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978 F.2d 1227
Helen L. BAIN,andE. Jeanne Post and Donna L. Johnson, Petitioners,v.OFFICE OF PERSONNEL MANAGEMENT, Respondent.
Nos. 91-3503, 91-3504.
United States Court of Appeals,Federal Circuit.
Oct. 30, 1992.Rehearing Denied Feb. 2, 1993.
Barbara A. Atkin, Sr. Appellate Counsel, Nat. Treasury Employees Union, Washington, D.C., argued for petitioners. With her on the brief was Gregory O'Duden, Director of Litigation. Also on the brief was Michael T. Lamb, Asst. Counsel, National Treasury Employees Union, Chicago, Ill., of counsel.
John P. Sholar, Atty., Commercial Litigation Branch, Dept. of Justice, Washington, D.C., argued for respondent. With him on the brief were Stuart M. Gerson, Asst. Atty. Gen., David M. Cohen, Director and Terrance S. Hartman, Asst. Director. Also on the brief were Vernon B. Park, Gen. Counsel, Thomas F. Moyer, Asst. Gen. Counsel and Earl A. Sanders, Atty., Office of the Gen. Counsel, Office of Personnel Management, of counsel.
Before ARCHER, CLEVENGER, and RADER, Circuit Judges.
RADER, Circuit Judge.
1
Petitioners, seasonal employees with the Internal Revenue Service (IRS), appeal two final decisions of the Merit Systems Protection Board. Both decisions denied petitioners retirement credit for off-season, non-work periods before October 1, 1980. Bain v. Office of Personnel Management, 49 M.S.P.R. 307 (1991); Post v. Office of Personnel Management, 49 M.S.P.R. 172 (1991). This court consolidated the appeals for briefing and argument. Because the Office of Personnel Management (OPM) did not abuse its discretion in denying retirement credit for non-work periods, this court affirms the Board's decisions.
BACKGROUND
2
IRS hired petitioners as seasonal "while actually employed" employees. IRS employed these seasonal employees on an "as needed" basis during peak tax filing months; IRS laid them off the remainder of the year. While on active duty, petitioners worked full-time--forty hours each week.
3
Before October 1, 1980, IRS gave its seasonal employees service credit toward retirement only for work periods. Bain, 49 M.S.P.R. at 308. Beginning October 1, 1980, IRS gave its seasonal employees retirement credit for non-work periods, treating those periods as leaves of absence. Id.
4
IRS appointed Ms. Bain as a seasonal employee in 1972. Ms. Bain worked an average of six months per year before October 1, 1980. Ms. Bain retired on February 28, 1990. Ms. Johnson received her seasonal appointment in 1966. Ms. Johnson's service record shows that she worked an average of over eight months per year before October 1, 1980. Ms. Johnson retired on March 31, 1989. Ms. Post began seasonal work with IRS in 1967. Ms. Post's record shows an average of over seven months per year of work before October 1, 1980. Ms. Post retired on August 31, 1988.
5
Petitioners elected to retire under the Federal Employees' Retirement System (FERS), 5 U.S.C. §§ 8401-8479 (1988 & Supp. II 1990). Petitioners transferred their service credits under the Civil Service Retirement System (CSRS), 5 U.S.C. §§ 8331-8351 (1988 & Supp. II 1990), to FERS. See 5 U.S.C. §§ 8402(b)(2)(B), 8411(b)(3); 5 C.F.R. § 842.104(b)-(c) (1992).
6
When computing petitioners' retirement annuities, OPM gave them service credit only for time actually worked before October 1, 1980. Petitioners requested reconsideration of OPM's denial of service credit for non-work periods before October 1, 1980. OPM affirmed its initial denial. Petitioners appealed to the Board.
7
An administrative judge consolidated Ms. Post's and Ms. Johnson's cases and found them entitled to retirement credit for off-season, non-work periods before October 1, 1980. Post, 49 M.S.P.R. at 174. Upon review, the Board reversed this decision. Id. at 173. In a separate proceeding, an administrative judge did not grant Ms. Bain service credit for non-work periods before October 1, 1980. Upon review, the Board affirmed. Bain, 49 M.S.P.R. at 308.
DISCUSSION
I.
8
Title 5 of the United States Code sets forth this court's standard for reviewing Board decisions. This court upholds a Board decision unless "(1) arbitrary, capricious, an abuse of discretion, or otherwise not in accordance with law; (2) obtained without [following] procedures required by law, rule, or regulation ... or (3) unsupported by substantial evidence." 5 U.S.C. § 7703(c) (1988). Petitioners bear the burden of proving entitlement to annuity benefits. Cheeseman v. Office of Personnel Management, 791 F.2d 138, 141 (Fed.Cir.1986), cert. denied, 479 U.S. 1037, 107 S.Ct. 891, 93 L.Ed.2d 844 (1987).
II.
9
Section 8411 of title 5 sets forth the type of service creditable toward retirement under FERS. Creditable service under FERS includes, with some exceptions, service creditable under CSRS. 5 U.S.C. § 8411(b)(3) (1988). Section 8332(f) makes leaves of absence creditable toward retirement under CSRS. Thus, an employee receives credit under that section for leaves to perform military service or to recuperate from an injury suffered while working for the Government. 5 U.S.C. § 8332(f) (1988). Section 8332(f) further provides:
10
[C]redit may not be allowed for so much of other leaves of absence without pay as exceeds 6 months in the aggregate in a calendar year.
11
Id. Thus, under CSRS, an employee receives no more than six months retirement credit per year for leaves of absence without pay. FERS contains a similar six-month limit on service credit for leaves of absence. 5 U.S.C. § 8411(d); see also 5 C.F.R. 842.304(a)(4) (1992).
12
In defining employees entitled to FERS benefits, 5 U.S.C. § 8401(11) adopts the definition in 5 U.S.C. § 8331(1). Section 8331(1)(A) in turn adopts the broad definition of "employee" at 5 U.S.C. § 2105.1 Although title 5 defines "employee" broadly, section 8331(1)(iii) gives OPM authority to exclude certain workers from CSRS benefits. OPM "may exclude" those employees "whose employment is temporary or intermittent." 5 U.S.C. § 8347(g). A similar provision allows OPM to exclude temporary or intermittent employees from FERS coverage. 5 U.S.C. § 8402(c)(1).
13
Taken as a whole, the statutory schemes of FERS and CSRS allow federal employees up to six months per year of retirement credit for leaves of absence, provided OPM has not excluded the employee from FERS or CSRS benefits as a "temporary or intermittent" worker. 5 U.S.C. §§ 8347(g), 8402(c)(1). Thus, this appeal presents the issue of whether OPM properly classified petitioners as intermittent workers during their off-season, non-work periods before October 1, 1980.
III.
14
Title 5 charges OPM with administering both FERS and CSRS. See 5 U.S.C. §§ 8347(a), 8461. The Civil Service Commission (CSC) preceded OPM in this capacity. See 5 U.S.C. § 8347(a) (1976) (amended in 1978, replacing the CSC with the OPM). Title 5, however, does not mention seasonal workers, nor give instructions on classifying seasonal employees as part-time or intermittent workers. Indeed, the Code does not define "intermittent."
15
OPM interpreted the term "intermittent" in title 5 to encompass seasonal workers and therefore denied petitioners retirement credit for annually recurring non-work periods before October 1980. In making this interpretation, OPM followed a 1965 policy. In an August 2, 1965, Memorandum of Understanding, CSC instituted the policy denying IRS seasonal employees retirement credit for leaves of absence.
16
In the 1965 Memorandum, the General Accounting Office (GAO), IRS, and CSC sought to place IRS seasonal employees within one of the three employment classifications--full-time, part-time, or intermittent.2 After weighing alternatives, CSC in the 1965 Memorandum instructed IRS to give seasonal employees service credit as full-time employees, except that non-work periods would earn no retirement credit. The Memorandum stated that this hybrid arrangement would equitably credit service to seasonal employees. This decision effectively placed seasonal employees in an intermittent status.
17
In computing petitioners retirement annuities, OPM followed the 1965 Memorandum and denied petitioners service credit toward retirement for their inactive periods before October 1, 1980. In the decisions on review here, the Board sustained OPM, finding that the record evidence showed that petitioners' service was "intermittent in nature." Bain, 49 M.S.P.R. at 310; Post 49 M.S.P.R. at 176.
18
In the absence of any statutory standards, this court reviews the Board's determination of whether the agency based its action on a "permissive interpretation of the statute." Chevron U.S.A. v. Natural Resources Defense Council, 467 U.S. 837, 842-43, 104 S.Ct. 2778, 2781-82, 81 L.Ed.2d 694 (1984). In other words, this court accords "substantial weight to an agency's interpretation of a statute it administers." American Lamb Co. v. United States, 785 F.2d 994, 1001 (Fed.Cir.1986) (citing Zenith Radio Corp. v. United States, 437 U.S. 443, 450-51, 98 S.Ct. 2441, 2445, 57 L.Ed.2d 337 (1978)). Moreover, the agency's statutory interpretation "need not be the only reasonable interpretation or the one which the court views as the most reasonable." Consumer Prods. Div., SCM Corp. v. Silver Reed Am., Inc., 753 F.2d 1033, 1039 (Fed.Cir.1985) (citing Fulman v. United States, 434 U.S. 528, 534-36, 98 S.Ct. 841, 845-47, 55 L.Ed.2d 1 (1978) (emphasis in original)); see also Federal Election Comm. v. Democratic Senatorial Campaign Comm., 454 U.S. 27, 39, 102 S.Ct. 38, 46, 70 L.Ed.2d 23 (1981).
19
This court finds no basis for reversing the Board's approval of OPM's classification. Petitioners did not show either that OPM's long-standing3 employment classifications were unreasonable or that OPM unreasonably classified IRS seasonal employees as intermittent workers.
20
Under OPM regulations, intermittent employees are non-full-time workers without a prearranged regular tour of duty. 5 C.F.R. § 831.703(b) (1992). Petitioners' service records show that their work periods varied from year to year. Petitioners worked different months and different lengths of time each year as dictated by the needs of IRS. This as-needed characteristic of petitioners' service conforms most harmoniously with the intermittent classification.
21
Moreover, neither the full-time nor the part-time classifications completely fit seasonal employees. The record shows that petitioners worked from six months to just over eight months a year--a schedule incompatible with full-time service. When on duty, petitioners worked full administrative work-weeks--a schedule incompatible with part-time service.
22
In sum, OPM reasonably classified petitioners as intermittent workers. The Board upheld that classification. This court discerns no reversible error in the Board's decision.
IV.
23
This court's conclusion does not conflict with the Court of Claims' decision in Dove v. United States, 161 Ct.Cl. 768 (1963). In Dove, this court's predecessor determined that one particular "while actually employed" worker was "in all respects save the paper record, a full-time, per annum employee." Id. at 785. Mr. Dove had worked more than ninety percent of the possible work days over a six-year period. Therefore, acknowledging substance over form, the Court of Claims looked past administrative records to classify Mr. Dove as a full-time employee. In determining a federal employee's employment status, Dove clarifies that the actual conditions of the worker's employment take precedence over administrative labels.4 See id. at 786. Under the exceptional circumstances of that case, the Court of Claims determined that Mr. Dove's 1935 appointment "contemplated full-time employment." Id. at 785.
24
Finding that petitioners in this case worked, at most, just over eight months per year, the Board declined to reclassify them as full-time. Substantial evidence supports the Board's finding.
V.
25
In October 1980, OPM reversed its longstanding policy and began giving all seasonal employees retirement credit for yearly inactive periods. Although the nature of petitioners' seasonal employment remained unchanged before and after this decision, the reversal does not show that either OPM's pre-1980 or OPM's post-1980 policy was arbitrary or capricious. Rather title 5 grants OPM discretion to give intermittent employees CSRS benefits. Section 8347 states: "[OPM] may exclude from the operation [of the CSRS] an employee ... whose employment is temporary or intermittent." 5 U.S.C. § 8347(g) (emphasis added).
26
In October 1980, OPM exercised its statutory discretion to ensure uniform treatment of seasonal employees within the Government. See Federal Personnel Manual Letter 296-75, July 10, 1980. Thus, OPM decided to no longer exclude intermittent employees, like petitioners, from CSRS benefits. That decision did not, however, obligate OPM to make its 1980 change retroactive. The Supreme Court notes:
27
Retroactivity is not favored in law. Thus, congressional enactments and administrative rules will not be construed to have retroactive effect unless their language requires this result.
28
Bowen v. Georgetown Univ. Hosp., 488 U.S. 204, 208, 109 S.Ct. 468, 471, 102 L.Ed.2d 493 (1988) (citations omitted). Furthermore, "[e]ven where some substantial justification for retroactive rulemaking is presented, courts should be reluctant to find such authority absent an express statutory grant." Id. at 208-09, 109 S.Ct. at 471. No provision of CSRS required OPM to give its discretionary change retroactive effect. Without a statutory requirement compelling retroactive application of the 1980 change, OPM properly expressed its intention to effect its change prospectively only.
CONCLUSION
29
Because this court upholds an agency's reasonable construction of a statute it administers, the Board's affirmance of OPM's classification of petitioners as intermittent is in accordance with law. Moreover, substantial evidence supports the Board's factual findings about the duration and timing of petitioners' work periods prior to October 1, 1980. Petitioners have not met their burden of showing that OPM's classification of them as intermittent was unreasonable or contrary to the actual nature of their employment. Lastly, OPM acted within its statutorily-granted authority when it granted all seasonal employees credit for leaves of absence without pay after October 1, 1980. For the reasons stated above, this court affirms both Board decisions.
30
AFFIRMED.
1
Section 2105(a) defines "employee" as an individual:
(1) appointed in the civil service by one of the following acting in an official capacity--
(A) the President;
(B) a Member or Members of Congress, or the Congress;
(C) a member of a uniformed service;
(D) an individual who is an employee under this section;
(E) the head of a Government controlled corporation;
(F) an adjutant general designated by the Secretary
concerned under section 709(c) of title 32;
(2) engaged in the performance of a Federal function under authority of law or an Executive act; and
(3) subject to the supervision of an individual named by paragraph (1) of this subsection while engaged in the performance of the duties of his position.
2
The Federal Personnel Manual (FPM) at that time defined full-time employees as "[r]egularly scheduled to work the number of hours and days required by the administrative workweek for their employment group or class. Most full time employees have an administrative workweek of five days of eight hours each." FPM Chapter 292, Subchapter 2. The FPM defined part-time employees as "[n]onfull-time with a prescheduled tour of duty such as four hours a day, five days a week." FPM Supplement 296-31, Appendix B, S210-4h. The FPM defined intermittent employees as "[n]onfull-time without a prescheduled tour of duty." FPM Supplement 296-31, Appendix B, S210-4d
OPM did not define seasonal employment until 1984. See 5 C.F.R. § 340.401(a) (1992) (defining seasonal employment as "annually recurring periods of work of less than 12 months each year. Seasonal employees are placed in a nonduty/nonpay status and recalled to duty in accordance with preestablished conditions of employment.")
3
Neither the CSC nor the OPM substantially changed the definitions for full-time, part-time, and intermittent employment since at least 1954. See Dove v. United States, 161 Ct.Cl. 768, 786 (1963) (citing Federal Personnel Manual, Chapter R1-33, revised March 13, 1954); see also 5 C.F.R. §§ 340.401(c), 831.703(b), 842.402 (1992)
4
Taking a false cue from an erroneous headnote, the Board and OPM seemed to cite Dove v. United States, 161 Ct.Cl. 768 (1963) for the proposition that full-time employment status is a requirement for retirement credit during leave periods. Bain, 49 M.S.P.R. at 312 (failure to prove full-time employment "controlling"); Post, 49 M.S.P.R. at 175 (citing Dove for the proposition that "the employee must establish that her appointment had been contemplated as constituting full time employment.") Dove held only that an employee who worked over ninety percent of the possible work days during a six-year period should properly be considered a full-time employee regardless of how the agency classified him administratively. 161 Ct.Cl. at 784-86. Dove did not make full-time employment a requirement for credit during leave periods
This misreading of Dove does not alter this court's analysis or judgment. The Board specifically found that record evidence showed petitioners' service to be "intermittent in nature." Bain, 49 M.S.P.R. at 310; Post, 49 M.S.P.R. at 176. This court reviewed that conclusion and its supporting findings.
