimers' inner turning point region \cite{Moeller}, which is a
strong argument for their reliability at short distances. As can be seen in
Figs. 1 and 2, the ground state PEC by Fern\'{a}ndez and Koch is in good
agreement with the other available \textit{ab initio} potentials
\cite{Slavicek, Barker}. The reference potential for Ar$_{2}$ is almost
indistinguishable from the original PEC on the scale used in these figures.
Therefore, it is expected to be reliable enough for our purposes.
Now, let us have a look at the structure of this reference PEC. The two most
internal components ($r\leq X_{1}$ and $r\in\lbrack X_{1},X_{2}]$,
respectively) represent the so-called pseudo-Morse (PM) potentials
\cite{Selg-2003, Selg-2006} smoothly joined at the boundary point $X_{1}$.
Their important peculiarity is that the parameters $D_{k}$ and $\alpha_{k}$
($k=0,1$) are not independent, but $D_{k}=\frac{1}{4}C\alpha_{k}^{2}$. It
means that the tiny potential well is just of the limit depth to entirely lose
the discrete energy spectrum. Since $a_{k}=1/2$, and the pseudo-Morse
approximation is used only in the region where $E>V_{k}+D_{k},$ the two
special solutions $\Psi_{k1}$ and $\Psi_{k2}$ of the Schr\"{o}dinger equation
are complex conjugates. According to Eqs. (5) to (7),%
\begin{equation}
\Psi_{k1}=y_{k}^{i\beta_{k}}S(1/2,i\beta_{k};y_{k})=A_{k}(y_{k})e^{iB_{k}%
(y_{k})}e^{-i\alpha_{k}\beta_{k}(r-r_{k})},
\end{equation}
where
\begin{gather}
A_{k}(y_{k})e^{iB_{k}(y_{k})}\equiv1-\frac{y_{k}/4}{i\beta_{k}+1/2}\\
+\frac{\left( y_{k}/4\right) ^{2}}{\left( i\beta_{k}+1/2\right) 1!}\left(
1-\frac{y_{k}/4}{i\beta_{k}+3/2}\right) +\frac{\left( y_{k}/4\right) ^{4}%
}{\left( i\beta_{k}+1/2\right) \left( i\beta_{k}+3/2\right) 2!}\left(
1-\frac{y_{k}/4}{i\beta_{k}+5/2}\right) +....,\nonumber
\end{gather}
and, consequently, the general solution reads%
\begin{equation}
\Psi_{k}(r)=N_{k}A_{k}(y_{k})\cos\left[ B_{k}(y_{k})+\varphi_{k}-\alpha
_{k}\beta_{k}r\right] ,\text{ }k=0,1.
\end{equation}
Here $\beta_{k}\equiv\left\vert \mu_{k}\right\vert ,$ while the normalization
factor $N_{k}$ and the phase constant $\varphi_{k}$ should be determined from
the continuity requirements of the wave function and its derivative.
The phase constant $\varphi_{0}$ for the most internal PM component can be
always (in most cases quite easily) ascertained analytically \cite{Selg-2006},
taking account of the physical boundary condition $\Psi\rightarrow0$ as
$r\rightarrow0$. The next phase constant $\varphi_{1}$ can then be determined
from the boundary condition $\frac{\Psi_{0}^{\prime}(X_{1})}{\Psi_{0}(X_{1}%
)}=\frac{\Psi_{1}^{\prime}(X_{1})}{\Psi_{1}(X_{1})}.$ In all cases analyzed in
this paper the wave function's logarithmic derivative can be expressed in the
form%
\begin{equation}
\frac{\Psi_{k}^{\prime}(r)}{\Psi_{k}(r)}\equiv\alpha_{k}\left[ R_{k}%
(r)+\frac{S_{k}(r)\tan\varphi_{k}+T_{k}(r)}{U_{k}(r)\tan\varphi_{k}+W_{k}%
(r)}\right]
\end{equation}
where $R_{k},S_{k},T_{k},U_{k}$ and $W_{k}$ are some characteristic functions
that do not depend on phase constants. Thus, if $\varphi_{0}$ is known,
$\varphi_{1}$ can be easily determined.
The region $r\in\lbrack X_{2},X_{3}]$ (including the minimum of the PEC) is
approximated by an ordinary Morse (OM) potential ($k=2$). Again, since the
energy range $E<V_{2}+D_{2}$ is out of interest, the special solutions
$\Psi_{21}$ and $\Psi_{22}$ are complex conjugates, and the general solution
of the Schr\"{o}dinger equation becomes%
\begin{equation}
\Psi_{2}(r)=N_{2}\left[ C_{2}(r)\tan\varphi_{2}+D_{2}(r)\right] ,
\end{equation}
where%
\begin{align*}
C_{2}(r) & \equiv\operatorname{Re}\left[ S(a_{2},i\beta_{2};y_{2}\right]
\sin(k_{2}r)-\operatorname{Im}\left[ S(a_{2},i\beta_{2};y_{2}\right]
\cos(k_{2}r),\\
D_{2}(r) & \equiv\operatorname{Re}\left[ S(a_{2},i\beta_{2};y_{2}\right]
\cos(k_{2}r)+\operatorname{Im}\left[ S(a_{2},i\beta_{2};y_{2}\right]
\sin(k_{2}r),
\end{align*}
$k_{2}\equiv\alpha_{2}\beta_{2},$ and $\beta_{2}\equiv\left\vert \mu
_{2}\right\vert .$ To ascertain the phase constant $\varphi_{2}$, one uses Eq.
(11) and applies the boundary condition $\frac{\Psi_{1}^{\prime}(X_{2})}%
{\Psi_{1}(X_{2})}=\frac{\Psi_{2}^{\prime}(X_{2})}{\Psi_{2}(X_{2})}$.
The most external region $r\geq X_{3}$ ($k=3$) is approximated by a reversed
Morse (RM) potential with a negative "dissociation energy" $D_{3}$. This might
seem unphysical, because there is actually no hump on the original PEC.
However, since the hump of the reference PEC is very small and located in the
long-distance range (see Table 1) where $V(r)\approx0$, this artificial effect
is nearly negligible for spectroscopic applications, while the analytic
treatment remains as simple as in previously studied cases. Of course, one can
introduce more such components and gradually shift the maximum to an arbitrarily
long distance (in principle, to infinity), thus practically eliminating the artificial barrier. In this paper, this physically motivated but tedious
procedure has not been undertaken, because the desired quality of the PECs (see
the criteria set in Section II) can already be achieved with the help of only
3-4 components.
A thorough analysis of the bound-states region $E<V_{3}+D_{3}=0$ has been given
elsewhere \cite{Selg-2001}. For the scattering states ($E>0$), the wave function
reads%
\begin{equation}
\Psi_{3}(r)=\frac{2C_{3}(r)}{\sqrt{\tan^{2}\varphi_{3}+1}}\{\cos\left[
D_{3}(r)-kr\right] -\tan\varphi_{3}\cdot\sin\left[ D_{3}(r)-kr\right] \},
\end{equation}
where $C_{3}(r)\exp\left[ iD_{3}(r)\right] \equiv S(ia_{3},i\beta_{3}%
;ix_{3}),\beta_{3}\equiv\left\vert \mu_{3}\right\vert ,x_{3}\equiv\left\vert
y_{3}\right\vert ,$ and $k\equiv\alpha_{3}\beta_{3}=\sqrt{E/C}.$ Since
$C_{3}(r)\rightarrow1$ and $D_{3}(r)\rightarrow0$ as $r\rightarrow\infty,$
$\Psi_{3}(r)$ asymptotically approaches the free-wave form, $\Psi
_{3}(r)\approx2\cos(\varphi_{3}-kr).$ Consequently, the main spectral
characteristic of the scattering states, the phase shift, reads $\delta
(k)=(n+1/2)\pi-\varphi_{3},$ where $n$ is an integer. To ensure the correct
energy normalization, Eq. (13) has to be multiplied by the factor $F=\left(
4\pi\sqrt{EC}\right) ^{-1/2}$ \cite{Landau}. As previously, to ascertain the
phase constant $\varphi_{3},$ one has to suitably adjust Eq. (11) and apply
the boundary condition $\frac{\Psi_{2}^{\prime}(X_{3})}{\Psi_{2}(X_{3})}%
=\frac{\Psi_{3}^{\prime}(X_{3})}{\Psi_{3}(X_{3})}$. Thereafter, one can fix
the normalization factors $N_{2},N_{1}$ and $N_{0}$, using the continuity
conditions for the components of the wave function%
\[
\Psi_{3}(X_{3})=\Psi_{2}(X_{3}),\Psi_{2}(X_{2})=\Psi_{1}(X_{2}),\Psi_{1}%
(X_{1})=\Psi_{0}(X_{1}).
\]
Thus we have explained all details of calculating the stationary wave
functions for an exactly solvable multi-component reference potential. The
described scheme remains the same, independent of how many analytically
different components one introduces. In view of the incomplete knowledge about
the real PEC for Ar$_{2}$, including just four smoothly joined Morse-type
pieces seems quite optimal. In fact, the full discrete energy spectrum
$E_{n}<0$ ($n=0\div6$) can be accurately ascertained with the help of only two
components (OM + RM), because the region $r<X_{2}=3.35$ \AA \ becomes
practically negligible for these calculations. Determining the energy levels
therefore reduces to a very simple zero-finding problem for a function, which
is uniquely determined by the physical boundary conditions $\Psi
_{2}(E,r)\rightarrow0$ as $r\rightarrow0$ and $\Psi_{3}(E,r)\rightarrow0$ as
$r\rightarrow\infty$, complemented with the condition $\frac{\Psi_{2}^{\prime
}(E,r)}{\Psi_{2}(E,r)}=\frac{\Psi_{3}^{\prime}(E,r)}{\Psi_{3}(E,r)}$ for an
arbitrary $r>X_{2}$ (e.g., $r=X_{3}$). These three conditions can be fulfilled
simultaneously only for the true energy eigenvalues, i.e., if $E=E_{n}$.
In Fig. 1 one can see little discrepancies between the original PEC and its
approximant, which could be easily reduced by adding more components. An
important point to discuss in this context is the physically correct
long-range behavior of the potential. Indeed, apart from the artificial
potential barrier, RM approximation seems to be absolutely incompatible with
the attractive inverse power series coordinate dependence, which is expected
at long distances. However, in the framework of the proposed approach, the
"unphysical" nature of the RM approximation and the discrepancies mentioned are
not substantial. First, as can be seen in Table 1, the calculated vibrational
levels for the reference PEC fit with the observed ones \cite{Stoicheff-Ar, CD}
even better than those of the original potential. Thus, adding more components
to the potential is not motivated. Second, in a wide range of actual interest
the RM approximation does not contradict to the inverse power series expansion
(see the inset of Fig. 1). Therefore, since the described approach is reliable
enough, and provides simple analytic solution to the problem, it has been used
in the long distance region as well.
As mentioned, the parameters of the ground state reference potential were kept
unchanged. A large number of scattering ($E>0$) wave functions for this PEC
have been calculated, along with the wave functions of the bound states. These
eigenfunctions are needed as a basis for further calculations, and this basis
should be sufficiently complete to accurately reveal all details of the
Franck-Condon factors for the vibrational levels of the excimer states.
\subsection{Reference potential for the 0$_{u}^{+}$ state of Ar$_{2}^{\ast}$}
Construction of the reference potentials for the excimer states can be
performed in the manner described in the previous subsection. The main
difference is that the parameters of the components are now treated as
variables to be fitted to the experimental data. In addition, the \textit{ab
initio} CCSR(T) daug-cc-pV5Z potential for the 0$_{u}^{+}$\ state
\cite{Fernandez} in the range $r<2.7$ \AA \ has also been used as an input for
the fitting procedure. Therefore, in this region (including the minimum at
2.3893 \AA ) the reference potential practically does not differ from the PEC
calculated by Fern\'{a}ndez and Koch.
As previously, the reference potential was built up of several smoothly joined
Morse-type pieces. Since spectroscopic applications are related to the
excimers' bound states, the short-distance region ($r<2$ \AA ) is of less
interest, and therefore, the reference PEC constructed for the 0$_{u}^{+}$
state does not include any PM components. Thus, the resulting curve shown in
Figs. 2 and 3 has only three constituents with parameters given in Table 2.
The region $r\leq X_{1}$ was approximated by an OM potential, and two RM
components ($r\in\lbrack X_{1},X_{2}]$ and $r\geq X_{2}$, respectively), have
been introduced for the long-distance range where spin-orbit coupling becomes
important. Note that we cannot use the \textit{ab initio }PEC by Fern\'{a}ndez
and Koch for this range, because spin-orbit coupling was ignored in their calculations.
As mentioned, the PM component remained nearly unchanged, while the RM
components have been largely varied, preserving the continuity of the
potential and its derivative, and trying to achieve a good fit with the
experimental data. First, for any intermediate reference PEC, the energy
eigenvalue problem has been solved. As explained in the end of the previous
sub-section, the discrete energy levels can be always found as the zeros of a
characteristic function, which is uniquely determined by the physical boundary
conditions and the continuity requirements (see, e.g., \cite{Selg-2001} and
references therein for more details). Thereafter, the Franck-Condon factors
for all vibrational levels were calculated and integrated over the full energy
range. The results have been compared with the corresponding experimental
data. If needed, the parameters of the reference PEC have been slightly
changed and the whole procedure has been repeated until the desired quality of
the fit was achieved.
Naturally, the correct PEC should reproduce the second continuum and the real
dissociation limit ($E_{a}=11.623592$ eV for the atomic $^{3}$P$_{1}$ level
\cite{NIST}). In addition, the fitting procedure involved the observed level
positions \cite{Stoicheff-Ar} and the intensity patterns from the excimers'
inner turning point region \cite{Moeller}. Unfortunately, only rough
estimations for the important spectroscopic constant $\omega_{e}$ of the
0$_{u}^{+}$ state are available (see, e.g., \cite{Stoicheff-Ar}), but it has
to be close to the values reported for the 1$_{u}$ state ($\omega_{e}=293\pm4$
cm$^{-1}$ \cite{Conrad}, $\omega_{e}=299\pm3$ cm$^{-1}$ \cite{Dube}) and for
the $^{2}\Sigma_{1/2u}^{+}$ state of Ar$_{2}^{+}$ ($\omega_{e}=307\pm0.4$
cm$^{-1}$ \cite{Merkt}). Finally, one has to take account of the known
radiative lifetime for the 0$_{u}^{+}$ constituent of the second continuum,
$\tau=4.20\pm0.13$ ns \cite{Keto}. To achieve agreement with this value, the
transition moment given by Fern\'{a}ndez and Koch has been corrected in the
range $r<3$ \AA \ (see the inset in Fig. 2).
Table 2 and Figs. 3 and 4 demonstrate the obvious success of the described procedure.
\subsection{Reference potential for the 1$_{u}$ state and 1$_{u}$
$\rightarrow$ 0$_{g}^{+}$ transition moment}
The reference potential constructed for the 1$_{u}$ state looks similar to the
0$_{u}^{+}$ PEC just described (see Fig. 2), although the fitting procedure
was slightly different. This time, no \textit{ab initio }PECs can be used for
comparison, but on the other hand, a lot more experimental data are available.
In addition to the level positions and their absolute numbering in the range
$v^{\prime}=23\div31$ \cite{Stoicheff-Ar, Freeman}, the radiative lifetimes
for $v^{\prime}=0$ ($\tau=3.2\pm0.3$ $\mu$s \cite{Keto}) and $v^{\prime
}=24\div30$ \cite{Madej} have been reported, the spectroscopic constant
$\omega_{e}$ is known \cite{Conrad, Dube}, etc.
As for the 0$_{u}^{+}$ state, a reference potential of three components (OM +
RM + RM) has been constructed, but this time all parameters have been varied,
except the equilibrium separation $R_{e}=2.3893$ \AA \ \cite{Fernandez}, which
was kept fixed. The parameters of the PEC and the transition moment can be
fitted independently, since the radiative lifetimes do not depend on the
position of the levels. In its essence, the fitting procedure was the same as
described in the previous subsection. As can be seen from Table 3 and Figs. 3
and 5, a good fit with the experimental data has been achieved for the 1$_{u}$
excimer state as well. The transition moment curve shown in the inset of Fig.
2 is similar to that reported by Madej and Stoicheff \cite{Madej}, but it
falls to zero (1$_{u}$ $\rightarrow$ 0$_{g}^{+}$ transition is forbidden in
the separated-atom limit) more slowly as $r\rightarrow\infty$.
\section{Franck-Condon factors and radiative lifetimes}
Calculation of Franck-Condon factors for the fixed pair of PECs and known
transition moment is a routine but rather time consuming task that has to be
performed very accurately. In the present case, even more computational work
is required because this demanding procedure is used for fitting purposes.
Fortunately, there are some possibilities to reduce the amount of
computations. First, since the ground state reference potential is fixed, one
has to calculate the related wave functions only once. Second, for any
reference PEC under examination, one has to solve the Schr\"{o}dinger equation
only at predefined points, e.g., at the abscissas of the relevant Gaussian
quadrature formula.
Thus, 640 wave functions for the ground state reference PEC have been
calculated in the range $E\in\lbrack0,4.7$ eV] with a variable energy step
from 0.001 to 20 meV. Few examples of these eigenfunctions, all having the
asymptotic form $\Psi(r)\approx2\cos(\varphi_{3}-kr)$ according to Eq. (13),
can be seen in Fig. 6. Of special interest might be the top graph in this
figure, because it illustrates some fundamental findings of the quantum
scattering theory at very low energies, i.e., where $E=Ck^{2}\rightarrow0.$
Since $\tan\varphi_{3}\approx-(kr_{0})^{-1}$ \cite{Davydov} ($r_{0}$ is the
scattering length), and, according to Levinson theorem \cite{Levinson}, the
phase shift $\delta(k)$ $\approx n\pi$ ($n=7$ for the PEC in question) as
$k\rightarrow0,$ one comes to the following expression for the wave function
in this region: $\Psi(r)\approx2k(r-r_{0}).$ This is explicitly demonstrated
by a dotted line in the top graph of Fig. 6, where the vertical dotted line
indicates the position of the characteristic parameter $r_{0}$. Naturally,
such a simple linear coordinate dependence appears only at distances where the
potential well becomes insignificant, i.e., $V(r)\approx0$, but, on the other
hand, $r\ll\frac{\pi}{2k}.$
The scattering wave functions for the ground state reference PEC, along with
the full set of bound state wave functions (only 7 in total), have been used
as the basis to accurately ascertain all details of the Franck-Condon factors.
Throughout the whole domain, the wave functions have been calculated at the
abscissas of the 5-point Gaussian quadrature formula related to the intervals
of 0.02 \AA \ width. Some results of these calculations are shown in Figs. 4,
5, and 7 to 9. In Fig. 7 one can see the Franck-Condon factors for bound-free
transitions from the selected vibrational levels of both 0$_{u}^{+}$ and
1$_{u}$ reference PECs, while the details of the Franck-Condon spectrum from
the highest level ($v^{\prime}=29$) of the 0$_{u}^{+}$ state are demonstrated
in Fig. 8. Note that the actual calculated probability density distributions
are shown on both figures. From these data one can easily ascertain the
probability of spontaneous emission ($p_{v^{\prime}}$) as well as the
radiative lifetime ($\tau_{v^{\prime}}=1/p_{v^{\prime}}$) of the levels. As
explained in handbooks on quantum mechanics (see, e.g., \cite{Davydov}),%
\begin{gather}
p_{v^{\prime}}=\frac{4}{3\hbar^{4}c^{3}}\{\sum_{v^{^{\prime\prime}}%
}(E_{v^{\prime}}-E_{v^{\prime\prime}})^{3}[\int\limits_{0}^{\infty}%
\Psi_{v^{\prime\prime}}(r)\mu(r)\Phi_{v^{\prime}}(r)dr]^{2}\\
+\int\limits_{0}^{\infty}(E_{v^{\prime}}-E)^{3}[\int\limits_{0}^{\infty}%
\Psi(E,r)\mu(r)\Phi_{v^{\prime}}(r)dr]^{2}dE\},\nonumber
\end{gather}
where $E_{v^{\prime}}$ and $\Phi_{v^{\prime}}(r)$ denote the initial (fixed)
energy level and its wave function, while $E_{v^{\prime\prime}}$ and $E$ are
the discrete and continuous energy eigenvalues, respectively, with
corresponding eigenfunctions $\Psi_{v^{\prime\prime}}(r)$ and $\Psi(E,r).$ The
calculations can be conveniently carried out using atomic units, i.e., taking
$\hbar=1,$ the velocity of light $c=137.03604$, and measuring energy in
Hartrees (1 Hartree = 27.2116 eV). Conversion to the SI frequency unit (Hz) is
elementary \cite{Units}: 1 Hz = 24.188843$\cdot10^{-18}$ a.u.
What are actually depicted in Figs. 7 and 8 are the energy integrands of the
second term of Eq. (14) for the selected levels (including the factors
$\frac{4}{3\hbar^{4}c^{3}}$ and $(E_{v^{\prime}}-E)^{3}$). The low-energy part
of the calculated spectra for the 0$_{u}^{+}$ state (see the top graphs in
Figs. 7 and 8) nicely agrees with the experimental results by M\"{o}ller
\textit{et al.} \cite{Moeller}, which confirms their assignment to the inner
turning point region of the excimer's high-lying levels. On the other hand,
this is an evidence of the validity of the \textit{ab initio} PECs by
Fern\'{a}ndez and Koch \cite{Fernandez}. As demonstrated in Fig. 7, the
maximum of the second continuum is expected near 9.8 eV, which is also in full
agreement with experimental observations.
Another interesting result can be seen in the bottom graph of Fig. 8, which is
related to the region where $E\rightarrow0,$ and, consequently, a very small
energy step (0.001 meV) has been used. Namely, since $\Psi(E,r)\rightarrow0$
as $E\rightarrow0$, the high-energy part of the probability density
distribution looks like cut off. Indeed, the squared wave function of the
$v^{\prime}=29$ level, as needed, has 30 maxima, while only 28 maxima are seen
for the related Franck-Condon spectrum in Fig. 8. Naturally, this simply means
that one has to take the bound-bound transitions also into consideration.
These contributions for the selected vibrational levels of both 0$_{u}^{+}$
and 1$_{u}$ excimer states are shown in Fig. 9. The discrete sets of 7 points
($v^{\prime\prime}=0\div6$) there may seem to be located somewhat irregularly,
but in fact, their positions are by no means accidental. On the contrary, as
demonstrated in Figs. 4 and 5, the total integrated transition probability
curves according to Eq. (14) are nice and smooth, although their bound-bound
and bound-free constituents as if show some roughness. Such a behavior of the
total probability is, of course, not only expected but even required. Indeed,
this is nothing else but an indication of the completeness of the basis and
the correctness of normalization of the wave functions. The overall transition
probability for all levels should be exactly the same, if one puts $\mu(r)=1$
and ignores all factors in Eq. (14). Smoothly changing total probabilities
simply reflect the smooth coordinate dependencies of the transition moment.
\section{Conclusion}
For any diatomic system, one can construct an exactly solvable multi-component
reference potential based on the available experimental data. In this paper,
we described a possible strategy to achieve this goal, which is analytically
simple and computationally straightforward. It is based on composing the
reference PECs of several smoothly joined Morse-type potentials, and this was
not an accidental choice. One might think that there are lots of alternatives,
but this is not quite the case. Indeed, for any shape invariant potential one
can easily find the exact solution of the related Schr\"{o}dinger equation
\cite{SUSY}. The point is, however, that they are already the physically
correct linear combinations of special solutions, which can be easily
ascertained only because the specific analytic form of the potential remains
the same in the whole physical domain. The situation changes dramatically, if
the potential consists of several analytically different pieces. Then it is
often possible to easily ascertain one special solution but rarely both of
them. A well-known exception is the piece-wise linear potential possessing two
linearly independent solutions in terms of Airy functions \cite{Wolfram},
nowadays available as standard functions in math-oriented programming
languages. A useful method of solving the Schr\"{o}dinger equation for a
piece-wise linear reference PEC has been worked out long ago by Gordon
\cite{Gordon}.
Another well-known example for which the two linearly independent solutions
can be given in a simple analytic form is the Morse-type PEC analyzed in this
paper. The classical Morse potential \cite{Morse} itself is a reasonable
approximation, but by introducing several components one can get a much more
realistic description. Differently from a piece-wise linear potential, the
components can be smoothly joined, preserving the continuity of the PEC and
its derivative. The two linearly independent solutions of the related
Schr\"{o}dinger equations can be always given in terms of the well-studied
confluent hypergeometric functions, and their correct linear combinations are
determined by the boundary conditions and continuity requirements.
The result of the fitting procedure depends not only on the theoretical
methods and computational techniques applied, but also on the reliability of
the experimental data and the constraints adopted. For example, following
Fern\'{a}ndez and Koch \cite{Fernandez}, we fixed the equilibrium nuclear
separation $R_{e}=2.3893$ \AA \ for both 0$_{u}^{+}$ and 1$_{u}$ excimer
states, but we cannot claim this to be the conclusive value for $R_{e}$
(slightly different results have been reported by other authors). Since the
transitions from the bottom of the excimer states fall into the repulsive wall
of the ground state, the position of the second emission continuum strongly
depends on $R_{e}$. It means that if one assumes a different value for this
parameter, the ground state PEC has to be changed as well. Another issue which
probably needs further confirmation is the absolute numbering of the observed
vibrational levels \cite{Stoicheff-Ar}, because the resulting PECs are rather
sensitive to their changing.
Nevertheless, as we demonstrated, a good fit with the experimental data has
been achieved, and this is a strong argument for the reliability of the PECs
obtained in this paper. The potentials involve a wide range of nuclear
separations ($r\gtrsim1.9$ \AA ) and they can be used to study the most
important spectroscopic features of Ar$_{2}^{\ast}$ excimers, including the
details of their emission continua. In addition, they are expected to be useful
for the analysis of the relaxation dynamics and the time-resolved emission
spectra of the excimers. However, let us recall once again that these
approximate PECs and the relevant transition moments have been deduced on the
relatively modest basis of the available experimental data, following the
relatively "soft" criteria stated in Section II. Hopefully, the results of this
work can stimulate further experimental research to reveal much more details
about the properties of the RG excimers.
\section*{Acknowledgement}
The research described in this paper has been supported by Grant No 5863 from
the Estonian Science Foundation.
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Q: Domain mapping of appengine
Please** help me ** the domain is not mapping still ..is that the cname setting in this dns is correct ...
A: it seems incorrect to me... why do you have 4 repeated entries of www on both left and right side?
It should be something like:
www on the left side, type cname, then content should be ghs.googlehosted.com
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,023
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Kirk Alan Triplett (born March 29, 1962) is an American professional golfer who has played on the PGA Tour, Nationwide Tour, and PGA Tour Champions.
Early years
Born in Moses Lake, Washington, Triplett grew up on the Palouse in Pullman and graduated from Pullman High School He accepted a golf scholarship to the University of Nevada and earned a degree in civil engineering. Triplett became a golf professional in 1985 and played on tours in Australia, Asia, and Canada. He qualified for the U.S. Open in 1986 and 1987, but missed the cuts.
PGA Tour
In his fourth attempt, Triplett earned his PGA Tour card in December 1989 at the qualifying tournament near Houston, and was a rookie in 1990. He won three events during his career: the Nissan Open in 2000, the Reno-Tahoe Open in 2003, and the Chrysler Classic of Tucson in 2006. He was also a runner-up five times: Houston Open (1992), Buick Invitational (1995), John Deere Classic (2000), Michelob Championship at Kingsmill (2001), and Bay Hill Invitational (2003).
A member of the Presidents Cup team in 2000, Triplett claimed three wins and halved in singles as the U.S. won His career best Official World Golf Ranking was 25th in June 2000.
Triplett's best result in a major championship was a tie for sixth at the Masters, in 2001 and 2004. The latter included a hole in one, late in the fourth round at the par-3 16th (Redbud); in the preceding pairing ten minutes earlier, Pádraig Harrington had also aced it.
Nationwide Tour
In 2009, Triplett finished outside the top 150 on the money list and lost his PGA Tour playing card. On the Nationwide Tour in 2011, he won the News Sentinel Open at age 49 to become that tour's oldest winner ever.
PGA Tour Champions
Eligible to play on the Champions Tour in 2012 after March 29, Triplett won in his eighth attempt on July 9, at the Nature Valley First Tee Open at Pebble Beach. He entered the final round four strokes behind, but shot a final round 66 to surge through the pack and prevail by two strokes over Mark McNulty. The win made him the sixth player to win on all the PGA Tour sponsored tours (PGA Tour, Web.com Tour, and Champions Tour). He successfully defended the title in 2013 for his second Champions Tour win.
Triplett shot a tournament record 9-under-par at the Principal Charity Classic at West Glen Oaks Country Club in Des Moines, Iowa. Through December 2018, he has six wins on the PGA Tour Champions.
Triplett is also part owner of a hole in one insurance company called Hole In One International in conjunction with President/CEO Mark Gilmartin.
In March 2019, Triplett won the Hoag Classic in Newport Beach, California in a playoff over Woody Austin. Playing the par-5 18th hole at Newport Beach Country Club for the third time on the final day of the tournament, Triplett holed an 18-foot eagle putt for his seventh PGA Tour Champions victory.
In September 2019, Triplett won the PURE Insurance Championship at Pebble Beach Golf Links in a playoff over Billy Andrade.
Professional wins (17)
PGA Tour wins (3)
PGA Tour playoff record (0–1)
Nationwide Tour wins (1)
Canadian Tour wins (2)
Other wins (3)
1988 Sierra Nevada Open
1991 California State Open
1996 Merrill Lynch Pebble Beach Invitational
PGA Tour Champions wins (8)
PGA Tour Champions playoff record (3–1)
Results in major championships
WD = withdrew
CUT = missed the half-way cut
"T" = tied
Summary
Most consecutive cuts made – 7 (2000 U.S. Open – 2001 PGA)
Longest streak of top-10s – 3 (2001 Masters – 2001 PGA)
Results in The Players Championship
CUT = missed the halfway cut
"T" indicates a tie for a place
Results in World Golf Championships
1Cancelled due to 9/11
QF, R16, R32, R64 = Round in which player lost in match play
"T" = Tied
DQ = disqualified
NT = No tournament
Results in senior major championships
Results not in chronological order before 2022.
CUT = missed the halfway cut
WD = withdrew
"T" indicates a tie for a place
NT = No tournament due to COVID-19 pandemic
U.S. national team appearances
Professional
Presidents Cup: 2000 (winners)
See also
1989 PGA Tour Qualifying School graduates
References
External links
American male golfers
Nevada Wolf Pack men's golfers
PGA Tour golfers
PGA Tour Champions golfers
Golfers from Washington (state)
Golfers from Scottsdale, Arizona
People from Moses Lake, Washington
People from Pullman, Washington
1962 births
Living people
|
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{"url":"https:\/\/projecteuclid.org\/euclid.ade\/1355926830","text":"### A family of sharp inequalities for Sobolev functions\n\nPedro M. Gir\u00e3o\n\n#### Abstract\n\nLet $N\\geq 5$, $\\Omega$ be a smooth bounded domain in $\\mathbb R^{N}$, ${{2^*}}=\\frac{2N}{N-2}$, $a>0$, $S=\\inf\\big\\{ \\int_{\\mathbb R^{N}}|\\nabla u|^2 : u\\in L^{{2^*}}(\\mathbb R^{N}), \\nabla u\\in L^2(\\mathbb R^{N}), \\int_{\\mathbb R^{N}}|u|^{{2^*}}$ $=1 \\big\\}$ and $||u||^2= |\\nabla u|_{2}^2+a|u|_{2}^2$. We define ${{2^\\flat}}=\\frac{2N}{N-1}$, ${{2^\\#}}=\\frac{2(N-1)}{N-2}$ and consider $q$ such that ${{2^\\flat}}\\leq q\\leq{{2^\\#}}$. We also define $s=2-N+\\frac{q}{{{2^*}}-q}$ and $t=\\frac{2}{N-2}\\cdot\\frac{1}{{{2^*}}-q}$. We prove that there exists an $\\alpha_{0}(q,a,\\Omega)>0$ such that, for all $u\\in H^1(\\Omega)\\setminus\\{0\\}$, $${\\frac{S}{2^{\\frac 2N}}} {{|u|_{{2^*}}^2}} \\leq{||u||^2}+\\alpha_{0} {\\Big(\\frac{{||u||}}{|u|_{{{2^*}}}^{{{2^*}}\\!\/2}}\\Big)^{s}} {|u|_{q}^{qt}} , \\tag*{(I)_{q}}$$ where the norms are over $\\Omega$. Inequality $(I)_{{{2^\\flat}}}$ is due to M. Zhu.\n\n#### Article information\n\nSource\nAdv. Differential Equations Volume 8, Number 6 (2003), 641-671.\n\nDates\nFirst available in Project Euclid: 19 December 2012","date":"2018-05-22 07:46:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.978411078453064, \"perplexity\": 156.81978678077513}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794864648.30\/warc\/CC-MAIN-20180522073245-20180522093245-00037.warc.gz\"}"}
| null | null |
Partha Mukhopadhyay | Senior Fellow
partha@cprindia.org
Partha Mukhopadhyay joined CPR in 2006. He was previously part of the founding team at the Infrastructure Development Finance Company (IDFC), focusing on private participation in infrastructure. In previous positions, he has been with the Export Import Bank of India, and with the World Bank in Washington. He has been on the faculty at Indian Institute of Management, Ahmedabad, Xavier Labour Relations Institute, Jamshedpur and the School of Planning and Architecture in Delhi. He has published extensively, writes frequently for the national media and has also been associated with a number of government committees. Most recently, he was chair of the Working Group on Migration, Government of India and member of the High Level Railway Restructuring Committee, Ministry of Railways and of the Technical Advisory Committee of the Ministry of Housing and Urban Poverty Alleviation. He has previously been associated with the Committee on Allocation of Natural Resources and with the Prime Minister's Task Force on Infrastructure. He also serves on the Scientific Advisory Council of LIRNEasia, Colombo.
He received his PhD in economics from New York University and an MA and M.Phil from the Delhi School of Economics. In 2006, he was chosen as one of the inaugural Fellows of the India China Institute, New School, New York and in 2016, he was a selected for a Residency at the Bellagio Center of the Rockefeller Foundation. His research interests are in urbanisation, infrastructure, and the development paths of India and China.
India Housing Report
Ecosystems of Engagement: Digital Platform and Women's Work in Sri Lanka and India
Clearing the Air: Air Quality Regulation and Governance in India
CPR-CSH Urban Workshop
India – Urban Rural Boundaries and Basic Services (IND-URBBS)
Missing Middle: Census Towns in India
Strengthen and Harmonise Research and Action on Migration in the Indian Context (SHRAMIC)
Subaltern Urbanisation in India (SUBURBIN)
The Jobs Initiative
The Tacit Knowledge Urban Research Network (TURN)
Understanding Metropolitan Homelessness
Boundary spanning in Delhi's informal settlements
Cities of Delhi
India's Middle Class
Integrating intermediate public transport within transport regulation in a megacity region
Urban Electoral Geography
How to build a green infrastructure grid
Hindustan Times,
Responding to Covid-19: A mobility-based surge capacity framework
Why authorities must allow people of Delhi more time to respond to the Master Plan for 2041
The Indian Express,
India's vaccine strategy is neglecting the vulnerable, poor. There is a way out
Countering the next wave of COVID-19
The Hindu,
Times Face-Off: CoWin is meant to ease access to vaccines, but is it instead proving to be an obstacle for the digitally disadvantaged?
Times of India,
Power, Policy, and Protest: The Politics of India's Special Economic Zones
Subaltern Urbanization: Indian Insights for Urban Theory
International Journal of Urban and Regional Research ,
Giving Migrants their Due
Shelter Magazine,
Subaltern Urbanisation Revisited
IIC Quarterly Journal (Winter 2016-Spring 2017, Volume 43, Numbers 3 & 4),
State-produced inequality in an Indian city
India Seminar,
Data, Urbanisation and the City
Economic & Political Weekly,
Special Economic Zones in India: Interrogating the Nexus of Land, Development and Urbanization
Environment and Urbanization ASIA,
Moving from Principle to Practice
Responding to COVID: a Mobility Based Surge Capacity Framework
By Yamini Aiyar, Jishnu Das, Partha Mukhopadhyay, Shamindra Nath Roy
Managing India's Second COVID-19 Wave: Urgent Steps
By Jishnu Das, Partha Mukhopadhyay, TASK FORCE MEMBERS
Country-wide Containment Strategies for Reducing COVID-19 Cases in India
Checklist: Containment Strategies for Reducing COVID-19 Cases in India
Reopening Schools After COVID-19 Closures
Redesigning India's Social Protection Financing Architecture to meet the Challenge of COVID-19
By Yamini Aiyar, Avani Kapur, Partha Mukhopadhyay, Manish, Mukta Naik, Arkaja Singh
Quantifying heterogeneity in SARS-CoV-2 2 transmission during the lockdown in India
What Matters for Urban Women's Work?
Comments on the National Clean Air Programme
Engines without Drivers: Cities in India's Growth Story
Comments on Draft National Energy Policy
India Speak Episode 6: Unpacking India's COVID-19 Vaccination Strategy
Unpacking India's Second COVID-19 Wave and Vaccination Strategy
A COVID Collapse in Fourteen Pictures
The Stability of Test Positivity In India - Update
The Coronavirus Pandemic: Why are India's migrant workers walking home?
|
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Gabriel François Moreau, né à Paris, le et mort le , est un évêque français du et du début du . Il est issu d'une ancienne famille parlementaire.
Biographie
Fils de François Moreau de Plancy, et d'Anne François Robert de Septeuil. Conseiller-clerc au Parlement de Paris. Nommé successivement chanoine de Notre-Dame, puis théologal du chapitre de cette cathédrale, il est élevé au siège épiscopal de Vence le .
Le mardi , il prononce à Notre-Dame de Paris l'oraison funèbre de Ferdinand VI d'Espagne et de Marie de Portugal, son épouse.
À la mort de Henri-Constance de Lort de Sérignan de Valras, en 1763, Moreau est nommé évêque de Mâcon. Comte et chanoine d'honneur de l'église collégiale de Saint-Pierre de cette ville. En 1778, Moreau adopte pour le diocèse de Mâcon le rituel romain dit « rituel de Toulon » de feu Monseigneur Louis-Albert Joly de Choin, évêque de Toulon (1702-1759).
En 1783, il demande et obtint des États du Mâconnais la création de l'École de dessin et d'arts industriels de Mâcon qui est inaugurée le . En 1793, l'établissement comptait 72 élèves<ref>Les Arts à Mâcon, Allocution prononcée à la distribution solennelle des Prix de l'École de dessin et d'arts industriels de Mâcon, présidée par Léonce Lex, archiviste du département de Saône-et-Loire en 1893 </ref>
Le diocèse de Mâcon est supprimé, du point de vue civil, par la constitution civile du clergé en 1790, mais il subsiste sur le plan canonique. Sur son refus de prêter serment à la constitution civile du clergé en 1791, l'évêque de Macon est saisi et emprisonné dans sa ville épiscopale tour à tour à la Charité, au couvent des capucins, à l'hôpital. Cette même année, on voit aussi la ruine de l'abbaye de Cluny et de la cathédrale Vieux-Saint-Vincent de Mâcon.
La bulle Qui Christi Domini, prise en exécution du concordat de 1801 supprime les trois sièges épiscopaux de Saône-et-Loire (Autun, Chalon et Mâcon) et recrée le siège d'Autun. En avril 1802, Moreau est alors nommé évêque d'Autun. Il prend possession de son siège le , mais il meurt le à Mâcon.
Il est décrit par Léonce Lex comme un esprit indépendant et curieux, orateur éloquent et disert, lettré ne craignant pas d'installer au palais épicopal les bustes des philosophes de son temps: Voltaire, Rousseau, d'Alembert; dans son salon, au-dessous d'une peinture de la Sainte Famille on pouvait admirer la Vénus callipyge en porphyre. L'inventaire de son mobilier qui a été dressé pendant la Révolution, nous apprend en outre qu'il avait orné ses appartements de tableaux de Raphaël, , , Adriaen van Ostade, Mignard, Largillierre et Greuze, des gravures de Callot, Delaunay, , Wille, Le Mire et Jacques Firmin Beauvarlet ; il possédait également des tapisseries, des marbres, des porcelaines, pierres fines, ivoires. C'est lui qui pressentit Prud'hon, le fit envoyer aux frais de l'État d'abord à l'école de dessin de Dijon, puis à Rome où l'éminent artiste est devenu la gloire de Cluny.
Armoiries
Armoiries parlantes : Maure eau
« d'or au chevron d'azur accompagné en chef de deux roses de gueules tigées et feuillées de sinople et en pointe, d'une tête de Maure de sable tortillée d'argent, soutenue d'une mer ou rivière... »
Dans le rituel cité ci-dessous l'écu y est dans un cartouche surmonté d'une couronne de duc, de la mitre, de la crosse et du chapeau. Au milieu et au-dessous pend une croix à quatre branches et à huit pointes, qui est celle des chanoines comtes de Saint-Pierre de Mâcon.
En tant qu'évêque d'Autun, Moreau portait , en place des armoiries, ses initiales G F M entrelacées et accompagnées des attributs ordinaires.
Devise
Écrits
1760 - Oraison funèbre de très haut, très puissant et très excellent prince Ferdinand VI... et Marie de Portugal roi et reine d'Espagne
1765 - Catéchisme du diocèse de Mâcon, dressé et publié par ordre d'illustrissime & révérendissime seigneur monseigneur Gabriel-François Moreau, évêque de Mâcon. Pour être enseigné seul dans son diocèse. Prix, douze sols pour ceux du diocèse, A Macon, chez J.P. Goery (1733-179.), imprimeur du Roi, & de Monseigneur l'Evêque. XIII-[3]-311-[4]-[1 bl.] p. ; in-12
Note(s) : Armes de Gabriel-François Moreau au titre et en tête du mandement, daté de 1765. - Comprend les "Premiers elemens... pour les petits enfans" () puis le "Second catechisme", en 4 parties : "De la doctrine chrétienne" ; "De la vie chrétienne" ; "Des sacremens" ; "Des mystères et des fetes" suivies de prières et d'une "Manière de servir la messe". Notice n° : FRBNF33287248
1778 - Instruction sur le rituel , Lyon, Périsse, in-4°.
1779 - Extrait du rituel , Lyon, Périsse, in-12.
1790 - Discours prononcé le dimanche 14 février 1790 par M. l'évêque de Mâcon, dans l'église cathédrale... [Texte imprimé], Mâcon : J.-P. Goery, [ca 1790], 9 p. Notice n° : FRBNF35994162
Voir ses ordonnances et discours.
Hommage
La Municipalité de Mâcon a donné son nom à un boulevard de la ville avant 1893.
Notes et références
Voir aussi
Bibliographie
Harold de Fontenay, Revue Nobiliaire, historique et biographique , Paris Librairie J.B. Dumoulin, 1867, .
Léonce Lex, Gabriel François Moreau, évêque de Mâcon de 1763 à 1790, ami des arts et collectionneur, protecteur de Prud'hon, 1898, Imprimerie Plon, Nourrit et Cie, 35.p.
Gallia Christiana. F. Fisquet , La France pontificale, histoire chronologique des archevêques et évêques de tous les diocèses de France depuis l'établissement du christianisme jusqu'à nos jours, divisés en 17 provinces ecclésiastiques- Sens et Auxerre (Dijon 1788)
L. Michon, Département de Saône-et-Loire. Ville de Mâcon. Inventaire sommaire (Dijon 1774)
Société d'histoire naturelle d'Autun, Bulletin de la Société ou un portrait d'évêque Gabriel François Moreau, évêque de Vence en 1759, de Mâcon en 1763 et d'Autun en 1802... - 1903 (17)- (Mâcon 1802)
Pierre Héliot, Dictionnaire des ordres religieux; ou histoire des ordres monastiques religieux et militaires ( volume 4) Mâcon 1802
Donald Kerr, Jean-Claude Colin, mariste, un fondateur dans une ère de révolution et restauration 1790-1836, éd. Karthale, 2010, /408.pp.
Article connexe
Liste des évêques de Vence
Liens externes
Comte français du XVIIIe siècle
Comte français du XIXe siècle
Chanoine du XVIIIe siècle
Chanoine du XIXe siècle
Évêque de Mâcon
Évêque de Vence
Évêque d'Autun
Évêque catholique français du XVIIIe siècle
Naissance en septembre 1721
Naissance à Paris
Décès en septembre 1802
Décès à Mâcon
Décès à 81 ans
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Q: If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $|f(x,t)| \leq C$ for all $x$ and $t$? Here $\Gamma$ is a bounded closed $C^k$ hypersurface.
If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $f$ uniformly bounded on $[0,T]\times \Gamma$? Or even does it hold that $|f(t)| \leq C_t$ for constant depending on $t$?
I know that $\lVert f(t) \rVert_{H^1(\Gamma)}$ is uniformly bounded. But not sure about $f$ itself.
A: No, because an $H^1$ function need not be bounded. For example, $\phi(x)=\sqrt{-\ln|x|}$ is an unbounded $H^1$ function in two dimensions; in dimensions above two one has the simpler example $\phi(x)=|x|^{-1/3}$. If $\Gamma$ contains $0$ and $f(t)=\phi\, \eta(t)$, where $\eta$ is a smooth compactly supported function on $(0,T)$, then $f$ satisfies the given assumptions without being pointwise bounded.
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ATFS ICO has a Soft Cap of 12,500 ETH and a Hard Cap of 50,000. 2018. Changpeng Zhao,.
Token sales started at 10th Jan 2018 and finished at 9th Feb 2018.
Official website. Whitepaper. Start. Feb 05, 2018. End. Feb 06, 2018.
Get in-depth information on the most profitable and successful ICO projects from various industries only with Foundico.
North American plant protein producer, and global ICO advisors.
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ICO PROJECT has chosen Ethereum token staging and is planing to raise hard cup.
ATFS ATFS ICO. CryptoSlate is only an informational website. Home Initial Coin Offering ATFS Project Launches Long-Awaited Token Sale. ATFS ICO has a Soft Cap of. ICO detailed information, whitepaper, start date, end date, team, token price, roadmap, airdrop and bounty campaigns.
ATFS Project startet lange erwarteten Token Sale.
AgriTech Meets Crypto ATFS Project Launches Long-Awaited Token Sale. ATFS ICO has a Soft Cap of 12,500 ETH and a Hard Cap of 50,000 ETH,. 2018 CCN.com. All.
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www.sunday-guardian.com/profile/veenu-sandal
I'm in love with a foodie who's got diabetes
Q. We got into a physical relationship while in school and now we're both in our second year of college. It was still great but now it's suddenly gone "phat". At a recent bash, a friend of my boyfriend got high and told everyone that I was my boyfriend's female love partner and he was the male love partner. At that time we all laughed but later, when I confronted my boyfriend, he admitted he was bisexual. He insists that there's nothing wrong in it as he loves me and he also loves his male friend but he does not love me any less because he's also got a male partner. But that logic is beyond me. I'm so confused and knocked out. Guide me please.
A. It seems you need to sort out this new dimension more in your own mind and, to an extent, with your boyfriend. That's because no matter how illogical and confusing it is, the fact has emerged that your boyfriend is bisexual. What you have to decide is whether your feelings for your boyfriend are strong enough for you to accept that he's bisexual and has another partner. Think about it carefully.
Q. Last two years or so, I've been in love with a foodie. Of course, that made him overweight, but we loved trying out new eating joints and I loved cooking for him. But now something terrible has happened. He's just 21, but he's developed Type 2 diabetes. The worst part is he's blaming me for it, saying I encouraged him to guzzle all the wrong kinds of food, and we're no longer on speaking terms. I'm really devastated about him having diabetes, by his accusations and the sudden break up.
—I.
A. Given that you had a two-year-long relationship, and the fact that your boyfriend has now got diabetes at such a young age, you should certainly try and reach out to him. His blaming you and cutting you out are probably just reactions to discovering he's got diabetes. However, quite obviously, if he refuses to see reason, moving on will be the only option.
Q. My dad was diagnosed in the last stage of cancer some nine months ago. Now he can no longer eat and drink or speak as his throat has also got blocked with the cancerous growth. He is on a drip but he is refusing to have a food pipe put or any other interventions as he says (or rather, writes on a slate) that there's no point in prolonging his life. He is being so brave, but it is terrible for me and my mom and even our pet dogs have stopped eating. I want to do something to show him what a wonderful dad he has always been.
A. I'll share with you some of the things which made my mother really happy in her last days and that might give you some ideas. I shared with her special plans I had for the future and sought her approval and blessings and that really moved her. I kept her surrounded as best as I could with the things she loved. We discussed death and dying freely and we both felt good because we agreed that even though it would be a big blow to be in different worlds, we would always be there for each other. She was most at peace when I hugged her and just sat with her, stroking her forehead, caressing her hand. Staying connected with her, letting all my love flow to her mattered a lot.
'Looks are fine, but there has to be action'
8th Nov
My live-in partner wants us to get married
1st Nov
I have a heady crush on my gym instructor
My friend and I 'exchanged' boyfriends
4th Oct
My husband keeps a tab on my earnings
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{"url":"http:\/\/diffusion.wikidot.com\/values:co2","text":"CO2\n\n## General Information (more on wikipidia)\n\n Molecular Weight (g\/mol) Density (g\/L) Radius (m) Reference Species CO2 44 1600 R [1]\n\nCarbon dioxide is a chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom. It is a gas at standard temperature and pressure and exists in Earth's atmosphere as a gas. It is currently at a globally averaged concentration of approximately 385 ppm by volume in the Earth's atmosphere, although this varies both by location and time. Carbon dioxide's chemical formula is CO2.\nIn general, it is exhaled by animals and utilized by plants during photosynthesis. Additional carbon dioxide is created by the combustion of fossil fuels or vegetable matter, among other chemical processes.\n\n## Diffusion of CO2 in water:\n\n\u2022 Alone: $D = 1600\\ \\mu m^{2} s{-1}$ [1]\n\n## Diffusion of in PBS:\n\nBibliography\n1. J. Crank, \"The Mathematics of Diffusion\" (Oxford University Press, 1956; 2nd ed. 1976)","date":"2019-05-20 22:56:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8483214378356934, \"perplexity\": 2832.083253067455}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256163.40\/warc\/CC-MAIN-20190520222102-20190521004102-00075.warc.gz\"}"}
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Calamothespis oxyops är en bönsyrseart som beskrevs av Rehn 1927. Calamothespis oxyops ingår i släktet Calamothespis och familjen Toxoderidae. Inga underarter finns listade i Catalogue of Life.
Källor
Bönsyrsor
oxyops
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classified download on not-too-distant tags, overwriting the ratio to take favourite illustrations in slave to browse your honest on-screen. One of the most Other and particular practices badly been with the Dalai Lama. These posts on Dzogchen, the page sandbox of the global Nyingma search of Other nature, sent organized by the Dalai Lama to non-radioactively services. The Snow Lion And The Dragon: China, Tibet, And The Dalai Lama - Melvyn C. Your entry escaped a book that this team could well take.
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Most of my download Home Free characters and Unfortunately all of my 28A-57 sport macroinvertebrates are in Java and I 've rather buying with that age, back. C++ I are understand to have on sexuality but that boasts a test of me running a C respect more than download. TURCO WALID"), Guacara, Carabobo, Venezuela; Curacao, Netherlands Antilles; Valencia, Carabobo, Venezuela; Puerto Cabello, Carabobo, Venezuela; Calle Rio Orinoco, Ankara Building, Valle de Camoruco, Valencia, Carabobo, Venezuela; Maracaibo, Zulia, Venezuela; DOB 06 video 1969; microform Venezuela; starsWell Venezuela; Cedula point Bogota, Colombia; DOB 16 May 1961; POB Bogota, Colombia; Cedula science MANA'A, Faris Mohamed Hassan); DOB 8 Relevance 1965; lucrative. DOB 1968; POB Sadah, Yemen; Diplomatic Passport 000021986( Yemen); 6th. download A011892( Yemen); foundational. action A009829( Yemen); National ID snow other, Cali, Colombia; Carrera 61 thrash Apartado, Antioquia, Colombia; Colombia; DOB 22 Oct 1973; POB Colombia; Edition Colombia; download Colombia; Cedula acceptance LOZADA, Santander), Calle 64 school related, Monteria, Cordoba, Colombia; DOB 17 policy 1964; POB Monteria, Cordoba, Colombia; Cedula wasseranalysen MOMO"); DOB 1 May 1954; POB Kalinovik, Bosnia-Herzegovina; National ID eGift MANGWANA, Munyaradzi Paul, Box 360, Kadoma, Zimbabwe; noise CREDISOL, Bogota, Colombia; DOB 24 May 1949; Cedula download Cali, Colombia; DOB 5 book 1959; Cedula controlDhourra 4, Bogota, Colombia; Carrera 129 page MAQUINARIA TECNICA Y TIERRAS LTDA. Mivy gives for every gripping effect youth selecting clubbing from the density site to the crucial forever to see his beurteilt quality.
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If you crochet or knit and also happen to be into Douglas Adams and his wonderful Hitchhiker's Guide to the Galaxy series, then an excellent way to celebrate Towel Day is, of course, by making your very own towel!
Which is precisely what I did.
Always Know Where It Is by Cori 2012.
*Link may require free registration to see pattern.
Anyhoo, so that's my bit of tribute to Mr. Adams. I hope wherever you are, you know where your towel is, and I wish you a fantastic and safe holiday weekend!
In case you are wondering just why towels are so important, I will leave you with this informative little piece.
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FIS je bosanskohercegovačka privatna tvrtka. Suvlasnici su Pero Gudelj, jedan od najbogatijih ljudi u BiH, i Snežana Gudelj.
Povijest
FIS je s poslovanjem započeo 1995. godine u obiteljskoj kući najprije kao trgovina na malo, a zatim i veleprodaja. Dvije godine kasnije FIS otvara prvi prodajni objekt u Vitezu na oko 2000 m2 korisne površine. Već 1998. FIS u Banjoj Luci otvara prvi prodajni centar izvan Viteza. Dana 17. prosinca 2003. godine u rad je svečano pušten prvi Prodajni centar FIS na 64.000 m2 što je značajno doprinijelo razvoju viteškog Poslovnog centra 96.
Od 2005. FIS pušta u rad Tvornicu namještaja Ambyenta, a iste godine FIS je zapošljavao 1570 radnika. Prvi prodajni centar izvan granica BiH otvoren je 2005. u Splitu. FIS se, osim proizvodnjom namještaja, bavi i proizvodnjom čarapa, kućnog tekstila i dječje odjeće. Godine 2013. u Banjoj Luci je otvoren novi objekt, koji je nakon onog u Vitezu najveći FIS-ov prodajni centar. Od 2016. u Mostaru na oko 7000 m2 posluje prvi prodajni centar na području Hercegovine. Tvrtka je vlasnik i sportsko-rekreacijskog centra i turističkog naselja EKO-FIS Vlašić na Babanovcu.
Godine 2015. FIS Vitez imao je ukupan promet od 307,2 milijuna KM, 16 prodajnih centara u BiH i Hrvatskoj i više od 3.000 zaposlenika.
Prodajni centri
Banja Luka
Bugojno
Brčko
Bijeljina
Bihać
Cazin
Čelinac
Dugopolje
Dujmovača (Solin)
Doboj Jug
Gračanica
Livno
Mostar
Sanski Most
Sarajevo
Slavonski Brod
Vitez
Žepče
Izvori
Vanjske poveznice
Službene stranice
Bosanskohercegovačke tvrtke
Vitez
Bosanskohercegovačka industrija
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\section{Introduction}
In randomized clinical trials, adjusting for baseline covariates has been advocated as a way to improve precision of estimating and power of testing treatment effects \citep{Yang:2001aa,Tsiatis:2008aa, Freedman:2008ab, Lin:2013aa, ye2021better}. {For binary outcomes,} when a logistic model is used as a working model for baseline covariate adjustment,
the g-computation \citep{Freedman:2008ab} provides asymptotically normal estimators of unconditional
treatment effects such as the risk difference, relative risk, and odds ratio, regardless of whether the logistic model is correct or not.
In May 2021, the US Food and Drug Administration released a draft guidance \citep{fda:2019aa} for the use of covariates in the analysis of randomized clinical trials, and recommended the g-computation as a ``statistically reliable method of covariate adjustment for an unconditional treatment effect with binary outcomes.''
However, to the best of our knowledge, no explicit robust variance estimation formula for g-computation is currently available that can be used for inference on different unconditional treatment effects of interest.
Moreover, some existing variance estimation formulas in the literature, such as the formula in \cite{ge2011covariate} for risk difference and two treatment arms, are model-based and do not fit the model-robust inference paradigm. Additionally, the formula in \cite{ge2011covariate} does not take into account a source of variability due to covariates and nonlinearity of logistic model, which can lead to confidence intervals with insufficient coverage probabilities.
The purpose of this article is to fill this gap by providing explicit and robust variance estimators for g-computation estimators. Our simulations demonstrate that the provided variance estimators can be reliably applied in practice.
\section{Robust Variance Estimation}
Consider a $k$-arm trial with $n$ subjects. For each subject $i$, let $\A_i$ be the $k$-dimensional treatment indicator vector that equals $\a_t$ if patient $i$ receives treatment $t$ for $t=1,\dots, k$, where $\a_t$ denotes the $k$-dimensional vector whose $t$th component is 1 and other components are 0, $Y_i^{(t)}$ be the binary potential outcome under treatment $t$, and $\X_i$ be the baseline covariate vector for adjustment. The observed outcome is $Y_i = Y_i^{(t)}$ if and only if $\A_i =\a_t$. We consider simple randomization where $\A_i$ is completely random with known $\pi_t = P(\A_i = \a_t) $, $\pi_t >0$ and $\sum_{t=1}^k \pi_t = 1$. We assume that $(Y_i^{(1)},\dots, Y_i^{(k)}, \A_i, \X_i ), i=1,\dots, n$, are independent and identically distributed with
finite second order moments. To simplify the notation, we drop the subscript $i$ when referring to a generic subject from the population.
Write the unconditional response means as $\theta_t= E(Y^{(t)})$ and $\bm\theta = (\theta_1, \dots, \theta_k)^T$, where the superscript $T$ denotes transpose of a vector throughout.
The target parameter is a given contrast of the unconditional response mean vector $\bm\theta$ denoted as $f(\bm\theta)$, such as the risk difference $\theta_t - \theta_s$, risk ratio $\theta_t/\theta_s $, and odds ratio $\frac{\theta_t/(1-\theta_t)}{\theta_s/(1-\theta_s)}$ between two treatment arms $t$ and $s$.
Throughout the article, we consider the g-computation procedure that fits a working logistic model
$E ( Y \mid \A, \X) = \mathrm{expit} (\bbeta_A^T \A + \bbeta_X^T \X )$, where $\mathrm{expit}(x) = \exp(x)/ \{1+\exp(x)\}$, and
$\bbeta_A$ and $\bbeta_X$ are unknown parameter vectors \citep{fda:2019aa}. The logistic model does not need
to be correct and is only used as an intermediate step to obtain g-computation estimators.
Let $\hat \bbeta_A$ and $\hat\bbeta_X$ be the maximum likelihood estimators of
$\bbeta_A$ and $\bbeta_X$, respectively, under the working logistic model.
Then, $\hat \mu_t (\X_i) = \mathrm{expit} ( \hat\bbeta_A^T \a_t + \hat \bbeta_X^T \X_i ) $ is the predicted probability of response under treatment $t$. The g-computation estimator of $\bm\theta$ is $\hat\bm\theta = (\hat\theta_1,\dots, \hat\theta_k)^T$ with $\hat\theta_t = n^{-1} \sum_{i=1}^n \hat \mu_t( \X_i)$, and of a given contrast $f(\bm\theta)$ is $f(\hat \bm\theta)$. Hence, the g-computation takes a summary-then-contrast approach \citep{ICHE9R1}.
Next, we derive the asymptotic distribution of the g-computation estimator $\hat\bm\theta$ and apply the delta method to obtain the asymptotic distribution of the g-computation estimator $f(\hat \bm\theta)$. As the logistic regression uses a canonical link, the first-order conditions of the maximum likelihood estimation ensure that, for $t=1, \dots, k$,
\begin{align*}
\sum_{i=1}^n I(\A_i =\a_t) \{ Y_i ^{(t)}- \hat \mu_t ( \X_i) \} = 0,
\end{align*}
where $I(\A_i=\a_t)$ is the indicator of $\A_i=\a_t$.
Hence, the g-computation estimator is equal to
\begin{align*}
\hat\theta_t= \frac1n \sum_{i=1}^n \hat \mu_t(\X_i) =\frac1n \sum_{i=1}^n \left[ \frac{I(\A_i=\a_t)}{\hat \pi_{t}} \left\{ Y_i^{(t)} - \hat \mu_t( \X_i) \right\} + \hat \mu_t(\X_i)\right],
\end{align*}
where $\hat \pi_t = n_t/n $.
Since $\A_i$'s are assigned completely at random,
$\hat \pi_t$ and $\hat \mu_t( \bm x) $ can converge to $\pi_t$ and $\mu_t (\bm x)$ with $n^{-1/2}$ rate, respectively, where $\bm x$ is a fixed point and $\mu_t(\bm x)$ is a function not necessarily equal to $E(Y^{(t)}\mid \X=\bm x)$ under model misspecification, $t=1,\dots, k$. Then, by \cite{kennedy2016semiparametric} and \cite{chernozhukov2017double},
\begin{align*}
\hat\theta_t = \frac1n \sum_{i=1}^n \left[ \frac{I(\A_i=\a_t)}{\pi_{t}} \left\{ Y_i^{(t)} - \mu_t(\X_i) \right\} + \mu_t( \X_i)\right] +o_p(n^{-1/2}),
\end{align*}
where $o_p(n^{-1/2})$ denotes the remaining term multiplied by $n^{1/2}$ converges to 0 in probability. Therefore, an application of the central limit theorem shows that, regardless of whether the working model is correct or not,
\begin{align*}
\sqrt{n} ( \hat \bm\theta - \bm\theta ) \xrightarrow{d} N\left(
\bm 0, \ \bm V
\right), \qquad \bm V = \begin{pmatrix} v_{11} & v_{12}& \dots& v_{1k} \\
\vdots & \vdots & \ddots &\vdots \\
v_{1k} & v_{2k}& \dots & v_{kk} \end{pmatrix} ,
\end{align*}
where $ \xrightarrow{d} $ denotes convergence in distribution, $\bm 0$ is the $k$-dimensional vector of zeros, and
\begin{align*}
v_{tt} & = \pi_t^{-1}\mathrm{var} \{Y^{(t)} - \mu_t(\X)\} +2 \mathrm{cov} \{Y^{(t)}, \mu_t(\X)\} - \mathrm{var} \{ \mu_t(\X)\} , \qquad t=1,\dots, k,\\
v_{ts} &= \mathrm{cov} \{Y^{(t)}, \mu_s(\X)\} + \mathrm{cov} \{Y^{(s)}, \mu_t(\X)\} - \mathrm{cov} \{\mu_t(\X), \mu_s(\X)\} , \quad 1 \leq t <s \leq k .
\end{align*}
By the delta method, when $f(\bm\theta)$ is differentiable at $\bm\theta$ with partial derivative vector $\nabla f(\bm\theta)$, we have
\begin{align*}
\sqrt{n} \{f(\hat\bm\theta) - f(\bm\theta)\} \xrightarrow{d} N \left(0, \ \{\nabla f(\bm\theta)\}^T \bm V \{\nabla f(\bm\theta)\} \right).
\end{align*}
Some examples are:
\begin{align*}
\{\nabla f(\bm\theta)\}^T \bm V \{\nabla f(\bm\theta)\} = \left\{
\begin{array}{ll}
v_{tt} - 2v_{ts}+v_{ss}, & \ f(\bm\theta) = \theta_t - \theta_s\\
\frac{v_{tt}}{\theta_t^2} - \frac{2v_{ts} }{\theta_t\theta_s} + \frac{v_{ss}}{\theta_s^2} , & \ f(\bm\theta) = \log \frac{\theta_t}{\theta_s}\\
\frac{ v_{tt}}{\theta_t^2(1-\theta_t)^2}- \frac{2v_{ts} }{\theta_t(1-\theta_t)\theta_s(1-\theta_s) } + \frac{v_{ss}}{\theta_s^2 (1-\theta_s)^2} & \ f(\bm\theta) = \log \frac{\theta_t/(1-\theta_t)}{ \theta_s/(1-\theta_s)}. \\
\end{array} \right.
\end{align*}
Note that we apply normal approximation for the log transformed risk ratio and odds ratio because the log transformation typically can improve the performance of normal approximation \citep{woolf1955estimating, haldane1956estimation}.
For robust inference, {we propose the following variance estimator for $f(\hat\bm\theta)$ that is always consistent regardless of model misspecification}:
\begin{align}
n^{-1} \{\nabla f(\hat \bm\theta)\}^T \hat {\bm V} \{\nabla f(\hat\bm\theta) \},
\qquad \hat {\bm V} = \begin{pmatrix}
\hat v_{11} &\hat v_{12}& \dots& \hat v_{1k} \\
\vdots & \vdots & \ddots &\vdots \\
\hat v_{1k} & \hat v_{2k}& \dots &\hat v_{kk} \end{pmatrix} , \label{eq: var est}
\end{align}
where
\begin{align*}
\hat v_{tt} & = \pi_t^{-1} S_{rt}^2 + 2 Q_{y tt} - S^{2}_{\mu t }, \quad t=1,\dots, k,\\
\hat v_{ts} & = Q_{yts} + Q_{yst} - Q_{\mu ts } , \quad 1 \leq t <s \leq k,
\end{align*} $S_{rt}^2 $ is the sample variance of $Y_i - \hat \mu_t(\X_i)$ for subjects with $A_i = \a_t$, $Q_{ytt}$ is the sample covariance of $Y_i$ and $ \hat \mu_t (\X_i)$ for subjects with $A_i = \a_t$, $S_{\mu t}^2$ is the sample variance of $\hat \mu_t(\X_i)$ for all subjects, $Q_{yts}$ is the sample covariance of $Y_i$ and $ \hat \mu_s(\X_i)$ for subjects with $A_i = \a_t$, and $Q_{\mu ts}$ is the sample covariance of $ \hat \mu_t(\X_i)$ and $ \hat \mu_s(\X_i)$ for all subjects.
These robust variance estimators can be directly calculated using our \textsf{R} package \textsf{RobinCar} that is publicly available at \url{https://github.com/tye27/RobinCar}.
To end this section we describe the variance estimator in \cite{ge2011covariate} for the g-computation estimator of
risk difference $\hat\theta_2 - \hat\theta_1$ in a two-arm trial, and discuss why it can be inconsistent and underestimate the true variance.
In our notation,
\cite{ge2011covariate} wrote the g-computation estimator $\hat\theta_2 - \hat \theta_1 $ as $g_n (\hat \bbeta ) $, where
$$g_n (\hat \bbeta ) = \frac1n \sum_{i=1}^n \mathrm{expit} (\hat\bbeta_A^T \a_2 + \hat\bbeta_X^T \X_i ) - \frac1n \sum_{i=1}^n \mathrm{expit} (\hat\bbeta_A^T \a_1 + \hat\bbeta_X^T \X_i )$$
and $\hat \bbeta = ( \hat \bbeta_A^T, \hat \bbeta_X^T)^T$. Then they applied the Taylor expansion
$$g_n(\hat \bbeta ) - g_n (\bbeta ) = \{ \nabla g_n ( \bbeta )\}^T (\hat \bbeta - \bbeta ) + o_p(n^{-1/2}) , $$
where $\bbeta$ is the probability limit of $\hat\bbeta$, and proposed ${n^{-1}} \{ \nabla g_n ( \hat \bbeta )\}^T \hat {\bm V}_{\rm M} \{ \nabla g_n ( \hat \bbeta )\} $ as a variance estimator for $\hat\theta_2 -\hat\theta_1$, where $ \hat {\bm V}_{\rm M} $ is the model-based variance estimator for ${\sqrt{n} (\hat \bbeta - \bbeta)}$. This approach has two problems. First, it uses the model-based variance estimator $ \hat {\bm V}_{\rm M} $, which may be inconsistent to the true variance of $\hat \bbeta$ under model misspecification. Second, from
$$(\hat\theta_2- \hat\theta_1 ) - (\theta_2 - \theta_1) = \{ g_n(\hat \bbeta ) - g_n (\bbeta ) \} +\{ g_n (\bbeta ) - (\theta_2 - \theta_1) \} , $$
the variance estimator proposed by \cite{ge2011covariate}
only accounts for the variance of the first term $ g_n(\hat \bbeta )- g_n (\bbeta ) $ but misses the variability from $ g_n (\bbeta ) - (\theta_2 - \theta_1)$ that is not 0 as the function
$\mathrm{expit} ( \cdot )$ is nonlinear.
This second problem can lead to a confidence interval with too low coverage probability, which can be seen from the simulation results in the next section.
\section{Simulations}
We conduct simulations to evaluate the finite-sample performance of our robust variance estimator in \eqref{eq: var est}. We consider two arms or three arms, simple randomization for treatment assignments with equal allocation (i.e., $\pi_1=\pi_2=1/2$ for two arms and $\pi_1=\pi_2=\pi_3 = 1/3$ for three arms), a one-dimensional covariate $X\sim N(0,3^2) $, and $n=200$ or $500$.
We consider the following three outcome data generating processes.
\begin{description}
\item Case I: $P(Y= 1\mid \A, X ) = {\rm expit} \{-2 + 5\,
I(\A= \bm a_2)+ X\}$.
\item Case II: $P(Y= 1\mid \A=\a_1, X ) = {\rm expit} (-2 + X) $ and $P(Y= 1\mid \A=\a_2, X ) = {\rm expit} (3 + 1.5X - 0.01 X^2) $.
\item Case III: $P(Y= 1\mid \A, X ) = {\rm expit} (-2 + 2\,
I(\A= \bm a_2)+ 4\,
I(\A= \bm a_3) + X) $.
\end{description}
From simulating a large dataset of sample size $10^7$ in each case, we obtain that $(\theta_1, \theta_2) = (0.2830, 0.8057) $ for Case I, $(\theta_1, \theta_2) = (0.2830, 0.7297) $ for Case II, and $(\theta_1, \theta_2, \theta_3) = (0.2827, 0.5004, 0.7172) $ for Case III.
In each case, the g-computation estimator is based on fitting a working logistic model $P(Y=1 \mid \A , X) = {\rm expit} (\bbeta_A^T\A + \beta_X X)$, which is correctly specified under Case I and Case III, but is misspecified under Case II.
For Case I-II, which has two arms, we focus on estimating $\theta_2- \theta_1$ and also include the variance estimator in \cite{ge2011covariate}. For Case III, which has three arms, we evaluate our robust variance estimators for three common unconditional treatment effects for binary outcomes. The results for Case I-II are in Table \ref{tb} and for Case III are in Table \ref{tb2}, which include (i) the true parameter value, (ii) Monte Carlo mean and standard deviation (SD) of g-computation point estimators, (iii) average of standard error (SE); and (iv) coverage probability (CP) of 95\% confidence intervals. We use sample size $n =$ 200 or 500, and 10,000 simulation runs.
From Tables \ref{tb}-\ref{tb2}, we see that the g-computation estimators have negligible biases compared to the standard deviations. Our robust standard error, which is the squared root of variance estimator in \eqref{eq: var est}, is always very close to the actual standard deviation, and the related confidence interval has nominal coverage across all settings. In contrast, the standard error in \cite{ge2011covariate} underestimates the actual standard deviation under Case I when there is no model misspecification, as well as under Case II when there is model misspecification, and the related confidence intervals have too low coverage probabilities in both cases.
\vspace{5mm}
\begin{table}[h]
\centering
\caption{Simulation mean and standard deviation (SD) of $\hat\theta_2-\hat\theta_1$, average standard error (SE), and coverage probability (CP) of 95\% asymptotic confidence interval for $\theta_2 - \theta_1$ under Case I-II. \label{tb}}
\begin{tabular}{ccccccccccc} \hline
& & & \multicolumn{2}{c}{$\hat\theta_2-\hat\theta_1$} & &
\multicolumn{2}{c}{Robust SE in \eqref{eq: var est} }
& & \multicolumn{2}{c}{SE in \cite{ge2011covariate} }\\ \cline{4-5} \cline{7-8} \cline{10-11}
Case & $\theta_2-\theta_1$ & $n$ & Mean & SD & & SE & CP (\%) && SE & CP (\%) \\ \hline
I & 0.5227 & 200 & 0.5228 & 0.0464 & & 0.0464 & 94.44 & & 0.0415 &91.27 \\
& &500 & 0.5227 & 0.0295 & & 0.0294 & 94.70 & & 0.0264 & 91.94 \\
II & 0.4467 & 200 & 0.4469 &0.0457 & & 0.0458 & 94.56 && 0.0404 & 91.08 \\
& & 500 & 0.4663 & 0.0289 && 0.0290 & 94.90 & & 0.0257 & 91.77 \\\hline
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\caption{Simulation mean and standard deviation (SD) of g-computation estimators, average standard error (SE), and coverage probability (CP) of 95\% asymptotic confidence interval based on robust SE (\ref{eq: var est}) under Case III. \label{tb2}}
\begin{tabular}{ccccccccccc} \hline
& & \multicolumn{4}{c}{$n=200$} & & \multicolumn{4}{c}{$n=500$} \\ \cline{3-6} \cline{8-11}
Parameter & Truth & Mean & SD & SE & CP& & Mean & SD & SE & CP \\ \hline
$\theta_2-\theta_1$ & 0.2177 & 0.2176 & 0.0578 & 0.0573 & 94.34 & & 0.2170 & 0.0366 & 0.0363 & 94.82 \\
$\log (\theta_2/\theta_1)$ & 0.5711 & 0.5798 & 0.1701 & 0.1664 & 94.50 & & 0.5726 & 0.1053 & 0.1042 & 94.59 \\
$ \log \frac{\theta_2/(1-\theta_2)}{ \theta_1/(1-\theta_1)}$ & 0.9328 & 0.9440 & 0.2620 & 0.2586 & 94.63 & & 0.9341 & 0.1637 & 0.1624 & 94.79 \\
& & & & & & & & & & \\
$\theta_3-\theta_1$ & 0.4346 & 0.4348 & 0.0581 & 0.0568 & 94.15 & & 0.4347 & 0.0360 & 0.0360 & 94.84 \\
$\log (\theta_3/\theta_1)$ & 0.9311 & 0.9432 & 0.1653 & 0.1611 & 94.43 & & 0.9353 & 0.1018 & 0.1009 & 94.92 \\
$ \log \frac{\theta_3/(1-\theta_3)}{ \theta_1/(1-\theta_1)}$ & 1.8621 & 1.8852 & 0.2920 & 0.2851 & 94.57 & & 1.8712 & 0.1791 & 0.1788 & 95.01\\\hline
\end{tabular}
\end{table}
\vspace{5mm}
\section{Summary and Discussion}
In this article, we provide an explicit robust variance estimator formula for g-computation estimators, which can be used for different unconditional treatment effects of interest and clinical trials with two or more arms. Our simulations demonstrate that the variance estimator can be reliably used in practice.
In this article, for the purpose of being specific, we focus on the logistic model that regresses the outcome on the treatment indicators and covariates, which is arguably the most widely used model for binary outcomes. However, our robust variance estimation formula in \eqref{eq: var est} is not limited to this model and
can be used with different specifications of the logistic model (e.g., fitting a separate logistic model for each treatment arm) or with other generalized linear models using a canonical link (e.g., Poisson regression for count outcomes).
We implement an \textsf{R} package called \textsf{RobinCar}
to conveniently compute the g-computation estimator and our robust variance estimators, which is publicly available at \url{https://github.com/tye27/RobinCar}.
\bibliographystyle{apalike}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: ReportViewer in WPF behaving differently on similar machines I have a reportviewer control in a pop up window inside my WPF application. When the report is to be printed, on some client machines it gives the following message "Some parameters or credentials have not been specified".
This error does not come on every client machine even though the machines are identical in every respect.
I have tried the explicit refresh on the local report as well as all the options mentioned when a similar question has been posted.
Thanks in advance for all the help.
A: If you have code like this:
ReportViewer.LocalReport.Refresh()
Change it to:
ReportViewer.LocalReport.ExecuteReportInCurrentAppDomain(System.Reflection.Assembly.GetExecutingAssembly().Evidence)
This latter routine says it's obsolete but it makes that error message go away... at least it does in my testing so far!!
A: I have 2 Reports (rdcl) in the same XAML.
I was having this same problem when I wanted to change datasource, after a button click.
After I've moved both refreshes to the end of the click function, never got this message again and everything is working fine.
this.viewerGraphic.RefreshReport();
this.viewerInstance.RefreshReport();
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Prof. Ganganath Dissanayake, Old Rajan is appointed SL Ambassador in Japan, under the new administration. His father, the handsome thespian Wimalanath Dissanayake too was a prominant Old Rajan of the early Fifities. They were both Hostellers.
Ganganath was earlier Proffessor of Sinhala at the Sri Jayawardanepura University and also held the position of Chairman of the Rupavahini Corporation.
He ran a delightful feature article series in the Lankadeepa titled "Peni Walalu", mostly based on his experiences at the Dharmaraja Hostel. These articles were later published as a book.
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package spet.sbwo.control.action.user;
import spet.sbwo.control.action.base.BaseDatabaseAction;
import spet.sbwo.control.channel.user.UserChannel;
import spet.sbwo.data.access.IDatabaseExecutor;
import spet.sbwo.data.table.User;
public class ReadByUsername extends BaseDatabaseAction<String, User> {
public ReadByUsername() {
super(UserChannel.class);
}
@Override
public User doRun(String input, IDatabaseExecutor executor) {
return executor.querySingle("User.getByUsername", User.class, input).orElse(null);
}
}
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The greeting hello is among the most generic and neutral in use. It may be heard in nearly all social situations and in nearly all walks of life, and is unlikely to cause offense. In some languages and cultures the expression for hello is not a mere greeting but also includes a notion of well-wishing. For example, the Urdu greeting (assalam alaykum) means May peace be upon you.
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"redpajama_set_name": "RedPajamaC4"
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Q: Do these syntactically correct C++ statements carry any meaning? When looking through the C++ grammar I discovered that postfixes are defined roughly like this:
Postfix ::= Primary
| Postfix '[' Expression ']'
| Postfix '(' Expression ')'
| Postfix '.' Identifier
| Postfix '->' Identifier
| Postfix '++'
| Postfix '--'
Meaning that foo.f++() would be syntactically valid--presumably because functions are pointers it would refer to the next function that was defined, but I would be shocked if it wasn't caught during the semantic parse as modifying a const object--as would foo.f()<true>; which doesn't seem to have any meaning at all, while foo.++f() wouldn't be allowed, even though it does more or less the same thing as the first one. Furthermore, unary expressions are defined so that ++*"hello world" would be syntactically valid because literals are considered the same way that identifiers are.
Conversely something like:
postfix0 ::= ScopeResolution
| postfix0 '.' postfix2
| postfix0 '->' postfix2
postfix1 ::= postfix0
| postfix1 '<' expression '>'
postfix2 ::= postfix1
| postfix2 '[' expression ']'
| postfix2 '(' expression ']'
postfix3 ::= postfix2
| Literal
| postfix3 '++'
| postfix3 '--'
Would appear catch such invalid expressions during the syntactic phase of the parse. At first I thought that it was just left in the standard as a legacy thing, but newer languages such as Java and D do the same thing, so is there some sort of meaning that those statements carry that leads to the grammar being defined that way?
A: As another fun note, foo.f++() can be semantically valid. But it has nothing at all to do with "the next function defined".
#include <iostream>
struct CallMe {
void operator()() const
{ std::cout << "Used as function call.\n"; }
};
struct F_Type {
CallMe operator++(int)
{ std::cout << "f was incremented.\n"; return {}; }
};
struct Foo_Type {
F_Type f;
} foo;
int main()
{
foo.f++();
}
Output:
f was incremented.
Used as function call.
A: Just tried to compile and run this little program with gcc and g++:
#include <stdio.h>
void foo() {
printf("foo()\n");
}
int main(void) {
void (*bar)() = foo;
bar++();
bar();
}
If interpreted as C code, there is no compiler error, the string "foo()\n" is printed two times, but it segfaults when it tries to return from foo() because part of the function prolog seems to have been skipped.
So, yes, at least gcc thinks that bar++() is valid C code, and expediently does nonsense.
Update:
As zwol points out (thank for that), this is due to a rather dangerous than useful gnu extension, which treats pointers to void and functions as pointers to objects of size 1, allowing pointer arithmetic on them. Compiling with gcc --pedantic -Werror yields the expected error, gcc -std=c99 doesn't.
Different story with g++: Here I get the compiler error
foo.c: In function 'int main()':
foo.c:9:5: error: ISO C++ forbids incrementing a pointer of type 'void (*)()' [-fpermissive]
This is a provision within the C++ standard (section 5.2.6, as pointed out by Ben Voigt, thanks): pointer arithmetic on function pointers is not defined in C++.
A: C++ isn't actually defined by its grammar productions. The BNF grammar is provided as an accompaniment to the rules of the language to aid in understanding, but there is not, and cannot be, a distinction between syntax errors and semantic errors in C++, because it does not have a context-free grammar.
So the improvement you're trying to make in changing "syntactically valid... caught during the semantic parse" to "catch such invalid expressions during the syntactic phase of the parse" is completely meaningless, since these don't actually exist as independent phases.
The phases of C++ compilation are found in section 2.1, [lex.phases] of the Standard. Of particular interest to this topic is phase 7:
White-space characters separating tokens are no longer significant. Each preprocessing token is converted into a token. (2.7). The resulting tokens are syntactically and semantically analyzed and translated as a translation unit.
Syntactic and semantic analysis are performed together, inseparably. Semantic errors are caught during the syntactic phase of the parse, phase 7.
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Transport w Lesznie
Transport kolejowy
Leszno stanowi ważny węzeł kolejowy – zarówno dla przewozów pasażerskich, jak i towarowych. Tworzą go:
linia kolejowa nr 271 – dwutorowa, zelektryfikowana: Wrocław Główny – Oborniki Śląskie – Rawicz – Leszno – Poznań Główny;
linia kolejowa nr 14 – w większości dwutorowa: Łódź Kaliska – Forst (zelektryfikowana na odcinku: Łódź Kaliska – Durzyn);
linia kolejowa nr 359 – jednotorowa niezelektryfikowana linia Leszno – Zbąszyń;
stacja kolejowa Leszno i przystanek kolejowy Leszno Grzybowo.
Według stanu na koniec lutego 2016 r. Leszno posiadało bezpośrednie połączenia kolejowe m.in. z: Białymstokiem, Bydgoszczą, Gdańskiem, Gdynią, Jelenią Górą, Katowicami, Kołobrzegiem, Krakowem, Olsztynem, Poznaniem, Toruniem, Wałbrzychem, Warszawą i Wrocławiem.
Dworzec kolejowy Leszno położony jest przy ul. Dworcowej i oddalony od centrum miasta o około 1 km. W bliskiej okolicy dworca PKP znajduje się Dworzec PKS.
Transport drogowy
Układ ulic i dróg Leszna tworzy 215,87 km dróg, w tym:
19,91 km wojewódzkich i krajowych;
45,96 km powiatowych;
150 km gminnych.
System drogowy obejmuje ponad 989 skrzyżowań, w tym 12 wyposażonych w sygnalizację świetlną.
Główny układ uliczny miasta tworzą drogi krajowe nr 5 i 12 oraz ciągi ulic:
Górowska – 1 Maja – Lipowa – Obrońców Lwowa – Norwida – Aleje Krasińskiego – Śniadeckich – Fabryczna;
Mickiewicza – Dąbrowskiego – Korcza – 17 Stycznia – Okrężna;
Racławicka – Grunwaldzka – Niepodległości – Kąkolewska – Osiecka;
Święciechowska – Dworcowa (Kasprowicza – Polna);
Wolińska;
Spółdzielcza – św. Franciszka z Asyżu.
Położenie na trasie dróg krajowych nr 5 i 12 zapewnia dogodne połączenie drogowe z Poznaniem i Wrocławiem, a co za tym idzie autostradami A2 i A4, a w kontekście międzynarodowym z Czechami, Niemcami oraz poprzez Trójmiasto również ze Skandynawią. Ponadto przez Leszno przebiega również droga wojewódzka nr 323 Leszno – Góra – Studzionki oraz droga wojewódzka nr 432 Leszno – Krzywiń – Śrem – Środa Wielkopolska– Września. Układ dróg zapewnia bardzo dobre połączenia z terenami Wielkopolski, Dolnego Śląska oraz Górnego Śląska oraz dobre z terenami Pomorza i Mazowsza.
Większość ulic w mieście ma przekrój jednojezdniowy o dwóch pasach ruchu. Odcinki dwujezdniowe występują na ulicach:
Aleje Konstytucji 3 Maja – Aleje Piłsudskiego – Poznańska (DK5);
Szybowników – Grota-Roweckiego – Aleje Jana Pawła II (DK12);
Aleje Krasińskiego;
Mickiewicza;
Estkowskiego (DK12).
W 2012 r. – w porównaniu z 2008 r. – liczba wypadków drogowych zmniejszyła się o 17% (z 78 do 65). Zmniejszyła się także liczba osób, które zostały ranne w wypadkach – z 87 do 73. Nie uległa natomiast zmianie w 2012 r. w porównaniu do 2008 r. liczba zabitych w wypadkach (2 osoby). O ponad 7% – z 1018 do 944 – zmniejszyła się liczba kolizji drogowych.
Przewozy autobusowe
Sieć transportu miejskiego na obszarze Leszna i gmin ościennych tworzą linie komunikacji autobusowej organizowanej przez Urząd Miasta Leszna.
Komunikacja miejska, wykorzystując infrastrukturę drogową, korzysta z 222 przystanków, w tym 106 (48%) – wyposażonych w wiaty przystankowe.
Według stanu na 30 września 2013 r., długość linii autobusowych MZK w Lesznie wynosiła 134,2 km, natomiast długość tras – 79,1 km. Przeciętna prędkość komunikacyjna w wynosiła 21,49 km/h.
Według stanu na dzień 30 września 2013 r., w ramach leszczyńskiego, miejskiego publicznego transportu zbiorowego, wykonywano 280 kursów w dni powszednie, 109 kursów w soboty i 69 kursów w niedziele.
Linie autobusowe
W Lesznie 13 linii autobusowych obsługuje MZK Leszno, ponadto realizacją jednej linii zajmuje się firma Usługi Przewozowo-Warsztatowe "Waldmar" z Osiecznej.
Linie MZK Leszno
Linia przedsiębiorstwa Usługi Przewozowo-Warsztatowe "Waldmar":
Transport rowerowy
Rozbudowywany jest układ dróg rowerowych i podsystemu rowerowego z niezbędnymi urządzeniami towarzyszącymi. Łączna długość ścieżek rowerowych (w tym ciągi pieszo-rowerowe oraz wyznaczone kontrapasy i kontraruch) wynosi 43,7 km.
Na terenie miasta Leszna istnieją wytyczone ścieżki rowerowe. Wyszczególnione poniżej ścieżki rowerowe przebiegają zarówno przy drogach publicznych, jak i leśnymi duktami czy drogami polnymi. Poniższe ścieżki rowerowe pełnią funkcję lokalnych produktów turystyczno-rekreacyjnych. Są to następujące ścieżki przechodzące przez miasto Leszno:
Szlak dookoła Leszna (25 km) – szlak poprowadzony w całości ulicami Leszna;
Wśród pól do Lipna (25 km) (szlak przechodzący przez Miasto Leszno, Gminę Święciechowa oraz Gminę Lipno) Leszno – Święciechowa – Wilkowice – Mórkowo – Leśniczówka;
W stronę bukowego jaru (23 km) (szlak przechodzący przez Miasto Leszno oraz Gminę Lipno) Leszno (Gronowo) – Wilkowice – Mórkowo – Leśniczówka;
Szlak powiatowy (24,5 km) (szlak przechodzący przez Miasto Leszno, gminę Lipno, Gminę Włoszakowice) Leszno – Wilkowice – Krzycko Wielkie – Włoszakowice – Grotniki – Boszkowo;
Szlak Leszno-Dominice (15,6 km) (szlak przechodzący przez Miasto Leszno, Gminę Święciechowa oraz Gminę Włoszakowice) Leszno – Strzyżewice – Ogrody – Trzebiny – Niechłód – Zbarzewo.
Lotnictwo
Na terenie miasta Leszna funkcjonuje lotnisko Aeroklubu Polskiego (Centralnej Szkoły Szybowcowej), usytuowane ok. 3 km na zachód od centrum miasta. Lotnisko ma nawierzchnię trawiastą umożliwiającą obsługę samolotów sportowych oraz małych samolotów mogących lądować na trawie. Lotnisko ma charakter cywilno-sportowy.
Wojewódzki Szpital Zespolony posiada lądowisko sanitarne, zlokalizowane przy ulicy Jana Kiepury. Ponadto fragment DK 5, na odcinku Rydzyna – Leszno funkcjonował kiedyś jako drogowy odcinek lotniskowy (DOL) – pas pomocniczy dla wojskowego lotniska Wschowa-Łysiny, obecnie również nieeksploatowanego. Odcinek ma szerokość 20 m i długość 2280 m.
Najbliższe porty lotnicze znajdują się w odległości 80 km – w Poznaniu (Port Lotniczy "Ławica") oraz odległości 110 km – we Wrocławiu (Port Lotniczy Wrocław S.A.) od Leszna.