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What holiday do they
celebrate at the beginning
of "High School Musical?"
Um--
Can I use Google?
I know this.
Hi guys.
Hey.
What's up, Morgan?
MORGAN TOMPKINS: Trevor, Ariel,
Phil, Izabela, and Siena,
how are you guys doing?
Doing really good.
MORGAN TOMPKINS: I got
a little something here.
So this is gonna be the
Ultimate Disney Channel Quiz.
OK.
All right, let's do it.
MORGAN TOMPKINS:
For Raven, what is
the name of Raven's
dad's restaurant
on "That's So Raven?"
Chill Grill.
And action!
If you like chilling--
And you like grilling--
Then come to the Chill Grill.
Started you off
with an easy one.
OK.
Miranda where are
Liv and Maddie from?
Wisconsin.
MORGAN TOMPKINS:
It was Wisconsin.
Good job.
Yeah, the cheese.
I remember that.
Ha ha, good morning,
Stevens Point.
I'm your old pal Johnny
Nimbus coming at you
live from the site
of the cheddar brat
fest where pride and pork
come together to celebrate
all things Wisconsin.
Cheddar brat fest, woo!
In "Zombies," what's
the name of the bracelet
that zombies must wear--
Z-band.
That one goes to Ariel.
Congratulations.
It was z-band.
For a long time, we were
forced to stay in Zombie Town.
But we've come a long
way since the outbreak
thanks to the z-band.
Who is the first person that
Hannah tells her secret to
in Hannah Montana?
Lilly?
I don't know that one.
You know that, Siena?
MORGAN TOMPKINS: Lilly.
It's Lilly.
MORGAN TOMPKINS: Nailed it.
I thought we were friends.
I thought we told
each other everything.
But I guess I was wrong
since you kept just
about the biggest
secret in the world,
Miley Stewart slash
Hannah Montana!
OK, well other than that,
did you enjoy the concert?
Raven, what is
Zenon and Nebula's
catchphrase in "Zenon--
Girl of the 21st Century?"
Cetus lupeedus.
MORGAN TOMPKINS:
You're two for two.
Is everything OK up there?
I mean, you guys aren't having
global meltdowns yet, are you?
Cetus lupeedus,
how did you know?
The attitude thrusters
are all messed up,
the whole base has been going
through these wild plunges
and scary dips.
It's been awful.
What holiday do they
celebrate at the beginning
of "High School Musical?"
Oh--
I know this.
Is it New Year's?
MORGAN TOMPKINS:
It's New Year's.
New year's.
New year's?
Doctor had New Year's.
Five, four, three, two, one!
[CHEERING]
I'm going to Phil,
specifically, on this one, OK?
Let's see.
OK.
In "Zombies," how are
Bucky and Addison related?
Can I use Google?
Can I--
You can phone a
friend, you can--
Wait, Bucky and who?
Who?
Bucky and who?
- Addison.
How are they related?
Are they brother sister?
Are they cousins?
You can phone a friend
if you get into trouble.
Text me.
Text me.
Can you text me?
What is it?
Can I phone a friend?
Can I call Ariel?
MORGAN TOMPKINS: Yes you can.
They're cousins.
OK, got it.
They're cousins!
MORGAN TOMPKINS:
Fantastic answer!
[INTERPOSING VOICES]
That's to your captain,
Bucky, My cousin.
What hotel do Zach
and Cody live at?
Tipton.
Tipton Hotel.
MORGAN TOMPKINS:
The Tipton Hotel.
I've changed my mind.
You're hired!
Mr. Tipton loves
my idea of a family
angle for the commercial.
I wish we'd thought of that.
You mean we're in?
Thanks Mr. Tipton!
Good job, you guys.
I had to do some
research as well.
OK.
It's all good.
We're brushing up on our deep
DCOM knowledge right now.
We look forward to having
you in the studio very soon.
Thank you so much.
Bye.
Bye.
Thank you.
Thank you.
- Thank you, appreciate it.
- Cheers, guys.
I'll see you soon.
Bye.
[MUSIC PLAYING]
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Westminster Announces Top 10 Opportunity Quest Finalists
College's annual business plan competition narrows down the race to the top 10 contestantsSALT LAKE CITY - Westminster's Center for Entrepreneurship has narrowed down its contestant pool for the 2012-13 Opportunity Quest business plan contest to 10 finalists. From businesses that focus on high-end fashion and art, to those that aim to help society, this year's Opportunity Quest has a myriad of unique business plan ideas.
"We are very excited to see such creative business ideas come across our campus this year," said Linda Muir, director of Westminster's Center for Entrepreneurship. "Westminster encourages its students to learn how to take concepts, validate market opportunities and develop sound business models."
Opportunity Quest is a business innovation competition that is open to all students at Westminster. The competition fosters and promotes business growth at the college and university level by giving students a chance to organize, direct and present a business idea to a panel of judges.
The top 10 finalists were selected from a group of 26 contestants. In January, they will make their final presentations to a panel of judges who will decide the competition winners. The first place team will receive a cash prize of $5,000, and will automatically be entered as a finalist in the Utah Entrepreneur Challenge where they will compete against teams from schools throughout the state. Cash prizes of $2,000 and $1,000 will be awarded for 2nd and 3rd place teams. This year, Westminster is also launching a new category dedicated to "Social Entrepreneurship" with a grand prize of $2,000.
This year's Opportunity Quest winners will be announced at the final awards ceremony on Jan. 25, 2013.
The 2012-13 Opportunity Quest Top 10 finalists are:
Social Category Finalists
Company Name
Company Overview
Daakye
(Team Leader: Abby Speicher)
Through Daakye, Abby Speicher contracts with Ghanaian seamstresses to create purses which are then sold in the United States. Daakye's profits go to provide scholarships to kids in Ghana.
Harmony Autism Center
(Team Leader: Karlie Harmon)
Karlie, hopes to build a facility for children with Autism Spectrum Disorders; the facility would help children develop sensory skills, release stored energy and introduce new social and behavioral skills.
Sugar Doodle Kids
(Team Leader: Nick Whatcott)
Sugar Doodle Kids is a company which sells children's sleepwear, the designs of which are from children and artists. A portion of the company's profits go to help hungry children.
General Category Finalists
Company Name
Company Overview
Art Addic
(Team Leader: Michael Odernheimer)
ArtAddic is an online community which allows artists to create and maintain digital portfolios, give and receive feedback and exposure.
Auto Source
(Team Leader: Jeremy Baker)
Auto Source is a web-based dealer management software system for small to mid-sized independent automotive dealers to manage their companies.
Daakye
(Team Leader: Abby Speicher)
Through Daakye, Abby Speicher contracts with Ghanaian seamstresses to create purses which are then sold in the United States. Daakye's profits go to provide scholarships to kids in Ghana.
Good Dog Brewing
(Team Leader: David Pang)
Good Dog Brewing is a registered independent craft brewery in Utah which brews ales and lagers while partnering with pet organizations to help ensure each pet has a home.
Harmony Autism Center
(Team Leader: Karlie Harmon)
Karlie, hopes to build a facility for children with Autism Spectrum Disorders; the facility would help children develop sensory skills, release stored energy and introduce new social and behavioral skills.
Liyen
(Team Leader: Chris Ciancone)
Liyen is a company that creates 'life devices;' currently they are working to sell their inhaler device, Slyder, which is a spacer that serves as a carrying container for the inhaler.
Moda Culture
(Team Leader: Meindert Kooistra)
ModaCulture is an American luxury fashion, shoe and accessory platform - providing high-end international fashion label
Munin
(Team Leader: Oeyvind Sandengen)
Munin is an online community where adventure motorists can access comprehensive resources in various communities.
Sugar Doodle Kids
(Team Leader: Nick Whatcott)
Sugar Doodle Kids is a company which sells children's sleepwear, the designs of which are from children and artists. A portion of the company's profits go to help hungry children.
Media Contacts
About Westminster
Westminster is a private, independent, and comprehensive college in Salt Lake City, Utah. Students experience the liberal arts blended with professional programs in an atmosphere dedicated to civic engagement. With the goal of enabling its graduates to live vibrant, just, and successful lives, Westminster provides transformational learning experiences for both undergraduate and graduate students in a truly student-centered environment. Faculty focus on teaching, learning, and developing distinctive, innovative programs, while students thrive on Westminster’s urban Sugar House campus within minutes of the Rocky Mountains.
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For the next six days, take advantage of huge savings throughout the Steam store on thousands of titles. Plus, discover new and recommended games on your personalized Exploration Sale page here. Check back each day for new Daily Deals.
Comparisons to Titanfall were inevitable after Call of Duty: Advanced Warfare's multiplayer was revealed last week, but Sledgehammer have been playing with Advanced Warfare's jump-thrust move shortly after finishing work on Modern Warfare 3 in 2011, and Tribes fans will tell you that Titanfall hardly has dibs on the jetpack shooter. While the new exoskeleton movement options are interesting, I'm more excited about what they can do to refresh the design of Call of Duty's multiplayer maps.
The Call of Duty multiplayer experience has remained largely unchanged since the dawn of the Modern Warfare era. The opportunistic run-and-gun through flat, crumbling maps started to feel rote back in Black Ops 2, a game that could have done with a map like Advanced Warfare's Riot. Granted, it's also a crumbling concrete facility a prison, in fact but it's built around tiered structures and guard towers that make use of the exoskeleton's new jumping abilities. Like a couple of the other maps Sledgehammer showed, there's a quirk: The prison's high tech inmate monitoring system is still active, which means sensors at the top of guard towers draw laser pointers to passing soldiers.
Riot was actually one of the plainer maps. The others showed at Gamescom move away from the bleached urban aesthetic to something shinier, reminiscent of some of CoD's quirkier, more colourful DLC packs. There's a map set in the terminal of a space elevator which is bright, airy and layered with gantries and intricate spaceport corridors. Advanced Warfare will include a mode that bans exosuits, but I wonder if these maps will still be fun if you're flightless. The need to cater to the legions of fans who love the standard CoD formula is understandable, but once you've experienced extra speed and super-jumps bursts of flight, why go back?
This year you can customise your soldier with with dozens of facemasks, armour plates and colour schemes and tailor your loadout with a "pick 13" system, mirroring Black Op's "pick 10". You get 13 points you can devote to weapon attachments, special exosuit abilities like cloaks and shields, and kill streak rewards. More options can be unlocked by ranking up.
Leveling systems in shooters can hide the game's most interesting weapons and strategies behind layers of grind (see Battlfield, Payday 2), but the sheer amount of stuff packed into Advanced Warfare's customisation system is encouraging. As long as the leveling curve is fast, the expanded unlock system could provide valuable context and rewards for each 10-20 minute battle for those who don't care about 'prestiging' the practice of reaching the CoD's level cap and then resetting for a badge.
I'm quite excited about Advanced Warfare this year, for the multiplayer, at least. The single player may have Spacey, but in spite of the high-tech gubbins the set pieces look samey and predictable. I've never had a fun game of Battlefield 4, Battlefield Hardline leaves me cold and Titanfall is great, but lacks staying power. CoD could sweep the board this year. On the multiplayer side, it's the most exciting entry in the series for years.
With the upcoming, stupidly pretty, Call of Duty: Advanced Warfare pouting its lips and batting its eyes on the horizon, I decided to give the last one some love. I've contained my exploits to the first mission, but even that's plenty variety given CoD's penchant for catapulting players between ludicrous scenarios. Here I go from outrunning an orbital strike on San Diego to floating around the very space station responsible for it, all while dodging bullets from jetpacking terrorists.
It's strange to hear developers boast about how small the maps in their newest map pack are, but I hear just that in this trailer for CoD: Ghosts' final piece of DLC, Nemesis. This adds some "small-to-medium"-sized maps based around the themes 'mine cart level', 'wintry submarine base' and 'please desecrate this lovely Chinese village', along with a remake of the "smallest map ever made for Call of Duty": Shipment (now called Showtime). This one's a futuristic, Smash TV-style game show, replete with a cheesy announcer commentating on the killy goings-on. The DLC also adds the final bit to the game's full-on sci-fi Extinction mode. Exodus will see you coming face-to-elongated-face with the Ancestors, ie XCOM-ish psionic aliens. The trailer is below.
Nemesis releases next week on Xbox-flavoured systems, but if it's anything like all the other map packs, we can expect it on PC a month later. It's part of the Ghosts season pass, but you can also stump up $14.99/ 11.59 to buy the DLC separately.
Outnumbered and outgunned, but not outmatched. Call of Duty®: Ghosts is an extraordinary step forward for one of the largest entertainment franchises of all-time. This new chapter in the Call of Duty® franchise features a new dynamic where players are on the side of a crippled nation fighting not for freedom, or liberty, but simply to survive.
Call of Duty®: Ghosts owners can now enhance their experience with these exciting packs:
Some would argue that paid-for personalisation packs are endemic in the games industry chronic, even. But is there anything wrong with showing off some style while high at the top of a leaderboard. Infinity Ward are no dopes, they know how to hit their target markets. Presented with a money making opportunity, they're not going to make a hash of it. And so, some Call of Duty developers embarked on a skunkworks mission to create the Blunt Force Character Pack a marijuana-themed DLC release.
The DLC, priced $1.99, is one of three new character packs now available for the Xbox version of the game. The other two character packs are themed as "Inferno" and "Bling". Infinity Ward definitely know their audience.
Each character pack contains a uniform and two helmets. In addition, the recent DLC drop also makes available six new 'Personalisation' packs and two new weapons. You can see the full round-up here. Expect all items to arrive on PC after the standard Xbox-exclusivity period expires likely next month.
Outnumbered and outgunned, but not outmatched. Call of Duty®: Ghosts is an extraordinary step forward for one of the largest entertainment franchises of all-time. This new chapter in the Call of Duty® franchise features a new dynamic where players are on the side of a crippled nation fighting not for freedom, or liberty, but simply to survive.
Call of Duty®: Ghosts owners can now enhance their experience with these exciting packs:
What do Hitler, Snoop Dogg, George Washington, and Shakespeare have in common? If you said they all appear in Rik Mayall’s autobiography Bigger Than Hitler, Better Than Christ*, you’re wrong. Washington and Snoop aren’t mentioned in that. The correct answer is that each has been digitised and reformed–Weird Science style–into your games as DLC.
There’s a lot of DLC out there, ranging from the mundane to the insane, and I think I know why. Games are increasingly serious business, with huge budgets and a cast amount of public scrutiny. DLC–well some of it–feels like the passion projects that don’t fit into the canon. An outlet for the stuff that gets cleared from the whiteboard for being too off message, or too niche. DLC is cathartic. I’ve been on a strange journey, readers. I’ve been looking through games catalogues and hunting down the sort of DLC that could be described as ’boutique’. I’ve been on a boutique call, ahahahahahahaha!
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February 8, 2017
Girl Gang Twitch Stream for ACLU
The American Civil Liberties Union (ACLU) has our backs. They defend our rights and our freedoms, and this happens to be a time when we need that more than ever. They have received large donations recently, but it's going to be a long fight. To that end, we're going to do the kind of fundraiser you can only do in this modern age: a Twitch stream!
I NEED A GENUINE SPELL CASTER TO BRING BACK MY EX BACK: CONATCT DORCTOR THOUGH HIS WEB SITE https://ultimatespellcast3.wixsite.com/dr-trust HE IS THE BEST OF ALL
It's a privilege to share this miraculous testimony to the world. I'm Elisa Wendy from England. My husband divorced me 4 months back and i have been filled with remorse for i didn't know what to do to amend issues with my husband. I searched for help on the internet on how i could get help in my marriage and i discovered great testifiers about Doctor Trust who has been progressive with his spells. I got in touch with him and behold, Doctor Trust told me that he will prepare a spell for me that will bring back my husband. I was skeptical but i had no other option but to work with him. 2 days after, my husband called me that he's coming back home and from that day till this moment, we have been living peacefully. He is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my husband,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love. I will highly recommend Doctor Trust to anyone one out there who needs help whatsoever.if you have any problem contact Doctor Trust, i give you 100% guarantee that he will help you!!.