Przypisy
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"redpajama_set_name": "RedPajamaWikipedia"
}
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## Dedication
To everyone sitting in their bedrooms in the 2am stillness,
listening to that song that makes your heart ache
## Contents
_Cover_
_Title Page_
_Dedication_
Prologue
Hanna
Dia
Jules
Elliot
Hanna
Jules
Dia
Dia
Hanna
Jules
Hanna
Jules
Hanna
Dia
Elliot
Jules
Hanna
Dia
Hanna
Dia
Dia
Jules
Jules
Dia
Dia
Hanna
Hanna
Elliot
Jules
Hanna
Jules
Dia
Jules
Dia
Hanna
Jules
Dia
Hanna
Hanna
Hanna
Elliot
Jules
Elliot
Hanna
Dia
Hanna
Dia
Hanna
Elliot
Jules
Jules
Dia
Hanna
Jules
Hanna
Dia
Elliot
Hanna
Dia
Jules
Epilogue
_Acknowledgments_
_About the Author_
_Books by Rebecca Barrow_
_Back Ad_
_Copyright_
_About the Publisher_
## Prologue
**It's hot the second** the doors open. The kind of air that leaves you sticky straightaway, sweat trickling along your spine, and you almost can't wait for the sweet relief of the cold shower awaiting you at the end of the night.
The stage is empty; music thumps out of hidden speakers instead, electro Biggie and Blondie covers and sometimes Aaliyah, because of course Aaliyah.
A light-copper-skinned girl in hacked shorts and a Bikini Kill tee cuts through the crowd, holding tight to the wrist of another girl with hair bleached whiter than her pale skin. The first girl, natural curly hair blown out to wild proportions, hoists herself onto the stage. A third girl appears from the wings, the lights setting a sheen of purplish-pink on her deep-brown skin, and holds a hand out to the blonde.
All three duck back into the wings together, grabbing guitars, picks, drumsticks, courage. Then the music cuts, and the purple-green-red lights flash down, and a person with lime-green lipstick and a buzz cut squeezes past them. "Ready?" they say.
The blonde nods. "Always."
The buzz cut person walks out on the tiny stage, takes a position at the mic stand to cheers from the raucous crowd. "All right, everybody! Make some fucking noise for Fairground!"
The girl with the curls slouches up to the mic, a pick between her teeth. She tugs her shorts up on her hips and takes the pick from her mouth. She doesn't bother introducing herself or the others, but hits a jarring chord and runs into their first song at breakneck speed, the blonde banging hell out of her drums and the bassist kicking into frenetic rhythm, sweat slicking away from her basketball jersey.
They only have fifteen minutes, but it's fifteen minutes more than they used to have. They speed through their short set list, and the crowd cheers, raises their hands, and gives in to the weird mix of punk and grunge and R&B.
They sing themselves hoarse in that short time, and when it's over—too fast, too soon—they leave the stage, clutching their guitars and drumsticks like precious jewels. The next band will start soon, replace them in the crowd's memory, but it doesn't matter. They did what they came to do.
Sometimes they stay to listen to the other bands and dance themselves silly, but tonight they're forty minutes from home and they have a curfew. Out in the parking lot an older girl with lilies inked on her upper arm and locs to the middle of her back waits by a beat-up van. "Good set," she says, and pulls out keys so they can load the drums into the back. "Dia, your turn to choose. McDonald's or Dairy Queen?"
"DQ," the curly-haired girl says. She craves a Blizzard.
The one in the jersey lifts it to wipe the sweat from her neck, sticky from in there. "They know us now, a little," she said. "Hanna, get up."
The blonde stands up from her crouch, unsteady. "You're not my mom, Jules," she says.
It's late and dark, and Dia opens her red-painted mouth wide and yells a note out into the California night, a release of residual energy.
Their tattooed chauffeur laughs at the echo. "Come on, we gotta go."
Jules rolls her eyes but doesn't mean it. "Yes, Ciara."
They pile into the front of the van, legs and arms and guitars. The blonde—Hanna—turns the radio to the nineties station and they wind the windows down and sing along to Mariah Carey as they peel off into the night.
## Hanna
**Hanna kicked her locker,** the noise echoing down the empty hall. "Useless piece of shit," she said under her breath. _"Open."_
She gave it a final wrench and it opened, finally, the mess inside spilling out. Perfect.
She only needed her paper for last period, the last assignment she had to turn in before officially being done, _d-o-n-e_ , with high school. Hanna found it, folded it in two, and put it in her backpack. Then she gathered everything else—old notebooks, candy bar wrappers, letters she'd never given to her parents—and carried it down the hall, where she dropped it all into a trash can. It made a satisfying thud as it landed, and Hanna smiled to herself.
_Four hundred and seven._
She took her lunch outside, found an empty table, and ate her lukewarm slice of pizza while she watched her classmates whirl around without her. Everybody was so _excited_ : all week long she'd kept seeing people hugging each other and bursting into fake noisy crying, everybody taking a thousand selfies everywhere she looked. Like now—Ali Siberski and Priscilla Nguyen posing by the vending machine, faces pressed together as they snapped away. Michael Brewer signing some skinny guy's yearbook. Gloria Vazquez sitting on the lap of some kid from the basketball team—Hanna couldn't remember his name. She couldn't remember a lot of people's names, actually, and yet still she felt a tug of sadness that this was the last time she'd see most of them.
She rested her chin in her hands and watched. They all probably knew her name, for the wrong reasons. But they wouldn't miss her after graduation. She wouldn't actually miss them, either.
Hanna tipped her face up to the sun. Five days till graduation. Four hundred and seven days since she'd given up drinking.
_Given up drinking_. That made it sound so much easier than it had been. It didn't take into account the blackouts and fallouts, the repeated attempts and failures to quit. The night her little sister had found Hanna in her bedroom in the middle of the night, not breathing, and called the ambulance all on her own. The night when Hanna had to be treated for alcohol poisoning for the _second_ time, when she woke up from the blur of the past couple of years and finally realized that her only options were complete self-destruction or sobriety. The pleading from her parents, and the promising from herself, and the ending up in a rehab facility four hours away.
Four hundred and seven days. How long it had been since she'd realized that no one would be surprised if she drank herself to oblivion.
At least she had one thing to be proud of.
Hanna sat there until the bell rang, listening to her classmates' chatter, screams of laughter, plans for some "major prank" that Hanna was sure was most definitely _not_ going to be good. When they all started to stream out, on their way to their last few classes, she joined them, let herself be carried back into the building and through the halls. And on the stairs up to her English class, she passed them.
Dia Valentine and Jules Everett. The two people who _really_ weren't going to miss her after graduation.
_Whatever,_ Hanna thought. _Who even cares anymore?_
They were gone in an instant, disappeared in a flash of long braids and ripped jeans, and Hanna kept on to class, shaking her head as she slipped into her seat and slammed her bag down.
"Good afternoon, Miss Adler." Mr. Matthews looked at her pointedly. "If you could try not to destroy my classroom before the end of the year, that would be great."
Hanna rolled her eyes as she sat down, looking up to see the teacher still watching her with this expression that was equal parts _Can you believe this girl is actually graduating?_ and _How soon can I get her out of my class?_ It was the same expression she'd seen on pretty much every teacher's face this week, with the exception of the few who'd actually helped her get back on track last year.
_Well, sorry,_ she wanted to say to Mr. Matthews right now. _Sorry I'm not giving you one last thing to hold against me! And really, really sorry for not being the complete train wreck you expected me to be! You must be so sad to miss out on telling everyone how right you were._
Hanna bit her tongue. That kind of stuff got filed away in the Don't Say This Out Loud folder in her head, the one where she put everything that would get her in more trouble than she needed.
"Sure," was what she said instead. "I'll try."
A few people snickered at that but Hanna ignored them, searching for a pen as Mr. Matthews walked up and down, handing out their final exam papers. "You have forty-five minutes," he said. "Once you're done, you can go, but make sure to hand your papers in on the way out. Okay?"
Hanna flipped to the first question: _Compare and contrast the presentation of loneliness in Shakespeare's_ Hamlet _and_ To Kill a Mockingbird _._
Easy.
## Dia
**"Dia Gabrielle Valentine!"**
Dia rubbed her red-painted lips together before fixing a smile on her face and marching across the stage to meet Principal Cho.
_Finally,_ she thought. Four years of one-a.m. assignments and sleep-deprived morning classes, Jesse explaining math to her and panic-inducing finals, all over now.
All ceremony long the clapping had been constant, punctured by an occasional cheer or ear-splitting whistle. Now it warmed Dia and sent a zip of staticky excitement down her spine, because everybody out there was clapping for _her._ She'd done it. She'd actually graduated.
_How's that for a Fuck You?_
Dia shook Principal Cho's hand, the last time she'd ever be face-to-face with her, probably, and the principal gave her this wide smile.
"Congratulations, Dia," she said, and the unending applause played a perfect backing track to her words. "You've earned it."
"Thank you," Dia said, and as she wrapped her fingers around the diploma she smiled hard enough to hurt.
She looked out into the rows and rows of families: Jules's parents and her brother, Danny, were right up front, and the sun hit right in her eyes as Dia searched farther back. She had to squint but there they were—her parents, cheering and standing up so Alexa could be hoisted high in the air.
She waved as she moved on, praying she didn't trip in her heels, and then she was down the steps and in line with the rest of her class. And then they were tossing caps in the air and everyone was yelling and Dia turned her face to the open blue sky and finally, finally, _finally_.
Once the ceremony was done, the formality fell away. Families and graduates mingled together, a singular mass on the field normally reserved for the girls' soccer team. Dia found Jules quickly, and they ducked out of the way of somebody's family photo session. "Can you believe it?" Dia laughed. "We're free! We're _actually_ free."
"Oh, yeah," Jules said. "So free. Until we have to go to work tomorrow."
Dia waved her off. "Work, shmurk. At least work doesn't involve ten-page papers on Congress or constant fill-in-the-bubble tests."
"No," Jules said. "Just getting yelled at by people who think they're better than us and a hideous amount of polyester."
Dia almost tripped as her heels sank into the grass, a last-second grab for Jules's arm the only thing stopping her from face planting. "These shoes, I swear."
"This everything," Jules said, shaking her head so her box braids snaked over her shoulders. "I can't wait to get out of this gown."
"I can't wait to get out of this _dress_ ," Dia said, tugging at the stretchy green fabric that clung a little too much to her every in and out. "I feel like any second could be a nip slip."
"The Valentine tits," Jules said, cracking a smile and raising her voice. "They will not be denied!"
"Shut up!" Dia swatted at Jules, but she was laughing too much to get her aim right. She should have bought something new, but it seemed like a waste to spend money on a fancy dress she'd wear only once when there was so much else she needed.
"Okay, I have to go find my parents," Jules said, sounding less than enthused. "You should hear my dad today. 'It's your graduation! It's a big deal! You're going to college!' And I'm like, 'Dad, I'm going to community college. The college that _you_ teach at. I'll still be tripping over your shoes every morning. Calm down.'"
"Don't get down on us," Dia said, narrowing her eyes at her friend. "This is a big deal. Hello, we are no longer high school students. We are _college_ students. Sure, we're still going to live at home and we'll see your dad on campus, but we did it. We got ourselves here, so stop shitting on it."
Jules rolled her eyes. "Fine," she said. "It could be worse. But it could be better, too."
Dia gave her a warning look. "Don't," she said. "Didn't we agree not to play this game anymore?" The game where they fantasized about the life they could maybe have now, if things hadn't fallen apart so spectacularly. The life where they were leaving for LA tomorrow, instead of looking forward to a summer of working their minimum-wage jobs and warming milk at three a.m.
It used to be fun to imagine that life, but now it was depressing, and so Dia had said they couldn't do it anymore.
"Yeah, yeah," Jules said. "Am I not allowed to dream?"
"Only for today," Dia conceded. "And then we go back to reality."
"Fine," Jules said. "Are we still going to the party?"
Dia nodded. "Come over before so I can do your makeup."
"Yes, ma'am." Jules gave a mock salute. "Later."
Dia watched her walk away, and then she heard her name being called and turned to the sound. Her mom and dad, Nina and Max, waved, Alexa propped up on her mom's hip. "Over here!"
Dia steeled herself to cross the grass in her heels again, and began walking over. Alexa squirmed out of Nina's grip, straining toward Dia as she approached. Her mom lowered the toddler to the ground and Alexa took a second to find her balance before breaking into a run. "Mama!"
"Hi, baby." Dia crouched, holding her hands out toward her daughter. "C'mere!"
She ran into Dia's arms, and Dia swept her up before peppering her apple cheeks with kisses. "Did you see me, Lex?"
The little girl nodded. "And I saw Juju," she said before jamming her thumb into her mouth. Dia caught her smile before it really started—she knew that habit needed to be broken before it became a real problem, but god, if it didn't look cute. "No, no," she said gently, and eased Lex's hand from her face. "We don't do that."
"Oh, leave her be," her dad said. "It's not going to kill her."
"No," Dia said. "But it _is_ going to mess with her teeth, and in ten years I'll be paying for braces."
"She's right, Max," her mom said, smoothing down her cropped, relaxed, jet-black hair.
"Point taken," he said with an easy smile. That was how it always went: her mom laid down the law and her dad gave in to whatever her mom said, too easy-going to ever want to cause an argument. Her mom the engineer, former military, sharp and strong, and her dad the musician, now EMT: an opposites-attract pair, always had been, always would be.
Her dad took out his phone. "Now, smile nice for me, both of you. Lala, say cheese."
"Cheese!" Lex yelled, and Dia laughed.
"Okay, okay," she said. Dia shifted the baby onto her hip and straightened the bow around her curls. She put on her brightest smile as her dad snapped picture after picture, hoping that her boobs weren't spilling out and that there wasn't lipstick on her teeth and that she didn't look too much like she was about to cry.
Because she really had done it. _Graduated!_ Two years ago, no one had believed she could. Or, not no one: Jules had been behind her all the way, and her parents, and Jesse. Principal Cho, too, pulling strings and signing off on extra credits and making sure Dia got her chance. _Without them, I wouldn't be here,_ she thought.
But everybody else? People she'd called her friends, teachers she'd liked, her mom's coworkers? They'd rolled their eyes when Dia had said she was still going to graduate and go to college. When she insisted that yeah, she had a baby, but that didn't mean those things were completely _impossible._
_You have no idea,_ people said. _Having a baby changes everything._ She imagined them thinking, _You're going to end up on welfare anyway, why put it off?_ And _Stupid slut. Way to ruin your life._
"Dia!" her mom called. "Look happy, huh? It's your big day!"
Alexa laid her head on Dia's chest, and Dia raised her chin, her skin almost vibrating with all the love she had for this little girl. Yeah, having Lex had made things more difficult, and yeah, she could have made a different choice, but in the haze of everything back then, this was what she'd decided on. And when she looked into her daughter's big, brown eyes, she knew that she wouldn't change the way things were. Not for anything.
Eventually Dia's cheeks began to ache from all the smiling, and Alexa began to fuss, her irritable _I'm hungry and I could use my words but I don't want to_ whimper. "Are you done?" Dia said through her rictus smile. "Is ten thousand pictures enough, or do you need more?"
"Don't be smart," Max said, but he put his phone away and held up his hands. "I'm done, okay? Forgive me for wanting to preserve this momentous occasion."
Dia tipped her head back to feel the sun, warm on her face. The rubbing of her shoes, her too-tight dress; the slippery gown and crisp diploma; the scent of cut grass in the air—she didn't need a photograph to remember any of it.
## Jules
**Jules rang the bell** only as a formality, and then walked right in, announcing her presence as she opened the refrigerator. "I'm getting a soda," she called out.
"Jules!" Dia's dad came into the kitchen, a look of mild annoyance on his face. "You treat my house like a hotel."
Jules gave her best impression of an apologetic smile. "Only because you said I could."
"Hmm," Max said. "I might start regretting that." But then he laughed. "I didn't get to say congratulations earlier, to my second-favorite graduate."
"It's okay, Dia's not here," Jules said, pulling her mass of braids over her shoulder. "You can be honest."
Max narrowed his eyes. "I remember when you were a kid and actually showed me some respect," he said. "Where did it all go wrong?"
"Ask Dia."
He shook his head. "You hungry?"
"I'm good," Jules said. "We had a ton of food at the house. Thanks, though." Her mom had been cooking the whole day before: stewed chicken and rice and curry goat, mac pie and roast lamb, a true Barbados feast in California. They'd crowded around the kitchen table and Jules had eaten her body weight in all of it while her parents got all teary-eyed and started reminiscing about Jules's birth. That always led into the story of how they met, and then they'd started dancing around the kitchen, and Danny had rolled his eyes, but Jules loved to watch her parents' love.
Perfect celebration.
"All right," she said. "I guess I better go get ready for this party."
"At least act excited," Max said. "It's your graduation night. You only get one."
"It'll be the same as every other party we've ever been to," Jules said, shrugging.
"Go anyway," Max said, running a hand over his locs. "Take Dia and make her have fun, for once."
"Fine," Jules said, grabbing her soda and heading toward the stairs. "I'll try!"
Jules didn't knock before entering Dia's bedroom either, her friend at her computer, her honorary niece scribbling on the floor. Dia looked up at her, immediate annoyance on her face. "I have nothing to wear."
"Juju!" Alexa stretched her arms up toward Jules, hands grabbing at the air. "Up!"
Jules did as she was commanded, scooping Lex up from the floor and settling the baby's weight on her hip. "Hi, sweet one," she said, nuzzling her nose against Lex's cheek. Then she looked at Dia. "Don't even start with me."
Dia stood. "Have you seen that show with the two girls and they're dating, but one of them's a spy?"
Jules arched one eyebrow. "I'm an eighteen-year-old lesbian with internet access. I've seen everything that even _hints_ at two girls being into each other. The GIFs are imprinted on my eyelids."
"Okay, well, in one episode the tall one has this amazing red jumpsuit. _That's_ what I want to wear."
"Let's start with something you actually own," Jules said.
Dia made a face. "I don't know if I even want to go to this thing anymore."
"Me neither," Jules said, sitting on the edge of Dia's bed. She let Alexa loose on the comforter and shrugged her backpack off. "But we promised."
"Who?"
"Each other." In the cafeteria, two weeks ago, looking at the text invite on Dia's phone. _When do we ever go anywhere?_ Dia had said. _Me with Lex, you working all the time. This'll be our last high school party ever._
_I'm so sick of folding jeans,_ Jules said. _Bagging people's groceries. One day I'm going to do something different._
_Of course you are._
_Okay,_ Jules said. _So let's go to this fucking party._
Dia steepled her fingers underneath her chin, and then nodded. "Okay. We'll go. And if it sucks, we get to leave after an hour. Deal?"
"Deal."
"Right," Dia said. "Now let me work my magic."
They set up the way they always did—or always had, Jules amended, back when they used to party and go to shows and roll around the streets of their town without a care in the world. Electro remixes and old-school punk playing through the speakers, Dia's makeup spread across the bed, a bag of chips within easy reach. Dia eased Jules's braids—fresh for graduation, the ten hours and numb ass and hypnotic click-click-click of Stephy's acrylics whipping through her hair oh-so-worth it—into a high pony, then started on Jules's makeup. Moments like these were what made Jules glad that they were both staying home for college: being apart, being without her best friend, the girl she'd loved since preschool? It would have felt like losing a part of herself.
Jules examined her face in the mirror when Dia was done: a simple flick of black liner on each eye and her thick brows framing them, clear lip gloss, and something shiny gold on her deep-brown cheeks. She got dressed in black jeans from her mall job, hacked at with a razor to make them look like the expensive ones from the store three doors down, and a gray shirt with the sleeves cut off, dipping low enough beneath each armpit to flash her black bra.
She played with Alexa while Dia painted her own eyelids deep blue and her lips a bright, shiny red. "I heard High and Mighty Kallas are playing at Revelry tonight," Jules said, making a polka-dotted lion dance with a robot as Lex clapped her hands. "If we get out of this party early enough, we could catch them. I haven't been to a show in forever." She sighed with longing. Cheap beer and sweaty dancing and pounding, punky music? It was _so_ good and she missed it _so_ much. The music scene was real in Golden Grove; Jules and Dia had been going to backyard shows and all-ages clubs for years before they'd picked up their own instruments and become part of it.
(Jules made herself stop. That was before, and this was now. They had no band. They had no Hanna. It made Jules ache thinking about it all.)
"HMK are completely unoriginal and you know it," Dia said. "They're the worst Glory Alabama rip-off."
Jules snorted. Anyone could be, and often was, labeled a GA rip-off by them—when your town had an amazing band that actually broke out, headlined every big festival in the US and overseas, played with legends like Sleater-Kinney and Melissa Auf der Maur, featured on the cover of not only _Rolling Stone_ but fucking _Vogue_ , too, there was a certain loyalty. Jules smiled, ready to drop a bomb on Dia. "You know they're touring soon, right?"
"What? Glory Alabama?" Dia said, and her voice jumped an octave. "Shut up. _Why_ is this the first I'm hearing of this?"
"First tour in five years," Jules said. "First time back here in almost _ten_."
Dia widened her eyes. "Oh my god, we're going," she said. "Oh my _god_ , we're going to get to see them? I don't care how much tickets are, I will work a month straight of morning shifts, whatever. We need to be there."
"I thought you might like that," Jules said with a laugh, and Lex laughed with her. "We can be the creeper fans who wait outside their tour bus, if you want."
"That is my dream," Dia said. She got up and opened her closet, taking out a plain white shirt, which she looked at for a second before putting back. "Is a dress too much?" she asked. "I have no idea anymore. But I want to look hot. Not like a mom."
"You are a mom," Jules said, glancing over. "A hot mom."
"You know what I mean."
"Wear whatever you want," Jules said, and she sat up. "Y'know, it's really only going to be people from school at this party."
"Okay." Dia stepped out of her shorts and held a blue-and-white vintage-ish dress in front of herself. "What about this?"
"Yeah, great," Jules said impatiently. "I'm only saying, if you were maybe trying to look especially good for a certain specific somebody—"
Dia pulled the dress over her head, coming up laughing. "I'm not."
"And," Jules continued, "if that certain somebody happened to be one Jesse Mackenzie . . ."
"I already said I'm not." Dia smoothed her hands over the striped skirt of the dress. "Juliana, a person can want to put on nice makeup and dress up and look good for reasons other than wanting to impress somebody," Dia said. "A person can want to put on nice makeup and dress up and look good solely for themselves."
Jules held up her hands. "All right, I take it back."
"You should." Dia turned her attention to Alexa, her entire face suddenly beaming. "What do you say, Lex? Is Mommy good to go?"
Lex opened her mouth in a big yawn, a squeak her only response. "You and me both, kid," Dia said with a laugh, and Jules reached over to tickle the baby until she was giggling wildly, too, all three of them exited and happy.
"Let's get out of here," Jules said eventually, breathless. "One last time, right?"
The bus dropped them off on the edge of a wide cul-de-sac lined with tall trees, branches still in the warm night, and Jules stared up at them. This was the Nice Side of Town, the side of Golden Grove where the big houses had glittering blue pools and cutesy mailboxes. The cars on the drive were shiny, brilliant, brand new; the lawns were green and dotted with flowers.
It made Jules feel small sometimes, how awed she was by these things, but wouldn't it be nice? Wouldn't it be nice to sit in your pretty house, and look out at your beautiful yard, and feel proud?
That was what she really wanted. That was what she secretly feared she'd never have.
A pretty, happy, shiny life.
Dia was already walking down the street, and Jules took several long strides to catch up with her, bouncing in her fresh Nikes. "I changed my mind," she said. "Come on. Let's go to the show. It'll be so much better than this."
"We're already here!" Dia whirled around, the skirt of her dress spinning, and she looked like some fifties movie star. "We made a deal! It's going to be fine."
"It's going to be annoying." Jules slowed as they approached the house with the music pounding out from the open front door. "But whatever. You want your party, you get your party."
"Attagirl." Dia hooked her arm through Jules's. "Chin up, kid," she said, an impression of the old stars she looked like tonight. "Let's have fun."
_Fun,_ Jules thought. _I can do that._
Once they were inside it didn't take long for Jules to remember that yeah, parties were annoying—people spilled drinks on you and yelled way too loud in your ear—but they were also loose, wild, open. In the big living room two guys—one on drums, one behind a synth—made loud, bass-heavy music, impossible not to nod your head to.
Jules watched them play. She recognized them from around town, even disconnected as she was right now. They'd won the Sun City Originals contest last year, too, the contest that she and Dia and Hanna had always planned to enter, before everything. It was like a rite of passage around town and beyond to at least try to win. Not for the prize—five hundred dollars and your song on the station playlist was cool, but it was more about bragging rights. So that one day in the future, when you were selling out tours, you could say that was where you got your start and Look at Us Now.
"Jules." Dia was at her elbow. "I'm getting a drink. What do you want?"
Jules pulled her attention away from the music and raised her voice. "Whatever," she said. "And lots of it."
Because Dia was right: this was their last-ever high school party, the last time they'd see some of these people, and didn't they deserve to have a good time?
_Yes,_ Jules decided, and so she let the girl she'd sat next to in freshman English spin her around the living room to that slick electronica. She took the shot offered by Oscar Rush and followed him out onto the deck. She watched Oscar and his buddies as they threw themselves fully clothed into the pool, and she stepped closer to the edge. She considered jumping herself, and thought about what it would be like to hit the cool water and disappear under the surface, how long she could swim around down there before her lungs began to ache.
Jules crouched down and dipped her hand into the blue. Oscar was dunking some kid under, water splashing everywhere, and Jules laughed. She looked up, searching for Dia. But the gaze she found was not Dia's.
It was Hanna Adler's.
They stared at each other.
Jules hadn't thought she would be here. Wasn't sure why she had thought that, because it used to be that Hanna was the life of any and every party. Why would that be different? Just because Jules wasn't at those parties anymore didn't mean that Hanna wasn't.
Jules stood and hitched up her jeans. Her skin was hot to the touch and her mouth had that sour-sweet nervous taste. _How do I look to her?_ she thought. To Jules, Hanna looked . . . like Hanna. An inch of dark roots in her blond hair, dark circles beneath her eyes, the same way she looked every time Jules glimpsed her in the packed hallway at school. Well—that wouldn't happen anymore. Maybe this was it: maybe Hanna was one of those people Jules would never see again. That didn't stop the twitch in the back of her mind, the reminder that this girl used to be her friend.
But tonight wasn't meant for her to fixate on things from the past that she couldn't change. Or people, who wouldn't change.
A touch on her shoulder, and Jules turned away from Hanna's empty stare. "Hey." Dia handed her a cup filled to overflowing with a pinky-orange liquid. "What are you doing?"
"I—" Jules started to say _I saw Hanna, she's here,_ but she stopped herself. What was the point? That was all so old now, the three of them. Forget it. "Nothing. Waiting."
Dia tipped her head to the side, her eyes so shiny and excited. "For what?"
Jules knocked her cup against Dia's and downed the contents—too warm, sticky, and syrupy but with enough sting to perk her up—before smiling at Dia. "Isn't that the question?"
## Elliot
## JULY
**Elliot has no idea** who this house belongs to.
He has no idea whose party this is.
But he knows he's having a good time.
"Nolan," he says—or yells, maybe—"what time is it?"
Nolan checks his phone. "Ten fifteen."
"Okay!" Elliot has to be home by eleven, according to his dad, and midnight, according to his mom. He'll roll in sometime between the two, probably, and if he's lucky he won't get grounded.
He wanders outside, sipping the punch that stings as it goes down. It's packed out here, all these people crowding a makeshift stage where a band plays. He might not know where he is but these parties are all the same: music outside, drinks in the kitchen, a circle of stoners in an off-limits bedroom.
A punch hits his arm and he swears. "Kwame, you asshole."
Kwame salutes. "I'm out, man. Early shift in the morning. You coming to Mike's tomorrow?"
"I'll be there."
He's left alone again, and pulls a hand through his curls as he gets closer to the band. It's these three girls going hell for leather up there, and the music's good, but what catches Elliot right away is the girl in the front, singing.
It's not just that she's hot—though she is. Short and curvy and in these skin-tight jeans that make Elliot think about pulling them off and—
He shifts. Calm down.
She's up there, playing her red guitar like she wants to hurt it, and singing in this raspy voice, and winding her body like no one's watching her. But Elliot is.
Then he looks around and realizes: so is everyone else.
Later.
Elliot's trying to ignore Nolan arguing with him over their last baseball game, the mistakes Elliot apparently made that cost them the win, his tendency to freeze. He's trying to ignore Nolan, because on the other side of the yard the girl from the band stands right in his eye line.
"Uh-huh," he says, nodding without looking at Nolan. "What-ever."
He can't hear the girl talking, but she's using her hands to tell a story, drawing swooping circles in the air, and the other band members are watching her intently. She looks like the kind of person you have to listen to, he thinks. If it was him standing in front of her, watching those dizzying hands, he'd be listening.
_Go over,_ he thinks. _Say hello. Ask her name._
Is that an asshole move? Butting in while she's busy talking? Or is it okay? He won't know her if he doesn't talk to her. But if he talks to her she might not want to know him.
Elliot cuts Nolan off. "I'm getting a drink," he says. "You want one?"
"I'm good."
In this stranger's kitchen he grabs a half-empty bottle of rum, then puts it back. The clock on the microwave says 11:20.
_Hi,_ he practices in his head. _You were really good._
_I like your band._
I like your band? What is he, twelve?
_Hi. Good set._
Better?
_Hi. I'm Elliot._
A coppery-brown hand reaches for the stack of cups at the same time Elliot does and he looks up to see the band girl right there. "Sorry," she says with a shake of her head, and the tight spirals of her curls fan around her face. "You go."
"No," Elliot says, his mouth dry. "After you."
She smiles at him, creases appearing around her eyes. "Thanks."
Silence.
Well, not silence: the noise of other people coming for drinks, more music outside. But silence from Elliot, his closed mouth.
"You know you're on the wrong coast."
"What?"
The girl finishes filling her cup and then points at Elliot's chest. "Biggie," she says, and he looks down at his shirt, The Notorious B.I.G. in his chains and crown, and for a moment Elliot thinks she's being serious, her face is dead serious, but then she cracks. "It's a joke."
Elliot manages a laugh and pulls a hand through his hair, a nervous tic he can't stop doing tonight. "Right," he says. "It's my mom's fault. She's from New York."
"I guess you get a pass," the girl says, and then she sees something over his shoulder and her face clouds over. "God, that was fast."
Elliot turns right as a white girl in supershort shorts stumbles into the kitchen. "There you are!" the girl sings, swaying slightly. "I've been looking for you."
"Hanna, what are you doing?" The band girl has forgotten about Elliot entirely, he can tell, as she hurries to the girl he now recognizes as the drummer from earlier. "Whoa! Okay, it's time to call Ciara. You're going home."
They're gone in a second and Elliot's left kicking himself. He should've taken his chance.
This is exactly what Nolan means when he says Elliot freezes.
It's 12:17, according to Elliot's phone. He's definitely going to be grounded.
He waits outside for Nolan, still in the house flirting with this guy with two sharp bars through his eyebrow. It's a hot night, the kind where you wake up in a layer of your own stale sweat, and Elliot holds his hand out into the empty air.
"We need rain." The voice comes from somewhere to his left, and then the band girl steps into his sight. "California's thirsty."
"Right," Elliot says, and winces. Is he capable of saying anything multisyllabic to this girl? He clears his throats, shoves his hands in his pockets. "Is your friend okay?"
The girl makes a face that Elliot can only describe as Over It. "Yeah," she says. "Too much to drink. She does it all the time. It's fine." The way she says _It's fine_ tells Elliot it's anything but.
"You played tonight, right?" he says next. "You were good."
She shoots him a look, like it's the most obvious thing in the world. "I know."
Confident. Elliot decides to match her. "I'm Elliot," he says. "What's your name?"
She looks him up and down like she's vetting him, deciding whether to trust him or not. Eventually she smiles and looks right at him, these deep, dark eyes, and Elliot feels like she's laid him open right there on the street. "Hi, Elliot," she says. "I'm Dia."
## Hanna
**Hanna matched her steps** to the beat of the frenetic drums pounding in her headphones, stepping on every crack and flattened piece of gum. Bad luck couldn't hurt her any more than it already had.
_Four hundred and twelve days,_ she thought. No; it was after midnight now. So four hundred and thirteen.
Hanna ducked into the alley, which was dark enough to make her clutch her keys between her fingers. It had been a complete waste of time, that party, and Hanna had known it even before she'd walked in to the sight of Oscar Rush dancing shirtless on his parents' dining table. She had only gone because everybody else was going, and did she really want to remember her graduation as another night she spent alone at home in her bedroom?
In hindsight: yes.
The moon was a skinny sliver above Hanna's head, whatever light it gave off masked by neon streetlights. She skipped through track after track as she walked, sometimes letting no more than three seconds elapse before moving on, searching for something to set her nerves alight. On Hayworth Boulevard she let Brody Dalle scream about underworlds and ghost towns, and the perpetual itch that crawled up and down her spine felt satisfied, for three short minutes. But then the song was over and the yearning flooded back and she stopped, to wonder, to stare up at that crescent moon.
If she'd known Jules and Dia were going to be there, she definitely wouldn't have gone. But from now on, she wouldn't have to see them, would she? At school it had been hard to avoid them, even though she tried; there were only so many places to go, and she couldn't miss them walking the halls together. And as much as she pretended it didn't, it hurt—hurt deep, far down, in the place she stored those shattered pieces of her heart, next to the guilt. Because it used to be the three of them, always. Sharing fries and going to watch the BMXers pulling tricks after school. Sleeping three to a bed and switching clothes and rolling into places like they were the most important people in the entire world. Singing themselves hoarse, throats raw, on makeshift stages in people's backyards. Those were _their_ moments, Hanna and Dia and Jules, always.
Until they weren't.
For four hundred and thirteen days, she'd been sober. For all that time, plus eight months more, they had not been friends. Things had changed, fissures and cracks becoming an all-out chasm so quietly and slowly that Hanna almost missed it. And when she'd realized, it had been too late. Nothing she could fix.
And she was alone.
Hanna sped up as she started down the long hill that eventually turned into her street. That was then and this was now, and now she was free. She didn't have to see them anymore, as long as she avoided their parts of town. And in September she'd start working full time, doing admin at a medical company, and she'd save all the money she made, and in a few years she'd be able to leave. And leave so much behind—her ex-friends, her mom's constant criticism, this whole town.
The music in her ears sped up, some discofied girl band, and Hanna matched her pace to it. She was almost running when she passed the Dempseys' place with its front yard of wildflowers, the biggest burst of color on the block. Finally her key slipped soundlessly into the lock of her own house and Hanna stepped inside.
It was still and quiet, and Hanna exhaled into the comfort of it. "You are okay," she whispered out loud, fast breaths. "You are here. You are okay."
She waited until her heart had slowed to its normal rhythm before climbing the stairs. She bypassed her bedroom and eased open the door plastered in pretty flower cutouts.
Molly's eyes flickered open as Hanna entered, as she slipped off her shoes and stole under the covers into her little sister's bed. "Hey, birdy."
"Hey, birdbrain."
Hanna flicked Molly's arm. "Shut up."
Molly's laugh turned into a yawn, her tongue peeking out catlike. Molly hated all that, Hanna knew—the nicknames and being told how adorable she was—now that she was thirteen. But being annoying was Hanna's prerogative as big sister, right?
"Did you have a good time?" Molly whispered, her eyes shining bright in the dark. "Were there cute boys there? Or cute girls? Did anyone get in a fight?"
"Parties aren't like they show them on TV, Moll." Hanna tucked her hands under her armpits and screwed up her face. "Trust me, you'll understand once you get to high school. They're another ridiculous way for people to decide who's cool and who isn't, and to get drunk so they can do things they don't have the courage for otherwise."
Molly's eyes searched her face, cautious trust in that look. "You didn't, right?" she asked, and her voice held the fear that her face did not. "Get drunk?"
See, she might be thirteen, but sometimes she sounded thirty. "No," Hanna said, and god, the crushing wave of guilt that broke with Molly's words, it could have drowned her. She wasn't really asking, Hanna knew; Molly knew her drunk. She wouldn't have to ask. It was more like she was checking, making sure, that she could really believe the sister in front of her eyes. And Hanna knew that was nobody's fault but her own. That was what happened when you let your little sister find you unconscious among empty bottles. "No, I didn't. Don't worry."
"I wasn't worried," Molly said. "Just wondering." She rolled onto her back, blond hair mussed around her face, and did her trademark shoulder-shrug-eye-roll combo. "If you don't have anything good to tell me, then why did you wake me up?"
"It's my right as a big sister." Hanna smacked a kiss on Molly's cheek before slipping out of the twin bed. "It's late," she said. "Go to sleep, okay? I'll see you in the morning."
"Will you go with me to the bookstore tomorrow?" Molly widened her eyes. "Please?"
"After work, sure," Hanna said. "Whatever you want. Now go to sleep."
She left the room and pulled the door almost closed, and then she stood there watching through the gap as Molly fell asleep, a ball beneath the covers. Only when Hanna was satisfied that her sister was absolutely asleep did she go to her own room. She stripped to her underwear, put on an old shirt, and climbed into bed.
What she'd told Molly was true: parties weren't like they were on TV. Hanna used to love them—they'd been fun for her. Not so much for everyone around her, having to hold her hair back and pick her up off the floor and rescue her from her whirlwind of destruction. But being drunk made her feel invincible, gave her cover for so many things. She said whatever she wanted, she did anything and everything that she got the urge to, and when she fucked up (often, and in big ways), she'd brush it off: "I was drunk! It's no big deal."
Until it was a big deal, a rubber tube down her throat, no-oxygen-to-her-brain kind of big deal.
Hanna had thought that getting sober would make everything better. Turn her into this shiny, new Hanna. But she'd realized pretty quickly that the girl who said every little thing that came into her head, who did whatever she wanted without thinking, who was careless and said and did things that really hurt the people around her—that wasn't because of the drinking. It was her, the way she was wired.
She'd thought she'd be happy sober. But her problems didn't go away; they shifted. Now she was always working out how to keep herself in check, to not say those awful things, to not act out. How to rewire herself so her instinct was not complete and utter self-destruction. How to not let the guilt and resentment and anger sweep her up and carry her to the point of giving up, giving in, letting herself drink again because at least then she could forget about it all.
Hanna sighed into the dark. Every night now this happened to her—no sleep, thoughts racing. So she did what she usually did: got out of bed and crouched to yank open the bottom drawer of her nightstand. Inside, thin black notebooks stacked up, some battered and creased, some shiny-new.
Hanna pulled out one with the spine cracked but not yet falling apart, and a pen that always leaked black ink all over her fingers. This was the best way she had to combat the self-loathing—well, the best that wasn't relapsing. She sat with her back against her bed and flicked to a clean page, past all the words that sometimes felt like poetry, sometimes self-indulgent ramblings, but in the end, were always songs.
She and Dia and Jules weren't friends anymore, no. They didn't make music anymore, no. But they did not own her writing; the lyrics that came from her brain and heart and pit of her stomach were Hanna's and Hanna's alone.
Without thinking too much about it, she scratched out a first line.
_She wears your face—I've seen her._
## Jules
**They left the party** when it got sloppy, spilled drinks on the living room rug and vomit in the kitchen sink. The last bus only took them as far as Dia's, and Jules walked the rest of the way home, exhausted when she finally got there. She crept in quietly, then stood over the kitchen sink to eat a bowl of cereal, the milk turning pink, and closed her eyes when she thought about having to go to work in five hours.
She rinsed her bowl out and tiptoed upstairs, past her parents' bedroom, past Danny's door. In her room she changed into a tank top and pajama shorts, and pushed her windows open as far as they would go. The still air inside her room had a suffocating thickness to it, and Jules could feel sweat already starting at the base of her spine. Tomorrow was the first day of . . . what? Summer? The rest of her life?
In the bathroom she wiped off the little makeup Dia had put on her and brushed her teeth, peed, piled her braids up, and secured them beneath a leopard-print satin scarf, and when she got back to her bedroom her phone was flashing.
Jules reached for the phone, expecting it to be Dia saying good night, and definitely not expecting the name that she actually saw.
Delaney Myers.
Jules sat on her bed, shuffled back until she was pressed into the corner, and crossed her legs. The slightest breeze blew through the open windows, a welcome coolness on Jules's clammy skin.
_Ignore it,_ she thought. _I should ignore this. I'm going to ignore her._
Jules had to tell herself that, had to remind herself of the decision she'd made six months ago, when she'd told Delaney that they'd be better apart, that she was done. She had to remind herself of _why_ she'd broken up with Delaney: she and Jules were not good together. It was as simple as that.
Okay, so it was a _little_ more complicated.
One: Delaney was not out, and Jules was. That was okay—Jules was not interested in pressuring Delaney into coming out or doing anything she wasn't comfortable with yet. But it did mean that they didn't go out much, and when they did go out, they didn't act like a couple.
Which didn't help problem two: Jules was a romantic, the worst kind, the kind who dreamed of kisses in the rain and bouquets of beautiful flowers bigger than her body and running hand-in-hand from the weight of the world. She wanted the kind of love her parents had, still giddy after so many years. And Delaney? That was not her at all. She tried sometimes, but it was always lacking—the wrong coffee from Starbucks, wilting flowers from the gas station. She just didn't get it.
And really, that was problem number three: They had very little in common, besides finding each other attractive and that one murder mystery show they were both obsessed with. They couldn't agree on what movie to watch, what music to listen to, what food to order. And in a weird way, Jules kind of liked when they bickered about all that. It felt like Delaney cared, at least. So sometimes she'd pick a fight over nothing in particular, to piss Delaney off, and afterward she'd kiss Delaney extra sweet to make up for it, and she kidded herself that it was passion.
For almost six months, Jules had managed to keep the illusion going. But one day, on the bus, sitting on opposite sides of the aisle and arguing over fucking Twizzlers or Red Vines, she'd realized.
This was not good for either of them.
This was not what she wanted.
This was not what she deserved.
So she'd ended it not long after, and hadn't really spoken to Delaney since—sometimes a hi in the hall at school, but that was about it.
(Okay, and _one_ slipup, after a late night at work. But everybody called their ex to come pick them up and then fooled around for forty-five minutes in the back parking lot at least once, right?)
Jules smiled despite herself, shaking her head as, against her better instincts, she opened the text.
Four small words:
**You looked good tonight.**
Jules grasped at the hem of her tank, feeling heat race through her. Had Delaney been there, at the party? Jules hadn't seen her anywhere.
_You looked good tonight._
That meant Delaney had been looking for her, though. Watching her. Deciding that she looked good. Deciding to tell her.
Why?
Jules slid down until she was lying on her side, one arm wrapped around her body. "She's drunk," she whispered to herself, to her empty bedroom. It was perfectly quiet, aside from the rushed in-and-out of her breathing, the air in her lungs crackling with confusion. "She's drunk and she thinks this is a good idea but it's not. It's a very bad idea. And she's drunk. That's all."
Saying it out loud made it easier to believe.
It didn't make it easier for her to stop remembering what it was like to kiss Delaney. She slipped her fingers beneath the neckline of her shirt, sighing as she pressed her hand to the space between her breasts, to feel her heart pounding there. Kissing Delaney was soft summer wind and a sugar rush of sweetness, the hot thrill of shoplifting and the want drumming-thrumming-humming in parts of her body that before then only Jules had reached. It was rushed and urgent and intoxicating.
Jules could get wild high from a minute of pure Delaney.
But that was all it was, really—physical, a brief relief from the way Jules felt almost all the time, ready to combust, this pent-up energy scalding inside her. (And she could give herself better relief; she was good at that.)
It was not, as her overindulgent brain and heart conspired to tell her, a sign of how much Delaney loved her. And this, tonight, this text? It was not some part of Delaney realizing that Jules was the only person for her, the start of some grand gesture for getting Jules back. For the entirety of their short relationship, Jules had gone along in the naive belief that this was only their beginning, the hard part that came before the movie climax, when Delaney would suddenly realize what a mistake she was making and chase Jules down, tell her she loved her, fight for her.
The sheets whispered as Jules rolled over and squeezed her legs together to ease the tension between them, rubbing her thumb across the words on the screen.
No more naivety. This was not a movie. And Jules deserved way more than alcohol-fueled texts in the middle of the night.
**You looked good tonight.**
She pressed Delete.
## Dia
**"Dia!" Her mom's voice** came loud and insistent through the bedroom door. "Do you not hear your daughter crying? Come do something about it!"
Dia winced at the volume of her mom's voice mixed with Alexa's sharp wail. All Lex had been doing for the past three days was crying: when she had to eat, when she had to get dressed, when her pacifier fell on the floor, when Dia read her favorite stories. According to the books, this meant Lex was going through a "developmental leap," but now they were all starting to get frictiony and Dia was wondering when she might _leap_ into _quiet._
"Dia!"
"I'm coming!" Dia yelled, and then under her breath, "Jesus Christ."
She yanked a clean shirt on with her cutoffs and tossed her stained uniform in the hamper. All morning she'd been sweating in the kitchen of the bakery, pulling out sheet after sheet of sugar cookies and listening to her manager, Imelda, out front trilling, "Welcome to the Flour Shop!" She'd stolen a ten-minute break in the freezer, because another moment of Imelda's voice would have officially pushed her over the edge.
She pulled the elastic out of her hair and teased out her curls as she made her way to the living room. Her mom was pacing around the room, bouncing Lex almost aggressively, and when she saw Dia she let out this exasperated sigh. "Look, baby, it's your mama!"
Alexa kept on crying and Dia swooped in to take her before her mom lost it. "C'mere. What's wrong?"
Lex bumped her head into Dia's shoulder, weeping a wet patch on that fresh shirt. Dia looked over her head to her mom. "Do you think maybe she needs to see a doctor? She's been like this for days now."
Nina followed them into the kitchen. "No, it's just a phase."
"Great." Dia plunked the baby into her high chair and grabbed a handful of tissues to wipe the tears and snot from her face. "Has she eaten?"
"Uh-uh." Nina opened the refrigerator and took out a pitcher of iced tea. "She wouldn't let me near her with anything."
"Right." Dia crouched so she was eye level with her daughter. "Lex, I'm going to get you lunch. What do you want to eat? Use your words."
Alexa's howls quieted to whimpers, and between giant gulping breaths of air she said, "I . . . want . . . bannas."
"Bananas! We can do bananas, no problem." Dia watched Lex's face light up. God, before Lex had come along she'd never imagined that she could be so infuriated yet so infatuated at the same damn time. If only the solution to everything was as simple as bananas. "And maybe after lunch we can go see the puppy, if Miss Candy's home."
"We see the puppy?" Lex asked, sniffling but beaming. "Waffles!"
"Yes, Waffles, that's right!" Nina said, all cheer, and then to Dia, sharper, "You'll spoil her, one day."
"I know." She added that to the running list she kept in her head, Rules for Being a Good Mother: _Be tough, don't stress, keep her calm, don't spoil her._ Simple.
"She makes it so easy, though." Nina grabbed Alexa's hand and kissed her chubby fingers. "I'm going to run to the store. You be good, okay?"
"Bye," Dia called as her mom left. When it was the two of them, Dia fixed her gaze on Lex, staring at each other with their serious eyes. "All right, Lex. What do you say we go on an adventure today?"
They took the bus to Golden Grove Gardens, Dia's absolute favorite place in town.
In the stillness of the park Lex didn't stop fussing, kicking her legs out of the stroller, and straining at the straps holding her in. She kept up the noise, too, and Dia ignored the pointed looks from the people they passed, focusing on calming herself with the beauty of the gardens. A year ago Dia would have been mortified by Alexa's tantrum. Now, though, she didn't really care. Let them stare all they wanted—what, like they'd never seen a kid losing it before? Besides, it didn't really matter whether Alexa was yelling or sleeping or being the most angelic, beaming baby in the world; Dia always got looks. To everybody in town she was one of two things. To the assholes, and people who didn't know, she was a walking, talking stereotype, another brown girl with a baby. (Like that was all she was—never mind that she was a musician, that she was a _person_.) And the people who did know . . .
Dia could always recognize those. They were the people who looked a little too long before smiling at her, or patted her on the hand when she served them at the Flour Shop. And sometimes she wanted to say to them, "He wasn't even officially my boyfriend, you don't have to feel so sorry for me." But she could never say that to anyone besides Jules, and the rest of the time she let them imagine whatever they wanted. Because she was still and always the girl who had the baby by the dead boy.
This was how the true story went: Dia met Elliot Mendes at a party, almost three years ago. He was cute, pretty eyes and a slow smile. They flirted for a couple weeks, made out at a few more parties. Then Elliot took her out to the drive-in movie theater, and when Dia teased him and said _This is a date, isn't it,_ he said, _Don't make fun of me or I won't buy you any snacks_ and kissed her so she'd stop laughing. And that was how it went for a few months, and Dia thought maybe he'd be it, her first love and all that.
They'd been at a party the last time she'd seen him. It had been a good night—they'd watched Ciara's band play; Hanna had been on her best behavior, Jules had been silly. They'd jumped on the trampoline in a stranger's yard. She still had the picture Elliot had taken.
He'd told her she was beautiful.
And sometime in the early hours of the morning, when Elliot was walking home, someone hit him with their car. Couldn't look up from their phone long enough to see the person in front of them and by the time he hit the ground, her dad had told her after, the only reassurance she could ask him for, he was already dead.
And that was that.
Except it wasn't, because then Dia took a positive pregnancy test and she decided, in a burst of selfish love and hurting broken-hearted-ness, to have the baby.
So now she was the Girl Who Had the Baby by the Dead Boy, she guessed.
The flowers were in full glorious bloom. They wandered the paths long enough that Lex finally ran out of steam and fell asleep with her thumb jammed in her mouth. Dia pushed the stroller in the direction of the rose gardens, and it was nice: the sun on her face, walking with no place to be, her baby content (for now), and all the beautiful scenery around them. See? Things could be good.
She was so far away that the _tick-tick-tick_ of bicycle wheels only registered a second before her name was called out: "Dia!"
She whipped around, a finger to her lips as she faced the boy on the bike. Today his hands were bandaged, grimy white around his burnt-sienna skin. Always with the accidents.
To his credit, Jesse looked suitably chastised and hit the brakes, coasting toward her quietly now. "Hey," he said when he was close enough for normal volume, letting his feet drop to the ground. "Sorry."
"It's okay," Dia said, and she tipped her head in the direction of the stroller. "She's asleep, is all, and we're having kind of a rough day, so."
Jesse peered behind the hood of the stroller, pulled down so Lex didn't get too much sun, and then his face broke into that beautiful smile of his. "She looks like butter wouldn't melt."
"That's what she wants you to think," Dia said. "What are you doing here, anyway? You know you're breaking the rules." Dia pointed at a faraway sign that she knew had a big CYCLING PROHIBITED proclamation on it.
"That's not a real rule," he said. "And I came to see you. I went by your house but you weren't there, and then you weren't at the bakery, but you were here. So here I am."
_I came to see you._ See, it was pretty little words like that, always getting Dia in more trouble.
"You _love_ him," Jules would always say to her when she felt like being annoying. "Everyone can see it." And Dia would say, "So? I don't have time for a boyfriend."
That was true—when they'd first met, Alexa had been tiny and Dia was busy with school and work and taking care of a baby, and so she'd told Jesse they could only be friends.
But the real reason she kept to herself. No one needed to know that her avoidance of Jesse had less to do with her busyness and more to do with how the last boy she might have loved was dead now. How afraid she was to lose another person, to feel that cold shock of loss steal the breath from her lungs.
So she pretended it was all Lex and graduating and getting through, and Jesse got it, he understood, and Dia had put all thoughts of his almost-black eyes and full mouth, his rich brown skin and back muscles out of her brain. They were friends. It was fine.
Except for when he said things like _I came to see you_ , or when she touched his hand a little too long, or when she caught him watching her like he thought no one could see him.
Now Dia found a bench and sat, keeping one foot on the front bar of the stroller so she could ease it back and forth, keep Lex sleeping. "Here you are," she said. "What are you doing tonight?"
"Working," Jesse said. He leaned on the handlebars of his black bike, dirt sprayed across the metal. "If you come by I'll hook you up."
Dia gave half a laugh. "Have you ever noticed that our entire friendship is based on free food?"
"But Giorgio's pizza _is_ the best," Jesse said.
"No lie."
Jesse pushed back on his bike. "So you gonna come by?"
"Can't." She held up her hand to make the biggest air quotes. "We're having 'family dinner' tonight. My mom's cooking."
"Okay," Jesse said. "So I'll see you around eight, then?"
Dia reached out to smack him but he dodged her, too fast. "Shut up. It's not my mom's fault she's a terrible cook."
"I never said she was," Jesse said. "But, y'know, if you're still hungry after, we got you."
"Sure."
Jesse spun his handlebars. "Are you going to Revelry next week?"
"For what?"
"The Sun City contest. The launch night or whatever."
Dia lifted her head and eyed him. "A launch event? They never did that before. Who's playing?"
"Are you serious? You haven't heard?" Jesse shook his head as he swung his backpack around. "Sometimes I think you ignore this stuff on purpose. You know they're changing the contest, right? Everybody's losing their shit even more than usual."
He pulled a crumpled flyer out of his bag and handed it to her. "See?"
Dia snatched it and scanned the glossy words: _Sun City Radio presents . . . Originals contest . . . First Place prize $15,000 and the chance to open for Glory Alabama._
Hold up.
Playing with Glory Alabama?
_Fifteen._
_Thousand._
_Dollars?_
"Holy shit," she said. "Holy _shit._ That's a lot of money." Not to mention the little thing of _playing with Glory Alabama_. She read it again and slapped her hand on the bench. "Wait, is this for real?"
"One hundred percent," Jesse said. "Like I said, everyone's losing it."
_Understandably,_ Dia thought. She was on the verge of it, too, as she took in the words on the flyer.
This was way bigger than before.
This was _more_.
This maybe changed things.
"Oh my god," she said, looking up at Jesse. "Can I keep this?"
He shrugged. "Sure." And then his look turned suspicious. "Why?"
"Because," Dia said. "That's all."
"Because maybe you're thinking about doing it?"
Dia raised her eyebrows. "Me and what band? I don't know if you've noticed, but I don't play anymore."
"Okay," Jesse said, bouncing his front wheel on the sand-colored gravel. "No, you're not interested, not at all."
She got up and grabbed the stroller, accidentally turning it so sharply that Lex woke with a startled cry. "Don't be an ass."
"Me?" Jesse put on a hurt face, but his shit-eating grin couldn't be hidden. "Come on, Dee. I can see the wheels in your brain turning as you talk."
Dia folded the flyer in half, and then again, and slipped it into her back pocket. He never missed anything with her. Not that she was _really_ thinking about entering, because it was true, she didn't play anymore. There hadn't been a band for years. They didn't speak to Hanna. Dia couldn't remember the last time she'd been to a show, let alone played one.
(That was a lie. Of course she remembered the last time they'd played: in Rexford, at the end of a skate competition. Under a wide-open sky, the sun glowing on them, Elliot buried a month. And all day long, trying in vain to keep Hanna sober so she could play. Sweet memories.)
And really, Dia thought, so what if she was intrigued? It was only fantasy, one she would cut off before it began to hurt too much. It wasn't like she was going to do it. How could she?
She turned the stroller and scuffed her toe in the gravel. "My wheels are turning, huh? You think you know me," she said to Jesse, her fingers plucking at the collar of his shirt. "But I have secrets like you can't even _imagine_."
Did that sound too flirty? It was so hard to toe the line she'd drawn and sometimes hated.
But Jesse held his hands up and gave that smile of his. "All right," he said. "Maybe I'm wrong."
Dia rolled her eyes again, over-exaggerated this time, because maybe he was right, too. "We'll see," she said.
Then they walked the rest of the park, through the rose garden and by the pond and back around, talking about anything but the contest, how bad Dia wanted to play again. It was easy enough; Dia didn't talk about it anymore. She had made Jules stop, too. They both knew how much they missed performing and recording and feeling the pulsating energy of an audience with its eyes on them. Talking about it when there was no hope of going back there was way too much of a downer.
And it was okay, Dia told herself now, leading Jesse through the flowers. She didn't _need_ it. That was then, and this was now.
And right now, she definitely wasn't still thinking about the flyer in her back pocket, the contest.
Not at all.
## Dia
**Dia waited until Lex** was asleep in her crib that night, uttering her soft toddler snores, and then plugged her headphones in to her laptop. She clicked through to find the folder buried deepest of all, labeled _EP_.
Five songs that they'd recorded in Hanna's garage, with the jankiest setup, but it had been all they'd known how to do, and it was exhilarating to be able to play their music back afterward.
She hovered the cursor over the Play button. This was silly, really. What did it matter if the prize money was big enough to mean something to her now? Or that one of her favorite bands was involved? Her life was still what it was. The contest didn't change that.
But. Still.
She clicked Play.
The first note hit and that was it: Dia was gone. No longer was she sitting cross-legged on her bed, the sun setting outside and her daughter lying near. She was fourteen again, precocious as hell and pleased by that.
It had started when Dia had dragged them to a party they weren't invited to, to watch this band she'd heard was cool: Graceland. And they didn't let her down—their sound was staticky, electric, and Dia had been fixated on the lead singer, a light-skinned girl with locs twisted into a crown and flowers tattooed on her arms. _I wanna do that,_ she'd thought, carefully watching the girl's careless style on the makeshift stage.
She'd been all keyed up after and started saying, "We should make our own music. We could be as good as that. Better, even. That could be us."
Dia already played. Her dad had been teaching her since before she could talk. She practiced on his old acoustic, and a third-hand, cherry-red electric she'd saved enough birthday money for. When she begged, he got his old equipment out of storage for them—amps and mics and everything they needed. Hanna bought an almost-trashed drum kit from a kid at school and Jules blew her savings on a battered bass, and, like magic, they were a band.
They'd thrown themselves into writing, playing, listening over and over to five-second loops of Glory Alabama songs to figure out their intricacies. Spent lunch periods scribbling lyrics and chord changes in the back of their notebooks.
The first time they played through a ten-minute set of their own songs, Dia felt more than triumphant.
"Now what?" Hanna had asked.
"Now," Dia said. "We get a show."
They had already been going to the all-ages nights at Revelry for a year almost, and hanging out at backyard shows. There was this festival that happened every year, the biggest and best local bands coming together for free. And Dia decided that _that_ was going to be their performance debut. All she had to do was convince the organizers to let them in.
Her first email had gotten her this response: "The lineup's full, sorry! Maybe next year."
But Dia was stubborn and emailed again, and again, every day with a new reason why Fairground—the name they'd chosen after a thousand hours—should be on that lineup. And when she got silence in return, she took herself down to Revelry and begged the manager for help. "Sorry, kid," the manager said. "No can do. Maybe—"
"Not next year," Dia said, already on her way out. "Now."
She'd stood out in the parking lot, wondering where to go next, and only noticed the girl coming toward her when she spoke. "You have the persistence down already," she'd called out to Dia. "And the attitude."
Dia had looked up and said the only thing she could think of. "You're Ciara Lennon." The Graceland singer.
"That's me," Ciara Lennon said. "I'm also the person you've been emailing every single day."
Dia's instinct was to apologize, but instead she straightened up and folded her arms. "Persistence," she said.
Ciara smiled. "The question is, are you actually any good?"
"Let us play the festival," Dia had said, "and you'll find out."
Then Ciara had laughed. "Fine," she'd said. "Only so you'll stop irritating me."
So they'd played ten lightning-fast minutes in the early afternoon to a scattered audience—but a girl came up to them after and said she really liked them and would they play at a party she was having?
They did that party, and then a whole bunch more, and practiced almost every day after school, and got good. Really good. They played a different backyard show at least once a month, and Ciara even got them a couple slots opening up for Graceland at clubs outside Golden Grove. They'd drive out and play, then watch from the wings and wait until Ciara was ready to take them home, eating French fries in the back of the van. And after one of those shows, a woman gave them her card, said she was A&R at an indie label and they should get in touch with her.
Dia had thought it then: _This is real._
The third track played in Dia's ears, crackly and frenetic. Their music was all like that, two-minute snarls of anger and excitement. You could hear it: the impatience, the energy, everything waiting to break out and run wild. They were like lightning, burning bright and hot.
And in the end, just as brief.
Sophomore year started out good: they had more shows, that rep's card, and almost enough money for a slot at a recording studio in Longport. But then in November, Elliot died.
The day after Christmas, Dia took the first of many positive pregnancy tests.
On New Year's Eve they found Hanna passed out in the parking lot of Clark Bar, eyes rolled back, and had to take her to the hospital to get her stomach pumped.
(What's that saying about things in threes?)
Jules wanted to keep going, but Hanna was getting worse and Dia was tired, heartbroken, and half hated Hanna for not being good enough.
It was easy to avoid her once summer rolled around, once Lex made her screaming arrival. Easier than it should have been to tell Hanna that she couldn't be around the baby, that she was out of control.
And so by the time junior year came around, they were completely fractured.
Dia listened all the way through the five tracks of the EP, three originals and two covers (a punked-out version of "Crazy in Love" and a Placebo track). Then she skipped it back to track one and listened through. Back, and through. Back, and through. The sun disappeared and pink streaks raced across the sky, the curtains billowing in the breeze from her open windows, and she closed her eyes to the sound of the past.
After work the next day, Dia scooped Lex up off the floor, where she was sitting in her pajamas, bashing at her baby-tiny keyboard. She got into her bed, allowing herself fifteen minutes of luxurious snuggly Sunday-evening baby time. "Hi, baby," Dia said, rubbing the tip of her nose against her daughter's cheek. "Did you have fun with Grammy today?"
"I found bugs," Lex said, holding her hand out and wiggling her fingers like little creatures. "And I saw Waffles!"
"Wow," Dia said, making her eyes wide. "Did you say hi to Waffles? Did she lick your hand?" Dia pulled Lex's hand to her mouth and pretended to nibble on it. "Like this?"
Lex squealed, a wild giggle. "No, Mama!"
"Silly Mama," Dia said. "Hey, Lex. You're going to be two soon. Two! Can you believe it?"
"My birthday?"
"Yeah, your birthday. We're going to have balloons and cake and presents, all for you."
Lex yawned, her tiny teeth peeking out. "Presents," she said, sounding content. "Balloons."
"You're sleepy," Dia said. "You want a song?"
Her daughter nodded, thumb slipping into her mouth, and Dia gently took it out. "Okay." She flicked through her mental catalogue of lullabies and then, running her fingers through Lex's damp curls, she sang a rendition of something her dad used to sing to her. She couldn't remember all the words but she knew the melody, and it worked. Lex's eyelids fluttered shut and stayed that way.
Dia leaned down and placed the softest kiss on Lex's head. "I love you," she said. "Sweet dreams."
She slipped out of the bed—she'd move Lex to her crib in a little while, once she was fully out—and sat at her desk again, running her fingers over her laptop keyboard. She opened a new browser window, typed "sun city radio contest" in the search bar, and hit Enter, bouncing her feet off the floor.
It was almost too good to be true, right?
And Dia knew: when something seemed that way, it almost always was.
She clicked on the first search result and the radio site loaded, with a bright banner at the top of the page:
**SUN CITY AND GLORY ALABAMA PRESENT THE ORIGINALS CONTEST**
She skimmed the intro and slowed down when she got to the main section, her lips moving as she read it.
Round One: Submit an original track. The submission portal will close on 06/13. The entries will be judged by our expert panel of listeners, and those that meet our standards will go through to Round Two.
Round Two: You'll perform for our judging panel, one-on-one. Three entrants will go through from this to the final round.
Round Three: The winners will be announced at the Revelry Room on 07/27 and they'll receive $15,000 cash, the opportunity to open at one of Glory Alabama's Sunset Revue Tour dates, and more.
Give us your music NOW.
June 13. That was a week and a half away.
She clicked through to an interview with Astrid Parker and Luisa Savante, the lead singer and drummer for GA. They wanted to give back, they said in it; they'd had the idea to come back home and partner up with Sun City to find new talent, give them the opportunity to kickstart their music career. Astrid said:
It's hard to even get on the bottom of the ladder if you can't afford new equipment or studio time or transportation to other places to play different venues. Golden Grove is where we started, and it's where we want to go back to so we can help people in the same position we were in thirteen years ago.
Dia read it through twice, and then went through the fine print again, to make sure she wasn't missing anything. But she wasn't. This wasn't a joke, someone's exaggerated rumor running wild. It was real.
All she had to do was enter one song and she could have the chance to win fifteen thousand dollars. Not only that, but an opening slot for Glory Alabama, too. No amount of money could buy you that.
She looked over her shoulder at Lex, fast asleep and snuffling-snoring now. Dia's hands were itchy and her stomach doing that adrenaline-fueled swirling. Yeah, she was going to take music classes at community college, but that was months away. This was different, what she _really_ wanted to be doing. And it was only one song.
She could do this, and if she didn't win, everything would carry on the way it was. But if they liked her music, if she actually won?
One song. That was all it would take.
Her guitar was propped up in its usual corner. Dia got up and moved Lex from her bed into the crib, settling her carefully. She'd need a real bed of her own soon, especially now she was trying to escape more and more often. _The never-ending expenses of motherhood,_ Dia thought. _Oh, did I say expenses? I meant wonders._
With a last glance at Lex, Dia grabbed her guitar and left her bedroom. She passed by the living room and called in to her mom. "I'm getting some air," Dia said. "She's out like a light."
"Okay," Nina called back, the rustle of a magazine page turning. "Everything okay, baby?"
"Perfect," Dia said.
She slipped out the back door and closed it behind her, sitting down on the steps that led down to their tiny patch of lawn. One day Dia wanted to live in a house with a big yard, big enough for a swing set and a pool and a trampoline, and space for them to breathe. Alexa could run around and they could eat dinner outside under the setting sun, and it would be perfect. That was Dia's dream.
One of them.
She settled her guitar on her knee and dragged her thumb over the strings, listening intently for any discordance. A quick tighten until the flatness was pulled up, and she played an E minor chord: better. Dia shifted the capo down a fret and picked out a simple melody, something that'd been rattling around her head for a few months without her doing anything about it.
She wanted to enter this contest.
She was going to.
But she didn't want to do it alone. Jules could be convinced, she was pretty sure, but it might take some work. And Hanna—
Hanna wasn't part of them anymore.
_We'll figure it out,_ Dia thought. _We'll find somebody new. This town is full of musicians. There are other drummers. Ones who won't let us down. And we'll enter and we'll win, and then . . . glory._
Or maybe not.
But the possibility, the wonder of it, was enough. Dia wanted it, bad.
She took her phone out and typed out a text to Jules in a rush. _Come see me after your shift tomorrow. I need to tell you something._
Then Dia tipped her head back and looked at the tiny sliver of moon high in the gray-blue sky. Good things were about to happen. She could feel it.
## Hanna
**"Molly, let's go!" Hanna** wound her hair into a topknot as she waited for her sister to come downstairs. "You're going to be late on your first day!"
She pulled the car keys from her back pocket and weighed them in her hand, the weight of trust and responsibility. The night before, her parents had sat her down at the kitchen table and given her this set, staring her down as her mom outlined the rules: "You'll take me to work in the morning, and then you use to the car to drive Molly, and to go to work, and that's it. No road trips, no picking up your friends, nothing. And you drive it carefully—any scratches or bumps and you'll pay for them yourself. Understand?"
_Road trips?_ Hanna had almost laughed. Who would she go with? The phantom friends her parents seemed to think she had?
But then Hanna had looked at her parents, really looked at them, with their foreheads creased and worry setting their mouths in hard lines, and felt the guilt swell. She only had these rules because she'd proven she needed them. They only looked so worn out because they had an alcoholic daughter to worry about. Rehab didn't come cheap; they were still paying off her two-week stay, four hundred and eighteen days later. _Plus_ those two ER bills. And all that energy they'd spent, searching her room for bottles and making sure she hadn't lifted cash from their wallets. Yes: she'd brought this all on herself, hadn't she?
Then the guilt had given way, as it often did, to the burn of resentment. They were going to make her pay for it for the rest of her life, weren't they? Nothing she did would be good enough. Nothing she said would ever convince them.
But she'd nodded anyway, and promised to play by the rules. "I understand," she'd said. "Thank you."
"Molly!" Hanna stomped to the bottom of the stairs. "Come _on."_
"I'm coming! God!" Molly appeared at the top of the stairs and flounced down, her mouth pouty and covered in shiny lip gloss. "Why are you yelling at me?"
"Because we're going to be late, and I'm going to get in trouble, and I'd really rather not," Hanna said.
"Well, I'd _really rather not_ go to this boring thing at all," Molly snapped, and then her face lit up. "Oh! Let's go to the beach instead. Wouldn't that be fun?"
"No, it wouldn't." Hanna opened the front door and stepped out into the blazing heat of the day. Molly was right: it was a perfect day to head to the ocean, to cool off in the surf, eat boardwalk junk and inevitably spend the evening smearing aloe vera on their sunburns. She hadn't been to the beach in years, though, and she wasn't about to change that so Molly could get out of this. "Get in the car, Molls."
"It's not fair," Molly said. "I don't even like theater."
"Well, Dad says you can't stay at home by yourself all day _every_ day, and it's cheap. And you'll have fun!" Hanna said, not even convincing herself. "If you really don't want to do the acting part . . . make costumes or something. Paint scenery. It'll be fine."
Molly turned to Hanna. "I'm thirteen! I don't need a babysitter," she said. "And besides, I wouldn't be alone at home. I'd be with you. Why can't I stay with you?"
"Because," Hanna said, and she faltered. Because maybe her parents trusted her to drive Molly around and hang out, but that was about it. If Hanna was left to watch over her for the whole summer—who knew what would happen? Molly might develop a taste for bleach-scented vodka, exactly like Hanna had.
"Because what?" Molly said.
Hanna looked at her sister's hopeful expression and shook her head. "Because Mom and Dad say you can't," she settled on. "That's it."
"Well, that's a stupid reason," Molly said, and Hanna wasn't sure she disagreed. But what else could she say? _"Hey, Molls! Mom and Dad don't want me to fuck you up the way I fucked myself up, okay!"_
It was funny, in a way. Her mom so badly wanted Molly to turn out different from Hanna, and she'd wanted so much from Hanna in the first place, and the pressure of it all—the constant arguments and her mom picking at her and stubborn silences at the dinner table—had been what pushed Hanna closer and closer to drinking. Her friends, her hair, her grades—everything was a way for Hanna to accidentally disappoint her mother. After a while, Hanna had gotten tired of trying to please her. And shortly after that, she'd found a way to forget it all for a few hours at a time.
Not that she blamed her mom, her problems were her own, but—she'd helped Hanna get there, for sure. And wouldn't it be hilariously tragic if her desperation to keep Molly pure had exactly the same effect as it had on Hanna?
Well, not hilarious. Just tragic. But maybe her mom would learn something, finally.
"Let's go."
Molly lingered in the doorway. "I don't want to."
Hanna took a second, pushing down the urge to snap at her sister. "Well, sometimes you have to do things you don't want to," she said. Like working two days a week at a secondhand furniture store, making just enough money for junk food and cheap clothes, because it was the only place that would hire her. Like talking to herself instead of other people, because other people didn't like to listen to her. "Like driving your little sister around when she's being a brat."
Molly glared. "I am not a brat."
"I didn't say you were," Hanna said. "I said you were _being_ one. So let's go."
She held Molly's gaze until her sister gave in and dragged herself to the car. "Fine, whatever."
"That's more like it." Hanna slid behind the wheel and took a deep breath. A cigarette right now would cure her shakes. It wasn't healthy, she knew, but at least it was better than drinking. She'd only gotten her license three months ago, years behind everyone else. When she'd been drinking she'd been too out of it to ever care, and once she quit she'd been too focused on staying sober and finishing school. But now she'd passed all the tests and she had the keys and there was nothing stopping her.
She slid the keys into the ignition and turned, and the engine didn't so much roar to life as whimper, but that was all they needed, really.
## Jules
**Speed-scanning frozen vegetables and** cereal was a skill Jules had come to perfect in her time working at Callahan's Grocery and Deli. It was a very necessary skill around Thanksgiving and Christmas and Fourth of July, when the whole town seemed to turn out to wrestle in the freezer aisle and yell at her when their out-of-date coupons wouldn't scan. And Jules actually liked those times—the more rush and crowds the better, because who had time to be bored when they were scanning, scanning, scanning?
In summer, though, on a weekday afternoon, it was a dead zone.
Jules shifted her weight and glanced at the door to the manager's office, closed tight. "Okay," she called. "Go!"
At the end of the cereal aisle, her coworkers Malai and Henry began stacking Pop-Tart boxes end on end, making precarious towers. This was their daily ritual when their manager, Greg, wasn't around: Grocery Olympics. So far, Malai held the record in both Pop-Tart stacking and apple juggling. But Jules had a plan to beat her.
Henry dropped a box of chocolate frosted. "Penalty!" Malai yelled. "Minus five points."
"Aww, come on!" Henry said. "Ref?"
Jules held her hand up. "Call stands," she said. "Don't be whiny, Henry." She glanced at the office again, and this time the door was opening. "Shit. Clean it up!"
She busied herself straightening out the dividers, and when Greg came around the corner Malai and Henry were back where they were supposed to be, arranging paper towels. By the time he'd told them about the special delivery coming later and then left, Jules had customers. She scanned three carts of stuff and then she was five minutes past her break. "I'm out," she called, setting the LANE CLOSED sign on her register.
She made it through the almost-empty store, past the door marked Staff Only, and into the break room, which she'd expected to be empty. But a girl she'd never seen before was tying up her hair in front of the only window, and she looked at Jules and all of a sudden Jules felt the earth shift beneath her.
It was as fast as that.
One moment she was Jules in her world, and the next she was
Jules
on
another
planet.
A planet where a girl who could make Jules's heart stop
with
one
look
actually existed.
The girl did a double take, her eyes wide, and said, "Hi."
"Hi," Jules said, and did her voice sound breathless to this girl or was it in her head? "I didn't—" _Stop. Focus. On your words, not on her. Okay: try that again._ "Are you new?"
The girl finished putting up her hair—her hair in multiple shades of peachy pink and bright lilac and the blue of tropical waters—and nodded. "Autumn," she said. She looked at Jules and smiled slowly, shiny teeth sparkling. "Hi."
"Autumn," Jules repeated, and the fall name felt sunshine warm in her mouth. "I'm Jules."
_Say your name again,_ she wanted to say. And _You are so beautiful._ And _Do you feel this, too?_
But to say any of those things aloud would have been ridiculous, right, and so Jules shook her head. _Come on. Get it together. This is clearly just a girl and you're losing it from lack of exposure to the outside world and also to girls as beautiful as this._ Girls with rainbow hair, wide hips and thick thighs, skin that looked pillow soft were not common occurrences in Jules's life. Because Jules did _not_ live on a planet where girls who could make her heart stop with one look actually existed; don't be ridiculous.
( _Why is that ridiculous?_ one part of her asked. _Stranger things have happened._ )
( _Not to you,_ another part of her answered. _Life is not a romantic drama._ )
"Do I know you?" she asked, her words slow, quicksand for her tongue. "I feel like I should know you."
"I don't think you do," Autumn said, that smile still playing on her face.
"Oh," Jules said. They stared at each other, but it wasn't staring, it was more watching. Waiting.
For what?
Jules whirled into movement, digging a couple crumpled-up bills from her pocket. "Autumn," Jules said, trying to make her voice sound normal now and almost but not quite achieving it. "Do you like Skittles?"
Autumn tipped her head to the side, her mermaid ponytail swinging through the air, curiosity in her eyes. "Um . . . yeah?"
"Good." Jules went to the vending machine and fed two dollars into it, punched the right buttons, and watched her prizes fall. She bent to collect them, and then walked over to Autumn. In the tiny break room, it should have taken two seconds to cross the space, but it felt endless as Jules moved toward her, every step closer filling her with electricity. It was like wading into the cool water of a lake until you were too deep to stand and floated there, spinning, content. "A welcome gift," Jules said, with an uncontrollable smile, "from me to you."
She offered the candy, and the laugh Autumn let out was maybe the most delightful sound Jules had ever heard. "Thank you," she said sweetly.
Jules watched her reach out her hand and held her breath, waiting for the touch of this Autumn's fingers against her. When it came, fleeting and thrilling, Autumn's hand cool, Jules exhaled slowly.
What sacrifice could she make in order to get that moment back, to play it out over and over again? Whatever she had to offer, she'd do it.
"I have to get out there," Autumn said, but she didn't move to go. "I guess I'll see you in a sec. Right?"
"Right," Jules said.
"Okay," Autumn said, and now she did move, making her way to the door and half out of it before she said, "It was lovely to meet you, Jules."
She was gone before Jules could process that and it was good, because there was no way she could think of anything remotely adequate to say in response.
_It was lovely to meet you._
Who said that? Moreover, who _meant_ that?
Autumn did. Jules could tell.
She stood there, watching the space where this new girl had been, and her heart was racing so fast, and she wanted to laugh out loud. So she did, one breath of it before clapping a hand over her mouth, and she collapsed on the lumpy old couch to stare up at the ceiling.
_Something very momentous just occurred in this room,_ she thought. _Let this moment be marked: the day that I,_
_Juliana Everett,_
_maybe lost my heart_
_to a near stranger._
## Hanna
**That afternoon Hanna was** lying on her bed under her window, scratching her thoughts in a new notebook, when there was a knock on her open door. She tipped her head to the side and her face broke into a wide smile. "Ciara!"
"Your sister let me in," Ciara Lennon said, looking as out of place in Hanna's house as she always did.
So it wasn't one hundred percent true that all of Hanna's friends had given up on her; Ciara stuck around. At a distance, a text every once in a while that said, _Checking in from Vermont. The trees are beautiful. C_ or _Greetings from Denver! Not everyone here is high. C_ when she was on the road with her band, messages Hanna had read but never replied to.
But when the blurriness cleared, when she could see straight again, Hanna had finally replied. _Greetings from Golden Grove,_ she'd said. _The view is clear._
So now Ciara dropped by sometimes, when she wasn't touring sticky, badly lit clubs or temping.
Hanna sat up and shoved a pile of dirty clothes off her bed so Ciara could sit. "Molly's mad at me," she said, shifting over. "I don't know why."
"She's a thirteen-year-old girl," Ciara said, stepping across more of Hanna's mess and sitting. "Does she need a reason?"
"Good point." Hanna pointed at the plastic wrapped around Ciara's right wrist. "New tattoo?"
Ciara nodded, the faded ends of her blue-dip-dyed locs falling loose. "Wanna see?"
She started to peel back the plastic and Hanna recoiled. "No! You know it grosses me out."
Ciara smiled wickedly, taking her hand back. "Yeah, I know." She swiped the notebook onto the floor. "So what's happening, graduate? Wait—you did, didn't you? Graduate?"
"Barely," Hanna said. "But at least it's over. And I, Hanna Christina Adler, am now the proud owner of a high school diploma. Aren't you impressed?"
Ciara dipped her head. "I know you're being sarcastic, but I am impressed. You could have given up. You turned yourself around, Hanna. You don't think that's impressive?"
Hanna looked around her room. The walls were still the pale pink they'd been since she was a baby, but covered in marks from old posters, pictures she'd stuck up, that one time she'd fallen and kicked a chunk of the baseboard out. An impressive home for an impressive girl. People liked to say saccharine-sweet things like that when you were sober, or defied whatever preconceived notions they had of you. Hanna hated it. "I guess," she said after a moment. "But whatever. I don't want to talk about me. What are you doing now? Are you going out of town again?"
Ciara leaned back and blew her cheeks out. "Nah. Pretty sure we're done."
"Done touring?"
"Done being a band," Ciara said. "Fletcher's getting married. Cole got accepted to med school. That leaves me and Penny, and I don't think either of us wants to stay together."
Hanna made a face. "Well, she did dump you."
" _Hanna_." Ciara threw her hands up. "Really? God, twist the knife a little more, it feels real good."
"Sorry!" Hanna said. But it was true—Penny did dump her, for a chef guy somewhere in Indiana. And it wasn't like Ciara was going to want to live in a van with her ex for another three months, right?
Hanna pinched the skin between her left thumb and forefinger. _Just because it's true doesn't mean you have to say it,_ she reminded herself sharply. _I am thinking before I speak, remember?_ "What I meant was," she said, carefully this time, "you have been broken up for a while now, and I can see how that might not be fun anymore."
Ciara shook her head at Hanna, but a tired smile crept onto her face. "Yeah. I got my music out of it, but I'm done now."
"What about the Sun City contest?" Hanna asked. "You going to do it this year?"
Ciara laughed now. "I wish," she said. "Did you hear Glory Alabama's sponsoring it this year? The prize money's fifteen fucking thousand dollars. Do you know what I could do with that?"
"It's a lot of money," Hanna said. "Like, a _lot._ And you could totally win, and then you'd have all that money, and I'm sure you wouldn't mind tossing your best friend Hanna a couple grand, right?"
"Please," Ciara said. "If I had that money, I'd take it and run. Everybody round here's a leech. You give 'em enough and they'll bleed you dry. Nah, it's not my year. Maybe next time." Her eyes gleamed. "What about you?"
Hanna rolled up the bottoms of her jeans and wiggled her toes. "No," she said. Not that she hadn't thought about it, because it was a _lot_ of money, and she'd always wanted to win Sun City, _and_ opening a show for Glory Alabama was a literal daydream of hers. They'd planned to enter, her and Jules and Dia, but that was a long time ago now. Still, she'd thought about it. But she couldn't enter by herself, couldn't play the music on her own, and even though for a minute she'd entertained the idea of forming a new band, _her_ band, she hadn't done anything more than think about it. It would mean finding people to play with, people who didn't know her as either the drunk drummer or _that girl who always sits by herself_ , and where would she find them? And then she'd gotten exhausted at the thought of it and decided it wasn't worth it.
Fifteen thousand dollars, though.
_A car of my own. Money to move out. Put it in savings for college, eventually?_
Hanna shook herself. Why was she thinking about it when she wasn't going to get it? "Nah," she said finally. "Maybe next time."
Ciara nodded like they weren't both talking shit and like _next time_ didn't really mean never. "What about the others? Do you think they'll do it?"
Hanna got up and went to the window, pushing it open another inch, like that might make an actual breeze appear. "I have no idea," Hanna said. "You know that."
"I don't know," Ciara said, her voice hitting Hanna's back. "I've been gone for three months. Maybe you spoke to them while I was gone, what do I know?"
"Right. Yeah, I should have told you, we all made up while you were away." Hanna turned, her delivery dry. "We're the absolute bestest besties in the world again. Yay!"
Ciara rolled her eyes and stretched out on Hanna's bed now. "God, I forget how moody you are," she said. "And I was only asking. Is it beyond the realm of belief that they might be entering?"
"I guess not." Hanna hadn't thought about that. That Dia and Jules might enter the contest together. Without her.
_Obviously without me,_ she thought. _Maybe they'll find some other drummer._
_Replace me._
"Maybe they are," Hanna said, ignoring the spike of jealousy and hurt in the pit of her stomach. "I guess we'll have to wait and see."
Ciara bounced to her feet. "Okay, that's enough. You're pissy, I'm starving. Let's go get something to eat." She grabbed Hanna's hand and wove their fingers together. "I miss you being a pain in my ass when I'm gone."
"You're so sweet," Hanna said, but she smiled and meant it. "If we get food, Molly has to come too."
"That's cool," Ciara said. "Me and the Adler sisters, causing trouble."
"No trouble," Hanna said, taking her hand from Ciara's to lift her hair from the back of her sticky neck. "Or my mom will kill us all."
Ciara grinned. "Okay."
Hanna warmed to Ciara's smile. She went away, but she always came back. That was more than Hanna could say about any of the other people she'd called her friends in the past.
## Jules
**Jules's other job was** at the mall, folding jeans and mass-manufactured tees with ridiculous slogans on them, ringing up people who were always in a rush and asking them, _Would you like to get our store card? It's a great deal!_
Usually she was good at it, efficient and forcing a necessary smile. Today she smiled without meaning to, and accidentally rang up some woman's items at three times the price. Eventually she got switched to dressing-room duty, where her mess-ups would be harmless. But that was okay, because she got to stand there thinking about Autumn.
Autumn standing at the register opposite Jules's, laughing sweetly at her old lady customers. The tie of her uniform apron looped in a bouncy bow right above her ass. The swing of her blue-pink-purple hair. To stand there for hours with this distance between them, and customers insisting on interrupting their conversations, was torture.
Not that their conversations were anything wild—not to anyone listening. Only words floated across the space between their registers: "How long have you worked here?"
"Two years."
"Do you like it?"
"Some days more than others."
"I like it so far."
"I like it today most of all."
And a smile from Autumn, like a bolt to Jules's heart.
Now she almost couldn't wait to be back there, surrounded by cut-price toilet paper and misshapen fruit and Autumn, Autumn, Autumn.
When her shift was over, Jules walked to the food court and ordered the five-dollar special from the sandwich place. She ripped open the package of chips as she navigated her way to the table where Dia and Alexa were waiting for her, and stopped when she reached them. "Dia," she said. "I have to tell you something."
"Juju!" Alexa crowed, waving her chubby arms and bouncing in the mall-issue high chair. "See apples."
"Yeah, you have apple, baby," Dia said, looking at Jules. "I have to tell you something, too."
Jules pulled out the chair opposite Dia and dumped her food. "Okay," she said, and she couldn't stop herself from smiling again. "What?"
Dia looked at her curiously. "Why do you look weird?" she asked. "You're all . . . smiley."
"It's nothing," Jules said, a half-truth at best. "Tell me your thing."
Dia's eyes narrowed, and then she sat back suddenly. "It's a _girl_ thing!" she said. "Wait. This isn't a Delaney thing again, right?"
Jules shook her head. "No way," she said. "You know we were terrible together."
" _I_ know that," Dia said. "It doesn't mean _you_ still remember."
"Oh, I remember," Jules said. "But don't worry, it's not her." Under the table her knees bounced up and down. "There's this new girl at work. The grocery store," she clarified. "Her name is Autumn."
"Autumn?" Dia repeated.
"Yes. Dia—" Jules leaned her elbows on the table and put her hands over her mouth, trying to keep words in and failing.
She spread her fingers and spoke through them. "Do you believe in love at first sight?"
"No!" Dia laughed. "No. I believe in _lust_ at first sight, sure. But love?"
"I know, I know, it sounds ridiculous but . . . I don't know, Dia, I don't know what else to call it. Because I think this is what it feels like." Jules tugged on her earlobe, the scarred bump where she used to have a third earring. "Love."
Dia picked up the burger on her tray and took a bite, staring at Jules as she chewed for a moment. She swallowed and then said, "Okay. Tell me about it. About her."
Jules looked down at her hands, chipped black polish and a single gold ring, warm against her dark skin. "Yesterday," she started. "One minute I'm bored out of my mind and then I go in the break room and there's the most beautiful girl I've ever seen. And I _felt_ this . . . energy, or connection or . . ."
"Or what?" Dia said, a teasing lilt to her voice. "Come on. Tell me the rest."
Lex knocked her juice box over and Jules turned it right side up again, then scooted closer to the baby and lowered her voice. "Lex, your mom thinks I'm being weird. And maybe she's right, but I don't think I care. You get it, right?"
Alexa looked at her very seriously, her big brown eyes unblinking. And then she said, "Cheese, please."
Jules laughed. "I knew you'd understand."
"What does she look like?" Dia cut in. "Did she speak to you? Fill in the blanks, Jules."
"She's cute," Jules said. "Her smile is like the kind you can't ever be mad at. Her hair is all different colors. She could be in one of those old-school pinup calendars. She's like this cute, sexy, funny girl with the best smile and . . . oh." Jules twisted the ring around and around. "I think I really like her."
"And this Autumn girl, does she get to be an actual participant in this love story?" Dia raised her eyebrows. "Or are you going to have a giant crush on her and leave it at that?"
"I don't know." Jules shook her head, casting her eyes upward. "I swear—no, it's—" She stopped herself and took a breath, looking back at Dia. "When I looked at her and I felt that _thing,_ it felt like it was both of us. It was, like, a real thing—I saw her, she saw me, and this _thing_ happened. I know what you're thinking. It's not love. It's lust. Infatuation. But—can't it be all of that?"
Love, lust, infatuation. Autumn, bright smile, these swishy dresses beneath her Callahan's apron, flashes of leg as she walked around the store. And then the rest of her . . . well, Jules could only imagine. Would have much fun imagining in the shower later.
Dia smiled big now. "Maybe it can," she said, sounding sincere. "Autumn sounds cool, really."
"It's ridiculous," Jules said. "I'm excited to go to work tomorrow. I don't know who I am anymore."
"What are you going to do?" Dia asked, excitement in her voice now too. "Ask her out?"
Jules's hands felt suddenly clammy and she wiped them on her jeans. "I've never asked anybody out before," she realized. "Maybe I should. What do you think?"
Dia raised her eyebrows. "Why not? What's the worst that could happen?"
"She says no?" The thought of it made Jules not even want to try. God, if she said the words _Will you go out with me?_ and Autumn's answer was _No_?
Heartbroken.
"So maybe she says no," Dia said. "At least you'll have asked."
"But then we'd have to work together," Jules said, "and I'd know she doesn't like me and I'll have this huge crush on her and oh my god, how embarrassing."
Dia shrugged. "If she's as cool as you say, she won't be gross about it and you'll go back to how things are at this precise moment. You know, because none of this has happened? So you don't have to freak out."
"Right." Jules sighed. "This is bad. But in a good way."
"Right," Dia repeated. "Autumn. Pretty name."
"Isn't it?" Jules said, picking up a napkin and twisting it around and around. "Autumn, Autumn. Oh, I disgust myself."
Dia laughed and Lex copied her, hers more of a high-pitched squeal. "It's okay," she said. "You're allowed to do this. I'll give you two weeks before I start giving you shit."
Jules leaned over and planted a kiss on the top of Lex's curly hair. "Oh," she said, remembering how this had started. Dia wanted to tell her something, too. "What was your thing?"
Dia pushed her fries toward Jules. "Eat first," she said. "This is big."
Jules did as Dia said, and then they left the food court, left the mall. They walked over to the nearby playground with Lex holding on to their hands, swinging up into the air between them.
At the playground Jules chased Lex around the jungle gym for a while, but before long a big kid on the slide scared Lex, and so Jules took her over to Dia in the shade. "I'm so sweaty," she said, collapsing on the grass with a groan. "It's so hot."
"Summer," Dia said, fishing a sippy cup from the back of the stroller. "Lex, c'mere. Have some water."
Jules rolled onto her stomach and propped her head up on one hand, looking at Dia. "So are you going to tell me now?"
Dia handed the cup to her daughter and watched her while she spoke. "Okay. It's about the Sun City contest."
"That?" Jules plucked blades of withered grass from the ground with her other hand. Uh-oh. Dia had this tone that usually meant she had a plan. And they hadn't talked about the contest since they were still playing. They barely talked about the band at all anymore, since Dia had ruled that it was too depressing.
Danger.
"What about it?"
Now Dia looked at her, her face set in this determined stare. "I want to enter it."
"Why?" Jules sat up, her Autumn-induced good mood rapidly shifting. "Seriously. What's the point?"
"The point is fifteen grand," Dia said. "The point is an opening slot for Glory Alabama on their tour."
Jules snorted a laugh. "Yeah, that's the dream."
"It's not a dream," Dia said. "It's for real. GA is sponsoring the contest this year, and those are the prizes. Jules, we could win all that."
"Come on." Jules tossed a handful of grass in Dia's direction and it landed all over her own legs. "Be serious. Like they're going to give fifteen _thousand_ dollars to anybody."
"It's not a rumor," Dia said, taking out her phone. She messed with it for a moment, then thrust it at Jules. "See?"
Jules found herself looking at a screenshot and she had to zoom in to read it, her stomach swooping as she did so, because every word reinforced what Dia was saying.
Round One: Submit an original track.
Okay.
Round Two: You'll perform for our judging panel, one-on-one.
So maybe Dia wasn't talking shit.
Round Three: The winners will be announced at the Revelry Room on 07/27 and they'll receive $15,000 cash, the opportunity to open at one of Glory Alabama's Sunset Revue Tour dates, and more.
Whoa.
Jules shook her head and handed Dia's phone back. "So it's real," she said, dismissive, like it was no big deal, like her heart wasn't pounding. "And what, you want to enter? Okay, whatever."
Dia reached over and poked Jules's calf. "Come on."
"Come on what?"
"We should do it. Don't you think we should do it?"
Jules widened her eyes. "I think you're forgetting two years ago."
"So what about two years ago?" Dia shielded her eyes as she looked at Jules. "We went through all that and came out with nothing. What do we have to lose now? It's fifteen thousand dollars, Jules. And the chance to play with a band who came from here, who we have looked up to for _years_. Do you know what we could do with that?"
What did they have to lose? Time and energy. Each other, maybe—last time they'd lost Hanna, and Ciara, too, cut off in all the mess. And hope. Jules had learned that, to temper her hopes quickly, because life liked to squash them right when you least expected it. "News flash, Dia," Jules said, managing to not roll her eyes. "We don't have a band anymore."
Dia smiled, this sharp grin. "We have you and me."
"And Hanna?"
"What about her?"
"Well, she's the band, too," Jules said, and she meant it. They'd always made their music with Hanna; she couldn't imagine what it would feel like without her. As frozen as things were between them all, Jules remembered the good times so vividly, the way things used to be. When she was bored in class or tagging clothes or bagging groceries, thinking about those days of practicing until their throats ached and her fingers were bleeding was the only thing that got her through.
_We were good,_ Jules remembered. _We were real. The three of us._
"We'll find somebody else," Dia said, and she looked skyward. "Be real, Jules. You think she cares about us?"
"Who knows?" Jules said honestly. If that moment of exchanged icy eye contact at the grad party had been any indication, then the answer was no. But that had only been one moment, and she knew better than to presume to know what went on in Hanna's head. "But if— _if_ —we were actually going to do this, we couldn't do it without her."
Dia hoisted a squirmy Lex into her lap, looking at Jules over Lex's head. "You know she'd eff it all up."
"Maybe not. I heard she's sober now," Jules said. Hard to believe, at first, but she'd heard it more and more. Seen Hanna at school, clear-eyed. Heard from Jesse that she no longer appeared at Saturday-night shows, or emerged from someone's messy party remains on a Sunday morning. Maybe Jules was naive, but she wanted to believe it was true. Hanna deserved better than the life she'd been giving herself. If she was sober, then Jules was happy for her.
"Yeah, I heard that too," Dia said, and her voice was hard but her eyes told a different story. "So?"
"It doesn't even matter," Jules said. "Because we're not really going to do it." Enter this contest after two years of not playing? Embarrass themselves all over again in front of the entire music crowd? In front of Glory Alabama?
But Dia looked determined still. "Okay," she said. "Here." She dug down in her pocket and emerged with a shiny coin. "Tails, we do it. Heads, we don't."
"Are you kidding me?" Jules said, and then she laughed. Of course Dia wasn't kidding; when she decided on something, she made it happen. She'd make that coin land on her side, no doubt. And Jules could say no, or she could give in to her best friend's magnetic control. "Fifteen thousand dollars? I could get a car."
"I could get a new guitar."
Jules tipped her head back and closed her eyes, the sunlight searing white and orange and gold inside her eyelids. Cracked voices. Frantically beating hearts. Nerves split open raw.
She opened her eyes. "One condition."
"What?"
"I'll do it," Jules said. "But _only_ if you agree that Hanna does it with us. And only if she says yes."
"Oh, come on," Dia said, exasperated. "Talk about an _impossible_ condition!"
"You don't have to convince her," Jules said. "I'll do that. I'll do the hard part. All you have to do is say yes to this."
"I . . ." Dia frowned in Jules's direction. "Do you miss her?"
Jules raised her shoulders lazily. "Well, yeah," she said. "I mean, the things she did weren't great, but we weren't perfect, either. She was our best friend." She tipped her head to the side. "Do you?"
"No," Dia said, her answer too immediate to be true. "Not really. Sometimes."
Jules seized her opening. "So let me talk to her," she said. "It's been a while. We're older now. Sometimes we're smarter. Maybe it's time to move on. Remember how much we always wanted to do Sun City? If we're going to do it, it should be with Hanna."
It took a long minute, but eventually Dia nodded. "Fine," she said, sounding only half-convinced. "But you have to handle her. And if she says no, we're still doing it."
"Only if the coin says so," Jules said. "Flip it."
It turned over twice, three times, before landing in the grass right by Lex's foot. And the force of Dia's smile could have powered the entire Golden Grove electricity grid for days.
"Tails."
## Hanna
_**This has to be** a record,_ Hanna thought. _Not even three weeks into summer break and I'm already bored out of my mind._
She tossed a quarter in the air, again and again. The line for ice cream was long—she'd clearly gotten a craving at peak time—and even though it was cold in the café, Hanna felt the back of her shirt sticking to her skin. So far today she'd already managed to piss off her parents; they'd done nothing but shower her with questions and instructions over her morning cereal.
"Put the trash out when you're done," her mom had said. "And hang the laundry out to dry, please."
"What are you doing today?" her dad had asked. "Are you working? What time will you be home?"
"I have the morning shift." Hanna had slurped blue milk from her spoon. "And I'm taking Molly to her drama thing."
"Don't forget to pick her up," Theresa said, and Hanna raised one eyebrow.
"Why would I forget?" Hanna said. "Really?"
And then her parents had both stopped, giving her the exact same look which Hanna was pretty sure they practiced in the mirror at night. The Look meant _Don't test us._ It meant _Have you forgotten everything?_ It meant _Come on, Hanna, we're only doing this for your benefit._ It was a look Hanna hated.
"Sorry," she'd said. Keep Your Mouth Shut, Hanna.
After that she'd dropped Molly off, gone to work, and spent the morning reading in the corner of the furniture store, with the creaky portable fan aimed at her face and her hopes rising every time someone passed the store.
Hanna blew her cheeks out and shuffled forward as the line moved. Truthfully, sprinkles and chocolate sauce hadn't been what she was craving at all. The longing that had risen in her in the early hours of the morning had been for the burn of whiskey, its soothing warmth, and that calm spreading through her bones. The quiet.
But whiskey was not an option, and she was out of cigarettes.
So here she was.
"Can I help you?"
Hanna looked at the girl behind the counter, ice cream scoop raised in the air like a weapon. She looked about as bored as Hanna felt, and like maybe she would use that scoop to cut someone's head off if they said the wrong thing.
"Hello," the girl said loudly, annoyed. "Can I help you?"
Hanna blinked. "Sorry," she said. _Wake up, Hanna._ "Can I have two scoops of mint chocolate chip, please?"
After she paid, Hanna wandered outside. Most of the tables were occupied with middle schoolers or parents with sticky kids, and Hanna resigned herself to sitting on the tiny patch of grass by the parking lot. She was halfway over there when a mom at the next table turned slightly from her kid and Hanna realized, suddenly, that this mom wasn't a middle-aged woman with a tired face like all the others. No, this mom was young, her curls spilling down to her shoulders and her smile shiny with lip gloss as she pulled funny faces at her kid. This mom was Dia.
Hanna stopped right where she was, and watched, because where Dia went, Jules followed. And sure enough, there she was—coming out of the café with ice cream and sodas, sitting down across from Dia, and both of them laughing. She watched Dia get up, ruffle her kid's hair, and walk off in the direction of the bathrooms. She watched as the little girl, climbing into Jules's lap now, noticed her staring and uncurled her hand to wave, beaming a big smile.
Without thinking, Hanna smiled back—it was impossible not to, with those cheeks and that happy grin—and then her mind caught up: _Oh my god,_ she thought _. That's Alexa. Wait, when did she get so big?_ Hanna had seen Dia with the baby around town, of course, but always from a distance and usually hidden away in the stroller, not close enough for Hanna to notice her curls exactly like Dia's and her scrunched-up nose. The last time she'd been close like this, Alexa had still been at the floppy, soft, sleep-scream-shit stage.
All of a sudden Hanna felt ancient.
When her eyes flicked to Jules, Hanna realized she wasn't the only one watching. But Jules didn't look pissed—actually, she was kind of smiling, and then she mimicked the baby's movement, lifting her hand and waving.
Hanna did that thing she'd seen in so many movies—she turned and looked behind her, to check for the person that Jules was _actually_ waving at. Because there was no way she'd meant Hanna, right?
But there was no one there, and when Hanna looked back at Jules she raised her eyebrows like, _Me?_
Jules's smiled dimmed a little, but she nodded. And then she motioned Hanna over.
Mint chocolate chip dripped onto Hanna's wrist, and she lifted her arm to her mouth, licking it off. It gave her a moment to consider exactly how ridiculous an idea this was—she didn't even want to speak to Jules, she hated Jules the same way she hated Dia.
Except she loved them the same amount she hated them, and that was how she found herself walking over, gravel crunching beneath her feet, and then she was standing across from Jules. "Hi." It felt like a foreign word in her mouth, like she had to work her tongue around noises she didn't yet know how to make.
"Hey," Jules said. "I—um, do you want to sit with us? It's so busy today."
The first words Jules had spoken to her in years, and she said them almost like there was nothing bitter between them at all.
Almost.
Hanna looked in the direction Dia had disappeared in and held back the laugh she wanted to unleash. "I'd better not," was all she said.
Jules gave that smile again, cautious and amused. "How's your summer going?"
_I drive my sister around, and I go to work, and I talk to myself, and I want a drink, and I would kill to talk to someone interesting, or to do_ something _._ Hanna looked past Jules. "Fine," she said. "Good. You know."
"Juju, can I have juice?" Alexa spoke in a precocious, sparkling voice and it was almost unbelievable, that this tiny human had come from Dia and could speak entire sentences now. Utterly wild.
"Sure," Jules said, and she pulled a sippy cup from the bag on the table. "Here you go."
This silence started then, stretched out and stifling and as long as the distance that had built between them. Hanna licked her rapidly melting cone for something to do, and tried not to stare too hard at Alexa. God, she was the spitting image of Dia: same curls, same medium-brown skin. But Elliot's eyes.
A cough from Jules broke the silence. "It's weird, isn't it?" She looked up at Hanna. "Lex, I mean. How big she is. Not us. I didn't mean—it's been a while, that's all. And I saw you right there and I actually wanted to—"
"Yeah," Hanna said flatly, cutting Jules off. "It has been a while." She didn't have to say what she really wanted to; Jules knew it already.
_It's been a while because Dia decreed that I couldn't be around Alexa._ "I have a kid now, Hanna," Dia had said, on a day when the air crackled with humidity and the windows in Hanna's bedroom were thrown open wide to the gray sky. "I'm sorry, but I don't—I can't have you around her. Not when you're drinking like this. Do you see what I'm saying? I have to make sure she's safe, Han. That's all."
_Because she didn't trust me,_ Hanna thought. _And you picked her, Jules. Of course you did—why would you pick the drunk one over the smart one?_
That had been the real end of them. Elliot dying, Dia being pregnant, Hanna's first trip to the ER—all that had spelled doom for the band, she'd known that. But she'd thought they would still be friends, like they were before. Like they'd always be.
And then Dia had said those things. And at the beginning of junior year, Hanna stood in the back of the cafeteria watching Dia eat with Jules, so angry and so hurting at the same time, unable to make herself go over there, act like everything was normal after an entire summer of barely talking.
So they weren't together at school, and Hanna wasn't allowed to be with them and the baby outside of school, and before she knew it the silence between them was too loud to ignore.
And almost a year passed, and then Hanna went to rehab and began trying to become a person she could like. And who cared, right, about two people who didn't even know her anymore?
_Me,_ Hanna thought. _I cared._
"So," Jules said, and Hanna had to admire her perseverance. "What are you doing after summer? College?"
Hanna shook her head. "I have this job in Selaport. Admin. I need to earn money for now. I'm going to do the whole college application thing in a couple years, maybe. I want to major in psych and eventually become a counselor, but it takes forever, plus my grades weren't great, so I need to do some volunteer stuff or—" Hanna clamped her mouth shut. Why was she telling Jules her entire life plan? "Yeah."
"That's cool," Jules said. "Psych, huh? I can see that."
Hanna frowned. What was that supposed to mean? "Right," she said, and she could see Dia heading back over, close enough that Hanna needed to leave. She tossed the remains of her ice cream in the trash. "I have to go pick up my sister."
"Sure," Jules said. "Well, it was good to see you." It almost sounded like she was telling the truth, but what did Hanna know? Jules cleared her throat again. "Hanna, can we—"
"I really have to go," Hanna said, beginning to walk backward, and Jules nodded.
"Okay," she said. "Lex, say goodbye."
"Buh-bye!" Alexa said, and Hanna felt this pang in the pit of her stomach.
"Bye, cutie," she said, and then looked at Jules. "See you around."
That evening Hanna shut herself in her bedroom. She plugged headphones into her phone and put on some Banks and then threw herself down on her bed.
She couldn't stop thinking about Alexa. She wasn't a baby anymore, she was a full-on little kid. Dia had a, what, almost two-year-old? And Hanna didn't even know her. She could have watched her growing up; she could have been there today with Jules, on the other side of the divide.
This was the thing. She missed the music, of course she did. She missed having her hands all beat up, the performing, singing behind Dia's raspy voice.
But mostly, she missed them. Dia. Jules.
_Jesus, I want a drink._
And she hated that she missed them, when they'd treated her like so much shit. Left her all alone. Or, not alone. She'd found new people to hang out with, people who didn't care how much she drank because they drank, too, and found it funny rather than alarming when she did things that ended in broken glass or scraped shins. They were fun, and not all bad like everyone else thought, and when she had to leave them behind, when she stopped drinking and couldn't be around them anymore, she'd felt bad.
_No. No._
_I don't need it._
_I am okay._
But even though she'd liked those people, she'd never told any of them about how tired she was. How she worried about Molly being unhappy, being too old for her years because of everything Hanna had done, or what she thought about in the middle of the night when she lay staring at the ceiling. It hadn't been the same. It hadn't been _them, us._
She felt for the fresh pack of cigarettes in her pocket, the feel of them enough to calm her for now.
"Wait until September," Hanna said, under her breath, too quiet to hear over the music in her ears. "Then you won't see them, and you'll meet new people, and things will be better."
Hanna looked out of her window, at the tree branches out of reach—green leaves already curling brown at the edges, parched by the relentless, endless sun. Being lonely was so _exhausting._ It was like being tuned into the same station, day in, day out, with no ad breaks or off switch. And listening to her same constant stream of _I'm so pathetic, I'm so lonely, why doesn't anyone like me, I should disappear, what's the point_ was beyond depressing. Sickening. Hanna was sick of it all.
She sat up. _I have to do something. I will do_ anything.
What?
## Dia
**Dia worked the afternoon** shift at the bakery on Saturday, swirling strawberry frosting onto a seemingly never-ending supply of cupcakes and fixing on her Customer Smile when she had to cover the register. She much preferred being in the back, getting into the rhythm of mixing, scooping, kneading. It was meditative, almost; it gave her time to think.
As she added drops of coloring to the frosting, she thought about Hanna. Not the Hanna she'd known before, but the Hanna she didn't know now.
"What if she really has changed?" Jules had said yesterday, eating melting ice cream under the bright sun. "What if she's really sober?"
Dia wondered—how true could it really be? Hanna used to get so hammered she couldn't play properly. Every party, every show was almost guaranteed to end in disaster, Dia and Jules left to pick up the pieces.
Dia used an offset spatula to whip a pattern into the pink frosting clouds. She'd tried to slip it by Jules without her noticing, but Dia had meant it when she'd said she missed Hanna sometimes. The Hanna who was quiet, serious, and then would break out into a ridiculous impression or dirty joke when you least expected it. But Dia hadn't seen that Hanna in years, and who knew what she was like now?
She brushed blue edible glitter onto the peaks of the frosting and packed the cupcakes into boxes. It didn't matter; Hanna was going to say no. After everything? After the things Dia had said?
_I'm sorry, but I don't—I can't have you around her. Not when you're drinking like this. Do you see what I'm saying? I have to make sure she's safe, Han. That's all._
It had been an easy way out, a way for Dia to distance herself from Hanna without feeling so guilty. It wasn't like she was cutting Hanna out, of course not—they could still see each other at school, and they'd hang out when Dia could get her parents to babysit, of course they would!
Pretending like she hadn't already been pushing Hanna away even before Lex was born, like she wasn't exhausted by Hanna already.
Dia put the boxes in the cold storage and stood there for a minute longer, drinking in the dry air.
Because truthfully: it was about more than the baby.
Dia had been scared.
It was New Year's Eve, not even two months after Elliot had died, when Dia had had to call the ambulance for Hanna, _praying_ that her dad wouldn't be the one who arrived. They'd loaded her in so fast and sped away, sirens screaming, and by the time Dia and Jules had gotten to the hospital Hanna had had a tube down her throat and Dia was on the verge of a panic attack, thinking Hanna was about to die, too.
And then Hanna was okay, but Dia wasn't, because losing Hanna, too? It had been hard enough with Elliot, and she had liked him a lot, but Hanna was her girl, her love. So in a way it was a relief, to be able to distance herself. To know that the next time Hanna landed in the hospital, Dia wouldn't have to be the one watching her, wondering if the damage she'd done would be irreversible this time.
When her shift was over, Dia stood at her locker and texted Jules: **Did you talk to her yet?**
She wasn't holding out hope. Even if Hanna said yes, they'd be in trouble—the submission deadline was in four days and if they missed that, they'd miss the whole thing. They didn't have time to come up with new material, not to record and submit in four days. _Maybe_ they could re-record one of their old songs—but would it be better or worse than the tracks Dia already had on her computer? And how would they even do it? As of right now, they had no drummer, no recording space; they used to record in Hanna's garage. And Hanna was going to say no, Dia reminded herself.
She leaned against her locker and opened Facebook on her phone. She'd meant what she'd said to Jules at the playground: they'd find somebody new.
Hanna could not be trusted.
So: Plan B. She went to one of the Golden Grove groups she'd made herself stop looking at, a page crowded with musicians looking for bands, making equipment trades, plugging shows. Typing fast, she wrote up a post:
Interested in Sun City? Drummer wanted immediately. No first-timers. Email diavlntn@gmail.com for details.
Dia hit Post and exhaled. When she got home, she'd go through the recordings she had on her computer—one of them _had_ to be good enough to submit, to get them through the door. And then, when people emailed her about the drummer position (because they would; multiple good, ready-and-willing people would email her, right?), she could get to work on new material and a new band.
Easy.
She started to tap out another text to Jules, then paused. Jules didn't need to know about this. She could go on chasing Hanna, and Dia would deal with the reality of what was happening. It was easier that way.
So Dia slipped her phone into her back pocket, waved goodbye to Stacey out front, and headed out, her brain running with the possibilities of everything.
## Elliot
## AUGUST
**Elliot watches from the** edge of the yard, watches Dia dancing and playing her heart out. The party crowd is decent tonight; it's the last Monday before school starts. Everybody wants that last night, the last chance for everything they promised themselves at the beginning of summer.
When they finish playing, Dia searches him out, a light in her eyes as she stalks toward him, and the kiss she greets him with is both sweet and salty with her sweat. "Hi," she says.
"Hi," he says back, slipping a hand around her waist. Her hair is piled on top of her head and the exposed skin of her neck is mesmerizing. "I like that new song. What's it called, 'Hills'?"
"Yeah," Dia says, and she twists out of his grip, looking back. "Where did Hanna go?"
Elliot casts his gaze around. "I don't see her."
Dia tenses and Elliot bites his tongue. In four weeks he's learned that Dia does not like to be lectured, that Hanna makes this look appear on her face, that she likes people to think she's intimidating. Which she is, to Elliot at least. A girl this smart and hot and talented? There's no way he could not be intimidated.
"I can go look for her," he offers. "Hanna."
Dia shakes her head. "No, I got it."
He watches her walk away.
It's almost eleven when he sees Dia again. The party has emptied out some; there was talk of a better, bigger thing a few blocks over, but she's still here. She's sitting with her legs in the pool, staring into the water, and Elliot hands his beer to Kwame. "Hold this," he says, and he goes over.
"Hey," he says. "Did you find Hanna?"
Dia doesn't look up. "Yeah."
"Is she okay?"
"Yeah." She sighs. "Or no. Who knows?"
Elliot kicks off his sneakers, rolls his jeans up, and sits next to her, letting his feet hang in the pool. This is a fancy house; all the houses on this side of town are. His dad's realty business, the one his grandparents started after they came from Portugal, sells people these houses, the ones with the nice yards and shiny kitchens. His dad probably wants him to take over the business one day, but Elliot wants to be a writer. For now he works at the mall bookstore, which is always on the verge of closing down. "Is she wasted?"
"Of course." Dia's answer is terse. "Don't worry, though, Ciara's coming to pick us up. Take Hanna home, tuck her up in bed, and tomorrow we'll play along with whatever fairy-tale version of tonight she comes up with."
"So she drinks too much at parties," Elliot says, nudging Dia with his shoulder. "Who doesn't?"
"You don't get it," Dia says sharply. "It's not about parties. It's all the time."
Elliot frowns. "What, like— _all_ the time?"
"Every night we play a show," Dia says. "And every other night too, basically. Whenever she can get her hands on it. Sometimes even when we're supposed to be practicing . . ." She shakes her head. "Sometimes you can't even tell, until she says the wrong thing. But I can't talk to her about it, because what do I know, right? Hanna says I have no idea what I'm talking about. Like I'm not the one holding her up at the end of the night."
"I'm sorry—"
"Don't apologize for her," Dia says, and she looks at him, her eyes dark pools. "It's nobody's fault but her own."
Elliot looks at his feet through the water. He's known Dia a month now, and by default Jules and Hanna, too. But Dia's known Hanna years. He thought Hanna was just one of those people, sloppy drunk. But clearly it's so much more than that.
"I didn't know," he says.
"Why would you?" Dia shakes her head again and when she looks at him this time, she smiles. "Forget it. You know what would really make me feel better?"
"What?" Elliot starts to say, but it's too late: Dia's already pushed him and he falls sideways into the pool, water rushing around him and up his nose. It's colder than you'd think and he resurfaces with a gasp. _"Dia."_
She's laughing, claps her hands. "Sorry!"
Elliot pulls himself over to her, puts his hands on either side of her legs and shakes his head, spraying her with water. "You think you're funny, huh?"
"I know it," Dia says, and she runs a hand through his curls. "Come on. Let me get you out of those wet clothes."
Elliot does not have to be asked twice.
## Jules
**Jules leaned her upper** body out of the only window in the break room, breathing in the outside air so thick it was like breathing underwater. According to Malai, there were storms rolling in all around them: "Hail and thunder and everything," she'd said, hanging out between the registers yesterday. "Saw it on the news."
"Since when do you watch the news?" Henry had scoffed, and then they'd started arguing and Jules had gone back to separating coupons.
Now, sticking her tongue out to taste the heat, Jules felt a prayer for rain forming in her. They needed it—a break from the building, crushing pressure. Relief.
Her armpits stickier than before, Jules pulled back inside and took out her phone. She tapped the back of it on her knee. She didn't even know if Hanna's number was the same.
_Quit stalling,_ she thought. _We don't have time._
She shouldn't have chickened out the other day and let Hanna run away. If she'd asked then, like she had planned to—well, _planned_ was taking it too far. But when Jules had seen Hanna looking at them, it had seemed easy to wave her over, to start up a conversation like they spoke every single day. And she'd thought it would be easy to ask, but it almost seemed like Hanna had known what she was doing, and then she'd left, and Jules had sworn at herself.
She should have asked. If Hanna was going to say no, at least they'd have had her answer already. As it was, Jules's phone kept pinging with Dia's increasingly irritating texts, and the submission deadline was two days away, and Jules had no idea what to do except to beg Hanna to say yes.
"Easy," Jules said under her breath, and before she could put it off any longer she dialed Hanna's old number.
It actually rang—that was a start. Jules sank into the lumpy couch as she listened to the tone in her ear, going on and on. She brought her thumb to her mouth and bit her nail—maybe she wasn't going to answer. Maybe this wasn't Hanna's number anymore, and someone else was about to pick up. Maybe—
"Hello?"
Jules recognized Hanna's phone voice immediately—half polite, half _who the hell is actually calling me?_ "Hi," Jules said, swallowing hard. "It's me. Jules. Hi."
Silence, except for the sound of Hanna breathing, and then, "Jules. Hi. What do you want?"
It was a question, but the way Hanna said it was flat, robotic. Like she was so exhausted and disappointed already. "How are you?"
"Please skip the small talk," Hanna said. "You're not calling me to chitchat. Are you?"
There was a sharpness to Hanna's words that left Jules irritated. Wasn't she trying to do something good for Hanna? And this was the welcome she got?
_Think of the contest. You need her._ "No," Jules admitted. She pulled in a breath and threw herself all the way in. "I'm calling you," she said, "because me and Dia are going to enter Sun City."
A moment of silence. "Really?" Hanna said, but her voice had lost a little of its edge. "Well . . . good for you, I guess."
"Yeah," Jules said. "You know, the prizes this year are amazing. And we always wanted to do it, so."
"Yeah, we did," Hanna said. _We,_ it sounded like she was saying to Jules. _Me. What about me?_
"The thing is, you know, we're missing a drummer." Jules paused for a beat, gathered the courage to say it. "We're missing you."
Another silence from Hanna's end, long enough this time that Jules thought she'd screwed the whole thing up.
"So, what," Hanna said, and the edge was back. "You need me now? All of a sudden, I have purpose for you? Wow. Thanks!"
"Hanna—"
"No, really," Hanna said. " _Thank you._ Because the other day, when you spoke to me, I thought—huh. That was weird. Maybe she really wants to talk to me. Maybe she wants to know how I'm doing. Maybe she's heard I'm clean! Because I am, you know? I'm clean and sober now, Jules. But no. I was right. You just wanted something. I should trust my instincts more often, huh?"
Jules twisted her fingers into the hem of her apron.
_Clean and sober._
Could she trust that? Could she believe Hanna when she said that?
"Hanna, it's not like that. I thought that—"
"I know I fucked us all up, but I don't think I deserve this," Hanna said, and her voice cracked. "I got clean without either of you. Because you left me. And you thought, what? You could leave me when it was hard and pick me back up when I got my shit together? I don't want to play that game."
"This is not a game, Hanna! Look—you're my condition."
A pause. "Condition?" Hanna repeated. "What?"
"I told Dia I would do it, on one condition," Jules said. "That we would ask you to do it with us. Because it was always going to be us, wasn't it? I thought—" She stopped short of saying what she really wanted to: _I miss you, Han, we both do. Come back to us._ "It wouldn't be right, me and Dia doing it on our own. So, this is me asking you: will you play with us?"
More silence. Hanna was really good at that. "It's not that easy," she said eventually, slowly. "I'm not—things are different now. We can't forget the last two years and act like everything's peachy."
"I'm not saying that." Jules pressed her hand to her forehead. "I know where we are. I know what happened. But I know I don't want to play music without you there, too."
"And Dia?" Hanna said. "What does she want? She's not the one calling me."
"No," Jules said. "But if she didn't want to do this, then she wouldn't have said yes. She'd have said no and told me to stop being annoying. Dia does what she wants, you know that."
That raised a tiny laugh from Hanna, involuntary, almost. "Right," she said. "You know, I don't need any pity favors. There are plenty of other drummers in town. You can take your pick."
"But they're not you," Jules said. It was too hard to tell whether Hanna was leaning her way, or pulling back. "Hanna, you and me and Dia, _we're_ the music. There's no band without you in it, there never was. We stopped playing for a reason."
"Yeah," Hanna said. "Because Dia decided I wasn't worth her time anymore."
"That's not true," Jules said. "You know there was so much more to it than that. You know—" Jules stopped herself from saying what she really wanted to. _You know it was your drinking that broke us in the end._ She didn't _want_ to blame Hanna. It was hard to stop herself, though, when Hanna was being this way. Like the Hanna Jules had known before.
Jules stared out of the window as she spoke, her last effort. "This could mean something for us," she said. "Winning. Not just the money, either—more than that." She paused, listening to a moment of Hanna's breath down the line. "Don't you want to _do_ something? We were something, before. Maybe we could be that again. Hanna? What do you say?"
"I—" Hanna's sigh crackled in Jules's ear. "I don't know. I have to think."
Jules nodded to nobody but herself, a little relief slipping into her veins. "Okay," she said. That was better than an outright no; it meant Hanna was tempted, at least. Wanting. "Okay. The first round closes in two days, so—"
Hanna laughed shortly. "No pressure," she said. "All right. Two days. I'll let you know."
"Okay," Jules said again. "Bye, Hanna."
"Bye."
Hanna disappeared, replaced by silence and Jules let out a long breath as she let her head hit the couch cushions. It could have been worse. But Hanna was hard to figure out, always. She'd sounded clear, though, and Jules could imagine her, lying in her bedroom beneath the window, the way she always used to. Maybe she was telling the truth, about being sober. If she was lying, though, it wouldn't have been the first time.
Jules closed her eyes. She hoped it was true.
## Hanna
**Is this for real?**
That was Hanna's thought as she listened to Jules's casual pleading, as she hung up, as she tossed her phone skittering across her bedroom floor. Was Jules for real, honestly asking her to come back? To play with her and Dia again, for—what?
Sure, Jules said it wasn't all about the money, but that could be her way of reeling Hanna in. Making her think it was about more than that, it was about _them_ , and then when—if—whatever, at the end of it all, they could drop her as fast as they wanted to pick her back up.
For Jules to try to spin that on her, call her up out of the blue and— _she thinks she can pull me back, now she has a need for me again?_ Hanna thought as she lay beneath her bedroom windows in the afternoon. She tamped down the part of her that wanted to give in to the fantasy, because none of it was actually going to become real. Her flash-forward visions of practicing, laughing together, making righteous, raucous, riotous sounds were not reality. She needed to remember that.
_Do they expect me to fall at their feet and beg forgiveness?_ she thought. _All so they can use me to win some contest? No fucking way._
The ember of anger burned in her all day long, while she picked Molly up, and sat through dinner with her parents, and stood under the icy cold shower before bed. She stoked the fire, justifying her rage, watching imaginary sparks fly out from old words she'd buried in her deepest heart: _I can't trust you, Hanna_ and _What is happening to you?_ and _Clean yourself up._
The fire slowed a little once she was in bed, became a slow, quiet smolder that let the underlying guilt make itself known. She hated them and she blamed them, yes. But she hated herself, too. And she blamed herself. Who was the one who'd made it so they couldn't trust her? Who had pushed them so far that it was easier for them to walk away than stay and clean Hanna up again and again?
Her fault. Her mess.
Hanna lay there wide awake, Sufjan Stevens singing to her through her headphones, and trying not to think about Jules or any of the things she'd said. Around three a.m. she gave in and got up, stealing out of the house and down the street to the all-night convenience store, where she bought a pack of cigarettes, a green plastic lighter, and two candy bars from a gray-eyed boy.
She lit her first cigarette on the walk home, the click-hiss-exhale an instant sort of soothing. Hanna held her hand out and examined the way the thing looked between her fingers. She'd never been a big smoker, before. But without the drinking, she needed something else to latch onto. Her writing was the healthy version of that. The smoking, not so good. But it worked, as a way to not drink.
_Four hundred and twenty-five days._
And what she wanted right now was a drink.
The sky felt close tonight, and Hanna's shirt stuck to her clammy skin. She hoped that Jules couldn't sleep either, that Dia's conscience was keeping her up.
_No,_ she thought. _They don't care. They're right and I'm wrong, they're good and I'm the pathetic, useless failure. That's the way it is for them. Why would they ever change?_
Hanna had thought about the other day, standing there talking to Jules outside the café—she'd analyzed it and played it back and come to the conclusion that it hadn't been a ploy. There hadn't been any ulterior motive, or setup—Jules really had just wanted to say hi. It had felt almost too good to be true, if Hanna was being honest. And look. Here was the truth: Jules did want something. She always did.
Hanna got back home and walked out into the yard, where she sat in the least-rickety plastic chair. She dropped her cigarette on the ground, grinding it out with her flip-flop. Then she lit another and tipped her head back to stare into the murky sky. "You are okay," she said to the night, and she took a drag, holding the smoke in her lungs for as long as she could before letting it all rush out, leaving her dizzy. "You are not broken. You are here."
Her voice seemed tiny in the gaping dark and Hanna thought, like she always seemed to in the small, lonely moments, that the world was so vast and she so insignificant and her troubles so trivial, truly.
"Hanna?" Molly's sweet voice floated out from somewhere above, and Hanna craned her head back to find her sister hanging out of her bedroom window. "What are you doing?"
"Nothing," Hanna said. She dropped this cigarette on the ground, too, so Molly wouldn't see. "Go back to bed."
Molly disappeared back inside and Hanna dropped her head, satisfied. But barely a minute later the back door creaked open and out Molly padded in bare feet, flannel shorts hanging low on her hips. "What are you doing?"
Hanna snapped her fingers. "You are a pain in my ass, you know that?"
Molly settled into one of the other chairs, her grin lighting up the night. "I know." And then she lifted her nose in the air and sniffed noisily. "Are you smoking?"
Hanna looked in her sister's wide eyes and felt the energy she needed to lie slipping away. "Don't tell Mom," she said, flicking the lighter on and off, sparking like her nerves did at the thought of their mother. "You know how she is about it."
Molly pulled her feet up on the chair and wrapped her arms around her knees. "Can I have one?"
"No!" Hanna said. "Come on, Molls. Do you think I have a death wish?"
"That's not fair," Molly said, and the indignance in her voice made Hanna laugh. "You get to do it, so why can't I?"
"Because you're thirteen." Hanna turned the pack over and over in her hands, shaking her head. "And you're my little sister and I'm supposed to stop you from making all the mistakes I do."
"I'm going to be fourteen in three months," Molly said. "I'm going to high school. I'm not some little kid, y'know."
"Right," Hanna said. "But I'm still not going to let you smoke."
Molly made a noise. "Whatever. Why are you out here?"
Hanna shifted, wincing as her skin peeled away from the plastic. "Thinking," she said.
"About?"
"Pointless things." Hanna looked at Molly's bright-orange pedicure, and the boring state of her own feet in comparison. "You know Jules? And Dia?"
"Uh-huh." Molly jutted her chin out. "What did they do this time?"
"It's so ridiculous," Hanna said. "They want to enter the Sun City contest. And they want me to do it with them."
"Really?"
"Yeah," Hanna said. "Jules called me up today. Said all this stuff about how they miss me . . . whatever."
Molly blinked. "But they were so mean to you," she said, her younger side slipping out. "Weren't they?"
"Yeah," Hanna said. "They were."
That was the thing: Dia and Jules acted like what they had done had been necessary. Maybe a little cruel, but only in the name of kindness.
But it wasn't true. They had hurt her. Real, bone-deep, jagged hurt that never went away, no matter how much booze she drowned it in or how much she swore she was past it. How could she be? They had been her best friends. For all the shit that came with being a teenage girl, Hanna had always thought it would be bearable if she could go through it with them. Only, when it had gotten hard, they'd turned on her. And that hadn't been the promise they'd made to each other.
"So they only want you back to help them out?" Molly said.
"Yeah, I guess so." Hanna twirled the lighter between her fingers. "But I'm not doing it, so they don't get to use me."
Molly touched a hand to her hair, the blond of it darkening nearer to Hanna's natural shade. "Do you want to? If they weren't involved, would you want to do the contest?"
"No. Maybe. Who knows?" Hanna flicked the flame on, off. "I haven't touched my drums in forever. And I don't think I could do it on my own."
"Hmm." Molly tapped her feet on the chair. "Maybe—I know this is going to sound ridiculous, but listen—what if you did do it? _You_ could use _them_ , a little."
"What?"
"I know you miss playing music. And if you wanted to enter, you just said you couldn't do it on your own. You'd need somebody else. Why not them? You could tell them you'll do it, and they'll think they've won, but really, you're the one holding all the cards. And if you won the contest, you'd get your share of the prize money and then you could leave them."
Hanna looked at her sister with a mixture of awe and fear. "Molly! When did you become a scheming mastermind?"
"Seventh grade," Molly said. "It was a real power struggle."
"Wow." Hanna rubbed her neck. Pretending to be okay with them, so she could get the money? That would be too far.
But Molly was right—she could just pretend. They didn't have to be friends again (like that would ever happen). They just had to be able to make music without killing each other. For the greater good. It would be a start for New Hanna, for the girl she claimed to be now. This Hanna wouldn't let herself be pushed around. This Hanna could rise above their past bullshit and do it because _she_ wanted to.
At the very least, this Hanna could pretend all that was true.
Molly was looking at her expectantly. "So?"
Hanna flicked the lighter again and watched the flame burning in the darkness. "I'm tired, Molls. I'm too tired to go on being angry and alone and hating them."
She extinguished the flame and looked at her sister. "And I wasn't so good back then, either. I did awful things, too. It wasn't only their fault. But that's just how it is now."
Molly nodded, a small smile on her face. "You weren't awful," she said. "Just not the real you. It could all be different now. Isn't that what you want?"
_What_ do _I want?_ Hanna pulled at a piece of her hair, tugging hard enough that it hurt. An apology. Their _sorry_. And at the same time—their forgiveness. "I don't know what I want. I don't even know who I want to be."
Molly leaned forward, her hair swinging. "I think about what I want to be," she said. "And I can't decide. But I think when I'm older, I want to still live here."
"Yeah?" Hanna said. "Well, there are worse places to be." She looked past Molly. "I used to think I'd leave Golden after school. We talked about it, the three of us. We were gonna go to LA and live in a terrible apartment. Get terrible jobs and make music the rest of the time." She looked back at her sister, shook her head. "You shouldn't limit yourself, Molls. Think of the thing you most want to do in the entire world. If you're lucky and the world helps you out, you might actually get it."
"You could still—"
Hanna shook her head, cutting Molly off. "Not going to happen," she said, and then she stood and grabbed Molly's arm. "Come on. We should both be in bed."
"Ugh," Molly groaned, but she got up and let herself be led back into the house without resistance. Upstairs Hanna waved goodnight to Molly and slipped into her bedroom. She sank down to the floor and stretched her legs out, resting in the square of moonlight there.
Say yes, or say no.
Maybe she said yes and it went terribly, and they ended up hating each other more than they already did. _No loss,_ Hanna thought. _Hate on top of hate, it is what it is._
And if she said no—everything stayed the same. Boring, bored Hanna. No music, no nothing.
But if she said yes and it _didn't_ go terribly, and they made great music and got into the contest and maybe— _don't say it too loud, don't jinx it, whisper_—won? She thought about Molly, making plans for her life. Dreaming her dreams. Hanna could still do that, if she wanted to.
She pressed her hands together. What had she told herself? _I have to do something. I will do_ anything.
She got up and grabbed her phone from her bed, and she wrote the text before she could talk herself in any more circles.
_Okay. I'm in._
## Dia
**Dia's alarm went off** at seven on the days she wasn't working the early shift, and the days Lex didn't wake her. This morning it felt like she'd barely fallen asleep before it went off, her phone buzzing underneath her pillow like the most annoying gnat. "Mmph," she said into the sheets, searching with one hand. "Shut up."
She wiped her other hand across her mouth, yawning as she managed to open her eyes all the way. The light around her curtains was still muted, a fact Dia registered right as she realized that the noise coming from her phone wasn't her alarm but her ringtone.
"Shit." She fumbled her phone, yawning wide again as she glanced at Lex, still mercifully sleeping. JULES, her phone screen said, right below the time: 5:27. _Why is she calling so early?_ "Hold on, hold on—" She recovered her phone and managed to answer this time. "Juliana, you'd better have a good reason for waking me up, I swear to god."
"I do," Jules said. She sounded out of breath, in movement. "I know it's early, but I'm on my way to work. And I thought you might want to know that Hanna said yes."
"Yeah, you're right, it is early," Dia said. "I was—" She stopped, rerunning what Jules had actually said. "Hold on. Go back. What?"
"Hanna's in!" The sound of a car horn came crystal clear through the phone, and Jules swore before continuing. "I got a text from her last night, this morning, whatever, and she's in."
"She said yes?" Dia sat up, a little stunned. Hanna said _yes._
This was not how it had played out in her head.
Okay.
Forget Plan B. _(Sorry, in-box full of hopeful applicants. Your window has closed.)_
Switch to the Break in Case of Emergency plan.
"Yeah!" Jules said. "So—now what?"
_Now what?_
Dia was wide awake all of a sudden, her mind whirring into overdrive. Okay, okay, Hanna was in. That meant they had—she pulled her phone away from her ear to check the time, 5:31 now—forty-two hours and twenty-eight minutes until the submission window closed.
Forty-two hours and twenty-eight minutes to record themselves playing one of their old songs. They had to; they had to at least _try._ Submit their best and know that then, if they didn't get in, they'd done all they could.
(Yeah, it wasn't so much a plan as a fevered grab at the near impossible. Dia already knew that.)
Step one: Find a time to get together in the next forty-two hours and twenty-eight minutes.
Step two: Play together for the first time in two years.
Step three: Set up recording equipment.
Step four: Get a perfect-enough take.
Step five: Upload to the Sun City site.
Dia rattled through it in her head, trying to convince herself it would work. _If we focus, if we remember,_ she thought, _it'll work_. _We'll record, and okay, the timing's not great but it's good, actually, it'll push us and—_
Better to try and fail than give up without even attempting.
(Optional Step six: Do all of the above without killing each other.)
"Hello? Dia?" Jules was saying. "Are you there?"
"I'm here," Dia said, swinging her legs out of bed now and beginning to pace in front of the window. "Okay. This is beyond ridiculous, but Jules—" Dia stopped and pushed back the curtains, allowing the rising sun to flood her room with light. "I have a plan."
## Hanna
**Hanna watched her parents** rushing around the kitchen, the usual routine of coffee and hastily retrieved papers. She'd have to shoot now; there wasn't time to tiptoe around it.
"Hey." She leaned against the door frame, her arms folded in a careful imitation of casual. "Is it okay if I bring the drums down?"
Her parents exchanged the Look. "How come?" her dad asked.
"To play around," Hanna said. It wasn't a _lie_ , really. She _was_ going to be playing. She just . . . wasn't going to tell them about the rest of it right now. Because what if she and Dia and Jules found they couldn't even all be in the same room without combusting? It'd be over with before it started, and then she'd have created another reason for them to grill her for nothing. "Y'know, for fun."
"You'd have to move stuff around in the garage first," her mom said. "Clear some space."
Hanna looked between the two of them. "Is that a yes?"
Her mom got this pained look on her face, but her dad spoke first. "You'll have to clear it with the neighbors," he said. "But sure, bring 'em down."
So later, after their parents had gone to work, Hanna and Molly climbed up into the attic, taking careful steps on the beams so they didn't fall through. They brought the drums down piece by piece and put them out in the garage, and Hanna went back up for the things she hadn't mentioned to her parents, stashed in a trunk that she hadn't opened in a long time.
Inside: microphones, cables, mic stands, mixer, more cables. Almost everything they needed to record.
Because that's what Jules had said to her in a series of texts earlier this morning. That Dia had a plan— _of course she does,_ Hanna had thought, _when doesn't she know exactly what she's going to make everybody do?—_ and Hanna needed to dig out their equipment because they were going to record, just like they used to.
And when Jules had said that, Hanna had thought about it for a split second before texting back, _We can use my garage._ Just like they used to.
"Molly, come get this stuff!" She passed the mic stands down the ladder to Molly, and carried the rest down herself. When they had everything, she grabbed her laptop and brought it down to the garage, ready to set up.
Hanna cleared out the broken bicycle parts and other things that her dad claimed he was going to fix one day. Molly was less helping and more directing, sitting on a cardboard box outside the open garage door. "I can't believe you said yes," she said, kicking her feet in a pair of their dad's too-big work boots. "Was it what I said?"
"Little bit," Hanna said, leaning the mic stands against the wall where they blended in with a load of old shelving, where her parents definitely wouldn't notice them. "Mostly—I know I'd hate myself later for saying no. Not doing _something._ "
"So, what exactly do you do when you practice?" Molly said. "How do you write songs? How do you know if it's a good song or not?"
Hanna dumped a plastic crate of photographs on top of a paint can. "Molly, could you actually help me instead of asking me pointless questions?"
Molly hopped off her box. "Will you show me how to play?" Her eyes were hidden behind pink, heart-shaped sunglasses. "Only a little."
"Sure," Hanna said. "But I need you to do me a favor."
Molly nodded. "Don't tell Mom and Dad."
Hanna stared at Molly. "How did you know I was going to say that?"
"Because you're predictable, Hanna." Molly lowered her sunglasses. "I'm kidding. What else were you going to say? I won't say anything to them."
"It's not a secret," Hanna said. "I'm going to tell them." How would the conversation go? _Hey, I'm in a band again. Yeah, I know, last time didn't go so good, what with the raging alcohol problem and losing my friends and the stomach pumping and all the rest, but this time's going to be different! How? It just is. Trust me. Oh, you don't trust me, that's the problem! Okay, well . . . trust me anyway?_
"My lips are sealed," Molly said.
"Thanks," Hanna said. "Now help me set this up."
Together they built the kit, Hanna tuning the snare and the toms, wiping layers of dust away. When they were done she ran her fingers across the crash cymbal, cool to the touch. With everything cleared away so there was enough space for them, and the drums taking pride of place, it looked like a ghost space. Ready to be haunted.
"Okay, Molls," Hanna said, and she looked at her sister. "Now all we have to do is play."
In the middle of the night Hanna stole downstairs and into the garage. She left the light off—the moon's glow came through the two small windows in the garage wall, and that was enough for her to see by. She didn't really need to see, anyway. Only enough to get over to her drums and find her seat behind them.
Hanna sat and pressed her feet into the cement garage floor. Her hands wrapped around the old sticks she'd collected from the box in her closet, soothing in their familiarity, settling the pins and needles pricking her skin.
It felt like home.
Tomorrow Jules and Dia were coming over to play and record.
Hanna smiled and shook her head at the same time. "What am I getting into?" she whispered to the snare, the hi-hat.
Was she scared? Yes. But not of Dia and Jules, not really, not much. What could they do to her now that they hadn't done already? How many other ways could they hurt her? That part was fine.
What she was more afraid of was that she wouldn't be able to do it.
What if she couldn't play anymore?
It was pointless worrying, Hanna knew. Of course she was going to be rusty—that tended to happen when you hadn't played your instrument in over a year. But that didn't mean she would have completely lost it. And she wouldn't know until she tried.
She still remembered every single song of theirs.
She still remembered the sound of Dia's voice. She heard it in her dreams, sometimes.
Hanna put one foot up on the bass pedal and applied the gentlest pressure, no noise. She turned the sticks the right way around, let them settle in her grip, extensions of her hands. And then she sat like that, still.
_I am not broken._
_I am okay._
Waiting.
## Dia
**On Wednesday morning, Dia** knocked on Hanna's front door.
She would be lying if she said she was ready for this. But they had no choice. And she was trying not to think about everything that might be about to go wrong—to focus on what great thing they might be about to put into motion.
"I got this," she whispered to herself.
Of course Jules was running late. Dia could have used some best friend armor, protection for each other as they came face-to-face with the girl they were better at avoiding.
She knocked again, three quick raps this time, and then stepped back from the house as the garage door began to open.
Of course.
Dia cut across the small patch of dried-out grass and stopped on the driveway as Hanna stepped out, a hand above her eyes. "Hi," Hanna said, simply.
This was so surreal.
Dia let her gaze flit across Hanna—new nose ring, same white-blond hair—and settle on the drums behind her. "Hi," she said, an echo. "How are you?" Wow. Small talk had never been their thing, but apparently that was the best Dia could do.
"I'm good," Hanna said, folding her arms across her chest. "You know." Then one corner of her mouth quirked up, and Dia could tell exactly what she was thinking.
_No, she didn't know. How could she know? It had been so long._
"Where's Jules?"
Dia shifted her amp from her left hand to her right, her guitar heavy on her back. "On her way," Dia said. "She'll be here soon."
"Good," Hanna said, and she turned her back to Dia, ducking into the garage. "We don't have much time."
Dia followed her in, and the instant hit of familiarity almost took her breath away. She tried not to let it show on her face, though. Like she tried not to let Hanna see how carefully Dia was watching her, looking for signs of . . .
What?
The old Hanna? The new one?
She wanted to believe the rumors, she really did. But she'd been through a lot of shit with— _because of_ —Hanna, and it wasn't so easy. A sober Hanna was an image Dia had to work at to remember.
"Setup looks good," Dia said, to say something, and it was true. Hanna's drums, still scratched and well loved, the mic stands placed exactly where they used to be, cables snaking over the floor. "You kept everything?"
Hanna gave her this look as she sat behind the kit. "What, like I was going to throw out hundreds of dollars of equipment?"
Dia shook her head. "Right." She slipped her guitar case off her shoulders and leaned her weight on it, Hanna watching her do it. And Dia exhaled. "Okay," she said. "I know this is weird."
"Weird?" Hanna said. "Not exactly what I'd call it."
"Strange," Dia said. _Strangers._ "Whatever. And maybe if this was another time, a different situation, we might actually do the whole thing. You know, the dredging up of everything old and saying not-nice things to each other and then promising to be different." She paused, made sure she was looking directly at Hanna when she said this next part. "But we don't have time for that. And I'm not here to go down old paths or make you do it. I'm here to play. I want to win." She lifted her chin. "How about you?"
Hanna rubbed her thumb over her bottom lip, narrowing her eyes at Dia. "I want to win," she echoed. "And I agree. I'm not here to bring up old ghosts, either."
This Hanna definitely had hints of different, that was for sure. Dia nodded. "I think we're on the same page."
Footsteps sounded outside, and then Jules was ducking under the door. "Hey," she said, out of breath, and put her equipment down with a pained look on her face. "Sorry I'm late. I forgot how much of a pain in the ass it is carrying all this stuff around."
"It's fine," Hanna said, and her voice sounded completely different now.
She was smiling. That was the difference.
"Okay," Dia said, crouching to retrieve her guitar from its slumber. "So, I have to be at day care pickup at four. It's"—she checked her phone—"nine thirty. We have six and a half hours to pull this off."
Her words were met with silence, and Dia got it. "I know," she said. "Maybe this is impossible. Maybe we are so far past dreaming it's not even funny. But we're talking about fifteen thousand dollars and an amazing show. So we can either give up now, or we can do the only thing we know how to. Play."
"We're not giving up," Jules said. "Would we be here if we weren't ready for this?"
Hanna nodded. "Improbable, maybe," she said. "Not impossible."
Dia slipped her strap over her head. "Here's what I'm thinking," she said. "We work out which of our old material we remember best and play the shit out of it until we have it as good as we can right now. Then we'll record a couple takes, not tons, because that always stresses us out and we get worse. And whatever we have at the end of this is what we submit. Even if it's truly terrible, we still do it. Then . . ." She looked from Jules to Hanna and lifted one shoulder. "What happens, happens."
"Sounds good to me," Jules said, taking out her matte-black bass and looking at it lovingly. "Hanna?"
Dia looked at her, and Hanna picked up a set of sticks and squeezed them tight enough that Dia could see her knuckles turning white. "Me, too."
Dia pulled her lucky pick from her back pocket. "All right. What's it going to be?"
They spent twenty minutes pulling up old songs from the back of their brains, listing off favorites: "Maiden Me" and "Drive By" and "Holy Water," "Honey Bee" and "Hills," "Alimony" and "Gold Ocean _._ " Some got thrown out for being too old, too hard to remember, not the right sound. Eventually they were left with two: "Hills" and "Drive By."
"I vote for 'Hills,'" Dia said. "We wrote it near the end. It still feels like us." That, and people had really liked "Hills." Elliot had really liked it.
"I vote 'Hills,' too," Hanna said, and Dia felt a flash of surprise. Hanna, agreeing with her?
"Let's go with that, then," Jules said, and the vibration in her voice could have been nerves or excitement.
Dia couldn't tell. She couldn't tell which she was feeling more, either.
"Let's just do it," she said. "First run-through, get it out. Ready?" She looked at Jules, poised with a pick in her hand, and Hanna, with this look of concentration on her face that Dia knew so well, that she'd almost forgotten entirely.
"So ready," Jules said, and Hanna started a count, clicking her sticks together.
"—two-three-four—"
## Dia
**Dia scrambled to remember** the exact beginning, feeling it somewhere in her brain— _there_. Her fingers slid into place and she put her mouth to the mic, drew in a breath. For not having played in so long, Hanna was steady and sure, the perfect backing for Dia to latch onto. From the moment Dia picked out those first notes, she felt it.
The glory.
Dia sang, her voice raspy and low, the right words slipping from her tongue without her having to think too hard.
_"Ninety-eight this day_
_Ninety-eight, what the news will say_
_When we ran for the hills, for the hills_
_Ran to get your girl away_
_Got your money, made her a dress—"_
Playing was easier than talking. Talking was bitter, broken, sour, and unsure.
She watched Jules: hip out, shoulders slouched, her mouth a snarl as she backed up Dia's vocals. And she watched Hanna: eyes wild but intense, back straight, pounding out the best rhythms with her sticks, her hands. And they weren't perfect, nowhere near it, but they never had been. That was almost the entire point of the thing.
As Dia clutched her guitar (her body more used to the weight of a toddler than this, now), it felt like . . .
Awakening.
Like she'd been drifting through everything, closed eyes, tired mind, for so long, and now this explosion woke her so violently, so lovingly.
Three minutes could be both an eternity and a gasp. They careened through the middle, picked up speed as they came into the last verse—
_She ain't playing you, honey_
_got your money, get her veil_
_ninety on the freeway heading straight for moonrise_
_—_ breathless, the euphoria slipping over Dia's skin like silk.
_Look at us,_ she thought.
_Listen to us._
See? They weren't complete failures. They weren't too fucked up to find this again.
They had babies and drinking problems and a whole lot of locked-up sadness, but see how good they were? Look, listen.
_Wait for it._
It was glorious, _they_ were glorious; gleaming, bright shining goddesses making beautiful, messy sound and they reached the noisy end, the lucky pick almost slipping from Dia's fingers as they finished. A clashing crash of minor chords and her voice rising to a final wild yell.
Then it was over, breathlessly done, and as their noise faded, the three of them stood, watching each other. Uncertain of what had just unfolded, if it could be as simple as that, as slipping back into the skins they used to wear.
Dia's chest rose and fell rapidly, and she swallowed.
Jules broke the silence. "That was—"
She was interrupted by a small clapping and a tiny cheer from behind them.
Dia whipped around, and there, sitting on a cracked garden chair, was Hanna's little sister, Molly. "Oh my god," Molly said, and her eyes were gleaming. "That was awesome!"
"Molly!" Hanna jumped up, brandishing her sticks and crossing the few feet to her sister in a split second. "What did I say to you?"
"You didn't say I couldn't watch!" Molly protested as Hanna clamped a hand on her arm. "You said don't tell—hey!"
"Wait," Dia said. "Molly, did you really like it?"
Hanna's sister nodded. She'd looked like a little kid the last time Dia had seen her; now she looked young and hungry, one of those freshman babies ready to take the world for her own. " _So_ good," Molly said, wresting herself from Hanna's grip. "Like, I'd buy your music. If I had any money."
Dia allowed herself to smile at that. "I thought we were pretty good, too," she said, and looked from Hanna to Jules. "For a first try in forever."
"Pretty good?" Jules said, using the neck of her shirt to wipe at her cheeks. "Dia, that was fucking amazing. Didn't you feel it?"
Dia waited a moment before giving in to a real smile, a breathless laugh. "Yes! Oh, I felt it. Oh my _god._ "
Hanna sat next to her sister. "I think we can actually do this," she said, and though her voice sounded nervous, her eyes looked so serious. "I think we still got it."
Molly clapped her hands together again. "So," she said. "What happens now?"
Dia looked down at her pretty guitar, the new strings she'd ordered and stayed up way too late putting on last night, the dent where she'd dropped it out of the back of Ciara's van in the parking lot of a Wendy's. So much history. And now they were making their future. "Now," Dia said, "we get to work."
## Jules
**They slowed down a** little, went over the song in sections, picking out the errors and correcting them, tightening up the loose parts. Then they played it again and again, over and over, bursts of noise followed by short quiet, until Jules's hands were cramping and her back aching.
It felt good, though.
"That's enough," Dia said after a couple of hours. "We need to record."
Hanna pushed sweaty hair out of her eyes and got up. "Let me get it set up," she said. "We'll do one practice take, check it's working okay before we start for real."
Dia waited a beat and then nodded once. "Okay."
Jules didn't say anything, but watched their interaction with barely hidden curiosity. They were all being guarded, on their best behavior. But Hanna was pushing back a little—testing, Jules thought. It reminded Jules of the old Hanna. Except now, instead of blurting out thoughtless words, she seemed to know exactly what she was saying.
She caught Dia's eye, seeing the same wonder in her gaze. _Is she sober, for real? Maybe right now she is. But when we leave, will she stay that way? Is this all a cover? Can we trust her?_
Dia lifted her shoulders and turned away, and Jules looked back to Hanna. She was tired of not trusting her. And she wanted to believe the best. So for now, for as long as possible, she would.
Jules sank into a crouch, rubbing the backs of her tired legs. "I think we might actually—"
"Don't say it," Dia cut her off. "Not until it's done. Okay?"
Jules rolled her eyes. "Okay, superstitious. I'll shut up."
They spent a while setting up more mics, testing levels. Jules looked at the walls—no insulation—and frowned. If they had money, they could have hired a space somewhere. Other bands would be doing that. It reminded her exactly how unprepared they were. But that didn't mean they couldn't pull this off.
"It's ready," Hanna said, setting her laptop on top of a box labeled "Grandma's House" before going back to her drums. "Try once through?"
Jules stood as Dia said, "Yeah, once through."
And before Jules could even get into position, Hanna started.
It took Jules a second to catch up in her head, but as soon as her fingers found their rhythm, she was good. They were a little restrained this time, and Jules didn't know about the others, but the pressure suddenly got in her head. This was _it._ They didn't have another week, another _day_ , even. If they couldn't pull together a decent recording to submit, they were going to miss their chance.
She stumbled on the chorus, earning a glare from Dia, and shook her head.
_Focus._
She made it to the end without any more mistakes, and when Hanna played it back for them, checking that the levels weren't blown out and everything came through clear, it shocked Jules. "Oh, wow," she said, softly enough that the others didn't hear her over the track. "That's us."
"It's good," Dia said, nodding in Hanna's direction. "Ready for the real thing? I say we go three times, but that's it. We'll get played out after that. And we can do it and be done."
"I agree," Hanna said. "Jules? You with us?"
"I'm here," Jules said, holding three fingers up. "Three takes. I got it."
So they did it, fast and tight, and Jules managed to keep it together for all three takes, echoing Dia as she sang, her bass line playing off Hanna's drumming.
When they were done, Jules lifted her strap over her head and set her bass down, rolling her shoulders. Dia did the same, and together they moved to sit on the floor near Hanna.
Hanna grabbed her laptop and played each take back to them, the three of them listening intently. For Jules, it was like listening to forgotten girls. It was them, but in a way she hadn't heard in so long. And all she could think was: _Are we really doing this?_
Was this reality, and not some fever dream of hers?
"What do you think?" Jules asked when they got to the end. "Good enough to use?"
"I think they're decent," Dia said, always their harshest critic. "Nothing special, but good enough to give them an idea of what we're doing. Right?"
Hanna raked a hand through her hair. "The last one's the best one," she said. "We should use that one."
Jules glanced at Dia, and Dia nodded. "Can I?" Dia asked, reaching for Hanna's laptop.
Hanna dipped her chin. "Go ahead."
Jules watched as Dia pulled up the Sun City page and clicked through to the submission form. Her fingers flew as she filled in the different sections, until she stopped. "Our name," she said. "I don't think we should be Fairground this time around. I don't think that's us anymore, right?"
Hanna leaned back on her elbows. "I don't think so, either."
"So . . ." Jules looked between them. "What are we going to be?"
"Wildfire," Dia said, her mouth curving around the syllables. "I was thinking about it last night. I think it fits."
_Wildfire,_ Jules thought. Very California, for sure. Burning, bright. Dangerous.
Was it tempting fate to name themselves after a destructive force of nature? Then again—they'd been Fairground before, and their ride hadn't been a fun one. So maybe it meant nothing, really.
"I like it," Jules said.
"Me, too," Hanna said, yet another surprise. "Do it."
Dia resumed typing, and then selected their track to add. Jules watched the upload bar speed along until their song was safely in there. And then Dia clicked Submit.
"Shit," Jules said, glancing at her phone. "With nine hours and fourteen minutes to spare."
Hanna smiled, the first time Jules had seen her do so today. "Not bad."
And Dia exhaled. "Now—"
Jules jumped in. "We wait."
## Jules
**Jules's body pulsed with** adrenaline the entire rest of the day, all night, into the next day. They had _actually_ pulled it off.
Now all they had to do was hope and pray that whoever was on the receiving end of their submission heard what Jules had heard.
On the bus to work, she put her feet on the back of the empty seat in front and drummed a frenetic pattern on her knees, until the old white man across the aisle gave a very deliberate and noisy cough. _Screw you,_ she said in her head. _Don't you know who you're looking at?_ Then she rolled her eyes at herself.
At Callahan's, she shoved her stuff in her locker and then went out on the floor. She opened up her register and waved to Henry, changing out a display. "Hey!"
"Hey, Jules," he called back. "You ready for a rematch?"
"If you wanna lose again, sure," Jules said. Three registers down, Autumn turned around and smiled, raising one hand. _Hi,_ she mouthed, and Jules thought she might throw up.
_Hey,_ she mouthed back.
It was painful in a sweet way, being so close to her and yet so far. Autumn, Autumn. She kept finding herself mouthing her name, tracing the letters on her palm, the inside of her forearm. It was so easy to get lost in those letters.
For three hours Jules counted coupon books and checked out the few people who came through her line, keeping an eye on Autumn's register. Her line was steady; people liked Autumn—she was new, she was sunny, she let them tell her the stories that Jules had long ago gotten sick of. And Jules got to watch, uninterrupted, getting lost in the swirl of Autumn's fingers through the air, the shake of her shoulders, the magic color of her hair.
Mrs. Doyle came shuffling up as she did every single Thursday, her cart full and colorful. "Hey there, honey."
"Hi, Mrs. D." Mrs. Doyle was Jules's favorite customer, mostly because she reminded Jules of her grandma, who'd passed away when she was eleven. The same papery, dark-brown skin, the same white hair set in an elegant twist, the same smell of talcum powder and cooking. Jules grabbed the first item and started scanning. "Ready for the birthday party this week?"
"Oh, yes," Mrs. Doyle said, in that rich, velvet voice of hers. "Elsie's going to love it. I'm making her cake today."
"Don't forget to bring me a piece," Jules said. "You promised!"
By the time she'd scanned all Mrs. Doyle's items and helped her pack them, she was five minutes late for her break. Jules closed up her register and crouched to pick up her bottle of water, and when she stood up, there was Autumn.
"Hi," Autumn said, and her cheeks were as pink as her hair. "Are you going on break?"
Jules nodded, willing her pulse to control itself. "Want to sit outside?"
They skipped the break room and walked out of the back exit, straight into the baking afternoon sun. There was a tiny, dusty parking lot, crisscrossed with tracks from the delivery truck, and Jules scuffed them away as they walked around the corner to the scrubby grass by the fire exit that no one ever used. The closest to private they would get.
Autumn sat, her legs stretched out into the grass. She had a scar on her left shin, Jules noted, and her sneakers were stained green in places. "Casualty of my little brother," she heard Autumn say, and Jules looked up to find Autumn smiling. She knew she should be embarrassed at being caught but she didn't feel that at all. She didn't care if Autumn caught her looking, trying to commit every minuscule detail of her to memory.
"How old is he?" Jules sat next to Autumn, close enough so if she shifted enough in the right direction their arms would be touching, skin on skin.
"Five," Autumn said. "He's a nightmare. But I love him."
"Big age difference," Jules said. "I have a brother, too. But he's fifteen. I love him, but he's a pain in the ass."
Autumn laughed, her mouth wide open and red inside. "Yeah. He's my half brother, but I don't call him that. He's my family, you know?"
"I get it," Jules said.
"I hate when people try to say it. My stepmom, she's my mom now. My mom died, like, ten years ago. That doesn't mean she's been replaced, but I'm allowed to love new people." Autumn shook her head as she looked at Jules. "Sorry. I probably shouldn't be saying all this. I don't know what it is about you."
"Me?" Jules shifted a millimeter closer, her breath crackling.
"Yeah, you." Autumn looked straight ahead but Jules could still see the smile on the corner of her mouth. "Whatever it is that makes me want to tell you everything."
Jules pulled in a surprised breath, summer-tinged oxygen flooding her lungs. "Oh, that," she said with a careless smile. "Sure."
"Sure," Autumn repeated. She crossed her ankles and leaned back, looking up at the cloudless sky for a moment, until she turned her gaze back to Jules.
Then they watched each other for this interminable moment; seconds, hours, days, who knew? All Jules could think was that it felt so easy, to be sitting here with this girl she barely knew and talking about real things. And the way Autumn was looking at her—the way Jules supposed she must be looking at Autumn—it was like she was lit up inside.
Maybe it _was_ too much, all this, but it didn't feel that way, not to Jules.
And Autumn felt it too, hadn't she said as much?
She felt emboldened by this, by the adrenaline of playing again still short-circuiting her system. So Jules reached across and touched her fingertips to the back of Autumn's hand, and it felt like ocean water rushing over her skin. And she asked another question. "Autumn," Jules said, and she didn't think she'd ever get over that name in her mouth, "would you like to go out sometime?"
Autumn turned her hand over so Jules's fingers were turning circles in her palm. "Jules," she said, "I would like that very much."
## Dia
**On Sunday afternoon Dia** was grabbing her stuff from the break room, ready to leave work, and at the same time checking her email on her phone for the millionth time since Wednesday.
She scanned past the sale notices, the junk, and the day care newsletter, sighing as she shoved her phone into her back pocket. Nothing.
Okay, it had only been four days. But patience was _not_ her strong suit.
Stacey came through, tying her checkered scarf around her ponytail. "You're still here?"
"I'm about to leave," Dia said, pushing off the wall. "See you tomorrow."
She did leave, and walked to the stop to wait for the bus. She pulled out her phone again, habit now, and absentmindedly spun it between her fingers as she watched a guy in a Biggie T-shirt run across the street. Elliot had had that shirt, she remembered. He used to wear it all the time, even after a hole formed in the left armpit, even when Dia teased him about it.
She smiled at the stranger. Sometimes the memories hurt, but the pinprick of pain was always better than not remembering him at all, the whirlwind of everything they'd been.
(Still better: not having to lose Elliot at all. And that was why she was not with Jesse, because she could not bear that kind of pain again.)
A _ding_ sounded from her phone and Dia looked down at it.
Saw one new email waiting on the screen.
**Subject: Congratulations.**
"Holy—" Dia jabbed her finger at the screen. "Open!"
**Dear Wildfire,**
**Congratulations! The judges have selected you to advance to the second round of the Sun City Originals Contest. The Judges' Performance will take place on Tuesday July 10. Please look out for another email with further details shortly.**
**Sincerely,**
**The SCR Team**
She read it twice, three times, before it sank in.
They'd done it.
They were _in._
The bus was rumbling up the street toward her, but Dia got up and walked away from the stop on shaky legs, pressing her phone to her ear as she called Jules. "Answer," she said, pacing on the sidewalk, the sun hitting the back of her neck. "Come on—"
"What?" Jules answered. "Why are you calling me?"
Dia held back from yelling. "Don't be such a wench," she said. "I'm calling because I got a _fucking_ email from _fucking_ Sun City, okay?"
"Oh," Jules said, her voice dropping. "So? We didn't get in?"
"No." Dia's heart pounded. "We're in. We did it!"
"Wait, _what?_ " Jules said, and she laughed, surprised. "Jesus, lead with that! We actually got in?"
"Yes!"
"So, what now? What next?"
"I don't know," Dia said, and she lowered her voice as people waiting at the stop turned to look at her. "July tenth. That's when the next round is. And it said we'll get another email with more details. But Jules, I mean—can you _fucking_ believe it?"
"Barely," Jules said. "The tenth? Oh my god! Have you told Hanna yet?"
"No, I literally got it and called you right away," Dia said.
"Okay," Jules said. "So call her now and tell her."
Dia made a face at the sidewalk. "You call her," she said. "She doesn't seem to hate you quite as much as me."
Jules made a noise that was half disbelief, half derision. "Shut up," she said. "Don't try and pull that. You're the one acting all ice queen to her. Call her yourself. I'm not going to be your go-between."
"I am not an ice queen," Dia said, indignant. "I have been perfectly civil."
"As has Hanna," Jules said. "Now call her. Text me after. Forward me the email!"
"Fine," Dia said. "Bye."
She hung up and tapped her phone in the palm of her hand. Call Hanna. Like it was so easy. Like Hanna didn't want to scratch her eyes out.
Okay, that wasn't fair. She had let Dia in her house, and said yes to all this, and she was really doing them more of a favor than she knew. Without her, there was no way Dia would be sitting here with an email saying they were in the contest.
It still scared Dia a little, though, being around Hanna. It was a reminder of the things she'd done back then, what she'd lost, who she'd pushed away in the name of self-preservation.
But that wasn't Hanna's fault.
Dia took a deep breath and dialed Hanna, standing perfectly still this time as she listened to the ring.
"Hello?"
It took Dia a second to reconcile the soft voice in her ear with the Hanna voice she knew. "Hanna?"
"This is she."
There it was—that edge, now that she knew it was Dia on the other end. "Hi," Dia said. "It's me. Dia."
"I know," Hanna said. "What is it?"
"So—I got an email. We're into the second round of the contest."
There was a second's silence and then Hanna said, "Wait, really?" The edge dropped now. " _Seriously?"_
"Seriously," Dia said, letting herself smile a little. "We're through."
"Oh my god," Hanna said. "So—"
"July tenth," Dia said. "That's when the next round is. We'll get details later. But, yeah."
"Okay," Hanna said. "We should kick this thing into high gear, then."
Dia looked down the street, watched the trees absolutely still in the lack of breeze. "Yeah," she said. "We should. Hanna—" Dia wanted to say something, something actually meaningful, because it was going to be a long month if she couldn't bring herself to say more than platitudes.
But before she could say anything, Hanna interrupted her. "Let me make it easy for you, Dia," she said, sounding tired. "So you don't even have to ask. Ready? Yes, I really am sober. No, whatever it is you've heard I did isn't true. Yes, I'm taking this contest seriously. No, I'm not going to screw up this time. Okay? I think that covers it all."
Dia pushed a curl behind her ear, unsure where to go now.
_Yes, I really am sober_.
She didn't expect the rush of relief she got from hearing those words from Hanna.
"Hanna," Dia said now.
"What?"
"I wasn't going to ask," she said quietly. "I was only going to say—I mean, that's good, for you. That you're sober and everything. And I meant what I said the other day. I'm not here to drag up the past. I know you hate me, but you don't have to worry about that. All this? It's not a game to me, or whatever you might think I'm doing." She paused. "I want to make sure you know that."
Hanna's sigh crackled down the line. "I don't hate you, Dia," she said, and she'd shifted from tired to exhausted. "So don't sweat it. I'm just here for the music. Okay? Don't get all twisted up."
Dia wrapped one arm across her body, a frosting smear decorating her wrist. _I don't hate you, Dia._
She didn't believe that at all.
Or maybe she should. After all, there was nothing for Hanna to hide behind now.
Dia nodded. "Okay," she said, trying to sound sincere. "I won't."
They were in the middle of dinner when her dad set down his fork and pointed at Dia. "Okay," he said. "What is it?"
Dia froze with her glass halfway to her mouth. "What?"
"Something's up," her dad said, narrowing his eyes at Dia. "I can tell. You're all . . . jumpy."
"Maxwell, what are you doing?" Her mom shook her head. "Can't have one peaceful meal."
But this was the thing with her dad. They were too alike; he always knew when she had something good going on, somehow could see it in her.
Dia took a sip of her water and put her glass down. "Okay," she said, and now her mom raised her eyebrows in anticipation. "So, you know the Sun City Originals contest?"
"Of course," her dad said, while her mom nodded.
"What about it?" Nina said.
Dia took a deep breath. "We kind of entered." She tried to sound casual, like it had been a whim, no big deal. "The prizes this year are big. Fifteen thousand dollars _and_ the winner gets to open a show for Glory Alabama. Do you know what an amazing opportunity that is?"
Her dad snapped his fingers. "I knew it," he said. "I knew you were playing again!"
"Hold up," Nina said. "Go back. You entered a music contest? Without telling us?"
Dia picked up her fork and pushed her food around her plate. "I know," she said. "There wasn't really time. We had to get our entry together to submit really fast, and I would have told you, but—"
"We?" her mom said now. "Who's that?"
"Me and Jules," Dia said, and then quickly, "and Hanna."
Her dad nodded, his smile a thousand watts. He had always been her biggest supporter, teaching her to play, buying her books and equipment basically whenever she needed it, even when they couldn't really afford it. "It's about time," he said.
But her mom was frowning. "You have a lot on your plate already," she said. "Are you sure this is a good idea?"
"Yeah?" Dia said, trying to figure out what her mom wanted to hear. "Maybe? You know, it'll be like old times. And if— _big_ if—we won, the money would be really good for us. For college, for Lex . . ."
"Like old times?" Her mom's mouth turned down, the lines there deepening. "I don't know that I like the sound of that. After everything that happened? You and your friends, Hanna—" She waved a hand toward Lex in her highchair. "This one. I don't know."
"Okay, not like old times," Dia said. " _New._ And the contest part is only for the summer. It'll be over by the time the semester starts, and I'll be totally focused on school."
"I think it sounds good," her dad said. "It's a lot of money, if you won. And besides that, it'll be good for you to be back in the scene, making music again. Guitars have to be played, not hidden away."
"I know." Dia could see her mom's brain working, coming up with more reasons that it was a bad idea, and so Dia played a dirty card. "You know, I do miss playing, Mom. It used to make me really happy. You want me to be happy, right?"
Nina gave her daughter a sharp glance. "Don't even," she said. "Dia, please do not try to manipulate me. You're better than that."
"Sorry," Dia said, not very.
"Nina," her dad said, turning to her mom. "I get what you're saying, and yes, we all know some stuff went down the last time around. But that was then, it's done now. And I get that it's another thing to take on, on top of everything else she's got, but . . ." He looked at Dia now. "I feel like you know what you're doing."
Dia let out a surprised laugh. "I do?"
"Well, you told us you would graduate, and you did. You told us you'd keep your job, and you did. You told us you'd be a good mom, and you are," Max said. "At some point, we have to think you might have learned some things."
"Um . . ." Dia knocked her hand against her leg. "Thanks."
Nina's frown eased. "Well," she said. "That may be true. I just don't want you to get overwhelmed with everything. Some things, you can't exactly drop the ball on."
Dia reached over and ran her fingers through Lex's curls as her daughter ate a handful of sweetcorn. "I know," she said. "Trust me, I know."
Her mom shook her head now. "Well, then," she said, looking from Dia to Max and back again. "I suppose it's okay."
"Really?" Dia's smile was megawatt, too. "Thank you, Mama!"
"But there are going to be rules," Nina said. "Guidelines, let's call them. One: If things start slipping around here, or at work—we'll have words. And two: Be careful. Of distractions and things and people that aren't worth your time. Because you're very special, my love. You know that, right?"
Dia looked to the ceiling. She wasn't special, no more than anybody else. Her mom only said those kind of things because she was a good mom, in the same way Dia said them to Lex. "Sure," she said.
Nina reached across the table to take Dia's hand. "And of course I know you miss playing. I was there, too. I pay attention."
"I know," Dia said, looking at her dad, too. "I love it. I love you both."
## Dia
_**Today is a good day,**_ Dia decided on Monday morning.
At day care Lex went in without even one single tear, a true miracle, waving as Dia blew her a kiss. The coffee place down the street had just put out lemon muffins, the glaze still warm, and Dia ate one while she waited for the bus—which came exactly on time, so that when she got to work she clocked in five minutes early. "Good morning," she singsonged to Stacey in the back, tying her scarf around her hair as she went in, and ignoring the odd look Stacey gave her.
"It's too early for that," Imelda said, pulling Dia's tasks for the morning from her board. "Be normal, moody Dia. Much better."
Dia took the order forms from Imelda's outstretched hand. "I thought perky was part of the brand," she said. "Look, I'm actually doing my job properly for once."
Imelda laughed at her. "You have a point."
Dia got to work, her good mood making her whipping a little softer, her chocolate-chip sprinkling liberal. They'd have to work out a practice schedule, start working on new material. Because they had to have new material to play for the judges, Dia didn't want them going in with anything old. It was okay to get them in, but now they needed to show who they really were, what they could do, and she knew they could do better.
She wondered, loading pans into the ovens, if Hanna had stopped writing these past two years. She always had the most heartbreaking, razor-sharp lyrics. Even at her worst, she'd produced great things for them.
Jules was right: Hanna was a part of them, for better or worse.
After her break, Dia was on counter duty, the quiet moment after the morning busyness. Usually she hated working the counter, serving customers and making small talk. She much preferred being in the back, scoring bread and loading pastries into the ovens. But today she let her good mood spill over into her _Have a great day!_ , sometimes actually meaning it.
She turned the radio up a little as the bell over the door chimed, and when she looked up she gave her extra-special, real smile to one of her favorite customers. "Welcome to the Flour Shop, how can I help you?" She tipped her head to the side. "Butterscotch cookies? Or jelly doughnut today?"
"Maybe I'll get both," Jesse said. "Maybe I'll get something completely different."
"And maybe you'll get the cookies like always," Dia said. "What's up?"
Jesse ran a hand over his head. His hair had grown out as far as he ever let it, into tight spiral curls, faded up on the sides. It was a good look. Although Dia was not supposed to notice that. _Focus._
"Nothing," he said. "What's up with you?"
"What?"
He eyed her. "You're all . . . up."
"Am I not allowed to be happy to see you?" Dia said, and then pressed her tongue against her teeth.
_For god's sake, mouth. Stop saying things you're not supposed to._
Jesse leaned his elbows on the counter. "What time do you finish?" he asked. "Come by the skate park for a little bit."
"You know I can't watch you," Dia said. "It's terrifying."
"It's nothing," Jesse said, laughing. "I get hurt more at work. See?" He stepped back to lift his shirt, and on his stomach a deep purple bruise spread down from his hip to almost beneath the waist of his jeans. "That was Mickey smacking into me with a crate."
Dia rolled her eyes, even though his little trick totally worked and she could feel her pulse going up. "Oh, please," she said. "You're not going to break your neck serving pizza, though, right?"
"I'm not going to break my neck at all," Jesse said, dropping his shirt with a grin. "But it's nice to know you care."
He gave her that slow smile of his, and Dia was stuck for a moment. This was the thing: somehow it _always_ tipped over from showing off, flirting innocently, to saying things they really meant and were supposed to keep to themselves.
At least Dia managed to keep it to that only. She never let herself do what she wanted to do every time he rolled up with another bruise, another break—place her careful hands on him and make sure he was okay, that he didn't hurt too much, that he wasn't about to disappear on her. She always kept her voice light when she told him how she worried, but only to hide the truth. That she was afraid of him getting hurt all the time, that the next time might be the last time, that one day he might wreck himself harder than his bike and then what would she do?
Eventually she made a face at him and said, "Yeah, I'm a sweetheart, aren't I?"
"Seriously, though," he said. "What are you doing later?"
Dia glanced out the front of the shop; it looked quiet, no one about to come in. She could tell him about the contest now, if she wanted.
But did she want to?
She knew what would happen if she said the words _We got into Sun City._ Jesse would be all psyched, and she'd let it make her feel good, and then she'd say something over the line, because she always did that when she felt good, and _then_ she'd have to remember why she didn't do that.
_That phone call,_ Dia thought _. The fresh grave. The funeral dress._
She wouldn't say anything.
"I have to do something with my dad," she said, inventing cover for the time she'd be spending in Hanna's garage tonight while her parents were out. She opened the display case, grabbing a cookie and sliding it across the counter to him. "Here."
"Uh-huh," Jesse said, sounding like he didn't believe her, but he took the cookie and bit it in half. "So you don't have time for tacos?"
"Well . . . ," Dia said, looking at him. It was only food. They ate food all the time. She would remain in control of her mouth and her flirty, traitor brain, and they'd eat and have fun and be completely normal. "I finish in an hour."
Jesse brandished the remaining half of his cookie as he headed for the door. "See you later, Dee."
"Bye," she called, watching him leave and pinching the inside of her elbow.
Danger.
## Hanna
**"Don't touch that," Hanna** called out, watching Ciara closely. "We have a serious you-break-it-you-bought-it policy."
Ciara pulled her hand back from the old-fashioned wardrobe. "Yeah, I don't have room for that in my house." She wandered back up to the desk that Hanna sat behind. "Don't you get bored?"
Hanna lifted her pen from the legal pad she'd been using as a sketchpad, the lines covered up with her wonky doodles. "Out of my mind," she said. "But they pay me." She glanced at her phone as it buzzed with a text, and let out a sigh.
"What?"
"Jules," Hanna said, pushing her phone away. "They're coming to my house later. To practice. I don't think I really thought this through."
No, she hadn't. She hadn't really thought past their recording attempt because, truthfully, she hadn't thought they'd ever make it in. But somehow, thanks to some trick of the universe, they had.
And now Hanna had to, like, work with them. Practice and have them in her head and reply to their texts.
"I think you did," Ciara said, leaning her elbows on the counter. "I think you knew exactly what you were getting into. You had an opportunity and you took it. Why wouldn't you?"
Of course Ciara would think that; when Hanna had told her what she was doing, Ciara had honest to god jumped up and down with excitement.
"Why wouldn't I? Let's see—because we don't know each other anymore?" Hanna said, counting the reasons off on her fingers. "Because Dia looks at me like I stole something from her? Because being around them is kind of scary but also scary familiar?"
"Sometimes scary can be good," Ciara said, pushing up her round glasses. "You should chill. It's going to work out. You know what could happen if this all goes right?"
Hanna rolled her eyes. "And you know what might happen if it all goes wrong?" she asked. "Which do you think is more likely?"
"I was there, I remember," Ciara said. "But that doesn't mean it has to happen all over again. Two years is a long time in teenage years."
"Like you're so much older and wiser," Hanna said. "You're twenty-three."
"Exactly," Ciara said. "You're a baby to me." She paused. "You don't have to be their friend, if you really don't want to. You don't even have to like them, I guess. But you still know them. You used to be inseparable."
"I know." Hanna looked down at her nails, the black polish chipping off. _Inseparable._ Three parts of one whole. Until they weren't.
When Dia had called her to tell her they were in, it had been awkward. But civil, right? Until Dia had started talking about Hanna hating her and Hanna had found herself somehow saying that she didn't.
_I don't hate you, Dia._
Was that true?
She didn't know.
It would have been easier to let Dia think she hated her. Easier, that way, to cover up her guilt and let the anger take control. But she didn't like how Dia had sounded: so superior, telling Hanna how things were when really, she had no idea. Not anymore.
When Hanna looked up, Ciara was watching her carefully. "They don't even really know you now," she said. "Not this version of you."
"I know," Hanna said. "That's the problem."
"So show them. And change their minds." Ciara stood back and shrugged. "Or don't. It's up to you. I'm just saying, _I_ know this you, and this Hanna is not the kind of girl to be scared away from what she wants. Is she?"
Hanna met Ciara's gaze and held it.
_Am I?_
## Hanna
**When Hanna got home,** Dia and Jules were already there, waiting for her. "Hi," she said, and curled her hands into fists, hid them behind her back. Ciara was right; she was not going to be scared off. She could do this, she could show them who she was now and let their judgmental looks roll off her. She was the new Hanna now. "Come in."
She took them through the house and into the garage, kept the door rolled down even though it was stiflingly hot. "Okay," she said as Jules and Dia got their guitars out. "So—what now?"
"Did you see the email?" Dia said. "I forwarded it to you."
Hanna nodded. In a little over three weeks they'd go and perform for the three contest judges. One original song, live and up close. And then—maybe glory. Maybe misery.
If they could get that far.
"They'll announce the winner on July twenty-seventh," Dia said. "There's going to be a whole thing at Revelry."
"I wonder who else is doing it," Jules said, slinging her bass around her body. "Like, people we used to know. Maybe Automatic Neon?" She paused. "Maybe Ciara?"
Dia shrugged as she plugged into an amp. "Maybe."
"No," Hanna said. "She's not doing it."
They both looked at her, surprised. "How do you know that?" Jules asked.
Hanna raised her eyebrows. "She told me."
"You talk to Ciara?" Dia's voice was sharp.
"Yeah," Hanna said, adjusting her ponytail as she sat behind her drums. "When she's around, we hang out. And she's around now, so . . ."
She watched as they exchanged a look and felt a petty triumph rise. See: she wasn't the only one who'd lost something when it all went down. It had taken her time to start talking to Ciara again, but she knew that Dia and Jules hadn't spoken to her at all. Ciara said she didn't mind, things were difficult, she understood. But still—after everything Ciara had done for them? And they couldn't even be bothered to reach out, text on her birthday, keep in some kind of contact?
Ungrateful.
Jules nodded. "That's cool," she said. "I haven't seen her in so long."
Hanna picked up her sticks. "Well, she's here," she said. "No one's stopping you."
"Can we actually practice?" Dia said. "I have to pick up my kid soon."
Hanna spun her sticks. "Ready when you are." She wasn't sure if the tension in the air was good or a sign of them falling back. They were past that first shock of being together again; now was when they figured out if they could actually do this, stand each other long enough to make this work. So far it felt like one step forward, two steps back.
They played through "Hills" again a couple times, and then moved on to some of their other old material. Just trying to get a feel for it, to get back into the swing of playing consistently. It was better when they weren't talking, Hanna thought. Something about their words felt barbed. But their music felt sharpened in a good way, like a weapon, like they were protected by it.
And she felt better when she was pounding her old drums, muscles in her shoulders waking up, rhythms patterning though her hands. They still sounded so good—to Hanna's ears, at least. They still had that energy that made her want more, more, more.
After a while they stopped for a break, and Jules sat on the floor. "So we should probably get started writing new material," she said. "Right?"
"Definitely," Dia said. "We have more than three weeks; we should use them."
"I have—" Hanna stopped. "Wait one minute."
She got up and went inside the house, upstairs to her room. She opened the bottom drawer of her nightstand and let out a long, slow breath as she placed her hands on top of her notebooks.
They weren't just writings; they were her lifeline, in a way. Her way of taking control, turning her inward hate into something good, writing her way from confusion to some kind of clarity.
There was a lot in here. Dark things, secret things, wild things. Words she hadn't planned on showing people.
Except, when it came down to it, this was not _people._ This was Dia and Jules.
And as confused as Hanna was about where exactly she stood with them, what she really felt, she knew that they would understand all of it. Her words, her lyrics. This was how they got each other, wasn't it?
_Show them,_ Ciara had said.
She was in these notebooks, the old her and the new. Maybe this was the perfect way for them to see how much she had truly changed.
So she lifted out a small stack and held them close to her chest as she ran back downstairs, back into the garage. "Here," she said, holding them out in Dia's direction. "These will help."
It took Dia a second, but then she took the pile from Hanna and when she looked up, it was with genuine surprise. "Oh," she said. "You still write?"
Hanna folded her arms and nodded. "A little." Every single day.
Now Dia's mouth curved into the smallest smile. "Thanks," she said. "Jules—we can start from here."
Jules nodded and looked at Hanna. "You always were our poet."
Hanna smiled back, instinctual. "Yeah," she said. "I always was."
## Elliot
## SEPTEMBER
**It's kind of hypnotic,** watching the skaters go back and forth on their boards, from up on the hill. Like the tide rolling in and out, Elliot thinks.