[email protected] or [email protected] Call or write him on WhatApp +2348156885231
POTENTIAL SPELL TEMPLE A PLACE WHERE ALL KIND"S OF PROBLEMS ARE SOLVEDWITH 100% GUARANTEE CALL: +2347051453019
I"m BILL ANN from United states, i want to use this opportunity to thank the Great [DR SAVIOR] for helping in getting my husband back to me, i have been in great pains until the day i contacted [DR SAVIOR] he casted a love spell for me and told me to wait for just 12 hours that my husband will call me and i did according to the instructions given to me by him and surprisingly, in 12 hours, my husband really called me and started apologizing for all he had caused me. I am the happiest Woman on earth today because [DR SAVIOR] has done a wonderful deeds in my life and i will continue to share this testimony, if you would love to contact [DR SAVIOR] and if you know you are encountering same situation, visit him today by contacting him via hisEmail:[email protected] [email protected]
I never knew people still have powers and make things happened this way. My name is Sandra Ruths am from London. my boyfriend Slay Scot left me for another girl for three months' ever since then my life have been filled with pains sorrow and heart break because he was my first love who dis virgin me when i was 21 years old. about two years ago, A friend of mine kido matthew told me he saw some testimonies of this great Dr osauyi that he can bring back lover within some few days, i laugh it out and said i am not interested but because of the love my friend had for me, she consulted the great priest on my behalf and to my greatest surprise after three days my boyfriend is going to call me for the very first time after three months that he is missing me and that he is so sorry for every thing he made me went through. i still can't believe it, because it highly unbelievable it just too real to be real. Thank you Dr osauyi for bringing back my lover and also to my lovely friend who interceded on my behalf, for any one who might need the help of this great priest here is the email address [email protected] or you can call the great man on +2347064294395
HOW TO GET YOUR EX BACK PERMANENTLY,DO NOT CRY ANYMORE CONTACT THIS POWERFUL SPELL CASTER [email protected]
How to get your ex back permanently? Is this what you are asking yourself? Have you been dumped? Can't you just move on? Did he/she chose someone else over you? Your lover shows no interest in you anymore? You might have been asking yourself how to get your ex lover back permanently and you have been looking for unanswered questions about it, Well today is the day for your worries to be eliminated for good. Dr Ogudu have assisted me in getting my ex lover back,Are you going through emotional torture contact him now so he can use his birth gift of casting strong love spells and his super natural magical powers that mean no harm but motivated by peace, wellness and love. The fact that you ask how to get your lover back permanently is the reason why you should no longer face sorrow on your own, and that is because you are in a perfect condition to win over your ex lover and make him or her fall in love with you all over again. It is a great pain to be in love with someone who does not show the same affection back to you as you hope for. However, it is never too late to have all that you wish for and deserve in a lover. Love spells by (DR OGUDU) are pure nature of magic, they are inspired by true mystery of the beauty of the planet earth and they will do well for you as well. CONTACT HIM NOW AND BE HAPPY AGAIN""""Email [[email protected]] Whatsapp number +2348106058254 Website...http://oguduspelltemple.wixsite.com/droguduDo not cry anymore, contact this powerful spell caster
I'm still shocked of the way Dr ODELI helped me, I am so proud and happy to be out here sharing this awesome testimony, I just can't believe this now my girlfriend is really back to me and she promised me that she will never let go of me ever again, she keep pleading to me to please accept her back and she was feeling regretful and sorry for leaving me and for causing me pains after the quarrel which occurred last year. And this whole miracle happened after i ordered an urgent 24hours Dr.ODELI powerful spell which he cast on me and my girlfriend. may God continue to bless your work for the lives of many people, I am the happiest woman today in this whole wide world. Dr. ODELI. still unbelievable the way i was back with my girlfriend.. I am so happy now and i don't know how much to thank you Dr ODELI. contact him if you need this same help like i did, just contact Dr ODELI and he will surely help you right away on his email: [email protected]
WOW i'm glad to be sharing with everyone how Dr Otoide the powerful spell caster helped me to get back with my wife. I am Jamie Watson, after me and my wife divorced after having a huge quarrel then. i searched for a spell caster that can help me then i found testimonies about Dr Otoide, how he helped many people to get their lovers and broken homes back, i contacted him through his email address :[email protected] because I was absolutely desperate to get my wife back. Life without my wife was a real mess for me and my children. i wanted a dramatic change and I thought casting a love spell could be the solution. After discussing the resolution with Dr Otoide. he gave me hope that he will restore my marriage. I felt confident that he will actually make my wife to return home and he did! It’s fantastic what this great spell caster have done for me, his help is priceless! I don't know what I would have done without Dr.Otoide. he does his job so well he is organized and highly functional. i believe he is the best spell caster i can count on when it comes to all kinds of spell. if you need help. contact him on his websites:https://drotoidesolutionte.wixsite.com/dr-otoide or his Mobile line:+2347089081873 or his email:[email protected]..
Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
This is the most wonderful thing i have ever experienced. I visited a forum here on the internet on the 17 October 2016, and i saw a marvelous testimonies on the forum about the good works Doctor SAM. Am Lopez Angelito from United States. I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until Doctor SAM did a marvelous work for me that restored my marriage of 4 years by getting back my divorced wife within 24 to 48 hours just as i read on the internet. I was truly shocked when my wife came home pleading for forgiveness to accept her back. Am really short of words to show my appreciation to Doctor SAM for he is a God sent to me and my entire family for divine restoration of marriage. Are you in need of any help whatsoever? Don't hesitate to get in touch with him now. He also specializes on the follows;
This is the most wonderful thing i have ever experienced. I visited a forum here on the internet on the 17 October 2016, and i saw a marvelous testimonies on the forum about the good works Doctor SAM. Am Lopez Angelito from United States. I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until Doctor SAM did a marvelous work for me that restored my marriage of 4 years by getting back my divorced wife within 24 to 48 hours just as i read on the internet. I was truly shocked when my wife came home pleading for forgiveness to accept her back. Am really short of words to show my appreciation to Doctor SAM for he is a God sent to me and my entire family for divine restoration of marriage. Are you in need of any help whatsoever? Don't hesitate to get in touch with him now. He also specializes on the follows;
WHAT A WONDERFUL LOVE SPELL, MY HUSBAND AND I ARE BACK TOGETHER AGAIN AFTER 2 YEARS OF SEPARATION AND DIVORCE, AM TRULY BLESSED BY DR.OGUDU SPELL TEMPLE, ALL THANKS TO HIM, YOU CAN CONTACT HIM NO VIA EMAIL:::[email protected]:::WHATSAPP NUMBER:::+2348106058254:::WEBSITE:::http://oguduspelltemple.wixsite.com/drogudu
I'm ROY LILY from BRONX NY, USA. I got separated from my husband 2 years ago. There was no communication between us. I was advised by families and friends to let go, forget the marriage and move on with my life. I didn’t want to marry someone else because deep down, I still love my husband. I was in so much pain and confused, until I read a testimony online on how Dr Ogudu reunion a broken marriage with the help of his spiritual spell power. I kept reading so many testimonies on how he helped to stopped divorce and brought back peoples ex-lover's back to them, and my faith was renewed. I have to contact Dr Ogudu immediately, few minutes later, he replied and instructed me on what to do, after meeting up with the necessary requirement, he sent me a parcel and instructed on how to use it, which I did. 48 hours later communication was restored between me and my husband. I saw him after 7 days and we were able to settle our differences, he actually knelt down and was begging me to forgive him. I was visiting so many website seeking for help, it looked hopeless, until I came in contact with DR OGUDU, who helped me to restore my joy, in fact, I'm the luckiest and happiest woman ever to have her husband back in her life. If you're having similar problem in your marriage, you want your husband or wife to love you again, you have someone you love and you want him or her to love you in return, you are having any challenge in your marriage. DR OGUDU SPELL TEMPLE is the solution to your problem, contact him today via email: EMAIL:::[email protected]:::WHATSAPP NUMBER:::+2348106058254:::WEBSITE:::http://oguduspelltemple.wixsite.com/drogudu
hello everybody,my name is stella scotte from U S A, i want to use this medium to thank Dr porosky for his help in restoring my marriage of three years which as almost gone under the rock.my husband and i have been having problem in our marriage for the past one year that he had to move out of the home,but with the help of Dr porosky,i have been able to work on my marriage and am back with my husband,we are having a happy home once again.All thanks goes to Dr porosky of [email protected]://drporosky.wordpress.com/
Happy to tell the world how wonderful Dr Otoide has restore my dignity as a woman back to me. it been 2years in marriage with my husband and i have never conceive i wonder what was wrong wih me, so i searched out for a remedy and fortunately i met with the great spell caster Dr Otoide who has helped severally others with the same issues and i was very glad when i wrote to him on his email he told me to calm down that everything will fine once he is done casting spell for me that i will be pregnant again, and h gave me the instruction on how he spell worked and i agreed with his terms and i gave him all that he needed to cast my spell, and indeed after 48hours he was done and told me that the next morning i will be pregnant and i believed him and that was how i got pregnant by the help of Dr Otoide a very great spell caster that can help you too on any challenges that you might be passing through. just contact him today on his Email:[email protected] or his phone number:+2347089081873 or his websites:https://drotoidesolutionte.wixsite.com/dr-otoide
Effective Love Spell To Get Back Your Ex Back Within 48 Hours: Fast And Read Through Dr. Trust Website Now. https://ultimatespellcast3.wixsite.com/dr-trust
Hi everyone, I was going crazy when my husband breakup with me and left me for another woman!! All thanks to Dr Trust the best love spell caster online that helped me to bring back my husband today and restore happiness in my marriage.. My husband breakup with me and left me to be with another woman, and i wanted him back. I was so frustrated and i could not know what next to do again, I love my husband so much but he was cheating on me with another woman and this makes him break up with me so that he can be able to get married to the other lady and this lady i think use witchcraft on my husband to make him hate me and my kids and this was so critical and uncalled-for, I cry all day and night for God to send me a helper to get back my husband!! I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Trust can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a spell for me. 48 hours later, my husband really called me and told me that he miss me so much, Oh My God! i was so happy, and today i am happy with my man again and we are joyfully living together as one big family and i thank the powerful spell caster Dr.Trust, he is so powerful and i decided to share my story on the internet that Dr.Trust is best and good spell caster online. if you are here and your lover is turning you down, or your husband or wife moved to another person, do not cry anymore, contact this powerful spell caster Dr.Trust on his email at: [email protected] OR [email protected] you can also call him or add him on Whats-app: +2348156885231 ,His website: https://ultimatespellcast3.wixsite.com/dr-trust.
I will always be thankful to the Great Dr.BEN. for many years now my family has suffered from HERPES DISEASE. it happened that after i got married to my Husband. i found out that i got HERPES VIRUS and when i did i was confirm positive and was completely so angry and then i had to convince my Husband to go test himself and when he did he was also found positive and then it was dawn completely on me that me and my husband is having HERPES DISEASE. i went out in search of a solution despite my doctor told me that there was no remedy to curing me and my husband HERPES DISEASE VIRUS. when i was searching i found out there so many testimonies about Dr.BEN helping others to be cured of HERPES VIRUS. i instantly added his email and asked for his cure and he instruct me on how his cure can get reached to me and my husband and i believe and did the little things he asked of me and that was i received his cure and told me that after me and my husband use it for some days i will come back and thank him and that was true. after we both use the herbal cure. we went to the doctor after the days which he instruct us. we were both confirmed negative. all thanks to Dr.BEN for his marvelous healing he gave to my family. contact him on his Email: [email protected] or [email protected].
I will always be thankful to the Great Dr.BEN. for many years now my family has suffered from HERPES DISEASE. it happened that after i got married to my Husband. i found out that i got HERPES VIRUS and when i did i was confirm positive and was completely so angry and then i had to convince my Husband to go test himself and when he did he was also found positive and then it was dawn completely on me that me and my husband is having HERPES DISEASE. i went out in search of a solution despite my doctor told me that there was no remedy to curing me and my husband HERPES DISEASE VIRUS. when i was searching i found out there so many testimonies about Dr.BEN helping others to be cured of HERPES VIRUS. i instantly added his email and asked for his cure and he instruct me on how his cure can get reached to me and my husband and i believe and did the little things he asked of me and that was i received his cure and told me that after me and my husband use it for some days i will come back and thank him and that was true. after we both use the herbal cure. we went to the doctor after the days which he instruct us. we were both confirmed negative. all thanks to Dr.BEN for his marvelous healing he gave to my family. contact him on his Email: [email protected] or [email protected].
My mouth is short of words, i am so happy because Dr voodoo has healed me from HIV ailment which i have been suffering from the past 5years now, i have spend a lot when getting drugs from the hospital to keep me healthy, i have tried all means in life to always i can become HIV negative one day, but there was no answer until i found Dr voodoo the parish of African who provide me some healing spell that he uses to help me, now i am glad telling everyone that i am now HIV Negative, i am very happy, thank you Dr voodoo for helping my life comes back newly without any form of crisis, may the good lord that i serve blessed you Dr voodoo and equip you to the higher grade for healing my life. i am so amazed. so i will announced to everyone in this whole world that is HIV positive to please follow my advice and get healed on time, because we all knows that HIV disease is a deadly type,contact Dr voodoo for your HIV healing spell today at: [email protected] or [email protected] He will be always happy to assist you online and ensure you get healed on time, contact Dr voodoo today for your healing spell immediately, you can also add him on whatsApp +2349061243458 thank you sir.