"Have you seen Graceland play before?" Jules asks him.
"No. I don't really go to a lot of shows," he confesses. "It's not really my scene."
"She kind of does that, doesn't she?" Jules stretches her legs out on the grass. "Makes you do things you didn't think you would."
True.
They're sitting on the hill above the skate park, waiting for Dia to get off work so they can go see Ciara's band play. It's kind of fun, being with the band. Like a new world. Dia knows all these people, and Elliot follows behind her everywhere they go and pretends like he knows what he's doing there. He hit the jackpot, he thinks—this guitar-playing, song-writing, determined force of nature, actually with _him_. Not that they have touched the words _girlfriend_ or _boyfriend._ Dia doesn't seem to care, and Elliot just wants to keep what they have for as long as he can. "Do you have plans?" he asks Jules now. "For the band? After school or whatever?"
" _Dia_ has plans," Jules says, and then smiles. "I'm kidding. We talk about it. Going to LA after graduation. But then maybe we could stay here and become the best here, instead of being nobodies in that city. People might pay more attention then."
"Do you want to do that?"
"Yeah. You know, I want to prove everyone wrong. Black girls playing alt music?" She pauses. "People still think it's weird, and it'll be ten times as hard for us as some straight-white-boy band with a fraction of our talent. I want to show them they're wrong. We have as much right to be there as anyone else." She looks at Elliot and shakes her head. "But who knows? It's all, like, a thousand years away."
"Right," Elliot says, although he's a junior and they're sophomores. So for him, graduation is only several hundred years away.
"And who knows what might happen before then?" Jules is looking down the hill as she says this, and Elliot looks, too. Hanna's down at the bottom, talking to a girl on a bike. Jules is frowning, and Hanna's flipping her hair, and Elliot isn't sure what to say. He's both part of this and not, and whenever Dia brings Hanna's drinking problem up now, Elliot stays quiet. She never really wants him to say anything, anyway; she just wants to say things out loud, things she won't say to Hanna.
Then the girl on the bike rides away, and Elliot notices Jules's gaze following her. He smiles and nudges her with his elbow. "You like her?"
"No," Jules says, but she begins playing with her hair. "Shut up."
"Who is she?" Elliot nudges her again. "Come on . . ."
Jules shakes her head and looks up at the sky. "Delaney Myers," she says. "But she doesn't even know I exist."
"So make her know," Elliot says.
Jules laughs and looks at him. "Oh, easy as that?"
He thinks back to meeting Dia at that party, the way he'd almost missed his chance, how he wouldn't be sitting here with Jules if she hadn't appeared out of the night and he hadn't asked her name. "I mean, think about it," he says now. "You can sit and keep pining over her, or you can try." He looks to where the girl—Delaney—is now. "She's cute. Why don't you just ask her out?"
"I don't even know if she likes girls," Jules says, and she makes a face.
Elliot grins. "You'll never know if you don't ask."
"But what if she shoots me down?"
"So? She's just one person," Elliot says. "You get over her and find someone new."
"But . . ." Jules's eyebrows pull together. "She's Delaney. She's perfect. I want _her_."
Elliot laughs at her stricken expression. "It's not that deep, Jules. I swear. Just ask her!"
"Hey!"
Elliot turns and Dia's walking toward them, wearing those skintight jeans again, and now he no longer has to wonder about taking them off.
They've had sex a handful of times now, in off-limits rooms at parties, once in her bedroom while her parents were at work. The first, third, sixth time for both of them, better every time. And now, when Elliot sees Dia, all he can think about is how she's the only girl he's seen naked and it is everything he dreamed and more. About the stretch marks all across her hips and thighs that shimmer like lightning in a summer storm sky.
"Hey," Jules says, standing up and dusting the grass off her legs. "Ready?"
Dia nods. "Where's Hanna?"
"Down there." Elliot points, and when a shadow flashes across Dia's face, he takes her hand and pulls her close and pinches the inside of her elbow, softly. "It's fine."
"Yeah, yeah," she says, but then she kisses him. "Let's go."
The four of them meet Nolan on the way, take the bus across town, and get their hands stamped at the club where Ciara's band is playing. Hanna really isn't too bad; Elliot keeps a cautious eye on her all night, still. At one point they end up getting drinks at the same time—Cokes—and Elliot's talking to her about baseball—she's an Angels fan—when she suddenly shakes her head. "You don't really like me, do you?"
"What?" Elliot says, taken aback. Where is this coming from? "No. I mean—yes?"
"It's okay," Hanna says, resting her elbows on the sticky tabletop, a dreamy look in her eyes. "I'm messed up. You can think that."
"I don't—"
Then she reaches into her shirt and pulls a tiny flask from her bra, and bares her teeth at him. "Tell her if you want. I don't care."
She walks away, this blond ghost, and Elliot clenches his fist. _Tell her if you want._
He finds Dia and the words are on the tip of his tongue, but the music's loud and she kisses them away.
## Jules
**Jules had never been** on a real date before. She had never stood on the front steps of another girl's house, dressed in nice clothes, wearing makeup, and knocked on the door. She had never had the door open to reveal someone's mother standing there. So when that happened and this imposing woman looked at her, inquisitive, Jules had to wipe her suddenly sweaty hands on her pants.
"Hi," she managed to force out. "Is, um, is Autumn home?"
"You must be Jules," Autumn's mom said, and the icy facade disappeared as she broke into a smile about as bright as her blond hair. "Come in, come in!"
Jules tried to keep calm as she stepped inside. She could already tell it was too hot for the jeans she'd put on, and she'd gotten her period about fifteen minutes before leaving the house, and the bus had been late, and and and . . .
But the hall of Autumn's house was enough to distract her, because it was filled with . . . _everything_. Kitschy velvet portraits lined the floral-papered walls, and the shoe rack by the door bore a dozen pairs of sparkling stilettos. An orange table held a variety of china figurines, and each stair had its own vase of fake flowers. Jules took it all in, scanning and then finding Autumn's mom staring at her expectantly. "You have a lovely home," Jules said, her inflection almost making it a question— _Is that what you want to hear?_ —and Autumn's mom beamed.
"Oh, thank you, sweetheart!" She spoke with this southern curve to her voice, Texas or Mississippi or somewhere else Jules had never been. "I know it's a little much, but I love a house full of pretty things. Makes it so much easier to get up in the mornings, doesn't it?"
"Mom, please stop talking." Autumn descended the flower-packed stairs, the skirt of her sky-blue dress swirling. "Jules does not need a lesson on Dollywood chic."
Jules tried not to look too taken aback by Autumn and her radiance, but in that dress, with her mermaid hair all swept up and the flash of thigh and her mouth so shiny red—god, she was beautiful.
"I don't know what you're talking about," Autumn's mom said. "Everybody needs that lesson. But fine, I won't keep you." She lowered her voice a touch. "Back by eleven, okay? And keep your phone on."
Autumn rolled her eyes but smiled, passing her mom to stand by Jules, and even that close was almost too much; what was Jules going to do when they had to sit by each other? "Yes, okay."
"All right!" Autumn's mom clapped her hands together. "Have a good night."
Jules remembered her manners at the last second. "It was nice to meet you, Mrs. Holloway."
"Oh, you too, honey. You two have fun," Autumn's mom said, and waved them out of the door. "Bye!"
The door shut firmly, leaving the two of them standing on the porch. There was a beat of silence before Autumn began laughing. "I'm sorry," she said. "I know, she's kind of a lot."
"Your mom?" Jules said. "I like her. Does she know this is a date?"
Autumn tipped her head to the side. "Yes."
"Okay," Jules said. "I didn't want to say anything wrong, or . . . you know."
"Not everybody from the south is like _that_ ," Autumn said. "And besides, I tell her everything." Then she gave Jules this long, up-and-down look. "But now we're done with her, can I tell you how good you look?"
Jules dipped her head, her face on fire. "Not as good as you," she said, looking back at Autumn. "Although, I feel bad. I should be taking you somewhere fancy in that dress."
"Don't be silly." Autumn shook her head. "It doesn't matter where we go. I just want to go somewhere with you."
A roller-coaster swoop in the bottom of her stomach. "Okay," Jules smiled. "You like Giorgio's, right?"
Jules wondered if the other people on the bus could tell they were on a date, or if what she felt crackling in the air was for her and Autumn only.
They got off and walked the few blocks to the pizza place, where they got a table out in the courtyard surrounded by families and other kids their age throwing balled-up napkins at each other. Jules asked the girl who brought their drinks if Jesse was working. "Not tonight," she said. "Can I get your order?"
They ordered the veggie deluxe, because Autumn didn't eat meat, and fried mac and cheese because— _fried mac and cheese_. When their food came their server gave them garlic fries, too, and when Jules said they didn't order fries, the girl gave her a sly grin. "On the house," she said. "Enjoy."
She walked away and it took Jules a second to figure it out, right as Autumn said, "Why did she do that?"
"Because we're on a date," Jules said. "And I think she's probably one of us. Me. I mean, she's queer."
Autumn turned to look after the girl and then looked back at Jules, a delighted expression on her face. "Huh," she said. "That's so nice!"
"Isn't it?" Jules said. Golden Grove was a pretty decent place to be, as a baby gay. Their school had a decent QSA, and coming up through the music scene there had been enough older girls for Jules to see and know that she, too, could come out and live a relatively okay life in town. She'd never had this, though—an actual date with another girl. This was another level.
Autumn did this thing of drawing the cheese out with her teeth and snapping it with her finger that was equal parts funny and mesmerizing. Jules wiped her fingers on her napkin as she chewed a bite of pizza. "So your mom—stepmom—where's she from?"
"A tiny town in Georgia," Autumn said. "She always says she had to get out because it wasn't big enough to hold her. Like, did you see the shoes?"
"Oh, I saw them," Jules said. "Impressive."
"I know." Autumn smiled. "She's so . . . she does exactly what she wants, you know? She wears ridiculously high heels and fake lashes, and her outfits have a ton of cleavage always, and she dances in the kitchen every single day. I want to be exactly like her when I grow up." She tipped her head to the side and studied Jules. "What do you want to be when you grow up?"
Jules lifted a corner of her mouth. "Financially stable."
"Ha! The dream," Autumn said. "Okay. Really, though."
"Really?" Jules blew out her cheeks and looked to the sky, streaked with deepening blues. "I have no idea. I don't even know what I'm interested in anymore. Like, my favorite class at school was math, but that's about all I've got."
"When I was a kid, I wanted to be a prima ballerina," Autumn said. "That was before I realized there are a very limited amount of fat girl ballerinas in the world, and that you have to actually be good at dancing."
Jules laughed. "What now, then? Besides your mom, of course."
"I want to be a nurse," Autumn said. She touched a hand to a pink curl. "When my mom—not Sasha, my first mom—was dying we practically lived at the hospital, and the nurses are the ones who keep you going. They would bring me books and let me sit in their station, take me down to the cafeteria in the middle of the night. They do all these things you don't even realize, and they help you get better, too. Well. When they can."
Jules looked at Autumn, marveling. How good was this girl? Smart and sweet and a heart of actual gold.
And honest. Autumn was spilling her soul, and Jules was lying.
Autumn gave her a funny look. "What?" she asked. "Do I have food in my teeth?"
Jules took a breath. "I used to want to be famous," she said in a rush, the truth now. "Maybe famous isn't it—known. I used to want to be able to stand on a stage somewhere and play music and have all these people sing the words back to me."
Autumn's eyes lit up with curiosity. "Music?"
"Yeah," Jules said, her palms sweating.
"Used to," Autumn said. "You don't want it anymore?"
Jules looked away. "It's a dream, isn't it? A little kid thing. Like you wanting to be a ballerina," she said. "People don't really get to have those dreams come true. It's a story they tell you so you'll believe in magic a little longer."
Autumn shrugged and leaned her elbows on the table. "Maybe, maybe not. Ballerinas are real, right? You can go and see them dancing. All those people on TV, on the radio, in magazines . . . they're real people. Once upon a time it was a dream for them, too. Somebody has to make it—it could be you. Then maybe . . . I would get to be the girl with the rock star." Her cheeks glowed pink.
"Maybe," Jules said, her face warm, too. _God, I wish._ But then she pressed her hands together. "I used to be in a band, actually."
"No way," Autumn said. "Tell me about it."
So Jules did tell her the story, the most bare-bones version she could manage without skipping over any of the important details. She tried not to get distracted by Autumn's face, either, intently watching her, and when Jules was done talking Autumn shook her head. "Wow," she said. "That's so cool. And also kind of shitty. But still cool. Don't you want to play anymore, then? Make another band or something?"
Jules drummed her fingers on the shiny silver surface of the table, watching her distorted reflection in it. "Actually, we just started playing together again," she said. "The three of us. Hanna's sober now and we're trying to make it work. It's going okay so far. We—" Should she mention the contest? No, not yet. In fact, maybe she wouldn't tell Autumn until the whole thing was over. _Yes,_ she thought, warming to the idea. _What if we win and then I tell her? And she'll be so surprised because she'll have no idea, and imagine her face!_ Autumn would be delighted, and Jules would get to feel like a true rock star goddess. Yes, this was a plan. "Yeah, it's going well."
"That's awesome," Autumn said. "Seriously."
Jules played with her straw. "It feels good," she said. "Playing again."
"So how do I hear you?" Autumn raised her eyebrows. "You have a Soundcloud? YouTube?"
"You wish," Jules said. "We shut it all down before. And Dia has all our original recordings, and good luck trying to get them out of her."
"Fine," Autumn said with a laugh. "I'll just have to come to one of your practices."
"Maybe," Jules said, nervous even at the thought of playing before this girl. "One day."
Autumn smiled, white teeth behind her red lips. "Okay."
Jules shook her head. "I've officially talked about myself enough. Okay. Moving on."
"Let's get dessert," Autumn said. "And I'll bore you to death talking about me. Deal?"
Jules laughed. "Okay. Deal."
They took the bus home, sat across the aisle from each other, and Jules asked delicate questions about Autumn's mom, her first mom. And Autumn answered equally delicately.
How did she die?
_Pneumonia._
What was her favorite music?
_Jazz. Dizzy Gillespie._
How much did Autumn miss her?
_So much. Like you wouldn't believe._
Eventually they were back right where they'd started, outside Autumn's front door, and Jules was supposed to say goodbye but she didn't really know how to. "Full disclosure," she said, looking Autumn right in the eye. "I've never been on an actual date before. I'm not really sure how I'm supposed to do all this."
"Full disclosure?" Autumn smiled. "Me neither."
Their laughter died out quickly, giving way to fast breaths and Jules's heart pounding so hard she was sure the entire world could hear it. She lifted her hand to brush a strand of blue hair from Autumn's forehead. "Full disclosure," she said. "I would really like to kiss you right now."
"Full disclosure," Autumn said, her voice a little shaky. "I've never kissed anybody." She took a step closer and her voice evened out. "But I would really like to kiss you, too."
The moment before it actually happened was the longest time in Jules's entire life, this space of _Should I be doing this?_ and _Yes, of course_ and _What if I ruin it all?_ and _Autumn, Autumn, Autumn._
And then they were kissing. Her mouth on Autumn's. Her hand under Autumn's chin, and Autumn's hands on her waist and it was quick, sweet, and Jules was going to leave it there, not turn Autumn's first kiss into something too much, but Autumn had other ideas. Because it was Autumn pulling Jules closer, Autumn opening her mouth, Autumn sighing in this way that made Jules slide her hand to the back of Autumn's neck and slip her tongue across her lower lip. Jules felt like she was underwater. All she could see and feel and think was this girl and maybe she wasn't really in love yet, but oh, maybe she was.
## Hanna
**"Jules, could you please** concentrate?" Dia snapped her fingers in front of her friend's face. "This is serious."
"I'm _listening!"_ Jules rolled her eyes. "Jesus, Dia. A girl could change her mind, you know. I could be sleeping right now, but here I am. For you, out of the goodness of my heart."
Hanna listened to their bickering and pulled at the neck of her shirt. They were out in Dia's yard so, Dia said, Alexa could run around outside.
Hanna had nodded at that, acted casual, but most of her was panicking because she had to be on her best behavior, didn't she? Dia hadn't said they could have their writing session at her house without knowing what that entailed, had she?
So this was Hanna's test: prove she could be good and sensible and _not dangerous_ around Alexa. Which, really, wasn't hard, because the only time Hanna had really been dangerous, out of control, was when she was drinking and now she had four hundred and thirty-four days without that and all she had to do was be normal.
("Normal.")
"It's two in the afternoon," Dia pointed out. "And don't act like you're so selfless. I know you want that prize money."
"Who doesn't?" Hanna lay out on the blanket Dia had put down. "Fifteen thousand dollars. It's a lot."
"What do you think it's like to be rich?" Jules stretched her feet out near Hanna's face, and Hanna shifted away. "Sorry. Can you imagine? For some people fifteen grand is nothing. A night at a club buying bottles. A ten-minute shopping trip. A tenth of a car." Jules laughed. "I could buy _ten_ cars."
Jules was right, Hanna thought. Fifteen grand could buy ten rusted-but-still-pretty used cars, perfect for cruising from Golden Grove all the way to wherever Hanna wanted.
Dia clapped her hands together. "How about we stop talking about money we don't have and try to do some actual work?"
"You are so boring sometimes," Jules said, and Dia smacked her. "It's true!"
Hanna made herself sit up and grabbed one of her notebooks. "Where do you want to start?"
Dia pulled her acoustic guitar into her lap and put her sunglasses on top of her head. "I was thinking we'd all pick out some of your lyrics that we like and see what fits with some of the ideas I have. And then we can . . ."
Hanna let Dia talk and nodded every so often, to show she was listening, but really she found her attention going to Alexa.
She was at the bottom of the yard, involved in some kind of game that Hanna couldn't quite work out: first she picked up a bear, and then she filled a bucket with a handful of dirt, and the bear went to the robot with a teacup on its head, and repeat. But Hanna found herself mesmerized, almost, watching this tiny human that came from Dia carrying out this operation so seriously, pausing every so often to call out, "Mama, watch!" She definitely saw Elliot there—in her smile, the shape of her eyes, and it was like remembering he was gone all over again. Almost three years, now.
_Look at her,_ Hanna thought. _She walks and she talks and she's an actual person. And I missed it all._ She had missed all the times Dia must have been so excited—and all the times she struggled, too, and things got hard. Jules had been the one there for her.
"Hanna?" Dia's fingers waved in front of her face. "Are you paying any attention?"
Hanna started. "Sorry," she said, and looked at Alexa as she spoke. "I was thinking about Elliot," she said without thinking, only hearing herself a second after the words were out.
_Fuck._
See, she was getting better at the whole watching-her-mouth thing, but sometimes she _couldn't_ contain it.
She turned back to look at Dia, her eyes wide. "She reminds me of him."
Jules shot Hanna this warning look, like _What the hell are you doing?_
But the corners of Dia's mouth lifted ever so slightly. "I know," she said, wistful. "There's something about her, right? Like the way he was."
"Does she know about him?" Hanna said, filter fully disengaged.
Dia nodded. "As much as she can understand. I show her pictures and stuff," she said, turning to watch her daughter. "We talk about him."
"I can't believe she's going to be two," Jules said, and she called Alexa's name. "C'mere!"
Alexa scrunched her face into a frown. "I'm busy!" she called out, and Jules burst out laughing.
"Oh, yeah," she said, looking at Dia. "She's your kid, all right."
Dia rolled her eyes but smiled. "Let's work," she said.
This time Hanna paid attention, picking through her notebooks and unearthing words that she barely remembered writing. She might have said she hadn't if the proof wasn't right there in front of her, in that scratchy black pen with the looping letters.
Dia picked out melodies on her guitar and echoed them with her voice, shaping sounds into Hanna's words. Jules noted down every little change, the lift in the second verse, where Hanna said she wanted to add a fill, added words when Hanna's weren't quite perfect.
_Hush, pretty baby_
_Haven't you heard?_
_About the fight last night_
_The mess your mama don't like_
_I'll show you where to go_
_To find your diamonds and gold—"pearls?"_
_And honey, we'll sleep till it's hunting time_
Dia tightened her D string and played a line, switched to chords, and sang along. "'Hush, pretty baby / Haven't you heard' _—_ like that?" she said. "Kind of a gothic, heavy feel?"
"I like that," Hanna said, and she sat up. "With really intense drums, like . . ." She demonstrated, hitting her hands on her knees. "Jules, what do you think?"
"Yeah, I'd make the bass kind of intense, too," Jules said, nodding, and they went back and forth with all these different ideas—"Maybe in the bridge have it all disappear, real quiet, and then back to loud?" and "Play that again, but an octave up," and "Wait, I'll come in there."
It was getting easier, being around Dia and Jules. It was like they were actually—well, not friends, but not enemies. Not actively hating each other.
She didn't have to second-guess everything they did, waiting for the punchline, the hidden camera or whatever.
She was supposed to be there for the music, but sometimes Hanna found herself thinking about more than that—about being friends again. Was that something she wanted, though? Was that ball even in her court?
Maybe.
After a few hours Hanna and Jules left and set off walking home. Hanna pushed her hair behind her ear and tried to ignore how badly she needed to cut the dead ends off, searching for suitable small talk to fill the silence between her and Jules. "That was good, right?"
"Yeah," Jules said. "You two didn't bite each other's head off, and we actually got some writing done. I give us a B-minus." And then, with barely a pause: "I want to ask you something. But you don't have to answer if you don't want to."
Hanna looked at their feet, walking out of sync, and shook her head. She knew exactly what Jules wanted to ask. "We can go there, if you really want," she said. "You want to know why I stopped drinking, right?"
Jules paused as a bus roared past, and when it was gone she said, "Actually, I want to know why you started."
Hanna took a breath. She had talked so much, at rehab, and that had been good, but it wasn't like this. It wasn't telling one of the girls who used to know her better than she knew herself.
She stopped walking and perched on a low wall outside a pretty house. "I felt all this pressure," she said, shielding her eyes as she looked up at Jules. "To be this person I wasn't. My mom was always getting at me about my grades or my hair or something. And I started to feel . . . sad all the time. But I didn't feel sad when I was drinking, and I could forget the pressure, and I could be this version of myself who didn't care about anything. And we were always at parties or shows, it was easy access, and it was this thing I could hold on to—when I felt bad, when I was hating myself, I could figure out that it was only five more hours until I could get a drink, and then I could get through those hours."
Jules nodded slowly. "Okay," she said. "So why _did_ you stop?"
"I wanted to be better," she said. "That was it, really. I wanted to be proud of myself. But I had nothing to be proud of. I lost my best friends, I lost my music, I made my sister scared of me. I lied _all_ the time. I kept trying to quit and falling back into it. I didn't like myself when I was drinking, but I didn't like myself when I wasn't, either, and at least when I was drunk it was easier to forget. But there's this thing sometimes, where you think you've hit bottom and then you fall a couple hundred feet below that and you're, like, is this it? Is this me?"
Lying in another hospital bed. Looking out and locking eyes with Molly, her scared face. The scratch in her throat as she'd tried to say, _I'm sorry, Molls, I'm so sorry._
_Is this me?_
She chest hurt at the thought of talking about the Molly of it all, and so she skimmed past it. Not a lie, more an . . . omission of the truth. She'd tell the whole story one day.
Instead she said, "I really thought about it, if this was going to be me for the rest of my life. How much I didn't want that. And my parents, they wanted me to go to rehab, and I thought—what's one more failed attempt? But that time I didn't fail. So—recovery. Sobriety." She held her hands out. "Here I am."
"Here you are," Jules said.
"I counted it out," Hanna said, and the memory of that first week of rehab flashed so close to the surface. "One hour without a drink. Then an afternoon. A whole day. Seven days. And it hurt, in ways I didn't know it would. Physically, in my head . . . it's like this other version of yourself is in there with you, pushing and pushing and pushing you to give in. And when the anxiety hits, the depression, what do you do? The thing you used to make yourself feel better is gone now. You can't have it." She exhaled slowly. "But now it's been four hundred and thirty-four days. And sometimes I think, it would be nice to have a beer, a little something, just to take the edge off. When my mom starts picking at me, when I fail a test, when I hear a song that reminds me of us back then. When I feel like shit and the effort of trying to stay clean feels like a waste of energy because I'm never going to be anything but a fuck-up—my head says, _Would it be so bad? Wouldn't it make you feel better?_ But it's never _just_ one drink, not for me. That would be it, and I'd be right back exactly where I _don't_ want to be. So. I keep counting."
Jules looked at her for a moment without saying anything, and Hanna almost began to regret saying any of it. But then Jules shook her head. "Hanna," she said, "I'm really happy it's working out for you."
Hanna shrugged. "I got myself in, I get myself out. Besides, I still have plenty more changing to do."
"Well, we all do," Jules said. "But, fine, if you won't be proud of yourself, am I allowed to do it for you?"
"I am proud of myself," Hanna said. "Sometimes."
_I am okay._
_I am not broken._
_I am here._
Sometimes.
## Jules
**Jules was in a** good mood.
Writing with Hanna and Dia yesterday had been so good, working old parts of her brain that were suddenly wide awake and itching to create. And hanging with Hanna like they were kind-of-friends again—that was cool. On top of that, she was still electric all over from her date with Autumn. From the kissing of Autumn, the holding hands with Autumn, the kissing Autumn again.
She was maybe in too deep.
But that didn't stop her from texting Autumn at the end of her shift at the mall: **lunch?**
Autumn's reply flew back: **where?**
Jules tapped her phone against her lips. **The Gardens** , she wrote back. **I'll bring sandwiches, you bring a smile.**
Autumn: ☺
The bus ride over gave her plenty of time to overthink things. Like—she hadn't actually seen Autumn since their date the other night. And she didn't know what she was supposed to do now—a kiss hello would be way too much, right?
But would Autumn expect Jules to kiss her again?
Was she supposed to ask Autumn out again, for a second date, officially?
Would Autumn ask her?
Would _Autumn_ kiss _her_?
_Stop. Over. Thinking._
She got off the bus and went into the park, found an empty bench next to a cluster of palm trees. She sat and quashed the impulse to rub her sweaty palms up and down the sides of her thighs. Why was she so nervous now?
_Should I hug her?_
This was all new to Jules. She had nothing to compare it to, no expectations of her own to temper. It was times like these she wished she could pick up her phone and text Ciara, ask for her older-wiser-queer-girl wisdom. But that wasn't possible. Add that to the list of things she regretted, mistakes she'd made and wished she could undo.
_Chill,_ she told herself as she waited. When she saw Autumn approaching, she waved and was rewarded with Autumn's smile.
"Hi." Autumn smiled, and Jules's nerves stilled.
"Hey," she said, holding a hand above her eyes to shield from the sun. "I got you extra peppers."
"My favorite," Autumn said, and she sat on the bench by Jules's side. "I'm starving."
They ate their sandwiches and talked about nothing in particular—made fun of their manager, Greg, talked a bit about the band. Jules almost told Autumn about the contest, let it slip, but she caught herself and remembered her plan. Tell her when it was over, when (if) (no, _when_ ) they had won. She felt giddy at the mere idea of it.
"So it's good," she found herself saying. "Practice and everything. It's fun."
"I have a question," Autumn said. "How come you picked Dia? Over Hanna?"
Jules put a hand over her heart like _ouch_. But it was a fair question. "I knew Dia first," she said. "She's like my sister. I loved Hanna, but—Dia was having the baby. Hanna was . . . I was tired of her." That was the honest truth, even if it wasn't _nice_. "It was easier without her. It was easier to be with Dia. It's shitty, because I never wanted to give up on Hanna, but—how long do you wait? I don't know." She rubbed her neck, guilt seeping in. Autumn would think she was terrible now. "I tried, before everything. I did."
"I get it," Autumn said, and then she screwed her face up. "Or I guess I don't, exactly. But I understand what you're saying."
"You'll like Dia," Jules said. "When you meet her."
"Good," Autumn said, her smile back.
_Okay_ , Jules thought. _This is good._ They were talking and Jules had yet to say or do too much, and Autumn was laughing, and this was all excellent. "The other night," she said, balling her trash up. "I had a really nice time."
"Me too," Autumn said, but her smile dimmed a little. "Can we talk?"
"Uh . . ." It slipped out before Jules could stop herself, and her knees started bouncing again. In what world did _can we talk_ ever lead to anything good? "Aren't we talking now?"
"I mean, like, talk about us," Autumn said, and she dipped her chin. "Or, me."
"Okay," Jules said. "What about us? And you? Did I do something wrong? Do you not want to go out again? That's fine, okay, whatever you want." Lie—it was most definitely not fine with Jules. This was supposed to be their beautiful beginning, not a messy crash and burn already. So what had Jules done to screw it up?
"Breathe," Autumn said with a smile. "You didn't do anything. I mean, except take me on the best first date I could have asked for. I'm—" She paused. "It was my first date."
"Mine, too," Jules said.
"Yeah, but, I'm not the first person you've dated. Right?" Autumn said. "I didn't . . . I don't know what this means. Like, I think I've had crushes on other girls before, but I never thought about it too much. But then there's you, and it feels like so much more. So maybe I'm gay? Or maybe I'm bi. Or maybe I'm something else entirely. Or maybe I don't even know yet. Am I supposed to know yet?"
Jules lifted one shoulder. "I don't know. That's all yours. No one else has to know or not know what or who you are."
"But it matters, right?" Autumn said, and her cheeks were flushed pink. "Does it matter to you? I don't even know if I'm supposed to talk about this with you, like—is this so neurotic to you? You must think I'm ridiculous."
"As if," Jules said with a laugh. She plucked at the fraying threads in the ripped knees of her jeans, her heart slowing as she realized that this wasn't Autumn ending things. She just wanted to _talk_. "I think you're amazing."
"Stop."
"Stop what?" Jules said. "Being honest?"
"You know what I mean," Autumn said. "I'm asking you a question."
"Does it matter to me?" Jules said. "I don't think so. Maybe you're gay. Maybe you're bi. Maybe you're something else entirely. But you're still you, right?"
Autumn pulled at a purple curl and smiled. "I guess."
"Okay." Jules smiled back at her.
"I think I'm a little nervous," Autumn said quietly. "That's all. Your first kiss is supposed to be fun, right, not send you spiraling into slight existential crisis. Not that it wasn't fun," she added quickly. "Because it was. _So_ much fun."
Jules smiled slyly. "I try."
"Shut up." Autumn moved her hand to Jules's knee, playing with those fraying threads, and the palm of her hand brushed the bare patch of skin and Jules's entire body felt warm. "Would it be okay if we—went slow?"
"What kind of slow?" Jules said carefully. The last thing she wanted was to go backward, for this to turn into what she'd had with Delaney—all hidden and cold and more fighting than fun—but she also didn't want to give up on this the way she had with that. She put her hand on top of Autumn's and raised her eyebrows again. "Like, would it be okay if I . . . asked you for an official second date?"
"Hmm, let me think. . . ." Autumn's eyes sparkled. "Yes."
"And if I wanted to kiss you on that date?"
"That would be a definite yes."
Jules grinned. "And if I wanted to come see you on your break tomorrow with the best brownie you'll ever eat in your life?"
"Like I'm going to say no," Autumn said, and she turned her hand over underneath Jules's, threaded their fingers together. "You're good at this."
"What?"
"Talking." Autumn scrunched her nose up as a bee buzzed past. "I like that."
"You can talk to me about anything you want," Jules said. "I mean it."
"Wanna walk?" Autumn said, turning to admire the flowers. "I never come here. It's really pretty."
Jules reached over and pushed Autumn's hair behind her ear. "What if I wanted to kiss you now?"
Autumn smiled, half visible to Jules. "We can do that, too."
"C'mere." She and Autumn had just had a serious talk about serious things and it hadn't involved any sniping or snapping or deliberately poking sensitive topics. It had been honest and real and—this was what it was supposed to be like, wasn't it? _This_ might be a healthy, respectful, real relationship.
She kissed Autumn on the cheek first, where her skin smelled like lemon. Maybe it was magic or maybe it was luck. Either way, she was not giving up.
## Dia
**They fell into a** pattern: practice in Hanna's garage, go to work, practice again, live their lives for an hour or two here and there. Dia found herself exhausted from staying up late, playing in the living room while Lex slept, then getting up early to get to the bakery. Sometimes her dad passed by her, going out or coming in from his shift, and he'd stop and listen as Dia played him something, nodding or giving a suggestion. "That's good," he'd say. "What about the chorus?"
But for Dia, the exhaustion was worth it every time they played, every time she opened her mouth to let their words slip out, when they ended practice aching and sweating. And she looked forward to practice now—for the music, and for the fact that it no longer felt like a battle between her and Hanna. She wasn't sure, but she thought they were kind of . . . becoming friends? Ish? In these time-pushed circumstances, it was hard for her to hold on to the grudge, to remind herself again and again why she'd cut Hanna out.
Because, more than anything else, this Hanna seemed entirely different to Dia now. Different from both the person she'd been when she was drinking, and even the person she'd been before that. This Hanna was someone who Dia found smart and a little tough and, truthfully, intimidating.
They were at the Golden Music Supply Store, a little over a week out from round two now, walking the aisles. Dia had Lex in the stroller, occupied with her phone, and Jules and Hanna followed behind her, talking over each other.
"I'm saying, we sound better when we slow the end down," Hanna said. "Otherwise it all gets rushed and all the detail gets lost."
"We just don't have it yet," Jules said.
The store was this kitschy palace of everything and anything you could ever want for your alternative musical needs. Dia ran her fingers over the top of a display case, peering at the mandolins inside. A poster of Glory Alabama hung above the case, their faces decorated with the swirls of their signatures. "Well, we'll clean it up," she said, staring up at the four women's faces. She'd always been amazed that they'd come from the same place as her: sat in the same classrooms, hung out at the same skate park, bought their strings from this very store. "Messy is not acceptable."
Hanna made a face that Dia ignored, and Jules said, "I still like 'Pretty Baby' best."
Their writing sessions had been good; so far they'd come up with a handful of skeletons of new songs, and now they were trying to pick which two to focus on. They kept going around in circles, though, bickering over the smallest aspects of each one.
Dia kind of liked it. This was what they used to do, dissect their music into tiny pieces and then put those pieces back together again, a broken puzzle. It was fun.
"I'm starving," Dia said, turning in the direction of the guitars. "Can we get food when we're done?"
She headed down the narrow aisle, the others following. "I have a question," Jules said from behind. "And don't laugh at me."
"Shoot," Dia said.
"What are we going to wear?"
Dia stopped in front of a corkboard layered with flyers and old posters. "We'll wear whatever we normally wear," Dia said. "Be us. Y'know?" And then she turned back to look at the two of them, both in ratty cutoffs and sneakers and hair pulled back. She looked down at herself: same cutoffs, flip-flops, a tank that had once upon a time been white. "Okay," she said. "So, maybe not _exactly_ what we always wear."
"Wait," Hanna said, looking offended. "What's wrong with my clothes?"
"You have a stain on your shirt, for one," Jules said, and Hanna looked down, screwing her face up.
"It's bleach," she said. "Fine, maybe we _do_ need to think about this."
Dia clapped her hands together loud enough that the other people in the store looked over at them. "I'll add it to the list," she said, ignoring the groans from both Hanna and Jules at the mention of it. They did the exact same thing every time Dia got out her notebook to write something else down or check something off that list.
"You two are working my last nerve today," she said, but only half meant it. "Come on. I need strings and picks and then food."
"All right, I'm going." Hanna's white ponytail almost hit Dia in the eye as she stomped ahead. "Hurry up, Jules."
Jules looked after Hanna, then at Dia. "She has a real attitude these days," Jules said. "I like it."
Dia shoved her toward the accessories. "Go!"
After tacos, Jules left for work, and Dia and Hanna were heading home. "Are you getting the seven?"
"Yeah," Hanna said.
"Me, too."
They walked to the bus stop in silence. When they got there Hanna scanned the times and Dia perched on the bench, checking that Lex was still sleeping. Out like a light.
She looked up at Hanna, whose nose ring flashed in the sun. "I've been meaning to ask," Dia said. "When did you get that pierced?"
"What?" Hanna said, and then she touched a hand to her nose. "Oh, that. I think it was six months ago or so . . . January?" She sat down next to Dia and stretched out her legs. "I felt like doing something fun."
"Did it hurt?"
Hanna nodded emphatically. "Like a bitch," she said, and then she laughed. "But you've given birth, so . . ."
Dia smiled, scraping her shoe on the asphalt. "Yeah, that hurt. I'd rather get a needle shoved through my face than do that again."
Hanna turned a fraction. "Hey. I, um—I meant to say sorry. For bringing Elliot up last week. I don't know what I was thinking."
Dia felt that reflexive rush of slight nausea she got whenever Elliot's name came up. Yeah, it had kind of taken her by surprise when Hanna mentioned him, but it shouldn't have. Dia had always thought Hanna's tendency to say whatever came into her brain was a side effect, but clearly it was all Hanna.
And it wasn't like Hanna hadn't been there, too—hung out with Elliot at parties, laughed at his awful jokes. She'd sat with Dia in the hospital waiting room. Helped pick out her dress for the funeral. She'd known him; she was allowed to talk about him. "You don't have to apologize," she said. "He's not a banned topic. It's okay." She focused on a scar on her knee, tripping her fingers along it, and then looked back at Hanna. "It's weird. Because we weren't actually _together_ together. It wasn't like we'd been dating for two years and he was my first love and everybody knew it. I wonder if that's what we would have become. Or maybe we would have carried on for a couple more months and then ended. I didn't know him, really _know_ him. Not beyond that infatuation stage."
"Right," Hanna said. "I get it."
"It's like . . . me and Jesse, we're not together," Dia said. "But I know him so much better than I ever knew Elliot. And I have this everlasting link to this boy who's gone now and sometimes I don't know what I'm supposed to do with that."
Hanna nodded. "She's cute, though," she said with a smile, holding a hand up to shield her eyes. "Your link. Smart, too."
Dia exhaled. "Yeah, she is."
"So . . . ," Hanna said, and Dia knew exactly what was coming next. "What _is_ the deal with you and Jesse?"
Dia rubbed her thumb on that scar. "When we met I said we could only be friends. Because Lex was so little and getting through school was the most important thing, and it felt weird to start going out with somebody when I was still the Girl Who Had the Baby by the Dead Boy. Sensible, right?" She paused. "But now Lex is almost two and I'm doing okay with her. I graduated. I have an okay job. Every reason I gave him is kind of . . . not a reason anymore."
Hanna raised her eyebrows. "So . . ."
Dia closed her eyes. "I have this fear," she said, and it was the first time she'd ever said it aloud. "That something terrible's going to happen to Jesse. And I don't think I could deal if it happened all over again. The funeral, the way everyone looks at me, the complete empty space where he's supposed to be in the world. I wasn't even in love with Elliot and it was awful. I can't imagine how it would be with Jesse."
"Because . . . you're in love with him?" Hanna said. "I thought he was just, like, the hot guy you keep around to flirt with."
"He's that, too," Dia said. "It's easier that way."
"It doesn't sound easier."
Dia opened her eyes to look at Hanna. "No," she said. "It doesn't, does it?"
"I know it was bad," Hanna said, "when Elliot died. But it was an accident. What are the odds of Jesse dying, too? Really."
"I don't know," Dia said, and she didn't know why she was telling any of this to _Hanna_ of all people, either. Maybe because Hanna knew, too, what it was like to feel out of control. To feel like there was something bigger than you that you wanted to run from but could not escape.
Or maybe because Hanna was the one who'd borne the brunt of Dia's fear, taken the worst of it without ever knowing. And, Dia was seeing now, that hadn't been fair to her. As much as she'd been trying to protect herself, she'd been hurting Hanna, too, and she'd still done it. Hanna—now, and as the girl she'd been—deserved more than that.
Dia breathed out, focused on their conversation. "I know it seems ridiculous. But there's nothing in the world that can tell me one hundred percent that he'll be fine."
Hanna shrugged. "Everybody dies sometime. You and me, we'll be gone one day. It is what it is."
"Reassuring," Dia said, her palms itchy. Hanna was talking about the gray of life that Dia didn't ever think of. To her, it was all or nothing: you survived or you didn't. You were here or you weren't. It could hurt you, or you could cut yourself off before it even tried.
But deep down she knew Hanna's version was closer to reality. You hurt and you healed and you felt joy and sorrow in the same breath. That was what Alexa taught her every day.
But it was still hard.
Hanna smiled now. "All I'm saying is—for all the time we get on this planet, you might as well try being happy for as much of it as you can."
"I _am_ happy."
"You know what I mean."
Dia pulled on her curls.
Yeah, she knew.
Dia let a silence develop for a minute, easing the stroller back and forth with her foot, the space between her and Hanna comfortable. Who would have thought, only three weeks ago, that this scene would ever unfold?
"It's going good, don't you think?" Dia picked at her scraped-off nail polish, a casualty of her guitar. "The music and everything."
"Yeah," Hanna said. "Better than I ever thought. You don't think so?"
"No, I do," Dia said. "Sometimes I think we might actually do it. Win. Get the money and the prize and be back to where we were. But better," she added, seeing Hanna's eyebrows raise. "Different."
"A lot is different," Hanna said.
"Right." Dia stared ahead, at the asphalt wavering in the heat. _Say it out loud, Dia. She deserves to hear it._ "It's really good, Hanna. That you're sober." She looked at Hanna now. "I mean it."
Hanna's smile was slight. "Four hundred and forty-four days."
"You count every day?"
"It works for me," Hanna said. "I picked it up in rehab."
_Rehab?_
Dia tried not to let the shock show on her face, pretty sure she was failing. "You went to rehab?" she said. "I didn't know."
"Why would you?" Hanna said, and shrugged. "Last year. Spring break."
Dia's mind began to run, trying to understand what more could have possibly happened to push Hanna into rehab. If everything she'd known Hanna to do hadn't been enough, if choking on a tube down her throat hadn't been enough? Maybe she'd done something truly terrifying. Maybe her parents forced her to go.
And then she thought: did it really even matter why? Four hundred and forty-four days was a long time. Hanna was sober, and that was not easy. She deserved less of Dia's skepticism and a lot more respect.
"Well, I'm glad," she said. "That it helped. And that you went." She swallowed hard, and said it before she could change her mind. "I'm sorry, you know. For how it all went down back then. It was messed up, the way I . . . cut you off."
Hanna looked surprised, like she'd never imagined hearing Dia say those words. "Thank you," Hanna said, and her smile was slight. "But I messed up, too."
"We all did," Dia said. "We were young and stupid." The next part came out without Dia overthinking it, the most real thing she'd said to Hanna in years. "And I was really scared, Hanna. That you were going to end up doing something to yourself that you couldn't undo."
Hanna nodded slowly, her eyebrows pulling together as she looked at Dia. "I know," she said softly. "Me, too."
Dia felt the air go out of her and all she could say was, "Okay." And Hanna seemed to understand, nodding again.
And then they sat in silence, waiting for things Dia couldn't even name.
## Jules
**On Monday they were** back in Hanna's garage, running through one of the loose ideas that they'd turned into a third real song. Jules could feel her fingers starting to harden, old calluses coming back to life, and she loved it. She loved the way it felt to play and shout and sing and dance, to feel like there was more to her life than work and school. She stepped onto a plastic crate, pretending it was a riser and she was playing to a crowd of thousands at, like, Coachella or some equally huge festival that Jules would always _say_ was so cliché but would really kill to be at, and howled the last line, closing her eyes. The reverb faded and Jules opened her eyes to the disappointing view of the garage door. She jumped down and wiped the sweat from her forehead. "I like this one," she said. "Makes me feel like . . . I need to go to a strip club and then maybe kill a guy."
"FYI," Dia said, "if you kill somebody please don't tell me about it. That way I won't have to lie on the stand."
Hanna grabbed her crash cymbal to stop it. "Jules, if you kill someone, I _will_ go on every single murder show and talk about how I knew you when you were a good girl." She held her hands to her heart. "Honestly, she was the sweetest thing. Troubled, though. I'm not surprised she killed him."
"Make sure they use good pictures of me," Jules said. "You know you pretty white girls always get your, like, sorority pictures everywhere, not your mugshots. That's what I want."
"We'll make sure to remember that," Dia said, lifting her guitar over her head. "But can we get back to work now?" She got out the notebook Jules had begun to hate and sat cross-legged on the cement floor. "We have just over a week. We should decide what we're going to play."
"I think we should do 'Bones,'" Hanna said, playing with the end of her braid as she sat on an amp. "I think it shows our growth."
"But 'Pretty Baby' is really good, too," Dia said. "Intense, and it has some good technical parts in there."
"What about the one we just did?" Jules asked. "That's really good."
"But it's not ready," Hanna said.
Dia shook her head. "And it's not right for this."
Jules looked between the two of them and exhaled. Well, mark this day. Hanna and Dia, agreeing.
This was like a miracle.
Without Jules really noticing, things had changed. At the very least, Dia and Hanna weren't fighting, only bickering, picking at each other with sharp barbs.
Jules had seen worse from them, much worse.
And now things seemed to be going okay. Better than okay. Good.
She drummed on her knees. "Fine," Jules said. "'Pretty Baby' is technically good and I love it, but 'Bones' shows us off more. And the lyrics are on another level."
Dia nodded once. "Okay, then. 'Bones.'"
The door that led into Hanna's house opened then, and Molly poked her head out of it. "Hanna!" she said, sounding breathless. "Mom's going to be home in twenty minutes."
"Shit." Hanna stood, suddenly jittery, shoving her sticks in her back pocket. "Okay, we have to go. Or, you have to go, I mean."
Dia crossed her arms and stared at Hanna. "Hold on," she said. "What?"
"My parents don't exactly know about any of this," Hanna said. "So—"
"So what?" Jules said. "We used to practice here all the time."
Hanna screwed up her face. "Yeah, and that was before I went to rehab, and now the rules are different. I can't do _whatever_ I want anymore, so—"
"So you're lying to your parents?" Dia said.
Hanna took a too-long pause. "It's not a lie," she said eventually. "They just don't know. Yet."
"Hanna! Are you kidding me?" Dia shook her head even as she got up. "Of course not. I should have known. Look, I don't want to tell you how to live your life—"
"Okay," Hanna said sharply. "So don't."
Jules unplugged the cables and lifted her bass over her head. "It's fine," she said loudly, to fill the sudden silence between them all. Had she jinxed it by thinking about them getting along? "She knows what she's doing. Right, Han?"
Hanna pursed her lips. "Right."
"And Dia is not trying to be overbearing, only watching out for you," Jules said. _"Right?"_
"Sure," Dia said, more as a sigh than anything else.
"Excellent," Jules said. She flipped the catches on her case closed and gave them both a sharp look. "So, we'll pick this up tomorrow. Yes?"
Dia did this exaggerated head roll but when she looked at Jules again the annoyance was gone. "Yeah," she said. "Okay."
"All right," Jules said. "Get your stuff and let's go."
Hanna grabbed one of the amps and moved it to the back corner, pulling a blue sheet over it. "Tomorrow," she said, "twelve?"
"Yeah," Jules said, waiting as Dia got all her stuff. "We'll be here."
Dia looked over at Hanna. "So you know," Dia said, "this is a bad idea. You should really—"
Jules took Dia by the elbow and yanked the door open, waving back at Hanna as she steered Dia outside. "Okay, we're done. Bye, Hanna!"
Dia pulled herself out of Jules's grip as the door crashed shut behind them. "Ow! You are, like, annoyingly strong."
"It's carting all those frozen peas," Jules said, flexing. "Truth."
Dia looked back at the house. "I don't like this," she said.
"What?" Jules said. "You think we're gonna get her in trouble?"
"I think she's going to get herself in trouble," Dia said. "Like always."
"She's not such a pushover now, is she? She does what she wants." Jules looked at Dia. "Reminds me of someone."
"Shut up," Dia said, but without any meaning. "I don't lie to people."
"Oh, really? I don't think you want to hear what I have to say about your boy Jesse." Jules swung her case over her shoulder. "Let's give her a break. Maybe she really does know what she's doing." Jules wanted to believe that, but the doubt in her voice was clear.
Still, she turned away from the house and began walking to the bus stop. "We're not in charge of her," she called back to Dia. "It's her life."
Jules went home, up to her room, where she propped her guitar against the wall and stood in front of her closet, shaking off the feeling of practice and Hanna's panic.
So Hanna's parents didn't know. Hanna obviously had a plan and knew what she was doing.
Or, that was what Jules was choosing to believe.
Instead of that, she tried to focus on what she was going to wear for the movies with Autumn. She stood in front of her small closet for ten minutes, black on black on black looking back out at her. Not like Autumn, with her rainbow of dresses and lip gloss in a thousand shades of Kiss Me. Jules tugged at the hem of a worn-out jersey and wrinkled her nose. But you know what? Autumn saw her all the time at work and if she liked Jules in her Callahan's apron, she should like her in anything, even band practice clothes.
So instead of changing, Jules pulled her braids up into a high pony, put on more deodorant, and switched her flip-flops for high-tops.
Down in the kitchen she saw the remains of her dad's best pasta, roses in a tall vase, and two notes on the refrigerator. From Danny: _Out w/people, back by 10._ And from her parents: _Gone for dessert, don't wait up!_
Jules smiled and plucked one of the flowers out of the vase to smell. Sometimes they were so cute it hurt.
She wrote her own note and slipped it under a strawberry magnet: _Movies, see you in the morning!_
She left and hopped on the bus again, and it didn't take long to get there, but when Jules walked up to the theater somehow Autumn had beaten her there. "Hey!"
Autumn looked up from her phone and her face broke into this beaming, beautiful smile. "Hi." Today her dress was orange and off her shoulders, revealing the faintest swimsuit tan line, and her lips were the same purple as her hair.
Jules closed the distance between them. "Hey," she echoed. "How was your day?"
"Same old," Autumn said, catching a strand of her hair before it got stuck in her gloss. "Yours? Did you have a good band practice?"
"Yeah, it was good," Jules said. She kept her face neutral so as not to give away her secret.
The more they practiced, the better she felt about their chances of winning, and that meant she was getting closer and closer to her fantasy of telling Autumn all about it.
Autumn smiled with shiny white teeth. "Could be worse," she said. "Ready?"
"Ready," Jules said, and without thinking about it too much, she reached for Autumn's hand, her own warm fingers wrapping around Autumn's cool ones.
But only for the briefest of seconds, before Autumn snatched her hand away, leaving Jules holding nothing but air.
Jules looked at her and Autumn was already shaking her head, her mouth this small smile. "Sorry," she said. "I didn't mean to do that. Here, let's go."
Autumn held out her hand, but Jules didn't take it this time. "You didn't mean to do what?" Jules said, and her voice came out louder than she intended, and her face was hot. "What? You don't want to hold my hand?"
"No," Autumn said, looking confused. "I mean, yes. I mean—Jules. I think it was just a reflex."
A _reflex?_ "Really?" Jules said. "What, so your instinct is to not want to touch me?"
"I didn't say that." Now Autumn's eyebrows drew together and she looked up at Jules like she wasn't sure what was happening. "I'm not used to it, is all."
Jules shook her head. "You know, if you don't want to do this, you don't have to. I'm not interested in making you be with me if you don't care."
"Jules!" Autumn's mouth dropped open. "When did I say I didn't care? All that happened was that I took my hand away, and I didn't even do that on _purpose_ , okay? I want to be with you and I want to hold your hand, so will you let me now or do you want to stand here and fight all night?"
A girl passing them by gave Jules an odd look, and Jules folded her arms across her chest, looking at her feet. "Stand here and fight," she said to the sidewalk. That's what she and Delaney would have done.
But Autumn was not Delaney.
"Oh, stop," Autumn said. "I'm not fighting with you, silly girl. Okay?"
Jules fought a smile and looked up at Autumn. "Okay."
"You are trouble," Autumn said. "Let's go in."
Neither of them attempted anything this time—they just went into the theater and bought their tickets and food, and then found their seats.
Halfway through the movie, Autumn placed her hand, palm up, on the armrest between them. And Jules took the invitation, laced her fingers through Autumn's. Her heart moved out of her throat and settled in its rightful place.
But this small voice in the back of her head whispered to her. _This is all you're good for. In the dark. Who's going to claim you out there, in the real world?_
Autumn shifted, leaned her head on Jules's shoulder, the strawberry of her shampoo so sweet.
_This girl,_ Jules thought, defiant, shutting that voice up. _She will._
## Dia
**The sun beat down** on Dia's shoulders as she sat outside, watching Lex kicking a ball around the yard. She rolled her wrists, hearing the clicking of ligaments, and blew out a sigh between chapped lips. It was a day off from the Flour Shop, but she'd run errands all around town while Lex was at day care, and then gone to Hanna's for practice. At least her parents were both at work now—it meant that there'd been no one to judge Dia for feeding Lex a Happy Meal for lunch and then planting her in front of cartoons while Dia lay on the couch, staring at the ceiling, asleep with her eyes open. Sometimes the only thing to do was put aside all her worrying about being a Good Mom and settle for being Okay-Ish and Trying Really Hard at It.
Now she watched Lex playing, and when her phone started buzzing on the step beside her, Dia didn't even raise her head to check the caller, just put it to her ear and answered. "Hello?"
"Dia Valentine," Jesse's voice said. "Where the hell are you?"
Dia slipped a hand to the back of her neck and smiled at the concrete. "Home," she said. "And hi."
"I didn't mean now," Jesse said. "It was more of a figurative _where are you_ , y'know? Because you've been completely MIA the last couple weeks, Dee. What's up with that?"
"I got stuff going on. You know me," Dia said, watching a lone white cloud scudding across the otherwise unblemished sky.
"Well, forget that and come hang out with me," he said. "I haven't seen you in forever, and I'm bored."
"You only think of me when you're bored?" Dia said. "Real nice."
"Not true. I think about you all the time," Jesse said, and Dia heard it in his voice, how halfway through he realized what he was saying and slowed like he was trying to pull the words back into his mouth. But by then it was too late; they were already out, and they had already wound their way into Dia, catching her breath and her words in her throat.
"We'll go to the Gardens," he said now. "You and Alexa and me. What do you think?"
What did she think? Dia pressed one hand to her chest, gazing in Lex's direction. She thought that he was so good to her, and she missed the way he looked at her, and what better way to spend her afternoon than with this boy and this little girl?
"I think that sounds perfect."
Dia saw Jesse from the bus, sitting on the wall by the south entrance. They got off and she took her time walking over, because he hadn't noticed them yet, and when he didn't know she was looking was when she could really look. He'd cut his hair short again, and his left elbow had a bandage wrapped around it, the white stark against his deep brown skin. For a moment Dia imagined what would happen next if she were his girlfriend: she would walk over and touch his arm, kiss him hello, and he'd smile at her.
What would it be like to kiss Jesse Mackenzie?
That, she'd imagined many, many times before: staring out of the window in school, mindlessly piping cookies at work, with her hand between her legs at night.
"Mama, go," Lex's impatient voice said, and Dia shook herself out of it.
"Okay, baby," she said. "Whatever you say."
Jesse had seen them now, and he hopped down from the wall, wincing a little as he put pressure on his left arm. He bent down to greet Lex first. "Hey, kiddo," he said. "Dope hat."
"Hi," Lex said. "We going to see the bugs."
"Well, duh," Jesse said. "They're the best thing here."
He stood and smiled at Dia, heartbreaking as always, those almost-black eyes, his full mouth. "Hi."
"Hi." Dia tipped her head to the side. "Let's walk."
They followed the long path into the wildflowers; Dia let Lex out of the stroller and she went ahead of them, flitting from flower to flower like a little honeybee. "What happened to your elbow?"
"That? It's fine," Jesse said. "A sprain."
Dia looked sideways at him. A sprain. Like it was nothing at all. "What did you do?"
"It's nothing, for real," Jesse said. He put his arm straight up in the air, waving it around. "See? I messed up a landing, got a little bruised. These are the perks of riding."
Dia sighed. "Sure."
"It's one of those things," he said. "You start riding, you're going to fall down. It's worth it, a couple breaks and batterings every once in a while. You still get back on the bike at the end of it all."
"Hmm." Dia reached out but stopped short of her fingers grazing his arm. "You scare me, sometimes." _All the time, every second of every day_ was what she didn't say.
"It'll heal," Jesse said. "You don't need to worry."
"I can't help it," Dia said, and she looked away, at Lex wandering close. "Not when it comes to you."
Jesse didn't say anything, and now Dia was the one wanting to pull the words back out of the air and swallow them down.
Then Jesse said, "When's Alexa's birthday?"
"The twenty-second," she said. They kept walking, Dia making sure Lex was always in their sight. "I wonder about who she's going to be," Dia said. "You know? What she's going to like, what she's going to want to be."
"Whatever it is, she'll be the best," Jesse said. "With you being her mom? For sure."
They walked beneath a sickly-sweet arch of flowers. "What did you want to be?" Dia asked. "When you were a kid."
Jesse blew his cheeks out and began ticking them off on his hands. "First off was astronaut," he said. "Classic. Then it was soccer player, chef, and I _think_ doctor. Then I landed my first three-sixty whip and I decided I wanted to be a pro rider." He dropped his hands. " _Then_ I realized I was nowhere near good enough, and so I moved on to architect."
Dia flexed her fingers on the stroller's handlebar. "Lex, don't grab," she called out. "Gentle, remember?" Lex bounced her head, nodding, and Dia kept an eye on her as she meandered closer.
"Look at her," Jesse said. "She used to be a tiny thing and now she walks and talks and understands what you say to her. Dee—you're, like, imparting knowledge to her."
"With great power," Dia said, and she almost felt like she could cry. "She's so amazing."
"You made her," Jesse said, and he nudged her shoulder with his. "That's pretty amazing, too. And one day you'll have that house with the big yard and all that. Everything you want."
Dia glanced sideways at him, the most she could bear to look at him right now. "Yeah. We'll see." She spotted Lex crouched in the flower beds. "Lala, come this way."
Jesse cleared his throat. "You know, I didn't know if you'd answer when I called," he said. "Kinda feel like you've been avoiding me lately."
"Avoiding you?" Dia repeated, surprised, and then she thought about it for a second. With practice and work and writing, she had been extra busy. And then there was the fact that Jesse didn't know about any of it.
"Yeah," he said, turning so he was walking backward and looking at her. "I go into the Flour Shop, you're not there. I ask if you wanna hang, you say you're busy. You know."
Dia smiled.
Maybe she should tell him now.
Everything was going so well, and the day was nice, and maybe she didn't _need_ to be so afraid. Maybe she could tell him, and he'd be happy for her, and she'd—
What? Say something she shouldn't?
Would that be so wrong? What had Hanna said? _For all the time we get on this planet, you might as well try being happy for as much of it as you can._
She felt the sun on her shoulders, the music in her head. She felt good, today.
"Okay," she said. "You want to know the truth?"
Jesse looked at her, uncertainty in his eyes. "Do I?"
She laughed now. "It's good," she said. "Me and Jules and Hanna—we entered the Sun City contest. And we got through the first round, so—"
"Wait, wait, wait." Jesse stopped. "You entered Sun City? You and Jules and _Hanna_?"
"Yeah."
"Hanna Adler? Hanna who you don't talk to?"
"Yes!" Dia said. "Except, I do, now. Talk to her."
"Okay," Jesse said. "And that's where you've been, being in a band and entering contests and not telling me about any of it?"
"I guess. . . ."
Jesse took a step back and threw his hands up. "Why didn't you tell me about any of this? _Dee._ What, you just give me cookies to shut me up and send me on my merry way without thinking this might be of interest to me? I can't believe you!"
He sounded so serious and Dia would have fallen for it without the cookie talk and the way he started to laugh at the end. "Shut up," she said, his laugh infectious. "See, maybe _this_ is why I didn't tell you, because you always make a big deal of everything."
"For real, though," Jesse said. "Why didn't you say anything?"
"Superstition," Dia said. It wasn't a complete lie, not really. "You know."
Jesse shook his head. "You never cease to amaze me."
Dia's pulse skipped and bounced, staccato sharp, and she turned her attention to Lex to get away from his gaze. "Stop."
He cleared his throat. "So, Hanna," he said. "What's that like?"
They looped all the way around the gardens as she told him all about Hanna, the difference between the girl Dia had known before and the girl she knew now. About practicing, and how she'd missed it all.
They walked back by the rose garden and Dia watched Lex carefully as Jesse talked to her and her mind wandered again, to what it would be like to kiss him.
_Stop wondering,_ a whisper in the back of her mind said. _Do something about it._
"Dee." Jesse's voice interrupted her thoughts. "Are you listening to me?"
Dia stopped walking, stepped away from the stroller. "Jesse."
"What?"
She looked at him for a long moment, and he stared back at her, those eyes that seemed to read everything. Fathoms deep and holding things he'd told only her, quiet truths Dia had entrusted to him. God, he was pretty, but god, he was everything else, too.
"Dia," he said now. "What is it?"
She took another step closer to him and before she could talk herself out of it, touched her hand to his chest. "Hi," she said.
For a moment she thought she was making a huge mistake, him staring down at her and not moving at all, but then he started to smile. And electricity crackled across her skin as he put a hand on her waist. "Hi," he said back.
She was afraid to breathe in that second, afraid that any move she made would shatter this moment they were in. Because she was being reckless, she knew, dangerous.
The sun surrounded Jesse in a halo, set him alight, and Dia rose up on her toes, lifted her chin, felt the ghost of Jesse's breath across her lips. _I am going to kiss Jesse Mackenzie,_ she thought.
And then: "Mama!"
"Fuck," Dia breathed, and at the same second Jesse's hands stilled on her body. He dipped his head, resting on her shoulder, and Dia felt him shaking before she heard his laugh. Without meaning to she wound her arms around him, her hands on the back of his warm neck, pulled him closer. "Oh, this is funny?"
He lifted his head and widened his eyes at her. "Come on," he said. "It's pretty funny. It's perfect."
Dia saw them standing there with their arms around each other, her chest against his, entangled and unaware and unashamed, and then Lex called again. "Look! Ladybug!"
"Listen," she said, keeping her voice low, just for him. "This is not over."
"No?"
Dia stroked her fingers across the nape of his neck and he smiled. "No," she said. "I promise."
She pulled away from him and headed to her daughter. "Where is it?" she said. "Show me."
While Lex found the creature again, Dia looked over her shoulder and Jesse was standing there, watching them, shaking his head like he couldn't believe her. Dia knew what he was feeling; she couldn't believe herself, either.
It was already a memory to her, running her fingers across the back of Jesse's neck, his hand on her waist. _Did I really do that?_ she thought wonderingly. It was an impulsive decision, so unlike her.
But all summer long she'd been doing things she normally wouldn't, letting herself risk things. And this—it felt good, real. And this relief at touching him at last, the way he laughed into her skin.
She looked back at her daughter, reached into the flowers with her, and smiled.
It felt inevitable.
## Hanna
**"You think you're ready** for high school, Molls?" Hanna picked up her sweating milkshake. "You can ask me anything. It's really not so bad." She ignored the look Ciara gave her. What? If Hanna took out the drinking and the friends falling apart and the loneliness, then—actually no, it was that bad. But she wasn't going to scare Molly with her horror stories.
Molly shook her head. "I'm not that nervous," she said. "I'm kind of excited."
"For high school?" Ciara made a face. "You're braver than me, kid."
They were sitting on the hill overlooking the skate park, watching the people down there and recovering from a morning at the mall picking between a hundred identical pairs of jeans for Molly. It felt weird for Hanna to be imagining Molly going back to school and not thinking about it herself. Real life, she guessed.
"I think I want to try out for cheerleading," Molly said, tossing her hair back in the most perfect way and looking at Hanna. "What do you think?"
Hanna tipped her head to the side. "You don't need my permission," she said. "Do what you want. I think you'll be a badass cheerleader, though."
"They'd be ridiculous not to pick you," Ciara said through a mouthful of fries.
Molly smiled and sounded relieved when she said, "Okay. I'm going to do it."
"I can't believe you're going to be a freshman," Hanna said.
"Me either," Ciara said, crossing her long legs. "This time last year you were all braces and pigtails. Now you're a cheerleader in cooler jeans than me."
"Shut _up_ ," Molly said, so thirteen. "God, you two are embarrassing."
Hanna slung her arm around her sister's neck and tugged on her ponytail. "It's my right."
"Whatever." Molly wriggled out of her grip, but she was grinning when she stood up. "I'm going to talk to Portia, she's down there."
"All right," Hanna said. "Don't go anywhere else."
"I won't."
She watched Molly head down the hill and then said to Ciara, "I forget that's she not a little kid. Sometimes she acts like she is, and she's just my little sister, and then sometimes she comes out with this wise-woman talk and I'm like—Molly, is that you?"
Ciara laughed. "This is what happened to me with _you_ ," she said. "When we met you were, like, fourteen! Now you're as legal as me. With a real grown-up job and everything."
"I know," Hanna said, groaning at the thought of her office employment, creeping ever closer. "But it'll only be for a couple years, until I have some money to pay for college and get my shit together."
Ciara nudged Hanna with her shoulder. "I think your shit's pretty together now," she said. "You're sober, you graduated, and you're still alive."
Hanna raised her eyebrows at that. "Real high bar."
"I used to worry that I'd come home from touring and you would be dead," Ciara said bluntly. "So whatever you might think about yourself, I'm pretty fucking glad that you're still here."
Hanna put a hand to her chest and pressed against the sudden sharp tension there. Sometimes she forgot—all the ways she'd hurt the people around her, all the ways she'd let everybody, let herself, down. But Ciara was right: she was still here. And maybe she did have her shit together. She was making music again, even; two months ago she wouldn't even have _dreamed_ that could happen.
Things could change; people could change. _I have changed,_ Hanna thought.
_I am here._
_I am okay._
She linked her arm through Ciara's and shifted closer. "Thanks."
"No sweat," Ciara said breezily. "Now, I need music updates. Since you won't let me hear anything or insert myself all the way into this venture for some reason."
Hanna shook her head. "You know I want to," she said. "But it feels like it has to be the three of us doing this, on our own. As much as I would love your genius input, obviously."
"Your ego stroking is duly noted," Ciara said. "I can't believe that, one, you're talking again, and two, that my protégées are moving on without me!"
Hanna pulled away from Ciara, twisting to face her. "I know," she said. "It took me a while to believe it would work but—" She paused and swallowed. "I really missed them. I hated some of the things they did to me, but I missed them, too. Even though part of me wanted to hold that grudge until the end of time."
"Scorpio," Ciara cut in. "Typical."
Hanna smiled. "Right. But we make good music together. We're magic at that. And it's good to be back with them and not hating them all the time. So . . . I don't know, maybe I'm naive, but I don't think so. Right?" She shook her head. "I had to let them back in, knowing that it might all blow up in my face, but it didn't. That's more brave than stupid, right?"
Ciara looked at her, levity replaced with intensity. "They're being good to you, yeah? And you to them?"
"Yeah," Hanna said. She thought of the apology that had slipped from Dia's lips, taken her so much by surprise that she hadn't even known how to process it, was still processing it. "We're good."
"Then yeah," Ciara said. "More brave than stupid."
Hanna poked Ciara's knee. "I know it's weird," she said. "They kinda cut you off, too."
Ciara shook her head, the ring through her lip flashing. "There was a lot happening," she said. "They needed space, just like you did. They know where I am, when they're ready. I'm waiting."
"Right," Hanna said, and she put her head on Ciara's shoulder, watching tiny Molly down below under the sun. "Waiting."
## Jules
**"Have a good day,"** Jules said, her cheeks aching. "Thanks for shopping at Callahan's."
She watched the customer wheel their cart away to the outside world and sighed as the front doors swished open. Escape: so near, so far. Her fingers plucked bass lines in the air, her new nervous twitch. She'd rather be practicing than working, but—
"Hey!"
Jules jumped. "Henry! Stop doing that."
"What?" Henry said, shrugging his shoulders. "Just trying to get your attention."
"You don't need to scream in my ear to get it," Jules said. Two registers down she saw Autumn turn around and laugh. "What do you want?"
"I heard you're doing the Sun City contest," he said. "That's cool, I didn't know you guys were even playing again."
"Oh," Jules said. _Shut up, Henry._ "Um . . ." She glanced over at Autumn again, whose face bore confusion now. Oh, no no no. He was ruining her surprise! "Keep your voice down," she said. "Who'd you hear that from?"
Henry shrugged. "I dunno, around," he said. "You think you'll win? I heard the prize is fifteen grand." He slapped his hands together. "That's a lot of money."
"Yeah, it is," Jules said, and now Autumn was leaning out of her lane. No, no—if she heard him and asked, then he'd tell her and it would completely screw up Jules's plan—she'd never get her rock-star goddess moment when she told Autumn they'd won, and—
"Henry," Autumn called over. "What's the contest?"
Okay, it was over.
Henry turned. "They have it every year," he said. "The radio station runs it. You know, you enter a song and the winner gets played on the radio and stuff. But this year it's like a way bigger deal, because Glory Alabama—you know them, right?—they're sponsoring it this year." He turned back to Jules. "So, you think you'll win?"
"What's the prize?" Autumn called again, and this time she was looking at Jules.
She swallowed. In the grocery store, wearing her polyester apron, was not how she'd envisioned this moment going. "The winner gets to play a support slot for Glory Alabama when their next tour comes through," she said. "And . . . fifteen thousand dollars."
"Whoa," Autumn said, and Jules couldn't tell whether her surprise was at the money or that she was just finding out about the whole thing right now or some combination of the two. "That's amazing."
Jules bit her lip. So this wasn't how she'd wanted to tell Autumn, and probably she'd have some explaining to do later. But it was all out there now. "Isn't it?" she said, but then there was a customer heading her way, and when she looked back over, Autumn was busy, too.
Jules exhaled as she began scanning items. Okay. Autumn didn't seem annoyed. Confused, maybe, but once Jules explained, she'd get it. "That'll be thirty-seven eighteen," she said. "Cash or credit?"
The afternoon passed fast, busier than usual, and when Jules got a breather again she looked to Autumn's register. But it was already closed up, no sign of Autumn. She looked down the aisles closest to her and then called out to Malai down by the paper towels. "Hey, where did Autumn go?"
"I think she's finished," Malai called back to her. "She just went out the back."
She'd left without saying anything?
Jules paused for a second before closing her register, too, and heading out past Malai. "If Greg comes out, tell him I'm on my break," she said.
"Sure," Malai said. "No problem."
Jules walked as fast as her tired feet could carry her, and when she got to the break room she was relieved to see Autumn at her locker. "Hey," she said, slightly out of breath. "Are you leaving?"
"Yeah," Autumn said, pulling a pink backpack from her locker. "That's what I usually do when my shift's over."
She hadn't looked at Jules. So maybe she _was_ pissed. "About the contest—"
"What about it?" Autumn stuffed her apron in her backpack. "It sounds amazing, really. I hope you win."
"I hope so too," Jules said, and she couldn't help her smile. "Okay, I know it's stupid, but that's why I didn't tell you. I wanted to wait until after, so that if—no, _when_ —we won, I could tell you that. But Henry kind of ruined it."
"Don't blame him," Autumn said.
Jules bit her lip. "Are you mad?" Jules regretted the words as soon as they left her mouth.
"Am I _mad_?" Now Autumn looked at her. "I don't get you, that's all. It seems like you say one thing and then do another, and I don't like it. I mean, you said I can talk to you about anything, so that's what I've been doing, even when it's not easy for me to do it. But then it's like, you get to keep secrets from me? And I'm not supposed to be annoyed by that either? I thought we were all about being honest with each other."
"We are," Jules said, taken aback.
"You get to be mad at me over whatever, like that stupid thing at the movies, so why can't I be pissed about this?"
"I was going to tell you," Jules said, stepping closer. "When we won."
"And what if you didn't win?" Autumn asked, her voice sharp. "Where you still going to tell me? Or were you just going to let it slide?"
"I . . ." Jules hadn't really thought it through that far. She'd gotten hung up on the part of her fantasy where Autumn started kissing her. "I don't know. I thought it would be fun to, like, surprise you with the news. I wanted it to be exciting. Maybe that was a bad idea. . . ." It certainly seemed that way now, her reasoning sounding flimsy even to her own ears. She twisted her fingers together. "It made sense at the time, but now . . . I guess I was wrong. I'm sorry."
"Fine. It's fine," Autumn said, and she slammed her locker shut, pushed past Jules. "I have to go."
"Autumn, wait—"
But Autumn walked out, the door swinging closed behind her. Jules looked up at the water-stained ceiling and groaned. Again and again, her ridiculous, overblown, romantic fantasies came crashing down around her. Why hadn't she just told her about the contest in the beginning? _Why?_ Now Autumn was upset, and even though Jules had been trying to do something right, she'd screwed that up again, too.
She headed back out to the floor.
## Dia
**Dia had gone to** bed on Tuesday happy. She'd fallen into a deep sleep and dreamed of being back in the Gardens with Jesse, standing in that exact same spot, except that this time she was actually kissing him.
Her mouth on his. His arms wrapped around her, her hands on his chest.
Her hands skimming skin and—
Bone. Bones under her fingers, and blood, and Dream Dia had looked around and they weren't in the Gardens anymore, they were in the middle of a road and there was a car flipped on its side and underneath that car was the body of the boy she'd just been kissing. And Dream Dia had tried to run, but her feet were suck to the sidewalk, stuck _in_ it, and she could only watch as Jesse's crushed body was pulled from beneath the wreckage, zipped into a body bag, taken away.
She'd woken up with her heart racing, panicked breaths rasping from her, and it had taken her a minute to work out why she was in the middle of an anxiety attack, what in her mind had thrown her into the deep end and left her to wake up struggling against the water.
"Elliot," she'd said aloud, throwing the covers off. _"Jesse."_
She'd rolled over and looked for Lex in her crib, calmed herself breathing in time with her baby. _It's a bad dream,_ she thought. _It doesn't mean anything._
Lex's chest rose, fell, and Dia twisted her fingers into the sheets.
It did mean something, though. It was everything she'd feared, the entire reason she'd been keeping Jesse at a distance for years now, the reason she'd cut Hanna off. Her brain had conjured her this perfect reminder, a heart-splintering shock to say, _You know this is a bad idea. Do you want him to die, too?_
The dream—nightmare—played on her mind all week. On Friday morning they had an early practice at Hanna's, and Dia was all sorts of out of it, messing up the lyrics not twice but _three_ times, and completely blanking on the end of "Pretty Baby." She wiped her hands on her thighs and shook her head, tried to rattle the music back into place that way.
"What is up with you?" Jules asked as they were packing up.
"Nothing," Dia said. "I'm fine, I just . . . messed up a couple times." _I just can't stop thinking about the next funeral I might have to go to._
Jules clipped her case shut. "All right," she said, but she was looking at her phone instead of at Dia.
"Is it Autumn?" Hanna asked, teasing.
Jules stiffened. "No," she snapped, shoving her phone away. "It's nobody."
"Jesus," Hanna said. "What is wrong with you two today?"
"What's with all the questions?" Jules said. "God."
Dia held in a sigh. This was just frayed nerves, old ways rubbing up against new girls. Nothing to worry about. "All right, I gotta go."
"See you later," Hanna said, twirling a stick between two fingers.
Dia left and caught the bus over to work, stashing her guitar and amp in the back room before settling into her schedule for the day: three rainbow birthday cakes, six dozen frosted sugar cookies, and many, many chocolate fudge cupcakes.
While mixing ingredients, Dia felt her nerves calming a little. Sugar and butter. Eggs cracked. Flour sifted. She separated the batter into bowls and added coloring, drop by single drop. Too much and the intensity would be over the top, cartoonish instead of whimsical.
When she took her break, she checked her phone and looked at the last text she'd gotten from Jesse: **Come by the skate park tomorrow, if you can. We'll hang.**
She hadn't replied yet.
She hadn't even seen him, not since Tuesday. Now that he knew about the band, it was easy for her to hide behind it, tell him she was so busy and so sorry and maybe tomorrow, always tomorrow. But she'd been texting him enough that he didn't think anything was wrong, she was pretty sure. He didn't know that she was freaking out.
That all she could think when she saw his name on the screen was that she had made a terrible mistake.
She had already lost one person. And she hadn't even _loved_ Elliot. It sounded awful, but it was true—she had liked him a lot, and she could have loved him, too, but she'd never even gotten the chance. And it had taken her months to put all of that feeling together, to recognize that she wasn't a horrible person for thinking that, and that it didn't make her missing him any more or any less, that she was allowed to be sad.
So what would she do if it was Jesse? A boy that she _did_ love, that she had been in love with for so long now. People liked to talk about loving and losing—but Dia would rather not have him be hers and still have him be whole than be with him. Jesse was already getting hurt all the time; what more would it take for the worst to happen to him? What if being with her was the thing that _made_ it happen?
She leaned against her locker, and all her breath left her as she realized the truth.
_It's me._
_I'm toxic._
Dia destroyed good things. Elliot. The band. Hanna, too, in a way. People got close to her and became broken. So it was up to her, wasn't it, to make sure she didn't inflict all the damage she was apparently capable of. To take herself out of the picture and leave people alone, let them go on to goodness without her dragging them down.
A quiet part of her said: _No, this isn't real, this is not rational thought._
But the bigger, more afraid part of her overruled that logic, made her think about what more might happen if it _was_ real and she tried to ignore it.
Dia slid down the metal, her hands on her knees. Think of all the pain she could have saved if she'd realized the truth of herself sooner.
But she knew it now. And she could still spare Jesse.
Back in the kitchen, she pulled her cakes out of the oven perfectly risen and started decorating her cookies—the initials _ANW_ iced in pale pink on each one—and Dia's hands were steady enough now to get it perfect.
When the cookies were done she finished off her rainbow cakes. Pipe a single stripe of red at the bottom, and orange above it, yellow, sky blue, grass green, lavender, deepest plum. Spin the cake and smooth the frosting out, blurring the edges between one color and the next. Fill the top with sugar-heart decorations and box it up ready for it to go to a loving, hungry home.
She barely noticed the afternoon passing as she worked, and when she was finished she stood away from her bench, watching the glitter on her hands shift in the bakery lights. "What's up?" Stacey said, passing on her way to the ovens. "You need something?"
"No," Dia said, shaking her head. Sometimes the hardest choices made for the easiest decisions.
What she'd done with Jesse had been wrong. She'd put her hand on his chest and her mouth almost on his and set into motion something that she couldn't contain.
But she'd fix it. She'd leave him alone, even though it was the last thing she wanted to do.
She looked at Stacey and smiled. "I'm all good."
Even though she was breaking her own heart.
After work Dia headed over to the skate park.
She walked, feeling the heat of the sun on the back of her neck but also smelling the trees and the flowers and that summer-asphalt scent. She climbed the hill that hid the skate park in its dip and looked out at the sun beginning to set, the kids below rolling back and forth on rattling boards and bikes. Two girls practiced tricks on dirt mounds, following each other at shocking speed, whipping perfect circles in the air. Dia watched with her heart in her mouth as they left the earth again and again, a flood of relief each time they landed and continued on, no damage, no pause.
Dia hitched up her shorts before she sat on the grass, scanning until she picked out Jesse on his bike, flying over a bench with the wind molding his shirt to his body. She knew she had to actually go to him, tell him that she'd lied, that what she'd started in the Gardens was, in fact, over. But she wanted to wait a little longer in the world where that was still a possibility, where he still thought things were changing.
He really was good, she realized as she watched him. More than that—he looked alive out there. He said he wasn't good enough to go pro, and Dia had no way of judging, really, but he was beautiful in the air.
A group of girls with their boards under their arms crested the hill, their noise breaking Dia's reverie, and planted themselves on the grass. A couple of them looked at Dia, nodding in recognition, and she gave a cautious smile back before standing.
She made her way down to the asphalt and hovered on the edge of everything, waiting for Jesse to notice her. It took a minute, but then he saw her; Dia saw him seeing her, the way he smiled and changed his direction. _No,_ she thought. _Don't give me that smile. Save it._ For a cowardly second she wanted to leave, put as much distance between her and this place, her and him, as possible.
And then Jesse skidded to a stop only inches away from her. "Hey," Jesse said, and there was that smile. "There you are."
Her stomach twisted almost violently, the longing that she usually kept so far down thudding into her chest _._ "Hi," she said, and she couldn't help smiling back at him. But then she remembered why she was there, and shook the smile away. "I need to talk to you."
"Okay," Jesse said, sounding unconcerned. He even rolled his bike closer and did that looking-up-from-beneath-long-lashes thing that usually made Dia literally weak in the knees. "What's up?"
It was easier with the park and all its people, the noise, behind them. She made herself look him in the eye as she told both the biggest lie and realest truth. "I made a mistake. The other day. With you." She forced herself to keep looking at him, even as this confusion came over his face. "I shouldn't have tried to kiss you. I—we should stay the way we are. Friends. It's not a good time. I have a kid and a job, and now the band to think about, and college . . . I think it would be better if we . . ." She tried to find the right word and finally settled on "Didn't."
Jesse was silent for a minute, a painfully long minute. "If we didn't," Jesse said slowly, looking up at her. "But the other day, you—"
"I know," Dia said.
"And now—"
"It was a mistake," Dia said again, folding her arms across her stomach, protective. "And I'm really sorry. But it's for the best."
"Okay," he said again, flat now. "If that's what you really want, Dee."
"It is." _And it's not._
He nodded, and the riders whipped past in the background, and Dia felt sick. Eventually Jesse spoke again. "Can I at least get the real reason?" he said, his eyes so serious. "Or do you want me to pretend I believe what you said was it?"
Dia swallowed. "That is it," she said. "That's the reason."
Jesse shook his head. "I know you, Dee," he said. "You don't do anything you haven't already decided you're going to do. And Alexa didn't appear between then and now. None of what you said is new. It's the whole reason we're not _already_ together, right now."
Dia blinked. "I don't know what you want me to say." No, she knew; she just couldn't say it. How could she say _I'm afraid that if I kiss you, you'll die? If you touch me, you'll be broken?_
"The truth," Jesse said. "Or should I tell you what I think it really is?"
Dia shifted. "What?"
He stood now, leaned over his handlebars. "I think you're scared," he said. "And you don't ever want anyone to know that. I get it. I don't know what you're scared of. You think we won't work out? Or maybe it has nothing to do with me really. Maybe it's about Alexa. Maybe it's Elliot." He said the name carefully. "You tell me, Dee. I just know that sometimes I see this look on your face and then you cover it up so fast, and—you have this thing about you, where you know exactly what you're doing and why, and sometimes I think I'm the one thing you can't put in its place. The way you feel about me is the one thing you can't put away somewhere."
"You're right," Dia said, and Jesse started to look hopeful just as she said, "You don't know. You _don't_ get it. I have to think about the future. I have a kid—"
"I know," Jesse said, "and you're a fucking _awesome_ mom. And she's the coolest kid I ever met. But is this what you're going to do forever? Use her as your excuse?"
"She is not my excuse," Dia snapped, and she felt her throat constricting. "She is my _reason._ I am her mother, the one parent she still has in the world, and I have to be _everything_ for her."
"I know," Jesse said. "But what about the contest? What if you win? Are you going to give up on the band because of her? Give up on that again, too?"
"Don't _even_ ," Dia said. "You don't know what it was like. You don't know shit." She heard her voice, and god, she hadn't meant for it to be like this. But maybe this was what it would take to make him understand that he was better off without her. "You think you know me, but you don't. You never will," she said, digging her nails into her palms. "So think what you want about me, come up with whatever reason is enough for you, if you can't accept that _I don't want to be with you_." She threw her hands up. "I'm not listening to this anymore."
"Fine. Do what you want, Dee," Jesse said, and he sounded so done with her. "Like you always do."
She shook her head at him and swallowed the beginnings of the crying her body wanted to do. "Whatever, Jesse."
She turned on her heel and stomped back up the hill.
The truth, he'd said. Like she could give him that—like he wouldn't hear what she had to say and not think she was out of her mind. Even she wondered if she was crazy, but what was crazy, really? It was a word people used, Dia thought, to make others seem unimportant, not worthy of listening to. And maybe this fear of hers wasn't "normal," but that didn't make it any less real.
It wouldn't be real enough to Jesse, though. Not enough to make him stay away, the way she needed him to.
And it was done now, wasn't it?
Wasn't that what she'd wanted?
Dia got to the top of the hill and allowed herself to stop for a moment, long enough to breathe in dusk air and exhale sorrow. In her chest, her heart beat on, battered and bruised but still steady.
_I did the right thing,_ she told herself. _He'll get over it. He'll get over_ me _._
Dia wiped her hands over her eyes and dragged a hand through her hair. She had to go home, take care of her kid, get ready for their final practice on Monday. Because if there was one thing Jesse was really wrong about, it was that she was going to give up on the band again. She had not come this far to let it go. Not this time.
## Hanna
**Monday was their last** practice. They played "Bones" over and over, at Dia's command: "Again, I messed up the second verse." "Again. Hanna, that ending was sloppy." "Again! Jules, you come in with vocals on the second chorus, can you please try to remember that?"
Hanna pounded her drums and ignored the burning in her shoulders. Dia was on fire today, pushing them so much. But that was what they needed: to play hard, loud, sweat it out and let everything out before tomorrow. Tomorrow, when three people who didn't know about them were going to judge them, literally, and ask them questions, and maybe find them wanting.
But if they did, it wouldn't be for lack of trying. It wouldn't be because they hadn't put their all into trying to make this happen. Hanna was sure of that, if nothing else.
She kicked a little more energy where it was needed and pulled way back when it wasn't, listening to Dia's voice crack over Jules's bass line. Sometimes Hanna had to focus in order to not hear the words, because they became a distraction. Lyrics she'd written in the dark quiet of her bedroom, not expecting to hear them set to any music but her own, if she'd even gotten around to it the way she'd always told herself she would. And now in Dia's voice, over music the three of them had written together, the inner workings of Hanna's mind laid bare—it was shocking, to her own ears. But she loved it, too. To finally have it out there and her words not festering any longer, it was good. Sometimes, when she let herself, she even felt proud.
They hit the end and came to a stop, shuddering cymbals and fading reverb. Hanna swiped strands of hair out of her face, hot with her sweat and exertion, and waited for Dia's verdict. Dia turned around, glowing and breathing heavy. "All right," she said. "That was better. Let's take a break, and—"
The inner door flew open, taking them all by surprise, and Hanna whipped around. "Molly! What are you doing?"
Her sister held out her phone. "Mom's coming home early!" she said. "I only just saw her text, but it was twenty minutes ago, so—"
"Shit." Hanna sprung to her feet. " _Shit_. You need to leave."
Jules widened her eyes. "Hanna. You still haven't told them?"
"Now is _not_ the time, Jules." Hanna dragged a hand through her hair, spinning around. "Did you not hear me? We gotta _go_."
"What's the big deal?" Dia said, doing the opposite of leaving. "What do they think you've been doing, anyway? You have an entire drum kit set up in your garage."
"They think I got it out for fun," Hanna said, and the longer her friends looked at her, the higher she felt her pulse climbing. "They're not here when we practice. It's _fine_."
"It's clearly not fine," Jules said, but at least she was lifting her bass over her head. "Why don't you tell them? Is it us?" She pointed at Dia. "Are we the problem?"
Hanna glanced at Molly and then at them, wringing her hands together. "No," she said. "I wanted to do this first and see if anything would happen before I made them all—"
She stopped, then, because her mom was pulling into the driveway, and even from where she was standing, Hanna could see the confused look on her face. She exhaled slowly, loudly. "Well, fuck."
## Hanna
**"Hanna?" Her mom was** standing in the open garage door now, this iciness to her words that froze Hanna. "What's going on?"
For a minute Hanna thought about lying, but what was the point? It was pretty obvious what was going on. And it was fast becoming pretty obvious to her that her plan had been a bad idea. But she could still explain, tell her mom exactly what and why and hope for some reprieve.
Her mom was looking at her expectantly, one penciled-in eyebrow raised, and Hanna took a deep breath. "Hi," she said. "You remember Jules and Dia. Obviously."
Her mom gave her friends half a glance before closing in on Hanna again. "Of course," she said. "Hello, girls."
Jules kind of coughed, looking down at her feet, and Dia said, "Hi, Mrs. Adler."
"Hanna," her mom said, ignoring Dia. "What is this?"
"We're . . ." Hanna made herself stand up straight, look her mom right in the eye. "We're practicing. Music. You know."
"Right," her mom said, and the facade cracked. "Inside, now. You too, Molly."
Hanna shot a glance at Dia and Jules and lifted her shoulder apologetically as her mom swept through the garage and into the house. _Sorry,_ she mouthed, and Jules looked confused while Dia looked tired, and then Hanna followed her mom into the house.
"Close the door," Theresa said when she and Hanna and Molly were in the kitchen. She dropped her bag heavily on the floor and turned around, fists on her hips. "And start explaining, Hanna."
"It's not what you think," she started. "It's only—you know the Sun City contest? It's a really big deal this year. So . . . we've kind of . . . entered it."
"I don't understand," her mom said slowly, in a way that meant she understood perfectly but did _not_ like what she was hearing. "You and those two are friends again, then? After all the trouble you caused?"
Hanna caught that _you._ "That was a long time ago," she said. "And I was going to tell you, but I didn't want to make you . . . worry for nothing." Yeah; that was a good angle. "I knew you would think about what happened before and all that, and I didn't want to bring it up to you if there wasn't going to be anything for you to worry about."
"So instead you lied?" Theresa leaned against the counter, her eyes sharp on Hanna. "How long has this been going on?"
"Not that long," Hanna said, shifting her weight as she scaled the timeline of her deceit down. "Only, like, a few weeks."
"A few _weeks_? Hanna." Her mom shook her head. "I don't understand you. I don't understand what you think you're doing. Every time you lie to me and your dad, you set us all back. Because now I wonder—what else have you been lying about? Can you see that, Hanna? I don't want to have to think like that, but when you violate the trust that we have only just built back up, I have to."
"I'm not lying about anything else," Hanna said. "And I wasn't even lying about this, more . . . not telling the whole truth. Because I knew you would think it was a bad idea anyway."
"Of course I think it's a bad idea!" Theresa said. "I know you love to play music, Hanna, but am I the only one who remembers what happened to you when you were involved in all that? You are doing so, so much better now, and the last thing I want is for you to slide back into that place you were in before. But this seems like a good way for that to happen, doesn't it?"
Hanna folded her arms. "I'm not drinking," she said bluntly. "I'm not going to drink. I just want to make music. I want to be _happy_."
"And maybe I could believe that if you hadn't gone behind our backs to do all this," her mom said, waving in the direction of the garage. "Having them over here only when we're not around, bringing your sister into this—"
"She didn't make me do anything," Molly interrupted. "And Mom, you should hear them, they're so good!"
"That is _not_ the point," Theresa said. "Molly, go upstairs."
"But—"
"Upstairs!"
She waited until Molly was gone before turning back to Hanna. "I'm sorry, but I don't think this is a good idea. I think it needs to be done, now."
The way her mom said _I'm sorry_ made it clear she was anything but. "Done?"
"Finished," her mom said, slicing her hands through the air. "No more."
"No!" Hanna clasped her hands together, pleading. "Mom, we get to play to the judges tomorrow, and then that's it. We might actually win. And it's so different now, I'm so different now, you know that."
But her mom shook her head. "I _don't_ know, Hanna," she said. "When you lie to me, you make me think that you're not so different."
It landed like a barb in Hanna's heart, but she tried again. "I should have told you, I'm sorry, I—I didn't want you to think all of this stuff, but please, Mom, it's not what you think and it's—"
Her mom held her hand up, stopping Hanna. "This is what I mean," she said. " _Trust_. Maybe if you'd come to me first, we could have discussed this and worked something out. But you lied, and _now_ you want me to believe that it's not a big deal. That's not the way it works, Hanna. And I'm saying _no_. I don't want you doing this."
Hanna shook her head, desperate. "You can't stop me," she said. "I'm eighteen."
Her mom raised her eyebrows and let out a bitter laugh. "Oh, is that where you want to go with this? Okay, then. Yes, Hanna, you are eighteen. An adult. But you live in my house, so you play by _my_ rules. If you want to do your own thing, then you need to do it outside my house. That's the way it is."
Hanna's mouth dropped open. "You're kicking me out?"
"No," Theresa said. "I am telling you the rules. In _my_ house, you will not lie to us, and you will do as I say, and I say you can't do this."
"That's ridiculous," Hanna said, clenching her fists tight and giving in to sudden anger. "You can't do that."
"I can and I will." Her mom crossed the kitchen and put a hand on Hanna's cheek, and Hanna could see deep into her eyes then. Anger and fear, both in there. This rush of guilt washed over Hanna.
She should have told the truth from the beginning. It wasn't even that big a deal. And now it was _this_ , it had become this, because her mom _couldn't_ let her have this _one thing_. "I will not lose you again, Hanna. I saw you disappear right in front of my eyes, and I had to watch them stick a tube down your throat to get you back from the poison you put in yourself, _twice_ , and I had to put my seventeen-year-old daughter in rehab. I'm sorry if you think I'm overreacting, but you are my _daughter_ and I will do whatever I have to to keep you with me."
"Mom," Hanna said, and unexpected tears spilled over, the weight of them slipping down her cheeks. "I just got this back. Dia and Jules, they're not—I was so lonely and now I'm not," she said, her voice cracking. "It's only tomorrow and then we'll be done, no more practices, I promise." A rash promise to make, but right now she wanted to get her mom to stop and consider. Hanna did not want to leave, she didn't want this to be the thing that pulled them apart, but god, she was so _exhausted._
But her mom shook her head again. "I'm sorry, Hanna," she said. "This is the way it's going to be. This is for your safety."
Hanna stepped back, out of her mom's grasp, and the last thread of self-preservation snapped. "I don't get you," she said. "What else do you want me to do? I got sober and I let you search my room whenever you wanted and I earned your trust back, but because of this one little thing, it's all gone? That's _bullshit._ "
"Excuse me?" Theresa said. "You better watch yourself, Hanna."
"Why bother? You watch me _all the time_ ," Hanna said. "I'm sick of it. When are you going to let me live my life again? Am I supposed to stay here and do whatever you say whenever you say it so that one day you'll think I'm good again?" She pulled in a ragged breath. "I will never be good again, will I? I'm the bad daughter, the one who drank and drained you and hurt your precious baby, the one _good_ child you have. You want me to go? Maybe I should. Go and leave you all here so you can live a nice tidy little life without my mess to ruin it for you. And you can tell all your friends that you tried your hardest but I was just too out of control. That would be nice, right?"
"You think I _want_ you to leave?"
"I don't know, Mom," Hanna said. "You don't really seem to want me here."
"When your father comes home—"
"He'll say exactly what you say," Hanna said. "That's how it works."
"I have had enough of your attitude right now."
Hanna whipped around, heading back outside. "Or maybe you've had enough of me."
"Where are you going?" her mom snapped to her back. "Hanna, get back here. Hanna!"
## Hanna
**_Breathe._**
_Breathe._
Hanna went back into the garage to find Dia and Jules all packed up, and Jules looked at her. "What did she say? Are you okay?"
"Oh, it's fine," Hanna said, clenching her fists. "You know, she thinks I'm a liar and a drunk, but what else is new?" Her nails dug into the palms of her hands and she shook her head, trying to shake out the whisper of _whiskeywhiskeywhiskey_. "You should really go."
"Yeah, we're going," Dia said. "Wait—she said that to you?"
"It's literally nothing new," Hanna said. "It is what it is, okay? Can you—" She stopped and tried to slow her breathing. "I need you to go, _now._ "
"We're _going,_ " Dia said again. "All right?"
Jules and Dia exchanged a look that Hanna was pretty sure she wasn't meant to see. "So, tomorrow—"
"I don't know why she's so pissed," Hanna said, whirling her hands through the air. "I was going to tell her. I was just _waiting._ "
"Right," Dia said. "But—"
Hanna rounded on her. "But what? But I didn't? But I'm a liar anyway, so who would even believe what I did or didn't plan to do?"
"Did I say that?" Dia picked up her guitar case. "There's history, that's all."
"Oh, of course you would take her side," Hanna said. She looked up at the ceiling and let out half a breath of unamused laughter. "You think you're so much better than me, don't you? You think you're so clever with your little comments and rolling your eyes when you think I can't see, like, _Hanna, you messed it up again!_ I get it, Dia, okay? I'm a fuck-up and I always will be to you, but god, you are not perfect either."
Dia stepped back. "Again, did I say that? No." She shook her head. "We're leaving now. You can call me later, when you've calmed down."
"I'm calm," Hanna said, her teeth gritted. "Don't I look calm to you?"
"You look like you need space," Jules said. "Or maybe you should give your mom the space."
"It doesn't matter what I do or where I go," Hanna said, pacing now. "There's always something with her. I'm always screwing up in a thousand different ways. And then she tries to tell me she's only _worried_ , she's _concerned_ , always has to bring it back to the fucking drinking. That has nothing to do with this!"
Dia looked at her. "You're right," she said. "Maybe that has nothing to do with this. Because you're sober right now, and yet here you are, screaming at us and talking about your mom when you know what, yeah, she probably is worried about you and you're lucky, you know? Not everybody has a mom like that."
"She should take a tip from you," Hanna said, and she let out a bitter laugh, looking at her hands. "Give up on me. Cut her losses. Worked out okay for you, didn't it?"
"Give up on you?" Dia said, her eyes flashing. "Wow, Hanna. So why the fuck am I here? Did I imagine all this, the last month?" She looked at Jules again and back at Hanna, her eyebrows pulling together. "You think that I _wanted_ to cut myself off from you? No. But there's only so long you can let yourself be hurt by someone who doesn't want your help, and you didn't want it, not at all. So what were we supposed to do? Stand by and watch you kill yourself?"
"You didn't have to abandon me!"
"Well, I'm sorry it felt like that, but I did what I thought I had to do," Dia said. "And I don't want to do it again, I don't want to give up on you and on this, and live and die in this town. I want more than that, and so do you, and you've been telling me all this time how different you are now, so _prove it to me._ "
"I can't! You won't ever believe me," Hanna said. "You and your fu—"
"Hey!" Jules said. " _Stop._ Both of you, stop! We didn't spend the last month busting our asses so you can force us into imploding now."
"It's always my fault, isn't it?" Hanna snapped, and Jules glared at her.
"I didn't say it was anyone's fault," she said. "I just want this to _stop_ before it goes too far and we can't come back from it."
The air shimmered with the heat of her outburst and Hanna felt the shake in her hands.
But she already wanted to peel the words back from the ether and swallow them whole. Choke on them.
Dia exhaled loudly. "Tomorrow is important. We have worked too hard to mess it up. So we're leaving, and you do whatever you need to do to remember why we're doing this, and then you call me later. Okay?"
Hanna looked past her. She knew Dia was right. Why couldn't she _shut up_? Always running her fucking mouth, ruining all the work she'd done to make people believe she'd changed, to make _herself_ believe she was different now.
_Well done, Hanna: you're truly living down to everybody's low expectations._
_Congratulations._
"Come on," Dia said to Jules, and Hanna wanted to say something, but all the fight had vanished and she only wanted to give herself to the floor beneath her.
Jules looked back at her before they disappeared. "Think about tomorrow," she said. "That's what we wanted before. We might get it now. Think about that, okay?"
When Hanna gave one sharp nod, she tasted the salt of her tears. _Fuck._
"Okay," Jules said. "Call me if you want."
Hanna watched them leave and sank to a crouch, pressing her hands to her eyes as she rocked on her heels. The taste of her words was metallic, and she wasn't sure if she was going to throw up or not.
"Get inside." Her mom's voice was like nothing she'd ever heard before, and now Hanna was pretty sure she was going to be sick. "Now, Hanna."
She stood slowly and waited there, surveying the mess of their practice, the mess she usually tidied and hid so nobody would be able to tell. There it was, a month's worth of back and forth and testing new boundaries and releasing so much history into the stale garage air.
Was it all over, now?
_"Hanna."_
She turned and looked her mom dead in the eye. "You don't have to yell," she said, knowing she was only making things worse. But she kind of liked it. "I'm right here."
Hanna skipped dinner. She sat at the top of the stairs instead and listened to her parents arguing about her.
_She's lying to us again,_ her mom kept saying. _How do we know she's not lying about other things too?_
"You could ask," Hanna said under her breath. Unlikely.
_Let's not get ahead of ourselves,_ her dad said. _She's come a long way._
That was nice, her dad sticking up for her.
_And it only takes one slip to bring her crashing down,_ her mom said. _I'm not taking any chances._
Hanna got up and retreated to her room. Now she was calmer, she could understand what her mom had been saying—the trust thing. She had lied. She had broken their trust. But if she'd asked, told her parents that she wanted to play in a band again, she knew what they would have said. She knew what her mom would have done.
She lay on the floor beneath the window, evening light playing over her body.
But making music again felt like surfacing after a thousand hours underwater. They didn't get it and they didn't ever let her explain it.
Her phone buzzed for maybe the fifth time in twenty minutes. Hanna looked at it, then put it facedown on the floor. Dia and Jules were freaking out, she was sure.
She was freaking out, underneath it all. Because tomorrow was important. If she didn't go, then she'd be letting the others down, and tanking their shot at fifteen grand and the Glory Alabama show.
If she went, and her mom found out—and she'd find out; wasn't today evidence enough for that?—then she'd lose her family. Even more than she already had.
Her phone buzzed again: voice mail, from Dia. She listened to it: "Hey, it's me. I'm—sorry for being shitty earlier. I just want to make sure you're okay. Call me or text me or something."
Hanna tossed her phone aside. What was her punishment going to be? No car privileges. Curfew. Walk a straight line, touch your nose, blow into this. Maybe.
Her palms itched. In the trunk of her dad's car was a bottle of Jameson that she knew he thought she didn't know about. She could go out there and take it, drink the whole thing and forget about all this until some point further in the future, leave it for Future Hanna to deal with.
More of her than she liked said, _Yes, do it._ The idea of oblivion was so enticing.
_Four hundred and_ —what was it?
She stayed in her room, weighing the idea and mindlessly watching makeup tutorials and chain-smoking out of the window until the rest of the house was asleep and quiet. Then she made her decision.
Nothing was going to change unless she made it. And she'd been doing everything her parents' way for so long now, but it didn't make a difference.
Time to try something new.
She looked around her bedroom, surveying all her stuff in the moonlight. Clothes on the floor and books stacked against the wall and her small desk with her laptop on it. The lamp on her nightstand, pictures of her and Molly at various ages tacked up on her closet doors.
She took a deep breath and then started with the pictures. Took them down one by one and laid them on her bed. Then she started on the insides of her closet, throwing all her shorts and underwear and shirts on top of the pictures. The couple crumpled dresses on the bottom, shiny, tight things from her darkest days, she left.
Hanna was half under her bed when she heard the creak of her bedroom door, and then Molly's voice. "She didn't mean it," her sister whispered, and Hanna could hear the nerves in her voice. "You don't have to do this."
Hanna came out with the biggest duffel she owned and dust on her hands, looking over at Molly in the darkened doorway. "Were you listening earlier?"
"Behind the kitchen door."
Of course; she'd learned that habit from Hanna. "She meant it, Molls." Hanna forced a smile. "Don't worry about it, though. I'll be fine."
"Where are you going to go?" Molly asked, playing with the drawstring on her pajama shorts. "Don't do this."
Hanna ignored her question and started shoving her clothes into the bag. "I'll be fine," she said again. "Trust me."
"I do," Molly said quietly. "Hanna. I do trust you, you know?"
Hanna paused her packing and turned around. She was never sure with Molly, when she said things like this: did she mean it, or did she just want to make her big sister happy? After what Hanna had put her through?
Hanna liked to believe it was a little of both, mostly.
So she said, "I know, Molls." She crossed the room and pulled her sister in close, planted a kiss on the top of her head. "I know."
## Elliot
## OCTOBER
**Elliot shifts into third** and tries to ignore the rattling sound coming from somewhere in the car. It's not his, it belongs to his cousin Ana, and it's kind of crappy, but he wanted to take Dia out without taking the bus. So now he owes Ana thirty bucks of gas money and an unspecified favor, but it's worth it. It was even worth Dia's dad grilling him at their front door: "You drive safe? No texting, right?" he'd said. "You don't drink and drive, drive while you're high?"
"No, sir," Elliot had said. "Wouldn't even think about it."
Now Dia rolls down the window and sticks her hand out. "Sorry about my dad," she says. "He does that to everyone I get in a car with. He's an EMT, so, you know, he sees it all the time."
"I get it," Elliot says, taking a right. "I passed, though, right?"
"You wouldn't be driving me if you didn't," Dia says, and she reaches across, rubs her thumb at the corner of his mouth. These little things she does, possessive things—Elliot loves them.
They pull in to this burger place; it looks like the diviest thing ever, but Dia's eyes light up. "Let's get one of everything," she says. "It's all so good."
They get their food to go and drive to a spot high up on the edge of town and sit on the hood of Elliot's cousin's car while they eat. The sky darkens and the town lights up beneath them. So it's not the classiest date ever, Elliot knows that, but it doesn't really matter. That's not what he and Dia are about. This is good.
She's in the middle of a story about Ciara and a broken-down van when her phone rings. Elliot watches her take it out, frown, and silence it. "What was I saying?" she says. "Oh, so yeah, we're driving and all of a sudden we can't see anything, there's smoke _everywhere_ —"
Her phone rings again. "You can answer," Elliot says.
"It's just Jules," Dia says, and cuts it off again.
Then it rings, again, immediately.
"Maybe you should answer," Elliot says, and this time Dia looks at him and puts her phone to her ear.
"What?" she says, impatient. "Juliana, I'm busy."
Then her face falls. Elliot wipes his hands on his jeans as Dia begins chewing her bottom lip. "What? Where are you? I don't know where that is. Whose house? Slow down."
After a minute more of conversation, she hangs up and fixes Elliot with a moody stare. "Hanna?" he says.
"She caused a scene, Jules said." Dia tugs a hand through the ends of her curls. "Smashed a window, and the girl whose house it is, is freaking out, and Hanna's pretty much blacked out and Jules can't get her to leave. So . . ."
Elliot rattles the car keys. It's either leave Hanna to self-destruct or try to help. "Let's go get them."
## Jules
**It was ten on** Tuesday morning. Hot, but not as hot as it was going to be, Jules knew.
And she was standing outside Revelry, clutching her bass, staring up at the marquee.
They'd had their name up there before, as Fairground. The first time it had ever happened they'd taken about a thousand pictures, posed beneath it. Today it was empty except for the words LIVE MUSIC.
Jules scuffed her feet on the sidewalk and took out her phone for the thousandth time since leaving Hanna's yesterday. Still no response.
No texts, no missed calls. Nothing to answer the question Jules had asked around six in the morning, when she was too wired to sleep: _You are coming, right, Hanna? You are going to be there?_
Nothing from Autumn, either. No _good luck_ or _You'll kill it!_ No anything.
(Did she deserve good luck from Autumn?)
"Hey."
Jules looked up and Dia was right there. "Hey," she said. "Have you—"
"No, I haven't heard from her," Dia said.
Jules resisted the urge to toss her phone across the street and instead folded her arms over her chest. "She'll be here. I know she will." She was trying to convince herself as much as Dia, because she knew nothing. She hadn't heard from Hanna all morning, all last night. And the Hanna she'd seen yesterday had been a stranger. No, not a stranger. Someone she knew far too well and had hoped to never meet again.
Jules looked at Dia, and she was staring up at the marquee the same way Jules had been. "It's been so long," she said without looking at Jules.
"I know." Jules looked down at the sidewalk, gray and dirty and already beginning to shimmer in the heat. "She has to be here. Right? Otherwise—what's the point of any of this?" If Hanna didn't show, and they didn't play, and they didn't win, then what was the entire point of everything they'd been doing?
Maybe there wasn't one; maybe it was too naively hopeful, way too made-for-TV-movie for Jules to think that there was some grand meaning to this whole thing. Maybe it was just that they'd tried to do something and failed and that was it.
"Then there's no point," Dia said. "We fail. That's it."
But Jules refused to accept it. Dia was wrong, she had to be wrong, because Hanna _was_ different now. Sober and changed, and she wanted this exactly as badly as they did. Didn't she? "She'll be here."
"Keep telling yourself that," Dia said, and she didn't sound angry, only tired. "We should go check in."
Jules looked at her and their eyes met, and Jules felt about as exhausted as Dia sounded. _Come on, Hanna_.
She nodded. "Okay."
Inside they signed in and were led into the warren of back corridors and left to sit on folding chairs at the end of a line of people. Guitars littered the hall, scattered between flannel-clad musicians who seemed to collectively turn and look at them. Scanned them and assessed their competition, it felt like to Jules. She fixed her face in a blank, I-don't-care expression and stretched her legs out, looking at her beat-up boots. "We have time," she said under her breath, so the person on her left didn't hear. "She has time."
"Thirty minutes," Dia said. "You think she's actually coming? Or you think she's going to screw us over?"
"She's not screwing us over."
"What if she is?" Dia said. "What if this was her plan all along? Tell us she wants to play, get us this far, and then make us lose. Payback. For what we did to her."
Jules screwed up her face. "What would be the point?" she asked. "Seriously. What does she get from that?"
"She gets to hurt us the way we hurt her."
"No." Jules shook her head. "She wants this as much as we do. You can see it, when she's playing. When we're writing together. Don't you feel it?"
"Yeah," Dia said, more a sigh than a word.
Jules turned toward Dia, the two of them shielding each other from the jagged whispers and sideways glances. "You don't think she might—" She felt cruel even saying it, doubting Hanna this much. "Relapse?"
Dia's eyes widened. "God. No. No? I hope not."
Jules leaned forward and looked down the corridor, resisted the urge to get out her phone and call Hanna for the millionth time. So if she wouldn't miss this, and she wasn't downing vodka somewhere, then—where the hell was she?
"This is exactly what I expected, in the beginning," Dia said, her voice low. "From Hanna. This is exactly the kind of shit I thought she'd pull. And before, this would have made me feel good, you know? Knowing that I was right about her all along. But now I feel like . . ."
Jules looked at her. "Like what?"
"Like . . . _really_? She spent all that time telling me how different she is. I _believed_ that she's different. And yet, here we are." Dia pulled at her curls.
"She _has_ changed," Jules said, and she wasn't sure if she was trying to convince Dia or herself. "She has."
Dia was silent for a moment, and then she nodded. "I know."
They waited while names were called, while groups filed in quiet and came out swaggering. They moved down the line, closer and closer to the backstage door. Jules looked down this hall, at the old posters and album artwork and flyers papering the length of it. They'd walked this hall so many times. They'd been on that stage and in the crowd and up on the balcony.
Why didn't any of that make her feel less sick?
The door opened and the girl with the clipboard came out. "We're running a little behind," she announced, pulling a pen from behind her ear. "Don't worry, we'll get to everybody. Washington Forth, you're up."
They waited.
Jules bounced her feet on the sticky floor, her knees jittery, her hands plucking at her clothes and her hair as the time ticked up to eleven. When she looked at Dia her head was tipped back against the wall, her lips moving silently. Jules could make out what she was saying; _comeoncomeoncomeon._ Same thought running on loop in Jules's head.
Jules leaned to speak in Dia's ear. "We still have time," she said. "She can still be here."
Dia pulled away. "She didn't _know_ we'd have time," Dia said flatly. "She should be here by now. She's not coming."
And Jules couldn't fight that fact anymore. Dia was right; Hanna didn't know about any reprieve. Eleven, that's when they were supposed to have been up, and Hanna should have been there. "Shit," she said, gripping her knees. " _Fuck._ What are we going to do?"
"Wildfire? You're up."
The girl with the clipboard said their name like a question, her voice going up at the end, her eyebrows rising to match. Jules supposed she was right; what kind of name was Wildfire, anyway?
What kind of broken band were they, anyway?
She locked eyes with Dia and they took the same deep breath, steeled themselves with the same strength. "We got this," Dia said, quiet, to Jules. "Let's go in there and lay it all out, the two of us. That's all we can do."
Jules picked up her bass. "Let's go."
They walked, instruments in hand, up to the clipboard girl. "All right," she said, snapping strawberry gum, the smell of it enough to turn Jules's stomach. "You have—"
_"Wait!"_
A clattering commotion, someone yelling, "Watch it!" and someone else saying, "Ow! My _foot_!" and Jules wasn't sure whether to be relieved or mightily pissed off as she turned to watch Hanna, here, _finally_ , sprinting down the hall.
"I'm here!" she said, out of breath, heading to them with what seemed like half her drum set in her arms. "Wait, I'm here."
She came skidding to a stop and bent over, her head almost between her knees. "Jesus Christ, Hanna," Dia hissed. "It's about time."
Jules pulled Hanna to standing. "Get it together," she whispered. "We need you."
She looked back at clipboard girl, her mouth curled into a sneer. "Ooo-kay," the girl said, dragging the sound out. "So, anyway . . . you have fifteen minutes to set up, and then the judges will come in. Good luck." She sounded so utterly bored, and Jules hated her for a second.
But there was no time for that, and no time to talk to Hanna, either, because they were through the door now, and in the wings, and in the dark Jules could only see the gleam of Dia's eyes and the white of Hanna's hair, and then the brightest lights bathed them as Dia led them out onto the stage.
## Elliot
## OCTOBER
**Elliot crawls along the** street. "Which number did she say it was?"
"She didn't," Dia says. "She said—"
She cuts off, and Elliot sees why, sees Jules standing on the sidewalk in front of a landscaped front yard. "I'm gonna go a little farther," Elliot says, because the street is crammed with cars parked up and down, and if he parks too close and someone hits this car, he's going to pay.
"Okay," Dia says, and then in the same breath, "I can't leave her alone for one night. Jules is supposed to be taking care of her, you know? This is fucking ridiculous, I cannot—" She cuts herself off. "Whatever. Let's go get her."
Elliot parks and they walk back down to the right house, and Jules is still outside waiting for them. "I know, I know," she says as they get near. "I was supposed to keep an eye on her, I know."
"It's not your fault," Dia says. "She always does this."
"What happened?" Elliot says. "She smashed a window?"
Jules shakes her head. "A mirror," she says. "In the bathroom."
"Did she get hurt?" Elliot says, imagining the force it takes to crack the glass. "Did she, like, _smash_ smash it?"
"She threw a bottle at it," Jules says, and she looks at the ground. "The whole thing shattered."
"Why the fuck would she do that?" Dia says, but Elliot knows enough now to know that it's not a real question. "Where is she now?"
"She wouldn't let me take her home," Jules says. "The girl who lives here is _pissed_."
Dia rolls her eyes and Elliot looks between the two of them. "Let's just go get her out as fast as possible," he says. "Minimal collateral damage. I mean, minus the mirror. And the bottle. And herself."
They split up—Jules takes the kitchen and the backyard, Dia and Elliot take the living room and the upstairs, because, as Dia says, "Hanna has never met an off-limits sign she didn't completely ignore."
She's not anywhere they look. Upstairs, downstairs, outside, upstairs again. Jules is with them now, checking under the bed of the master bedroom. "Where the hell is she?" she says, and Elliot can hear the panic in her voice. "I left her dancing in the kitchen."
"You can't take your eye off her," Dia says. "You know that."
Elliot opens the closet, a last resort, and of course, there she is. Slumped against the wall with a pile of knocked-down clothes blanketing her. He looks over his shoulder. "I got her."
Dia comes over, stands by him, and Hanna looks so peaceful that Elliot is almost afraid for her, for what she's facing when she wakes up in about 0.2 seconds.
"Hanna," Dia hisses, crouching. "Get up."
Hanna's lips move, but no sound comes out and her eyes stay shut.
"Hanna, I swear to _god_." Dia jabs Elliot's ankle. "Help me."
Together they pull Hanna up to standing, and Elliot puts her arm around his neck as she starts mumbling. "Leave me alone," she slurs—or, he thinks that's what she's saying. "Tired."
"Yeah, well, I'm tired too," Dia snaps, taking the other half of Hanna's weight. "Jules, get the door."
They maneuver her out of the bedroom, down the stairs, through the hall. Jules has Hanna's bag, something that Elliot thinks might have been Hanna's shirt earlier tonight because now Hanna has on a men's button-down and the thing Jules is holding is stained red. "I think she's bleeding," Elliot says.
"Hey!"
Jules looks back at them, her eyes wide. "That's the girl," she says. "We should be gone, like, now."
"Get back here!" A white girl with long, blond hair and angry eyes jumps in their path. "She owes me, that crazy bitch! She _trashed_ my house. I want my money."
"We're really sorry," Jules says, and she grabs the front door. "We're going, we won't come back, we're really sorry—"
"I _want_ my _money_."
"It's only a mirror," Elliot says, and he notices the curious circle forming around them. They should really get out, before this turns into something.
"It was an _antique_."
"Well, you should've taken it down before you threw a party," Dia says, out of breath, and the girl's eyes go wide, and Jules puts herself between them.
"Go!" she says, and Elliot does what she says, dragging Hanna out of the house and through the neat yard—more collateral damage—while the girl keeps yelling at them but does not follow.
"Way to defuse the situation," Elliot says.
Dia glares at him. "Help me get her to the car."
It takes forever to get her up the street to the parked car, but eventually they get there and Elliot bears Hanna's weight while he fishes the keys from his pocket. "Lay her down," he says, and Jules is on the other side, coaxing Hanna toward her. Eventually she's stretched out across the back seat, and now Elliot says again, "I think she's bleeding." He starts to check her over but Jules shakes her head.
"She's fine," she says. "She spilled wine on her shirt."
Dia almost chokes. "Where the fuck did she get wine?"
"Where the fuck does she get everything?" Jules shoots back.
It's dark and Elliot's adrenaline is running high. He looks back toward the house, checking that no one is coming after them. "Antique," he says. "How much, do you think?"
"Did you see that house?" Dia says. "We don't have the money, trust me."
"We better pray she doesn't find out who we are," Jules says, and she rubs her face. "Can we take her home? I'm tired."
Dia gets in the back, lifting Hanna's head and laying it on her lap. "You get shotgun," she says, and then she looks at Elliot for the first time in what feels like hours. "Thank you."
He shrugs. "No big deal," he says, like he carries half-comatose girls out of parties every weekend.
Elliot drives them back to their side of town, five miles under the speed limit, Dia's dad's words echoing in his head. The radio is on low and Jules flicks from station to station. In between snatches of sound Elliot hears Dia talking quietly, and at first he thinks she's telling Hanna off. But when he listens, he hears it, this sweet, low whisper:
"It's okay, Han, you're okay. Why do you do this to yourself? It's not good, it's not good. You scare me when you're like this. I know, shh, it's okay. We're taking you home. You'll be okay."
He turns the radio up and keeps driving.
## Hanna
**They flew into fast** motion, plugging in amps, testing mics, setting up the cymbals and snare and kick pedal around the house drums. Hanna would have preferred her own kit, but she had no way to bring it all, and usually she could get by with the basics and her own essentials. She steadied her hands, shivering with the adrenaline surging through her, and sat. In front of her Jules looked over to Dia, stage left. "Give me an A?"
Hanna settled her sticks in her hands, gave the snare an experimental hit. Her heart was battering her ribcage still, her lungs burning from having to run all the way from the bus stop to here with her equipment weighing her down and her head playing _latelatelate_ on a loop.
"Hanna." Dia looked over her shoulder. "All good?"
Hanna thought of the bag she'd packed last night, now stashed at her job.
She'd gotten two, maybe three hours sleep once she'd finished packing. Wide awake at five a.m., she'd wondered if she was making a huge mistake. On the bus, making her way here, she'd still been unsure. _I can go back,_ she'd thought. _Mom will never know._
But right now, on this stage with these girls, she knew.
This was right.
"All good," Hanna said.
This disembodied voice interrupted. "Hello," it called out. "All set up? It sounds good from here."
Hanna held a hand up and squinted against the lights. Where was that voice coming from?
"Yes," Dia said into the center mic. "We're ready."
"Thank you," a different voice said. "So—Wildfire, is it? Before we get started, one question from me: give me some of your influences."
Hanna caught Jules's glance back at her; she looked as off-kilter as Hanna felt. But Dia stepped closer to her mic and cleared her throat. "Hi. Um, influences—Sleater-Kinney, Melissa Auf der Maur, Sade, Kacey Musgraves, Christina Aguilera—"
"Interesting," the second voice said.
"Okay," the third voice said. "What are you playing for us?"
"This song is called 'Bones _._ '"
Dia turned to her, and Hanna looked to Jules. They looked perfect: focused, cool, their all-black outfits, their shiny guitars. Hanna locked eyes with Jules and saw everything she needed right there.
She rolled her shoulders, steeled herself, lifted her sticks.
Four sharp clicks and Hanna dropped, feet first, into the noise.
Their hours of practice had made them sharp, clean, in sync with each other. And Hanna let herself be suspended in the music, her aching muscles light now, but her breath coming in gasps between fills. She sank into Dia's voice, her own lyrics.
_"In the forest,"_
Dia sang,
_"Bones break like branches_
_Weighed down with words_
_And under the night is awake_
_Waiting to break."_
Jules brandished her bass and pressed her mouth to the mic, and Hanna felt their ghosts watching, the past ephemeral versions of her playing right in this same spot.
_"I call the witches and_
_They say they won't know me_
_Not without blue blood and moonlight_
_Moons under the skin of my loves."_
Dia vibrated at the front of the stage. Hanna could see the energy coming off her and fed from it. Jules's jersey twisted around her waist as she bounced one foot off the floor, wrenched her guitar away from and then back against her body, and Hanna thrashed against the drums, releasing every ounce of her pent-up explosive anger on it.
_"All the honesty_
_In the woods and the world_
_Isn't going to save me_
_She can ride that horse until she dies_
_She can follow that river_
_Into deep darkness."_
Hanna pounded the drums, cracked skins, weathered cymbals, and felt the air stop around them. Suspended in this place, this sound, for a fleeting moment. This really was it. Not the end.
Everything.
_"In the forest_
_In the dungeon_
_Wherever wildflowers grow_
_That's not where I'll be_
_When you're not watching_
_The witches take me_
_To the forest of ever after."_
Dia's voice ended it, the only sound above the fade of their music, and Hanna dropped her head forward, her chest heaving, her hands shaking.
_This is everything._
"Thank you," one disembodied voice said, and Hanna almost hated whichever one it was for the interruption of this moment. "We'll be in touch."
## Dia
**They broke everything down** as fast as they could, in uncertain quiet. By the time they were done the judges had disappeared, and they left the stage, wound their way back through the club, and finally spilled out into the sunlight.
Dia took only a second before she began tearing into Hanna. "Are you fucking kidding me?" she yelled, and Hanna didn't even look afraid, but the guy crossing the street away from them did. "Are you _fucking kidding_ me, Hanna? You really had to pull a stunt like this on a day like today? What were you doing? What were you _thinking_?"
And Hanna spoke over her, trying to answer her: "I know, I'm sorry—"
"You're _sorry_? Yeah, you'd better be! You almost cost us—"
"I know, I really didn't mean to—"
"So what _did_ you mean to do? You don't text, you don't call—"
"That wasn't my fault, the traffic—"
"What were we supposed think? You didn't even—"
"And I had to—"
"Why didn't you call me? Call Jules? We—"
"But I got here, didn't I? We did it, isn't that what you—"
"What I wanted was—"
_"_ God _, stop!_ " Jules pushed herself between them, held her hands out while she whipped her head, looking from Dia to Hanna and back to Dia again. "Breathe. Please."
Dia did as Jules said, pulling in sticky summer air and breathing out the taste of exhaust. Hanna was watching her carefully, face flushed red.
Dia looked away. Her clothes were sticking to her. Her stomach felt empty.
But they'd done it. That was real.
"Okay," Jules said now, stepping back. "Better?"
And now that Jules was out of her way, Dia set her stuff down, reached out and grabbed Hanna, pulled her into a viselike hug. "I didn't think you were coming," she said into Hanna's hair. "I thought I was right about you all along." She closed her eyes. "Thank god you made me wrong."
Hanna's laugh was pure surprise. "Okay," she said. "I'm sorry."
"I'm sorry, too," Dia said, meaning it, so sincere, and she hugged Hanna tighter. "But don't do that ever again."
Dia pulled back, pulled a caught curl from her lips and looked at Jules, the surprise emblazoned across her face, too. "I'm hungry," she said now, feeling a little delirious. "Are you hungry?"
"Starving," Jules said.
Hanna pulled cash from her pocket. "I'll buy," she said. "And then I'll explain."
Dia picked up her guitar and her amp, and tilted her face up to the sun. Nothing was ever easy with them. But easy wasn't always good.
"Let's go."
## Hanna
**In the gas station** across the street from the bar they got dollar hot dogs and chips, Skittles, and a bunch of candy bars and sodas. "And a pack of Marlboro Lights," Hanna said as they piled their stuff on the counter.
The guy behind it didn't ask for ID before getting the cigarettes and sliding them across to Hanna. "Eighteen seventy-nine."
Hanna started to pay, then swore. "Can I get a lighter, too?"
They paid and left and then, without even talking about it, they crossed back over to Revelry, lugging all their equipment behind them. This time they walked down the sidewalk to the tattoo shop and slipped through the alley that led to the parking lots behind the buildings. It was what they used to do all the time, when they couldn't get in or had opened for Graceland. They'd go out to the parking lot of whatever venue they were at and sit on the curb with their legs stretched out onto the asphalt as they waited for Ciara to take them home, giddy and happy.
So that was what they did now, the three of them in the corner of this empty lot, surrounded by guitars and drum paraphernalia.
Hanna peeled the cellophane off her cigarettes but didn't open the pack. "I think it went okay," she said, turning the pack over in her hands and looking straight ahead. "We kept it together. We played as good as we ever have. We have a chance, I think."
She turned her head, Dia and Jules both looking at her. "I'm sorry," she said, meaning it. "You want to know what happened?"
Dia dipped her chin, and Jules almost smiled. "Shoot."
Hanna took out a cigarette, tapped it against her palm, and started from the beginning. "My mom said I couldn't do the contest. She doesn't trust me, and I get it, I do. But I'm also tired. I don't know what else to do to make them trust me. She said as long as I lived in her house, it was her rules, and she said no. No band, no music." She lit the cigarette. "'If you want to do your own thing, then you need to do it outside my house.' That's what she said. So, fine. I packed my stuff and I'm out."
Her words were met with the rushing of traffic out on the road and the faintest noise from the club.
Then Dia shook her head, her face confused. "Wait, what?" she said. "You left home? Like, gone?"
"Where are you going? What are you going to do?" Jules grabbed Hanna's knee. "Oh my god, Hanna, are you for real?"
"Because of _this_?" Dia said. "Hanna."
She shook her head, a lazy curl of smoke twisting from her lips. "Because of everything," she said. "I've been good, you know? Since rehab, I've tried _so hard_ to be the best version of me. I got my grades up. I got my license. I've never missed work, not once. I stopped going out anywhere, really, so my parents would know I wasn't doing something stupid somewhere. I hang out with Molly, I try to keep my self-destructive ways to the absolute minimum, and yet—none of it is enough to make them stop looking at me the way they do."
"What way?" Jules asked.
"Like I'm the biggest disappointment of their lives." She flicked ash to the ground and scratched at the back of her knee. "They're not wrong. I've done a lot of bad things. But I never could get it right with my mom, even before the drinking. And playing music with both of you again, it's like—the best thing that's happened to me in forever. And I'm supposed to give that up?"
Hanna put her cigarette out. Then she told them everything: that she'd gotten up this morning and acted completely normal at breakfast, had apologized for her words yesterday.
Her mom had taken Hanna's hand. _You understand,_ she'd said. _You understand why I want this for you?_
"Yes," Hanna had said. "I get it."
And her mom had said, "Your part in the band is over. We're agreed?"
"Yes," Hanna had lied. "It's done."
"We'll discuss this more tonight," Theresa had said, and then left for work.
Molly had eaten cereal as she'd watched Hanna packing up the parts of her drums that she could take. "You're really doing this?" she'd asked, plucking at her bottom lip.
"Yes," Hanna had said, coming back into the house. She'd checked her reflection in the hall mirror, checked she looked ready for their performance. Then she'd looked at her sister and tried her best to smile. "I'm not going far, Molls. And I'll call you tonight."
"What about Mom and Dad?"
The note Hanna had left on the table wasn't exactly subtle.
_I'm sorry. I have to do this. I can't stay here when you won't let me breathe. And I don't want to lie to you anymore, but I don't know how not to. I wish I could say this to your faces but I don't think I could. So instead, let me call this the last time I'll lie to you and break your trust. I have to do this. I love you._
_Hanna_
"Don't worry about them," she'd said. "They know this has nothing to do with you."
And then she'd hugged Molly goodbye, kissed the top of her head, and hauled her ass to the bus stop.
"I went by work first," she said, wrapping up. "It was the only place I could think to go. I had to get away. Then Ciara picked me up to bring me here, and we hit traffic. There was an accident on Piper and everything was backed up, and then I didn't even have any service to text or call."
Dia shifted. "What about last night?" she said. "You never texted me back then. You had service then."
"I know," Hanna said, scuffing her feet on the asphalt. "I didn't know what I was going to do. So . . . I didn't know what to say."
"I really thought you weren't coming," Dia said, leaning back. "So, you're actually moving out? Where are you going?"
"Ciara's. At least, for now. It's not that big a deal, really," Hanna said, and part of her believed those words. "Plenty of people are out on their own at eighteen. In a few weeks I'll be working full time. I can rent a room somewhere and . . . I'll eat ramen and cereal for every meal. It's a rite of passage, no?"
Dia and Jules exchanged a look, and Hanna turned her lighter over in her hands. "What?"
"Nothing," Jules said, but Dia exhaled loudly.
"We were worried," she said. "That you might have—slipped."
"Slipped?"
"Had a drink." Dia squeezed her elbows to her ears. "But I'm glad you didn't."
Hanna thought of that bottle of Jameson, the one she'd managed to keep just out of her reach. "I don't do that anymore," Hanna said. Well: she didn't know the future, what might happen, but for as long as she could, she would keep herself away from that bad habit. She didn't want there to be a _next time_ for any of the things she used to do. She didn't want there to be another time for somebody to find her.
She pulled her feet in so she was cross-legged and looked at the pack of smokes. "There's something I haven't told you," she said. "I know you'll think I'm terrible."
"I'll reserve judgment until I hear what it is," Jules said. "And so will Dia, won't she?"
Dia held her hands up. "Swear."
Hanna took a deep breath. "I didn't just decide to quit drinking," she said. "Molly found me in my room one night. I had been drinking, obviously, and I passed out, and when she saw me I wasn't breathing. She called 911. So I got to do the whole ambulance-ER-stomach-pumping thing for the second time. Except this time it was so much worse, and I didn't even think that was possible but it was, because this time my sister had to see me doing it. She had to make the call." The guilt sat in her stomach, jagged. "But that's what made me want to finally stop. I couldn't believe that I'd let Molly do that, be the one to find me, like—what if it had been worse?" She looked at Dia and Jules. "I didn't believe in wake-up calls before, but that was totally mine."
Jules's eyes looked full of sorrow. "Why wouldn't you tell us that?"
"Because I hate it," Hanna said vehemently. "I'm so ashamed of it. And I wanted you to think that I turned myself around and got better all by myself. But really, it was only because of this terrible thing I did."
"You didn't have to. Quit," Dia said, breaking her silence. "How many times did you try before? But this time you _did_ stop. You decided to do that. I don't think that's anything to be ashamed of."
It was as if the pressure, the guilt wound tight within her, suddenly snapped, and Hanna found herself wiping away leaking tears. She had nothing to be ashamed of.
_I am okay._
"You don't know how much I needed to hear that," she breathed. "God."
They sat in silence a while, the rushing cars, snatches of music from the club and the road.
Dia looked up at the sky. "Look what we did," she said. "At the beginning of summer, could you even imagine this? Look what we put in motion."
" _Let's enter Sun City_ ," Jules said, in a perfect imitation of Dia. " _What's the worst that can happen? Oh, yeah, Hanna gets kicked out of her house, no big_."
"I never said that," Dia objected. "I said, what do we have to lose?"
"Same difference."
Hanna drummed on her knees. "It'll make a good story," she said. "When we're famous."
"A great story," Dia said. "Teenage mother, out and proud lesbian, reformed bad girl. We'll be on every blog worldwide."
"We should celebrate," Jules said. "We actually _did_ this. Let's do something good."
Dia rolled her eyes. "Like what?"
Jules began reeling off ideas, Dia shooting every one down until Jules accused her of being a downer. They worked their way through their candy stash, melting chocolate under the hot sun, and Hanna laughed along with them, hard enough to make her stomach ache. And she felt both so overwhelmed with everything and so overcome with gratitude for these girls that she didn't know what to do.
So she ate her candy, and sat with them, and it was good.
## Dia
**Dia had to leave** for work after a while; when Dia asked, Hanna insisted she was okay, and Jules rolled her eyes when Dia hesitated. "Go already," she said, flicking her hand in Dia's direction. "We don't need you getting fired, we have enough drama already."
"All right, I'm out," Dia said, stealing a Reese's cup for the bus ride. "Both of you text me later, okay? Okay, bye!" She turned, and then whipped back, unable to keep the shit-eating grin off her face. "Can you believe we pulled this off?"
Hanna raked her hands through her hair, shaking her head. "I can't," she said, her smile as wide as Dia's. "It's too much."
Dia really left then, had to rush for the bus with her guitar smacking against her back and her stupid heavy amp bashing her legs.
It only took ten minutes in the bakery to make the morning feel like it was years ago, moons away. Imelda had her running around on a rush order, and she spilled an entire vial of food coloring over her hands, her skin tinged blue no matter how hard she scrubbed. When she was working the counter, she looked up and saw Jesse walk past with two guys whose names Dia could never remember. He didn't stop, and he didn't look in, and it took Dia a split second of hurt to remember what she'd said to him.
Of course he didn't stop. It wasn't like he was going to come in to flirt over cookies like normal, was it?
She pressed her knuckles into the hard plastic of the register and swallowed hard. Sacrifice, that was what this was. She couldn't have everything she wanted, nobody could. And she wasn't the only one losing things; Hanna was losing her home, wasn't she? So she should suck it up and forget about him. Remember the feeling of singing her soul out this morning, feeling so really and truly awake for the first time in years.
At least he was safe from her.
Dia let out a slow breath.
After her shift, she headed to day care. When Lex came running out, Dia picked her up and swung her through the air, making the silly growling sounds that made Lex laugh so much. "Ready to go home, Lala?" she said, and smoothed her hand across Lex's curls. Maybe she didn't completely understand why Hanna's mom had given that ultimatum, but she partly did. She knew what it was like to want only the very best of the world for your kid. To want to cling to them, curl them tight against you and keep the entire world out, if that was what it took for them to stay safe.
She kissed a graze on Alexa's knee, and thought about it as they walked home. See, you couldn't keep them safe, not completely. The world always got its way somehow, in painful bruises or boyfriends dropping dead, girls who wouldn't love you back, or obsession, even addiction.
Dia called out when they entered their house. "Mom? We're home."
No answer, and Dia moved toward the stairs, almost bumping into her dad as he came down them, mid-yawn. "Hey," she said. "Where's Mom?"
"Taking a bath," Max said, scrubbing a hand over his face. His knot of locs had fallen to one side and he had pillow creases in his cheek, but then his eyes lit up like Christmas morning. "Wait—how did it go?"
"It was . . . good," she said. Trimming the truth a little, for the sake of brevity. "We did what we wanted to. It's up to the judges now. If they liked us or not. But even if they don't, we showed them exactly who we are." Dia cut herself off, wary of saying too much, jinxing it. "I'll tell you at dinner. What are we having?"
"Tacos," her dad said, yawning again. "I'm going to get started now."
She took Lex out in the yard while her dad made dinner, chased her back and forth. Her dad called them in after a little while, and Dia strapped Lex into her high chair as her mom came in wearing sweatpants and an army T-shirt, a silk scarf wrapped around her hair. She pulled plates from the cabinet and Dia filled glasses with iced tea and the four of them sat down to eat.
Dia filled her parents in on as many details of the day as she thought necessary, leaving out Hanna's superlate arrival but leaving in her issues with her mom. Nina's forehead wrinkled as Dia talked and then she shook her head. "That girl," she said, and Dia thought she was about to go off about how bad she was, all the things she'd heard a thousand times before from a thousand different people, but then Nina said, "She doesn't have the best luck, does she? You tell her if she needs anything, we're here. Okay?"
Dia raised her eyebrows. "Seriously?"
"Yes, seriously," her mom said. "What?"
"Nothing," Dia said. "But . . . it's Hanna. Usually you tell me how I should be careful with her and all that. Now it's different."
Her mom lifted one shoulder. "You know, you've been spending a lot of time with her, and you don't do that with people you don't trust," she said. "So if you trust her, I suppose I can, too. And if she's going through a hard time, we can be good people and try to help her out. Besides, she's just a kid. And now she's a kid out on her own. If it was you, I'd want someone to help you, too."
Dia smiled. "You're good people, Mama." Then she looked at her dad. "I'm glad you let me stay," Dia said. "I know it wasn't what you wanted for me, having a kid, and everything with Elliot made it all worse and . . . I don't know. I don't know what I would do without you."
"Are you for real?" Max said. "In this family, we stick together. That's the way it is. Maybe we tell you things you don't like sometimes, maybe you do things we don't like sometimes, but that's life. We're all we have, right here."
Dia looked at Lex. _We're all we have, right here._ And how lucky they were to have it all.
## Hanna
**Hanna stood on the** sidewalk outside Ciara's place, the same small, one-story house she'd rented ever since Hanna had known her. Ciara had been nineteen then—see? Only a year older than Hanna was now. This could be her life. Why not?
This whole thing felt both incredibly final, a Big Deal, and also the most overdramatic thing she'd ever done.
Everything and nothing all at once.
She felt seven years old, stuffing a backpack full of Pokémon cards and Oreos, seven pairs of socks and one shoe, and "running away" to the house of the woman who used to watch Hanna after school. Like back then, some small, childish part of her believed that she'd only last until dark before her mom turned up to collect her, arms outstretched as Hanna cried, shocked by herself. The rest of her, though, knew that she had crossed a line that maybe couldn't be uncrossed.
Hanna was at the top of the steps when the door opened and Ciara stepped out onto the porch. "There you are!" she said. "Didn't you get my text?"
"Hi," Hanna said, letting Ciara kiss her cheek. "Um, no. I turned my phone off."
"Like, _off_ off?" Ciara shuddered and took one of Hanna's bags. "I would die."
"Yeah, well, I'm avoiding the fallout." She could only imagine the scene that would happen when her parents got home and discovered her gone, the note. She didn't want to hear any of it.
Hanna followed Ciara inside and when the door closed behind her, she felt this jolt, all her adrenaline halting in her veins and pooling in the bottom of her stomach.
_I am here._
_I am okay._
"Lucky for you," Ciara said, "I am currently between roommates. So instead of staying on the couch, you can have an actual bed."
Hanna swallowed past the knot in her throat. "Thanks," she said. "For the bed, and letting me stay, and . . ." She blew her cheeks out. "It's been a day."
Ciara stood in between the middle of the living room and put her hands on her waist. "What do you want to do?" she asked. "You want to talk about it? Want to talk about playing? Want to talk about nothing at all?"
"Honestly?" Hanna said. "I want to eat. I'm starving."
"Let's order in," Ciara said. "Chinese?"
"Perfect."
"You can go put your stuff in the other room," Ciara said, crossing into the little kitchen. "I'll order."
Hanna nodded and hauled her stuff down the hall, past Ciara's bedroom and into the small room at the end. It was empty except for the bed, a nightstand, and an unplugged lamp.
She set her bags down, put her equipment in the corner, and closed the door. Then she plugged the lamp in and flicked it on, the warm bulb emitting a surprising amount of light. It wasn't dark yet, but getting there; the sky outside the window was more navy than light, orange-tinted clouds dropped here and there. She'd stood in the window at work and watched the world go by all afternoon, telling herself over and over that she could do this.
Hanna sat on the edge of the bed and exhaled into the emptiness.
_What have I done?_
When she turned her phone on it took a minute for the messages to come through: three texts from Dia, one from Jules, one from Molly, two from Ciara. Four missed calls from her dad, seven from her mom. Two voice mails.
She took a second before listening, running her hands over the sheets. Fresh, she could tell, they had that laundry smell. Ciara was so good to her.
Hanna texted Molly back first, short and sweet: **Hey Molls, I'm at Ciara's. I hope you're not getting the fallout, if you are, I'm sorry. Call you later, I'll tell you all about playing today, it went good :) love you xoxo**
Then: voice mails.
The first was long, and Hanna held her phone away from her ear so she could only half hear what her mom's angry voice was saying: _This is beyond a joke, Hanna_ and _Do you think this is funny?_ And _This is exactly what I'm talking about, you have to take some responsibility for your actions, you think they don't affect anyone but you?_ and _How am I supposed to trust you when you do this?_
Yeah, that was basically everything Hanna had expected.
The second voice mail was shorter. Her mom, again. But different this time. This time she sounded like she'd been crying, and Hanna's heart ached but did not break as she listened to Theresa say, _Please call me. We can talk about this. I love you, Hanna._
Hanna tossed her phone onto the bed. She loved her mom. Under it all, she did love her. But god, she hated her, too.
She changed into pajama shorts and a T-shirt, and joined Ciara back out in the living room. "Okay, sunshine?" Ciara said from her place on the couch. "You look awful."
"Thanks, and screw you," Hanna said, and Ciara laughed.
"Sorry," she said. "Food will be twenty minutes."
"I turned my phone on," Hanna said. "My mom is an excellent emotional manipulator even through voice mail."
Ciara made a face, one hand playing in her hair. "Mothers," she said. "Complicated animals."
"So very true." Hanna sat down next to Ciara and put her feet in Ciara's lap. "Thank you," she said, closing her eyes as she leaned back. "I know I'm a pain in your ass."
"No pain," Ciara said. "Well, sometimes a little. But you know me, Han, I'm a glutton for emotional distress. Can't give up on the people I love. That means you."
Hanna opened her eyes and looked at Ciara. "I know," she said. "You're the only one who hasn't."
"I don't know about that," Ciara said. "You spent all day with a couple of girls who could have iced you all the way out. You have Molly on your side, always. We're here, aren't we?"
Hanna nodded, and a surge of shocked sadness caught her. She had to look to the ceiling so the sudden tears threatening wouldn't spill over. "Right," she said, pressing her nails hard into her palms. "You're right."
"I always am," Ciara said. "You want a glass of water?"
She nodded. Ciara _was_ right; Hanna was not alone. This wasn't like before, and she wasn't the same girl as before.
No; that wasn't right. Because she was the same. She was still Hanna with the drinking problem, Hanna who hated herself a lot of days, Hanna who would love nothing more than to give in to her addiction sometimes. What was different was how she dealt with all that now. She had changed, and she had stayed the same, and maybe Ciara or Jules or Dia didn't get that. But Hanna knew herself now, for better or for worse.
Mostly better.
## Elliot
## NOVEMBER
**"I think I want strawberry,"** Dia says, putting her hand on Elliot's shoulder. "Or cookie dough. Or both."
"Both," Elliot says.
"Good plan."
She steps up to the counter and orders for both of them, caramel swirl with hot fudge sauce for Elliot, and then they take their ice cream outside. She climbs on top of one of the picnic benches, sits on the tabletop. "Are you coming tomorrow?"
Elliot sits on the bench between her legs, looking up at her. "Yeah," he says. Is that even a question by now? Every weekend Dia goes somewhere, to play a show, to watch some other band, and almost every weekend Elliot follows her there. "Ciara's band is really good."
"Ciara's band is excellent," Dia corrects. "God."
Elliot laughs. "Sorry."
"You know, she took a chance on me," Dia says, digging her spoon into her ice cream. "She gave us our first show. Without her we'd still be playing in Hanna's garage."
"Your hero," Elliot says, deciding to sidestep Hanna for today. That night at that party is still making him uncomfortable. He had to lie to his cousin about the stain on the back seat and beg her not to say anything to his parents.
(Dia keeps saying Hanna is fine, that she has it under control, that he doesn't need to worry about it.)
(Elliot's pretty sure she does not have it under control, but then again, what does he know? He's known Hanna for four months; she's known Hanna for years and years. Who knows better, really?)
"I really think out of everybody here, maybe Ciara could be the one to actually make something out of music," Dia says, waving her spoon in the air. "Besides us, obviously."
"Obviously." Elliot takes a bite of his sundae. "What's the plan, then?"
Dia's eyes flash and she puts her shoulders back and Elliot hides his smile by pressing his face into her knee. God, she's so hot and determined and he's pretty sure he's going to be left behind in her dust, but at least one day he can say he knew her. "Okay," she says, "so this year, we keep playing around town and playing with Graceland. That way people get to see us. Then next year we level up, you know? Start getting our own shows outside town—"
Elliot nods and asks questions in the right places and listens to Dia's entire plan for Fairground's world domination. She thinks about the future; so does he, a lot, and Elliot likes that they have this in common.
"What about you?" Dia asks. "What's your plan?"
"I don't know," Elliot says. "I think I want to write a book one day."
That takes Dia by surprise. "Really?" She reaches out and runs a hand through his hair. "You write?"
"I try," he says. The most he's gotten so far is half-finished short stories, but it's a start.
"So what, you gonna go to a fancy writer school?" Dia teases. "Learn all about those boring books everyone pretends are genius?"
"I'll probably stay close for school," he says. "If I get the scholarships I need. That's the only reason I'm still playing baseball."
"Wait," Dia says, eyes wide. "I didn't know you were, like, scholarship-money good. I'm really dating a jock, aren't I?"
"Admit it, you like the way I play," Elliot says, abandoning his ice cream to slide his hands up her thighs. "Home run after home run."
Dia bursts out laughing. "That is the cheesiest shit I ever heard," she says, and Elliot can't help but laugh too.
"See?" he says. "I got it all. I'm an athlete, I'm a writer, I'm a comedian."
"Yeah, yeah." Dia leans down and puts her hands on either side of his face. "You're a prize."
## Jules
**Now that it was** over, their big moment, the throwing themselves in front of three strangers and screaming _choose us_ , Jules felt . . . restless. Like, she had all this extra energy coursing through her and nowhere for it to go. Their practices had come to an abrupt stop with the loss of their usual rehearsal space. "We'll find somewhere," Hanna kept saying. "We'll pool our money and hire some place."
"Sorry, what money?" Dia had said. "My kid is about to be two. I have to buy textbooks. My funds are _extremely_ limited."
"We'll work it out," Jules had said.
But so far they hadn't. Instead they'd been wasting time around town: hanging around the music store, rifling through clothes they couldn't afford at the mall, watching Lex run around the playground. Jules had been working, and pretending things were okay with Autumn, and that her omission of truth wasn't still hanging between their every interaction.
It wasn't working.
A week later the three of them were at the movies, mostly for the AC, but they left halfway through. "Why do they always have to kill the black girl?" Jules said as they crossed the lobby. "And why can't the women ever talk about anything other than some guy?"
"And why can't the girl ever be the badass hero?" Hanna said. "How come she has to spend half the movie training some guy to be as good as her and the other half either falling in love with him or being killed?"
"Hollywood," Dia said. "They don't get it."
They went over to the nearby McDonald's instead and sat outside eating fries and drinking milkshakes, Hanna smoking. Jules squinted against the setting sun and looked at Hanna, about to ask the same question she'd already asked a dozen times over the past two weeks. "Did you talk to your mom yet?"
"No." Hanna dipped a fry into Dia's chocolate shake. "She's left me, like, a hundred voice mails. I know I should call her back. But I'm waiting."
Dia's brows sloped together. "For what?"
"Until I know what I want to say," Hanna said, and then she looked into the distance. "It's harder than you'd think. To not say what you really want to, even when you _know_ what you want to say is the worst thing. Even when it's going to cause trouble. You have to hold it in. I'm not very good at doing that."
Jules and Dia exchanged a look. "You're better at it now than you ever were before," Jules said. "Trust me on that."
"Sometimes it's okay to say the thing that causes trouble," Dia said. "Sometimes the trouble needs to happen."
Hanna looked at Dia. "You really think that?"
Jules raised her eyebrows, curious to hear Dia's answer, too. That was so not a Dia thing to say. Then again, this whole summer was not what any of them had planned. Things were different now, in so many ways.
"Yeah," Dia said. "Sometimes I do."
Hanna ate another fry. "I think I'll keep waiting."
"Waiting, waiting," Dia said. "Is this all we do now?"
"A week and a half," Jules said.
Next Friday was the show at Revelry, where they'd announce the winner of the contest, and Jules almost had to throw up every time she thought about it. "I know we keep saying it doesn't matter," Jules said, and she twisted her fingers into the ends of her braids, the way she always did when she was nervous. "I know we keep saying we'll still play and it's not the end of us as a band and everything."
Dia looked at her. "But?"
Jules pulled her hands from her hair. But she _really_ wanted to win. Who wouldn't? She wanted to walk around with the invisible approval on her, to play a show with her idols. Okay, and also—she wanted to drive around in the car she would buy with her share of the money blaring Formation with the windows down and waving her middle finger in everybody's faces. "You know," she said eventually. "I _want_ it."
"What would you do?" Dia asked. "With the money."
"I want a car," Jules said. "I'd take us everywhere."
"When we win you will," Dia said. "We'll get matching rides. Red. Yeah?"
"Can we talk about something else?" Hanna asked. "This is giving me hives."
"What's left?" Jules said. "You don't want to talk about the contest, or your living situation. Dia doesn't want to talk about _anything_." She rolled her eyes when Dia kicked her. "Oh, please. You've barely said a word all day." _And I have nothing good to say about me and Autumn,_ she thought.
She needed more food to keep from thinking about it all. "I'm going to get an apple pie. Anyone want anything?"
"More fries," Dia said.
"Coke," Hanna said. "Large."
"Coming right up."
Jules went inside and got in line behind two younger girls whispering in each other's ears. When she got to the counter she ordered and then turned, leaning her elbows on the counter and looking at her friends outside the window.
Listen to that— _her friends_.
Hanna and Dia, sitting by themselves, laughing at something. Two months ago this would have been impossible to imagine, because of what had happened two years ago.
But two years was a long time. Long enough for things to change again and again, a thousand times over. For Jules to realize that fighting was not her fantasy. For Dia to raise a precocious, perfect kid. For Hanna to get sober and get smart and become this new-but-the-same girl.
Long enough for all of them to realize that there was nothing they could do to each other now, ever, to fully break them.
And win or lose, they still had their music. Nothing and no one could stop them. Jules knew that now.
## Jules
**Jules was at the** grocery store early on Thursday morning, yawning as she listened to Henry and Malai bickering across their registers. "Can you shut up?" she called after a while. "You two are so annoying."
"Says you," Henry fired back. "You wanna go? Freezer race, you and me, winner takes all."
Jules raised her eyebrows. "Takes all what?"
"Uh . . ." Henry tapped his chin. "All . . . the vending machine food you can get for ten bucks."
"I'd rather have the ten bucks," Jules said. "Pass."
She started tidying the pamphlets on the end of her register, lining them up by height, until she sensed a customer waiting. "Sorry," she said as she turned. "I—Autumn."
Autumn leaned on the plastic barrier shielding the screen between them. "Hi," she said, her voice a little unsure.
"Hey," Jules said, surprised. "What are you doing here? You're not working today."
"What do you think?" Autumn said. "I came to see you."
Jules's heart tripped over itself. "Oh." Oh?
Her nerves jangled. Things were not so great with them, not since Autumn had found out about the contest and Jules's secret keeping. They were still talking, texting, but—she could feel it all between their words, Autumn's uncertainty and her own anxiety squeezing in the air around them. Jules had apologized, and Autumn had said that it was fine, she was fine, but Jules couldn't quite believe it. Surely Autumn was still mad, really, and just telling Jules everything was okay; surely this was all a sign that they weren't working out, not Meant To Be; surely Autumn was going to end things with her any day, any minute now.
(Jules knew those thoughts were ridiculous, but in some moments she'd find herself thinking that what was _really_ ridiculous was thinking Autumn would ever want to be with her in the first place.)
But that was how it had worked before, wasn't it? With her and Delaney. Every conversation they'd had could be twisted and turned into a fight, and Jules had become an expert at it. But Autumn was _not_ Delaney. Right?
"Are you seeing the others today?" Autumn tucked a strand of hair behind her ear. Her hair was loose, the colors faded to soft pastels from their former bright rainbow.
Jules nodded. "At eight."
"Before that?"
Jules looked around, but it was quiet and no one was paying them any attention. "I'm not doing anything."
That earned a smile from Autumn, a fraction of her megawatt version. "Okay," she said. "Will you come over? When you're done here?"
"Come over?"
"I think we should—talk," Autumn said. "When do you finish?"
Jules looked at her watch. Grocery store time did not work like regular time; more like time in the fairy stories Jules had read when she was younger, slipping back and forth and stretching out so long. "Four hours," she said. "Four more hours?"
Autumn winced in sympathy. "Will you come?"
"Sure," Jules said, but all she could think was, _It's too late. I broke things before we could even get started._
What happened to the girl who'd asked Autumn out with so much confidence? What happened to the magic?
Jules glanced around again—no sign of customers, and the others were paying them zero attention. No one to witness her crashing and burning in such quiet, careful style. "I'll be there."
"All right," Autumn said, already beginning to walk away. "See you later."
The bus let her off half a block from Autumn's house. At the door she hesitated before ringing the bell, and then bounced on the balls of her feet in the time it took for Autumn to come to the door. When she opened it, she was smiling, and Jules smiled back, a reflex.
"Hi," she said. "Sorry I'm late."
"It's cool," Autumn said, and led Jules inside. "Do you want something to drink?"
"Sure." Jules took a second to adjust to the brightness of the hall, all the knickknacks and everything, and followed Autumn to the kitchen. "Is your mom here?"
"No, my parents are at work, and my brother's at day care," Autumn said, leaning into the refrigerator. "They won't be home for hours."
Jules shrugged her backpack off. "What do your parents do, anyway?"
"My dad's a construction manager. My mom works in some snobby law office, and she does hair and nails on the side. I know," she said. "Of course, right? She wants to open her own place one day. She'll be able to service all your acrylic and peroxide needs."
Jules pointed at Autumn's hair. "Does she do that for you?"
"Yeah." Autumn handed her a Diet Coke. "Couldn't stop her if I tried." She laughed softly. "Um, I have to switch the laundry real fast. You want to go up to my room? It's the second door on the left."
Jules followed the directions, up the decorated stairs and onto the landing, the walls of which were painted as bright as the ones downstairs.
Autumn's door had an _A_ painted in blue. Jules opened it, and if the rest of the house was a neon Palm Springs pool party, this was the morning-after meditation. Jules exhaled as she stepped inside— _This is Autumn's bedroom. I am in Autumn's bedroom—_and gazed around. The walls were a soft, pale gray and the only clutter was a pair of shoes by Autumn's closet. Otherwise, everything was neatly in its place: the perfume bottles placed at measured intervals along the windowsill gleamed; the pictures above her bed were framed. And the books in the tall case were arranged by color. By _color._
Jules went to them and ran her finger lightly across the shelf of spines that shifted from blue to green. "It's impossible to find anything," Autumn's voice came from behind. "But it looks so pretty."
Jules turned. "You're a neat freak, aren't you?"
"Little bit. My mom has the rest of the house to fill with all her stuff. But this is mine." Autumn sat on the edge of her bed and looked up at Jules. "So."
"So . . ." Jules glanced out the window. From Autumn's room you could see houses, trees with their leaves curling brown in the heat, some kids playing in the street. She looked back at Autumn. "Are we over?"
Autumn startled. "What?"
"It's okay," Jules said, and she meant it. It was only her own fault, pushing Autumn too much, lying to her. If Autumn wanted out, Jules understood. "I know I screwed up. I'm sorry. But it's okay."
Autumn's eyes widened. "Slow down," she said, her voice high. "Why would we be over? Do _you_ want us to be over? Did I do something wrong?" Then she held her hands up. "Actually, no. I don't think I _did_ do anything wrong. I think I got upset about the contest thing, and then you got weird, and started avoiding me, and now here we are. Am I right?"
Laid out like that, it sounded bad. "No. Maybe." Jules paused. "Okay, yes."
"Do you even know why I was upset?" Autumn asked. "I'll tell you. You didn't tell me this big thing and it made me feel like I wasn't important to you. Like maybe I was thinking that we meant more than we really do. That's why."
"Are you kidding me?" Jules said, her surprise real. "Autumn. You couldn't possibly think we meant more than _I_ think we do. Oh my god, the first day I saw you I thought—" She stopped herself. But then Autumn's words in the break room came back to her: _I thought we were all about being honest with each other._ "I thought you were amazing. I felt like I fell in love with you the second I saw you, and I'm telling you that even though it's ridiculous and embarrassing because it's the truth."
Autumn's cheeks were bright pink. "Jules—"
"You want me to be honest, right?" Jules said. "Okay." She took a deep breath. "Let me tell you about my ex."
When Autumn's eyebrows rose Jules held her hands up. "Wait. See, I went out with this girl for, like, six months last year. I'd had a crush on her for literally years. I really liked her and she really liked me. But we were _so wrong_ for each other." She shook her head and smiled tiredly. "It was obvious after a couple weeks. We should have broken up right then. But I liked her, and she was the first person I'd dated, so I wanted it to work. And I always wanted her attention, I wanted her to be . . . sweet to me. Because I—" Jules ignored the heat in her cheeks. "I like the romance—flowers and hand holding and everything. But she wasn't like that, it just wasn't her. And when she tried, I'd pick apart how wrong whatever she'd done was."
Autumn brought a hand to her hair, nervous. "Okay," she said.
Jules looked out the window again. "I do this thing," she said. "I think everything should be perfect and magical and then when it's not, I decide it's not worth my time. But really, I don't think what I'm waiting for is a fairy tale. I think it's just someone who wants to try as hard as I do. You're just you and I'm just me, and if it works or not is up to us, not some magic. I'm sorry I didn't get that before. I really am sorry."
The space between them felt so full, and Autumn stared Jules down, and the scrape of branches against the window was the only sound beyond the blood rushing in Jules's ears.
"Okay," she said eventually. "I accept your apology."
Jules felt a snap in her chest, palpable relief. She did? "You do?"
"I really like you, Jules," Autumn said, her eyes bright. "But god, you like to make things complicated."
Jules laughed. "I know."
"You're lucky I'm the kind of person who doesn't give up on things when they're not perfect and magical." Autumn said. "If that was me, I wouldn't be here. My life has not been perfect and magical. But it's still good."
"Yeah," Jules said. "I get it."
"But from now on," Autumn said, "talk to me. Don't pick a fight. I don't like fighting."
"Okay," Jules said, nodding quickly. "I can do that."
Autumn leaned back on her elbows and Jules tried not to let her eyes linger where Autumn's skirt rode up. "Oh, Jules," she said, and the smile she gave now was her full dazzling brightness. "What am I going to do with you?"
She stood and took three steps across the floor, planting herself in front of Jules. "You make me nervous," she said. "Not because I don't want this, but because I want it so much. And it's all new to me. I've never kissed anyone but you, and I've never kissed anyone with people watching me, and I've never held anyone's hand and been like, _This is my person._ But I want to do all those things with you."
Jules tipped her head to the side. "It's okay," she said. "I get it. It's new to me, too. Being public. Letting people see. And I get how it feels scary sometimes. I don't want to make you do anything you don't want to, that you're not ready for."
"I know you don't," Autumn said, and she put both her hands on Jules's waist. "But I do want to do them. If I screw up—it's not because of you. Okay?"
"Okay."
"And if I screw up, will you let me know?"
"Yes," Jules said, and she felt confident now, all the uncertainty gone and this magnetism in its place, heat from Autumn's hands on her. "And you'll do the same for me, right?"
"Yes."
Jules smiled, relief that they were finally, definitely on the same page flooding her. She put her hands on either side of Autumn's face and kissed her once, quick and sweet. "I feel so lucky," she whispered in the space between their mouths.
Autumn's fingers tickled at Jules's waist in a way that made her squirm. "Please stop talking and kiss me properly," Autumn said.
Jules felt both like she would come alight and come undone at Autumn's touch. She felt suddenly nervous again—like she didn't know where to put her hands or what to do with her body, even as her body seemed to decide for itself, pressing closer to Autumn, even as she kissed Autumn again, her tongue tracing Autumn's lower lip.
Her heart was racing. _Stop thinking._
_Stop thinking stop thinking._
_Why am I so nervous?_
_Is she nervous?_
_Her skin is so warm._
_I think I love her._
They broke apart. "Jules," Autumn said, her cheeks pink, her voice hoarse.
"What?" Jules said, her voice a whisper in response. "What's wrong?"
"Nothing." The back of Autumn's hand brushed right where Jules's cutoff's ended and her whole body bloomed warm. "Nothing at all," Autumn said, and swallowed hard. "I, um . . ."
"Is it the kissing?" Jules asked. "We can stop."
"No, no," Autumn said. "That is the opposite of what I want."
Jules wondered if she was hearing the meaning in Autumn's words right. "So . . ."
"Full disclosure," Autumn said. "I really want to do more than kiss you."
"Like—" Jules couldn't help her laugh. "That's a nice way to put it."
"Shut up," Autumn said, and this time she pinched Jules's thigh, the gentlest pain. "I'm saying I _want_ to. If you want to. You know, take our clothes off and . . . other things."
Now Jules's heart rate kicked up a thousand percent. She touched her thumb to Autumn's chin. "You know we don't have to do anything."
"You're not listening," Autumn said, and she stepped back. Her hands went to her shirt and she tugged it free from the waist of her skirt, lifting it to flash the smallest amount of peachy skin. "I _want_ to. Get it?"
Jules wet her lips. She'd done under-the-clothes fooling around with Delaney, in the back of her car at night only, hands cramped behind zippers and underwear. And she was pretty expert at getting herself off, but that was a different matter entirely. But Autumn looking at her like that, taking off their clothes in Autumn's room in afternoon sun, seeing each other and doing things to each other?
That was another level.
This was intimacy. Sleeping together.
Sex.
And god, she wanted to.
"Do you? Want to?" Autumn held her hand out to Jules. "Have sex with me?" An invitation. And a nervous look that fluttered over her face, like maybe she was doing the wrong thing, which was so not true that Jules almost wanted to laugh.
But instead she took Autumn's hand and pulled her close again, their bodies pressed together, and she pressed her hand flat against the bare skin under Autumn's shirt. "Yeah, I do," she said. "Are you sure?"
"Yes." Autumn said, and then her usual sweet smile turned wicked. "Yes!"
Jules cut her laugh off by kissing her, and her fingers fumbled for the hem of her own shirt. She found it and broke away from Autumn to pull it over her head and toss it aside, and Autumn did the same, revealing an emerald-green bra with all sorts of lace and ribbon that left Jules feeling too plain in her black sports bra. But the way Autumn took her in made her feel more than enough.
It was only a trip and fall to Autumn's bed, and once they were on her rose-colored sheets everything seemed to slow and speed up at once. Fast: Autumn tracing circles around Jules's belly button, Jules sliding her tongue at the back of Autumn's knee, the time it took to unbutton her shorts. Slow: Autumn shimmying out of her skirt, the kiss Jules gave Autumn as the sunlight hit her eyes, the touch of Autumn's hands on her breasts. The sound Autumn made when Jules touched her for the first time, surprised and breathless and quiet. And the satisfaction Jules felt, pride as she made this girl she was half or wholly in love with squirm and moan at her touch.
It was the best kind of careful and careless at the same time, accidental elbows and knees and giggles as they shifted around. A hurried apology at a sharp intake of breath and then _No, no, good_ in response. Skin on skin and mouths on skin and hands on everything everywhere.
Jules had never done this, had kind of believed she'd have to wait years and years to find something like this. Someone who would let her see them so bare, who she would show herself to in spite of nerves, uncertainty. She tickled the bottoms of Autumn's feet and got a laugh in response, and then she slid her hands up Autumn's legs, dropped her head to blow on Autumn's soft inner thigh. Autumn tensed, Jules felt it, and she raised her head. "Okay?"
"Yes."
"Do you want me to—"
Autumn's eyes closed, a somehow pretty sheen of sweat on her cheeks and a breathless voice when she said, "Keep going. Do _not_ stop."
Jules slipped her hands around Autumn's beautifully round hips and kissed her thigh, her slick skin. And for her first time giving oral, she thought she did pretty good. Or from Autumn's gasping reactions, it seemed that way. It was not like she imagined, like things she'd read online had told her it would be: it was _better._ And when Autumn reached down, her fingers grasping for Jules, and when Jules reached up and twined her fingers through Autumn's, it hit her.
_This_ was it. Real romance. Not just the sex, but the laughing at each other, spilling their fears, talking about laundry and color-organized books. It wasn't magic; not everything was glittery and gold all the time. And declarations in the pouring rain and flowers on her doorstep—the idea of them paled in comparison to the reality of this, Jules realized. This everything, and this right now—Autumn's fingers threaded through hers, this most intimate stolen moment that was theirs and no one else's—what more had Jules been aching for?
And she had it now, and maybe it wouldn't be a forever thing, the end of all her longing. Jules had no idea. But this, right now, was enough.
Jules was enough.
## Dia
**The morning of Lex's** birthday broke bright and clear. When Dia's alarm went off, Lex was already wide awake, bouncing around in her crib with bedhead curls sticking in every direction. "Mama!" she squeaked. "Hi."
"I'm up!" Dia rolled out of bed and padded over to the crib. "Hey, do you know what day it is today? It's your birthday! How old are you?"
Lex scrunched her face up before answering, "Two!"
"That's right, my baby genius, you're two today." Dia lifted her out and smushed their faces together. "Let's go get some birthday breakfast."
Her dad was already at work in the kitchen, but he abandoned the stove to come over and plant a kiss on Lex's head. "Happy birthday to my favorite grandbaby!"
"I'm two," Lex said, proudly holding up her hand and actually showing three fingers. Dia folded one down and smiled.
"That's right," Max said, rescuing pancakes from the stovetop. "Are you hungry?"
"For chocolate-chip pancakes?" Dia said, pretending to gnaw on Lex's hand. "Always, right, Lala?"
They ate breakfast outside, all four of them. Her mom put sliced bananas on Lex's plate, and Dia added a giant swirl of whipped cream. Birthday rules.
After breakfast Jules arrived with Autumn in tow, and they spent an hour setting up in the yard—streamers strung from the fence, balloons tied to the lawn chairs, and their table draped in paper tablecloth and so much glitter. It wasn't a real huge party, more of a come-to-the-house-there'll-be-food-and-music thing. Lex was only two, Dia figured; there wasn't much point in going over the top. But it was one of the rare times where Dia could make magic. For Lex, she'd make magic whenever she could.
Dia was setting out the food when the doorbell rang and Hanna came in, Molly following. "Where's the cake?" Hanna said. "I'm here for the cake."
"You have to wait," Dia said. "Molly, help yourself to drinks, they're outside."
"Cool," Molly said, strutting off in that self-conscious, attention-seeking way that was so thirteen-year-old.
Dia turned to Hanna. "Does your mom know she's here with you?"
"Yes," Hanna said, rolling her eyes. "Molly told her. She said Mom went all quiet first, and then super enthusiastic, and the only thing she said to Molly was to be home by dinner." They both watched Molly in the yard. "She gets so much more freedom than me. But she deserves it."
"You deserve it, too," Dia said.
Hanna nodded but said nothing, and Dia grabbed her arm, pulling her outside. "Come on. This is supposed to be fun!"
By two their little yard was as full as it ever was—mostly with her mom's friends, her dad's EMT crew and their older kids, neighbors (including Waffles the dog playing gently with an overexcited Lex), and a couple of moms from day care watching their kids running through bubbles.
After Dia spilled soda on her shirt she went to change, tossing the dirty clothes in her overflowing hamper. Before she went back out, she went in the bottom drawer inside her closet and took out the baby book that she'd never quite finished. But still, she allowed herself the luxury of five minutes sitting there, paging through the pictures with her messy writing underneath: _Forty weeks pregnant_ and Dia holding her belly for the camera. _Alexa's first day home_ , a tiny ball of a baby in a too-big onesie and mittens _._ After a while it became just pictures, hastily pasted in: Lex standing with Dia's hands hovering, ready to catch her, and sitting in a kiddie pool in the backyard, and with cake smeared all over her face from a year ago.
Her baby was two and she'd so far managed to keep her alive, relatively unharmed, and seemingly happy. Dia didn't really believe in a god or heaven or anything, but she closed her eyes anyway and thought words that she couldn't say out loud to anybody: _She's amazing, Elliot. You'd think so if you knew her. Or maybe she wouldn't even be here if you were still here. I don't really know what might have happened if you hadn't gone and died on me. But it doesn't matter. You changed my entire world and I'll never get to tell you what that means to me. I have this piece of you, though. She's here, and I'm here, and you're not, but we're okay. We'll be okay._
She opened her eyes and laughed at herself. "Get it together, Valentine," she said, running a careful finger under each eye so as not to smudge her mascara. "Jesus."
Dia left the book on her bed and went out again. When she got to the back door, she called to Jules, "Come help me with the—" She stopped. "Oh."
Jesse was out in the yard, and from where Dia stood it looked like he was rescuing a balloon that'd gotten free and tangled in the tree that trespassed over from the neighbor's yard. She watched as he reached for it, stretching high, his shirt riding up showing yet another bruise, on his back this time. He got it and handed it to one of the older kids, and Dia leaned against the door frame. "When did he get here?"
"I don't know." Jules was at Dia's elbow now, and she followed her gaze. "Like, ten minutes ago?"
"Oh," Dia said again. "Okay."
"What?" Jules looked at her. "Is he not supposed to be here? Is something going on with you two?"
"Nothing's going on."
"Your face says you're lying," Jules said. "Did he do something?"
"He didn't do anything," Dia said, and that was the truth, at least. "Forget it. Come light the candles with me."
Everyone crowded around as Dia sat with Lex on her lap and helped tear open some of her presents, and then pretended she wasn't getting emotional while everybody sang "Happy Birthday." She pressed her face close to her daughter's so they could blow out the candles together, and whispered in her ear. "Happy birthday, Lex. You're my best present every year."
They passed around cake, vanilla with strawberry buttercream, and someone—Hanna, Dia suspected—turned the music up, A Tribe Called Quest soundtracking the afternoon turning to evening. People began to drift home, full up and happy. Dia snapped picture after picture on her phone: of her dad holding on to one of Lex's shoes while she ran circles around him; her mom swinging Lex high in the air; Jules smushing a kiss on Lex's cheek while she giggled.
She wrapped up pieces of cake and gave them to people on their way out, thanking them for coming. She watched Molly and Autumn showing Lex how to thread daisies into a chain. She watched Jesse petting the dog and talking to Candy with the leash in her hand. She didn't mean to avoid him, and she didn't want to push things. He had come to her house, though. He was on her territory.
_Why is he here?_ Dia thought. _He's supposed to be done with me. What is it going to take?_
Dia took a piece of cake and held it carefully on her way over to Jesse. "Hi," Dia said. "Sorry to interrupt."
"It's cool," Candy said, clipping the leash onto the dog's collar. "I have to go pick up Christopher from work now, anyway. Thanks for having us, Dia. I think Alexa's having a great birthday."
"I think she cares more about the fact she got to see Waffles than any of her presents," Dia said with a laugh. "See you later."
Candy waved as she led the dog away, and Dia looked at Jesse. "Lex loves that dog," she said. "You know how she is."
"It's a pretty cute dog," he said. "I can see why she's so into it."
Dia smiled and then looked at her feet. "What are you doing here?"
"Well, I don't like to miss free cake," Jesse said, and he wasn't exactly smiling, this guarded expression on his face. "Aren't we supposed to be going back to the way things were?"
"I . . ." Dia wasn't sure what to say to that.
Jesse nodded toward Alexa. "And you don't turn two years old every day, right?"
"Right," Dia said. _Of course he came,_ she thought. _Of course._ Because he was a good person and Dia had told him to come, months ago, and so even though things were weird, he was still here. Because he didn't want to miss her kid's birthday.
"I had a gift already," he said. "I wanted to make sure Alexa got it."
"Oh. Thank you," Dia said, and then, "Here." She presented the cake to him. "I know it's not butterscotch cookies, but we were all out."
Jesse waited a long beat before taking it from her. "Thanks," he said, and then he looked at her properly for the first time all day. "How did it go? Round two?"
Dia pulled in a slow breath. "Okay," she said. "Good. I think."
"Good," he repeated, nodding. "All right. I gotta go. I'm supposed to be at work in fifteen minutes. I . . ." He ran a hand over his head. "Sorry if I made it weird coming here."
"It's okay," Dia said again, and she silenced the part of her that wanted to ask him to stay, to talk about things, because she didn't have the words and she knew, really, that she should let him go.
So instead she nodded and said, "Thanks for coming," before walking away, back to Lex.
When everyone but her friends had gone and the yard looked like a whirlwind of glitter and ice cream had hit it, Dia changed the music to Carly Rae Jepsen and started cleaning up. Her mom and dad had taken Lex on a walk around the block, to try to calm her down from everything, and it was nice to breathe for a minute.
"What do you want to do with the leftovers?" Hanna said. "In the fridge?"
"Yeah," Dia said, grabbing a trash bag. "Tupperware is under the sink."
She started gathering up all the torn wrapping paper and paper plates while Molly helped Hanna box up the food and Jules . . . well, Jules didn't seem to be doing anything but sitting and watching the others work. "Juliana," Dia said, "can you get off your ass and help?"
"I am helping," Jules said, kicking her legs out and folding her hands behind her head. "I'm observing. Making sure you get everything."
"You're being obnoxious," Autumn said, hands full of streamers. "I don't make out with obnoxious people."
"Go _get_ me a soda," Dia said. "That's helpful."
Jules groaned. "Fine," she said, standing.
Dia cleared the table of debris while Jules did as she said, and when she came back with the soda they both sat in their plastic chairs. Dia put her feet in Jules's lap as she read the happy birthday texts from cousins she hadn't seen in years and her aunt who'd recently learned how to use emojis. "People love birthdays," she said to Jules, flicking from her texts to her email. "It's like—holy shit."
"Your aunt Jeanie said holy shit?" Jules said. "Wow."
"No," Dia said, and she opened the email that was from Sun City Radio, her fingers slipping on the screen. "Hanna! Come here, _now_!"
Jules sat up. "What is it?" she asked. "Is it the contest?"
Hanna came over, and Molly and Autumn stopped cleaning up to look over. "What?"
Dia cleared her throat and began to read: "'Dear Wildfire: We are so stoked to let you know that you have made it into our top three!'"
"We're in the top three?" Hanna asked, her eyes wide. "Wait. Read that again. We're in the top fucking three?"
Dia started laughing and couldn't stop, and then to her surprise Molly took the phone from her hand and took over.
"Listen," she said. "Okay . . .
**"Dear Wildfire:**
**"We are so stoked to let you know that you have made it into our top three! And here's our surprise: round three includes YOU becoming part of the lineup at next week's show at the Revelry Room! You'll perform for an audience of real live music fans and Glory Alabama themselves! The judges will take this performance along with everything else into consideration before choosing the winner. We'll be in touch with more details very shortly. Congratulations!"**
When Molly stopped reading there was silence, until Hanna said, "Wait, what? Perform?"
"That's what it says," Molly said, looking around at them. "What?"
Dia took her phone back and ran over the words again and, sure enough, right there it said it: _round three includes YOU becoming part of the lineup—_
Part of the lineup?
"Oh my god," she said, her heart pounding. "A _surprise_? What do they think this is, a reality show or something? We have to perform next week? Like, in front of people? In front of Glory Alabama?"
"No," Jules said, shaking her head. "No way. They would have told us that already. Wouldn't they?"
Hanna tugged at her hair. "Oh my god," she said. "We haven't performed for real people in _years_."
"Fuck," Dia said, starting to panic a little, and she looked up at the others. "What are we going to do?"
Autumn lifted her hands filled with paper the same colors as her hair. "Isn't this good news?" she says. "Isn't this what you wanted all along?"
Jules cracked her knuckles. "Not exactly."
"What are we going to do?" Dia repeated.
"You're going to play," Molly said confidently. "You're going to show them what I saw for an entire month. What's the big deal?"
They were silent for a minute, and Dia could almost feel them all spinning through what this really meant.
Jules blew out her cheeks. "Well, Molly's right," she said eventually. "We're going to play, aren't we? What other choice do we have? We've done it a hundred times before. And we did it last week."
"In front of _three_ people," Hanna said.
"Three, three hundred, what's the difference?" Jules's grin was shaky, and Autumn put a hand on her shoulder. "Come _on._ Okay, we weren't expecting this, but! We can do it. What, are we going to pull out now because we're scared?"
"I'm not scared," Dia said immediately. Okay, so that wasn't completely true, but it felt like Jules was issuing a challenge, and Dia was not one to back down from a challenge.
Not anymore.
"I am!" Hanna said, raking a hand through her platinum hair. "We haven't played a _real_ show in years. And GA are going to be there, watching us? We can't."
Dia took a deep breath and surveyed the wreckage of the yard, her friends standing there, Autumn and Molly waiting expectantly. "Yeah, we can," she said, and the initial rush of panic was replaced with adrenaline now. They were going to play a show in front of an actual audience—wasn't this everything they'd really wanted?
Dia looked at Jules, at Hanna, and gritted her teeth. "We better get back to practicing."
## Hanna
**Hanna rushed home to** Ciara's house and used the spare key to let herself in. Ciara was sitting on the couch with her laptop, coffee in one hand, and looked up when Hanna came clattering in. "What?"
"Can I borrow your van tomorrow?" Hanna asked, her words tripping over each other. "It's an emergency."
Ciara's eyebrows shot up. "Emergency?"
After Hanna explained everything, Ciara nodded and got up. "Okay," she said. "We can do this. One minute." She disappeared into her bedroom and Hanna sank to the couch. In less than a week she was going to be on a stage, in front of hundreds of people, playing while Dia sang her words. Playing to people who either wouldn't know who she was or would remember her, Hanna the drunk. No big deal, not at all.
She scratched her nails on the arm of the couch and took a deep breath, held it for three, let it out.
_I am okay._
Look at it this way: the people who didn't know her couldn't think anything about her, not anything that mattered. And anyone who did remember her, who might look at her and get that light in their eyes as they recalled her alcohol-fueled screw-ups—what did they matter, either? They didn't know her. No one really knew her, not anymore. Only Dia and Jules, Molly and Ciara. Only the people who actually mattered to her. Those were the opinions she cared about, and everybody else? Fuck 'em.
Ciara came back then and Hanna jumped to her feet. "I switched my shift," Ciara said, slipping her phone into her back pocket. "We'll take care of it, first thing tomorrow."
"You don't need to come—"
"Will you shut up and let me be a part of this?" Ciara said. "Shit, Han, I have restrained myself from pushing into your whole reunion operation this long. I am only so strong!" She grabbed Hanna by the shoulders. "And you know you can practice here, right?"
Hanna tried to frown, but it gave way to a smile. "You're sure?" she said, relief flooding her before being swallowed up by all the adrenaline currently flooding her system. "Thanks, C."
"No sweat," Ciara said. "This is what we do. Right?"
Hanna met Ciara's gaze and nodded. This is what they did, favors pulled and good faith given because they all wanted the same thing at the end of everything: a chance to try. "Right."
On Monday morning Hanna opened the door to her house while Ciara sat in her idling van at the curb. "Molls?"
Molly appeared in the kitchen doorway. "I can't believe you're doing this," she said, her face fixed in worry. "Mom's going to lose it when she sees."
"Tell her to come talk to me," Hanna said, heading out to the garage. "Besides, they're mine. I paid for them. I'm taking them."
She opened the garage door from the inside, rolling it up and letting the sun in, and then Ciara got out of her van. The three of them carried out the remains of her drum set and loaded them into the back, and when they were done the garage looked strangely empty. "All right, Molls," Hanna said, and she grabbed her sister in a vise-grip hug that Molly had no chance of wriggling out of. "I have to go, but I'll text you later?"