Hi everyone, there is good news, a perfect and greats one. My daughter of 17years who has suffered of Ovarian Cancer for 2 and half years has been cure by a Herbal doctor called Dr voodoo, who uses herbal and traditional medicine to cure people. I have spent so much on my daughter's health for years but no improvement. A friend told me about this herbal man who cure people with herbal medicine. he cures all sickness, diseases, viruses, and infections with his different herbal curing medicine.The result that declares my daughter janet healed was from a hospital here in USA, And the results of the test says my daughter was okay. Dr voodoo directed that janet would be heal in 7days if she uses the medicine as directed. And 7days later, we went to confirm from the central hospital and several test that was done says my daughter is not sick or anything. So she was cured by this herbal doctor in 7days. This amaze me though. I would advice everyone to meet this Dr voodoo his email is [email protected] for his herbal medicine to cure your illness because am happy to see my daughter healthy again.Meet Dr voodoo and set yourself and your love ones will free.Dr voodoo email again is [email protected] you can also add Dr voodoo on whatsApp +2349061243458
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
God bless Dr.Osauyi for his marvelous work in my life, I was separate with my husband for the pass 3 year now and I wasn't satisfied i needed to get my husband back because the life of loneliness was so terrible for me and my 2 kids I searched about some possible spell caster that can bring back lover and i then come across Dr Osauyi i saw a comment about Dr.Osauyi,how he bring lovers together with his spell caster I contacted him and he guided me. I asked for solutions, he started the remedy for my spell he ask for some information which i sent to him and also told me that we need to buy some items for the spell to work for me for good and which i sensed the money to him after 48 hours my husband call me and start bagging me for forgiveness for all that happens i am so happy right now that we back again as one family for any one that may need the help of the great spell caster you can contact [email protected] or call +2347064294395
I did not believe is possible that HIV can be cured because i have been scammed so many times not until i came across this great man dr agbebaku who helped me at first I never believe all this comment and post about him and I was very sick because i have been infected with HIV for the past seven years,just last year I keep reading the testimony about this man named dr agbebaku they said that the man is so powerful he have cured different type of diseases, I keep monitor the post of some people about this man and I found out that he was real so i decided to give him a try I contacted him for help and he said he was going to help me get my cure that all i needed to do was to send him money to prepare the medication after which it will be sent to me via dhl service which i did to my greatest suprise the medication was sent to me he gave me instructions on how to use it that after three weeks i should go for check up at first i was shocked when the doctor told me that i was hiv negative i ask the doctor to check again and the result was the same that i was now free from hiv at of the shock i decided to come and share my testimony with you for those who think there is no cure for hiv the cure as finally come stop doubting and contact dr agbebaku via email [email protected] he will help you out you can also wattssap him via +2349053099479 he said he also as cure for DIABETICS,HERPES,INFERTILITY;ALS,ANCE PIMPUS,LUPUS VIRUS AND PENIS AND BREAST ENLARGEMENT
Hi My name is 'Niall Christopher' just want to share my experience with the world on how i got my love back and saved my marriage... I was married for 7years with 2kids and we lived happily until things started getting ugly and we had fights and arguments almost every time... it got worse at a point that she filed for divorce... I tried my best to make her change her mind & stay with me cause i loved her with all my heart and didn't want to loose her but everything just didn't work out... she moved out of the house and still went ahead to file for divorce... I pleaded and tried everything but still nothing worked. The breakthrough came when someone introduced me to this wonderful, great spell caster who eventually helped me out... I have never been a fan of things like this but just decided to try reluctantly cause I was desperate and left with no choice... He did special prayers and used roots and herbs... Within 7 days she called me and was sorry for all the emotional trauma she had cost me, moved back to the house and we continue to live happily, the kids are happy too and we are expecting our third child. I have introduced him to a lot of couples with problems across the world and they have had good news... Just thought I should share my experience cause I strongly believe someone out there need's it... You can email him via [email protected] Don't give up just yet, the different between 'Ordinary' & 'Extra-Ordinary' is the 'Extra' so make extra effort to save your marriage/relationship if it's truly worth it. website http://eromosalspiritualtemple.webs.com or call him with +2348161850195
With so much in my heart i am here to express myself on how Dr. Ozil saved my marriage from divorce. Myself and my husband were having some misunderstanding and it was tearing our marriage apart to the extend my husband was seeking for a divorce. So i have no option than to go to the internet to seek for solution to my problem it was there i came across Dr.Ozil details and about how he has helped a lot of people by restoring there relationship. I contact Dr.Ozil and in less than 48 hours my husband cancelled the divorce papers. Now myself and my husband live together in peace and harmony all thanks to Dr.Ozil for saving my marriage from breaking up. Dr.Ozil contact information are via email: [email protected] or [email protected] call his phone number: +2348130092712 or view and contact him on his website: http://drozilsolutionhome.webs.com/
With so much in my heart i am here to express myself on how Dr. Ozil saved my marriage from divorce. Myself and my husband were having some misunderstanding and it was tearing our marriage apart to the extend my husband was seeking for a divorce. So i have no option than to go to the internet to seek for solution to my problem it was there i came across Dr.Ozil details and about how he has helped a lot of people by restoring there relationship. I contact Dr.Ozil and in less than 48 hours my husband cancelled the divorce papers. Now myself and my husband live together in peace and harmony all thanks to Dr.Ozil for saving my marriage from breaking up. Dr.Ozil contact information are via email: [email protected] or [email protected] call his phone number: +2348130092712 or view and contact him on his website: http://drozilsolutionhome.webs.com/
I want to testify a great testimony of how Dr.SABU help me,i was heartbroken because i had very ( SMALL PENIS ) not big enough and I couldn't satisfy a real lady in bed and the worse of my problems was that i cum on the sight of a ladies vagina. I battled this problems since i was teen and still couldn't get any good victory. I was online searching for guaranteed results providing products to enlarge the size of my penis when i saw a review of Dr.SABU Penis Enlargement Skills posted by so many poeple that he has help to enlargement penis,i read through the comment and emailed the contact ([email protected]) on it. The doctor called me some few moments later and attended to me more better than i expected. I was convinced and i made purchase of his herbal medicines for Penis Enlargement and Quick Ejaculation. The medicines was sent to me BY UPS DELEVERY and i received it at my home, i applied it as instructed and just few weeks of using the herbal medicines i got a perfect good size of penis. I now measure about 16.0 cm (8.4 inches) when flaccid and 24.3 cm (9.1 inches) when erect. Am so happy and comfortable with the new size of my penis, i feel no discomfort or whatever. Am so happy with this fast results. You too can reach Dr.SABU for same Penis Enlargement medicines on [email protected] or his WHATSAPP/CALLS +2347057624993 or his websites: https://drsabuspelltemple.wixsite.com/dr-sabu
GET YOUR EX BACK WITH THE HELP GREAT SPELL CASTER CALLED DR SAMBO VISIT HIS WEBSITE BELOW; http://divinespellhome.wixsite.com/drsambo
Good dey to you all , my name is moore kathleen i want to share a brief testimony on how Great Dr Sambo helped me re-unite my long gone relationship . I thought I have lost it all until I saw a post abort this great spell caster called Dr Sambo , I contacted him and told him about my story “few years ago , my husband and I were deeply in love I thought life was that easy , because I was with the man of my dream , until his secretary took him from me , he stopped loving me ,we separated because of the incidence , I couldn’t take it anymore , I count live with him know he has someone else out there , but I still loved him . until few week back I contacted Great Dr Sambo who helped me bring back my long gone husband , he apologyzed to me and each day that passes he love me more . if you are out there , because I know a lot of people are experiencing difficulties in their marriage or any form of relationship , just contact Dr Sambo and he will definatily help you as he has done for me here are his details .
Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
OMG!!! I am so proud and happy to be out here sharing your work Dr. Zaba. I just can't believe this now my ex Husband is really back to me on his knees presenting a ruby rose begging me to take him back and he was feeling regretful and sorry for leaving me and for causing me pains after the divorce which occurred last year. And this whole miracle happened after i ordered an urgent 24hours Dr.Zaba powerful spell which he cast on me and my husband. Sir I am the happiest woman today in this whole wide world. Dr. Zaba you really did it..Yes.. Its a miracle and everlasting pleasure and cheerfulness for me and my family today.. I am so happy now and i dont know how much to convey my thankfulness and appreciation to you sir. And to the whole world, contact him if you need urgent help now because its guaranteed that he will help you. website http://zaba24hoursspell.wixsite.com/zabaspell or email him [email protected] OR [email protected] or call +2347087462033
OMG!!! I am so proud and happy to be out here sharing your work Dr. Zaba. I just can't believe this now my ex Husband is really back to me on his knees presenting a ruby rose begging me to take him back and he was feeling regretful and sorry for leaving me and for causing me pains after the divorce which occurred last year. And this whole miracle happened after i ordered an urgent 24hours Dr.Zaba powerful spell which he cast on me and my husband. Sir I am the happiest woman today in this whole wide world. Dr. Zaba you really did it..Yes.. Its a miracle and everlasting pleasure and cheerfulness for me and my family today.. I am so happy now and i dont know how much to convey my thankfulness and appreciation to you sir. And to the whole world, contact him if you need urgent help now because its guaranteed that he will help you. website http://zaba24hoursspell.wixsite.com/zabaspell or email him [email protected] OR [email protected] or call +2347087462033
I read some testimonies about a love spell caster called Dr.OBILUJESU on how he has helped lots of people in bringing back their ex lovers within 24 hours, Sincerely I was just thinking if that was real and if this man could really help bring back my lover whom I love so much. I decided to contact him because I love my boyfriend very much and we have been apart for a couple of months I really missed him so much, I have tried all other means to get him back but I couldn't. I contacted Dr.OBILUJESU and he told me that my ex will come back to me in the next 48 hours,Dr.OBILUJESU released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. My ex is now back to me again.As I`m writing this testimony right now I`m the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up.All thanks goes to Dr.OBILUJESU for the excessive work that he has done for me by helping me to get back with my ex boyfriend. I would like to drop Dr.OBILUJESU mail address and hope you see this testimony and contact him if you have a lover that you really want back so badly, His mail: [email protected] or [email protected] web sit: http://drobilujesuspellca.wixsite.com/drobilujesu OR call him +2348143481000
I read some testimonies about a love spell caster called Dr.OBILUJESU on how he has helped lots of people in bringing back their ex lovers within 24 hours, Sincerely I was just thinking if that was real and if this man could really help bring back my lover whom I love so much. I decided to contact him because I love my boyfriend very much and we have been apart for a couple of months I really missed him so much, I have tried all other means to get him back but I couldn't. I contacted Dr.OBILUJESU and he told me that my ex will come back to me in the next 48 hours,Dr.OBILUJESU released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. My ex is now back to me again.As I`m writing this testimony right now I`m the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up.All thanks goes to Dr.OBILUJESU for the excessive work that he has done for me by helping me to get back with my ex boyfriend. I would like to drop Dr.OBILUJESU mail address and hope you see this testimony and contact him if you have a lover that you really want back so badly, His mail: [email protected] or [email protected] web sit: http://drobilujesuspellca.wixsite.com/drobilujesu OR call him +2348143481000
My name is Bart Cruz am from U.S.A. I want to share a testimony of how Dr Okusu herbal oil saves me from shame and disgrace, my penis was a big problem to me as the size was really so embarrassing,I wasn't able to satisfy my wife any more and she became so tired of my sex life issues, and she was getting angry every time we make love, because i always cum quickly in 10minutes and it was very frustrating,but i never loose hope, i went out to search on a way to make my penis big and lastly in bed. miraculously i saw a testimony of how Dr Okusu has helped my fellow men, on getting help and making their penis bigger and stronger, i wrote to Dr Okusu for his herbal oil and he told me once i used his oil in 5days my penis will be 9inches bigger be able to satisfy my wife longer in bed and also be able to control my ejaculation to cum whenever i want to. to my shocking surprise after using the herbal oil 5days after i was shocked when my penis grow bigger to 9inches exactly as i wanted and now my wife enjoy me on bed and even crave for more. My fellow friends suffers no more, if you need your penis issue to be solved you can contact Dr Okusu today and right now on his email:[email protected] you can as well visit his websites:https://drokususolutiontem.wixsite.com/dr-okusu or call him or what-app him through Mobile:+2349037835860
I am very grateful to Dr. Believe The Mighty spell caster from Africa for helping me in my problem, I was almost losing hope in life when my lover of 4 years ended our relationship without any reason and I consulted an old friend of mine and explain my situation to her and she said she was in same situation 2 weeks back but a wonderful man who was sent to help people helped her out of the problem and I thought it was all a joke and I ask her if it was real and she said if it wasn't real so how did her lover come back to her, then I decided to give Dr. Believe a try because I had nothing to loose in trying and I contacted him on his email address which is [email protected] and he replied to my email very well.. and to cut the long story short Dr. Believe brought back my Lover in 48 hours.. I'm very happy he brought my man back to me and my lover promise that he will never and ever leave me again. Thanks to Dr. Believe contact him now for any kind of via email: [email protected] or [email protected] or view his website: https://authenticspellcast8.wixsite.com/authenticspellcast and call his mobile number: +2348159740630. Sherry Gibbs from Canada.
I Met a Powerful Spell caster and Psychic from Africa by name Dr Believe, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on [email protected] or [email protected] or check him directly on his website at http://authenticspellcast8.wix.com/authenticspellcast or call +2348159740630 he is the answer to your problem..
I Met a Powerful Spell caster and Psychic from Africa by name Dr Believe, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on [email protected] or [email protected] or check him directly on his website at http://authenticspellcast8.wix.com/authenticspellcast or call +2348159740630 he is the answer to your problem..
What a wonderful and a straight forward spell caster that has brought back joy and happiness into my life after i saw a post on how he helped a lady called Nicole Morgan; i decided to contact him for help, when i told this God sent man Dr BEN on how my lover left me for 2 years without calling nor texting me, When i shared this my sad experience with Dr BEN he said everything would be okay within 3 days i was like am i sure what this man is saying is real, So i decided to give him a try and at first i was thinking he was a scam and i taught he was like other spell casters who come online to add pain to people's life not knowing there feelings but to make money, this great man Dr BEN is never like that because he is for good and to make people happy with the one they love, am just so happy, Even before the 3 days i just got a call from a man who has left me for 2 years saying that he his sorry and that he wants me back to his life i was so happy, He invited me for a dinner which i met with him there and we both talked, he said he wants to prove that he would never leave me for any other lady he engaged me and also made me had access to all his account am so happy all thanks goes to this great man Dr BEN a man who has brought back joy to my life, friends that need help in getting their lover's back i would advice you contact Dr BEN email: [email protected] website http://benherbaltemple.webs.com or call him +1(310)-220-0075
What a wonderful and a straight forward spell caster that has brought back joy and happiness into my life after i saw a post on how he helped a lady called Nicole Morgan; i decided to contact him for help, when i told this God sent man Dr BEN on how my lover left me for 2 years without calling nor texting me, When i shared this my sad experience with Dr BEN he said everything would be okay within 3 days i was like am i sure what this man is saying is real, So i decided to give him a try and at first i was thinking he was a scam and i taught he was like other spell casters who come online to add pain to people's life not knowing there feelings but to make money, this great man Dr BEN is never like that because he is for good and to make people happy with the one they love, am just so happy, Even before the 3 days i just got a call from a man who has left me for 2 years saying that he his sorry and that he wants me back to his life i was so happy, He invited me for a dinner which i met with him there and we both talked, he said he wants to prove that he would never leave me for any other lady he engaged me and also made me had access to all his account am so happy all thanks goes to this great man Dr BEN a man who has brought back joy to my life, friends that need help in getting their lover's back i would advice you contact Dr BEN email: [email protected] website http://benherbaltemple.webs.com or call him +1(310)-220-0075
hello everyone in his forum i am little bit ashamed to share this marvelous story about a great spell caster who helped me enlarge my small penis size through his herbal mixture cream,i was heartbroken before because i have a very small penis about 3.5 inches which was so annoying and shameful i could not satisfy my wife in bed ,,my marriage was really breaking and i needed help urgently,i have used pills,vaccines,drugs,surgery but none worked...so one faithful day as i was browsing through the internet i saw few comments on a forum about how Mr CARLOS HENRY from USA testified of how he helped him enlarge his penis and also cured his dad diabetes type 2 disease,,i contacted him through his website link http://drharrysolutio.wixsite.com/dr-harry he replied and gave me steps to follow and i did just as he said ,and he sent me the herbal cream to me through UPS and i received it within 4 days and used the herbal cream for just 14 days to my greatest surprise my penis that was 3.5 inches before enlarge to 10 inches long when fully erected ,for just two weeks of using his herbal mixture cream ,,wow my wife loved me more and was so happy that i can satisfy her very well now in bed ,,i even last longer than before thank you so much DR HARRY for making life a better one to live in ..so if you are out there having similar problems please contact him now on his email [email protected] or visit his website http://drharrysolutio.wixsite.com/dr-harry or whatsapp and call him on +2348143240563 thanksnote he also have cure for DIABETES 1/2,HIV/AIDS,HERPES,CANCER,PREMATURE EJACULATION,WEAK ERECTION,ERECTILE DYSFUNCTION ,PREGNANCY HERBAL MEDICINE,ALS,HPV,SPELL OF ALL KINDS ETC
Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
My name is Elizabeth Shirley and I base in USA Florida...My life is back!!! After 2 year of Broken marriage, my husband left me with four kids . I felt like my life was about to end i almost committed suicide, i was emotionally down for a very long time. Thanks to a spell caster called Dr OCUSODO, which i met online. On one faithful day, as I was browsing through the internet,I came across allot of testimonies about this particular spell caster. Some people testified that he brought their Ex lover back, some testified that he restores womb,cure cancer,and other sickness, some testified that he can cast a spell that can help you get pregnant for your husband and so on. i also come across one particular testimony,it was about a woman called Sandra Denilson she testified about how he brought back her husband back in less than 48 days, and at the end of her testimony she dropped Dr OCUSODO e-mail address. After reading all these,I decided to give him a try. I contacted him via email and explained my problem to him. and he give me some instruction i follow it within 48 days, my husband came back to me. We solved our issues, and we are even happier than before Dr OCUSODO, is really a gifted man and i will not stop publishing him because he is a wonderful man... If you have a problem and you are looking for a real and genuine spell caster to solve all your problems for you. you can contact him on his email [email protected] or [email protected] contact number +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
Seriously i was flattened when my Husband of 8 years left to be with another woman in Texas. I cried and sobbed every day, until it got so bad that I couldn't bear it anymore. and i reached out to the Internet for help, until i hit on the real thing, and that is you Dr. Trust. I almost gave up trying to get my Ex Husband back and having a contented family again.. I had tried the whole lot I knew, and with your spells, blessings and extraordinary magical powers, you did all the work, and immediately after 48 hours, my Ex Husband came back to me and he was penitent for everything that he has done. And now my life is balanced and i am happy again. Dr. Trust you do a great service to people, and I don't think many people had known about you. You are the diamond in the rough. Thank you Dr. Trust You are talented and you give off yourself so freely like you did to me. Thank you for weaving your magical love spells for me and Sergio. He is back to me just the way it was when we first met.. from the depths of my soul! I am immensely happy now. Thank you and God bless you so much sir. Now my man is back to me just in 48 hours, as you have said it..wow.. i am fully rewarded. Thank you so much sir, Lots of appreciations.. Here is Dr. Trust Website: website: https://ultimatespellcast3.wixsite.com/dr-trust and his Email: [email protected] or [email protected]
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
GET YOUR EX BACK WITH THE HELP GREAT SPELL CASTER CALLED DR SAMBO VISIT HIS WEBSITE BELOW; http://divinespellhome.wixsite.com/drsambo
Good dey to you all , my name is moore kathleen i want to share a brief testimony on how Great Dr Sambo helped me re-unite my long gone relationship . I thought I have lost it all until I saw a post abort this great spell caster called Dr Sambo , I contacted him and told him about my story “few years ago , my husband and I were deeply in love I thought life was that easy , because I was with the man of my dream , until his secretary took him from me , he stopped loving me ,we separated because of the incidence , I couldn’t take it anymore , I count live with him know he has someone else out there , but I still loved him . until few week back I contacted Great Dr Sambo who helped me bring back my long gone husband , he apologyzed to me and each day that passes he love me more . if you are out there , because I know a lot of people are experiencing difficulties in their marriage or any form of relationship , just contact Dr Sambo and he will definatily help you as he has done for me here are his details .