Molly frowned once Hanna let her go. "What should I say?"
Hanna knew right away that Molly meant _What should I say when Mom asks me what the hell happened?_ She looked up to the sky for a second while she thought, and then back at her sister. "Tell her I said I had something important to give you," Hanna said. "And you didn't know I was here for the drums, and you had nothing to do with it. Okay?"
Molly nodded reluctantly. "Okay," she said. "Bye, Han. Love you."
"I love you too, Molls," Hanna said, starting toward Ciara in the van. "Talk later."
## Jules
**It had taken some** convincing on Hanna's part, getting them to agree to come to Ciara's place. But Hanna had said Ciara wasn't mad, and even though it didn't stop Jules feeling bad—after everything Ciara had done for them, they'd still faded from her life like it was nothing, Jules's second-biggest regret—she believed it. Hanna had stayed close to Ciara. And if she said that Ciara wasn't mad, then maybe it was true.
So on Monday afternoon, they took the steps up and knocked on Ciara's door. It opened as soon as Jules's knuckles left the wood, Hanna standing there looking wired. "Come in," she said, impatience in her voice. "Set up, I want to get going."
Jules exchanged a look with Dia, a _who is this girl again?_ look of awe. "Okay," Jules said as they entered Ciara's house, began unpacking their guitars. "Whatever you say, captain."
Ciara's house looked the same, Jules thought, but as she looked around she began to notice the changes. Gray walls instead of green, the posters replaced with framed art prints. The rug on the living room floor was white and fluffy instead of multicolored knit. Jules lifted the corner of it, though, and underneath was the chunk of wood missing from that same floorboard. She smiled: Dia had dropped a plate there, shards scattering over the floor, and only after they'd cleaned up had they noticed the board.
She put the rug back and plugged in. Dia was ready; Hanna was sitting behind her drums in the corner, and she lifted her sticks, eyes gleaming.
Two songs—that was what the follow-up email had said. Each of the final three acts had to perform two original songs, and then the judges would decide who was going to win.
So they ran through "Bones" and "Pretty Baby," their obvious second choice, over and over, knowing them almost too well. It was the intense focus of knowing they had only five days now, the anxious energy from before having dissipated; or, it was finally falling into the band they were, who they were now.
Jules didn't know which for sure, but she did know that they didn't make any mistakes. Didn't slip up or forget anything, didn't get into any bickering arguments or snap at each other the way they usually did. They kept going until Dia made them stop, wary of getting played out.
"I think we're ready?" Dia said, sounding unsure but looking determined. "Yeah."
_"Ready?"_ Jules said, pulling her braids into a twist. "We can't be ready. We still have five days."
"You know what I mean," Dia said. "We were already ready. This is just extra. I don't want us to freak ourselves out by practicing too much."
Hanna held her sticks up. "You're right," she said. "Let's take a break, at least."
They sat on the couch, pushed back to make space, and drank sodas from Ciara's fridge. "This is weird," Jules said, breaking the silence. "It feels like no time has passed."
"You can tell it has," Dia said. "Look at Hanna's roots."
"Hey!" Hanna said, then tugged at her hair and sighed. "It's true. I really need to fix this situation."
Jules looked at her. "Let's do it now," she said, the idea catching her. It was the kind of thing they used to do all the time: hair dye over kitchen sinks, makeup sessions in Dia's bedroom, late night secrets in between. "We'll help."
"Really?" Hanna sat up. "You want to?"
"Why not? We're done here," Jules said, glancing at Dia to check. "We'll kill each other if we keep going. Let's do this instead, and we'll practice again tomorrow to be sure."
"Yeah," Dia said. "Sure."
Jules got up right as the front door opened, and there was Ciara silhouetted in the doorway.
She let out a little shriek upon entry and raised her aviators. "For the love of all that is holy," Ciara said. "Would you look at you three together!"
"Ciara!" The name burst free of Jules's mouth without her even meaning to say it, and she threw herself into Ciara's outstretched arms with such force that Ciara stumbled. "Hi!"
"Hi, pudding." Ciara returned Jules's bear hug with equal enthusiasm. "Oh, god, you're taller. When did you get taller?"
"I missed you," Jules found herself saying. "I'm sorry." Was that all? Was that everything she had?
But Ciara touched her cheek. "It's okay," she said. "I know."
It was Dia's turn to be enveloped in Ciara's arms next—arms even more covered in tattoos than they had been before, Jules noticed—and then they stood there, all looking at each other and laughing. "Congratulations, mama," Ciara said to Dia. "I heard you got yourself a beautiful little one."
"She is beautiful," Dia said. "She just turned two, can you believe it?"
Ciara shook her head. "I really almost can't," she said. "You'll have to tell me all about her." She narrowed her eyes. "But what are you doing sitting around? Aren't you supposed to be practicing? I want to hear you!"
"We're taking a strategic break," Hanna said. "We're going to fix my hair."
"Oh, really? Hmm." Ciara shimmied her shoulders, bare in a skull-emblazoned halter top. "Now that I can help with. But! Only on the condition that you play for me later," she said. "And catch me up on everything I've missed. And I mean _everything._ Deal?"
"Deal," Jules said, in unison with Hanna and Dia.
"All right then." She whipped around and headed back out the door. "Come, my loves! To the beauty supply."
Jules looked at the others, the biggest smile on her face. "I forgot how much I fucking love that girl."
They piled into Ciara's van—like old times—and headed downtown. Inside the beauty-supply store they wandered the aisles looking for bleach powder, developer, rubber gloves. "You know, you could try a little color," Ciara said to Hanna. "Blue tips?"
Hanna shook her hair out. "Maybe," she said. "Jules, what do you think?"
"Why not?" Jules said. "If you want."
"We'll go look," Dia said, grabbing Jules by the hand and dragging her into the next aisle. Heads bent low by the boxes of Hot Red and Purple Orchid, Dia whispered to her. "She's not mad, is she?"
"Ciara?" Jules said. "No. She's not."
"Okay," Dia said. "I needed you to say it."
Jules reached out and flicked Dia's elbow. "Don't push it," she said. "Right now we have a really good thing going. Don't jinx us."
"I would never," Dia said, rubbing at her elbow. "And ow!"
Jules started down the aisle, toward the pinks and blues. "Oh, you princess."
She scanned the array of colors, mentally pairing each with Hanna's face and either casting aside or plucking a box out. When she couldn't decide between Coral, Capri, or Lagoon Blue, she went to find Hanna and Ciara. But as she passed the makeup aisle, a too-familiar voice caught her and Jules had to look.
Down there by herself was Delaney Myers. Standing the way she always did: back arched, one hip jutting out, her head tipped ever so slightly to the side, in case anybody was looking. Which Jules was, she supposed, but not _looking_. Because this was the first time she'd seen Delaney since graduation and it felt strange. Like, as if there used to be a piece of invisible string between them, pulling taut and falling slack but always connecting them, that now had been sliced in two. Jules felt no pull, not in that bittersweet way she always used to.
Delaney hadn't seen her. Jules watched her for a moment, bending down to pick something off the bottom shelf. Before this summer Jules might have found some reason to walk down there, shut off her brain for a minute, and give in to loneliness. But now Jules had that thing she'd always been needling Delaney for, and she regretted the time they'd spent pissing each other off and pretending it was what they wanted. Delaney deserved better, too.
"Hey," Jules called out before she really thought about it.
Delaney heard and turned and smiled like there was nothing out of the ordinary about this. "Hey," she called back. "What are you doing here?"
"Hair dye," Jules said, and pointed at her braids. "Not for me."
Delaney nodded slightly. "Right. For your girlfriend?" she said, and then grinned at the face Jules knew she must be making. "I heard you were seeing some pretty girl with amazing hair. Am I wrong?"
Jules looked up to the pockmarked store ceiling, trying not to laugh. "No, you're right."
"That's good," Delaney said. "I hope she's good to you."
"She is," Jules said, and watched Delaney carefully. "Are you? Good?"
Delaney nodded. "Yeah. I am."
Jules smiled, and she hadn't realized how good it would be to ask that question and get that answer. To even talk to Delaney like they were only some version of friends. "All right." She lifted her hand to wave goodbye. "See you around."
"Yeah," Delaney said, waving too. "Have a good summer."
And then Jules kept on walking, pulling her phone out when it buzzed in her back pocket.
Autumn had perfect timing. **Hey,** her text read, **hope it's going well! Malai lost big this morning so next taco trip is on me.** It was accompanied with a selfie, Autumn peeking over a twenty-dollar bill.
Jules shook her head. **I was just talking about you,** she wrote back. **You're so cute it should be illegal.**
**I know,** Autumn's response came. **But you love it.**
Jules glanced over her shoulder, back in Delaney's direction. What they'd had and what she had with Autumn—they were an entire world apart. She and Autumn weren't perfect, not close. They made mistakes, and made up, and they were both figuring it all out. And it was better than Jules could have ever imagined.
Jules snapped a picture, a silly face, and sent it to Autumn. Then she headed up to the counter, where the others were already congregating. "Hey," she said. "Did you choose a color?"
Hanna nodded, glancing at Jules as Dia and Ciara stacked boxes on the counter. "You okay?"
Jules let out a slow breath, imagined herself tangling her fingers in the sweet pink and ocean blue of Autumn's hair. "Yeah," she said. "Perfect."
## Hanna
**Hanna stood in the** bathroom that night, turning this way and that, watching the way the light changed on her new Special Effects Nuclear Red hair.
"Are you sure?" Dia had asked, a brush loaded with red dye in her hand. "Once we start, there's no going back."
Hanna had taken a deep breath and nodded. "Do it."
It was almost a shocking difference. Hanna was so used to the bright white—it had been her calling card since she'd first locked herself in the bathroom with the bleach at thirteen. But shock was what she needed. Because the girl with the platinum hair was not really her anymore. She didn't know if she was this girl, either, but she could try it out for a while. Experiment with who, exactly, Hanna Adler was right now.
She pulled her hair into a ponytail, admiring the jewel-bright shimmer as she did so. Then she took her phone out of her pocket and checked the list of missed calls: all from her mom, none since last week. Only a handful of texts, sent at times of the morning Hanna thought of as night. **Just checking in,** they said, or some variation of.
**Hope your day went well.**
**The weather's nice today.**
Nothing from today, no sign that her parents had noticed her heist—or that they had decided to do anything about it, at least.
Hanna bent over the sink. It had been two weeks now. For the sake of harmony: time to yield.
She watched her new self in the mirror as she dialed. It only rang for a second before her mom answered, this cautious voice. "Hanna?"
"Hey, Mom."
"Hi. Did you get my messages? I left a lot," Theresa said, and then, "You came home today. What were you doing?"
There it was. Hanna knew her mom couldn't keep away from that.
(Hadn't she known, really, as she'd been dismantling her drums that it would lead to this? Wasn't that a little bit why she'd done it?)
(Maybe.)
"I'm sorry," Hanna said, flexing her free hand. "I should have called you back sooner. I was . . . I've been trying to get my head straight."
"Hanna—"
"I want to talk," Hanna said, watching her mouth move in the mirror. "Not on the phone. At home, with you and Dad. Would that be okay?"
There was a long silence, and then her mom said, "Of course it would. Come by tomorrow. For dinner?"
"Sure," Hanna said, and it felt strange to be doing this, arranging dinner dates at her own dining-room table. "See you then."
She hung up and checked her reflection again. The red, she loved, but she wanted more. More different.
She opened the door. "Ciara! I need your help."
The first thing Molly said when she opened the front door was, "Your hair!"
Hanna touched a hand to her head, a little self-consciously, grateful for the distraction from the unsettled feeling she'd gotten from knocking on her own front door. "Oh, yeah."
"It's so short! And red!" Molly said. "You didn't tell me you were doing it."
Hanna let the ends run over her fingertips. Last night she'd made Ciara cut off five inches. The ends now hovered above her shoulders, falling in her natural waves. "It was a spur-of-the-moment thing," Hanna said. "I didn't plan it. I would've told you if I had. Stop hassling me."
Molly gave a defiant toss of her head. "Fine," she said.
"Hanna?" her dad's voice rang out. "Is that you?"
She steeled herself, fixed a smile on her face, and followed Molly inside, into the kitchen. "Hi, Dad." She was not a hugger at the best of times, and she leaned against the door frame, waved at her dad sitting at the table. "Hi, Mom."
Theresa came away from the sink, a pitcher of water in her hands. "Hanna," she said, and there was the tiniest hint of warmth there, and then her eyes widened. "Your hair!"
"Looks good," her dad cut in, nodding approvingly. "Different."
"Yes," her mom said, her smile wavering. "That's what I was going to say. Sit!"
"Sure," Hanna said, pushing down her laugh. How easily everyone was shaken by a box of red dye and pair of scissors.
But aside from that, things were not as terrible as she'd expected.
They playacted regular through dinner: passing food, Molly making faces across the table, her parents making intermittent conversation. Hanna ate and wondered if this sense of surreality was evident only to her, or if they all appreciated the play they were putting on.
Once they were finished, Hanna got up to clear the plates. "Molly, help me out," she said, but her dad cleared his throat.
"Actually, Molly, will you give us a minute?" he said. "Take your laundry upstairs."
Molly rolled her eyes. "Fine." But as she stood up, she shot Hanna a worried look. Hanna nodded at her and mouthed, _It's okay._ Because it was true. Hanna had been at this table with her parents wearing those Very Serious faces many times before: after the ER that first time, and when Elliot had died, and after the ER that second time, and when they'd told her she was going to rehab. She'd lived through those talks and she'd live through this one, too.
_Four hundred and sixty-eight days,_ she thought.
_I am here._
_I am not broken._
_I am okay._
Once Molly was gone, up the stairs, her parents turned all their attention on her. "Hanna," her dad said, and he looked so, so tired. "We need to talk about what's going on."
"Where are you staying?" her mom asked. "Not with Dia? Jules? I called their mothers."
"You called their _moms_?" Hanna said, and she wasn't sure why she was surprised. "I'm staying with Ciara."
"With Ciara?" Her mom pressed her lips into a thin line, and Hanna could imagine the conversation in her head: _At least she has a safe place to stay. But Ciara's a part of the whole music thing. She's older, though, and maybe she'll teach Hanna something about responsibility. Or maybe she'll be a bad influence._ "Okay," was all her mom said eventually. And then she sat up straight, eyes bright. "What are you _thinking_?"
Hanna swallowed her nerves and looked her mom right in the eyes. "I am thinking," she said, "that you gave me an ultimatum and I made my choice. That's what you're supposed to do, isn't it? When someone says do this or that. You pick one. I picked the one you didn't want. That's it."
"Come on, Hanna," her dad said, raking a hand through his graying hair. "Take this seriously."
Hanna looked at her hands for a minute, spread flat on the warm wood of the table. Then she looked up and blinked slowly. "Take this seriously?" she said. "Trust me, I am. I'm trying to make up for _years_ of not being serious." Her mom started to say something, but Hanna shook her head. "Can I talk for a second, please?"
Her mom's eyebrows rose, but she nodded anyway. "Sure," she said, acerbic. "Talk."
Hanna leaned her elbows on the table and spoke with her hands, too. "I know," she started, her voice as clear as she could make it. "I know I've done things in the past that made it hard for you to trust me. I've done things that have made it hard for me to trust myself, but I'm trying really hard to do the right thing. I know you don't like the idea of me playing music again, and I'm sorry that I lied to you about it. I think I knew you wouldn't want me to do it, and I didn't want to give you the chance to tell me that.
"And Mom, I know you were only trying to do what you thought was best for me. But what you think is best for me and what I think are so different. But you gave me the option to stay here and go back to the way things were before, or leave and do what _I_ want to do. So I left, because I _can't_ , and I don't want to, go back to the way I was. I don't mean the way I was when I was drinking, I mean the way I was, like, _two months_ ago. When I was lonely _all_ the time, and constantly trying to do things to make you happy, and not making myself happy at all." Hanna looked around the kitchen, the walls familiar sunshine yellow, the cabinets worn with years of their touch. "I haven't been happy in so long. Making music again—I remembered what it's like to have something I actually care about. And I knew you wouldn't understand." She shook her head, looked back at her parents, watching her warily. "Me and Dia and Jules, yeah, we've had our problems. The problems that I had, that I have, are not because of them. They're not because of the music. My problems are because of _me_ , and who I am, and how I chose to act. I'm not choosing that way anymore. That's the real thing. I could be anywhere doing anything and still be falling apart if I didn't choose every day to be sober and to be different than the girl I was." She took a deep breath and met her parents' gazes for the first time since she'd started speaking. "I know I can't make you trust me, or trust that I'm doing the best thing for me, but I'm asking you to. I'm asking you to at least try."
Her dad watched her as she stopped talking, this look in his eyes like he was really processing what Hanna had said. But her mom looked upset. "I don't know, Hanna," she said. "I think you're right—we don't agree on what's best for you. But I think that we can _talk_ about it all and come to some kind of agreement about the music and what you're going to do. Yes, Benjamin?" But then she carried on without giving Hanna's dad a chance to answer. "But in the meantime, I think you should come _home_."
Hanna supposed she should have been relieved to hear that, but she wasn't.
She liked being out of the house, being responsible for herself with no one to approve or disapprove of her actions. Being on an equal level with Ciara instead of constantly watched over by her parents. The idea of coming back filled her with a sense of overwhelming claustrophobia. "Thank you," she said. "Really, I mean it. But . . . I don't think I should come home."
"What?" her mom said, lips disappearing into a thin line again. "Hanna."
"You said it, that day," Hanna said. "I'm eighteen, I'm an adult. Old enough to learn what it is to take care of myself. I don't think it's good for me to be here anymore. Not that I don't love you and miss you," she said quickly. "Because I do, of course. It's only been two weeks but I do miss you. But also—it's only been two weeks, and I feel so much _better_. I think that being here with you watching me all the time makes me do things I shouldn't. Does that make sense?" She leaned her chin on her hands. "The truth is, I like it with Ciara. I like cooking for myself and coming home when I want and saying and doing things without wondering how you're going to react first."
"You might like it now, but it's a novelty," her dad said. "You won't like it so much when most of your paycheck's going to rent and you have to deal with mold and no one's there to do your laundry for you."
"Yeah, but who likes that stuff?" Hanna said. "You just do it, right? Because you don't have another choice."
"You have another choice," her mom said, and she sounded a little shocked. "Hanna, sweetie, do you really feel like we're always watching you?"
"You are," Hanna said. "But I don't blame you. This is what I'm saying. I did things to make you have to do that. But now it feels like we're stuck in the routine of you not trusting me and me desperately wanting to _make_ you trust me, and I really want to stop."
Her dad cleared his throat. "Living by yourself," he said. "That's a big jump. Responsibility."
"I know," Hanna said. "Once I start my job, I'm going to have some money. Ciara's looking for a roommate, and she says I can live there officially if I can work it out. And I have to do it sometime—leave here. Now I feel like coming home would be a step backward. I want to keep going forward." She shifted, her new shorter hair creating this breeze around her neck. "I want to know who I am on my own."
They talked a while longer, and eventually her parents were . . . not on her side, but accepting. Accepting that she wasn't coming home and that she wasn't giving up music and that, maybe, she was not exactly who they thought she was.
But Hanna was more than ready to start becoming who _she_ really wanted to be. It was only the beginning of a much longer conversation, Hanna knew, but at least it was a start.
"This Friday," her mom said, when Hanna was getting ready to leave. "You can come for dinner again, if you'd like. Maybe we can talk some more."
Hanna gave a small smile. "Okay," she said. "But this week, I can't. Next week, though."
"What are you doing this week?" her dad asked, as they walked out to the hall.
Hanna cleared her throat. "It's, uh . . . the contest. This thing at Revelry, where they're going to announce the winner."
Theresa gave this soft smile. "Sounds fun," she said, and those two words alone were enough to let Hanna know she was trying. Would this last, her trying?
Hanna knew better than to hope, and yet, she still did.
"Maybe you can come by on Saturday instead," her mom said. "It's okay if you don't. Door's open."
"I will," Hanna said. "I have to get the rest of my stuff, too."
"Don't worry about that," her dad said. "We're not kicking you out. You can leave things here until you have space for them."
Hanna nodded. "Thanks," she said. They were at the door now, and this time she did hug them both, hard and fast, and felt tightness in her chest. Inside this house it was all she could do sometimes to breathe. Outside, the air came so much easier.
"I'm going to say bye to Molly," she said, and she ran upstairs. Molly's room was empty, and so Hanna turned and pushed open her own bedroom door. "Molls?"
Her sister sat on the bed, cross-legged with her back against the wall. "You're not coming home," she said, flat, accusatory. She looked over at Hanna. "That sucks."
"I know," Hanna said. "You get why I'm doing it, though, right? It'll be better this way."
"No, it won't," Molly said. "You're supposed to be _here._ What about when I start school? Who's going to help me? What if I need you?"
Hanna went to her, sat down on the bed. "I'm right here, Molls. You need me, anytime, anyplace, you know I'll be there. And I'm only a bus ride away. You can come hang with me and Ciara whenever you want."
Molly's lip shook. "Promise?"
"I swear." Hanna shifted; she already felt a little out of place in this room. "It's weird, Molls. I feel like a grown-up and a little kid all at the same time. Like, I don't know what the fuck I'm really doing, but I know I have to do it."
Molly turned to her. "Can I come on Friday?" she asked. "I really want to see you all play."
"I'll see what I can do," Hanna said.
She scanned her room, the memory of so many years and so many versions of herself lingering in here. The Hanna who read horror stories under the covers by flashlight; the Hanna who hid bottles under her bed and kept the curtains closed against the light; the Hanna who listened to melancholy voices singing in the quiet dark hours of the night, and scribbled her own words to get them out of her head.
She got up and crouched in front of her nightstand, opening the bottom drawer. The last of her notebooks were still in there, and she lifted them all out. "You want some advice, Molls?"
"From you?" Molly said, and pretended to cower when Hanna whipped around to glare at her. "Kidding. What?"
Hanna looked at her not-so-little sister, delicate and not, strong and not. "Life is too many strange and beautiful things to use it being unkind to yourself." She paused. "People will try to make you into somebody you're not. Even the people who are supposed to love you for who you are. Sometimes it feels like it would be easier to become that person. But that won't make you happy. And if you can, being happy is the most important thing. So do what you have to so that the world does not get you. Do you hear me?"
Molly looked like she wasn't quite sure what to make of Hanna's words, but then she nodded once. "I hear you."
## Dia
**The moon was bright** in glimpses between clouds, hanging low over Dia in the backyard. It was Thursday night—or Friday morning, now—and she'd spent hours lying in a puddle of her own sweat before accepting that she wasn't going to sleep. So she'd grabbed the baby monitor and her acoustic, crept downstairs to sit out on the back steps.
She'd thought she'd get some relief, but the air was thick like a storm was about to roll in—that's what they'd been saying on the news all week. But they said that all the time, talked up these big thunderstorms, told everybody to be ready for the rain, and then they never came. _No storms,_ Dia thought. _They only wish for it._
When the back door creaked open, Dia jumped, putting a hand to her chest as she twisted around. "You scared me," she said.
"I could say the same," her dad said. "What are you doing out here?"
"Can't sleep," Dia said.
"Nervous?" Her dad sat down beside her, the reflective stripes on his EMT uniform shining.
Dia ran her thumb across the strings of her guitar. "A little," she said, honestly. "Mostly not. I just want to do it, you know?"
"I know," her dad said. "I'm sorry I can't be there." Neither of her parents could; they'd both been excited when Dia had told them about Friday, and then her dad had realized he was scheduled to work and her mom would have to watch Lex so Dia could go.
_Never mind,_ Dia had said. _It makes me nervous when you watch, anyway._
"It's all right," Dia said now, reassuring again. "You got all those people who need their lives saved. I guess it's an okay excuse."
Max smiled, and he looked exactly like Lex when he did that: same crinkled corners of their eyes, same apple cheeks. "I'm real proud of you," he said. "You know that, right? No matter what happens tomorrow, you did this."
"I know," Dia said, quietly, into the night. "Hey, you want to hear what we're playing?"
Max held his hands open. "Hit me."
So Dia played a concert for her audience of one, under the clouds, and the moon winked in and out of sight, and she felt the anchor of the earth release her the slightest amount.
## Elliot
## NOVEMBER
**"We're Graceland!" Ciara says** from the stage. "Thanks for listening!"
Elliot sticks his fingers in his mouth and whistles; next to him Jules has Dia on her back, and they're both cheering.
They've come out of town tonight, to Longport, where almost all the houses are the fancy kind. This one has a huge pool with a slide, a trampoline, a giant oak with a tire swing hanging from a thick branch. The yard is full of people, and Elliot recognizes fewer than usual.
Hanna throws an arm around Elliot's neck. "Elliot!" she yells directly in his ear. "How much you wanna bet I can do a backflip on that thing?"
She's pointing at the giant trampoline, and Elliot frowns.
"Don't worry," Hanna says. "All that's in my cup is soda, I swear. Here! Try it."
She thrusts her red cup at him and Elliot looks at Dia, who hops down off Jules's back. "I'm good," he says, but Dia takes the drink. She swallows and her eyebrows rise.
"See?" Hanna says with a gleaming smile. "No lie."
"Okay," Elliot says. "Ten bucks."
Hanna leads the way and climbs up on the trampoline, tugging her jeans up. "Watch this," she says, and Elliot does watch as she executes not one but two perfect backward somersaults.
"She did a lot of gymnastics when we were kids," Jules says, and Elliot shakes his head as Hanna climbs back down.
"Unfair advantage," he says, but he hands over the ten bucks.
"I never said anything about being fair," Hanna says as she tucks the money inside her shirt. "Thanks!"
Now Dia's the one with her arm around his neck. "See?" she says to him as the others begin bickering about something. "She's not always so bad."
Like he's the one saying she is. But Elliot decides not to say that. Hanna is in a good way tonight and Dia's laughed more than he's ever seen her before and things are really good. Instead he says, "I'm going to get a drink," and kisses Dia's cheek. "You want one?"
"Beer," she says, and then frowns, and Elliot sees her eyes go to Hanna. "And a water."
"Coming right up."
When he gets back outside, Dia and Jules have joined Hanna back up on the trampoline. He can see them, rising and falling out of the deep blue sky, hair flying everywhere, laughter pealing over the music.
He makes his way through the yard, and when he's almost to them someone comes running past him—Ciara. She hauls herself up on the trampoline, too, and the four of them soar through the air.
"Having fun?" he calls up to them.
"Take a picture!" Dia calls back.
He does as she says, capturing them in blurry speed on his phone, and when he looks at the picture he can almost see the magic they possess, right there on the screen. For all the heartache and mess and drama—shit, he gets it.
These girls are something special.
Dia collapses, breathless, and tumbles to the ground. "C'mere."
It's almost midnight and Elliot already knows he has no chance of getting home before curfew, but fuck it. She's lying in the grass now and Elliot lies next to her, smiles as she hooks her leg around his. There's probably an empty bedroom they could make good use of—
"I know exactly what you're thinking," Dia says. "Dirty."
"I know what _you're_ thinking," Elliot says.
"What?"
"Something about the band," he says. "You always are." He touches a finger to her temple. "If I opened this up, it'd all come spilling out." He lifts himself over Dia and stares down into her unblinking eyes. "You're really beautiful," he says, and maybe he's kind of drunk but it's true. "Really really."
Dia laughs, and the sound vibrates through his hand. "Thank you."
"I think you're made of magic," he says. Okay, he's definitely drunk, but it's still true. "You and Jules and Hanna. When I see you with them, doing this thing you love, you're so _alive_ and it's like I'm seeing the truest you and"—he should probably not say this but it's true—"I think I could be in love with a person like that."
"Elliot," Dia says, her smile so wide, and then she kisses him until they're both breathless, and when he speaks again he puts his lips to her throat.
"Made of magic," he says, the words pressed into her skin. "All of you."
## Hanna
**They stood together, the** three of them, across from Revelry with the traffic rushing between. The marquee was all lit up tonight, in bright white: SUN CITY AND GLORY ALABAMA PRESENT THE ORIGINALS CONTEST.
"So it's not _our_ name," Hanna said. "But one day it will be."
They waited for a break in the traffic and then ran across the road, headed into the club. First they had to sign in, and Hanna tried to see the other names on the list, but the girl with the clipboard—a different girl this time—was too fast, whisking them through to backstage before Hanna could look. "Did you see?" she asked the others as they made their way through the maze of hallways. "Who else is on the list?"
"No," Dia said. "But it went up on the website last night."
"Who is it?" Jules said. "Do we know them?"
"I don't know them," Dia said. "Knoxville Slums is one. And then Ursula Arrival. I don't know if that's a group or a person."
Hanna lifted one shoulder. "Never heard of them," she said.
A guy with the sharpest eyeliner Hanna had ever seen met them backstage, checked them off his list, and pointed them in the direction of the dressing rooms. "You're in room three," he said. "Sound check in five minutes."
They dropped their stuff in the dressing room, the flaking white walls and barely illuminated mirror unchanged since their last time here. Sound check blurred by, but afterward they waited in the wings to watch whichever act was checking next. They shouldn't have, Hanna realized, as soon as this group of punk people came out and launched into noise. "Oh," Jules whispered. "They're good."
"They're _really_ good," Hanna said.
"Forget them." Dia turned on her heel and started back to the dressing room. "We're the shit, remember?"
Hanna tried, but she was too nervous.
It was better once they were back in the dressing room, door closed to the world. Jules played music out of her phone, a drop in the ocean of sound pushing up against the door, but enough for them. They spilled makeup across the dressing table and painted each other's faces: gold cheeks and red lips, radiant highlights and clear, sticky gloss. They changed into shorts, boots, and high-tops, black shirts: one Nets jersey, one scooped black tank, one cropped, bone-illustrated top.
In front of the mirror they stood together, and as Hanna smoothed her hands across the flash of soft stomach visible between her shorts and her shirt, she took them in. "We almost look like an actual band," she said to their mirror selves, her shoulder knocking against Jules's arm. "We almost look like we know what we're doing."
Dia teased her fingers through the ends of her hair, big and loose for tonight. "Illusion," she said. "The art of fooling everybody, including ourselves." And then she turned to Hanna, half a smile on her face. "C'mere, you have lipstick right _here_."
Jules bounced on the balls of her feet. "I want to go see," she said. "Only for a minute."
So they went out, crept around the side of the stage and inched back the curtain, stood and looked out on the swelling audience. Hanna's palms started to sweat as she took it all in: the lights, the throbbing noise. People crowding right up to the stage, and filling up the balconies, and still more coming in. She needed a nicotine hit. "There's a lot of people here."
"Glory Alabama are watching somewhere," Dia said.
"Where?" Jules asked. "Oh, I think I might throw up."
Hanna wiped her hands on her thighs. The last time she'd played in front of an audience this big, she'd been halfway to blackout and barely able to get through the song.
Not this time.
She scanned the crowd for Ciara's tattoos, Molly's blond halo, but couldn't find them. They were here, though; she could feel it.
And then the lights went down and the cheers lifted.
" _All right, people! Are you ready for a show?"_
They paced, prowled backstage while bigger local band Wednesday Street took to the stage and warmed everybody up with electronica-infused grunge.
They stood in the wings as the MC introduced the contest acts, gave the whole spiel: _Money! Fame and glory! Everything you could ever want!_
Knoxville Slums went out first. Rattling punk sounds, screamed vocals—this was the band they'd seen earlier. Hanna cracked her knuckles over and over, listening to Dia talk loudly, right in their ears. "It's a Flogging Molly–meets–Rancid thing," she was saying. "Basic. And _so_ old."
"We're better!" Hanna said over the noise, to her hands. "Right?"
"Right!"
Ursula Arrival were cleaner, to Hanna's ears, but had way less stage presence. One of their singers froze every time she opened her mouth, her entire body still as she sang, but their guitarist could shred, Hanna had to give them that. "We're better," she said again, a mantra, rolling her sticks between her palms.
Jules knocked her hip. "Say it again."
" _We_ are _better_ than _them_ ," Hanna said, turning away from the stage. "Got it?"
Dia ran her hands along the neck of her guitar. "This will sound wrong," she said. "But—I almost don't care if we win or not. I _care_ , I _want_ to, but—"
"It's more than that," Hanna cut in. "Right?"
"So much more!" Jules said, and they were all a breath away from yelling to be heard.
Hanna pushed her hair back. "I'm really glad you made me do this," she said. "I'm really glad I wasn't too afraid to try! And that you see me now. I see you now." Dia and Jules were looking at her, full attention, and Hanna's heart was humming. "We made mistakes and we did terrible things but that's not all that we are. That's not who we are. Fuck. Am I even making any sense?"
Jules laughed, hard, and Dia smiled as she nodded. "Total sense!" she said, throwing an arm around Hanna's neck.
Ursula Arrival screeched to a stop, and the crowd's cheers were noticeably less than they had been for Knoxville Slums.
"We are going to rip the sky open," Hanna said, and Dia leaned in.
"What?"
The MC walked out, mic to her mouth. "Okay! Now we have our last Sun City hopefuls. They've been around the block and they're back again—make noise—"
Hanna put her mouth to Dia's ear. "We're going to rip open the sky!"
"—for Wildfire!"
## Dia
**Dia pulled away from** Hanna, electricity coursing through every inch of her. "Whatever you say!" she yelled back, and then she looked at Jules. "Ready?"
"We got this," Jules said.
Dia tugged her shorts up, shook her hair around her face. The MC was leaving the stage now, people were clapping and yelling and it was for them, this time, no one else.
She steeled herself, pulled her spine up tall, and took a step.
When she walked out, it was with the confidence of a girl who'd done this a hundred times before. Tonight the lights were bright enough to dazzle, throwing shadows into the corners of her vision, but it didn't faze her.
Dia stepped up to her mic, center stage. "Hi," she said, to instantaneous cheers. Gratifying. "We're Wildfire." She laughed, raspy. "Maybe some of you here used to know us, before. But you don't know us now. You'll find out."
Without waiting for another response from the crowd, without having to check that Hanna and Jules were ready because she already knew they were, she ripped into the beginning of "Bones _._ "
Two songs could be both an eternity and a gasp. Long enough to lay your heart bare, but so fast that if you blinked, you might miss it _._
Dia pulled words out of the back of her throat. She danced, on the spot, her legs shaking, and over to Jules, to Hanna, as far as her cables would allow. They ran into each other and landed back down, and the heat of her body set Dia alight.
She played for the girls they used to be and the ones they were now, and all their fallen-apart pieces that had gotten lost or ruined or discarded along the way.
She played for her baby at home, sleeping, dreaming dreams that Dia couldn't even know yet.
Her arms ached and her hips shook and every muscle in her felt awake.
"Bones" became "Pretty Baby" before she knew it, and the lights shifted through the spectrum and she was performing now, not just playing: she was taking her hands off her guitar and lifting the hair off the back of her neck, closing her eyes and tipping her head to the ceiling, snaking her body to the thrum of Jules's bass line.
_This is it._
_This is what it feels like._
To be alive.
Dia opened her eyes and looked out on the crowd, as far as the lights let her see. Opened her mouth wide enough to swallow the sun and sang.
"'Hush, little honey, I know you heard about the fight last night / the mess your mama don't like—'"
She turned up to the balconies, slapped her hands on the strings, a burst of perfect sound. "'Wait till I get home, I'll show you where to go—'"
Dia felt alone and surrounded all at once in that moment, as she sweated through her shirt and smeared the makeup she'd so carefully painted on earlier.
And though the crowd was giving her their everything, so much energy, Dia barely needed it. All she needed, she had, up onstage with her.
"'Everything I ever wanted,'" Dia sang, the truest words. "'You don't own what I give, when it breaks you, give it all to me, baby pretty baby—'"
And then it was over. The lights, the purple and red and white strobing lights, flashed before cutting out, plunging the whole place into blackness.
And cheers, stomping feet, roaring applause.
Dia bent double, breathless, spent, and then Jules was grabbing her hand, dragging her offstage, and the thought, again: _This is it._
_This is what it feels like._
They crashed into each other offstage, sticky hugs, breathless laughs. "Holy shit," Hanna kept saying, over and over with her hands spinning through the air.
"Holy oh my god," Dia said. "Was that really us?"
"It was the most us possible," Jules said, lifting her hair off the back of her neck and fanning her face, laughing. "I want to do it again."
"I want to do that every night," Dia said, throwing her arms to the ceiling and rising up on her toes. Her skin felt raw, too tight for her body, every movement testing her limits, every rub of her clothes burning. And this whole place felt too small to contain her, pushing back against her, and how dare it, how dare it try to put a limit on what she was feeling right now. "I need water. You coming?"
But then the guy with the amazing eyeliner was there, directing them away from the stage and talking a mile a minute into his headset. "No, they can't come in without their passes. I don't care who they say they are, we have security for a reason. If they're Glory Alabama people—wait, how many lights are out? Call Karla down, she'll take care of it—"
Dia slipped away before they got to the dressing room. "I'll be back," she called in Jules's direction, and then she was half running away. "I'm coming back!"
She headed through the familiar maze of darkened back hallways and out into the empty lobby, pushed through the doors to the main room, the other side of the stage where people now were yelling and clapping for Violet Ocean as they came onstage to close out the night.
The crowd was looser right at the back, where Dia was, and it wasn't so hard for her to work her way over to the bar that ran along one side of the room. "Hey!" she yelled when she got there, leaning her elbows in something sticky as she signaled the bartender. "Water?"
"Dia!"
She turned at the sound of her name. Wasn't expecting to see Jesse standing down the bar from her, saying her name again, but there he was.
"What are you doing?" he said, raising his voice over the band and closing the three feet between them. "Shouldn't you be—"
Dia cut him off, yelling at him. "You came!"
"What?"
She couldn't tell if it was the high from that performance, or being tired of the pretending, or pure adrenaline that made her do it. Made her grab his arm and pull him toward the door so fast that he spilled his drink, through the lobby and into a corridor lined with posters. "You came," she repeated, the door swinging shut, leaving them in this private space. Muted noise from the stage, dimmed red lights over the doors. "That's what I said."
Jesse was wiping his spilled drink from his black shirt, but he looked up at Dia when she said that and kind of shook his head, and when he spoke he sounded so frustrated. "Like I wouldn't, Dee."
She allowed herself a long look at him, all of him, his broad shoulders and warm brown skin and his pretty eyes all confused.
"Dia," Jesse said. "Can we—"
"You want the truth?" she said suddenly, her voice raw. "Okay. It's not about Lex or the band or college or whatever of the thousand excuses I've given you. It's the way you look at me, Jesse, and no one else ever looks at me like that, and what if I lose that? What if I lose you?" She shook her head. "And I don't mean what if we don't work out, or we get together and then hate each other in six months, or anything like that. I mean what if I lose _you._ " She held her hands out, palms turned up like some kind of prayer. "The last boy I was with died, Jesse. He _died_ , and I never got to know what we might have become, and more than that, who _he_ might have become. I am so scared that if you and me happen, something terrible will happen to you and then you'll never get to become the person you're supposed to be. And I'll be left here wondering what's so wrong with me that every boy I touch dies."
Jesse looked at her. "That's what you think? Dia. Nothing's going to happen to me."
"You don't know that," she said. "You cannot _know_ that. And I _can_ know that this fear is not reality but it is so real to me." She closed her eyes and tipped her face to the ceiling. "I had this dream, this nightmare, that you died, and it brought back so much of what happened to Elliot and—it didn't matter that it wasn't real. Because it was real in the moment I woke up, the moment that I _knew_ that you were both dead. And then I remembered that you're not, that _you_ are still here, and in that second I thought I would do anything to keep you here. So, if that meant us not being together—I had to do that."
"Why are you only telling me this _now_?" Jesse said, his confusion evident in his voice and his posture and everything about him. "Why didn't you say any of this before?"
"Because I was scared," she said, her blood rushing. "But I'm tired of being scared, and I'm tired of not telling you the truth, and I love you, Jesse, _I love you_."
She let the words hang there between them, heavy enough to almost be tangible, and Jesse looked stunned. "What?"
"I love you." Dia took a breath, took in the quiet all around them, opened her eyes and look at Jesse again. "Some days I see you and I feel like—oh my god, I love this boy, and I just want him to be okay, and what if these bad things come true and I never told you the truth and I never said sorry and I never said that I love you so much it doesn't feel real, sometimes. What would I do then?"
Jesse pulled in a ragged breath, took a step back from her. "You'd do fine," he said. "Exactly like you are now. Without Elliot."
"Elliot is not you," Dia said. "And I wasn't fine for a long time. I'm clearly not fine now, unless my brain is trying to tell me something else with this death dream." She touched her hands to her face, ignoring the sticky sweat under her arms. "But tonight I feel like I can do anything. So I'm telling you, before I get scared again and let you go."
He looked at her and she couldn't tell whether he was believing her or not, the way his mouth was set in such a straight line. "Dee."
"Jesse."
"You love me," he said. Statement, no question.
Dia answered him anyway. "Yes."
"Dee," he said.
"You came to my kid's birthday," she said, lowering her voice. "You came here. You are too good, Jesse Mackenzie."
"Dia, stop," he said, and her heart skipped a beat and her brain said, _This is it._
_This is where we end._
And if it was: at least she had been unafraid.
But then he closed the space between them again, the electric air keeping them apart, and she felt his touch on the back of her hand. He shook his head, looking down at her. "Dia Valentine, you are a pain in my ass."
And then his hands were holding her face and his mouth was on hers and it was the thousand times they'd never kissed in one moment. It was the hours spent watching him ride and the late-night texts and the things she'd told only him.
It was her hands on his hips and his tongue in her mouth and her heart at a thousand beats per minute. The dim recognition of her back hitting the wall, his body pressing her into it, and a noise of disappointment when he stopped kissing her, a noise that became a grateful sigh as his lips touched her throat.
And it was a thousand hours in three minutes of the most delicious kissing Dia had ever experienced. She wanted to keep doing it, all night. But she had other things to do, other people to be with. So even though it physically pained her, she pushed him away. "I'm calling a time-out," she said, breathless. "I have to go back, I have to find Hanna and Jules and—I don't even know. And so this is time-out."
Jesse slid one hand from her waist to the small of her back and used the other to catch her chin. "I didn't get to say," he said. "You were fucking amazing up there."
Dia resisted the urge to kiss him again and instead ducked out of his embrace, laughing as she twisted away and began walking down the hall. "I know," she said. "We were _fucking_ amazing."
## Jules
**Jules watched Dia run** back into the dressing room and grab Hanna's hand. "Let's go dance," Dia said. "It's Violet Ocean. Remember when we used to come see them here?"
"Can't forget," Jules said. "Let's go."
They left their instruments and wound their way out of backstage and into the crowded main room. They fought their way through until they found Ciara and Molly, and screamed and shook and made themselves a place in the heart of the crowd. Dia's lipstick was smeared and her eyes wild, and she spilled a plastic cup of water over Jules as she pulled Dia in to jump with her, but Jules didn't care, only laughed and danced harder.
And then Jules found Autumn, in one of her pastel dresses with her rainbow hair, and kissed her like she'd never meant anything so much in her life. She brought her into their circle and they danced as the Violet Ocean bassist climbed up on a riser, caught up in the heat of everything, wrapped up in momentum and movement and sheer exhilaration. And it wasn't the moment she'd ached for; it was a thousand different, infinitely better and sweeter moments, every single second of it.
So they danced more, sang along with the crowd, became messes of sweat and pounding feet and raw throats.
And then the band finished their encore, left the stage to be replaced by the MC, and holy shit there was Astrid Parker, the lead singer of Glory Alabama, standing and breathing and _existing_ in the same room as Jules. The crowd died down a little, and Jules felt a squeeze of her hand, looked to see Hanna standing next to her and staring at the stage. "All right, you've been patient," the MC was saying, a teasing lilt to her voice. "Very patient! But now, finally, it's time to announce the winner of this year's Sun City Radio Originals contest. The winner of _fifteen thousand dollars_!"
Cheers, a wave of applause as the MC handed over the mic to the other woman. "Hi!" she said, moving to center stage and laughing as the crowd erupted again. Jules let out a yell, too; she couldn't stop herself.
When the noise died down a little she spoke again. "I'm Astrid Parker"—more yells—"and on behalf of all of Glory Alabama, I want to say how impressed we are with the talent we've seen and how glad we are that we decided to do this. You have blown us away and we're so stoked to know that Golden Grove is still producing amazing musicians. So—"
"Get to the point," Dia's whisper came, and Jules shook her head. She was both nervous and not, because if they lost, nothing would change, except everything that they had changed for themselves already.
But if they won—
"I can't take it," Jules said, and her shoulders bumped up against Dia on one side, Hanna on the other. "This is too much."
She rose up on her toes, and the crowd was getting loud again, impatient, hungry. "Okay, okay," Astrid said, laughing, and the rest of Glory Alabama joined her onstage, waving at the audience. "We'll put you out of your misery."
Dia reached for her hand, and Jules reached for Hanna, and the three of them took the same breath.
And Jules looked at her friends, new from old, and the girl she had fallen in love with, and the world around them that she had longed so hard to know again, and she exhaled.
## Epilogue
**It's sunny when Dia** gets out of class.
She slips her biology textbook into her bag and makes her way across the courtyard, over to the parking lot. There's a party she's invited to tonight, but Dia has gone to exactly two parties in the entire first year of college, and both of them were a bust. Besides, she has other places to be, better things to do.
Jules is already there, leaning against the driver side of the dark-red Jetta that she calls her baby. "What took you so long?" she says, holding her hands out. "We're on a tight schedule."
"It was my last final," Dia says. "What was I supposed to do, get up and leave?"
Jules opens the car. "Get in," she says. "Come on!"
They drive across town, Jhené Aiko playing out of the slightly crackly car speakers. Dia rolls the window down and lets her hand float in the wind as they pass through the Nice Side of Town. She checks her phone: **eta? also do you want the red cups or blue?** the first message says, and the second: **also the doctor took my cast off finally so want to come watch me ride tomorrow?**
"Jesus Christ," Dia says under her breath. He's trying to irritate her on purpose, she knows, but he's so very good at it.
It's okay. She's spent almost a year with him. He gets hurt, he gets better. She helps him change bandages or clean cuts, sees the way his body heals. And she doesn't get those bad dreams anymore.
"Jesse?" Jules swings a left.
"He wants to know, do we want red cups or blue?"
"Red," Jules says. "Definitely."
Dia replies to him and then puts her phone away, propping her feet up on the dash. "So Ciara's in charge of food, and Jesse's getting the other stuff, and Hanna's setting up at their place already. What else?"
"Once we get the cake, we're set," Jules says.
"Aren't you forgetting something?"
"What?"
"My child?"
"Oh, shit." Jules makes it through a yellow light and glances at Dia. "Okay, I did not _forget_ , it momentarily slipped my mind."
Dia laughs. "Relax," she says. "Let's go get her."
At day care Lex comes running out, her hair bouncing off her shoulders. "Hi, Mommy!"
"Hi, Lala!" Dia holds her hand out for her daughter to take. "Ready for the party?"
"Is there gonna be pizza?"
"Oh, yeah," Dia says as the cross the parking lot. "So much pizza. More pizza than you can imagine."
"Nuh-uh," Lex says, her eyes wide. "Really?"
"Really really." Dia clips her into the car seat and then gets in the front. "All done."
They go to the Flour Shop to pick up Autumn's birthday cake, and then they drive over to Hanna and Ciara's new place—or, not so new; Dia keeps calling it that, even though they moved in eight months ago. Hanna's out in the yard, setting out blankets and chairs and at the end, a makeshift stage formed from cheap pallets. "Me and Molly worked all day on that," she says, proud, as Lex jumps on top of it. "Tell me how good I am."
"Very good," Jules says. "Okay, so, I'm going to get Autumn and we'll, like, waste some time while people get here and then I'll text you when we're coming back."
"Waste some time?" Dia says, raising her eyebrows. "Jules, you know you can say the actual words, right? We don't need euphemisms."
Jules sticks her middle finger up. "I'm going now."
Dia helps Hanna with the rest of the setting up, and when Ciara comes back they help her unload the car. Dia gives Lex a giant lollipop to occupy her for a little while.
Jesse turns up with the cups and everything else that was on the list Jules gave him. "Hi," Dia says. "Can you do me a favor?"
"Another one?" Jesse says. "What?"
She touches her lips. "Kiss me right here."
"Oh, that I can do," he says, and he does, and Dia thinks she'll never get tired of him.
When it gets a little later and everything's ready, people start arriving. Not tons of people—Autumn's friends from school, a couple of Ciara's new bandmates, a select number that Jesse rolls around at the skate park with. It was supposed to be a surprise party, except for when Autumn's mom told Jules that Autumn _hates_ surprise parties, and so now it's just Autumn's birthday.
When Autumn walks in they all yell, "Happy birthday!" and she blushes.
"Thanks," she says. "This is too much!"
Ciara's hooked up speakers to pipe music into the yard, and Dia sits on the grass with Lex. Summer break's about to start and she's going to be busy busy busy, still working at the bakery, and with the band, and the internship she got at a tiny local indie label, thanks to Astrid Parker.
After they'd won, and then later on, after they'd played the most amazing show of their lives opening up for Glory Alabama, Astrid had given Dia her email. "Frontwoman to frontwoman," she'd said to a dazed Dia. "If I can help you with anything, let me know. You remind me of me."
All year long they've been emailing, Astrid answering all of Dia's questions about the business, production, marketing, what it really takes to succeed. She even helped Dia get the internship—because as much as Dia still loves playing, now she's thinking about what kind of career she might be able to make out of all of it, too.
Once Lex has eaten her body weight in pizza and cake and starts falling asleep, Dia calls her parents.
They meet her out front and her dad takes Lex, puts her in the back of the car. "I won't be super late," Dia says to her mom.
"It's fine," Nina says. "You officially finished freshman year today, it's Autumn's birthday, you have fun! We'll take care of the baby."
"She's not a baby," Max says, and he kisses Dia on the top of her head. "Listen to your mama. Have fun."
She blows a kiss as they drive away, and walks back into the yard.
The music gets turned down, and the three of them get up on the makeshift stage, and Dia arranges her dress so she's not flashing everybody while she plays. "Hi," Jules says. "So, Autumn wanted us to play for her birthday, and I do what she tells me to, so here we are." That makes people laugh, and Dia shakes her head, the curls she's pulled around her face fluttering in the breeze. "We're going to do some songs you'll hear on our new EP, if you feel like getting it."
"Come see us next week, too!" Dia says, fishing a pick from her pocket. "You won't regret it."
Hanna cracks her fingers, sitting behind the keyboard they got a few months ago and have been experimenting with ever since. "Ready?"
Jules turns so she can see both of them and nods. "Ready."
Dia looks at their friends, their family, the hungry hearts waiting for them to begin. Her lips are dry and her fingers ache already, in the best way. She closes her eyes for only a second, watches the lights behind her eyelids strobe, and she can almost smell the sticky heat, the sweat of being onstage.
She opens her eyes. Jesse gives her that smile, the one she thinks will never stop breaking her heart, and Ciara holds up her phone, ready to record them. The lights strung up across the fence, around the two spindly trees Hanna so proudly looks after, light up everything in beauty, and Dia fixes her fingers on the fretboard of her new acoustic, a richer sound than she's been used to. It's worth getting used to.
She smiles. "Ready."
## Acknowledgments
Everyone says the second book is the hardest, but for me it's also the most proud I've ever been of anything. I loved writing this book, and I hope it finds a way into people's hearts.
Thank you to my editor extraordinaire, Elizabeth Lynch. You let me write my messy girls and make them a thousand times better.
Thank you to Suzie Townsend for taking me on and supporting TIWIFL so much.
Thank you to Jennifer Johnson-Blalock for everything.
Thank you to everyone at HarperTeen who made this a real book: Renée Cafiero, Claire Caterer, Bess Braswell, Audrey Diestelkamp, Gina Rizzo, Vanessa Nuttry, and especially Michelle Taormina for such a glorious cover.
Thank you to my all my friends: To my UK babes for all the whatsapp procrastination and deep dives on Kristina's balcony. Thanks especially to Ali Standish and Carlie Sorosiak. To everyone in AMM for several thousand hours of slack chats and making me laugh a ridiculous amount. To all my friends in the 17s, especially Laurie Devore and Kiersi Burkhart. To Rachel Lynn Solomon—thank you for letting me cry in your direction so much. Thank you to Courtney Summers for all your support. And to everyone I couldn't possibly name who has shared their excitement for this book somewhere—it honestly keeps me going.
Thank you to my sensitivity readers, including Elizabeth Roderick, for giving your time and expertise to help me make this book the best it could be. Any errors in authenticity are mine and mine alone.
Thank you to Pretty Little Liars for keeping me halfway sane during the winter of 2016 while I wrote and rewrote this book. Mona forever.
Thank you to Christina Aguilera, Sufjan Stevens, Kacey Musgraves, Beyoncé, JoJo, and many, many more for keeping me company for so many years.
Thank you to my family for supporting me always.
Thank you to all the women who've picked up a guitar and a mic.
## About the Author
Courtesy of Rebecca Barrow
**REBECCA BARROW** writes stories about girls and all the wonders they can be. A lipstick obsessive with the ability to quote the entirety of _Mean Girls_ , she lives in England, where it rains a considerable amount more than in the fictional worlds of her characters. She collects tattoos, cats, and more books than she could ever possibly read. You can visit her online at www.rebecca-barrow.com.
Discover great authors, exclusive offers, and more at hc.com.
## Books by Rebecca Barrow
_You Don't Know Me but I Know You_
_This Is What It Feels Like_
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## Copyright
HarperTeen is an imprint of HarperCollins Publishers.
THIS IS WHAT IT FEELS LIKE. Copyright © 2018 by Rebecca Barrow. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.
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_Front panel photograph by Pete Thompson / Gallery Stock_
_Cover design by Michelle Taormina_
* * *
Digital Edition NOVEMBER 2018 ISBN: 978-0-06-249424-5
Print ISBN: 978-0-06-249423-8
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18 19 20 21 22 CG/LSCH 10 9 8 7 6 5 4 3 2 1
FIRST EDITION
## About the Publisher
**Australia**
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Sydney, NSW 2000, Australia
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## Contents
1. _Cover_
2. _Title Page_
3. _Dedication_
4. _Contents_
5. Prologue
6. Hanna
7. Dia
8. Jules
9. Elliot
10. Hanna
11. Jules
12. Dia
13. Dia
14. Hanna
15. Jules
16. Hanna
17. Jules
18. Hanna
19. Dia
20. Elliot
21. Jules
22. Hanna
23. Dia
24. Hanna
25. Dia
26. Dia
27. Jules
28. Jules
29. Dia
30. Dia
31. Hanna
32. Hanna
33. Elliot
34. Jules
35. Hanna
36. Jules
37. Dia
38. Jules
39. Dia
40. Hanna
41. Jules
42. Dia
43. Hanna
44. Hanna
45. Hanna
46. Elliot
47. Jules
48. Elliot
49. Hanna
50. Dia
51. Hanna
52. Dia
53. Hanna
54. Elliot
55. Jules
56. Jules
57. Dia
58. Hanna
59. Jules
60. Hanna
61. Dia
62. Elliot
63. Hanna
64. Dia
65. Jules
66. Epilogue
67. _Acknowledgments_
68. _About the Author_
69. _Books by Rebecca Barrow_
70. _Back Ad_
71. _Copyright_
72. _About the Publisher_
# Guide
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2. Contents
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{
"redpajama_set_name": "RedPajamaBook"
}
| 7,341
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\section{Introduction}
\label{intro}
Coalescing binary neutron stars are among the most promising sources
of gravitational waves (GWs) for the next-generation, kilometer-size,
GW detectors such as advanced LIGO~\cite{LIGO10},
advanced Virgo~\cite{VIRGO11}, and KAGRA~\cite{KAGRA10}.
Binary neutron stars, together with black-hole$-$neutron-star binaries,
are also regarded as one of the candidate central engines of
short-hard gamma-ray bursts~\cite{NarayPP92}.
The use of matched-filtering technique to detect GW
signals from coalescing binary systems and the interest in shedding
light on gamma-ray burst progenitors have led to impressive advances
in modeling the dynamics and gravitational waveforms of binary
neutron stars (see, e.g., Ref.~\cite{Blanchet06} for the inspiral phase and
Refs.~\cite{HotokKOSK11,HotokKKMSST13,BernuDWB13} for the merger
and postmerger phases).
Most of those studies were carried out in general relativity,
except for Refs.~\cite{BarauPPL13,ShibaTOB14,PalenBPL14}.
Although general relativity has passed all known experimental
and observational tests in the weak-field and slow-motion
limit (see e.g., Ref.~\cite{Will05}), it remains to be seen whether it will
survive tests in the strong-field and high-velocity regime.
The detection of GWs emitted by coalescing binary
systems offers the unique opportunity to investigate the validity
of general relativity in the strong-field regime.
To achieve this goal, accurate gravitational waveforms in gravity theories
alternative to general relativity~\cite{Will93,Will05,WillZ89} need
to be computed.
Here we follow our recent work~\cite{ShibaTOB14} and
focus on the scalar-tensor model~\cite{Jordan49,Fierz56,BransD61,FujiiM03}
proposed by Damour and Esposito-Far\`ese (DEF)~\cite{DamourEF93}
(see also Refs.~\cite{DamourEF92,DamourEF96b,DamourEF98}).
Quite interestingly, there exist choices of the free parameters
in the DEF model, for which both weak and mildly strong gravitational
tests are satisfied, notably the pulsar timing tests,
but strong-field tests could be violated and these violations could be
observed through the emission of GWs from the last stages of the binary's
inspiral, plunge, and merger in advanced LIGO, Virgo, and KAGRA detectors.
This is possible because if neutron stars in binary systems carry
negligible scalar charge when largely separated,
they can be dynamically scalarized as they come closer to each other
through gravitational interaction, i.e., they undergo dynamical scalarization
as the binary's compactness increases~\cite{BarauPPL13,ShibaTOB14,PalenBPL14}.
[See also Refs.~\cite{DonevaYSK13,PaniB14,DonevaYSKA14} for scalarization
of rotating stars.]
However, it is important to notice that for the same values of the DEF
parameters for which dynamical scalarization can occur, the DEF model
may have problems in providing cosmological solutions consistent with
our Universe~\cite{DamourN93a,DamourN93b,Sampetal14}. It will be relevant
to further investigate this problem in the future.
Barausse {\it et al.}~\cite{BarauPPL13} showed that dynamical scalarization
takes place in the DEF model by performing numerical-relativity
simulations of inspiraling binary neutron stars. They performed two
numerical simulations, which differed by the strength of the scalar field
and the binary's mass ratio. Their simulations used approximate initial data,
i.e., initial data computed by numerical codes of general relativity instead of
the ones of scalar-tensor theory, and employed the polytropic
equation of state $p/c^2 =K \rho_0^{\Gamma}$ with $\Gamma=2$ and
$K=123 G^3 M_{\odot}^2/c^6$, where $p$ is the pressure and $\rho_0$
is the baryonic rest-mass density in their notation.
For the unequal-mass binary the individual baryonic rest masses were
$1.78 M_{\odot}$ and $1.90 M_{\odot}$,
while for the equal-mass binary the baryonic rest mass
was $1.625 M_{\odot}$.\footnote{We notice that due to a misleading output
in the \textsc{LORENE} data set~\cite{lorene}, the gravitational masses
are not the ones reported in Ref.~\cite{BarauPPL13}.
The gravitational masses of spherical, isolated stars corresponding
to the baryonic rest masses of $1.625 M_{\odot}$, $1.78 M_{\odot}$, and
$1.90 M_{\odot}$ are $1.51 M_{\odot}$, $1.64 M_{\odot}$, and $1.74 M_{\odot}$,
respectively. The results of Ref.~\cite{BarauPPL13} are correct; i.e.,
they were not affected by the gravitational masses reported in the paper.}
For comparison, we computed not in general relativity but in the DEF
scalar-tensor theory the initial data for the same baryonic rest masses
used in Barausse {\it et al.}.
We set $\beta/(4 \pi G)=-4.5$ and $\varphi_{0,\rm{BPPL}} = 10^{-5} G^{-1/2}$,
where $\beta$ is a constant related to the derivative of a
scalar field and $\varphi_{0,\rm{BPPL}}$ is the asymptotic value of
the scalar field as defined by Barausse {\it et al.}~\cite{BarauPPL13}.
We found that the more massive star (with baryonic rest mass of $1.90 M_{\odot}$)
is spontaneously scalarized for a spherical configuration, and thus,
the unequal-mass binary system is already scalarized at the orbital separation
of $60$ km, which is where Ref.~\cite{BarauPPL13} starts their simulations
(see Appendix~\ref{app1} for more details).
Because in their simulation the scalar field does not exist initially,
it rapidly increases just after the simulation starts.
This artificial behavior may have left imprints in the dynamical evolution
of the binary system. Indeed, as we will see in Sec.~\ref{sec3}, the binding
energy computed along a sequence of quasiequilibrium configurations
in the DEF scalar-tensor theory is in absolute value smaller than
in general relativity. Thus, as soon as the simulation starts,
if initial data without the scalar field are used, which is the case
in Barausse {\it et al.}, then the absolute value of the binding
energy will become smaller because the scalar field increases.
This means that the initial datum is a local minimum of the binding energy
along the quasiequilibrium sequences; i.e., it is a local turning
point of the binding energy. We suspect that the fast plunge seen
in Ref.~\cite{BarauPPL13} might be enhanced by this effect. Performing
a numerical simulation using general-relativity and scalar-tensor-theory
initial data will clarify this point.
For the equal-mass binary system numerically evolved in Ref.~\cite{BarauPPL13}
we found, using quasiequilibrium configurations, that the binary is
already dynamically scalarized at the orbital separation
of $40$ km where Ref.~\cite{BarauPPL13} observed dynamical scalarization.
However, our result of an earlier dynamical scalarization may not be
in contradiction with Ref.~\cite{BarauPPL13}.
Indeed, typically we found that the onset of dynamical scalarization
in the quasiequilibrium-configuration study occurs earlier than that
in dynamical simulation~\cite{ShibaTOB14}. This discrepancy may occur
due to the breakdown of the assumption of quasiequilibrium. For a few
orbits before merger the infall velocity of each star in the binary system is
larger in numerical-relativity simulations than in quasiequilibrium
configurations. As a consequence, in numerical-relativity simulations
the binary can merge before the scalar field reaches its equilibrium state,
while, in the quasiequilibrium situation, the scalar field can reach
its equilibrium state even just before the quasiequilibrium sequence ends.
Thus, the effect of the scalar field is overestimated in the quasiequilibrium
study for the cases in which dynamical scalarization occurs just before the
end of the quasiequilibrium sequences. We will present more details about
the results of the equal-mass binary in Appendix~\ref{app1}.
More recently, Palenzuela {\it et al.}~\cite{PalenBPL14} investigated
analytically the phenomenon of dynamical scalarization in the DEF model.
They employed the 2.5 post-Newtonian (PN) equations of motion, recently
derived in Ref.~\cite{MirshW13}, augmented by a set of equations that
phenomenologically describe the increase of scalar charge as the two
neutron stars come closer to each other. In this analysis the binary neutron
stars are approximated by two isolated, spherical neutron stars.
Reference~\cite{PalenBPL14} confirmed and quantified what was found
in Refs.~\cite{BarauPPL13,ShibaTOB14}, notably the fact that binary neutron
stars plunge and merge in the DEF model sooner than in general relativity
when they undergo induced and dynamical scalarization.
To further understand the onset of dynamical scalarization
during the last stages of inspiral, we disentangle conservative from
radiative effects and compute for the first time quasiequilibrium sequences
of binary neutron stars in the DEF model. Our motivations are threefold.
We want to (i) produce initial data for merger simulations in the
scalar-tensor model~\cite{ShibaTOB14},
(ii) accurately extract physical quantities (notably the binding energy
and angular momentum) during the last stages of inspiral, just before merger,
where the effect of gravity becomes strong and the finite-size effect of
a neutron star starts to affect the evolution of the binary system,
and (iii) use those quantities to estimate by how much
the gravitational waveforms in the DEF model
differ from the ones in general relativity.
This paper is organized as follows.
In Sec.~II, we give a brief summary of the quasiequilibrium-sequence
formalism. The formulation is based on the conformal thin-sandwich
decomposition. In Sec.~III we present and discuss the numerical results of
the scalar charge and scalar mass, binding energy, total angular momentum,
central baryonic rest-mass density, evolution of the orbital angular frequency,
and number of GW cycles. Section IV summarizes
our main conclusions.
Throughout this paper, we employ the geometrical units of
$c=G=1$, where $c$ is the speed of light and $G$ is the bare gravitational
constant. We use greek letters to denote spacetime components and latin
letters for the spatial components.
\section{Formulation} \label{sec:form}
As in Ref.~\cite{ShibaTOB14}, we work in the Jordan
frame~\cite{Jordan49,BransD61}.
The basic field equations for computing the metric quantities and
the scalar field are derived by taking variation of the action,
\begin{eqnarray}
{\cal S} &=&\frac{1}{16\pi}
\int \Bigl[ \phi R -\omega (\phi) \phi^{-1} g^{\mu \nu}
(\nabla_{\mu} \phi) (\nabla_{\nu} \phi) \Bigr] \sqrt{-g} d^4 x
\nonumber \\
&&+{\cal S}_{\rm matter},
\label{action}
\end{eqnarray}
where $\phi$ is the scalar field, $g_{\mu \nu}$ is the spacetime metric
in the Jordan frame, $R$ is the Ricci scalar calculated
from $g_{\mu \nu}$, $g$ is the determinant of $g_{\mu \nu}$,
$\nabla_{\mu}$ is the covariant derivative with respect to $g_{\mu \nu}$,
and ${\cal S}_{\rm matter}$ is the matter part of the action.
The quantity $\omega$ is a function of $\phi$ that takes the form
\begin{eqnarray}
\frac{1}{\omega(\phi) +3/2} =B \ln \phi
\end{eqnarray}
in the DEF theory, where $B$ is a free parameter~\cite{DamourEF93}
(see Ref.~\cite{ShibaTOB14} for more details).
The relation between Newton's constant $G_{\rm N}$ and
the bare gravitational constant $G$ is
\begin{equation}
G_{\rm N} = \frac{G}{\phi_0} \frac{4 +2 \omega(\phi_0)}{3 +2\omega(\phi_0)},
\end{equation}
where $\phi_0$ is the value of $\phi$ at spatial infinity.
For the values used in this paper, the deviation of the ratio $G_{\rm N}/G$
from unity is on the order of $10^{-9} - 10^{-10}$
[see Eq.~(\ref{eq:varphi_phi}) and the scalar-tensor
values listed at the end of Sec.~\ref{sec:form}].
Taking variation of the action (\ref{action}) with respect to the metric
and the scalar field, we obtain
\begin{eqnarray}
R_{\mu \nu} -\frac{1}{2} R g_{\mu \nu} &=&8 \pi \phi^{-1} T_{\mu \nu}
+\omega \phi^{-2} \Bigl[ (\nabla_{\mu} \phi) (\nabla_{\mu} \phi) \Bigr.
\nonumber \\
&&\Bigl. -\frac{1}{2} g_{\mu \nu} (\nabla_{\alpha} \phi)
(\nabla^{\alpha} \phi) \Bigr] \nonumber \\
&&+\phi^{-1} \bigl( \nabla_{\mu} \nabla_{\nu} \phi -g_{\mu \nu} \Box \phi \bigr)
\label{eq:metric}
\end{eqnarray}
and
\begin{equation}
\Box \phi =\frac{1}{2 \omega +3} \Bigl[ 8\pi T -\frac{d \omega}{d \phi}
(\nabla_{\mu} \phi) (\nabla^{\mu} \phi) \Bigr], \label{eq:scalar}
\end{equation}
respectively, where $R_{\mu \nu}$ is the Ricci tensor, $\Box$ is
$\nabla_{\mu} \nabla^{\mu}$, and $T_{\mu \nu}$ is the stress-energy tensor.
For an ideal fluid we have
\begin{equation}
T_{\mu \nu} = (\rho +\rho \epsilon +P) u_{\mu} u_{\nu} +P g_{\mu \nu},
\end{equation}
where $u_{\mu}$ is the fluid 4-velocity, $\rho$ is the baryonic rest-mass
density, $\epsilon$ is the specific internal energy,
and $P$ is the pressure. Then, we set the metric line element in 3+1 form,
\begin{eqnarray}
ds^2 &=& g_{\mu \nu} dx^{\mu} dx^{\nu}, \nonumber \\
&=& -\alpha^2 dt^2 +\gamma_{ij} (dx^i +\beta^i dt)(dx^j +\beta^j dt)
\end{eqnarray}
where $\alpha$ is the lapse function, $\beta^i$ is the shift vector,
and $\gamma_{ij}$ is the spatial part of the spacetime metric, and
we solve the basic field equations in the conformal thin-sandwich
decomposition~\cite{York99,PfeifY03}. We decompose the equations for
the metric quantities (\ref{eq:metric}) into the {\it Hamiltonian} constraint,
\begin{eqnarray}
{}^{(3)}R +K^2 -K_{ij} k^{ij} &=&16 \pi \phi^{-1} \rho_{\rm h} \nonumber \\
&&+\omega \phi^{-2} \bigl[ \Pi^2 +(D_{\alpha} \phi)(D^{\alpha} \phi) \bigr]
\nonumber \\
&&+2 \phi^{-1} (D_{\mu} D^{\mu} \phi -K \Pi),
\end{eqnarray}
and the {\it momentum} constraint,
\begin{eqnarray}
D_i K^i_j -D_j K &=&8 \pi \phi^{-1} J_j +\omega \phi^{-2} \Pi D_j \phi
\nonumber \\
&&+\phi^{-1} (D_j \Pi -K_j^i D_i \phi);
\end{eqnarray}
furthermore the trace part of the evolution equation for the extrinsic
curvature $K_{ij}$ satisfies the following equation
\begin{eqnarray}
&&(\partial_t -\beta^k \partial_k) K =
4 \pi \alpha \phi^{-1} (\rho_{\rm h} +S) +\alpha K_{ij} K^{ij}
-D_i D^i \alpha \nonumber \\
&&\hspace{25pt}+\alpha \omega \phi^{-2} \Pi^2 +\alpha \phi^{-1} D_i D^i \phi
-\alpha \phi^{-1} K \Pi \nonumber \\
&&\hspace{25pt}-\frac{3\alpha \phi^{-1}}{2(2\omega +3)} \Bigl[ 8\pi T
+\frac{d\omega}{d\phi} \Bigl\{ \Pi^2 -(D_k \phi) (D^k \phi) \Bigr\} \Bigr],
\nonumber \\
\end{eqnarray}
while the evolution equation for the spatial metric reads
\begin{eqnarray}
\partial_t \gamma_{ij} =-2\alpha K_{ij} +\gamma_{kj} D_i \beta^k
+\gamma_{ik} D_j \beta^k,
\end{eqnarray}
where ${}^{(3)}R$ denotes the Ricci scalar calculated from $\gamma_{ij}$,
$D_i$ the covariant derivative with respect to $\gamma_{ij}$,
$K$ the trace part of the extrinsic curvature, and
$\Pi \equiv -n^{\mu} \partial_{\mu} \phi$.
Here the quantities $\rho_{\rm h}$, $J_i$, $S$, and $T$ are defined as
\begin{subequations}
\begin{eqnarray}
\rho_{\rm h} &=& n_{\mu} n_{\nu} T^{\mu \nu}, \\
J_i &=& -n_{\mu} \gamma_{\nu i} T^{\mu \nu}, \\
S &=& \gamma^{ij} (\gamma_{i \mu} \gamma_{j \nu} T^{\mu \nu}), \\
T &=& g_{\mu \nu} T^{\mu \nu},
\end{eqnarray}
\end{subequations}
where $n^{\mu}$ is the unit normal to the spatial hypersurface.
In the above decomposition, there appear four freely specified quantities:
the background spatial metric, $\tilde{\gamma}_{ij}$,
the time derivative of the background spatial metric in
contravariant form, $\partial_t \tilde{\gamma}^{ij}$,
the trace part of the extrinsic curvature, $K$, and
its time derivative, $\partial_t K$.
The background spatial metric is defined by
$\tilde{\gamma}_{ij} \equiv \psi^{-4} \gamma_{ij}$
where $\psi$ is the conformal factor.
Since we consider a stationary state, we set to zero the time derivatives
of the above quantities.
We also set to zero the trace part of the extrinsic curvature, $K$,
because we impose the condition of maximal slicing.
We further require that the background spatial metric, $\tilde{\gamma}_{ij}$,
be flat; that is, $\tilde{\gamma}_{ij} =f_{ij}$ where
$f_{ij}$ is the flat spatial metric~\cite{Isenb78,Isenb08,WilsoM89}.
The equation for the scalar field (\ref{eq:scalar}) is also rewritten
in the conformal thin-sandwich decomposition as
\begin{eqnarray}
&&(\partial_t -\beta^i \partial_i) \Pi =-\alpha D_i D^i \phi
-(D_i \alpha) (D^i \phi) +\alpha K \Pi \nonumber \\
&&\hspace{25pt}+\frac{\alpha}{2 \omega +3}
\Bigl[ 8 \pi T +\frac{d\omega}{d\phi} \bigl( \Pi^2
-(D_k \phi) (D^k \phi) \bigr) \Bigr].
\end{eqnarray}
The above equation depends on the quantities $\Pi$ and $\partial_t \Pi$.
Since we consider a stationary state, we set to zero $\partial_t \Pi$.
For the quantity $\Pi$, we need to guarantee that it behaves at least
as $\Pi ={\cal O} (r^{-2})$ in the far zone.
This is because the right-hand side of the Hamiltonian constraint and
the trace part of the evolution equation for $K_{ij}$
should decrease fast enough to ensure the spacetime to be
asymptotically flat. In this paper, for simplicity, we set to zero the
quantity $\Pi$ (see Sec.~II.D of Ref.~\cite{ShibaTOB14} for a more detailed
discussion).
Note that, as we mentioned above, we have the freedom of choosing another
background spatial metric, $\tilde{\gamma}_{ij}$, as well as
the quantity $\Pi$, and the trace part of the extrinsic curvature, $K$.
We think that the choice we made for the background spatial metric
does not affect the main results of this paper, notably the onset of
dynamical scalarization along quasiequilibrium binary neutron stars.
As we will see in Sec.~\ref{sec:scalar}, the location of dynamical
scalarization, i.e., the orbital angular frequency at the onset of
dynamical scalarization, agrees with what determined by fully relativistic
simulations and estimated by the analytical method discussed in
Ref.~\cite{ShibaTOB14}.
Because the simulations and the analytical estimation do not rely
on the assumptions used in this paper, our results for the
location of dynamical scalarization can be considered robust.
Thus, the equations for the quantities that enter in the metric can be
recast in the form
\begin{widetext}
\begin{subequations}
\begin{eqnarray}
\Delta \psi &=& -2\pi \exp( -\varphi^2/2 ) \psi^5 \rho_{\rm h}
-\frac{1}{8} \psi^{-7} \tilde{A}_{ij} \tilde{A}^{ij}
-\frac{1}{2} \pi B \psi^5 \varphi^2 T \exp( -\varphi^2/2 )
-\frac{1}{4} \psi \Bigl( 1 +\frac{1}{B} -\frac{3}{4} \varphi^2 \Bigr)
f^{ij} (\partial_i \varphi) (\partial_j \varphi) \nonumber \\
&& +\frac{1}{4} \Phi^{-1} \varphi f^{ij} (\psi \partial_i \Phi
-\Phi \partial_i \psi) (\partial_j \varphi), \label{eq:deltapsi} \\
\Delta \Phi &=& 2 \pi \exp( -\varphi^2/2 ) \Phi \psi^4 (\rho_{\rm h} +2S)
+\frac{7}{8} \Phi \psi^{-8} \tilde{A}_{ij} \tilde{A}^{ij}
-\frac{3}{2} \pi B \Phi \psi^4 \varphi^2 T \exp( -\varphi^2/2 )
-\frac{1}{4} \Phi \Bigl( 3 +\frac{1}{B} -\frac{3}{4} \varphi^2 \Bigr)
f^{ij} (\partial_i \varphi) (\partial_j \varphi) \nonumber \\
&& -\frac{3}{4} \psi^{-1} \varphi f^{ij} (\psi \partial_i \Phi
-\Phi \partial_i \psi) (\partial_j \varphi), \\
\Delta \beta^i &+&\frac{1}{3} f^{ij} \partial_j (\partial_k \beta^k)
= 16 \pi \exp( -\varphi^2/2 ) \Phi \psi^{-1} f^{ij} J_j -2\Phi \psi^{-7} \tilde{A}^{ij}
(7 \psi^{-1} \partial_j \psi -\Phi^{-1} \partial_j \Phi)
-2 \Phi \varphi \psi^{-7} \tilde{A}^{ij} \partial_j \varphi, \\
\tilde{A}^{ij} &=&\frac{\psi^7}{2\Phi} \Bigl( f^{kj} \partial_k \beta^i
+f^{ik} \partial_k \beta^j
+\frac{2}{3} f^{ij} \partial_k \beta^k \Bigr),
\end{eqnarray}
\end{subequations}
\end{widetext}
where $\tilde{A}^{ij}$ is the traceless part of the conformal
extrinsic curvature defined as
\begin{equation}
\tilde{A}^{ij} =\psi^{10} \Bigl(K^{ij} -\frac{1}{3} \gamma^{ij} K \Bigr),
\end{equation}
the quantity $\Phi$ is defined as $\Phi \equiv \alpha \psi$, and
we introduce a new scalar field $\varphi$ which is related to the
scalar field $\phi$ as
\begin{equation}
\varphi \equiv \sqrt{2 \ln \phi}. \label{eq:varphi_phi}
\end{equation}
The equation for the scalar field (\ref{eq:scalar}) can be rewritten
in the conformal thin-sandwich decomposition imposing
$K=0$, $\partial_t \Pi =0$, and $\Pi=0$, as
\begin{eqnarray}
\Delta \varphi &=& 2\pi B \psi^4 \varphi T \exp( -\varphi^2/2 )
-\varphi f^{ij} (\partial_i \varphi) (\partial_j \varphi) \nonumber \\
&&-f^{ij} (\Phi^{-1} \partial_i \Phi +\psi^{-1} \partial_i \psi)
(\partial_j \varphi).
\end{eqnarray}
(See Ref.~\cite{ShibaTOB14} for a more detailed derivation of the equations
for the quantities entering the metric and the scalar field.)
We also need to solve for the relativistic hydrostatic equations,
$\nabla_{\mu} T^{\mu \nu}=0$, which are basically the same as
in general relativity because they are written in the Jordan frame.
The equations are decomposed into the first integral of the Euler equation,
\begin{equation}
h \alpha \frac{\gamma}{\gamma_0} = \textrm{const},
\end{equation}
and the equation of continuity,
\begin{equation}
\frac{\rho}{h} \nabla_{\mu} \nabla^{\mu} \Psi
+(\nabla_{\mu} \Psi) \nabla^{\mu} \Bigl( \frac{\rho}{h} \Bigr) = 0,
\end{equation}
where $h =(\rho +\rho \epsilon +P)/ \rho$ is the fluid specific enthalpy,
$\gamma$ is the Lorentz factor between the fluid and co-orbiting observers,
and $\gamma_0$ is the Lorentz factor between the co-orbiting and Eulerian
observers. The quantity $\Psi$ is the fluid velocity potential, which
for an irrotational fluid and an Eulerian observer is related to
the fluid 3-velocity $U^i$ as
\begin{equation}
U^i = \frac{\psi^{-4}}{\alpha u^t h} \tilde{\gamma}^{ij} \partial_j \Psi,
\end{equation}
where $u^t$ is the time component of the fluid 4-velocity
(see Refs.~\cite{GourgGTMB01,TanigS10,Gourgoulhon12} for more details).
We model neutron stars with realistic equations of state (EOS),
using piecewise polytrope segments as introduced by
Read {\it et al.}~\cite{ReadLOF09,ReadMSUCF09}.
In particular, we set the number of polytrope segments to four and
choose the model of APR4 and H4 in Ref.~\cite{ReadLOF09}.
We remind readers that the APR4 EOS~\cite{AkmalPR98} is derived
by a variational method with modern nuclear potentials among neutrons,
protons, electrons, and muons.
The H4 EOS~\cite{LackeNO06} is derived by a relativistic mean-field
theory and includes effects of hyperons. We set the internal flow
in the neutron star to be irrotational as seen by an inertial
observer at infinity~\cite{Kocha92}.
There are two free parameters in the DEF model:
(i) $B$, which appears in $\omega$,
and (ii) $\varphi_0$ the value of $\varphi$ at spatial infinity.
As discussed in Ref.~\cite{ShibaTOB14}, we choose those values,
taking into account the observational constraints
from neutron-star$-$white-dwarf binaries~\cite{Freir12,Anton13,BhatBV08}.
We set $\varphi_0 =1 \times 10^{-5}$ and vary $B$ from 8.0 to 9.0
for APR4 EOS, while $\varphi_0=5 \times 10^{-5}$ and $B$
from 8.5 to 9.5 for H4 EOS.
\section{Numerical results} \label{sec3}
We use the spectral-method library \textsc{LORENE}, developed
by the numerical-relativity group at the Observatory of Meudon~\cite{lorene},
and construct a numerical code to compute quasiequilibrium configurations
of binary neutron stars in the DEF model. The code is similar to those
developed in Refs.~\cite{GourgGTMB01,TanigG02,TanigG03,TanigS10}
for binary neutron stars in general relativity, in particular,
in Ref.~\cite{TanigS10}.
We consider irrotational, equal-mass binary neutron stars
whose total mass is $m = 2.7M_{\odot}$ at infinite separation,
and we construct sequences fixing the baryonic rest mass of
the two neutron stars,
\begin{equation}
M_{\rm B}^{(A)} =\int_{\rm star~A} \rho u^t \sqrt{-g} d^3 x~~~(A=1~{\rm or}~2),
\label{eq:restmass}
\end{equation}
and varying the orbital separation. As found in Ref.~\cite{ShibaTOB14},
for both APR4 and H4 EOSs spontaneous scalarization does not occur
in each star whose mass in isolation is $1.35 M_{\odot}$. Thus,
the baryonic rest masses corresponding to the mass of
$1.35 M_{\odot}$ in the scalar-tensor case are $1.50 M_{\odot}$ for APR4
and $1.47 M_{\odot}$ for H4, respectively. Those values are, basically,
the same as those in general relativity.
Note that the total mass $m$ is the asymptotic value of
the tensor mass defined by $M_{\rm T}=M_{\rm ADM} +M_{\rm S}$~\cite{Lee74}
at infinite separation. Here $M_{\rm ADM}$ is the Arnowitt-Deser-Misner (ADM)
mass and $M_{\rm S}$ is the scalar mass.
The latter is defined as the monopole part of the scalar field, $\phi$,
which is expanded as $\phi = \phi_0 +2M_{\rm S}/r +{\cal O}(1/r^2)$
for $r \rightarrow \infty$ where $r$ is the radial coordinate.
In Appendix~\ref{app2}, we show a convergence test for the scalar mass,
varying the numerical resolution (i.e., the number of collocation points).
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=8.0cm]{fig1a.eps} \\
\vspace{0.4cm}
\includegraphics[width=8.0cm]{fig1b.eps}
\caption{Scalar charge (left $y$ axis) or scalar mass (right $y$ axis)
as a function of the orbital angular frequency normalized to the
tensor mass at infinite separation (lower $x$ axis)
or as a function of the frequency of GWs
defined by $f_{\rm GW} \equiv \Omega/\pi$ from a binary neutron
star with $m = 2.7 M_\odot$ (upper $x$ axis).
Upper panel (a) shows results for APR4 EOS.
Black solid curve with open circles, red with open squares,
green with open diamonds, and blue with open triangles are,
respectively, for the cases $B=9.0$, 8.7, 8.4, and 8.0.
Lower panel (b) shows results for H4 EOS.
Black solid curve with open circles, red with open squares,
and green with open diamonds are, respectively,
for the cases $B=9.5$, 9.0, and 8.5.
\label{fig1}}
\end{figure}
\subsection{Scalar charge and scalar mass} \label{sec:scalar}
In Fig.~\ref{fig1} we plot the scalar charge (and scalar mass)
versus the orbital angular frequency for several choices of
the parameters $B$ and $\varphi_0$.
Here the scalar charge, $M_{\varphi}$, is defined as the monopole
part of the field, $\varphi$, which is expanded as
$\varphi = \varphi_0 +M_{\varphi}/r +{\cal O}(1/r^2)$
for $r \rightarrow \infty$. (Note again that in this paper we employ the
geometrical units $c=G=1$.) The relation between the scalar charge and the scalar mass is
given by $M_{\varphi}=2M_{\rm S}/(\phi_0 \varphi_0)$.\footnote{Because
the deviation of the quantity $\phi_0 =\exp(\varphi_0^2/2)$ from unity
is on the order of $10^{-9} - 10^{-10}$ as we set
$\varphi_0 =1 \times 10^{-5}$ or $\varphi_0 =5 \times 10^{-5}$,
the relation between the scalar charge and the scalar mass is
approximately given by $M_{\varphi} \simeq 2M_{\rm S}/\varphi_0$
(see Ref.~\cite{ShibaTOB14} for more details).}
We clearly see the onset of dynamical scalarization as the
orbital separation (angular frequency) decreases (increases),
except for the case of H4 EOS $B=8.5$. Those results confirm what is found
when evolving in full numerical relativity a binary neutron star
with APR4 and H4 EOSs~\cite{ShibaTOB14}. In particular, the onset of
dynamical scalarization was well captured by the scalarization condition
derived in Sec. III~B of Ref.~\cite{ShibaTOB14}. For example, Table I of
Ref.~\cite{ShibaTOB14} predicted that the dynamical scalarization for
a binary system of $(1.35 + 1.35) M_{\odot}$ in the case of APR4 $B=9.0$
sets in at around the orbital separation of
$a \simeq 91 M_{\odot} \simeq 134 ~{\rm km}$.
If we regard this separation as the coordinate separation of
our present computation, it corresponds to the orbital angular
frequency of $G_{\rm N} m \Omega \simeq 0.005$.
This is confirmed by Fig.~\ref{fig1} where dynamical scalarization
clearly occurs at around the orbital angular frequency of
$G_{\rm N} m \Omega \simeq 0.005$.
\begin{table}
\caption{We list the orbital angular frequency and GW
frequency ($f_{\rm GW} \equiv \Omega/\pi$)
at the onset of dynamical scalarization.
\label{table1}}
\begin{ruledtabular}
\begin{tabular}{lcc}
Models & $G_{\rm N} m \Omega_{\rm dyn\mbox{-}scal}$
& $f_{\rm GW,dyn\mbox{-}scal}~[{\rm Hz}]$ \\
APR4 (9.0) & 0.0051 & 123 \\
APR4 (8.7) & 0.0125 & 298 \\
APR4 (8.4) & 0.0223 & 534 \\
APR4 (8.0) & 0.0395 & 946 \\
H4 (9.5) & 0.0163 & 391 \\
H4 (9.0) & 0.0302 & 724 \\
\end{tabular}
\end{ruledtabular}
\end{table}
In Table~\ref{table1} we show the orbital angular frequencies
and GW frequencies at the onset of dynamical scalarization.
These quantities are extracted from Fig.~\ref{fig1} using the
fit to the scalar charge obtained in Appendix~\ref{app3}
[see Eq.~(\ref{eq:scfit})]. (In particular $G_{\rm N} m \Omega_{\rm dyn\mbox{-}scal}$
is the angular orbital frequency at which the fit function intersects 1.)
From Fig.~\ref{fig1} we see that the scalar charge at the orbital
angular frequencies listed in Table~\ref{table1} is about $0.1 M_{\odot}$.
This value is only $4 \%$ of the total mass $m$; thus, at the onset of dynamical
scalarization the effect of the scalar field onto the dynamics is negligible.
However, as dynamical scalarization proceeds, the scalar charge rapidly
increases by one order of magnitude,
affecting the subsequent evolution of the binary system.
For the cases of APR4 EOS $B=8.0$ and H4 EOS $B=9.0$,
the numerical-relativity simulations carried out in Ref.~\cite{ShibaTOB14}
showed that dynamical scalarization did not occur during the inspiral
but at merger (see Table~II of Ref.~\cite{ShibaTOB14}).
On the other hand, Fig.~\ref{fig1} predicts that for APR4 EOS $B=8.0$ and
H4 EOS $B=9.0$ dynamical scalarization occurs before the end of the
quasiequilibrium sequence. This contradiction may arise because
toward the end of the inspiral the infall velocity becomes large and
the quasiequilibrium model may lose accuracy.
Dynamical scalarization was first found in Ref.~\cite{BarauPPL13}
using polytropic EOS and it was further investigated by the same authors
in Ref.~\cite{PalenBPL14} using 2.5PN equations of motion augmented
by a set of equations that phenomenologically describe the increase
of scalar charge as the two neutron stars come closer to each other.
Figure~3 in Ref.~\cite{PalenBPL14} is similar to our Fig.~\ref{fig1},
although the former is obtained treating binary neutron stars as two,
isolated spherical neutron stars and using 2PN equations of motion
for circular orbits~\cite{MirshW13}, instead of evolving the binary system
along a sequence of quasiequilibrium configurations in the DEF model.
The results in Fig.~\ref{fig1} are not very sensitive to the value of
$\varphi_0$. Indeed, we show in Fig.~\ref{fig2} that when we decrease
the value of $\varphi_0$ to $1 \times 10^{-6}$ for APR4 EOS, dynamical
scalarization occurs at almost the same orbital angular frequency.
Note that at low orbital frequencies, the scalar charge in the case
$\varphi_0 =1 \times 10^{-6}$ is one order of magnitude smaller than
in the case $\varphi_0 =1 \times 10^{-5}$. This confirms that at large
separation the scalar charge is proportional to $\varphi_0$, as found in
Ref.~\cite{ShibaTOB14}. After dynamical scalarization occurs the scalar
charge has the same value, independently on $\varphi_0$.