I read some testimonies about a love spell caster called Dr.OBILUJESU on how he has helped lots of people in bringing back their ex lovers within 24 hours, Sincerely I was just thinking if that was real and if this man could really help bring back my lover whom I love so much. I decided to contact him because I love my boyfriend very much and we have been apart for a couple of months I really missed him so much, I have tried all other means to get him back but I couldn't. I contacted Dr.OBILUJESU and he told me that my ex will come back to me in the next 48 hours,Dr.OBILUJESU released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. My ex is now back to me again.As I`m writing this testimony right now I`m the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up.All thanks goes to Dr.OBILUJESU for the excessive work that he has done for me by helping me to get back with my ex boyfriend. I would like to drop Dr.OBILUJESU mail address and hope you see this testimony and contact him if you have a lover that you really want back so badly, His mail: [email protected] or [email protected] web sit: http://drobilujesuspellca.wixsite.com/drobilujesu OR call him +2348143481000
hello everyone in his forum i am little bit ashamed to share this marvelous story about a great spell caster who helped me enlarge my small penis size through his herbal mixture cream,i was heartbroken before because i have a very small penis about 3.5 inches which was so annoying and shameful i could not satisfy my wife in bed ,,my marriage was really breaking and i needed help urgently,i have used pills,vaccines,drugs,surgery but none worked...so one faithful day as i was browsing through the internet i saw few comments on a forum about how Mr CARLOS HENRY from USA testified of how he helped him enlarge his penis and also cured his dad diabetes type 2 disease,,i contacted him through his website link http://drharrysolutio.wixsite.com/dr-harry he replied and gave me steps to follow and i did just as he said ,and he sent me the herbal cream to me through UPS and i received it within 4 days and used the herbal cream for just 14 days to my greatest surprise my penis that was 3.5 inches before enlarge to 10 inches long when fully erected ,for just two weeks of using his herbal mixture cream ,,wow my wife loved me more and was so happy that i can satisfy her very well now in bed ,,i even last longer than before thank you so much DR HARRY for making life a better one to live in ..so if you are out there having similar problems please contact him now on his email [email protected] or visit his website http://drharrysolutio.wixsite.com/dr-harry or whatsapp and call him on +2348143240563 thanksnote he also have cure for DIABETES 1/2,HIV/AIDS,HERPES,CANCER,PREMATURE EJACULATION,WEAK ERECTION,ERECTILE DYSFUNCTION ,PREGNANCY HERBAL MEDICINE,ALS,HPV,SPELL OF ALL KINDS ETC
Hey guys, Get your ex back fast with the help of a real and genuine spell caster.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
I NEED A GENUINE SPELL CASTER TO BRING BACK MY EX BACK: CONATCT DORCTOR THOUGH HIS WEB SITE https://ultimatespellcast3.wixsite.com/dr-trust HE IS THE BEST OF ALL
It's a privilege to share this miraculous testimony to the world. I'm Elisa Wendy from England. My husband divorced me 4 months back and i have been filled with remorse for i didn't know what to do to amend issues with my husband. I searched for help on the internet on how i could get help in my marriage and i discovered great testifiers about Doctor Trust who has been progressive with his spells. I got in touch with him and behold, Doctor Trust told me that he will prepare a spell for me that will bring back my husband. I was skeptical but i had no other option but to work with him. 2 days after, my husband called me that he's coming back home and from that day till this moment, we have been living peacefully. He is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my husband,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love. I will highly recommend Doctor Trust to anyone one out there who needs help whatsoever.if you have any problem contact Doctor Trust, i give you 100% guarantee that he will help you!!.
[email protected] or [email protected] Call or write him on WhatApp +2348156885231
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: [email protected] or you can whatsApp his line:+2348107155060 or see more at http://azaspellcaster.wixsite.com/spell-caster
My life is devastated! I’m going through hell right now or from the past one month.One month ago,i broke up with my bf.We both were completely in love.But i started having some issues with myself.My mood swings were a lot to handle for him an that point i just felt the need to break up and so i did.And i hurt him to the core which i never wanted to.He kept on begging me but i didn’t hear.Later on he started avoiding me and kind of did everything as you said in your video.it felt bad.I always loved him but i had to break up.I regret breaking up so much.But after that he said no to me.He said he’ll never come back as he cannot put faith in me.I have tried my level best but he is adamant that he cannot trust me.Even though after i realised what i did.I changed myself completely.I dealt with my mood swings and now i don’t get any.He cannot trust me.what should i do?I love himand he does too but he says he don’t have that much courage to come back. one day in my office going through the internet i saw a post of a lady giving testimony on how she got her husband back,i also come across a blog site on how to reunite and get your ex back with a love spell it was really amazing to see this so i copy his email id on http://happyspelltemple.webs.com/ i explain my situation to him and gave him all the details he requested from me, he give me 3 days assurance that my boyfriend will come back after the spell i was so overwal to hear that so i just keep calm and follow up his order coz i really love my boyfriend and want him back. on the third day after the spell as he promise me, my ex call me also show up in my office begging me to forgive him and accept him back with out no delay, i lift him up with a kiss and was so very happy to have him back all my happiness was from Dr happy he is indeed a true man a father and a god sent. plz contact Dr happy on his email id direct at [email protected] or call his mobile number on +2348133873774
I never knew people still have powers and make things happened this way. My name is Sandra Ruths am from London. my boyfriend Slay Scot left me for another girl for three months' ever since then my life have been filled with pains sorrow and heart break because he was my first love who dis virgin me when i was 21 years old. about two years ago, A friend of mine kido matthew told me he saw some testimonies of this great Dr osauyi that he can bring back lover within some few days, i laugh it out and said i am not interested but because of the love my friend had for me, she consulted the great priest on my behalf and to my greatest surprise after three days my boyfriend is going to call me for the very first time after three months that he is missing me and that he is so sorry for every thing he made me went through. i still can't believe it, because it highly unbelievable it just too real to be real. Thank you Dr osauyi for bringing back my lover and also to my lovely friend who interceded on my behalf, for any one who might need the help of this great priest here is the email address [email protected] or you can call the great man on +2347064294395
Hey guys, Get your ex back fast with the help of a real and genuine spell caster called Dr.Unity.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
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Hello every one my name is Amanda Villa am sharing these because am so happy for what these man has done for me i have been married to my husband 8 years now i don't know what came over him last year July 17 he divorced me saying his is tired of me as a wife he traveled to Mexico and leave me and my kids i was confused in my life i was praying for him to come back home because it was hell for me taking care of two kids alone on one faithful day i was in my house sharing tears when my friend here in usa who visited me told me about these man called shankiki that he can help me that he has see him solve problems like my situation for people close to her that he can also help me i was left with no choice because i needed him back i contacted him and he told me not to worry he collected my details and pray for me i did every thing he ask me to do he told me he is going to call me when its time after three days i contacted him my husband called me and he started apologizing to me that he miss me and my kids that he is coming home soon after one week he called me i was verysurprise to see him am very happy shankiki saved my marriage we are now happily together now all praise be to DR Skankiki he is great and powerful i believe him so much his email is [email protected] if you have issues in your relationship he can help you i believe him he is real.
Hello Viewers My Name is Ashley Martin form Texas United State.. i want to say a very big thanks and appreciation to Dr.SABU for bringing back my EX LOVER who left me..My ex Lover was always picking up an argument with me and was always beating me. He started acting strange and funny and left me for no cause for another Girl. I thought all hope was gone. One day as i was browsing searching on the internet for recommend spell caster who will help me to bring my ex lover back home,Finally i met a writing how so many testimony talking about Dr.SABU who restore relationship back,so i email him and he cast the love spell,luckily within 24hours my ex Lover really called me and started apologizing for all he had caused me.and begging me to accept him back and we are living happily and our relationship is now very strong, I am the happiest woman on earth today because Dr.SABU has done a wonderful deeds,my Lover now love and respect me.Today we both live together and we are even more in love than how we use to be before he left,contact to help you email: [email protected] or his whatsapp number+2347057624993 or contact his web/site: https://drsabuspelltemple.wixsite.com/dr-sabu
I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: [email protected] or you can whatsApp his line:+2348107155060 or see more at http://azaspellcaster.wixsite.com/spell-caster
Hello every one out here.... I am form USA, am here to testify to the good work of the Best African Traditional Doctor Dr. Ogbefi. 4 years ago, my husband left home, he never returned, no phone calls, no letters, no emails, no sign of him anywhere. my daughter got sick with multiple sclerosis, things were so tough for me. I had lost hope, 2 years ago, i met a caster on the internet, he said he would help me,bring back my happiness to me of at least saving my daughter's life and getting a job that i was contesting for, and still nothing happened, i lost hope completely, my daughter's situation got worse each day. last month, i saw a posting concerning the good works of the A most powerful traditional doctor in Africa, Africa has always been my hope in spells but i never trusted any spell caster from the one who just eat my money. then i decided to try Dr. Ogbefi. i gave him a try...For all three spells (Bring Lover back, Healing spell and Career spells). In a matter of weeks, my husband called me and told me he was sorry and that he wants to come back to me and that he would explain everything when he comes back, three days later, i got a new job with a loan from a finance company, right now, my daughter's condition is getting better each day and i trust she would be well in a matter of days. I want to thank Dr. Ogbefi for his effort and for bringing my life back to normal and so close to perfection. When i talk about this man i don't want to finish he is my hero of the century thanks Dr. Ogbefi his email is [email protected] And also Reach him on WhatsApp Number: +2348056932763 Thanks Dr Ogbefi
Hello everyone i`am caroline from Uganda, i am very happy today and satisfied, This is certainly a shocking and a genuine Testimony..I visited a forum here on the internet on the 10TH OF JANUARY 2017, And i saw a marvelous testimony of this powerful and great spell caster called Prophet Abulele on the forum..I never believed it, because i never heard nor learnt anything about magic before.. Not a soul would have been able to influence me about magical spells, not until Prophet Abulele did it for me and restored my marriage of 5 years back to me and brought my husband back to me in the same 3 days just as i read on the internet..i was truly astonished and shocked when my husband knelt down begging for forgiveness and for me to accept him back.. I am really short of expressions, and i don't know how much to convey my appreciation to you Prophet Abulele you are a God sent to me and my entire family.. And now i am a joyful woman once again.. here is his Email: [email protected]
I'm still shocked of the way Dr ODELI helped me, I am so proud and happy to be out here sharing this awesome testimony, I just can't believe this now my girlfriend is really back to me and she promised me that she will never let go of me ever again, she keep pleading to me to please accept her back and she was feeling regretful and sorry for leaving me and for causing me pains after the quarrel which occurred last year. And this whole miracle happened after i ordered an urgent 24hours Dr.ODELI powerful spell which he cast on me and my girlfriend. may God continue to bless your work for the lives of many people, I am the happiest woman today in this whole wide world. Dr. ODELI. still unbelievable the way i was back with my girlfriend.. I am so happy now and i don't know how much to thank you Dr ODELI. contact him if you need this same help like i did, just contact Dr ODELI and he will surely help you right away on his email: [email protected]
WOW i'm glad to be sharing with everyone how Dr Otoide the powerful spell caster helped me to get back with my wife. I am Jamie Watson, after me and my wife divorced after having a huge quarrel then. i searched for a spell caster that can help me then i found testimonies about Dr Otoide, how he helped many people to get their lovers and broken homes back, i contacted him through his email address :[email protected] because I was absolutely desperate to get my wife back. Life without my wife was a real mess for me and my children. i wanted a dramatic change and I thought casting a love spell could be the solution. After discussing the resolution with Dr Otoide. he gave me hope that he will restore my marriage. I felt confident that he will actually make my wife to return home and he did! It’s fantastic what this great spell caster have done for me, his help is priceless! I don't know what I would have done without Dr.Otoide. he does his job so well he is organized and highly functional. i believe he is the best spell caster i can count on when it comes to all kinds of spell. if you need help. contact him on his websites:https://drotoidesolutionte.wixsite.com/dr-otoide or his Mobile line:+2347089081873 or his email:[email protected]..
Hey!I was in a relationship for quite along time almost 7 years, he cheated with a girl almost 11 years younger than him I took him back because I love him so much. He promised me the world never to leave me never to let me go to make me happy and to make it up to me everyday. 7 weeks later he has broke up with me after a few arguments .. he told me he can’t do it that I’ve ruined everything I deserve it all I’ve treated him bad he gave up his job overseas but he only gave it a week afrer he quit. he has left me in bits and is 1 week later out with other girls please help I don’t know what to do he is telling me he will never ever get back its over and he will never give it a go I have begged and I don’t know why I don’t no why he has done this to me I feel so lifeless… one day in my office going through the internet i saw a post of a lady giving testimony on how she got her husband back,i also come across a blog site on how to reunite and get your ex back with a love spell it was really amazing to see this so i copy his email id on http://happyspelltemple.webs.com/ i explain my situation to him and gave him all the details he requested from me, he give me 3 days assurance that my boyfriend will come back after the spell i was so overwal to hear that so i just keep calm and follow up his order coz i really love my boyfriend and want him back. on the third day after the spell as he promise me, my ex call me also show up in my office begging me to forgive him and accept him back with out no delay, i lift him up with a kiss and was so very happy to have him back all my happiness was from Dr happy he is indeed a true man a father and a god sent. plz contact Dr happy on his email id direct at [email protected] or call his mobile number on +2348133873774
Hey guys, Get your ex back fast with the help of a real and genuine spell caster called Dr.Unity.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
WOW i'm glad to be sharing with everyone how Dr Otoide the powerful spell caster helped me to get back with my wife. I am Jamie Watson, after me and my wife divorced after having a huge quarrel then. i searched for a spell caster that can help me then i found testimonies about Dr Otoide, how he helped many people to get their lovers and broken homes back, i contacted him through his email address :[email protected] because I was absolutely desperate to get my wife back. Life without my wife was a real mess for me and my children. i wanted a dramatic change and I thought casting a love spell could be the solution. After discussing the resolution with Dr Otoide. he gave me hope that he will restore my marriage. I felt confident that he will actually make my wife to return home and he did! It’s fantastic what this great spell caster have done for me, his help is priceless! I don't know what I would have done without Dr.Otoide. he does his job so well he is organized and highly functional. i believe he is the best spell caster i can count on when it comes to all kinds of spell. if you need help. contact him on his websites:https://drotoidesolutionte.wixsite.com/dr-otoide or his Mobile line:+2347089081873 or his email:[email protected]..
Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
This is the most wonderful thing i have ever experienced. I visited a forum here on the internet on the 17 October 2016, and i saw a marvelous testimonies on the forum about the good works Doctor SAM. Am Lopez Angelito from United States. I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until Doctor SAM did a marvelous work for me that restored my marriage of 4 years by getting back my divorced wife within 24 to 48 hours just as i read on the internet. I was truly shocked when my wife came home pleading for forgiveness to accept her back. Am really short of words to show my appreciation to Doctor SAM for he is a God sent to me and my entire family for divine restoration of marriage. Are you in need of any help whatsoever? Don't hesitate to get in touch with him now. He also specializes on the follows;
Hello everyone , I want to quickly drop my life stories on how a specialize-healer called Priest Moses who cure me from HERPES SIMPLEX VIRUS. A great man who have the root and herbs to all kind of deadly disease.i was infected with HERPES SIMPLEX VIRUS disease by a man called FRED. I never stop blaming my self for falling in love with him .i never think HERPES SIMPLEX VIRUS can be cure with the medication Priest Moses send to me until I was cure ,I saw his email on you-tube by a patient he have cure from herpes called MRS RAY which drop the email and phone contact of him that make me get cure by contacting him.i really thank MRS RAY for sharing her testimony on how she was cure. When I contact him.he prepare the herbal medication for my HERPES CURE and also send it to me in UK via DHL, that get me cure within two weeks I use it. he also cure all type of disease you can still contact him via [email protected] or for easy communication you can WhatSAPP him via his mobile number +2348151084488
Hey guys, Get your ex back fast with the help of a real and genuine spell caster called Dr.Unity.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
Hey guys, Get your ex back fast with the help of a real and genuine spell caster called Dr.Unity.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
I am Lord Ogbomola the high chief of Ogbomola palace, am a great herbalist and powerful spell caster with the great powers to get ex back, make people love you, win cases, get attracted to someone, get financial help and so many others.I am well known for the great powers I have and the fast results of my experienced works.Are you looking for a good spell to get back your ex lover, husband, wife, boyfriend or girlfriend?Do you want to win court cases?Do you want a spell to make yourself very attractive to men or women?Do you need financial assistants?Do you need powers to control anyone?Do you need the perfect spell to get pregnant of have babies of your desire?Do you want powers to get rich or famous?Do you need a cure to any disease such as HIV/Aids, Diabetes, Parkinson, ALS etc….Do you need Penis, Hips, Breast or Butt Enlargement?Do you need solution to Weak erection/premature ejaculation in men?Do you need connections to become a member of any occult group or society to get more richer?Do you want powers to make your business or any organization grow faster and higher? ETC……Then consider the Ogbomola Palace as the best place to be for the results and solutions you seek.For consultations on any marital or other problems, reach me immediately on my social email: [email protected] …
How this great man helped me to bring back my lover within 24hours,my lover who broke up with me since last three year,i love him so much that any time i see him with another girl i will be angry,so one day i was going through the internet when i saw a testimony of a woman how this great man name dr Onima helped her to get back with her husband, so lucky for me he dropped the contact of the man,so i quickly called the number he did not pick i emailed him after about 4hours time he called me back and he replied my email,i filled the form and he told me everything about me and my lover i was surprise so he told that he need some items to do the prayers for me,which i sent him the money to get and that was the only money he collected from me,within 24hours my lover came back to me pleading to me asking for forgiveness,he now love me, cherish me and he promise that he will never leave me again all thanks to dr Onima,you can also call +2348119178323 or email [email protected]
WHAT A WONDERFUL LOVE SPELL, MY HUSBAND AND I ARE BACK TOGETHER AGAIN AFTER 2 YEARS OF SEPARATION AND DIVORCE, AM TRULY BLESSED BY DR.OGUDU SPELL TEMPLE, ALL THANKS TO HIM, YOU CAN CONTACT HIM NO VIA EMAIL:::[email protected]:::WHATSAPP NUMBER:::+2348106058254:::WEBSITE:::http://oguduspelltemple.wixsite.com/drogudu
I'm ROY LILY from BRONX NY, USA. I got separated from my husband 2 years ago. There was no communication between us. I was advised by families and friends to let go, forget the marriage and move on with my life. I didn’t want to marry someone else because deep down, I still love my husband. I was in so much pain and confused, until I read a testimony online on how Dr Ogudu reunion a broken marriage with the help of his spiritual spell power. I kept reading so many testimonies on how he helped to stopped divorce and brought back peoples ex-lover's back to them, and my faith was renewed. I have to contact Dr Ogudu immediately, few minutes later, he replied and instructed me on what to do, after meeting up with the necessary requirement, he sent me a parcel and instructed on how to use it, which I did. 48 hours later communication was restored between me and my husband. I saw him after 7 days and we were able to settle our differences, he actually knelt down and was begging me to forgive him. I was visiting so many website seeking for help, it looked hopeless, until I came in contact with DR OGUDU, who helped me to restore my joy, in fact, I'm the luckiest and happiest woman ever to have her husband back in her life. If you're having similar problem in your marriage, you want your husband or wife to love you again, you have someone you love and you want him or her to love you in return, you are having any challenge in your marriage. DR OGUDU SPELL TEMPLE is the solution to your problem, contact him today via email: EMAIL:::[email protected]:::WHATSAPP NUMBER:::+2348106058254:::WEBSITE:::http://oguduspelltemple.wixsite.com/drogudu
Happy to tell the world how wonderful Dr Otoide has restore my dignity as a woman back to me. it been 2years in marriage with my husband and i have never conceive i wonder what was wrong wih me, so i searched out for a remedy and fortunately i met with the great spell caster Dr Otoide who has helped severally others with the same issues and i was very glad when i wrote to him on his email he told me to calm down that everything will fine once he is done casting spell for me that i will be pregnant again, and h gave me the instruction on how he spell worked and i agreed with his terms and i gave him all that he needed to cast my spell, and indeed after 48hours he was done and told me that the next morning i will be pregnant and i believed him and that was how i got pregnant by the help of Dr Otoide a very great spell caster that can help you too on any challenges that you might be passing through. just contact him today on his Email:[email protected] or his phone number:+2347089081873 or his websites:https://drotoidesolutionte.wixsite.com/dr-otoide
Hello everyone, I am proud to testify this testimony. i saw a post on how a lady got her husband back and i decided to try the spell caster that helped her because my relationship was crashing. although i never believed in spell work i reluctantly tried him because i was desperate but to my greatest surprise the spell caster Dr uzoya helped me get my husband back,after 5 month of divorce and my relationship is now perfect just as he promised, my husband now treats me like a queen even when he had told me before he doesn't love me anymore. well, i cannot say much but if you are passing through difficulties in your relationship contact him on his email: [[email protected]] OR call his number +13019098775. you can also go through his website on https://driraborspellcaste.wixsite.com/spell-caster
How can I explain this to the world that there is a man called Dr uzoya who helped me bring back my ex lover and restored my womb..when all hope was lost..Me and my lover have been married for the past 5 years with no child we were living happily..on till one day that we have an understanding because i had not be able to have a child for him, so he decided to live me because of another women, i was so sad and frustrated and i could not even eat on till one day i was browsing through the internet where i saw a lady called Jessica testifying about spell caster call Dr uzoya who helped her brought back her husband back home, so i decided to take a try in contacting the spell caster Dr uzoya so i explain all my problem to Dr uzoya and he told me that i should not worry that he is going to help me solve my problem,that my lover will come back home to my greatest surprise in just 24 hours my husband start calling me and begging for forgiveness i was so surprise now we are back again as husband and wife,all my thanks goes to DR,UZOYA, worry no more cause DR,UZOYA is here to help u all,i will do u favor my drooping his email; [email protected] OR his telephone number is +13019098775 https://driraborspellcaste.wixsite.com/spell-caster
I promise to share this testimony all over the world once my husband return back to me Thing don't just work out out until you make the right choice in your life, In my life i made the right choice when i contacted Dr.UZOYA the great spell caster who is specialized in restoring broken relationship or marriages. My name is Ramond rose from Albany New York City USA, and i here on this site to thank DR.UZOYA for a job well done, Because i never believe that i would have gotten my husband back if not for the sake of Dr.UZOYA that is why i won't be leaving this site without dropping the contact details of Dr.UZOYA which are via email: driraborspellcastergmail.com do contact Dr. UZOYA in order to restore your broken marriages or relationships you also call him on +13019098775 you can see more of Dr,UZOYA on his website https://driraborspellcaste.wixsite.com/spell-caster THANK YOU ONCE AGAIN Dr,UZOYA
Effective Love Spell To Get Back Your Ex Back Within 48 Hours: Fast And Read Through Dr. Trust Website Now. https://ultimatespellcast3.wixsite.com/dr-trust
Hi everyone, I was going crazy when my husband breakup with me and left me for another woman!! All thanks to Dr Trust the best love spell caster online that helped me to bring back my husband today and restore happiness in my marriage.. My husband breakup with me and left me to be with another woman, and i wanted him back. I was so frustrated and i could not know what next to do again, I love my husband so much but he was cheating on me with another woman and this makes him break up with me so that he can be able to get married to the other lady and this lady i think use witchcraft on my husband to make him hate me and my kids and this was so critical and uncalled-for, I cry all day and night for God to send me a helper to get back my husband!! I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Trust can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a spell for me. 48 hours later, my husband really called me and told me that he miss me so much, Oh My God! i was so happy, and today i am happy with my man again and we are joyfully living together as one big family and i thank the powerful spell caster Dr.Trust, he is so powerful and i decided to share my story on the internet that Dr.Trust is best and good spell caster online. if you are here and your lover is turning you down, or your husband or wife moved to another person, do not cry anymore, contact this powerful spell caster Dr.Trust on his email at: [email protected] OR [email protected] you can also call him or add him on Whats-app: +2348156885231 ,His website: https://ultimatespellcast3.wixsite.com/dr-trust.
I have come to give Imam Ali all the glory to everyone on this paltfrom, for destroying 10 years of career stagnation in my life.
I am Francisca Cain and ever since i contacted Imam Ali,My life has been from glory to glory.In November 2016,I got a miracle job in London by Imam Ali favor but due to the long distance and challenges of child care,I decided to quit the job in December. That same December 2016, I went for an interview with a reputable organization that was 10 mins walk from my house and just opposite my children's school.The job ticked all the boxes i wanted to make things easy for me in caring for the kids,doing my business,e.t.c while still working but i was not granted the job in spite of them saying i did well. On the day i went for the interview, I told Imam Ali and he said that he will give me the job and asked me to possessed that organization by faith. So based on my faith i have in Imam Ali, I never accepted the negative report of the rejection,but kept on the trusting Imam Ali and decreeing that they will call me back and grand me the job. While believing Imam Ali for my miracle job,Weeks later I received a call from the recruitment officer that interviewed me, stating if i was still available for the position with their organization as the role is still available. The contract letter has been signed and i am to resume my dream job in February, 2017. I give Imam Ali all the glory, who is not a robber of men but a rewarder of them that diligently trust him. Imam Ali is a covenant keeping spell caster, to him alone be all the praise. For any problem you may encounter in life do contact Imam Ali for he is the solution you been looking for okay..(email: [email protected]. OR Text and call me on +1561-231-9522).Please do let him know that i directed you to him,thanks....
I feel so blessed again in my marriage after Doctor Oku brought back my husband that separated with me for good 3 mounths. Am Jessica Butler by name from USA. Even though i have mouths all over my body, it won't be enough to thank Doctor Oku for his help upon my life. My husband separated with me for the 3 months and have been in pains and agony without him. So, i searched for help everywhere but nothing worked out not until i meant Doctor Oku who i contacted online. I explained my situation to him and he promised that my husband will get back to me within 24 to 48 hours as far that my heart still beats for him. I believed in him and he prepared a spell for me and my husband called me exactly when Doctor Oku said. He pleaded and said he needs me back and now we are living happily again for the past 9 months. Everyone out there reading my article that needs help should contact him... Email: [email protected] or call +2347053113465
I am MARIE KATHLEEN from USA SOUTH CAROLINA MY ADDRESS IS (GUMTREE RD HILTON HEAD ISLAND SC 29926) I and my husband have been sick for 5 months a disease called Genital herpes is a common viral infection caused by the herpes simplex virus (HSV) There are two types of the virus, types 1 and 2 (HSV-1 and HSV-2). As well as genital herpes, HSV can infect the mouth and cause cold sores. HSV-1 and HSV-2 lesions look the same and can only be distinguished by laboratory testing and my family doctor confirmed that we have this disease .i have seek help from various doctor but got scammed i was stuck with this same sickness without cure and my husbands own was getting worst each passing day i keep on praying to God for cure as God will have it all i came across a Testimony about (DR SHELTER NGOZI IWEALA) a very honest woman who have the fear of God a woman who doesn't disappoint i told her what i and my family is going through she told me to wipe away my tears that no condition is permanent she will help me as God will have it she sent me the package (Medicine) after three days we went for check up and today we are totally free from Genital herpes a common viral infection caused by the herpes simplex virus (HSV) and my husband healthy now Please i don't know what your situation is or what you might be going through okay if you must contact anybody for Genital herpes cure please (DR SHELTER NGOZI IWEALA) on her ([email protected]) she will wipe away your tears and put a smile in the face of your family.
I am MARIE KATHLEEN from USA SOUTH CAROLINA MY ADDRESS IS (GUMTREE RD HILTON HEAD ISLAND SC 29926) I and my husband have been sick for 5 months a disease called Genital herpes is a common viral infection caused by the herpes simplex virus (HSV) There are two types of the virus, types 1 and 2 (HSV-1 and HSV-2). As well as genital herpes, HSV can infect the mouth and cause cold sores. HSV-1 and HSV-2 lesions look the same and can only be distinguished by laboratory testing and my family doctor confirmed that we have this disease .i have seek help from various doctor but got scammed i was stuck with this same sickness without cure and my husbands own was getting worst each passing day i keep on praying to God for cure as God will have it all i came across a Testimony about (DR SHELTER NGOZI IWEALA) a very honest woman who have the fear of God a woman who doesn't disappoint i told her what i and my family is going through she told me to wipe away my tears that no condition is permanent she will help me as God will have it she sent me the package (Medicine) after three days we went for check up and today we are totally free from Genital herpes a common viral infection caused by the herpes simplex virus (HSV) and my husband healthy now Please i don't know what your situation is or what you might be going through okay if you must contact anybody for Genital herpes cure please (DR SHELTER NGOZI IWEALA) on her ([email protected]) she will wipe away your tears and put a smile in the face of your family.
Hello everyone, I am proud to testify this testimony. i saw a post on how a lady got her husband back and i decided to try the spell caster that helped her because my relationship was crashing. although i never believed in spell work i reluctantly tried him because i was desperate but to my greatest surprise the spell caster Dr uzoya helped me get my husband back,after 5 month of divorce and my relationship is now perfect just as he promised, my husband now treats me like a queen even when he had told me before he doesn't love me anymore. well, i cannot say much but if you are passing through difficulties in your relationship contact him on his email: [email protected] OR call his number +13019098775. you can also go through his website on https://driraborspellcaste.wixsite.com/spell-caster
Hey guys, Get your ex back fast with the help of a real and genuine spell caster called Dr.Unity.I'm so excited my broken Marriage has been restored and my husband is back after a breakup, After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
Are you going through a break-up or divorce? Do you need your Ex back fast? Dnot cry anymore, contact Dr .Goodluck powerful Love spell that work fast now!! I was hurt and heart broken when my Husband breakup with me, and i could not know what next to do again, I love my Husband so much but he was cheating on me with another woman and this makes him break up with me so that he can be able to get marred to the other lady and this lady i think use witchcraft on my Husband to make him hate me and our kids and this was so critical and uncalled-for,I cry all day and night for God to send me a helper to get back my man until i went to Texas in USA to see a friend and who was having the same problem with me but she latter got her husband back and i asked her how she was able to get her husband back and she told me that their was a powerful spell caster called Dr .Goodluck that help with love spell in getting back lost lover back, and i decided to contact the same Dr .Goodluck and he told me what is needed to be done for me to have my man back and i did it although i doubted it but i did it and the Dr told me that i will get the result after 17hours, and he told me that my husband was going to call me and i still doubted his word, to my surprise my husband really called me and told me that he miss me so much, Oh My God! i was so excited, and today i am happy with my man again and we are joyfully living together as one big family and i thank the powerful spell caster Dr .Goodluck for bringing back my man and i decided to share my story on the internet that Dr .Goodluck is best spell caster online who i will always pray to live long to help his children in the time of trouble, If you have any problem contact Dr .Goodluck and i guarantee you that he will help you,Email him at: [email protected] ,you can also call him or add him on whats-app: +2348138654465 ,http://goodluck05spellcaster.yolasite.com
I promise to share this testimony all over the world once my husband return back to me Thing don't just work out until you make the right choice in your life, In my life i made the right choice when i contacted Dr.UZOYA the great spell caster who is specialized in restoring broken relationship or marriages. My name is Ramond rose from Albany New York City USA, and i here on this site to thank DR.UZOYA for a job well done, Because i never believe that i would have gotten my husband back if not for the sake of Dr.UZOYA that is why i won't be leaving this site without dropping the contact details of Dr.UZOYA which are via email: [email protected]] do contact Dr. UZOYA in order to restore your broken marriages or relationships you also call him on +13019098775 you can see more of Dr,UZOYA on his website https://driraborspellcaste.wixsite.com/spell-caster THANK YOU ONCE AGAIN Dr,UZOYA
Email;[email protected] number; +2349067051818 if you have any of the below problems and you are in need for a real and genuine spell caster to solve all your problems. Kindly contact Dr Gozza via his email. or his whats app number +2349067051818 anytime, he is the answer to yourproblems.1 HIV /aids cure2 herpes cure3 stroke4 disable5 cancer6 GETTING YOUR LOVER OR HUSBAND BACK7 als8 dick enlargement9 hepatitis B10 Hpv Virus11 DIABETES12 IMPOTENCE,13 BARENESS/INFERTILITY14 DIARRHEA15 ASTHMA....
Hello to you all on this page, am Helena, i want to use this opportunity to share with you all on this page how i was able to get back with my man Jackson, after 2 years of breakup, it all started 2014 when my husband traveled from our city texas to Florida for work where he met this other lady since then my husband started to treat me like am was nobody until one day he left the house to start living with this other lady, me and our daughter of a year and 2 months then, were left alone, i cried and tried everything possible to plead with he to make he return home, but nothing worked, and i loved my husband so much and i knew he loved me as well, at this time i noticed all was not in it right senses with my husband, i made up my mind not to rest until am able to get my family back on track again, September 23 2015 my friend Darcy from Houston told me about this powerful spell caster, called Dr Gozza of [email protected], at whom she claim has helped her and some others with relationship issues,at first i was skeptical about this thing called spell casting, but after thinking for days i said to give it a try because i loved my husband and i want my child to grown up with both parents , i contact this Dr Gozza on his email, and also texted him on the phone, and explained everything that has happened between me and my husband to he , and he told me what to do, i did it all and he told to wait for three days after his work i will get a text from my husband, after three days of his work i actually got a sweet and apology text from my man to me it was like wow, and it was like a dream come through, so me and Jackson my lovely husband started again and till this moment we are still together in love okay, so with this great work done for me by this man , i vowed to share to the world my experience with he so whosoever in such situation, marriage, heartbreak, any kind of relationship problems/issues. can also contact him, his contact details below Email;[email protected] number; +2349067051818
Are you into any kind of marriage or relationship problems and you need help to sort it out here comes your helper contact Dr. Ikhine today through his email: [email protected] to get quick response to bring your long awaited problems to an end also call for fast response +2347038832903.
How can I explain this to the world that there is a man called Dr uzoya who helped me bring back my ex lover and restored my womb..when all hope was lost..Me and my lover have been married for the past 5 years with no child we were living happily..on till one day that we have an understanding because i had not be able to have a child for him, so he decided to live me because of another women, i was so sad and frustrated and i could not even eat on till one day i was browsing through the internet where i saw a lady called Jessica testifying about spell caster call Dr uzoya who helped her brought back her husband back home, so i decided to take a try in contacting the spell caster Dr uzoya so i explain all my problem to Dr uzoya and he told me that i should not worry that he is going to help me solve my problem,that my lover will come back home to my greatest surprise in just 24 hours my husband start calling me and begging for forgiveness i was so surprise now we are back again as husband and wife,all my thanks goes to DR,UZOYA, worry no more cause DR,UZOYA is here to help u all,i will do u favor my drooping his email; [email protected] OR his telephone number is +13019098775 https://driraborspellcaste.wixsite.com/spell-caster
I will always be thankful to the Great Dr.BEN. for many years now my family has suffered from HERPES DISEASE. it happened that after i got married to my Husband. i found out that i got HERPES VIRUS and when i did i was confirm positive and was completely so angry and then i had to convince my Husband to go test himself and when he did he was also found positive and then it was dawn completely on me that me and my husband is having HERPES DISEASE. i went out in search of a solution despite my doctor told me that there was no remedy to curing me and my husband HERPES DISEASE VIRUS. when i was searching i found out there so many testimonies about Dr.BEN helping others to be cured of HERPES VIRUS. i instantly added his email and asked for his cure and he instruct me on how his cure can get reached to me and my husband and i believe and did the little things he asked of me and that was i received his cure and told me that after me and my husband use it for some days i will come back and thank him and that was true. after we both use the herbal cure. we went to the doctor after the days which he instruct us. we were both confirmed negative. all thanks to Dr.BEN for his marvelous healing he gave to my family. contact him on his Email: [email protected] or [email protected].
Seriously i was flattened when my Husband of 8 years left to be with another woman in Texas. I cried and sobbed every day, until it got so bad that I couldn't bear it anymore. and i reached out to the Internet for help, until i hit on the real thing, and that is you Dr. Trust. I almost gave up trying to get my Ex Husband back and having a contented family again.. I had tried the whole lot I knew, and with your spells, blessings and extraordinary magical powers, you did all the work, and immediately after 48 hours, my Ex Husband came back to me and he was penitent for everything that he has done. And now my life is balanced and i am happy again. Dr. Trust you do a great service to people, and I don't think many people had known about you. You are the diamond in the rough. Thank you Dr. Trust You are talented and you give off yourself so freely like you did to me. Thank you for weaving your magical love spells for me and Sergio. He is back to me just the way it was when we first met.. from the depths of my soul! I am immensely happy now. Thank you and God bless you so much sir. Now my man is back to me just in 48 hours, as you have said it..wow.. i am fully rewarded. Thank you so much sir, Lots of appreciations.. Here is Dr. Trust Website: website: https://ultimatespellcast3.wixsite.com/dr-trust and his Email: [email protected] or [email protected]
I am Natasha Sproles, i am here to give testimony of how i got back my boyfriend, together for more than 4 years and everything were going well with us and we are always happy. until one day my boyfriend started to behave in a way i could not understand, i was very confused by the way he treat me and later that month he did not call me again and he call me sister and told her he has done with me, I asked him what have i done wrong to deserve this from him, all he was saying that he want me out of his life and do not want to see me again in his life, i was mad and also frustrated do not know what to do, i was sick for more than 2 weeks because of the way he treated me. i love him so much he was everything to me without him my life is incomplete. i told my friend and she told me to contact a spell caster, i never believe in all this spell casting of a thing. but i decided to try if something will come out of it due to the fact that i don't want to loose him. I contacted Dr. Iyere, at ([email protected]) for the return of my boyfriend to me, he told me that my boyfriend have been taken by another woman that she cast a spell on him that is why he hates me suddenly. then he told me that he need to cast the spell out from him that will make him return to me, he demanded for some items to cast the spell after 3 days my boyfriend called me and he told me that i should forgive him, he started to apologize on phone and said that he still love me that he did not know what happen to him that he left me. it was the spell that the Dr. Iyere, casted out on him that made him come back to me, today me and my is about to do our wedding. thank to Dr. Iyere for what he done for me i would have been nothing today if not for your great help. i want you my friends who are passing through this kind of love problem of getting back their husband, wife , or ex boyfriend and girlfriend to contact ([email protected]) and you will see that your problem will be solved.. call +2348083221034
GET YOUR EX BACK WITH THE HELP GREAT SPELL CASTER CALLED DR SAMBO VISIT HIS WEBSITE BELOW; http://divinespellhome.wixsite.com/drsambo
Good dey to you all , my name is moore kathleen i want to share a brief testimony on how Great Dr Sambo helped me re-unite my long gone relationship . I thought I have lost it all until I saw a post abort this great spell caster called Dr Sambo , I contacted him and told him about my story “few years ago , my husband and I were deeply in love I thought life was that easy , because I was with the man of my dream , until his secretary took him from me , he stopped loving me ,we separated because of the incidence , I couldn’t take it anymore , I count live with him know he has someone else out there , but I still loved him . until few week back I contacted Great Dr Sambo who helped me bring back my long gone husband , he apologyzed to me and each day that passes he love me more . if you are out there , because I know a lot of people are experiencing difficulties in their marriage or any form of relationship , just contact Dr Sambo and he will definatily help you as he has done for me here are his details .
i cant still believe that there are people out there dying of heartbreak or scammed by other spell casters,i contacted so many spell caster but non could help me bring my ex boyfriend back until i contacted Dr.SABU and explained my situation to him and then seek his help but to my greatest surprise he told me that he will help me with my case and here i am now celebrating because my Boyfriend has change totally for good. He always want to be by me and can not do anything without my present. i am really enjoying my marriage, what a great celebration. i will keep on testifying on the internet because Dr.SABU is truly a real spell caster.his email:[email protected] any heartbreak i will advice you to contact Dr.SABU Because in my own situation he did a very great job for me by bringing back my ex Boyfriend without wasting time and since then my relationship has been perfect and protected by his great power.or cell whatsapp number+2347057624993 So better contact him now web/site:https://drsabuspelltemple.wixsite.com/dr-sabu
After being in relationship with markiss for seven years,he broke up with me, I did everything possible to bring him back but all was in vain, I wanted him back so much because of the love I have for him, I begged him with everything, I made promises but he refused. I explained my problem to someone online and she suggested that I should rather contact a spell caster that could help me cast a spell to bring him back but I am the type that never believed in spell, I had no choice than to try it, I mailed the spell caster, and he told me there was no problem that everything will be okay before three days, that my ex will return to me before three days, he cast the spell and surprisingly in the second day, it was around 4pm. My ex called me, I was so surprised, I answered the call and all he said was that he was so sorry for everything that happened, that he wanted me to return to him, that he loves me so much. I was so happy and went to him, that was how we started living together happily again. Since then, I have made promise that anybody I know that have a relationship problem, I would be of help to such person by referring him or her to the only real and powerful spell caster who helped me with my own problem and who is different from all the fake ones out there. Anybody could need the help of the spell caster, his email: [email protected] you can email him if you need his assistance in your relationship or anything. I CAN NEVER STOP TALKING ABOUT HIM, HIS EMAIL ADDRESS IS: [email protected] CONTACT HIM NOW FOR SOLUTION TO ALL YOUR PROBLEM
Greetings to everyone reading this testimony i Met a Powerful Spell caster and Psychic from USA by name Dr UZOYA, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on; [email protected] or check him directly on his website at https://driraborspellcaste.wixsite.com/spell-caster or call +13019098775 he is the answer to your problem..
My name is Alicia,from UK.I contacted Mr. Femi okuns, after wasting time on other so called witches & spell-casters. I spent thousands of dollars with no results. Up to the point where I contacted via email: [email protected] I had No contact with my lost love,my husband whom he helped to bring back to me permanently.I realized Dr was special, because,the day and moment i sent our details,He asked me to call my husband, after two months of No contact at all.from that point,everything changed,now my husband is back to me. Mr. Femi has since cast 3 different spells for my friends and all are working fine.the days whilst he is performing the spells, He is there, as he says on his site 24/7. I live in the UK, and the time difference makes it difficult to keep in touch within reasonable hours, but he is always willing to listen and cast spells. I am confident that the results you wish for will manifest when you call or whatsap Mr. Femi okuns,+2347050775844 also via email: [email protected] . I want to share my testimony he is more than a wizard, he is a friend, mentor and spiritual spell-caster, He also specializes on the follows;*Divorce reconciliation*Marriage reconciliation*Business growth*Promotion in your place of work/office*Making banks to grants loans*Cures of diseases such as; Cancer, Hiv Aids, Zika virus, Herpes Disease, Sexual Infection E.T.C.*Pregnancy or Fertility spell*Getting your EX/lover back*Getting your wife/husband back*Getting your boyfriend/girlfriend back
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Read Up My Story Today as my story is a very special and unique one as is not like the numerous advert you always see online and also I want to be 100% sincere and truthful to you that there can never be any other online spell caster that can help you bring back your EX BACK to you other than DR OTIS DARKO as he alone is 100% effective,dynamic and very reliable as 99.9% of other spell casters are not for real and can never help you solve your problems at all instead they will complicate your present condition OK SO BE WISE AND ALSO BE WARNED AND CONTACT DR OTIS DARKO TODAY... MY OWN LIFE STORY / EXPERIENCES IS BELOW:
My Name is Alan Pratte, From USA. I wish to share my testimonies with the general public about what this man called Dr OTIS DARKO has just done for me , this man has just brought back my lost Ex WIFE to me with his great spell, I was married to this woman called Sharon we were together for a long time and we loved our self’s but when I was unable to make her pregnant for me and also give her al she needs she left me and told me she can’t continue anymore then I was now looking for ways to get her back until a friend of mine told me about this man and gave his contact email: [email protected] then you won't believe this when I contacted this man on my problems he prepared this spell cast and bring my lost WIFE back, and after a month she became pregnant for me because he gave me some herbs to take also and when she went for a test and the result stated that she was pregnant am happy today am a FATHER of a baby girl, thank you once again the great DR OTIS DARKO for what you have done for me, if you are out there passing through any of this problems listed below:
(1) If you want your ex back.(2) if you always have bad dreams.(3) You want to be promoted in your office.(4) You want women/men to run after you.(5) If you want a child/PREGNANCY SPELL.(6) You want to be rich.(7) You want to tie your husband/wife to beyours forever.(8) If you need financial assistance.(9) How you been scammed and you want to recover you lost money.(10)Stop Divorce(11) CURE TO ALL KIND OF SICKNESS/DISEASES HERE (12) Winning of lottery(13) Cure To Hiv/Aids(14) LOTTERY/LOTTO SPELL WINNING(15) CURE TO HERPES AND ANY OTHER SEXUALLY TRANSMITTED DISEASES AT ALL...(16) HAVE YOU BEEN SCAMMED BEFORE AND NEEDS TO RECOVER ALL THE DOLLARS/MONEY YOU LOST TO THESE FRAUDSTER ???.. CONTACT OTIS DARKO AS HE HELPED MY FRIEND CALLED LEONARD CAPTAIN FROM USA TO RECOVER BACK THE SUM OF $300,000.00 DOLLARS HE LOST TO SCAMMERS ONLINE
Email... [email protected]...... YOU CAN ALSO ADD HIM UP ON WHATSAPP USING THIS MOBILE NUMBER:+2348077526136
I'm Eunice Mabel , I contracted HIV in 2007, I was told by my doctor that there's no possible cure for HIV. I started taking my ARV's, My CD4 was 77 and viral load was 112,450. I saw a website of Dr ihibor, also I saw a lot of testimonials about him on how he uses herbal medicine to cure HIV. I contacted him and told him my problems, He sent me the herbal medicine and I took it for 7 days after then I went for check-up and I was cured. The medicine has NO SIDE EFFECT, there's no special diet when taking the medicine. He also cure ALS, HEPATITIS B, CANCER, HERPES, DIABETES and lots more. heres my contact +2349050141449... You can reach him on [email protected]. or call/Whatsapp [+2349050141449].....
I NEED A GENUINE SPELL CASTER TO BRING BACK MY EX BACK: CONTACT DOCTOR THOUGH HIS WEB SITE https://ultimatespellcast3.wixsite.com/dr-trust HE IS THE BEST OF ALL
It's a privilege to share this miraculous testimony to the world. I'm Elisa Wendy from England. My husband divorced me 4 months back and i have been filled with remorse for i didn't know what to do to amend issues with my husband. I searched for help on the internet on how i could get help in my marriage and i discovered great testifiers about Doctor Trust who has been progressive with his spells. I got in touch with him and behold, Doctor Trust told me that he will prepare a spell for me that will bring back my husband. I was skeptical but i had no other option but to work with him. 2 days after, my husband called me that he's coming back home and from that day till this moment, we have been living peacefully. He is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my husband,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love. I will highly recommend Doctor Trust to anyone one out there who needs help whatsoever.if you have any problem contact Doctor Trust, i give you 100% guarantee that he will help you!!.