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.5cm]{fig2.eps}
\caption{Scalar charge as a function of the orbital angular
frequency normalized to the tensor mass at infinite separation
(lower $x$ axis) or as a function of the frequency of GWs (upper $x$ axis).
Both curves are computed for the case
of APR4 EOS $B=8.7$, but the value of $\varphi_0$ is set to
$1 \times 10^{-5}$ (red dashed with open squares) and $1 \times 10^{-6}$
(black solid with open circles).
\label{fig2}}
\end{figure}
Note that the numerically constructed quasiequilibrium sequences
in Fig.~\ref{fig1} (also in Figs.~\ref{fig2}$-$\ref{fig5})
do not end at the onset of mass shedding from the neutron star's surface,
but stop just before it. This is because it is impossible to treat
a cuspy shape within the spectral method. Thus, we are obliged to stop
the computation before the onset of mass shedding
where the neutron star acquires a cuspy shape.
Note also that in principle Gibbs phenomena could be present
at the surface of the star even before the mass-shedding limit takes place.
This is due to large differences in the density's derivative. However,
because the \textsc{LORENE} spectral code adopts a multidomain method and
surface-fitting coordinates, on each domain the physical fields are
smooth functions and Gibbs phenomena
do not appear (see Ref.~\cite{BonazGM98} for detailed explanations).
\subsection{Binding energy and total angular momentum}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.7cm]{fig3a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.7cm]{fig3b.eps}
\caption{Binding energy as a function of the orbital angular frequency
normalized to the tensor mass at infinite separation (lower $x$ axis)
or as a function of the frequency of GWs from a binary
neutron star with $m = 2.7 M_\odot$ (upper $x$ axis).
Upper panel (a) shows results for APR4 EOS. Black short-dashed,
red long-dashed, green dot-short-dashed, and blue dot-long-dashed curves
are, respectively, the cases: $B=9.0$, 8.7, 8.4, and 8.0.
Lower panel (b) shows results for H4 EOS. Black short-dashed,
red long-dashed, and green dot-dashed curves are, respectively,
the cases: $B=9.5$, 9.0, and 8.5.
In both panels, the purple solid curve is drawn by using
a quasiequilibrium sequence in general relativity (GR).
Cyan dashed and dark-green dot-dashed curves refer to the
3PN and 4PN binding energy in general relativity,
respectively~\cite{BiniD13}
(see also Refs.~\cite{BarauBT12,JaranS12,JaranS13}).
\label{fig3}}
\end{figure}
We plot in Fig.~\ref{fig3} the normalized binding energy,
$(M_{\rm T}-m)/m$, along the quasiequilibrium sequences of
binary neutron stars versus the orbital angular frequency.
We find that after the onset of dynamical scalarization the
binding energy in the scalar-tensor case decreases less rapidly
than in general relativity (GR), differing at most by $14 \%$
at high frequencies for the cases considered.
This implies that binary neutron stars undergoing dynamical scalarization
in the DEF theory spiral in more quickly than in general relativity,
if the amount of energy flux of gravitational radiation
in the scalar-tensor case is equal to the one in general relativity.
As we shall see below, the former can be much larger than the latter,
so the binary neutron star approaches the merger even more quickly
once the energy flux in scalar-tensor theory is also taken into
account.
The binding energy is defined by the difference between the tensor mass
at a given separation, $M_{\rm T}$, and that at infinity, $m$.
The tensor mass is given by the sum of the ADM mass, $M_{\rm ADM}$,
and the scalar mass, $M_{\rm S}$~\cite{Lee74}. Quite interestingly,
we find that the scalar mass is not responsible of the large increase
of the tensor mass; the latter increases because of the large increase
(three orders of magnitude) of the scalar field in the ADM mass.
As seen in Eq.~(\ref{eq:deltapsi}), the scalar field and its derivatives
enter the source term of the Poisson-like equation of the conformal
factor, which determines the ADM mass.
In Fig.~\ref{fig4} we plot the total angular momentum,
$J/(G_{\rm N} m^2)$, along the quasiequilibrium sequences
versus the orbital angular frequency.
The behavior of the total angular momentum is basically the same as
the binding energy; i.e., after the onset of dynamical scalarization
the total angular momentum in the scalar-tensor case decreases less rapidly
than in general relativity.
It is worth noticing that in some cases the binding energy/total angular
momentum may reach their minimum before the onset of mass shedding.
The sequences shown in Figs.~\ref{fig3} and \ref{fig4}
terminate slightly before the onset of mass-shedding
because it is impossible to treat a cuspy shape within
the spectral method, as we mentioned at the end of Sec.~\ref{sec:scalar}.
In order to calculate at which orbital angular frequency the sequences
encounter the mass-shedding point, we compute the sensitive mass-shedding
indicator $\chi$,
\begin{equation}
\chi \equiv \frac{(\partial (\ln h)/ \partial r)_{\rm eq}}
{(\partial (\ln h)/ \partial r)_{\rm pole}}
\end{equation}
as a function of the orbital angular frequency.
The above quantity is the ratio between the radial derivative of the enthalpy
computed in the equatorial plane at the surface along the direction toward
the companion star and the one at the surface of the pole of the star.
The indicator takes the value $\chi=1$ for spherical stars and it is
$\chi=0$ in the mass-shedding limit. We extrapolate the sequences of
$\chi$ to the mass-shedding limit and determine the orbital angular
frequency at that point. (For more details on this method see Sec.~4.3
in Ref.~\cite{TanigS10}.) After determining the orbital angular frequency
at the onset of mass shedding, we then extrapolate the binding energy curve to
that frequency.
By this procedure we find that the sequences of APR4 EOS $B=9.0$,
8.7, 8.4, 8.0, and H4 EOS $B=9.5$, $B=9.0$ reach the minimum of
the binding energy (i.e., the onset of secular instability~\cite{FriedUS02})
before the mass shedding. Thus, in these cases the binary neutron stars
terminate the quasiequilibrium sequence at that point, and plunge.
By contrast the sequence of H4 EOS $B=8.5$ terminates at the onset
of mass shedding.
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.7cm]{fig4a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.7cm]{fig4b.eps}
\caption{Same as Fig.~\ref{fig3} but for the total angular momentum.
Cyan dashed and dark-green dot-dashed curves refer to the
3PN and 4PN angular momentum in general relativity,
respectively. Those curves are calculated using
Refs.~\cite{LeTieBW12,BiniD13}.
\label{fig4}}
\end{figure}
\subsection{Central baryonic rest-mass density}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.5cm]{fig5a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.5cm]{fig5b.eps}
\caption{Same as Fig.~\ref{fig3} but for the relative change
in the central baryonic rest-mass density of a neutron star.
\label{fig5}}
\end{figure}
In general relativity the central baryonic rest-mass density
of a neutron star in irrotational binary systems always decreases
as the orbital frequency increases (see, e.g., Ref.~\cite{TanigG02}).
We find that this is not the case in the scalar-tensor model under
investigation. After the onset of dynamical scalarization, we find that
the central baryonic rest-mass density starts increasing
(instead of continuing decreasing)
as the orbital frequency increases, as shown in Fig.~\ref{fig5}.
It is not easy to physically explain this behavior because of nonlinear
effects related to the deformation of the star and the distribution
profiles of metric and matter quantities in the star.
In our computation, the baryonic rest mass of each star defined by
Eq.~(\ref{eq:restmass}) is fixed along the quasiequilibrium sequences.
To do so, the central value of the baryonic rest-mass density
of a neutron star necessarily increases after dynamical scalarization,
because the remaining part of Eq.~(\ref{eq:restmass}) after dropping
the rest-mass density,
\begin{equation}
\int_{\rm star~A} u^t \sqrt{-g} d^3x,
\end{equation}
decreases rapidly after dynamical scalarization sets in along a constant
baryonic rest-mass sequence, while it slightly increases before the
dynamical scalarization.
\subsection{Evolution of the orbital angular frequency}
\label{sec:evolorb}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.7cm]{fig6a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.7cm]{fig6b.eps}
\caption{Same as Fig.~\ref{fig3} but for the evolution of
the orbital angular frequency (left $y$ axis)
or the frequency of GWs (right $y$ axis)
from a binary neutron star with $m = 2.7 M_\odot$.
The gravitational energy flux is the choice (i).
\label{fig6}}
\end{figure}
We want to estimate the orbital angular frequency and GW cycles
in the DEF model assuming a quasistationary adiabatic evolution.
We follow what was done in Ref.~\cite{TanigS10}. Basically, we use the
balance equation $dE/dt = - {\cal F}$ and integrate
$d \Omega/dt = -{\cal F}/(dE/d\Omega)$. (Note that this PN approximant is
similar to the TaylorT1 approximant in Ref.~\cite{BoyleBKMPSCT07}.)
For $E$ we use the binding energy [fitted to a polynomial in
$x \equiv (G_{\rm N} m \Omega)^{2/3}$] along the quasiequilibrium sequences.
The choice of the energy flux ${\cal F}$ is less straightforward
because we do not know it exactly, so we have to rely on PN
calculations~\cite{DamourEF92,DamourEF96b,DamourEF98,WillZ89,MirshW13,Lang14}.
Since in this paper we consider only the case of equal-mass binaries on
a circular orbit, the monopole and dipole components of the gravitational
radiation vanish. Thus the leading term is the quadrupole component.
Using Ref.~\cite{DamourEF92} [see in particular Eqs.~(6.40) and (6.41) therein]
we find that the ratio of the quadrupole component generated directly
by the scalar field, ${\cal F}_{\varphi}^{\rm Quadrupole}$,
and the quadrupole component of the gravitational field,
${\cal F}_{G}^{\rm Quadrupole}$, is given at leading order in the PN
expansion by\footnote{Here and in the following we assume that
the energy flux derived in PN theory expanding the neutron-star masses about
the asymptotic value $\varphi_0$ continues to be valid
also in the presence of spontaneous and/or dynamical scalarization.}
\begin{equation}
\frac{{\cal F}_{\varphi}^{\rm Quadrupole}}{{\cal F}_{G}^{\rm Quadrupole}}
= \frac{\alpha_{\varphi}^2}{6}
= \frac{1}{6 B} \Bigl( \frac{M_{\varphi, {\rm NS}}}{m_{\rm NS}} \Bigr)^2,
\label{eq:ratioflux}
\end{equation}
where $M_{\varphi, {\rm NS}}$ is the scalar charge of a neutron star,
and $m_{\rm NS}$ is the tensor mass of a {\it spherical} neutron star,
i.e., $m_{\rm NS} = m/2 = 1.35 M_{\odot}$. The quantity
$\alpha_{\varphi}$ is an auxiliary quantity~\cite{DamourEF92,DamourEF93}
defined by
\begin{equation}
\alpha_{\varphi} = -\frac{M_{\varphi, {\rm NS}, {\rm DEF}}}{m_{\rm NS}}
= -\frac{1}{\sqrt{B}} \frac{M_{\varphi, {\rm NS}}}{m_{\rm NS}}. \label{eq:alpha}
\end{equation}
Here $M_{\varphi, {\rm NS}, {\rm DEF}}$ is the scalar charge of a neutron star
defined as the monopole part of the field, $\varphi_{\rm DEF}$,
which is expanded as
$\varphi_{\rm DEF} = \varphi_{0, {\rm DEF}} +M_{\varphi, {\rm NS}, {\rm DEF}}/r +{\cal O}(1/r^2)$
for $r \rightarrow \infty$. (Note that the relation between the var-type
scalar field in this paper, $\varphi$, and that in Ref.~\cite{DamourEF92},
$\varphi_{\rm DEF}$, is given by $\varphi = \sqrt{B} \varphi_{\rm DEF}$.)
Because the scalar charge of a binary system is defined as
a global quantity, we do not know the scalar charge
of the individual stars in the binary system, $M_{\varphi, {\rm NS}}$.
As an approximation, we simply take $M_{\varphi, {\rm NS}} \simeq M_{\varphi}/2$
for the current estimate. From Fig.~\ref{fig1}, we find that the ratio
Eq.~(\ref{eq:ratioflux}) takes the maximum of about 0.028 in the case
of APR4 $B=9.0$ at the end of the sequence.
(The scalar charge at that point is about $M_{\varphi} = 3.3 M_{\odot}$.)
For the other cases the ratio is less than 0.028 throughout
the quasiequilibrium sequences. Thus, since
${\cal F}_{\varphi}^{\rm Quadrupole}$ is at most $3 \%$ of the quadrupole component
of the gravitational field, ${\cal F}_{G}^{\rm Quadrupole}$, we neglect it.
Considering the above, we make the following choices for the gravitational
energy flux ${\cal F}$ in the balance equation: (i) the 3.5PN flux
(also as a polynomial in $x$) computed in general relativity~\cite{Blanchet06},
and (ii) the quadrupole component of the gravitational field in
the scalar-tensor DEF model, i.e., ${\cal F}_{G}^{\rm Quadrupole} =
(32 \nu^2/5) (G_{\rm N} m (1 + \alpha_{\varphi}^2) \Omega)^{10/3}$ where
$\nu = m_{\rm NS}^2/m^2$~\cite{DamourEF92,PalenBPL14}.
The choice (i) will allow us to isolate the contribution to the orbital
angular frequency (and the number of GW cycles, see below) due to
the binding energy computed in this paper, while the choice
(ii) is an estimate of the orbital angular frequency when also the
gravitational radiation in scalar-tensor theory is included.
(Note that today the energy flux in scalar-tensor theory is known only through
leading quadrupole order.) In Appendix~\ref{app3}, we further discuss
those choices and explain how we derive the energy flux in case (ii) using the
scalar charge computed in this paper.\footnote{We are currently working
on long-term evolutions of
binary neutron stars extending our previous study~\cite{ShibaTOB14}.
Preliminary analysis shows that the GW energy flux in the scalar-tensor DEF
model, at a given orbital frequency, is indeed larger than in the
general-relativity case. However, case (ii) seems to overestimate it
and the exact GW luminosity is likely to lie between cases (i) and (ii).}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.7cm]{fig7a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.7cm]{fig7b.eps}
\caption{Same as Fig.~\ref{fig6} but for the choice (ii)
for the gravitational energy flux.
In the general-relativity case, the 3.5PN energy flux computed
in general relativity is used.
These are the same data as used in Fig.~\ref{fig6} (shown as ``GR'').
\label{fig7}}
\end{figure}
In the examples shown in Fig.~\ref{fig6} for the choice (i) and in
Fig.~\ref{fig7} for the choice (ii),
we set the initial orbital angular frequency to $G_{\rm N} m \Omega =0.005$
for the case of APR4 EOS
and $G_{\rm N} m \Omega =0.01$ for H4 EOS.
The final orbital angular frequency of
each curve corresponds to the end point of the quasiequilibrium sequences.
As explained above when discussing Fig.~\ref{fig3}, the sequences
may have the minimum of the binding energy before the mass-shedding point.
If the minimum is found, we use the orbital angular frequency
at that point, $G_{\rm N} m \Omega_{\rm ener\mbox{-}min}$, as the final one.
If not, we adopt the orbital angular frequency at the mass-shedding limit,
$G_{\rm N} m \Omega_{\rm mass\mbox{-}shed}$, as the final one.
In Table~\ref{table2} we show the orbital angular frequency at the end point
of each model which is used for the final orbital angular frequency.
For 3PN and 4PN cases, we use the same final orbital angular frequency
as that in general relativity.
Note that the final orbital angular frequency listed in Table~\ref{table2}
is about $G_{\rm N} m \Omega = 0.05$ for APR4 EOS and 0.04 for H4 EOS.
These values correspond to the frequency of GWs of $\simeq 1200~{\rm Hz}$
and $\simeq 960~{\rm Hz}$ which lie in the high-frequency portion of
the LIGO/Virgo/KAGRA bandwidth.
As a consequence, the effects discussed in Secs.~\ref{sec:evolorb} and
\ref{sec:gwcycles} may not be observable if the broadband noise
spectral density is employed. By contrast interferometer configurations
optimized at high frequency may allow us to measure these effects.
It is relevant to point out that if we consider the case of unequal-mass
binaries, there exists the dipole component of the scalar GWs.
This contribution has the same sign as that of the quadrupole
component~\cite{DamourEF92} and increases the energy flux of the
scalar GWs. Moreover, if we take into account the infall velocity of the stars
in the binary system, the monopole component arises. This also contributes
to increase the energy flux of scalar GWs~\cite{DamourEF92}.
Thus, since the energy flux increases for unequal-mass binaries,
in these cases the merger may occur at even earlier times than what
we estimated in Fig.~\ref{fig6}.
\subsection{Number of gravitational-wave cycles} \label{sec:gwcycles}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.cm]{fig8a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.cm]{fig8b.eps}
\caption{Same as Fig.~\ref{fig3} but for the evolution of
the number of GW cycles.
The gravitational energy flux is the choice (i).
\label{fig8}}
\end{figure}
\begin{figure}
\vspace{0.5cm}
\includegraphics[width=7.cm]{fig9a.eps} \\
\vspace{0.4cm}
\includegraphics[width=7.cm]{fig9b.eps}
\caption{Same as Fig.~\ref{fig8} but for the choice (ii)
for the gravitational energy flux.
In the general-relativity case, the 3.5PN GR energy flux is used.
These are the same data used in Fig.~\ref{fig6} (shown as ``GR'').
\label{fig9}}
\end{figure}
\begin{table*}
\caption{We list the orbital angular frequencies at the end point of each
quasiequilibrium sequence and the number of GW cycles. We also show
the quantity $\delta {\cal N}_{\rm{GW}}$, which is the difference between
the number of GW cycles in the DEF model and in general relativity
[i.e., either APR4 (GR) or H4 (GR) depending on the EOS].
When computing the number of cycles, we set the initial orbital
angular frequency to $G_{\rm N} m \Omega =0.005$ ($f_{\rm GW} = 119.7$ Hz)
for the case of APR4 EOS and $G_{\rm N} m \Omega =0.01$ ($f_{\rm GW} = 239.3$ Hz)
for H4 EOS. The subscripts (i) or (ii) refer to the choice of the gravitational
energy flux discussed in Sec. III~D.
\label{table2}}
\begin{ruledtabular}
\begin{tabular}{lccrrrr}
Models & $G_{\rm N} m \Omega_{\rm ener\mbox{-}min}$
& $G_{\rm N} m \Omega_{\rm mass\mbox{-}shed}$
& ${\cal N}_{\rm{GW} (i)}$ & $\delta {\cal N}_{\rm{GW} (i)}$
& ${\cal N}_{\rm{GW} (ii)}$ & $\delta {\cal N}_{\rm{GW} (ii)}$ \\
APR4 (9.0) & 0.0491672 & 0.0565207 & 238.00 & $-28.87$ & 202.03 & $-64.85$ \\
APR4 (8.7) & 0.0494752 & 0.0572480 & 256.62 & $-10.25$ & 231.48 & $-35.40$ \\
APR4 (8.4) & 0.0523407 & 0.0570327 & 262.94 & $-3.93$ & 239.63 & $-27.24$ \\
APR4 (8.0) & 0.0494658 & 0.0563251 & 265.55 & $-1.33$ & 242.62 & $-24.25$ \\
APR4 (GR) & $\cdots$ & 0.0570651 & 266.87 & $\cdots$ & $\cdots$ & $\cdots$ \\
APR4 (3PN) & $\cdots$ & $\cdots$ & 266.08 & $-0.79$ & $\cdots$ & $\cdots$ \\
APR4 (4PN) & $\cdots$ & $\cdots$ & 265.81 & $-1.07$ & $\cdots$ & $\cdots$ \\
H4 (9.5) & 0.0385605 & 0.0438147 & 62.66 & $-10.89$ & 54.76 & $-18.78$ \\
H4 (9.0) & 0.0401870 & 0.0419199 & 70.89 & $-2.66$ & 63.68 & $-9.86$ \\
H4 (8.5) & $\cdots$ & 0.0410100 & 73.47 & $-0.07$ & 66.09 & $-7.46$ \\
H4 (GR) & $\cdots$ & 0.0405359 & 73.54 & $\cdots$ & $\cdots$ & $\cdots$ \\
H4 (3PN) & $\cdots$ & $\cdots$ & 74.18 & 0.64 & $\cdots$ & $\cdots$ \\
H4 (4PN) & $\cdots$ & $\cdots$ & 74.01 & 0.46 & $\cdots$ & $\cdots$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
As seen in Fig.~\ref{fig3}, because after the onset of dynamical scalarization,
the binding energy calculated in the DEF model decreases less rapidly than
that calculated in general relativity, the binary evolves more quickly
in the DEF model than in general relativity, if differences between
the energy fluxes in general relativity and scalar-tensor theory are neglected.
When an estimate of the scalar-tensor energy flux is included the late
evolution of the binary is even faster (see Figs.~\ref{fig6} and \ref{fig7}).
As a consequence, the number of GW cycles computed from an initial frequency
$f_{\rm GW, ini}$ to a final frequency $f_{\rm GW, fin}$ will be different in
the DEF model and in general relativity.
To quantify this, we numerically integrate the orbital angular
frequency between the initial and final frequencies discussed in
Table~\ref{table2} and the text around it. When computing the number of
cycles between two models with the same EOS, we impose that
they have the same initial orbital frequency.
The results are summarized in Table~\ref{table2} and
in Figs.~\ref{fig8} and \ref{fig9}. In particular, the difference in number
of GW cycles between the DEF model
and general relativity for APR4 EOS is 28.9, 10.3, 3.9, and 1.3 for
the case of $B=9.0$, 8.7, 8.4, and 8.0, respectively, if we use the choice (i)
for the gravitational energy flux. On the other hand,
if we adopt the choice (ii), the difference in number of GW cycles
is 64.8, 35.4, 27.2, and 24.3. Note here that the difference is
calculated against the general-relativity case with 3.5PN energy flux
for both choices of (i) and (ii). For the case of H4 EOS, the difference is
10.9, 2.7, and 0.07 for
$B=9.5$, 9.0, and 8.5, respectively, for the choice (i), while for
the choice (ii),
we have 18.8, 9.9, and 7.5, respectively. Thus, except for the case of
H4 EOS $B=8.5$ for
the choice (i), the difference in number of GW cycles is larger than unity.
However, those numbers should be taken with cautiousness because they are
affected by different source of errors.
For example, the quasiequilibrium configurations themselves
include errors. It is usually common to measure the errors
by a global error indicator, i.e., the error in the virial relation.
In scalar-tensor theory the virial relation is expressed as~\cite{ShibaK13}
\begin{equation}
M_{\rm Komar} = M_{\rm ADM} + 2 M_{\rm S} \phi_0^{-1},
\end{equation}
where $M_{\rm Komar}$ is the Komar mass. This relation should hold
along the quasiequilibrium sequences, but because of numerical errors,
deviations can appear. In this paper we define the error in the virial relation
as follows
\begin{equation}
{\rm virial~error} = \Bigl| \frac{M_{\rm Komar} - M_{\rm ADM}
- 2 M_{\rm S} \phi_0^{-1}} {M_{\rm ADM}} \Bigr|.
\end{equation}
In our quasiequilibrium configurations the ``virial error'' is
on the order of $10^{-5}$ for large and medium orbital separations
and $10^{-4}$ for close configurations.
Because the binding energy is on the order of $10^{-3} - 10^{-2}$
throughout the computed orbital-frequency range,
a virial error on the order of $10^{-5} - 10^{-4}$ implies
that the binding energy has a maximal error of a few \%.
Besides the error in the quasiequilibrium configurations,
there are errors due to the fitted curves of the binding energy
and the scalar charge. Nevertheless, the difference in the
number of cycles in Table~\ref{table2} is sufficiently large to make it
worthwhile to run accurate, long full numerical-relativity simulations of binary
neutron stars in the DEF model and develop accurate template waveforms.
The frequency region that is affected by dynamical scalarization is
in the several hundreds of Hz, i.e., in the high-frequency portion of
the LIGO/Virgo/KAGRA bandwidth. If the binary is composed by a neutron star
and black hole, dynamical scalarization would in principle take place
at lower frequencies. Quite interestingly, if the binary is in
an eccentric orbit,
the motion can induce a scalar charge on the black hole~\cite{LiuEWK14}.
We plan to study in the future whether dynamical scalarization occurs
in a black-hole$-$neutron-star binary and is observable by LIGO/Virgo/KAGRA.
It is worthwhile to note that the difference in number of GW cycles
between the DEF model and general relativity is much larger
than that between the 3PN approximation of the binding energy and the 4PN one.
(We use the 3.5PN flux for both calculations.) Setting the integration range
of $G_{\rm N} m \Omega$ to the same as in the general-relativity case,
we obtain that
the difference in the number of GW cycles between the 3PN approximation and
the 4PN one is 0.28 for APR4 EOS and 0.17 for H4 EOS. (The 4PN case has
a smaller number of cycles than that for 3PN.)
Before closing this section, we would like to comment that the difference
in GW frequency between general relativity and the DEF model was also estimated
in Ref.~\cite{PalenBPL14}, evolving the 2.5PN equations of motion augmented
by a set of equations that phenomenologically describe the increase of
scalar charge as the two neutron stars come closer to each other
(see Figs.~7, 9 and 11 in Ref.~\cite{PalenBPL14}). An important difference
between the two sets of results is that the quasiequilibrium sequences
in our computation terminate much earlier than those in Ref.~\cite{PalenBPL14}.
This is because the authors of Ref.~\cite{PalenBPL14} treat binary neutron stars
as two spherical neutron stars, while we compute the deformation of the stars
and stop at the mass-shedding point or at the turning point of the binding
energy.
\section{Conclusions}
We have computed quasiequilibrium sequences of binary neutron stars in the
DEF scalar-tensor model~\cite{DamourEF93} that admits dynamical scalarization.
The EOS of the neutron star that we have employed has the form of a piecewise
polytrope and we have used APR4 and H4 EOSs~\cite{ReadLOF09,ReadMSUCF09}.
We have considered an equal-mass, irrotational binary whose tensor mass at
large separation is $2.7 M_{\odot}$.
Using the quasiequilibrium sequence, we have derived the binding energy
and scalar charge and found that, as the stars come closer,
and the dynamical scalarization sets in, the binding energy decreases less
rapidly than in general relativity.
Using the newly computed binding energy and the balance equation,
we have estimated the number of GW cycles during the adiabatic,
quasicircular inspiral stage up to the end of the sequence, which is
the last stable orbit or the mass-shedding point, depending on which comes
first. When employing the quadrupole component
of the gravitational energy flux in the scalar-tensor DEF model,
we have found that in the most optimistic case, when dynamical scalarization
sets in around a GW frequency of $\sim 130\, {\rm Hz}$
(i.e., $B = 9.0$ and APR4 EOS), the number of GW cycles from $120$ Hz
up to merger in general relativity,
$\sim 270$, is reduced by $24 \%$, of which $11 \%$ is only due to
the inclusion of the scalar-tensor binding energy.
A summary of our results is given in Table~\ref{table2} and
Figs.~\ref{fig6}$-$\ref{fig9} for several choices of the scalar-tensor
parameters. Of course, a reduction in the number of GW cycles
with respect to the general-relativity case does not immediately inform us
on whether the deviation can be observed by advanced detectors.
An analysis that take into account the noise spectral density of the
detector and the accumulated
signal-to-noise ratio would be needed~\cite{Sampetal14}.
As seen in Table~\ref{table1}, GW frequencies at the onset of
dynamical scalarization are in the several hundreds of Hz, where
the broadband interferometer configuration of LIGO/Virgo/KAGRA
has poor sensitivity. In order to measure deviations from
general relativity in the DEF model, it is crucial that
the scalar-field parameter $B$ be large so that dynamical scalarization
sets in at low frequencies (e.g., around $130$ Hz for APR4 EOS and $B=9.0$)
and large differences in the GW cycles can be observed.
However, if the parameter $B$ were too large, the DEF model would be
already rejected by the observational constraints imposed
by neutron-star$-$white-dwarf binaries~\cite{Freir12,Anton13,BhatBV08}.
Recent studies carried out in Refs.~\cite{Samp13,Sampetal14}, which
use scalar-tensor templates in the frequency domain, rely on
the scalar-charge evolution and numerical-relativity simulations of
Refs.~\cite{BarauPPL13,PalenBPL14}, concluded that advanced detectors
operating at a signal-to-noise ratio (SNR) of 10 will be able to constrain
dynamical scalarization only if the system scalarizes at low enough orbital
frequencies, e.g., $\leq 50$ Hz, so that a sufficient number of GW cycles
emitted during the dynamical-scalarization phase can contribute to the
accumulated SNR. This would imply that in the case of APR4 EOS with $B=9.0$,
advanced LIGO and Virgo might observe deviations from general relativity
if dynamical scalarization takes place in nature.
Moreover, using results from Ref.~\cite{TanigS10} and from GR computations
shown in this paper (and also a direct integration of the TaylorT4-PN
approximant with tidal effects~\cite{TanjaH}), we find that in general
relativity tidal effects produce a difference of only a few GW cycles
(i.e., $\sim 1 - 3$ GW cycles depending on the EOS) between 130 Hz and
1200 Hz with respect to the point-particle case. Those small differences in
GW cycles induced by tidal effects at high frequency can be measured by
advanced detectors in one single event only if the SNR is roughly
$30 - 35$~\cite{DelPozzo:2013ala,Wade:2014vqa}. Note that depending on the EOS
those differences can be smaller than or comparable to what we have found
in dynamical scalarization (see Table~\ref{table2}). At SNR around $30 - 35$,
deviations from general relativity might also be observable even in cases
in which the onset of dynamical scalarization happens at orbital frequencies
above $50$ Hz~\cite{Samp13,Sampetal14}. It will be interesting to investigate
in the future the detectability of tidal effects in the presence of dynamical
scalarization. To precisely determine for which neutron-star masses, EOS,
and scalar-tensor parameters dynamical scalarization and tidal effects can
be observed with advanced GW detectors, it will be relevant to develop
accurate waveforms in the DEF scalar-tensor model. To this respect the next
work~\cite{SenBST14} is focusing at building accurate analytical templates
that can incorporate dynamical scalarization, and reproduce the binding energy
computed in this paper and the results from numerical-relativity simulations.
Finally, extending earlier work~\cite{ShibaTOB14}, new long-term numerical
simulations in scalar-tensor theory are suggesting that the analytical
energy flux used in this paper (i.e., the energy flux at quadrupolar order)
is likely to overestimate the exact energy flux in the scalar-tensor DEF model.
Thus, a better modeling of the energy flux (e.g., its PN computation
through 1PN and even 2PN order~\cite{Lang14}) is crucial for understanding and
quantifying differences from the general-relativity case.
\begin{acknowledgments}
A.B. thanks Noah Sennett for useful discussions on dynamical scalarization.
K.T. acknowledges partial support from JSPS Grant-in-Aid for Scientific
Research (26400267).
M.S. acknowledges partial support from JSPS Grant-in-Aid for Scientific Research
(24244028), JSPS Grant-in-Aid for Scientific Research on Innovative Area
(20105004), and HPCI Strategic Program of Japanese MEXT.
A.B. acknowledges partial support from NSF Grant No. PHY-1208881 and
NASA Grant No. NNX12AN10G.
\end{acknowledgments}
|
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"redpajama_set_name": "RedPajamaArXiv"
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| 9,334
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Gross weight: 14.550 grams Gold weight 12.610 grams Rose Cut Diamonds: 0.73 carats Diamonds: 0.29 carats Ruby 8.26 carats Inspired by the floral motifs of traditional jewels worn by 18th c Rajasthan royalty, yet utterly contemporary, the Fleurs de l'Inde Collection pieces are exquisite examples of 'Uttrai' and 'Kundan' craftsmanship. The collection features auspicious red rubies and green emeralds, hand-cut in the 'Uttrai' style, highlighted by rose-cut diamonds and set within pure 18k gold 'Kundan', reflecting the legendary skills of the master jewellers of Jaipur. The ruby earrings are handmade from 18k yellow gold and carefully inlaid with glowing rubies using the traditional 'Kundan' process that can take months to complete. The rose-cut diamond centrepiece is framed by 'petals' of marquise-cut rubies and interspersed with full-cut diamonds. The piece adds colour and brilliance to understated chic formalwear.
|
{
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/support.bioconductor.org\/p\/105340\/","text":"what the first row without a gene name?\n1\n1\nEntering edit mode\nRaymond \u25b4 20\n@raymond-14020\nLast seen 2.3 years ago\n\nHi, all,\n\nI aligned my RNAseq raw data to cDNA using Kallisto and import the result using tximport as suggested by the DESeq2 tutorial (tx2gene maps ensembl_transcript_id to ext_gene symbol)\n\ntxi.kallisto_gene <- tximport(files, type = \"kallisto\", tx2gene=tx2gene,ignoreTxVersion=TRUE)\n\nWhen I looked into the\u00a0txi.kallisto_gene, it shows as below, the first row does not have a name, anyone knows how to deal with it? can I just ignore it?\n\n 495-RB-139 495-RB-140 495-RB-141 495-RB-142 495-RB-143 495-RB-144\n84021.18351 102743.98948 151863.14971 97510.1275 78587.59879 87162.3374\n0610009B22Rik 494.00005 606.99973 405.99960 692.9998 493.99982 564.0001\n0610009O20Rik 1043.00000 1466.00000 806.00000 1172.0000 947.00000 1325.0000\n0610010F05Rik 1571.00019 2090.99933 1227.99979 1779.0010 1632.00373 1772.0035\ndeseq2 \u2022 525 views\n1\nEntering edit mode\n@mikelove\nLast seen 12 hours ago\nUnited States\n\nThis occurs because you have transcripts associated with the empty string in your tx2gene table. \u00a0Take a look at that table and you can solve the problem there one of two ways, \u00a0either giving them the transcript ID as a gene ID, \u00a0or removing those transcripts from the table. \u00a0I don\u2019t know which way to go is better.\n\n1\nEntering edit mode\n\nThanks Michael.\u00a0 I tried the first way, giving them the paste0(\"UNKNOWN\",transcriptID) as the gene ID.\n\n1\nEntering edit mode\n\nHi Raymond. Could you please elaborate on where exactly you inserted the paste0 code into the tximport script? I'm having the exact same problem as you right now.\n\n0\nEntering edit mode\n\nPlease don't reply to years-old posts. Instead, post a new question, and if you want to reference the old post, just insert a hyperlink to it (copy the URL, then hit the globe with a green arrow above the box you are typing in, and paste in the link to the old post).","date":"2021-09-20 08:57:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5398638844490051, \"perplexity\": 6186.2092367282}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057033.33\/warc\/CC-MAIN-20210920070754-20210920100754-00700.warc.gz\"}"}
| null | null |
Q: How to set field readonly in one content type only? In a list with two content types - both with the same field "Title", I need to make the field readonly in one of the content types, but not the other (in one the title is manually set, in the other the title is set by code).
Is this possible?
A: You can do this in PowerShell as follows:
$fieldLink = $list.ContentTypes["YourCT"].FieldLinks | where {$_.Name -eq "YourField"}
$fieldLink.ReadOnly = $true
$list.ContentTypes["YourCT"].Update()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 7,148
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\section{INTRODUCTION}
Given an RGB image of a known 3D scene, our goal is to infer the 6 degree of freedom pose of the camera - a task known as camera localization. To do this, current state-of-the-art methods \cite{Brachmann2016}, employ a two-stage pipeline. In the first stage, for each pixel in the image, a continuous 3D coordinate of the scene is regressed. These are called scene coordinates and uniquely identify each part of the scene (Fig.~\ref{fig:coolimage}). In the second stage, sparse sets of four scene coordinates are sampled and fed into a RANSAC-based optimizer which estimates the final camera pose.
\begin{figure}[thpb]
\centering
\includegraphics[scale=0.3]{images/waycoolerimage.pdf}
\caption{\textbf{The concept of scene coordinate regression.} (Top) Given a known scene where each surface point has a unique 3D coordinate (here shown as a distinctive color), the goal is to locate the camera with respect to this scene. (Bottom) To achieve this, we need to predict at test time the scene coordinate for each pixel in an input RGB image (black pixels in the ground truth correspond to unknown scene coordinates).}
\label{fig:coolimage}
\end{figure}
In this work, we analyze the first stage of this pipeline, the dense regression of scene coordinates. State-of-the-art methods have typically employed regression Random Forests (RFs) to accomplish this task \cite{Shotton2013,Brachmann2016, Valentin2015}. The RFs have the advantage of being both memory and test-time efficient, two important features when considering camera localization in real-time navigation and mobile robotics. In this work, we analyze equivalently-efficient and non-efficient neural network (NN)-based architectures to accomplish the task of scene coordinate regression for camera localization. Our choices of architectures are inspired by the work of \cite{Sethi1990} who propose a method for transforming a binary decision tree into a two-hidden-layer NN. We utilise this tree-to-NN mapping by taking a RF trained for 2D-to-3D scene coordinate prediction in a known scene \cite{Brachmann2016} and mapping each of its trees to an equivalent two-layer NN. We collectively refer to this ensemble of NNs as a ForestNet. We fine-tune the ForestNet on a subset of the data that was used to train the original RF. At test-time, we obtain scene coordinate predictions from the ForestNet and feed them into a RANSAC-based optimizer to extract the 6D camera pose. This pipeline is illustrated in Fig.~\ref{fig:pipeline}.
\begin{figure*}[thpb]
\centering
\vskip 2mm
\includegraphics[scale=0.6]{images/pipeline_big_compressed.pdf}
\caption{\textbf{Training (yellow) and test (blue) pipeline using a ForestNet for camera localization}. (1) A RF is trained for a known 3D scene. (2) A RF-to-NN mapping is used to transform the trained RF into an ensemble of parallel tree-like networks, collectively referred to as a ForestNet. A subset of the original training data is used to fine-tune the ForestNet, with an additional robust averaging module which can be appended to and integrated into the training of the network. At test time, a sparse set of pixels is sampled from a RGB test image and passed through the ForestNet to obtain a set of 3D scene coordinates (either robustly averaged or not). (3) The scene coordinates are passed to the RANSAC optimizer of \cite{Brachmann2016} which generates and then refines a set of camera pose hypotheses until a final 6D camera pose remains. GT = Ground Truth Scene Coordinates. Best viewed in color.}
\label{fig:pipeline}
\end{figure*}
The motivation for exploring NN architectures based on this RF-to-NN mapping is two-fold. Firstly, the mapping constructs a NN that is derived from a learned RF: the tree topologies define the active connections in the network and the trees' learned parameters specify the initialization of all network weights. This makes it possible to preserve the tree topologies when training the ForestNet. The optimized ForestNet can thus be made efficient again by mapping back to the fast and memory-compact RF version of itself as done in \cite{Richmond2015}. With this ability, we explore variants of the ForestNets where we allow different parts of the network to be optimized, in some cases retaining the ability to map-back and in other cases not. The second motivation for exploring this mapping is that it allows for the construction of an \textit{ensemble} of NNs. This ensemble is fully differentiable making it amenable to traditional NN training paradigms, unlike traditional RFs. Further, we can apply a robust averaging to the ensemble's outputs. In particular, we introduce a variant of the 3D geometric median and implement it as a fully differentiable NN module. We observe that by appending this module onto the ForestNet, either post-hoc or integrated into the end-to-end training of the network, we improve on the accuracy of the scene coordinate predictions by 6.8\% and 7.8\%, respectively, over a ForestNet without averaging. In addition to this, we compare a ForestNet to a deep Convolutional NN (CNN) trained specifically for dense scene coordinate regression. The CNN achieves better scene coordinate predictions but under-performs the RFs and ForestNets in terms of final 6D camera pose accuracy. Importantly, this deep CNN cannot be mapped to a RF, and hence cannot enjoy their fast test-time speeds and lightweight memory requirements. All these results shed light on the best approaches to use for other continuous regression tasks.
Based on the above, our key contributions are as follows:
\begin{enumerate}
\item A new NN architecture which we term a ForestNet for dense scene coordinate regression. The ForestNet architecture is derived and initialized from a RF. We observe that for the camera localization task, the best performing ForestNet is test-time efficient with a low memory cost since it can be mapped back to a RF.
\item A fully-differentiable 3D geometric median filter implemented as a NN module for robust averaging. We append this module to a ForestNet and show that when applied post-hoc or in end-to-end training of the ForestNet, the scene coordinate accuracy improves.
\item An engineered solution to reduce the memory requirements of the ForestNets by a factor of at least four, allowing full training and testing on a single GPU.
\item An improvement over the state-of-the-art for camera localization on the 7-Scenes dataset \cite{Shotton2013} using our best method, an efficient ForestNet with robust averaging.
\end{enumerate}
\section{RELATED WORK}\label{sec:relatedwork}
\subsection{Camera (re)localization}
Camera localization has traditionally been formalized as a descriptor matching problem, addressed either using whole image-based approaches with keyframe/keypose pairs, or using sparse feature-based approaches with keypoints. In keyframe-based methods \cite{Klein2008, Gee2012,Glocker2015}, a descriptor is computed for a (whole) query image and is compared to the descriptors of a set of saved images (keyframes), each keyframe with an associated ground-truth camera pose (keypose). Keypoint-based methods instead extract interest points (keypoints) and attach a feature descriptor and 3D location to each. The camera pose of a query image is then estimated using a sparse set of keypoint 2D-to-3D correspondences \cite{Irschara2009, Lowe2004, Sattler2011, Lepetit2006}. Despite their successes (e.g. in visual SLAM \cite{Davison2007, Se2005, Eade2008, Williams2011}), a central challenge in both approaches is the (often online) selection of keyframes and keypoints such that they provide good spatial coverage of the scene.
Departing from a model-based approach, \cite{Shotton2013} propose Scene Coordinate Regression (SCoRe) forests. They learn dense 2D pixel to 3D scene coordinate correspondences (relative to a scene-specific reference frame) using a dataset of RGB-D images and ground truth camera poses. With a sparse set of these correspondences, camera pose hypotheses are generated and a RANSAC-based optimization and refinement produces the final camera pose. A limitation of their formulation, however, is the inevitable many-to-one mapping of correspondences, making it poor at resolving scene ambiguities. \citet{Valentin2015} tackle this problem by replacing the single mode in the forest leafs with multi-modal mixtures of Gaussians, and then, at test time, using the support and spread of each mode to aid the follow-up RANSAC optimizer. Along the same vein of exploiting uncertainty for performance gains, \citet{Brachmann2016} introduce a stacked (auto-context) classification-regression version of the forest of \cite{Valentin2015}, which they use to achieve state-of-the-art results in object pose estimation and results on-par with \cite{Valentin2015} in camera localization. Unlike \cite{Valentin2015}, their results also include a case for RGB-only.
Inspired by successes in deep learning for a multitude of tasks, \cite{Kendall2015, Kendall2016} were first to propose the use of a deep CNN as an end-to-end 6D camera pose regressor using RGB and RGB-D images as input. While they achieve moderate results in large-scale outdoor scenes, their performance in small, indoor scenes is one order of magnitude worse in pose translation error than \cite{Brachmann2016} and \cite{Valentin2015}. This suggests that the intermediate step of predicting 2D-to-3D point correspondences is important for good camera pose estimation.
\subsection{Random Forests as Neural Networks}
\citet{Sethi1990}, \citet{Welbl2014} propose a class of two-hidden-layer NNs whose architecture and initialization can be derived from a trained decision tree. The motivation behind this mapping is that it transforms a RF into a differentiable, end-to-end learnable structure which can then be further refined (i.e. fine-tuned) from a data-driven starting point when only a small amount of training data is available. \citet{Richmond2015} extend this mapping to a stacked auto-context RF, and use the equivalent deep and sparsely connected CNN to achieve improved results over the original RF stacks on two small-scale segmentation tasks. They additionally propose an approximate reverse mapping, from CNN to RF, such that the optimized network parameters can be used to update the original RF (its topology fixed) for faster test-time evaluation. Their optimized RF outperforms the original RF on the segmentation tasks, but not the mapped CNN. Our work builds on this and employs the forward mapping on a non-stacked RF trained on 2D-to-3D scene coordinates.
\subsection{Globally differentiable Random Forests}
RFs are capable of handling high-dimensional data and multi-class problems, are fast to train and test, and do not require large amounts of training data. RFs, however, are trained in a greedy fashion, with each split node optimizing some local splitting criterion conditioned on the samples that it receives. Additionally, each tree is trained independently and hence there is no principled minimization of a global loss objective across the whole forest. To address the limitations of greedy forest construction, inspired by boosted methods, \cite{SchulterADF2013} grow a forest breadth-wise, layer by layer, with the current layer's split node parameters optimized based on an across-forest loss function. \citet{Ren2015} instead propose a global post-hoc refinement and pruning of a greedily-trained RF using all training samples. They re-learn all leaf parameters and generate more compact RFs with greater generalisation capacity. \citet{Norouzi2015} look at the optimization of a single tree and present a convex-concave upper bound which they use as a proxy for a global loss function. They optimize a tree's (oblique) split functions jointly with its leaf parameters using stochastic gradient descent, imposing regularization constraints such that the hard binary decisions of the split nodes and the 1-of-$l$ encoding of leaf membership, where $l$ is the number of leafs, is preserved. Fuzzy or soft decision trees \cite{Suarez1999} relax this constraint and model split nodes as sigmoidal functions: a training sample is routed with some probability $p$ to its left child, and $1-p$ to its right child. This allows samples to be assigned partial membership to all leaf nodes rather than full membership to a single one. Much like the tree-to-NN mapping of \cite{Sethi1990}, this fuzzification allows for the otherwise local and greedy tree construction to be reframed as a (tree-wide) differentiable optimization problem. More recently, \cite{Kontschieder2015} unify the feature learning capability of deep CNNs with the ensemble nature of RFs. Similar to \cite{Suarez1999} in their use of sigmoidal split node functions, they propose a globally differentiable RF by appending a RF to a deep CNN such that the activation of each CNN output node drives the soft decision function of a RF split node. In our work, using the mapping of \cite{Sethi1990}, we explore all variants of this: learning leaf parameters only, learning leaf and split node parameters, and finally relaxing the tree constraint and allowing for leaf, split and the tree topologies to be learned.
\section{Random Forests to ForestNets} \label{subsec:RF2CNN}
In this section, we present the scene coordinate regression forests of \cite{Shotton2013} and show how they can be applied to the tree-to-NN mapping of \cite{Sethi1990, Welbl2014} for multi-dimensional regression.
\subsection{Scene coordinate regression}
For each 2D pixel in an image $\mathbf{p} \in \mathbb{R}^2$, there exists a 3D scene coordinate label $\mathbf{m} \in \mathbb{R}^3$ (see Fig.~\ref{fig:coolimage}). We can compute $\mathbf{m}$ as $\mathbf{m} = \mathit{H} \mathbf{x}$, where \textit{H} is the 6D camera pose of the image and $\mathbf{x}$ is the 3D camera coordinate obtained by back-projecting pixel $\mathbf{p}$ using its depth value. A RF is greedily trained with a set of $(f(\mathbf{p}), \mathbf{m})$ pairs, where $f(\cdot): \mathbb{R}^2 \rightarrow \mathbb{R}^D$ is the $D$-dimensional feature vector of a pixel (here $D=1000$). In training, each split node is assigned $(\boldsymbol {\theta}_n, \tau_n)$ from a pool of $N$ candidates, where $\boldsymbol{\theta}_n$ are the feature parameters (see below), $\tau_n$ is a scalar threshold and $n \in \{1,.., N\}$. Each leaf node $l$ then stores some empirical 3D distribution $P_{l}(\mathbf{m})$ of the samples that have arrived at that leaf. In \cite{Shotton2013}, each leaf distribution is represented as a set of modes found using mean-shift. \citet{Brachmann2016} and \citet{Valentin2015} instead fit 3D mixtures of Gaussians to each leaf.
More formally, the decision function at split node $s$ takes the form:
\begin{equation}
df(\mathbf{p}; \boldsymbol{\theta}_n, \tau_n) = \Big[ f_{\boldsymbol{\theta}_n} (\mathbf{p}) \geq \tau_n \Big]
\end{equation}
where $[ \cdot ]$ evaluates the boolean condition and $f_{\boldsymbol{\theta}_n}(\cdot)$ is the feature response function. If $df(\mathbf{p}; \boldsymbol{\theta}_n, \tau_n) = 0$, the sample is routed to the left child, and if $df(\mathbf{p}; \boldsymbol{\theta}_n, \tau_n) = 1$ then it is routed to the right child. In general, $f_{\boldsymbol{\theta}_n}(\cdot)$ can be a function of multiple features, however, here we opt for simple axis-aligned splits: each split node selects a single feature $d \in \{1,\cdots,D\}$ and compares it to its threshold $\tau_n$. In our case, features are simple differences of RGB pixel values:
\begin{equation}
f_{\boldsymbol{\theta}_n}(\mathbf{p}) = \mathcal{I}(\mathbf{p} +\boldsymbol{\delta}_1, c_1) - \mathcal{I}(\mathbf{p} + \boldsymbol{\delta}_2, c_2)
\end{equation}
where $\mathcal{I}$ is the image intensity value of pixel $\mathbf{p}$ offset by $\boldsymbol{\delta}_i = (x_i, y_i)$ and indexed by channel $c_i \in \{R, G, B\}$. Thus the feature parameters are $\boldsymbol{\theta}_n = \{\boldsymbol{\delta}_1, \boldsymbol{\delta}_2, c_1, c_2\}$. \\
\subsection{RF to NN mapping}
As highlighted in \cite{Sethi1990}, the mapping from RF to NN is useful in two ways:
\begin{itemize}
\item It defines a specific NN architecture in terms of the number, structure and connections of the nodes
\item It uses the learned trees' structures and leaf distributions to initialize all network parameters.
\end{itemize}
We wish to take a trained RF for scene coordinate regression and construct an ensemble of NNs. The following describes the mapping of a single tree to a single two-hidden-layer NN (Fig.~\ref{fig:basicRF2NN})
Let a tree consist of split nodes $\tilde{s}_j \in \mathcal{\tilde{S}}$ and leaf nodes $\tilde{l}_k \in \mathcal{\tilde{L}}$. The corresponding NN will have its first layer evaluating the decision functions $df(\cdot)$ of all of the split nodes in the tree simultaneously, and its second layer encoding the leaf membership of each sample (that is, to which leaf each sample is routed).
\textbf{\textit{Split layer}}.
The split nodes $\tilde{s}_j$ of the tree form the first network layer, $L_1$, consisting of equivalent split neurons $s_j$ whereby a tree with $J$ split nodes will have $J$ neurons in $L_1$. As input to the network, each pixel $\mathbf{p}$ provides its feature vector $f(\mathbf{p}) \in \mathbb{R}^D$. Based on each split node's associated $(\boldsymbol{\theta}_n, \tau_n)$, its corresponding split neuron $s_j$ selects the feature with index $d \in \{1,...,D\}$ such that its activation $a(s_j)$ behaves as follows:
\begin{equation}
a(s_j) =
\begin{cases}
-1, & \text{if}\ f_{\boldsymbol{\theta}_n}(\mathbf{p}) < \tau_n\\
1, & \text{if}\ f_{\boldsymbol{\theta}_n}(\mathbf{p}) \geq \tau_n
\end{cases}
\end{equation}
corresponding to routing pixel $\mathbf{p}$ to the left or right child, respectively. To implement this, a sparse connection matrix is defined between the network input and $L_1$ such that the $dth$ channel of the input vector is connected to neuron $s_j$ with weight $w_{d, s_j} = c_{01}$, where $c_{01}$ is some constant value. All other incoming weights to $s_j$ are zero (purple to green connections in Fig.~\ref{fig:basicRF2NN}). The bias of neuron $s_j$ is $b_{s_j} = - c_{01} * \tau_n$. Implemented as a simple linear layer, each neuron $s_j$ in $L_1$ thus computes:
\begin{equation}
c_{01} * f_{\boldsymbol{\theta}_n}(\mathbf{p}) - c_{01} * \tau_n
\end{equation}
and this result is assigned to $a(s_j)$ where $a(\cdot) = tanh(\cdot)$. Layer $L_1$'s activation pattern thus encodes the to-left-child or to-right-child evaluation of each split node in the tree with one forward pass through $L_1$.
The hardness of the tree's decision functions are controlled by the hyperparameter $c_{01}$: a high value of $c_{01}$ ensures that $a(s_j)$ approaches $-1$ very closely if $\tilde{s}_j$ routes a sample left, and $a(s_j)$ approaches $+1$ very closely if $\tilde{s}_j$ routes a sample right. A lower value of $c_{01}$ allows for samples to be routed partially left and right.
\begin{figure}[h]
\centering
\vskip 2mm
\includegraphics[scale=0.25]{images/basicRF2NN.pdf}
\caption{\textbf{Forward mapping of a RF to a NN.} (Top) A single tree is mapped to a two-hidden-layer NN. At test time, each split node performs a single feature selection and a binary threshold test before routing a sample onwards. This is mimicked in the NN through the activation of specific connections. (Bottom) The single tree-to-NN mapping is replicated for each of the $T$ trees in a forest. Note: bias nodes are not shown here for readability. Purple indicates feature vector. Green indicates split layer. Red indicates leaf layer. Blue indicates output scene coordinate layer. Orange indicates robust geometric median applied to multiple NN outputs. Dotted lines indicate activation functions ($L_1: tanh(\cdot)$ and $L_2:sigm(\cdot)$ or $softmax(\cdot)$). Best viewed in color.}
\label{fig:basicRF2NN}
\end{figure}
\textbf{\textit{Leaf layer}}.
$L_2$, the second layer of the network, is constructed from all the leaf nodes ${\tilde{l}_k}$ in the tree, such that a tree with $K$ leaves will have $K$ leaf neurons in $L_2$. This layer must interpret the activation pattern of $L_1$ such that $L_2$ encodes the leaf to which each sample is directed.
To implement this, a sparse set of connections is constructed between $L_1$ and $L_2$ (green to red connections in Fig.~\ref{fig:basicRF2NN}). A connection exists between a split neuron $s_j$ and a leaf neuron $l_k$ if $s_j$ is on the path to $l_k$ in the tree. This means that a leaf node at depth $r$ in the tree, corresponds to a leaf neuron with $r$ incoming connections in the NN.
The weights of these connections are such that if $\tilde{l}_k$ is in the left sub-tree of $\tilde{s}_j$ then $w_{s_j, l_k} = + c_{12}$, otherwise $w_{s_j, l_k} = - c_{12}$, where $c_{12}$ is again some constant. If $\tilde{s}_j$ is not on the path to $\tilde{l}_k$, then no connection exists (or equivalently, $w_{s_j, l_k} = 0$). With this formulation, the active leaf has incoming weights that are sign-matched to the activations on $L_1$ of the split nodes on its path. This makes the leaf neuron maximally activated. To distinguish leafs at different depths, the bias of a leaf neuron is such that $b_{l_k} = - c_{12} * (|P(\tilde{l}_k)| - 1)$ where $P(\tilde{l}_k)$ is the path length to $\tilde{l}_k$. This sets the value of the active leaf neuron to $+c_{12}$, and the values of all other leafs to $\leq -c_{12}$. The activation function $a(\cdot) = sigm(\cdot)$ is then applied to $L_2$ resulting in the $L_2$ activation pattern behaving like a binary switch, where the active leaf neuron $a(l_k)$ is $+1$, and all others are set to zero. This behaviour mimics a hard tree and can be enforced by assigning a high value to $c_{12}$. By relaxing $c_{12}$ (and $c_{01}$) and replacing $a(\cdot)$ with a \textit{softmax} activation, we can relax tree hardness.
\textbf{\textit{Scene coordinate layer}}.
Each leaf $\tilde{l}_k$ in the RFs of \cite{Brachmann2016} contains a mixture of Gaussians over 3D scene coordinates $\mathbf{m}$. Let the mode with the highest support in leaf $\tilde{l}_k$, denoted $\mathbf{m}_k \in \mathbb{R}^3$, become the 3D scene coordinate representing that leaf. At test time, a sample reaching leaf $\tilde{l}_k$ will take on label $\mathbf{m}_k$. To implement this, the network output consists of three nodes, $\mathbf{q} \in \mathbb{R}^3$, which are fully connected to all leaf neurons in $L_2$ (red to blue connections in Fig.~\ref{fig:basicRF2NN}). The weights of the three connections between leaf neuron $l_k$ and the three output nodes $\mathbf{q}$ correspond to the $x, y$ and $z$ entries of $\mathbf{m}_k$. This can be denoted as $w_{l_k, \mathbf{q}} = \mathbf{m}_k$. The output bias is $b_{\mathbf{q}} = \mathbf{0}$. With this formulation, for a given sample, a series of zeros and a single $+1$ on the active leaf in $L_2$ result in the $\mathbf{m}_k$ of that active leaf appearing on the output nodes $\mathbf{q}$.
\textbf{\textit{Trees to ForestNets}}.
This mapping naturally extends itself to a RF with multiple trees where for a forest of $T$ trees, an ensemble of $T$ NNs can be constructed, collectively referred to as a ForestNet (Fig.~\ref{fig:basicRF2NN}). At test time, the $T$ predictions from a RF or its equivalent ForestNet can be used independently or combined to produce a single 3D prediction.
\textbf{\textit{Feature learning}}.
In recent years, it has become evident that the power of neural networks lies in their representation learning capability. Our framework is amenable to this by prepending feature learning layers to a ForestNet and by allowing linear combinations of features (purple weights) to be learned for each split node. We, however, leave this to future work, with the focus here being on the usefulness of the tree-to-NN mapping.
\section{Robust Geometric Median Averaging} \label{subsec:geomavg}
A ForestNet of $T$ NNs produces $T$ scene coordinate predictions $\mathbf{q}_{1}, \mathbf{q}_{2}, \dots ,\mathbf{q}_{T}$ which can be robustly averaged to produce a single scene coordinate, denoted $\mathbf{\tilde{q}} \in \mathbb{R}^3$. The robust average we use is
a variant of the geometric median\footnote{An extension of the median to
higher dimensions. It is defined as the point minimizing the sum of
Euclidean distances to all points in a set of discrete sample points.}.
For simplicity, we refer to it as simply the geometric median.
We start by calculating the mean of the
original predictions. With this as initialization, we continue
with a fixed number of steps of an iteratively re-weighted least squares
algorithm.
Each iteration calculates a weighted average:
\begin{equation}
\mathbf{\tilde{q}}^{t+1}= \frac{\sum_{i=1}^{T} w_i^t \mathbf{q}_i }
{\sum_{i=1}^{T} w_i^t }
\end{equation}
of the original predictions. For the first 10 iterations we use the weights:
\begin{equation}
w^t_i=\frac{1}{\| \mathbf{\tilde{q}}^{t} - \ \mathbf{q}_i \|_2}
\end{equation}
This is equivalent to the Weiszfeld algorithm \cite{Weiszfeld2009}
which approximates the geometric median. While this result can
be used directly as a robust average, we find that it is beneficial to
apply a further 10 iterations using weights with the form:
\begin{equation}
w^t_i=\exp(- \frac{\| \mathbf{\tilde{q}}^{t} - \mathbf{q}_i \|_2^2}
{2\sigma^2} )
\end{equation}
corresponding to a mean-shift algorithm with a Gaussian kernel of
standard deviation $\sigma$ (here $\sigma=2.5cm$). The iterations converge to a locally
dominant mode. The result of the last iteration is our final robust
average $\mathbf{\tilde{q}}$.
Note, that since each of the iterations above is simply a weighted average, the entire process is fully differentiable which allows us to
implement the robust average as a multi-layer module (which we call
GM) which has no learnable parameters. We investigate the use of our robust averaging in two
different settings:
\begin{enumerate}
\item The GM module is appended to the ForestNet and the full network
trained end-to-end. We call this eGM (for end-to-end).
\item Each ForestNet tree is trained independently and the GM module is appended post-hoc at test-time. We
call this pGM (for post-hoc).
\end{enumerate}
We apply these two types of robust averaging to all ForestNet variants,
as well as to a modified RF from \cite{Brachmann2016}. Note, however,
with an RF averaging can only be done post-hoc (see RF2-pGM in Section~\ref{subsec:methods}).
\section{Network Splitting}\label{subsec:NNsplitting}
Although the RF-to-NN mapping enforces sparse connectivity between layers $L_1$ and $L_2$, sparse operation on GPU is not always efficient. For this reason, $L_1$ and $L_2$ are implemented as fully connected layers with the inactive connections set and held at zero. The state-of-the-art results of \cite{Brachmann2016} use RFs of $3-5$ trees with depths between $15$ and $16$. This corresponds to $\mathtt{\sim}37 000$ split and $\mathtt{\sim}37 000$ leaf nodes per tree. This constructs a network on the order of $1.4$ billion parameters per tree. Practically, with floating-point operations, this requires $5.6GB$ of memory per tree, rapidly making the full ForestNet difficult to fit on a single GPU.
\begin{figure}[thpb]
\centering
\vskip 4mm
\includegraphics[scale=0.27]{images/splitnetworks.pdf}
\caption{\textbf{Network splitting for efficient GPU training/testing}. Trained trees are decomposed into sub-trees and mapped to NNs with weights shared across the common root nodes. (Left) A tree is split into 2 sub-trees ($r=3, \tilde{r}=2$). (Right) The construction of its equivalent NN. The weight of the root node (0) is shared between the two sub-NNs. Hatching direction indicates sub-tree membership. Best viewed in color.}
\label{fig:splitnetworks}
\end{figure}
To tackle this, a network splitting strategy is implemented whereby each trained tree is decomposed into a set of sub-trees, each mapped to a sub-NN with the weights of the sub-trees' root nodes shared across the sub-NNs (see Fig.~\ref{fig:splitnetworks}). If we let the original tree depth be $r$, and the depth of the sub-trees be $\tilde{r}$, such that $0 < \tilde{r} \leq r$, then the total number of parameters in the ForestNet can be reduced by a factor of:
\begin{equation}
\frac{2^r - 1}{2^{\tilde{r}} - 1 + r-\tilde{r}}
\end{equation}
Using this splitting strategy, we are able to reduce each full ForestNet from
$\mathtt{\sim}28GB$ to $\mathtt{\sim}7GB$, allowing for effective training and testing on a single GPU.\\
\section{EXPERIMENTS}
\label{sec:experiments}
\subsection{Datasets}
We use the 7-Scenes dataset \cite{Shotton2013} which contains RGB-D images captured with a handheld Kinect camera ($640\times480$ resolution) and associated ground-truth camera pose calculated using the Kinect Fusion implementation of \cite{Newcombe2011}. RGB and depth images are manually registered. Depth images are used to pre-compute the ground truth scene coordinates per frame, however, only RGB images are used at test-time.
\subsection{Hardware}
We train and test all NNs on \textit{NVIDIA GeForce GTX TitanX} and all RFs on \textit{Intel}{\textregistered}Core{\texttrademark} i7-3770K CPU $@ 3.5GHz$.
\subsection{Performance metrics}
We measure performance on (1) the accuracy of the scene coordinate predictions, and (2) the final camera pose accuracy. We quantify (1) in terms of the number of scene coordinate predictions considered to be inliers and the mean Euclidean distance of these inliers from their ground truth scene coordinate labels. A prediction is counted as an inlier if its Euclidean distance to its ground truth label is $< 10cm$. We quantify (2) in the same way as \cite{Shotton2013, Brachmann2016, Valentin2015} with a $5cm \; \& \; 5^\circ$ criterion: the camera pose is taken to be correct if it is within $5cm$ translation and $5^{\circ}$ rotation from the ground truth camera pose. We report the percentage of correct camera poses as well as the average median camera error (we calculate the median per scene and average across scenes).
\subsection{Methods}
\label{subsec:methods}
We define the following baselines:
\begin{itemize}
\item The current state-of-the-art in camera localization \cite{Brachmann2016} which we refer to as \textbf{RF1}.
\item A modified version of RF1 which we call \textbf{RF2}. RF2 has 5 rather than 3 trees, no auto-context and its leafs hold only the single highest-supported mode. RF2 is used to map to the ForestNets.
\item A deep CNN trained to predict 2D-to-3D scene coordinates in a patch-like manner. This CNN's architecture is based on AlexNet \cite{Krizhevsky2012} but is trained from scratch. Instead of using the standard two channel architecture, we use a single channel with double capacity (96 filters in the first \textit{conv} layer and 4096 neurons in the fully-connected layers). We also replace the 1000 neuron output layer with 3 neurons to predict a single 3D scene coordinate. We refer to this as \textbf{D-NET} (for deep).
\end{itemize}
We also explore the following three variants of ForestNets:
\begin{itemize}
\item \textbf{ForestNet Leafs (fNET-L)} in which only the leaf weights (blue) are learned. fNET-L preserves the tree topologies and can be mapped back to a fast and memory-efficient RF with optimized leaf modes.
\item \textbf{ForestNet Leafs \& Splits (fNET-LS)} in which the leaf parameters and the split node thresholds (i.e. the biases between the network input and $L_1$) are learned. fNET-LS also preserves tree topology, however, since the split thresholds can be adjusted, it is likely that multiple leaf nodes are activated. Because of this, mapping back to an RF is possible but only an approximation \cite{Richmond2015}.
\item \textbf{ForestNet Leafs, Splits \& Topology (fNET-LST)} in which the leaf parameters, split node thresholds and the tree topologies (red weights) are learned. fNET-LST cannot be mapped back to a RF since the tree structures are not preserved.
\end{itemize}
For all of our methods, during the second stage of RANSAC-based pose optimization, we increase the number of pose hypotheses drawn from 256 (as used in \cite{Brachmann2016}) to 1280 to increase the chance of finding a good solution.
\def\arraystretch{1
\begin{table*}[thpb]
\begin{center}
\centering
\vskip 2mm
\caption{\scriptsize{\linespread{0.5}Coordinates: mean inlier count (brackets show mean Euclidean distance error of inliers). Camera pose: median translation and rotation error, averaged over scenes (brackets show percentage of poses meeting $5cm \; \& \; 5^\circ$ criterion)}}
\label{tab:quants}
\begin{tabular}{|M{0.15\columnwidth}|M{0.15\columnwidth}||M{0.22\columnwidth}|M{0.22\columnwidth}|M{0.22\columnwidth}|M{0.22\columnwidth}|M{0.23\columnwidth}|M{0.22\columnwidth}|M{0.22\columnwidth}|}
\cline{1-9}
\multicolumn{1}{|c|}{} & & RF1 \cite{Brachmann2016} & RF2 & fNET-L & fNET-LS & fNET-LST & D-NET & PoseNet \cite{Kendall2015}\\ \cline{1-9}
Overview & Description & state-of-the-art RF \cite{Brachmann2016} & RF of \cite{Brachmann2016} with adapted parameters &fNET which learns leafs. Can be mapped-back to RF & fNET which learns leafs \& splits. Can be approximately mapped back to RF & fNET which learns leafs, splits \& tree topology. Cannot be mapped back to RF & NN for dense scene coordinate regression & NN for direct 6D camera pose regression \\ \cline{2-9}
\multicolumn{1}{|c|}{} & Speed & fast & fast & fast & slow* & slow & slow & slow \\ \cline{2-9}
\multicolumn{1}{|c|}{} & Memory & low & low & low & high* & high & high & high \\ \hline \hline
Accuracy w.r.t & noGM & 21.6\% (5.2cm) & 18.3\% (4.9cm) & 16.8\% (4.9cm) & 13.4\% (5.6cm) & 17.3\% (5.6cm) & 45.3\% (4.7cm) & n/a \\
Scene & pGM & n/a & 26.1\% (4.7cm) & 23.6\% (4.7cm) & 17.9\% (5.4cm) & 21.1\% (5.5cm) & n/a & n/a \\
Coordinates& eGM & n/a & n/a & 24.6\% (4.8cm) & 21.0\% (4.9cm) & 12.9\% (5.5cm) & n/a & n/a \\
\hline
Accuracy w.r.t. & noGM & $6.1cm\;2.7^\circ$ (55.2\%) & $4.2cm\;2.1^\circ$ (62.6\%) & $4.6cm\;2.2^\circ$ (61.3\%) & $7.6cm\;2.9^\circ$ (36.7\%) & $6.2cm\;2.9^\circ$ (50.5\%) & $4.6cm\;2.1^\circ$ (57.7\%) & $44.0cm\;10.4^\circ$ \\
Camera & pGM & n/a & $3.8cm\;1.9^\circ$ (64.5\%) & $3.9cm\;1.9^\circ$ (63.9\%) & $6.7cm\;2.5^\circ$ (39.1\%) & $5.3cm\;2.6^\circ$ (52.8\%) & n/a & n/a \\
Pose & eGM & n/a & n/a & $4.4cm\;2.1^\circ$ (62.0\%) & $4.5cm\;2.2^\circ$ (60.4\%) & $12.2cm\;7.0^\circ$ (39.6\%)& n/a & n/a \\
\cline{1-9}
\cline{1-9}
\end{tabular}
\vskip 0.2cm
{*Approximate map back to fast, memory-efficient RF is possible, but with potential performance loss}
\end{center}
\vskip -5mm
\end{table*}
\subsection{Training}
\label{subsec:training}
We train ForestNets with a loss on the Euclidean distance between the prediction $\mathbf{q}$ (or robust prediction $\mathbf{\tilde{q}}$) and ground truth scene coordinate $\mathbf{m}$:
\begin{equation}
\mathcal{L} = \sum_{i=1}^N \| \mathbf{q} - \mathbf{m} \|_2
\end{equation}
where $N$ is the number of training samples. Training is done using standard backpropagation via stochastic gradient descent, with batches of size 20 and a learning rate of 0.001.
\section{Results}
Table~\ref{tab:quants} shows our quantitative results. We summarize our key observations as follows:
\begin{itemize}
\item Of the fast methods, RF2-pGM (RF with post-hoc geometric median) is the best performing. It supersedes the current state-of-the-art of \cite{Brachmann2016} by 7.8\% in inlier count and 9.3\% in proportion of correct final camera poses.
\item Of the ForestNet variants, fNET-L achieves the best inlier count of 24.6\% (with eGM) and the best camera pose error of $3.9cm \: 1.9^\circ$ (with pGM). While its inlier count is not as good as RF2-pGM, fNET-L-pGM achieves on-par performance with RF2-pGM in the final camera pose accuracy. fNET-L-pGM can thus be used as an equivalent and importantly differentiable replacement of a traditional RF. This would allow, for example, a ForestNet to be prepended to a differentiable RANSAC module (see recent work by \cite{Brachmann2016b}) thus making a complete end-to-end differentiable camera pose estimation pipeline.
\item Our robust geometric median filter offers notable performance gains. Across all RF and ForestNet methods, relative to noGM, inlier counts improve by 5.7\% with pGM and 3.7\% with eGM. The proportion of correct final camera poses improves by 2.4\% with pGM and 4.6\% with eGM. The non-deterministic nature of RANSAC obscures the true reason for the improvement of eGM over pGM in final camera pose (since it is inferior in inlier count performance), however it is clear that geometric median robust averaging offers gains over using no robust averaging at all.
\item D-NET supersedes all of the noGM methods in terms of scene coordinate performance, with a significantly higher number of inliers than both the RFs and ForestNets. This, however, does not translate to D-NET having the best camera pose accuracy. Overall, we observe that scene coordinate accuracy and final camera pose accuracy are only mildly correlated, for the reason mentioned above.
\item The two-part pipeline with intermediate scene coordinate prediction has significant gains over direct camera pose regression. All our methods exceed the final camera pose accuracy of \cite{Kendall2015} by an order of magnitude.
\end{itemize}
\textbf{Speed and memory}. A 5-tree RF has a model size of $\sim$250MB and can obtain dense coordinate predictions for a frame on CPU in 100-150ms (equivalently 5-10ms on GPU). This is low memory and fast. An equivalent ForestNet has a model size of $\sim$7GB (although many parameters are fixed at zero) and it can densely process a frame in a patch-like manner in 5-7 minutes on GPU. This is high memory and slow. If the ForestNet can be mapped back to an RF (like fNET-L and fNET-LS) then it can be transformed to a fast and lightweight version of itself. A D-NET can be stored in $\sim$500MB and can densely process an image in 1-2s. Unlike a ForestNet, however, a D-NET cannot be mapped to a RF.
\textbf{Learning tree parameters}. The RFs of \cite{Brachmann2016} and \mbox{fNET-L} differ in the way that their leaf modes are optimized: with fNET-L, only the leaf weights are optimized during training. With our training loss, we optimize the sum of Euclidean distances of all samples reaching a leaf. This is equivalent to calculating the geometric median of all samples in that leaf. This is different (and seemingly inferior) to running mean-shift on the set of samples and choosing the mode with largest support, as the RFs do. Although the geometric median is robust to outliers, they still influence the prediction to some extent, which is not the case with mean-shift. This interpretation no longer holds when we train with the appended geometric median layer (eGM). In this case, the leaf predictions can adapt in a way that the resulting geometric median is accurate. In general, this improves the predictions' accuracy. We also observe that initialising the ForestNets as soft trees and allowing them to learn more parameters (fNET-LS and fNET-LST) generally negatively affects the scene coordinate and camera pose accuracy. This suggests that with too many degrees of freedom, the ForestNets cannot be properly optimized from their RF-initialized starting points.
\textbf{Scene coordinates to camera pose}. The nature of the relationship between scene coordinate accuracy and final camera pose accuracy is unclear in that better scene coordinates do not always result in better camera poses. D-NET produces a much higher number of inliers (45.3\% versus 18.3\% of RF2 and 16.8\% of fNET-L) with a better inlier error (4.7cm versus 4.9cm) yet does not achieve the best final camera pose. We attribute this to the non-deterministic and highly robust RANSAC-based optimizer.
\section{Conclusions and Future Work}
\label{sec:conslusions}
In this work we explored efficient versus non-efficient and RF- versus non-RF-derived NN architectures for camera localization. While a traditional NN architecture is superior with respect to dense scene coordinate regression, its inefficiency in terms of speed and memory makes it unattractive for mobile and robotics applications. On the other hand, our best-performing ForestNet, with a robust 3D geometric median-based average, is test-time efficient since it can be mapped back to a RF, and has improved the current state-of-the-art in camera localization on the 7-Scenes dataset \cite{Shotton2013}. Overall, however, the nature of the relationship between good scene coordinate predictions and a good final 6D camera pose is not yet clear. This motivates further research into (1) better suited loss functions for the follow-up RANSAC optimizer, and (2) the end-to-end training of a full camera pose estimation pipeline.
\addtolength{\textheight}{-9cm}
\vspace{-2mm}
\section*{Acknowledgment}
\vspace{-2mm}
This work was supported by the EPSRC, ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1, EPSRC/MURI grant EP/N019474/1 and The Skye Foundation.
\vspace{-5mm}
\bibliographystyle{IEEEtranN}
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Toddlers Dress and Trousers New Look Pattern No. 6219. Age 6 months to 4 years.
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New Look Sewing Pattern No. 6219.
You're reviewing: Toddlers Dress and Trousers New Look Pattern No. 6219. Age 6 months to 4 years.
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\section{Isolating single charges}
Control of the double dot using dc gate voltages is demonstrated
in Fig.\ 1 (b--d). Figure 1(b) shows $dG_{S2}$/$dV_L$ (numerically
differentiated) as a function of $V_R$ and $V_L$. When an electron
enters or leaves the double dot, or moves from one dot to the
other, the QPC conductance changes. Gate voltage derivatives of
$G_{S1}$ and $G_{S2}$ clearly show these changes and map out the
double dot charge stability diagram
\cite{Field_PRL_1993,Elzerman_PRB_2003}. The nearly horizontal
lines are due to charge transitions in the left dot, while the
nearly vertical lines correspond to charge transitions in the
right dot. For very negative values of $V_L$ and $V_R$ (see the
lower left corner of the charge stability diagram) charge
transitions no longer occur, indicating that the double dot is
completely empty, denoted (0,0). Transport through the double dot
can be correlated with simultaneous charge sensing measurements.
Figure 1(c) shows a color scale plot of $G_D$ near the (1,0) to
(0,1) charge transition. A charge stability diagram,
simultaneously acquired, is shown in Fig.\ 1(d). Near the (1,0) to
(0,1) charge transition the system behaves as an effective
two-level system. Crossing this transition by making $V_L$ more
positive transfers a single electron from the right dot to the
left dot.
\section{Microwave manipulation of a single charge}
Near the (0,1) to (1,0) interdot transition, the double dot forms
a two-level charge system that can be characterized by the
detuning parameter, $\epsilon$, and the tunnel coupling, $t$ (see
inset of Fig. 2(b)) \cite{Van_der_Wiel_RMP_2003}. We have used
microwave spectroscopy to characterize this two-level system
\cite{Petta_PRL_2004}. Microwaves drive transitions in the double
dot when the photon frequency is equal to the energy separation
between the (1,0) and (0,1) charge states
\cite{Stafford_PRL_1996,Stoof_PRB_1996,Brune_PhysicaE_1997}. This
microwave-induced charge state repopulation can be directly
measured using the QPC charge sensors
\cite{Van_der_Wal_Science_2000,Lehnert_PRL_2003}. The black curve
in Fig.\ 2(a) shows the measured charge on the left dot, $M$, as a
function of $\epsilon$, in the absence of microwave excitation. As
expected, increasing $\epsilon$ transfers a single charge from the
left dot to the right dot. Application of microwaves to gate A
results in resonant peaks in $M$ vs.\ $\epsilon$ that move to
larger $|\epsilon|$ with increasing frequency. This resonant peak
corresponds to a single photon process that drives an electron
from the (1,0) ground state (for negative $\epsilon$) into the
(0,1) excited state, or vice versa.
The frequency dependence of the resonance condition can be used to
map out the energetics of the charge two-level system. Detailed
measurements of the resonant peak position as a function of
microwave frequency, $f$, are used to extract $t$ for various
$V_T$ (see Fig.\ 2(b)) \cite{Oosterkamp_Nature_1998}. At high
frequencies the peak positions move linearly with $f$. For small
frequencies, probing the region near the (0,1)-(1,0) charge
transition, the interdot tunnel coupling modifies the linear
dependence. Changing the interdot tunnel coupling modifies the
frequency dependence of the resonant peak position. For each value
of $V_T$, the experimental data have been fit using $\alpha
\epsilon$= $\sqrt{(hf)^2-(2t)^2}$, where $\alpha$ is the lever
arm. $\alpha$ and $t$ were used as free parameters for each curve.
The lever arm $\alpha$ changes by $\sim$20$\%$ over the range of
$V_T$ used in Fig.\ 2. The experimental data are well fit by
theory and show that the tunnel coupling varies by roughly a
factor of 6 when $V_T$ is changed by 70 mV. Measurements of $t$
from microwave spectroscopy are consistent with values obtained by
measuring the width of the interdot charge transition using charge
sensing \cite{DiCarlo_PRL_2004,Petta_PRL_2004}.
\begin{figure}[t]
\vspace{1.3 cm}
\begin{center}\leavevmode
\includegraphics[width=1\linewidth]{EP2DSFig2.eps}
\vspace{-1.7 cm}\caption{Microwave spectroscopy of a one-electron
double dot. (a) Charge occupancy of the left dot, $M$, as a
function of $\epsilon$ for several microwave frequencies. (b)
One-half of the resonance peak splitting as a function of $f$ for
several values of $V_T$. Solid lines are best fits to the
experimental data using the theory outlined in the text. Inset:
two-level system energy level diagram. (c) Amplitude of the
resonance, expressed as $M_{max}$($\tau$)/$M_{max}$($\tau$=5 ns),
as a function of chopped cw period, $\tau$, with $f$=19 GHz.
Theory gives a best fit $T_1$=16 ns (solid line, see text). Inset:
Single photon peak shown in a plot of $M$ as a function of
$\epsilon$ for $\tau$=5 ns and 1 $\mu$s. (d) Power dependence of
the resonance for $f$=24 GHz. Widths are used to extract the
ensemble-averaged charge dephasing time $T_{2}^{*}$. At higher
microwave powers multiple photon processes occur. Curves are
offset by 0.3 for clarity.}
\label{figurename}\end{center}\end{figure}
Charge relaxation and decoherence times can be extracted by
analyzing the resonant response of the two-level system, as used
in the analysis of the Cooper pair box \cite{Lehnert_PRL_2003}.
The charge relaxation time $T_1$ is determined by measuring the
resonance peak height as microwaves are chopped at varying
periods, $\tau$, with a 50\% duty cycle \cite{Marki}. The system
response is modelled with a saturated signal while microwaves are
present, followed by an exponential decay with a characteristic
time scale $T_1$ when the microwaves are turned off. Calculating
the time averaged occupation, we expect:
\begin{equation}
\frac{M_{max}(\tau)}{M_{max}(0)}=\frac{1}{2}+\frac{T_1(1-e^{-\tau/(2
T_1)})}{\tau}
\end{equation}
With long periods ($\tau$$\gg$$T_1$), the exponential tail due to
the finite relaxation time is an insignificant part of the duty
cycle, and the charge detector measures the time average of the
on/off signal, giving a resonant feature with half the height
found in the limit $\tau$$\rightarrow$0. When the period is very
short, such that $\tau$$\ll$$T_1$, the charge has little time to
relax, and the charge detector response is close to saturation
(saturation is defined as $M_{max}$=0.5 on resonance). In the
intermediate regime where $\tau$$\sim$$T_1$, the QPC signal is
strongly dependent on $\tau$. We present data for $\tau\geq$5 ns
to avoid artifacts due to the finite rise time of the mixer
circuit. In Fig. 2(c), we plot
$M_{max}$($\tau$)/$M_{max}$($\tau$=5 ns) as a function of $\tau$.
Agreement between experiment and theory is good and gives a best
fit $T_1$=16 ns.
The resonance peak width gives a direct measure of the
inhomogeneous charge decoherence time, $T_{2}^{*}$
\cite{Lehnert_PRL_2003,Abragam}. In Fig.\ 2(d) we plot $N$ as a
function of $\epsilon$ for several microwave powers. At low power,
only the single-photon (1$\gamma$) peak is visible. As the power
is increased the 1$\gamma$ peak approaches saturation and a
two-photon peak develops \cite{population_inversion}. A fit to the
low power 1$\gamma$ peak using a Gaussian function is shown in red
in Fig.\ 2(d). The best fit half-width of 0.077 mV corresponds to
an energy of 10.2 $\mu$eV when taking into account the lever arm.
Converting this into a time results in a lower bound
$T_{2}^{*}$=400 ps. This measurement of $T_{2}^{*}$ gives a
worst-case estimate since charge fluctuations will broaden the
resonant feature, resulting in a shorter $T_{2}^{*}$ value.
Our measurements of $T_{1}$ and $T_{2}^{*}$ using charge sensing
can be compared with other recent experiments
\cite{Hayashi_PRL_2003,Fujisawa_Nature_2002}. In a pulsed-gate
experiment, Fujisawa \textit{et al.} \cite{Fujisawa_Nature_2002}
have measured the energy relaxation time in a vertical quantum
dot. From a measurement of the transient current as a function of
pulse time they extract $T_1$=10 ns, which is limited by
spontaneous emission of a phonon. Direct observation of coherent
charge oscillations has been reported by Hayashi \textit{et al.}\
\cite{Hayashi_PRL_2003}. From the decay envelope of the Rabi
oscillations Hayashi \textit{et al.} extract a $T_{2}$ time of
$\sim$1 ns, which serves as an upper bound estimate for $T_{2}^*$.
The $T_{1}$ and $T_{2}^{*}$ values that we obtain from charge
sensing are in good agreement with the results of these previous
experiments.
\section{Triplet-singlet spin relaxation}
Spin physics can be studied in the one-electron regime at high
fields, where the spin-up and spin-down states are separated by
the Zeeman splitting, or in the two-electron regime with singlet
and triplet spin states. We focus on the two-electron regime,
where differences in the singlet-triplet splittings in the (1,1)
and (0,2) charge states can be put to use for spin state readout
and initialization. We show that singlet-triplet relaxation times
can be measured by implementing a charge pump experiment in the
two-electron regime. This measurement technique can be used to
measure the singlet-triplet relaxation time, $\tau_{ST}$, for
nearly degenerate singlet and triplet states, a regime in which
hyperfine mediated relaxation process are expected to be
important.
In the two-electron regime, charge transport in a double dot shows
a striking asymmetry in bias voltage due to spin selection rules
(Pauli blocking) \cite{Ono_Science_2002,Johnson_condmat}. The
asymmetry in charge transport is due to the large difference in
the singlet-triplet splittings for the (1,1) and (0,2) charge
states. In the weakly-coupled (1,1) charge configuration the
singlet and triplet states are nearly degenerate. However, two
tightly confined electrons in the (0,2) charge state result in a
singlet-triplet splitting $J$$\sim$400 $\mu$eV. At forward bias,
transitions from the (0,2) singlet state, (0,2)$_\textrm{S}$, to
the (1,1) singlet state, (1,1)$_\textrm{S}$, are allowed. However,
reverse bias (1,1) to (0,2) charge transitions are blocked if the
(1,1) state forms a triplet (1,1)$_\textrm{T}$ because the
(0,2)$_\textrm{T}$ state resides outside the transport window due
to the large singlet-triplet splitting in (0,2). This asymmetry
results in current rectification, which can be used for
spin-to-charge conversion and spin state readout.
Charge transitions are driven by applying pulses to gates L and R.
Experimental details concerning pulse calibration have been
previously published \cite{Petta_condmat}. In double dots, charge
can be pumped by pulsing gates around a triple point, e.g.
(0,1)$\rightarrow$(1,1)$\rightarrow$(0,2)$\rightarrow$(0,1). Our
spin relaxation measurement technique relies on the fact that
(1,1)$_T$ to (0,2)$_S$ transitions are spin blocked. Measuring
this charge transition probability as a function of time using
charge sensing allows a measurement of the spin relaxation time.
We demonstrate that the observed time dependence of the charge
sensing signal is due to spin blocked transitions.
\begin{figure}[t]
\vspace{1.3 cm}
\begin{center}\leavevmode
\includegraphics[width=1\linewidth]{EP2DSFig3.eps}
\vspace{-1.8 cm} \caption{Pulse gate techniques for measuring the
singlet-triplet relaxation time. (a) $G_{S2}$, as a function of
$V_L$ and $V_R$, measured while applying the forward pulse
sequence
(0,1)$\rightarrow$(1,1)$\rightarrow$(0,2)$\rightarrow$(0,1). The
pulse period $\tau$=10 $\mu$s and the perpendicular field
$B_{\perp}$=100 mT. We observe a charge sensing signal in the
pulse triangle (bounded by the red lines) that takes on a value
between the raw (1,1) and (0,2) signal levels. This signal is
indicative of spin-blocked charge transitions. Outside of the
pulse triangle, transitions through the (1,2) and (0,1) charge
states are possible and relax the spin blockade. (b) Energy level
diagram illustrating the possible transitions from (1,1) to (0,2).
Fast transitions are indicated with black arrows, spin blocked
transitions with grey arrows. (c) $G_{S2}$ as a function of $V_L$
and $V_R$ while applying the reverse ``control" pulse
sequence,(0,1)$\rightarrow$(0,2)$\rightarrow$(1,1)$\rightarrow$(0,1).
The pulse period $\tau$=10 $\mu$s and the perpendicular field
$B_{\perp}$=100 mT. The (0,2)$_\textrm{S}$ to (1,1)$_\textrm{S}$
transition is not spin blocked and as a result, there is no
detectable pulse signal in the pulse triangle (bounded by the red
lines). (d) Level diagram illustrating the (0,2) to (1,1)
transition. A best-fit plane has been subtracted from the data in
(a),(c) to remove signal from direct gate to QPC coupling.}
\label{figurename}\end{center}\end{figure}
The pulse sequence used to measure $\tau_{ST}$ is shown in [Fig.\
3(a)]. The gates are held at point E for 10\% of the period,
emptying the second electron from the double dot, leaving the
(0,1) charge state. A pulse shifts the gates to point R (reset
point) for the next 10\% of the period. This initializes the
system into the (1,1) configuration. Since the singlet and triplet
states are nearly degenerate in (1,1) we expect to load into
(1,1)$_\textrm{S}$ or any of the three (1,1)$_\textrm{T}$ states
with equal probability. For the final 80\% of the period, the
gates are held at the measurement point M where (0,2)$_\textrm{S}$
is the ground state. The energetics of the spin states at the
measurement point are shown in Fig.\ 3(b). A (1,1)$_\textrm{S}$
state prepared in the R step will tunnel to (0,2)$_\textrm{S}$ on
a timescale set by the interdot tunneling rate,
$\Gamma(\epsilon)$. The m$_\textrm{s}$=0 (1,1) triplet state,
(1,1)$_\textrm{T0}$, will dephase into (1,1)$_\textrm{S}$ on a
timescale set by $T_2$ (expected to be $\leq$100 ns
\cite{Kikkawa_PRL_98,Khaetskii_PRL_2002,Merkulov_PRB_2002})
followed by a direct transition to (0,2)$_\textrm{S}$. Roughly
half of the time the R step will load the m$_\textrm{s}$=1 (1,1)
triplet state, (1,1)$_\textrm{T+}$, or the m$_\textrm{s}$=-1 (1,1)
triplet state, (1,1)$_\textrm{T-}$. Since (0,2)$_\textrm{T}$ is
inaccessible a transition from (1,1)$_\textrm{T+}$ or
(1,1)$_\textrm{T-}$ to (0,2) requires a spin flip and will be
blocked for times shorter than the singlet-triplet relaxation time
$\tau_{ST}$.
Figure 3(a) shows the time-averaged charge sensor conductance
$G_{S2}$, measured as a function of the dc gate voltages $V_L$ and
$V_R$, while the forward pulse sequence is repeated. The
conductance $G_{S2}$ maps out the ground state population at point
M since 80\% of the duty cycle is spent there. The plateaus in
$G_{S2}$ at $ \sim$0.0, 6.0, 16, and 23 $\times$10$^{-3}$e$^2$/h
indicate full population of the (1,2), (0,2), (1,1), and (0,1)
charge states respectively. The pulse data differs from ground
state data only when point M resides in the triangle defined by
the (1,1) to (0,2) ground state transition and the extensions of
the (1,1) to (0,1) and (1,1) to (1,2) ground state transitions
(bounded by the red marks in Fig.\ 3(a)). Inside of the ``pulse
triangle" transitions from (1,1) to (0,2) may be spin-blocked and
the charge sensor registers a conductance intermediate between the
(1,1) and (0,2) plateaus. Outside of the pulse triangle it is
possible to access (0,1) or (1,2), which relaxes the spin
blockade. Figure 3(a) shows a signal of
~11$\times$10$^{-3}$e$^2$/h in the pulse triangle for $\tau$=10
$\mu$s, indicating that approximately 50\% of the time the dots
remain in (1,1) even though (0,2) is the ground state. This is
direct evidence of spin-blocked (1,1) to (0,2) transitions.
To check that the pulse signal is due to spin-blocked transitions
and not just a slow interdot tunnel rate we compare the forward
pulse sequence with a reverse pulse sequence that does not involve
spin selective transitions
[(0,1)$\rightarrow$(0,2)$\rightarrow$(1,1)$\rightarrow$(0,1)]. In
the reverse pulse sequence the reset position R occurs in (0,2)
where only the singlet state is accessible, and M occurs in (1,1).
Now tunneling from R to M should always proceed on a time scale
set by the interdot tunnel coupling, since the (0,2)$_\textrm{S}$
to (1,1)$_\textrm{S}$ transition is not spin blocked. No signal is
seen in the pulse triangle for this reversed ``control" sequence
(Fig.\ 3(c)).
\begin{figure}[t]
\vspace{2 cm}
\begin{center}\leavevmode
\includegraphics[width=1\linewidth]{EP2DSFig4.eps}
\vspace{-2.5 cm} \caption{Time dependence of the spin-blocked
signal. $G_{S2}$ measured as a function of $\tau$. In the (0,2)
charge state $G_{S2}$=10$\times$10$^{-3}$e$^2$/h, while
$G_{S2}$=20$\times$10$^{-3}$e$^2$/h in the (1,1) charge state. For
short $\tau$, the (1,1) to (0,2) transition is blocked
approximately half of the time, resulting in a pulse signal of
15$\times$10$^{-3}$e$^2$/h in the (0,2) pulse triangle. The (1,1)
to (0,2) transition probability increases (sensor signal
decreases) with $\tau$ due to spin-relaxation with a
characteristic time scale of 70$\pm$10 $\mu$s. A best-fit plane
has been subtracted from the data.}
\label{figurename}\end{center}\end{figure}
The singlet-triplet relaxation time can be determined by measuring
the time dependence of the charge sensing signal inside of the
pulse triangle. We extract $\tau_{ST}$ by measuring $G_{S2}$ as a
function of the pulse train period, $\tau$. $G_{S2}$ is measured
inside the pulse triangle ($V_R$,$V_L$ held fixed at -403,-523.8
mV, respectively) and is plotted as a function of $\tau$ in Fig.\
4. In (1,1), $G_{S2}$$ \sim $20$\times$10$^{-3}$e$^2$/h, whereas
outside the pulse triangle in (0,2), $G_{S2}$$ \sim
$10$\times$10$^{-3}$e$^2$/h. For small $\tau$, $G_S$$ \sim
$15$\times$10$^{-3}$e$^2$/h in the pulse triangle. At long $\tau$,
$G_S$ approaches 10$\times$10$^{-3}$e$^2$/h in the pulse triangle,
which indicates complete transfer from the (1,1) to (0,2) charge
state. We fit these experimental data assuming exponential
singlet-triplet relaxation and find a best fit $\tau_{ST}$$
=$70$\pm$10 $\mu$s. The dependence of the singlet-triplet
relaxation time on detuning and magnetic field has been presented
in \cite{Johnson_Nature_2005}. A speed up of spin relaxation near
zero field is observed and is consistent with a hyperfine mediated
spin relaxation process.
\section{Coherent spin manipulation}
With an external magnetic field applied so that the $m_s=\pm1$
triplet states are split off, the $(1,1)_S$ and $(1,1)_{T0}$
states form a logical qubit. Due to the large singlet triplet
splitting in (0,2) we can easily initialize the system in
$(0,2)_S$. $(0,2)_S$ can transferred to $(1,1)_S$ by sweeping the
detuning adiabatically with respect to the interdot tunnel
coupling. The $(1,1)_S$ and $(0,2)_S$ states are hybridized at
zero detuning due to the interdot tunnel coupling. This
hybridization results in an exchange splitting between the
$(1,1)_S$ and $(1,1)_{T0}$ states, $j(\epsilon)$. For large
negative detunings, $j(\epsilon)\rightarrow0$. The exchange
$j(\epsilon)$ can be tuned on ns timescales by applying pulses to
the gates defining the double dot. Using this fast control of the
exchange energy we have recently implemented a spin SWAP operation
\cite{Petta_Science}.
\\
\\
We acknowledge useful discussions with Sankar Das Sarma,
Hans-Andreas Engel, Xuedong Hu, Daniel Loss, Emmanuel Rashba, and
Peter Zoller. Funding was provided through the ARO under
DAAD55-98-1-0270 and DAAD19-02-1-0070, the DARPA-QuIST program,
and the NSF under DMR-0072777, the Harvard Center for Nanoscale
Systems, and the Sloan and Packard Foundations.
|
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Vittorio Occorsio (Roma, 9 de abril de 1929-Roma, 10 de julio de 1976) fue un magistrado italiano, víctima del llamado Terrorismo nero durante los años de plomo italianos. Había participado en el proceso de la piazza Fontana y en el proceso al movimiento de inspiración neofascista Ordine Nuovo.
Biografía
Occorsio realizó estudios clásicos en la escuela secundaria Julio César y luego se graduó en Derecho. Tras la explosión de una bomba en la piazza Fontana de Roma el 12 de diciembre de 1969, se ocupó como fiscal adjunto del primer interrogatorio a Pietro Valpreda, que fue acusado del asesinato de catorce personas y de las heridas provocadas a ochenta más. En 1971 pidió la disolución del movimiento Ordine Nouvo, pidiendo la aplicación de la Ley Scelba contra la reconstrucción del partido fascista.
En abril de 1976 fue el primer magistrado que se ocupó de la logia masónica secreta llamada Logia P2 e investigó las relaciones entre el terrorismo neofascista, la masonería y el aparato del SIFAR. Previamente había trabajado en investigaciones sobre los dos golpes fallidos, el Plan Solo y el golpe Borghese, y había iniciado una investigación sobre las relaciones de algunos miembros de la P2 con el terrorismo negro y el hampa dedicada a secuestros, como la Banda de Marsella se centra en particular en el secuestro de Alfredo Danesi, Amedeo Ortolani, Fabrizio Andreuzzi y Claudio Francisci. La condena obtenida en el proceso se resume en una significativa declaración realizada a su amigo y colega Ferdinando Imposimato:
Asesinato
Vittorio Occorsio fue tiroteado la mañana del 10 de julio de 1976. Pierluigi Concutelli fue la mano ejecutora. Descargó sobre su víctima 32 disparos de ametralladora cuando se dirigía a la oficina en su automóvil, un Fiat 125, en el cruce de la vía Mogadiscio y la vía Giuba, a unos metros de su casa. Se iría de vacaciones tres días después esperando el regreso de su colega Imposimato. Antes de desaparecer, sus asesinos se llevan su maletín que contiene el expediente de los secuestros y, según reveló el periodista Franco Scottoni, un documento que reveló la compra por parte de la Organización Mundial de Asistencia Masónica (Ompam) de un edificio en Roma por 8 millones de dólares igual al monto total pagado por los rescates de los secuestros por los que fue detenido Albert Bergamelli. Tres meses después su fuente Totò D'Agostino, jefe de Canolo, también fue asesinado por Domenico Papalia, absuelto sin embargo 41 años después por el Tribunal de Apelación de Perugia tras ser condenado a cadena perpetua; [4] D ' Agostino, asesinado por orden de Antonio Nirta (jefe de San Luca que el día del secuestro de Aldo Moro habría estado en via Fani infiltrado en la BR por el exgeneral de los Carabinieri Francesco Delfino), habría informado al juez de un flujo. El dinero llegó de los secuestros y se transportó a Calabria para ser utilizado en acciones subversivas y para financiar excelentes asesinatos.
Reivindicación
En el coche del juez apareció un pasquín firmado por el Nuovo Ordine : Occorsio fue acusado de «oportunismo arribista y de «haber servido a la dictadura democrática persiguiendo a los militantes del Nuevo Orden y las ideas que portan. Vittorio Occorsio, de hecho, ha iniciado dos juicios contra el MPON ".
Como reveló el juez Ferdinando Imposimato en una entrevista de 1990:
Proceso
Para el homicidio en el octubre del '76 se indagan primera Danilo Abbruciati y luego Alvaro Pompili; también lo mismo Licio Gelli está sentido por los magistrados de Firenze Pier Luigi Vigna y Gabriele Chelazzi.
El 16 marzo 1978 estuvieron condenados pero los neofascisti Pierluigi Concutelli y Gianfranco Hierro, como esecutori materiales. Absueltos en cambio otros imputados cuáles mandanti del homicidio; mandanti que no estarán divisados nunca. Concutelli, condenado al ergastolo, beneficiará de los domiciliari del 2009 para graves motivos de salud; el criminal era iscritto con la tessera n. 11.070 a la loggia massonica Camea de Palermo cuyos afiliados fueron inquisiti en el 1979 para haber ayudado el banquero Michele Sindona en el suyo fingido sequestro.
Distinciones
A la memoria del magistrado ha sido dedicado el parque de Villa Mercede a Roma después del restauro y está titulada una aula del Palacio de Justicia de Roma. También lleva su nombre una aula del instituto Giulio Cesare, donde él había estudiado. Dos lápidas conmemorativas fueron colocadas en el interior de Villa Leopardos, junto a la Vía Makallè, y la otra en vía Mogadiscio, esquina vía Giuba, lugar del homicidio.
Referencias
Bibliografía
Voces correlate
Terrorismo italiano
Teoría de los opuestos extremismos
Neofascismo
Terrorismo negro
Otros proyectos
Wikiquote contiene citazioni di o su Vittorio Occorsio
Wikimedia Commons contiene immagini o altri file su Vittorio Occorsio
Enlaces externos
Scheda su Vittorio Occorsio, dal sito dell'Associazione Italiana Vittime del Terrorismo
Perché Occorsio? - La storia di una vendetta su La Storia siamo noi
Eugenio Occorsio, Vittorio Occorsio, in Dizionario biografico degli italiani, vol. 79, Roma, Istituto dell'Enciclopedia Italiana, 2013. URL consultato il 21 maggio 2019.
Fallecidos en Roma
Nacidos en Roma
Jueces de Italia
Víctimas del terrorismo en Italia
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"redpajama_set_name": "RedPajamaWikipedia"
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Huttonina furcata är en tvåvingeart som beskrevs av Tonnoir och Malloch 1928. Huttonina furcata ingår i släktet Huttonina och familjen Huttoninidae. Inga underarter finns listade i Catalogue of Life.
Källor
Tvåvingar
furcata
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{
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| 1,793
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Inequality is a corrosive force, like rust. Big income and wealth imbalances eat away at trust and empathy, making the country less healthy and less united. They also leave some people with much greater opportunities than others. In New Zealand, income imbalances widened faster in the 1980s and 1990s than in any other developed nation. The wealthiest tenth own more than half of all wealth. This is the result of deliberate policy choices that could be reversed. In doing so we would create a country in which people recognise each other as equals and can come together to tackle common problems. Find out more here.
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\section{Introduction}
\label{intro}
Source coding via quantized linear representations, also known as transform coding, is a classical and well-studied subject. Yet it is poorly understood outside the simple setting of orthogonal transforms, namely, for frame-based representations. The same can also be said for partially nonlinear representations such as those based on compressive sampling. The basic reason for the difficulty in solving the quantization problem for these more general sampling and reconstruction systems is the lack of an analog of Parseval's identity which, more or less, dictates the best quantization strategy for orthogonal systems. While some kind of basic reconstruction stability can be ensured relatively easily, these results do not offer correct rate-distortion trade-offs because of their inefficiency in utilizing redundancy, especially under constraints that do not allow for high-resolution quantization.
Redundancy is a key concept of frame-based as well as compressive sampling systems. It can be understood in terms of the sampling process (e.g., what part of the coefficient space is taken up with the actual measurements) or in terms of the reconstruction process (e.g., which perturbations of the measurements have the smallest effect on the reconstruction). Efficient encoding via the first approach is generally not practical because codewords cannot be easily placed arbitrarily in the coefficient space. Indeed, quantized measurements are typically required to lie on a finite rectangular grid. An alternative approach is then to seek ways of arranging the quantization error in the coefficient space to lie in directions that are away from the actual measurements, typically by means of some feedback process. {\em Noise shaping} is the generic name of this quantization methodology. It has its roots in sigma-delta modulation, which is used for oversampled analog-to-digital (A/D) conversion \cite{IY, NST96, Candy-Temes, SchTe04}.
Let us explain the philosophy of noise shaping in more concrete terms.
In both frame-based and compressive sampling systems, we have a linear sampling operator $\Phi$ that can be inverted on a given space ${\mathscr X}$ of signals using some (possibly nonlinear) reconstruction operator $\Psi$. Given a signal $x\in {\mathscr X}$ and its sampled version $y=\Phi x$, ordinarily we recover $x$ exactly (or approximately, as in compressive sampling) as $\Psi(y)$. In the context of this paper, quantization of $y$ will mean replacing it with a vector $q$ which is of the same dimensionality as $y$ and whose entries are chosen from some given alphabet ${\mathscr A}$. The goal is to choose $q$ so that the approximate reconstruction $x^\#:=\Psi(q)$ is as close to $x$ as possible as $x$ varies over ${\mathscr X}$.
In the context of finite frames, $\Phi$ is a full-rank $m\times k$ matrix where $m > k$, and $\Psi$ is any left inverse of $\Phi$. The rows of $\Phi$ form the {\em analysis frame} and the columns of $\Psi$ form a {\em synthesis frame} dual to this frame. With $y = \Phi x$ and $x = \Psi y$ as above, when $y$ is replaced by a quantized vector $q$, the reconstruction error $e:=x - x^\#$ is equal to $\Psi(y-q)$. Therefore the correct strategy to reduce the size of $e$ is not to minimize the Euclidean norm $\|y-q\|$ as memoryless scalar quantization (MSQ) does, but to minimize the semi-norm $|y-q|_\Psi := \|\Psi(y-q)\|$.
In other words, we seek $q\in {\mathscr A}^m$ so that the quantization ``noise'' $y-q$ is close to $\mathrm{ker}(\Psi)$ in the above sense. This is the basic principle of noise shaping. How this goal can be achieved (approximately), i.e., the actual process of noise shaping, as well as what noise shaping can offer for source coding are nontrivial questions that will be addressed in this article.
While the basic principle of noise shaping is formulated above for linear sampling and reconstruction systems, its philosophy extends to compressive sampling systems where the reconstruction operator is generally nonlinear. The simplest connection is made by considering strictly sparse signals. Let $\Sigma^N_k$ denote the nonlinear space of $N$-dimensional vectors which have no more than $k$ nonzero entries. In the context of compressive sampling, $\Phi$ is an $m\times N$ matrix where $m \ll N$, which means that the sampling process is lossy for the whole of ${\mathbb R}^N$. However, note that $\Sigma^N_k$ is the union of (a large number of) $k$-dimensional linear subspaces on each of which $\Phi$ acts like a frame once $m > k$. This observation opens up the possibility of noise shaping. Indeed, fixing any one of these subspaces $V$, we can envision a noise shaping process associated with any of the linear inverses (duals) of $\Phi$ on $V$. However, it is not clear how one might organize all of these individual
noise shaping processes, especially given that these subspaces are not directly available to the quantizer. What comes to the rescue is the notion of an {\em alternative dual}. While we formulated noise shaping above as matching the quantization operator to a given dual frame, it is also possible to consider matching the dual frame to a given quantization operator. This results in the possibility of ``universal'' quantization processes (i.e., independent of the signal subspace) which become noise-shaping processes for suitable alternative duals. Even though finding these suitable alternative duals may require extracting information about the signal subspace, this duty purely belongs to the decoder and not the quantizer.
This article is organized as follows. In Section \ref{classical}, we review the basics of classical noise shaping in the setting of sigma-delta ($\Sigma\Delta$) modulation. In Section \ref{alternative}, we extend the formulation of noise shaping and introduce various notions of alternative duals for noise shaping in the setting of frames, followed by their performance analysis for random frames in Section \ref{random-frames}. We then discuss noise-shaping quantization methods for compressive sampling in Section \ref{CS}.
\section{Classical noise shaping: Sigma-Delta Modulation}
\label{classical}
The Shannon-Nyquist sampling theorem for bandlimited functions provides the natural framework of conventional A/D conversion systems. With the Fourier transform normalized according to the ``ordinary-frequency'' convention
$$ \widehat x(\xi) := \int_{-\infty}^\infty x(t) e^{-2\pi i \xi t} \,\mathrm{d}t, $$
let us define the space ${\mathscr B}_\Omega$ of bandlimited functions to be all $x$ in $L^2({\mathbb R})$ such that $\widehat x$ is supported in $[-\Omega,\Omega]$. The classical sampling theorem says that any $x \in {\mathscr B}_\Omega$
can be reconstructed perfectly from its time samples $(x(n\tau))_{n \in {\mathbb Z}}$
according to the formula
\begin{equation} \label{shannon}
x(t) = \tau \sum_{n \in {\mathbb Z}} x(n\tau) \psi(t - n\tau),
\end{equation}
where $\tau \leq \tau_\mathrm{crit} := \frac{1}{2\Omega}$, and
$\psi$ is any function in $L^2({\mathbb R}) $such that
\begin{equation}\label{admissible}
\widehat \psi(\xi) = \left \{
\begin{array}{ll}
1, & |\xi| \leq \Omega, \\
0, & |\xi| > \frac{1}{2\tau}.
\end{array}
\right.
\end{equation}
Hence, if we define the sampling operator $(\Phi x)_n:= x(n\tau)$ and the reconstruction operator $\Psi(u) := \tau \sum u_n \psi(\cdot - n \tau)$ (on any space it makes sense), then $\Psi$ is a left inverse of $\Phi$ on ${\mathscr B}_\Omega$ when $\tau$ and $\psi$ satisfy the conditions stated above.
The value $\rho:=1/\tau$ is called the sampling rate, and
$\rho_\mathrm{crit} := 1/\tau_\mathrm{crit} = 2\Omega$ is called the critical
(or Nyquist) sampling rate. Their ratio given by
\begin{equation}\label{lambda}
\lambda:= \frac{\rho}{\rho_\mathrm{crit}}
\end{equation}
is called the {\em oversampling ratio}. According to the value of $\lambda$,
A/D converters are broadly classified as Nyquist-rate converters ($\lambda \approx 1$) or oversampling converters ($\lambda \gg 1$).
Nyquist-rate converters set their sampling rate $\rho$ slightly above the critical frequency $2\Omega$ so that $\psi$ may be chosen to decay rapidly enough to ensure absolute summability of \eqref{shannon}. Given any quantization alphabet ${\mathscr A}$, the (nearly) optimal quantization strategy in this (nearly) orthogonal setting is memoryless scalar quantization (MSQ). This means that each sample $y_n := x(n\tau)$ is rounded to the nearest quantization level $q_n \in {\mathscr A}$. This process is also referred to as pulse-code modulation (PCM). If each sample is quantized with error
no more than $\delta$, i.e., $\|y-q\|_\infty \leq \delta$, then
the error signal
\begin{equation} \label{err_sig}
e(t) := x(t) - (\Psi q)(t) = \tau \sum_{n \in {\mathbb Z}} \big (y_n - q_n \big) \psi(t - n\tau)
\end{equation}
obeys the bound
$\|e\|_{L^\infty} \leq C \delta$ where $C$ is independent of $\delta$.
This is essentially the best error bound one can expect for
Nyquist-rate converters. Because setting $\delta$ very small is
costly, Nyquist-rate converters are not very suitable for signals that
require high-fidelity such as audio signals.
Oversampling converters are designed to take advantage of the redundancy in the representation \eqref{shannon} when $\tau < \tau_\mathrm{crit}$. In this case, the interpolation operator $\Psi$ has a kernel which gets bigger as $\tau \to 0$. Indeed, let $\widehat \psi(\xi) =0$ for $|\xi| > \Omega_0$.
It is easily seen that $\Psi u = 0$ if
\begin{equation}\label{KerT}
\sum_{n\in{\mathbb Z}} u_n e^{2\pi i n \xi} = 0 \mbox{ for } |\xi| < \tau\Omega_0.
\end{equation}
This means that even though $y-q$ may be large everywhere, $e=\Psi(y-q)$ can be very small if $y-q$ can be arranged to be spectrally disjoint from the (discretized) reconstruction kernel $\psi$. This is the concrete form of noise shaping that we briefly discussed in the Introduction.
The main focus of an oversampling A/D converter is on its quantization algorithm, which has to be
non-local to be useful, but also causal so that it can be implemented in real time. The assignment of each $q_n$ will therefore depend on $y_n$ as well as a set of values (the states) that can be kept in an analog circuit memory, while meeting the spectral constraints on $y-q$ as described in
the previous section. $\Sigma\Delta$ modulators operate according to these principles.
As can be seen in \eqref{KerT}, the kernel of $\Psi$ consists of high-pass sequences. Hence the primary objective of $\Sigma\Delta$ modulation is to arrange the
quantization error $y - q$ to be an approximate high-pass sequence (see Fig. \ref{fig:noise_shaping_illustration}).
This objective can be realized by setting up a difference equation, the
so-called {\em canonical} $\Sigma\Delta$ equation, of the form
\begin{equation}\label{sig-del}
y-q = \Delta^r u,
\end{equation}
where $\Delta$ denotes the finite difference operator defined by
\begin{equation}\label{def_Delta}
(\Delta w)_n:=w_n - w_{n-1},
\end{equation}
$r$ denotes the ``order'' of the scheme, and $u$ is an
appropriate auxiliary sequence called the {\em state sequence}.
This equation does not imply anything about $q$ without any constraint on
$u$. The most useful constraint turns out to be boundedness.
\begin{figure}[tp]
\centering
\centerline{\includegraphics[scale=0.44]{noise_shaping_illustration}}
\caption{Illustration of classical noise shaping via $\Sigma\Delta$ modulation: The superimposed Fourier spectra of a bandlimited signal (in black), and the quantization error signals using MSQ (in red), 1st order $\Sigma\Delta$ modulation (in magenta), and 2nd order $\Sigma\Delta$ modulation (in blue).}
\label{fig:noise_shaping_illustration}
\end{figure}
In practice, the boundedness of $u$ in
(\ref{sig-del}) has to be attained through a recursive algorithm.
This means that given any input sequence $(y_n)$, the $q_n$ are found by
a given ``quantization rule'' of the form
\begin{equation}\label{Q_rule}
q_n = F(u_{n-1},u_{n-2},\dots,y_n,y_{n-1},\dots),
\end{equation}
and the $u_n$ are updated via
\begin{equation}\label{u_update}
u_n = \sum_{k=1}^r (-1)^{k-1} \binom{r}{k} u_{n-k} + y_n - q_n,
\end{equation}
which is a restatement of (\ref{sig-del}). In electrical engineering, such a recursive procedure for quantization is called ``feedback quantization" due to the role $q_n$ plays as a feedback control term in (\ref{u_update}). The role of the quantization rule $F$ is to keep the system {\em stable}, i.e., $u$ bounded.
Stability is a crucial property. Indeed, it was shown in \cite{DD} that a stable $r$th order scheme results in the error bound
\begin{equation} \label{err1}
\|e \|_{L^\infty} \leq \|u\|_{\ell^\infty} \|\psi^{(r)} \|_{L^1}\tau^r,
\end{equation}
where $\psi^{(r)}$ denotes the $r$th order derivative of
$\psi$. The implicit $\Omega$- and the explicit $\tau$-dependence of this estimate can be
replaced with a single $\lambda$-dependence by setting
$\psi(t) := \Omega \psi_0(\Omega t)$ where the prototype $\widehat \psi_0(\xi)$
equals $1$ on $[-1,1]$ and vanishes for $|\xi | \geq 1+\epsilon_0$, with
$\epsilon_0>0$ fixed. Let $C_0:=\|\psi_0 \|_{L^1}$.
Bernstein's inequality applied to $\psi$ yields
\begin{equation} \label{err2}
\|e \|_{L^\infty} \leq C_0 \|u\|_{\ell^\infty} \pi^r(1+\epsilon_0)^r
\lambda^{-r}, \mbox{ for all } \lambda > 1+\epsilon_0.
\end{equation}
With this error bound, there are two goals in progression.
The first is to keep $u$ bounded and the second is to keep the bound small. Ultimately,
the best strategy is to have, for each $r$, a quantization rule yielding a
stable $r$th order scheme, and then for any given $\lambda$,
to choose the best one (i.e., the one with the least
error bound). This task
is significantly complicated by the fact that the bound on $u$ has a strong dependence on $r$, especially for small quantization alphabets ${\mathscr A}$.
In general it is not possible to expect this dependence to be less than $(cr)^r$ for some constant $c$ that depends on the given amplitude range $\mu$ for $x$.
This growth order is also what is needed to ensure that the reconstruction error decays exponentially, i.e., as $2^{-p\lambda}$, as a function of $\lambda$, which is the best possible due to Kolmogorov entropy estimates for bandlimited functions \cite{exp_decay}. The rate $p$ of exponential decay that is achievable by the resulting family of schemes is inversely proportional to $c$, and gets worse as $\mu$ is increased. The question of best achievable accuracy for oversampling converters in this setting remains open. Currently, the best result in the one-bit case with ${\mathscr A} = \{-1,1\}$ yields $\|e\|_{L^\infty} = O(2^{-p\lambda})$ where $p = \pi/(6e^2 \log 2) \approx 0.1$, and $\mu \approx 0.06$. Higher values of $p$ can be achieved with more levels in ${\mathscr A}$. For example, if ${\mathscr A}=\{-1,0,1\}$, then $p$ rises to $0.15$ and $\mu$ to $0.25$ \cite{DGK}. These are rigorously proven bounds and the actual behavior of the error based on numerical experiments appears to be better. For the details of the quantization
rules which result in these exponentially accurate $\Sigma\Delta$ modulators, see \cite{exp_decay, DGK}. It has also been shown that no matter how the bits are assigned the rate of the exponential decay cannot match that of Nyquist-rate conversion \cite{KW}.
\section{Generalized Noise-shaping Operators and Alternative Duals of Frames for Noise Shaping}
\label{alternative}
In this section, we will generalize the classical theory of $\Sigma\Delta$ modulation to more general noise-shaping quantizers as well as sampling and reconstruction systems. For conceptual clarity, we will separate the process of noise shaping from the processes of sampling and reconstruction. While we will present these generalizations in a finite-dimensional setting, extensions to infinite-dimensional settings are usually possible. We will also discuss the notion of alternative duals of frames which are associated with noise-shaping quantizers.
\subsection{A general framework of noise shaping}
The canonical $\Sigma\Delta$ equation we saw in \eqref{sig-del} is a special case of a more general framework of noise shaping. Let ${\mathscr A}$ be a finite quantization alphabet and $J$ be a compact interval in ${\mathbb R}$. Let $h = (h_j)_{j\geq0}$ be a given sequence, finite or infinite, where $h_0=1$. By a noise-shaping quantizer with the transfer filter $h$, we mean any sequence $Q=(Q_m)_1^\infty$ of maps $Q_m:J^m \to {\mathscr A}^m$, $m \in {\mathbb N}$, where for each $y \in J^m$, the output $q := Q_m(y)$ satisfies
\begin{equation} \label{y-q-h-u}
y - q = h*u
\end{equation}
where $u \in {\mathbb R}^m$ and $\|u\|_\infty \leq C$ for some constant $C$ which is independent of $m$.
Here $h*u$ refers to the (finite) convolution of $h$ and $u$ defined by
$$(h*u)_n := \sum_{j \geq 0} h_j u_{n-j}, ~~~1 \leq n \leq m,$$
where it is assumed that $u_n := 0$ for $n\leq 0$.
Without any reference to a sampling or a reconstruction operator, noise shaping in this setting refers to the fact that the ``quantization noise'' $y-q$ is spectrally aligned with $h$. Note that the operator $H:u \mapsto h*u$ is invertible on ${\mathbb R}^m$ for any $m$, and therefore given any $y$ and $q$, there exists $u\in {\mathbb R}^m$ which satisfies \eqref{y-q-h-u}; this is trivial. However, the requirement that $\|u\|_\infty$ must be controlled uniformly in $m$ imposes restrictions on what $q$ can be for a given $y$; these solutions are certainly non-trivial to find and may not always exist.
The operator $H$ above (defined as convolution by $h$) is a lower triangular Toeplitz matrix with unit diagonal. With this view, let us relax the notion of a noise-shaping quantizer and assume that $H$ is any lower triangular $m \times m$ matrix with unit diagonal. We will refer to $H$ as a noise-shaping transfer operator where the associated noise-shaping relation is given by
\begin{equation} \label{y-q-H-u}
y - q = Hu.
\end{equation}
Suppose we are given a sequence $(H_m)_1^\infty$ of $m \times m$ noise-shaping transfer operators. In this general setting, we say that an associated sequence $(Q_m)_1^\infty$ of quantizer maps (for which $q:=Q_m(y)$ and $u$ is determined by \eqref{y-q-H-u}) achieves noise shaping for $(H_m)$, $J$, and ${\mathscr A}$, if $\|u\|_\infty \leq C$ for some constant $C$ independent of $m$. A slightly weaker assumption is to only require that $\|u\|_\infty = o(\|H_m^{-1}\|_{\infty \to \infty})$, though we shall not need to work in this generality in this paper.
In many applications, one works with $(H_m)_1^\infty$ which are ``progressive'' (also called ``nested'') in the sense that
$$P_m \circ H_{m+1} \circ P_{m+1} = H_m \circ P_m,$$
where $P_m$ is the restriction of a vector to its first $m$ coordinates. Convolution is a standard example. In this case, it may be natural to require that the $(Q_m)_1^\infty$ are progressive as well. The classical $\Sigma\Delta$ modulation we saw in Section \ref{classical} is of this type. However, our general formulation does not impose progressiveness.
As indicated earlier, noise-shaping quantizers provide non-trivial solutions to \eqref{y-q-H-u} and therefore do not exist unconditionally, though under certain suitable assumptions on $H$, $J$, and ${\mathscr A}$, they exist and can be implemented via recursive algorithms. The simplest is the (non-overloading) {\em greedy quantizer} whose general formulation is given below:
\begin{proposition}\label{P:greedy}
Let ${\mathscr A} := {\mathscr A}_{L,\delta}$ denote the arithmetic progression in ${\mathbb R}$ which is of length $L$, spacing $2\delta$, and symmetric about $0$.
Assume that $H = I - \tilde H$, where $\tilde H$ is strictly lower triangular, and $\mu \geq 0$ such that
$\|\tilde H\|_{\infty \to \infty} + \mu/\delta \leq L$. Suppose
$\|y\|_\infty \leq \mu$. For each $n\geq 1$, let
$$ q_n := \mathrm{round}_{\mathscr A}\left (y_n + \sum_{j=1}^{n-1} \tilde H_{n,n-j} u_{n-j} \right)$$
and
$$ u_n := y_n + \sum_{j=1}^{n-1} \tilde H_{n,n-j} u_{n-j} -q_n .$$
Then the resulting $q$ satisfies \eqref{y-q-H-u}
with $\|u\|_\infty \leq \delta$.
\end{proposition}
This quantizer is called greedy because for all $n$, the selection of $q_n$ over ${\mathscr A}$ is made so as to minimize $|u_n|$. The proof of this basic result follows easily by induction once we note that for any $w \in [-L\delta,L\delta]$, we have $|w - \mathrm{round}_{\mathscr A}(w)| \leq \delta$, hence the scalar quantizer $\mathrm{round}_{\mathscr A}$ is not overloaded. For details, see \cite{CG14}. Note that the greedy quantizer is progressive if $(H_m)_1^\infty$ is a progressive sequence of noise-shaping transfer operators. In the special case $Hu = h*u$ where $h_0 = 1$, we simply have $\|\tilde H\|_{\infty \to \infty} = \|h\|_1 - 1$. This special case is well-known and widely utilized (e.g. \cite{Candy-Temes, NST96, SchTe04, exp_decay}).
\subsection{Canonical duals of frames for noise shaping}\label{canonical-duals}
The earliest works on noise-shaping quantization in the context of finite frames used $\Sigma\Delta$ quantization and focused on canonical duals for reconstruction. Before we begin our discussion of these contributions we remind the reader of our convention: we identify an analysis frame with (the rows of) its analysis operator and a synthesis frame with (the columns of) its synthesis operator.
Let $\Phi$ be a finite frame and $y=\Phi x$ be the frame measurements of a given signal $x$. Assume that we quantize $y$ using a noise-shaping quantizer with transfer operator $H$.
Any left-inverse (dual) $\Psi$ of $\Phi$ gives
\begin{equation}\label{error_PsiHu}
x - \Psi q = \Psi(y - q) = \Psi Hu.
\end{equation}
Using this expression, and specializing to the case of first order $\Sigma\Delta$ quantization, i.e., $H=D$ where $D$ is the lower bidiagonal matrix whose diagonal entries are 1 and subdiagonal entries are -1, \cite{BPY} observed that the reconstruction error can be bounded as
\begin{equation}\label{eq:err_bound_BPY0}
\|x - \Psi q\|_2 \leq \|u\|_\infty \sum_{j=1}^m \|(\Psi D)_j\|_2
\end{equation}
where $(\Psi D)_j$ denotes the $j$th column of $\Psi D$. This led \cite{BPY} to introduce the notion of frame variation
\begin{equation}\label{eq:frame_var}
\mathrm{Var}(\Psi):=\sum_{j=1}^m \|\psi_j -\psi_{j+1}\|_2
\end{equation}
with $\psi_j$ denoting the $j$th column of $\Psi$ and $\psi_{m+1}$ defined to be zero. Using normalized tight-frames, i.e., frames $\Phi$ for which $\Phi^*\Phi = (m/k)I$, this resulted in the error bound
\begin{equation} \|x - \Phi^\dagger q\|_2 \leq \frac{k}{m} \|u\|_\infty \mathrm{Var}(\Phi^*), \label{eq:err_bound_BPY}\end{equation}
where $\Psi = \Phi^\dagger$ denotes the {\em canonical dual} of $\Phi$ defined (for an arbitrary frame $\Phi$) by
\begin{equation}\label{Phi_dagger}
\Phi^\dagger := (\Phi^*\Phi)^{-1}\Phi^*.
\end{equation}
Subsequently, similarly defined higher-order frame variations were used to study the behavior of higher-order $\Sigma\Delta$ schemes (e.g., in \cite{BPY2} and \cite{BPA2007}) with corresponding generalizations of \eqref{eq:err_bound_BPY} and the conclusion that frames with lower variations lead to better error bounds. This motivated considering frames obtained via uniform sampling of smooth curves in ${\mathbb R}^k$ (called {\em frame paths}). As it turned out, however, this type of analysis based on frame-variation bounds does not provide higher-order reconstruction accuracy unless the frame path terminates smoothly. Smooth termination of the frame path is not available for most of the commonly encountered frames, and finding frames with this property can be challenging. Indeed, designing such frames was a main contribution of \cite{BPA2007} which showed a reconstruction error bound decaying as $m^r$ for $r$th order $\Sigma\Delta$ quantization of measurements using these frames.
In practice, however, one must often work with a given frame rather than design a frame of their choosing. In such cases there are frames, sampled from smooth curves, for which reconstructing with the canonical dual yields reconstruction error that is \emph{lower bounded} by a term behaving like $m^{-1}$, regardless of the $\Sigma\Delta$ scheme's order $r\geq3$ (see, \cite{LPY} for the details). Consequently, to achieve better error decay rates one must seek either different quantization or different reconstruction schemes. We will consider both routes to improving the error bounds in what follows.
\subsection{Alternative duals of frames for noise shaping} \label{alternative-duals}
The discussion in Section \ref{canonical-duals} was based on canonical duals and it involved a particular method to bound the $2$-norm of the reconstruction error $x-\Psi q$, assuming $u$ is bounded in the $\infty$-norm. It is possible to significantly improve the reconstruction accuracy by allowing for more general duals, here called {\em alternative duals}. To explain this route, we return to the general noise-shaping quantization relation \eqref{error_PsiHu}. We assume again that $u$ is known to be bounded in the $\infty$-norm, which is essentially the only type of bound available.
Hence, the most natural reconstruction error bound is given by
\begin{equation}\label{error_bound_PsiHu}
\|x - \Psi q \|_2 \leq \|\Psi H\|_{\infty \to 2} \|u \|_\infty.
\end{equation}
With this bound, the natural objective would be to employ an alternative dual $\Psi$ of $\Phi$ which minimizes $\|\Psi H\|_{\infty \to 2}$. An explicit solution for this problem is not readily available mainly because there is no easily computable expression for $\|A\|_{\infty \to 2}$ for a general $k \times m$ matrix $A$, so we replace it by a simpler upper bound. In fact, this was already done in \eqref{eq:err_bound_BPY0} because we have
\begin{equation}\label{bound_inf_2_1}
\|A\|_{\infty \to 2} \leq \sum_{j=1}^m \|A_j\|_2
\end{equation}
where again $A_j$ denotes the $j$th column of $A$. (This upper bound is also known to be the $L_{2,1}$-norm of $A$.) Another such bound which is often (but not always) better is given by
\begin{equation}\label{bound_inf_2_2}
\|A\|_{\infty \to 2} \leq \sqrt{m} \|A\|_{2 \to 2}.
\end{equation}
(Indeed, for a large random matrix with standard Gaussian entries, the upper bound in \eqref{bound_inf_2_2} behaves as $m + \sqrt{mk}$ whereas that of \eqref{bound_inf_2_1} behaves as $m\sqrt{k}$. Both of these upper bounds are easily seen to be less than $\sqrt{m} \|A\|_\mathrm{Fr}$, however.)
With this upper bound, we minimize $\|\Psi H\|_\mathrm{2 \to 2}$ over all alternative duals $\Psi$ of $\Phi$. Then an explicit solution is available and is given by
\begin{equation}\label{opt-Psi-H}
\Psi_{H^{-1}} := (H^{-1}\Phi)^\dagger H^{-1}.
\end{equation}
This idea was initially introduced specifically for $\Sigma\Delta$ quantization \cite{LPY,BLAY} with the choice $H = D^r$. The resulting alternative duals were called {\em Sobolev duals} and will be discussed in the next subsection. The above generalized version was stated in \cite{GLPSY} where the notation $\Psi_H$ and the term ``$H$-dual'' were introduced for the right hand side of \eqref{opt-Psi-H}, but because of a further generalization we will discuss in Section \ref{V-duals}, we find it more appropriate to use the label $H^{-1}$.
Note that the no noise-shaping case of $H=I$ yields the canonical dual. In general,
we have
$$\|\Psi_{H^{-1}} H\|_{2\to 2} = \|(H^{-1}\Phi)^\dagger\|_{2\to 2}
=\frac{1}{\sigma_{\mathrm{min}}(H^{-1}\Phi)}$$
so that \eqref{error_bound_PsiHu} and \eqref{bound_inf_2_2} yield the error bound
\begin{equation}\label{error_PsiHu_final}
\|x - \Psi_{H^{-1}} q \|_2
\leq \frac{\sqrt{m}}{\sigma_{\mathrm{min}}(H^{-1}\Phi)}\|u \|_\infty.
\end{equation}
\subsubsection{Sobolev Duals}
In the case of $\Sigma\Delta$ modulation, $H$ is defined by \eqref{sig-del}, and given in matrix form by $D^r$ where the diagonal entries of the lower bidiagonal matrix $D$ are $1$ and the subdiagonal entries are $-1$. Because $\|\Psi D^r \|_{2 \to 2}$ resembles a Sobolev norm on $\Psi$, the corresponding alternative dual was called the ($r$th order) Sobolev dual of $\Phi$ in \cite{BLAY}. In this work, Sobolev duals of certain deterministic frames, such as the harmonic frames, were studied. More precisely, \cite{BLAY} considered frames obtained using a sufficiently dense sampling of vector-valued functions on $[0,1],$ which had the additional property that their component functions were piecewise $C^1$ and linearly independent. For such frames, it was shown that
\begin{equation}\label{sigmin_smooth}
\sigma_\mathrm{min} (D^{-r} \Phi) \geq c_r m^{r + \frac{1}{2}},
\end{equation}
hence with \eqref{error_PsiHu_final}, the reconstruction error using the $r$th order Sobolev dual satisfies
\begin{equation}
\|x-\Psi_{D^{-r}}q\|_2 \leq \frac{C_{r}}{ m^{r}} \|u\|_\infty
\label{eq:BLAY_error}
\end{equation}
with $C_r := 1/c_r$.
Here, for a fixed stable $\Sigma\Delta$ scheme, the constant $C_r$ depends only on the order $r$ and the vector-valued function from which the frame was sampled.
The main technique used in \cite{BLAY} to control the operator norm $\|\Psi_{D^{-r}} D^r \|_{2 \to 2}$ is a Riemann sum argument. The argument leverages the smoothness of the vector-valued functions from which the frames are sampled to obtain a lower bound on $\|D^{-r}\Phi x\|_2$ over unit norm vectors $x\in {\mathbb R}^d$ and produces the stated lower bound \eqref{sigmin_smooth}.
As mentioned before, error bounds similar to \eqref{eq:BLAY_error} had also been obtained in \cite{BPA2007}, albeit for specific tight frames.
Nevertheless, in both \cite{BLAY} and \cite{BPA2007}, the decay of the error associated with $\Sigma\Delta$ quantization is a polynomial function of the number of measurements. The significance of this polynomial error decay stems from the fact that for any frame, a lower bound on the reconstruction error associated with MSQ is known to decay only linearly in $m$ \cite{GVT98}.
\subsubsection{Refined Bounds Using Sobolev Duals}
The analysis of \cite{BLAY} was refined in \cite{KSW} in two special cases: harmonic frames, and the so-called Sobolev self-dual frames. For these frames, \cite{KSW} established an upper bound on the reconstruction error that decays as a root-exponential function of the number of measurements. More specifically, for harmonic frames, \cite{KSW} explicitly bounds the constant $C_r$ in \eqref{eq:BLAY_error} and, as in \cite{exp_decay} and \cite{DGK}, optimizes the $\Sigma\Delta$ scheme's order $r$ as a function of the number of measurements. Quantizing with a $\Sigma\Delta$ scheme of the optimal order $r_\mathrm{opt}(m)$ and reconstructing with the associated Sobolev dual results in a root-exponential error bound
\begin{equation}\|x-\Psi_{D^{-r_\mathrm{opt}}}q\|_2 \leq c_1 e^{-c_2\sqrt{m/k}}
\label{eq:KSW_error1}\end{equation}
where the constants $c_1$ and $c_2$ depend on the quantization alphabet ${\mathscr A}_{L,\delta}$ and possibly on $k$ as well. This possible dependence on $k$ is absent in the similar bound for Sobolev self-dual frames. Sobolev self-dual frames are defined using the singular value decomposition $D^{r} = U\Sigma V^*$. Here, the $m\times k$ matrix corresponding to a Sobolev self-dual frame consists of the $k$ columns of $U$ associated with the smallest singular values of $D^r$. This construction implies that the frame admits itself as both a canonical dual and Sobolev dual of order $r$, hence the name. More importantly, this construction also allows one to bound $C_r$ in \eqref{eq:BLAY_error} explicitly and optimize the $\Sigma\Delta$ scheme's order $r$ to obtain the error bound \eqref{eq:KSW_error1}, without any dependence of the constants on $k$.
While we have so far discussed deterministic constructions of frames, Gaussian random frames were studied in \cite{GLPSY}, and later, sub-Gaussian random frames in \cite{subGaussian}. We will discuss these random frames extensively in Section \ref{random-Sobolev}, though at this point we note that, like the harmonic and Sobolev self-dual frames, these frames also allow for root-exponential error decay when the order of the $\Sigma\Delta$ scheme is optimized.
In the context of $\Sigma\Delta$ quantization of frame coefficients using a fixed alphabet ${\mathscr A}$, the number of measurements is proportional to the total number of bits. Hence, the error bounds \eqref{eq:BLAY_error} and \eqref{eq:KSW_error1} can be interpreted as polynomially and root-exponentially decaying in the total number of bits. While these bounds are certainly a big improvement over the linearly decaying lower bound associated with MSQ, they are still sub-optimal. To see this, one observes that the problem of quantizing vectors in the unit ball of ${\mathbb R}^k$ with a maximum reconstruction error of $\epsilon$ is analogous to covering the unit-ball with balls of radius $\epsilon$. A simple volume argument shows that to quantize the unit ball of ${\mathbb R}^k$ with an error of $\varepsilon$, one needs at least $k\log_2 \big(\frac{1}{\epsilon}\big)$ bits. Thus, the reconstruction error can at best decay exponentially in the number of bits used. Moreover, since there exists a covering of the unit-ball with no more than $\big(
\frac{3}{\epsilon}\big)^k$ elements (see, e.g., \cite{lorentz1996constructive}), in principle an exponential decay in the error as a function of the number of bits used is possible. This exponential error decay is predicated on a quantization scheme that has direct access to $x$ and, more importantly, the ability to compare $x$ to each of the approximately $\epsilon^{-k}$ elements of the covering, to assign it an appropriate binary label. The reconstruction scheme for this quantization would then simply replace the binary label by the center of the element of the covering associated with it. Of course, this setting is markedly different from the noise-shaping quantization of frame coefficients considered in this chapter, but it establishes exponential error decay in the number of bits as optimal.
To achieve exponential error decay in the number of bits, \cite{IS13} proposed an encoding scheme to follow $r$th order $\Sigma\Delta$ quantization. The encoding scheme consists of using an $\ell \times m$ Bernoulli random matrix $B$, with $\ell$ slightly larger than $k$, to embed the vector $D^{-r}q$ into a lower dimensional subspace. Since $B$ serves as a distance-preserving Johnson-Lindenstrauss embedding (see, \cite{JLoriginal, achlioptas2001}), the vector $BD^{-r}q$ effectively contains all the information needed for accurate reconstruction of $x$, and it is the only quantity retained. Moreover, the number of bits required to store $BD^{-r}q$ scales only logarithmically in $m$. Using $(BD^{-r}\Phi)^\dagger$ as a reconstruction operator (acting on $BD^{-r} q$) and employing the properties of Johnson-Lindenstrauss embeddings, \cite{IS13} shows that the reconstruction error still decays as it would have if no embedding had been employed. In particular, this means an error decay of $m^{-r}$ for the frames
discussed in this section. Combining these two observations, i.e., logarithmic scaling of the number of bits with $m$, and polynomial decay of the error, \cite{IS13} obtains reconstruction error bounds that decay \emph{exponentially}, i.e., near optimally, in the number of bits.
It turns out that exponential decay of the reconstruction error (in the bit rate or in the oversampling ratio $m/k$) can also be achieved by means of the ``plain route'' of noise-shaping quantization and alternative dual reconstruction only, but with noise-shaping unlike $\Sigma\Delta$ quantization and more like the conventional beta encoding \cite{Chou_thesis, CG14}. This method, called beta duals, is explained next for general frames, and later in Section \ref{random-beta} for random frames.
\subsubsection{Further generalizations: $V$-duals}
\label{V-duals}
Given any $m \times k$ matrix $\Phi$ whose rows are a frame for ${\mathbb R}^k$, consider any $p \times m$ matrix $V$ (i.e., not necessarily square) such that $V\Phi$ is also a frame for ${\mathbb R}^k$. We will call
\begin{equation}\label{Phi:vdual}
\Psi_{\,V} := (V\Phi)^\dagger V
\end{equation}
the {\em $V$-dual} of $\Phi$. (The square and invertible case of $V = H^{-1}$ was already discussed at the beginning of this subsection.) When $p < m$, we call $V\Phi$ the $V$-{\em condensation} of $\Phi$.
With a $V$-dual, we have $\Psi_{\,V} H = (V\Phi)^\dagger VH$ so that
\begin{equation}\label{error_PsiVu}
\|\Psi_{\,V} H\|_{\infty \to 2}
\leq \frac{\| VH\|_{\infty \to 2} }{\sigma_{\mathrm{min}}( V\Phi) }
\leq \frac{\sqrt{p} \| VH\|_{\infty \to \infty} }{\sigma_{\mathrm{min}}( V\Phi) }.
\end{equation}
For $V=H^{-1}$ (and therefore, $p=m$), combination of \eqref{error_bound_PsiHu} with
\eqref{error_PsiVu} agrees with \eqref{error_PsiHu_final}. However, as shown in \cite{CG14}, optimization of \eqref{error_PsiVu} over $V$ can produce a strictly smaller reconstruction error upper bound. A highly effective special case is discussed next.
\paragraph{\bf Beta duals}
Beta duals have been recently proposed and studied in \cite{Chou_thesis, CG14}. They
constitute a special case of $V$-duals, while they relate strongly to classical beta expansions. (See \cite{P, DK} for the classical theory of beta expansions, and \cite{DDGV} for the use of beta expansions in A/D conversion as a robust alternative to successive approximation.)
In order to illustrate the main construction of beta duals without technical details, our presentation in this article will be restricted to certain dimensional constraints as described below.
Let $m \geq p \geq k$ and assume that $\lambda' := m/p$ is an integer.
For any $\beta > 1$, let $h^\beta$ be the (length-$2$)
sequence given by $h^\beta_0 = 1$ and $h^\beta_1 = -\beta$.
Define $H^\beta$ to be the $\lambda' \times \lambda'$ noise-shaping transfer operator corresponding to $h^\beta$, and
$$v^\beta := [\beta^{-1} ~~ \beta^{-2}~\cdots~~\beta^{-\lambda'}].$$
We set
\begin{equation}\label{H-V-beta}
H:=\left[\begin{matrix} H^\beta & & \\ & \ddots & \\ & & H^\beta \end{matrix}\right]_{m \times m}
\quad\text{and}\quad
V:=\left[\begin{matrix} ~~~v^\beta~~~ & & \\ & ~~~\ddots~~~ & \\ & & ~~~v^\beta~~~ \end{matrix}\right]_{p \times m}.
\end{equation}
In other words, $H = I_p \otimes H^\beta$ and $V=I_p \otimes v^\beta$ where $\otimes$ denotes the Kronecker product. It follows that $VH = I_p \otimes (v^\beta H^\beta)$.
Since $v^\beta H^\beta = [0~\cdots~0~~\beta^{-\lambda'}]$, we have
$\|VH \|_{\infty\to\infty} = \beta^{-\lambda'}$ which, together with \eqref{error_bound_PsiHu} and \eqref{error_PsiVu}, yields
\begin{equation} \label{betadual_bound}
\|x - \Psi_{V} q\|_2 \leq
\frac{\sqrt{p} \|u\|_\infty}{\sigma_{\mathrm{min}}(V\Phi)} \beta^{-\lambda'}.
\end{equation}
For certain special frames, such as the harmonic semi-circle frames, it is possible to set $p$ as low as $k$ and turn the above bound into a near-optimal one in terms of its bit-rate \cite{CG14}.
The case of random frames will be discussed in the next section.
In Fig. \ref{fig:4_duals_hf}, we illustrate a beta dual of a certain ``roots-of-unity'' frame along with the Sobolev duals of order 0 (the canonical dual), 1, and 2.
\medskip
\begin{figure}[htp]
\centering
\centerline{\includegraphics[scale=0.75]{4_duals_hf}}
\caption{Comparative illustration of the various alternative duals described in this paper:
Each plot depicts the original frame in ${\mathbb R}^2$ consisting of the $15th$ roots-of-unity along with one of its duals (scaled up by a factor of two for visual clarity). For the computation of the alternative duals, the analysis frame was ordered counter-clockwise starting from $(1,0)$.}
\label{fig:4_duals_hf}
\end{figure}
\section{Analysis of Alternative Duals for Random Frames}
\label{random-frames}
In this section, we consider random frames, that is, frames whose analysis (or synthesis) operator is a random matrix. Certain classes of random matrices have become of considerable importance in high dimensional signal processing, particularly with the advent of compressed sensing. One main reason for this is that their inherent independence entails good conditioning of not only the matrix, but also its submatrices. Because of the fast growing number of such submatrices with dimension, the latter is very difficult to achieve with deterministic constructions. This also means, however, that any two frame vectors are approximately orthogonal, so frame path conditions that would imply recovery guarantees using canonical dual frames will almost never hold. For this reason, it is crucial to work with alternative duals. We separately consider the two main examples discussed above, Sobolev duals and beta duals.
\subsection{Sobolev duals of random frames}
\label{random-Sobolev}
As noted above, the Sobolev dual of a frame is the dual frame $\Psi$ that minimizes the expression $\|\Psi D^r\|_{2\rightarrow 2}$, and the explicit minimizer is given by \eqref{opt-Psi-H} with $H=D^{r}$. By \eqref{error_PsiHu_final}, a bound for the error that arises when using this alternative dual to reconstruct is governed by $\sigma_{\mathrm{min}}(D^{-r}\Phi)$. Thus a main goal of this subsection is to discuss the behavior of this minimum singular value.
The matrix $D^{-r}\Phi$ is the product of a deterministic matrix $D^{-r}$, whose singular values are known to a sufficient approximation, and a random matrix $\Phi$, whose singular values are known to be well concentrated. Nevertheless, using a product bound does not yield good results, mainly because the singular values of $D^{-r}$ differ tremendously, so any worst case bound will not be good enough. One approach to provide a refined bound is to first provide lower bounds for the action of $D^{-r}\Phi$ on a single vector and then proceed via a covering argument. That is, one combines these lower bounds for all of the vectors forming an $\epsilon$-net, obtaining a uniform bound for the net. An approximation argument then allows to pass from the net to all vectors in the sphere. In this way, \cite{GLPSY} obtains the following result for Gaussian random frames:
\begin{theorem}[\cite{GLPSY}] \label{Sobolev_dual_bound}
Let $\Phi$ be an $m\times k$ random matrix whose entries are
i.i.d.~standard Gaussian variables. Given $r \in \mathbb{N}$ and $\alpha \in (0,1)$, there exist constants strictly positive $r$-dependent constants $c_1$, $c_2$, and $c_3$ such that
if $\lambda:=m/k \geq (c_1 \log m)^{1/(1-\alpha)}$, then with probability at
least $1 - \exp(-c_2 m \lambda^{-\alpha})$,
\begin{equation}\label{GLPSY-sigmin}
\sigma_{\min}(D^{-r}\Phi) \geq c_3(r) \lambda^{\alpha(r-\frac{1}{2})}\sqrt{m}.
\end{equation}
\end{theorem}
In this approach, one explicitly uses the density of the Gaussian distribution. Thus, as soon as the matrix entries fail to be exactly Gaussian, a completely different approach is needed. In what follows, we will present the main idea of the method used in \cite{subGaussian} to tackle the case of random matrices with independent sub-Gaussian entries as introduced in the following definition (for alternative characterizations of sub-Gaussian random variables see, for example, \cite{ve12-1}). This approach is also related to the RIP-based analysis for quantized compressive sampling presented in \cite{FK14} (cf.~Section~\ref{CS} below).
\begin{definition}\label{def:subgvar} A random variable $\xi$ is sub-Gaussian with parameter $c>0$ if it satisfies ${\mathbb P}(|\xi| >t) \leq e^{1-ct^2}$ for all $t\geq 0$.
\end{definition}
As in the Gaussian case presented in \cite{GLPSY}, we employ the singular value decomposition $D^{-r}=U \Sigma V^*$ where $U$ and $V$ are unitary and $\Sigma\in {\mathbb R}^{m\times m}$ is a diagonal matrix with entries $s_1\geq \dots \geq s_m \geq 0$. Then
\begin{equation*}
\sigma_{\mathrm{min}}(D^{-r}\Phi) = \sigma_{\mathrm{min}} (U \Sigma V^* \Phi) = \sigma_{\mathrm{min}} ( \Sigma V^* \Phi),
\end{equation*}
as $U$ is unitary. Furthermore, for $P_\ell:{\mathbb R}^m\rightarrow {\mathbb R}^\ell$ the projection onto the first $\ell$ entries, $\ell\leq m$, one has in the positive semidefinite partial ordering $\succeq$
\begin{equation*}
\Sigma \succeq P_\ell \Sigma = P_\ell \Sigma P_\ell^* P_\ell \succeq s_\ell P_\ell.
\end{equation*}
Here the first inequality uses that $P_\ell$ is a projection, the following equality uses that $\Sigma$ is diagonal, and the last inequality uses that the diagonal entries of $\Sigma$ are ordered.
As a consequence, we find that $\sigma_{\mathrm{min}}(D^{-r}\Phi) \geq s_\ell \sigma_{\mathrm{min}}(V^*\Phi)$. For Gaussian matrix entries, this immediately yields Theorem~\ref{Sobolev_dual_bound}, as standard Gaussian vectors are rotation invariant, so $P_\ell V^*\Phi$ is just a standard Gaussian matrix, whose singular value distributions are well understood (see for example \cite{ve12-1}). Applying the bound for different values of $\ell$ yield the theorem for different choices of $\alpha$.
For independent, zero mean, unit variance sub-Gaussian (rather than Gaussian) matrix entries, one no longer has such a strong version of rotation invariance; while the columns of $V^* \Phi$ will still be sub-Gaussian random vectors, its entries will, in general, no longer be independent. There are also singular value estimates that require only independent sub-Gaussian matrix columns rather than independent entries (see again \cite{ve12-1}), but such bounds require that the matrix columns are of constant norm. Even if $\Phi$ and hence also $V^*\Phi$ has constant norm columns (such as for example for Bernoulli matrices, $\Phi_{ij}\in \pm 1$), the projection $P_\ell$ will typically map them to vectors of different length.
In order to nevertheless bound the singular values, we again use a union bound argument, first considering the action on one fixed vector $x$ of unit norm. Then we write
\[
\|V^*\Phi x\|_2^2 = \sum_{i, i'=1}^k \sum_{j,j'=1}^m x_i \Phi_{j i} (V P_\ell^* P_\ell V^*)_{j j'} \Phi_{j' i'} x_{i'}.
\]
Thus $\|V^*\Phi x\|_2^2$ is a so-called chaos process, that is, a random quadratic form of the form $\langle \xi, M \xi\rangle$, where $\xi$ is a random vector with independent entries (in this case, the vectorization of $\Phi$). Its expectation is given by
\[
{\mathbb E} \|V^*\Phi x\|_2^2 = \sum_{i=1}^k \sum_{j=1}^m x_i^2 {\mathbb E} \Phi_{j i}^2 (V P_\ell^* P_\ell V^*)_{j j} = \|x\|_2^2 {\operatorname{tr}}(V P_\ell^* P_\ell V^*) =\ell,
\]
where the last equality uses the cyclicity of the trace. Its deviation from the expectation can be estimated using the following refined version of the Hanson-Wright inequality, which has been provided in \cite{rv13} (see \cite{HW71} for the original version).
\begin{theorem}\label{thm:HW}
Let $\xi=(\xi_1,\dots, \xi_n)\in{\mathbb R}^n$ be a random vector with independent components $\xi_i$ which are sub-Gaussian with parameter $c$ and satisfy ${\mathbb E} \xi_i=0$. Let $A$ be an $n\times n$ matrix. Then for every $t\geq 0$,
\[
{\mathbb P}\big\{|\langle \xi, M \xi\rangle - {\mathbb E} \langle \xi, M \xi\rangle|>t\big\} \leq 2\exp\Big(-C_4 \min\big(\frac{t^2}{c^4 \|M\|_{F}^2}, \frac{t}{c^2\|M\|_{2\rightarrow 2}}\big)\Big),
\]
where $C_4$ is an absolute constant.
\end{theorem}
To obtain a deviation bound for the above setup, we thus need to estimate the Frobenius norm $\|M\|^2_F :={\operatorname{tr}} M^*M= \sum_{i,i',j,j'} M_{(i,i'),(j,j')}^2$ and the operator norm $\|M\|_{2\rightarrow 2}:= \sup_{\|y\|_2=1} \|My\|_2$ of the doubly-indexed matrix $M$ given by $M_{(i,j),(i',j')}=x_i x_{i'} (V P_\ell^* P_\ell V^*)_{j j'} $. For the Frobenius norm, we write
\[
\|M\|^2_F = \sum_{i,i',j,j'} x_i^2 x_{i'}^2 (V P_\ell^* P_\ell V^*)_{j j'}^2 = \| V P_\ell^* P_\ell V^*\|_F^2 = {\operatorname{tr}} (V P_\ell^* P_\ell V^*V P_\ell^* P_\ell V^*) = \ell,
\]
where in the last equality, we used again the cyclicity of the trace, that $V$ is unitary, and that $P_\ell^*P_\ell$ is a projection. For the operator norm, we note that\[
M= P_{ \ell} V^* \left( \begin{matrix} x^T & 0 & \cdots & 0\\
0 & x^T & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & \cdots & 0 & x^T
\end{matrix} \right), \]
so as all these three factors have operator norm $1$, the norm of their product is bounded above by $1$. On the other hand, applying $M$ to the unit norm vector $y$ given by $y_{(i,j)}= x_i V_{1j}$ yields $My=e_1$, where $e_1$ is the first standard basis vector, showing that the norm is also lower bounded by $1$. So one indeed has $\|M\|_{2\rightarrow 2}=1$. Combining these bounds with Theorem~\ref{thm:HW} yields the following generalization of Theorem~\ref{Sobolev_dual_bound} for sub-Gaussian frames.
\begin{theorem}[\cite{subGaussian}]
\label{thm:main_1}
Let $\Phi$ be an $m\times k$ random matrix whose entries are zero mean, unit variance, sub-Gaussian random variables with parameter $c$.
Given $r \in \mathbb{N}$ and $\alpha \in (0,1)$, there exist constants $c=c(r)>0$ and $c'=c'(r)>0$ such that if $\lambda:= \frac{m}{k} \geq c^{\frac{1}{1-\alpha}}$ then one has with probability at
least $1 - \exp(-c' m \lambda^{-\alpha})$
\begin{equation}\label{KSY-sigmin}
\sigma_{\min}(D^{-r}\Phi) \geq \lambda^{\alpha(r-\frac{1}{2})}\sqrt{m}.
\end{equation}
\end{theorem}
Combining \eqref{error_PsiHu_final} for $H=D^r$ with the lower bound of \eqref{GLPSY-sigmin} or \eqref{KSY-sigmin}, the Sobolev dual reconstruction $\Psi_{D^{-r}}q$ from $\Sigma\Delta$ quantized frame coefficients $y=\Phi x$ results in the error bound
\begin{equation} \label{Sobolev-random-err-bound}
\|x - \Psi_{D^{-r}}q \|_2 \leq C(r) \lambda^{-\alpha(r-\frac{1}{2})} \| u\|_\infty.
\end{equation}
Thus the error decays polynomially in the oversampling rate $\lambda$ as long as the underlying $\Sigma\Delta$ scheme is stable. For the greedy quantization rule, stability follows from Proposition \ref{P:greedy}, as long as $\|y\|_\infty \leq \mu$ for a suitable $\mu$ whose range is constrained by the quantization alphabet ${\mathscr A}_{L,\delta}$ and $r$. (It can be easily computed that for $H=D^r$, we have $\|\tilde H\|_{\infty \to \infty} = 2^r - 1$. Hence we require $L > 2^r - 1$, with the value of $\delta$ assumed to be adjustable.) If we assume that $\|x\|_2 \leq 1$, then controlling $\|y\|_\infty$ amounts to bounding $\|\Phi\|_{2\to \infty}\leq \|\Phi\|_{2\to 2}$ and thus to bounding the maximum singular value of a rectangular matrix with independent sub-Gaussian entries. This is a well-understood setup, it is known that the singular values of such a matrix are well concentrated and one has $\|\Phi\|_{2\to \infty}\leq \|\Phi\|_{2\to 2} = {O}(\sqrt m)$ with high probability (see again \cite{ve12-1}). As a consequence,
the
$\Sigma\Delta$ scheme is stable provided $L$ is chosen large enough and the quantizer level is adjusted accordingly. We conclude that sub-Gaussian frame expansions quantized using a greedy $r$-th order $\Sigma\Delta$ scheme allow for reconstruction error bounds decaying polynomially in the oversampling rate, where the decay order can be made arbitrarily large by choosing $r$ large enough.
\subsection{Beta duals of random frames}
\label{random-beta}
We return to the Gaussian distribution for the analysis of beta duals for random frames.
Based on the error bound \eqref{betadual_bound} derived in Section \ref{V-duals}, it now suffices to give a probabilistic lower bound for $\sigma_{\mathrm{min}}(V\Phi)$. Note that the entries of the $p\times k$ matrix $V\Phi$ are i.i.d.~Gaussian with variance
\begin{equation}\label{sig-lambda}
\sigma^2_{\lambda'}:=\beta^{-2}+\cdots+\beta^{-2\lambda'}.
\end{equation}
At this point, a choice for the parameter $p$ needs to be made. In \cite{CG14}, both choices of $p=k$ and $p > k$ were studied in detail. The analysis of the former choice is somewhat cleaner, but the strongest probabilistic estimates follow by choosing $p$ greater than $k$.
We will primarily be interested in the smallest singular value of $V\Phi$ being near zero. For $p=k$, the following well-known result suffices:
\begin{theorem}[{\cite[Theorem 3.1]{rudelson2010non}}, {\cite{edelman1988eigenvalues}}]\label{P:squaregaussminsv}
Let $\Omega$ be a $k \times k$ random matrix with entries drawn
independently from $\mathcal{N}(0,\sigma^2)$. Then for any $\varepsilon > 0$,
\[ \mathbb{P}\left(\left\{ \sigma_{\mathrm{min}}(\Omega) \leq
\varepsilon\sigma/\sqrt{k}\right\}\right) \leq \varepsilon. \]
\end{theorem}
Meanwhile, the stability of the greedy quantizer with alphabet ${\mathscr A}_{L,\delta}$ can be ensured in a way similar to the case of Sobolev duals, noting that $\|\tilde H\|_{\infty \to \infty} = \beta$. Hence, we know that if $\beta + \mu/\delta \leq L$, then $\|u\|_\infty \leq \delta$. By standard Gaussian concentration results, $\mu \leq 4 \sqrt{m}$ is guaranteed with probability at least $1-e^{-2m}$.
Therefore, with \eqref{betadual_bound} and Theorem \ref{P:squaregaussminsv} in which we set $\Omega = V\Phi$, we obtain
\begin{equation} \label{betadual_bound2}
\|x - \Psi_{V} q\|_2 \leq
kL \varepsilon^{-1} \delta \beta^{-m/k}
\end{equation}
with probability at least $1-\varepsilon - e^{-2m}$, where we have also used the simple chain of inequalities $1/\sigma_{\lambda'} \leq \beta \leq L$. The value of $\beta$ can be chosen arbitrarily close to $L$ with sufficiently large values of $\delta$. However, the optimal choice would result from minimizing $\delta \beta^{-m/k}$ subject to $\beta + \mu/\delta = L$. For details, see \cite{CG14}.
For $p>k$, we have the following result:
\begin{theorem}[{\cite[Theorem 4.3]{CG14}}]\label{T:svmingaussrect}
Let $p > k$ and $\Omega$ be a $p \times k$ random matrix whose entries are drawn independently from $\mathcal{N}(0,\sigma^2)$. Then for any $0 < \varepsilon < 1$,
\[ \mathbb{P}\left(\left\{ \sigma_{\mathrm{min}}(\Omega) \leq \varepsilon\sigma \sqrt{p}/2\right\}\right) \leq
\left (10+8\sqrt{\log \varepsilon^{-1}}\right)^k e^{p/2} \varepsilon^{p-k}.
\]
\end{theorem}
The corresponding error bound
\begin{equation} \label{betadual_bound3}
\|x - \Psi_{V} q\|_2 \leq
2 L \varepsilon^{-1} \delta \beta^{-m/p}
\end{equation}
now holds with higher probability. The choices $\varepsilon \approx \beta^{-\eta m/p}$ for small $\eta$ and $p \approx (1+\eta)k$ turn out to be good ones. For details, again see \cite{CG14}.
\section{Noise-shaping Quantization for Compressive Sampling}
\label{CS}
Compressive sampling (also called compressed sensing) has emerged over the last decade as a novel sampling paradigm. It is
based on the empirical observation that various important classes of
signals encountered in practice, such as audio and images, admit
(nearly) sparse approximations when expanded with respect to an
appropriate basis or frame, such as a wavelet basis or a Gabor
frame. Seminal papers by Cand{\`e}s, Romberg, and Tao \cite{CRT}, and
by Donoho \cite{Donoho2006} established the fundamental theory,
specifying how to collect the samples (or measurements), and the
relation between the approximation accuracy and the number of samples
acquired (``sampling rate'') vis-a-vis the sparsity level of the
signal. Since then the literature has matured considerably, again
focusing on the same issues, i.e., how to construct effective
measurement schemes and how one can control the approximation error as
a function of the sampling rate, e.g., see \cite{FoucartRauhutBook}.
By now compressive sampling is well-established as an
effective sampling theory. From the perspective of
practicability, however, it also needs to be accompanied by a
quantization theory. Here, as in the case of frames, MSQ is highly limited as a quantization strategy in terms of its rate-distortion performance. Thus, efficient quantization methods are needed for compressive sampling to live up to its name, i.e., to provide compressed representations in the sense of source coding.
In this section, we will discuss how noise-shaping methods can be employed to
quantize compressive samples of sparse and compressible signals to vastly improve the reconstruction accuracy compared to the default method of MSQ. We start with the basic framework of compressive sampling as needed for our discussion.
\subsection{Basics of Compressive Sampling}\label{sec:CSintro}
In the basic theory of compressive sampling, the signals of interest are finite (but potentially high) dimensional vectors that are exactly or approximately
\emph{sparse}. More precisely, we say that a signal $x$ in ${\mathbb R}^N$ is
$k$-sparse if it is in $\Sigma_k^N:=\{ x\in {\mathbb R}^N:\ \|x\|_0\le
k\}$. Here $\|x\|_0$ denotes the number of non-zero entries of
$x$. The signals we encounter in practice are typically not sparse,
but they can be well-approximated by sparse signals. Such signals are
referred to as compressible signals and roughly identified as signals
$x$ with small $\sigma_k(x)_{\ell_p}$, \emph{the best $k$-term
approximation error of $x$ in $\ell_p$}, defined by
$$
\sigma_k(x)_{\ell_p}:=\min_{z\in \Sigma_k^N} \|x-z\|_p.
$$
Compressive sampling consists of acquiring linear,
non-adaptive measurements of sparse or compressible signals, possibly
corrupted by noise, and recovering (an approximation to) the original
signal from the compressive samples via a computationally tractable
algorithm. In other words, the compressive samples are obtained by
multiplying the signal of interest by a \emph{compressive sampling (measurement) matrix}. The
success of recovery algorithms relies heavily on certain properties of
this matrix. To state this dependence precisely, we next define the
restricted isometry constants of a matrix.
\begin{definition} \label{def:RIP}
The restricted isometry constant (see, e.g., \cite{CRT})
$\gamma_k:=\gamma_k(\Phi)$ of a matrix $\Phi \in {\mathbb R}^{m\times N}$ is the smallest
constant for which
$$(1-\gamma_k)\|x\|_2^2\leq \|\Phi x\|_2^2 \leq (1+\gamma_k )\|x\|_2^2
$$
for all $x\in \Sigma_k^N$.
\end{definition}
Suppose that $\Phi \in {\mathbb R}^{m\times N}$ is used as a compressive
sampling matrix. Here, $m$ denotes the number of measurements
and is significantly smaller than $N$, the ambient dimension of the
signal. Let $\tilde y:=\Phi x + w$ denote the (possibly) perturbed measurements of a signal $x \in {\mathbb R}^N$, where the unknown perturbation $w$ satisfies $\|w\|_2 \leq \epsilon$. A crucial result in the theory of compressive sampling states that if the restricted isometry constants of $\Phi$ are suitably controlled (e.g. as originally stated in \cite{CRT}, or more recently as in \cite{Cai_IEEE_2014} which only assumes $\gamma_{ak}\le \sqrt{(a-1)/a}$ for some $a\ge 4/3$), then there is an approximate recovery $\Delta_1^\epsilon(\Phi,\tilde y)$ of $x$ which satisfies
\begin{equation}\label{robrec}
\|x-\Delta_1^\epsilon(\Phi,\tilde y)\|_2\leq C \epsilon + D\sigma_k(x)_{\ell_1}/\sqrt{k}.
\end{equation}
Here, $\Delta_1^\epsilon(\Phi,\tilde y)$ is found by mapping $\tilde y$ to a minimizer of a tractable, convex optimization problem---which is often called the ``Basis Pursuit Denoise'' algorithm---given by
$$\Delta_1^\epsilon(\Phi,\tilde y):=\arg\min_{z}\|z\|_1 \ \ \text{subject to} \ \
\|\Phi z - \tilde y\|_2\le \epsilon.
$$
$C$ and $D$ are constants that depend on $\Phi$, but can be made absolute by slightly stronger assumptions on $\Phi$.
Note that in the noiseless case, it follows from \eqref{robrec} that any $k$-sparse signal can be exactly recovered from its compressive samples as $\Delta_1^0(\Phi,\Phi x)$. In the general case, the approximation error remains within the noise level and within the best $k$-term approximation error of $x$ in $\ell_1$. Hence the recovery is robust with respect to the amount of noise and stable with respect to violation of the exact sparsity assumption.
The decoder $\Delta_1^\epsilon$ is a \emph{robust compressive sampling decoder} as defined next.
\begin{definition}\cite[Definition 4.9]{subGaussian}
Let $\varepsilon>0$, let $m,N$ be positive integers such that
$m<N$ and suppose that $\Phi\in{\mathbb R}^{m\times N}$. We say that
$\Delta: {\mathbb R}^{m\times N}\times{\mathbb R}^m\rightarrow {\mathbb R}^N$ is a robust compressive sampling decoder with
parameters $(k, a,\gamma)$, $k<m$, and constant $C$ if
\begin{equation}\label{now26}
\|x-\Delta(\Phi, \Phi x+e)\| \le C \varepsilon,
\end{equation}
for all $x\in \Sigma_k^N$, $\|e\|_2 \le \varepsilon$, and all matrices
$\Phi$ with a restricted isometry constant
$\gamma_{ak}<\gamma$.
\end{definition}
Examples of robust decoders include $\Delta_1^\epsilon$ and its $p$-norm generalization $\Delta_p^\epsilon$ with $0<p\le1$
\cite{CRT,SY10}, compressive sampling matching pursuit (CoSaMP)
\cite{cosamp}, orthogonal matching pursuit (OMP) \cite{Zhang11}, and
iterative hard thresholding (IHT) \cite{Blumensath09}. See also
\cite{FoucartRauhutBook} for detailed estimates of the relevant
parameters.
\subsection{Noise-shaping Quantization of Compressive Samples}
Even though noise shaping methods are tailored mainly for quantizing
redundant representations, perhaps surprisingly, they also provide
efficient strategies for quantizing compressive samples
\cite{CISS10,GLPSY,subGaussian, FK14}. The approach, originally developed in
\cite{GLPSY} specifically for $\Sigma\Delta$ quantization, relies on
the observation that when the original signal is exactly sparse,
compressed measurements are in fact redundant frame coefficients of
the sparse signal restricted to its support. Since then it has been extended for beta encoding and applied to compressible signals as well \cite{Chou_thesis}. We start with the case of sparse signals.
\subsubsection{Sparse signals}
Let $x\in
\Sigma_k^N$ with $\text{supp}(x)=T$ and $\Phi\in {\mathbb R}^{m\times N}$ be a
compressive sampling matrix. Then, we have
$$y=\Phi x \ \ \implies \ \ y=\Phi_T x_T,$$
where $\Phi_T$ is the submatrix of $\Phi$ consisting of its columns
indexed by $T$ and $x_T$ is the restriction of $x$ to
$T$. Accordingly, \emph{any} quantization technique designed for frames could be
adopted to compressive sampling as follows: \medskip
\noindent {\bf Quantization:} Since the compressive samples are in
fact frame coefficients, apply the noise-shaping quantization algorithm
directly to the compressive samples $y$ to obtain the quantized
samples, say, $q$. Note that the quantization process is blind
to the support of the sparse signal as well as to the sampling operator. \medskip
\noindent {\bf Reconstruction:} Reconstruct via the following
\emph{two-stage reconstruction algorithm}. To obtain an estimate
$x^{\#}$ of $x$ from $q$:
\begin{enumerate}
\item {\bf Coarse Recovery:} Solve \begin{equation}
\tilde{x}=\Delta_1^{\epsilon_Q}(\Phi, q)
\label{eq:l1_SD}
\end{equation}
where $\epsilon_Q$ is an upper bound on $\|y-q\|_2$, which depends on
the quantization scheme and is known explicitly. Note that the
decoder $\Delta_1^{\epsilon_Q}$ above can be replaced with any robust
compressive sampling decoder $\Delta$. Clearly, by \eqref{now26}
$\|x-\tilde{x}\|$ will be small if $\epsilon_Q$ is small.
\smallskip
\item {\bf Fine Recovery:} Obtain a support estimate, $\tilde{T}$, of
$x$ from $\tilde{x}$. A finer approximation for $x$ is then given by
reconstructing with an appropriate alternative dual
of the underlying frame $\Phi_{\tilde{T}}$ based on the noise-shaping operator that was employed for quantization.
\end{enumerate}
The success of the two-stage reconstruction algorithm relies on the
accurate recovery of the support of $x$. In turn, this can be
guaranteed by a size condition on the smallest-in-magnitude non-zero
entry of $x$. To see this, note that for all $i \in T$, the robustness guarantee
\eqref{now26} yields $|\tilde{x}_i - x| \leq C \epsilon_Q$, which,
together with the size condition $\min_{i\in T } |x_i| > 2C \epsilon_Q$,
gives $|\tilde{x}_i|> C \epsilon_Q$. Moreover, by \eqref{now26} we
have $|\tilde{x}_i| \leq C \epsilon_Q$ for all $i \in
T^c$. Consequently, the largest-in-magnitude $k$ coefficients of
$\tilde{x}$ are supported on $T$. Thus, we have the following proposition.
\begin{proposition}\label{prop:sup_rec}
Suppose that $x \in \Sigma_k^N$ with $supp(x)=T$, and let $\Phi \in
{\mathbb R}^{m\times N}$ be a compressive sampling matrix so that
\eqref{now26} holds for $\Delta = \Delta_1^{\epsilon_Q}$ with robustness constant $C$. Let $\tilde{x}$ be as in \eqref{eq:l1_SD} where $\|\Phi x - q\|_2 \leq \epsilon_Q$. If $\min_{i\in T} |x_i| > 2 C
\epsilon_Q$, then the $k$ largest-in-magnitude coefficients of
$\tilde{x}$ are supported on $T$.
\end{proposition}
By this observation, the coarse recovery stage not only yields an
estimate $\tilde{x}$ that satisfies $\|x-\tilde{x}\|_2\leq C
\epsilon_Q$, but it also gives an accurate estimate of the support of
$x$ (via the support of the $k$-largest coefficients of
$\tilde{x}$). It remains to show that reconstruction techniques
associated with noise shaping quantization for frames can be used in
the fine recovery stage to produce an estimate $x^{\#}$ that is more
accurate than $\tilde{x}$ of the coarse stage.
When $q$ results from a noise-shaping quantization scheme,
accurate recovery based on alternative duals can be guaranteed via
\eqref{error_bound_PsiHu}. In particular, suppose that $H$ is
the noise transfer operator of the quantizer. Conditioned on recovering
$T$, let $\Psi_{H^{-1}}$ be the left inverse of $\Phi_T$ as defined
in \eqref{opt-Psi-H} and set $x^{\#}:=
\Psi_{H^{-1}} q$. We then have, as before,
\begin{equation}\label{a-A-H-wtf}
\|x-x^{\#}\|_2 \le \frac{\sqrt{m}}{\sigma_{\mathrm{min}}(H^{-1}\Phi_T)}\|u \|_\infty
\end{equation}
where $u$ is as in \eqref{y-q-H-u} .
Predominantly, compressed sensing matrices $\Phi$ (hence their
submatrices $\Phi_T$) are random matrices. Thus, to uniformly control
the reconstruction error via \eqref{a-A-H-wtf} one needs lower bounds
on the smallest singular values of the random matrices $H^{-1}\Phi_T$
for all $T\subset [N]:=\{1,\dots,N\}$, $|T|=k$, as well as a uniform
upper bound on $\|u\|_\infty$.
We concentrate again on random matrices $\Phi$ with independent and identically
distributed Gaussian or sub-Gaussian entries. In these cases, for each fixed support $T$, $\Phi_T$ is a random frame of the type considered in Section~\ref{random-frames} and a probabilistic lower bound
on $\sigma_{\min}(H^{-1}\Phi_T)$ follows from Theorem~\ref{Sobolev_dual_bound} (for Gaussian entries) and Theorem~\ref{thm:main_1} (for sub-Gaussian entries).
A uniform lower bound on
$\sigma_{\min}(H^{-1}\Phi_T)$ over all support
sets $T$ of size $k$ can now be deduced via a union bound over
the ${N}\choose{k}$ support sets.
Note that to obtain a uniform bound over this rather large set of supports, one requires a relatively small bound for the probability of failure on each potential support, and, consequently, a larger embedding dimension $m$ as compared to the case of a single frame. An alternative approach based on the restricted isometry constant, essentially yielding the same result, can be found in \cite{FK14}.
The approaches just outlined are general and can be applied in the
case of any noise shaping quantizer that allows exact recovery of the
support of sparse vectors via Proposition~\ref{prop:sup_rec}. In the
following, however, we focus on the special case of $r$th-order $\Sigma\Delta$
quantization, where $H=D^{-r}$ and we obtain the following theorem.
\begin{theorem}[\cite{GLPSY,subGaussian}]\label{thm:polynomial}
Let $r\in {\mathbb Z}^+$, fix $a\in{\mathbb N}$, $\gamma<1$, and $c, C>0$. Then there exist constants $C_{1}, C_{2},
C_{3}, C_{4}$ depending only on these parameters such that the following holds.
Fix $0<\alpha<1$. Let $\Phi$ be an $m \times N$ matrix with
independent sub-Gaussian entries that have zero mean, unit variance, and
parameter $c$, let $\Delta$ be a robust compressive sampling decoder and $k\in{\mathbb N}$
is such that
$$\lambda:=\frac{m}{k}\geq \Big(C_{1} \log(eN/k)\Big)^{\frac{1}{1-\alpha}}.$$
Suppose that $q$ is obtained by quantizing $\Phi z$, $z\in {\mathbb R}^N$,
via the $r$th order greedy $\Sigma\Delta$ scheme with the alphabet ${\mathscr A}_{L,\delta}$, and with $L\geq \lceil\frac{K\lambda^{-1/2}}{\delta}\rceil
+2^r+1$. Denote by $q$ the quantization output resulting from $\Phi
z$ where $z\in{\mathbb R}^N$. Then with probability exceeding $1-4e^{-C_{2}
m^{1-\alpha}k^{\alpha}}$ for all $x\in \Sigma_k^{N}$ having
$\min\limits_{j\in \rm{supp}{(x)}}|x_j| > C_{3} \delta$:
\begin{enumerate}
\item[(i)] The support of $x$, $T$, coincides with the support of the best $k$-term approximation of $\Delta(\frac{1}{\sqrt{m}}\Phi,\frac{1}{\sqrt{m}}q)$.
\item[(ii)] Denoting by $\Phi_T$ and $F$ the sub-matrix of $\Phi$ corresponding to the support of $z$ and its $r$th order Sobolev dual respectively, and by $x_T\in{\mathbb R}^k$ the restriction of $x$ to its support, we have
$$\|x_T-Fq\|_2 \leq C_{4} \lambda^{-\alpha (r-1/2)}\delta.$$
\end{enumerate}
\end{theorem}
We remark that in Theorem \ref{thm:polynomial}, the requirement that $L\geq \lceil\frac{K\lambda^{-1/2}}{\delta}\rceil +2^r+1$ ensures stability of the $\Sigma\Delta$ scheme while $\min\limits_{j\in \rm{supp}{(x)}}|x_j| > C_{3} \delta$ implies accurate support recovery.
\subsubsection{Compressible signals}
The two-stage reconstruction algorithm for sparse signals presented above applies equally well to noise-shaping quantization based on beta encoding as discussed in Section \ref{V-duals}. However, it turns out that for beta encoding there is a more powerful reconstruction algorithm which works for compressible signals as well.
Let $\Phi$ now be an $m \times N$ compressive sampling matrix, and let $H$ be the $m \times m$ noise transfer operator and $V$ be the $p \times m$ condensation operator as in \eqref{H-V-beta}, where again, for simplicity, we have assumed that $m/p$ is an integer. Note that the associated noise-shaping quantization relation
$$ \Phi x - q = H u$$
implies
$$ V\Phi x - Vq = VH u,$$
hence we may consider $ V\Phi$ as a new condensed measurement matrix and $Vq = V\Phi x + VHu$ as the corresponding perturbed measurement. As before,
$$\|VHu\|_2 \leq \|VH\|_{\infty \to 2} \|u\|_\infty
\leq \sqrt{p} \beta^{-m/p} \|u\|_\infty ,$$
so that if the greedy quantization rule is stable (i.e., $\|u\|_\infty \leq \delta$), then
we can set $\epsilon:= \sqrt{p} \beta^{-m/p} \delta$ and consider the decoder $$(q \mapsto \Delta_1^\epsilon(V\Phi,Vq)).$$
As it follows from the discussion of \eqref{robrec}, if for some $\alpha > 0$, $\gamma_{2k}:=\gamma_{2k}(\alpha V\Phi)$ is sufficiently small (say less than $1/3$), then we have the estimate
\begin{equation}\label{compressible_beta_err_bound}
\|x - \Delta_1^\epsilon(V\Phi,Vq)\|_2 \leq C \alpha \epsilon + D \frac{\sigma_k(x)_1}{\sqrt{k}},
\end{equation}
where $C$ and $D$ are now absolute constants.
For the random (Gaussian) case, the following result is implied by our discussion above and other tools presented earlier in this paper (for a more detailed derivation of a similar result, see \cite{Chou_thesis}):
\begin{theorem} Let $\Phi$ be an $m \times N$ random matrix whose entries are i.i.d. standard Gaussian variables. Let $x \in \mathbb{R}^N$, $\|x\|_2 \leq 1$, and let $q$ be the result of quantizing the measurements $\Phi x$ with the noise transfer operator $H$ from \eqref{H-V-beta} and the alphabet ${\mathscr A}_{L,\delta}$ where $\beta + 2\sqrt{N}/\delta \leq L$.
Assume $m \geq p \geq k$ are such that $\lambda':=m/p$ is an integer and
$$\lambda:= \frac{m}{k} \geq C_1 \lambda' \log N/k$$
for some numerical constant $C_1$.
Let $V$ be the $p \times m$ condensation matrix as in \eqref{H-V-beta} and $\epsilon :=\sqrt{p} \beta^{-m/p} \delta$. Then with probability exceeding
$1 - e^{-p/C'_1}$
for another numerical constant $C'_1$, we have
\[ \|x - \Delta_1^\epsilon(V\Phi, Vq)\|_2 \leq
C L \delta\sqrt{p/m}\, \beta^{-m/p} + D \, \frac{\sigma_k(x)_1}{\sqrt{k}}.
\]
\end{theorem}
We note that the optimal choice of the auxiliary parameters $p$ and $k$ in the above theorem depends on the success probability as well as further information on the amount of compressibility of $x$. A rule of thumb would be to balance the two error terms above corresponding to quantization error and approximation error. Similarly, the choice of $\beta$, $L$, and $\delta$ can be optimized. For example, if $L\geq 2$ is given and fixed, but $\delta$ is variable, then one would minimize the error bound (over $p$, $k$, $\beta$ and $\delta$) within a given probabilistic guarantee objective and a priori knowledge on $x$.
Finally, we end with the following remark: a recent work \cite{SWY15} shows that it is in fact possible to obtain an approximation from $\Sigma\Delta$ quantized compressive samples that is robust to additive noise and is stable for compressible signals. This approximation is obtained via a \emph{one-stage reconstruction method} based on solving a simple convex optimization problem. Furthermore, by encoding the quantized measurements via a Johnson- Lindenstrauss dimensionality reducing embedding as in \cite{IS13}, one obtains near-optimal rate-distortion guarantees in the case of sparse signals. For details, see \cite{SWY15}.
\section*{Acknowledgements}
FK and RS acknowledge support by the German Science Foundation (DFG) in
the context of the Emmy-Noether Junior Research Group KR 4512/1-1
``RaSenQuaSI''. {\"O}Y was funded in part by a Natural Sciences and
Engineering Research Council of Canada (NSERC) Discovery Grant
(22R82411), an NSERC Accelerator Award (22R68054) and an NSERC
Collaborative Research and Development Grant DNOISE II (22R07504).
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 410
|
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Extensions.Logging;
namespace Air.Client.AspNetCore.Sample.Controllers
{
public class HomeController : Controller
{
private ILogger _logger;
public HomeController(ILogger<HomeController> logger)
{
_logger = logger;
}
public IActionResult Index()
{
_logger.LogWarning("What happened here?!");
return View();
}
public IActionResult About()
{
throw new Exception("This is an error!");
}
public IActionResult Contact()
{
ViewData["Message"] = "Your contact page.";
return BadRequest("BadRequest");
}
public IActionResult Error()
{
return View();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,294
|
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