[email protected] or [email protected] or write him on What-app +2348156885231
My name is Tina Morgan , From USA ,and I’m happily married with a lovely husband and three children.I had a very big problem with my husband few months ago,to the extent that he even packed his things away from our house. He left I and and my kids for almost 1 year, and i tried all my possible best and effort to bring him back. I discussed it with a very good friend of mine,and she gave me an advice concerning a spell caster, that he is the only one that can handle my situations and problem,that he’s always ready and able to do anything related to spell casting and helping of the needy, Please every one i would like you all to contact him with his email address ”[email protected]". I never believed in spell casting,but My friend convinced me and i had no choice than to follow my friend advice,because i never dreamed of loosing my lovely Husband. And i contacted him with his email address,and i discussed with him all my problems and worries and so surprisingly,he told me that I’ll get my husband back "1week" after. He asked me some details concerning me and my husband and made some applying items for me to use for one week, I didn't believed Him, until after using the items as instructed by him for one week and the next day,my husband called me to inform me that he is coming back home ….. i was instructed when he call that he is coming home i should not complain or quarrel or fight with him... and i was instructed to cook his favorite food with the love potion he also gave to me... and he became forever mine...So Amazing!! That’s how i got my husband back through spell casting and our relationship was stronger than ever. One of the price i was asked to pay was to tell it to the people around me that problems like this, can always be solved by Dr. olodumare So! my advice to you out there is to visit this same E-mail address,and tell him your problems too, if you are in any condition related to love issue or getting your ex back or and problem at all, please Contact him and have a happy life. you can contact him via email ( [email protected] )
CONTACT HIM TODAY VIA THIS EMAIL ADDRESS: ( [email protected] ) AS HIS POWERS ARE SO STRONG AND VERY EFFECTIVE AND HAS NO BAD EFFECT INSTEAD IT HAVE A VERY GOOD RESULT AFTER CASTING THE SPELL.
I have been through hell and pain,looking for a good andreal spell caster who can help me get my husband backhave been scammed so many times,by someone who claimedto be real spell casters.until i found the real and great spellcaster at the NAVEEN LOVE SPELL who helped me,andsolved all my problems concerning my boyfriend who leftme since ten months ago .and after that i also took myfriend along,who was also having the same problemconcerning her husband,who left her since five monthsago,and the problem was also solved by the sameDR.ODUDU. Can't you see! the real and great spellcaster is here,all you need to do now is to contact thissame address whenever you are in any problemrelated to spell casting.It took me a very long period oftime,before i could get this real and great spell caster.Soright now [email protected] or [email protected] is here,and the bestfor you to solve your problems.......Contact him today on: [email protected] can also add him on whatsapp with this number (+2348169616855)
(1) If you want your ex back(2) if you always have bad dreams.(3) You want to be promoted in your office.(4) If you want a child.(5) Herbal care, like HIV, AIDS, CANCER, KINDLY.. ETC(6) You want to be rich(7) You want to tie your husband/wife to be yours forever.(8) If you need financial assistance.(9) Let people obey your words and do your wish(10) Case solve E.T.C
TESTIMONY OF HOW I GOT CURED FROM GENITAL HERPES SIMPLEX VIRUS WITHIN 7DAYS[with Herbal Traditional medicine]
Hello Everyone out there, I'm Whitney Brookes, from Texas USA. I am here to give my testimony about a Herbalist called Dr GUNDEDE, who helped me in my life. I was infected with HERPES SIMPLEX VIRUS 2 in 2010, i went to many hospitals for cure but there was no solution, so I was thinking how can I get a solution out so that my body can be okay. One day I was at the river side thinking where I could get solution. so a lady walked to me asking me why I was so sad and i opened up all to her telling her my problem, she told me that she could help me out, she introduced me to a doctor who uses herbal medication to cure HERPES SIMPLEX VIRUS 2 and gave me his email, so i mailed him. He told me all the things I needed to do and also gave me instructions, which I followed properly. Before I knew what is happening after two weeks the HERPES SIMPLEX VIRUS that was in my body got vanished . so if you are also heart broken and also need a help, you can also email him via; [email protected]
He also have a herbal cure for HIV,CANCER,ASTHMA,IMPOTENCE,BARENESS/INFERTILITY ...
CURE YOUR SELF FROM THAT ILLNESS TODAY,DON'T SPREAD YOUR INFECTION OR DISEASES TO OTHER PEOPLE.Contact him today and you will have a testimony...Good luck
I am very grateful to this great spell caster who brought my ex husband back to me. This testimony is a true story and my name is renato Sanchez. When i came in contact with this man was also through a testimony written about him and i have also encountered many testimonies about how he has been helping others with their life. To get back with an ex is one of the most inner most feeling many people would love to experience especially as those memories with our ex always cloud our mind when someone else does some of those things our ex used to do. I was a single parent for almost 6 years and though my ex husband was far away from my kids i still wish someday he will return back to me. This spell doctor i encountered known as Doctor UZOYA helped me with my wishes and i am happy to say that i am back with my ex husband and i am very happy sharing the testimony with everyone so that they too can meet this great doctor and solve their problems. I don't know what others might feel about getting their ex back in their life but i always know there is a blessing in disguise with just a single reconnection with an ex. If you want to successfully get back with your ex, contact this great spell caster at [email protected] you can also call him on +13019098775 and share an awesome testimony too just like me.
Hi every one i am Janet Jordan and i am from USA had i fight with my husband about three years ago and since then he has refuse to talk to me or even care about our family affair he abandoned us and went after another woman leaving me with three kids two girls and a boy,so last weeks a friend of my introduced me to a spell caster called Dr. Ozil who helped me brought my husband back many think he is a scam but i promise you that he is nothing compare to that not even close so if you have some problems in having or getting your ex back you can contact the great and wonderful spell caster Dr. Ozil in this address and he be willing to help you via; [email protected] or [email protected] or view and contact him on his website: http://drozilsolutionhome.webs.com/ or call: +2348130092712
What a wonderful and a straight forward spell caster that has brought back joy and happiness into my life after i saw a post on how he helped a lady called Sara,i decided to contact him for help when i told this Godsent man called Dr Grant on how my lover left me for 2years without calling or texting me,When i shared this my sad experience with Dr Grant he said everything would be okay within 3days i was like am i sure what this man is saying is real,So i decided to give a try and what even surprised me the most at first i was also thinking he was a scam i taught he was like other spell caster who come online to add pain to peoples pain not knowing there feelings but to make money,But this great man called Dr Grant is never like that his own is for good and make people to be happy with the one they love,Am just so happy,Even before the 3days i just got a call from a man who has left me for 2years saying he his sorry and that he wants me back to his life i was so happy,He invited me for a dinner which i meant him there and we both talked and he said he wants to prove that he would never leave me for any other woman he engaged me and also made me had access to all his account am so happy all thanks goes to this great man called Dr Grant a man who has brought back joy to my life,Please friends that needs help i would advice and swear that Dr Grant is the right man and not those fake ones who are online to make money and not to help here is Dr Grant’s private mail is [email protected]
My name is Mary Silva, I just split up with my boyfriend of 7 years. We were great together, completely in love and talked about marriage and growing old together, we traveled the world together, everyone knows us as a perfect couple, we never fought always laughed together… however the last 2 years have been very difficult. We have been living together for over 5 years but the last 2 years have been a stressful living situation as I had to take in a friend that was in need. I also started my own business which kept me extremely busy. I am completely devastated, hurt, lost and just want him back. I love him completely. We had a few discussions before he left about trying to work things out, and he felt I took him for granted…I had stopped treating him with appreciation and love. I agreed that I have to be more romantic and be an equal partner in the relationship. Over the course of the following days I showed him love and appreciation, treated him like I had done in the past however he still wanted out. he said he needed time and space to figure out what he wants. I told him I would not give up on us but I would give him space that he wanted. he said he still loved me and care a lot about me but was just hurting.i was totally down because i need my man back one day a friend of my introduce me to a spell caster online who lift me up gave me hope and with-in two days after his spell my man who told me he needed space, call me and come back home to me i must say today we are happily married not only married but have two lovely kids all thanks to Dr happy a father and a real spell caster.
Contact email address [email protected] and Whats-App him +2348133873774Also contact and know more about his service at http://happyspelltemple.webs.com/
Now i believe high priest tokubo is a real and sincere spell caster because he has helped me to bring my ex back withing 24 hours of casting the spell for me, my ex break up with me two years ago December 28 2015 and ever since then i have been trying all my possible best to get him back i have tried many ways to get him back but there is no positive result. i came across high priest tokubo contact when a man testify how he helped him to get his wife back and i also saw another comment saying high priest tokubo helped him to get his ex girlfriend back and another woman testify that he also prepare spiritual charm to bring ex back and also prepare spiritual charm to strive for success in business. you can reach high priest tokubo on([email protected])once again a big thank to high priest tokubo and i hope you see this my testimony online thank you for your great work.
God bless Dr.Osauyi for his marvelous work in my life, I was separate with my husband for the pass 3 year now and I wasn't satisfied i needed to get my husband back because the life of loneliness was so terrible for me and my 2 kids I searched about some possible spell caster that can bring back lover and i then come across Dr Osauyi i saw a comment about Dr.Osauyi,how he bring lovers together with his spell caster I contacted him and he guided me. I asked for solutions, he started the remedy for my spell he ask for some information which i sent to him and also told me that we need to buy some items for the spell to work for me for good and which i sensed the money to him after 48 hours my husband call me and start bagging me for forgiveness for all that happens i am so happy right now that we back again as one family for any one that may need the help of the great spell caster you can contact [email protected] or call +2347064294395
I did not believe is possible that HIV can be cured because i have been scammed so many times not until i came across this great man dr agbebaku who helped me at first I never believe all this comment and post about him and I was very sick because i have been infected with HIV for the past seven years,just last year I keep reading the testimony about this man named dr agbebaku they said that the man is so powerful he have cured different type of diseases, I keep monitor the post of some people about this man and I found out that he was real so i decided to give him a try I contacted him for help and he said he was going to help me get my cure that all i needed to do was to send him money to prepare the medication after which it will be sent to me via dhl service which i did to my greatest suprise the medication was sent to me he gave me instructions on how to use it that after three weeks i should go for check up at first i was shocked when the doctor told me that i was hiv negative i ask the doctor to check again and the result was the same that i was now free from hiv at of the shock i decided to come and share my testimony with you for those who think there is no cure for hiv the cure as finally come stop doubting and contact dr agbebaku via email [email protected] he will help you out you can also wattssap him via +2349053099479 he said he also as cure for DIABETICS,HERPES,INFERTILITY;ALS,ANCE PIMPUS,LUPUS VIRUS AND PENIS AND BREAST ENLARGEMENT
TESTIMONY OF HOW I GOT CURED FROM (HERPES SIMPLEX VIRUS) BY THE GREAT HEALER DOCTOR ADEMUSHI.Hello everyone my name is Julia farbio I am here to testify of the great work Dr Ademushi did in my life with the help of his herbal traditional medicine. I have been suffering from Herpes since 2015 and ever since then I have been looking for solutions on how to cure it. I have being to different hospitals but I discovered none of them was working out for me that I was just wasting money. So I decided to live everything to God, then I was taking my medications to sustain my life and reduce the pains. So on this particular Sunday morning after coming back from service I sat down and begin to ask myself questions, then my phone rang behold it was my sister calling which I answered the call and she started telling me that a friend of hers who has been suffering from same herpes has just been cured by Dr Ademushi the herbal medicine man. Then I was over whelmed, even though I was a little bit confused so she encourage me to also give Dr Ademushi a trial and see for myself then I told her to help me get his contact and forward them to me, So immediately she got the contact of Dr Ademushi and send them to me, immediately I contacted Dr Ademushi and he told me everything I needed to do to enable me get the herbs for my medications and immediately i did that which he instructed me to do without wasting anytime. And here I am today sound and healthy without any symptoms of herpes or any sickness in my body system. And he also told me that there is no sickness on this planet earth that doesn’t have a cure that with the help of herbs he can cure any sickness or disease you can ever think off. So if you are suffering from HERPES or any other sickness or disease and you need a cure you can also contact Dr Ademushi with this Email Address at [email protected] or whatsapp or call him:+2348131652960 so you can also receive your healing and share your testimony.
My name is Monica, and I base in USA...My life is back!!! After 1 years of Broken marriage, my husband left me with two kids . I felt like my life was about to end i almost committed suicide, i was emotionally down for a very long time. Thanks to a spell caster called Dr DONKBABA, which i met online. On one faithful day, as I was browsing through the internet,I came across allot of testimonies about this particular spell caster. Some people testified that he brought their Ex lover back, some testified that he restores womb,cure cancer,and other sickness, some testified that he can cast a spell to stop divorce and so on. i also come across one particular testimony,it was about a woman called Sonia,she testified about how he brought back her Ex lover in less than 2 days, and at the end of her testimony she dropped Dr DONKBABA e-mail address. After reading all these,I decided to give it a try. I contacted him via email and explained my problem to him. In just 48hours, my husband came back to me. We solved our issues, and we are even happier than before Dr DONKBABA, is really a gifted man and i will not stop publishing him because he is a wonderful man... If you have a problem and you are looking for a real and genuine spell caster to solve all your problems for you.contact him email [email protected] CELL PHONE +2348151795598
Hi My name is 'Niall Christopher' just want to share my experience with the world on how i got my love back and saved my marriage... I was married for 7years with 2kids and we lived happily until things started getting ugly and we had fights and arguments almost every time... it got worse at a point that she filed for divorce... I tried my best to make her change her mind & stay with me cause i loved her with all my heart and didn't want to loose her but everything just didn't work out... she moved out of the house and still went ahead to file for divorce... I pleaded and tried everything but still nothing worked. The breakthrough came when someone introduced me to this wonderful, great spell caster who eventually helped me out... I have never been a fan of things like this but just decided to try reluctantly cause I was desperate and left with no choice... He did special prayers and used roots and herbs... Within 7 days she called me and was sorry for all the emotional trauma she had cost me, moved back to the house and we continue to live happily, the kids are happy too and we are expecting our third child. I have introduced him to a lot of couples with problems across the world and they have had good news... Just thought I should share my experience cause I strongly believe someone out there need's it... You can email him via [email protected] Don't give up just yet, the different between 'Ordinary' & 'Extra-Ordinary' is the 'Extra' so make extra effort to save your marriage/relationship if it's truly worth it. website http://eromosalspiritualtemple.webs.com or call him with +2348161850195
It took me 24 hours to confirm the spell power of HIGH PRIEST TOKUBO SPELL TEMPLE which brought back my ex lover to me till today. I never believed spell casting could bring my ex back after been fooled by other spell casters not until i went on a research on-line about spell-ism and i got a testimony from a lady who high priest tokubo help to get her husband back and another guy who got his ex back by High preist tokubo, no matter what happen between you and your ex that lead to break, he will bring your ex back to you and he also prepare spiritual charms to help get divorce lovers back and to help make more profit in business. His E-mail address: ([email protected] or whatsapp him on; +2349037990322)Am really full of happiness, may God continue to Bless this man for his hardwork and sincerity Please, once your ex is back, also share the good news to other people. Together we must fight to stay happy.
It took me 24 hours to confirm the spell power of HIGH PRIEST TOKUBO SPELL TEMPLE which brought back my ex lover to me till today. I never believed spell casting could bring my ex back after been fooled by other spell casters not until i went on a research on-line about spell-ism and i got a testimony from a lady who high priest tokubo help to get her husband back and another guy who got his ex back by High preist tokubo, no matter what happen between you and your ex that lead to break, he will bring your ex back to you and he also prepare spiritual charms to help get divorce lovers back and to help make more profit in business. His E-mail address: ([email protected] or whatsapp him on; +2349037990322)Am really full of happiness, may God continue to Bless this man for his hardwork and sincerity Please, once your ex is back, also share the good news to other people. Together we must fight to stay happy.
My name is Monica, and I base in USA...My life is back!!! After 1 years of Broken marriage, my husband left me with two kids . I felt like my life was about to end i almost committed suicide, i was emotionally down for a very long time. Thanks to a spell caster called Dr Mohammed, which i met online. On one faithful day, as I was browsing through the internet,I came across allot of testimonies about this particular spell caster. Some people testified that he brought their Ex lover back, some testified that he restores womb,cure cancer,and other sickness, some testified that he can cast a spell to stop divorce and so on. i also come across one particular testimony,it was about a woman called Sonia,she testified about how he brought back her Ex lover in less than 2 days, and at the end of her testimony she dropped DR DONKBABA e-mail address. After reading all these,I decided to give it a try. I contacted him via email and explained my problem to him. In just 48hours, my husband came back to me. We solved our issues, and we are even happier than before DR DONKBABA, is really a gifted man and i will not stop publishing him because he is a wonderful man... If you have a problem and you are looking for a real and genuine spell caster to solve all your problems for you.contact him email [email protected] CELL PHONE +2348151795598
My name is Monica, and I base in USA...My life is back!!! After 1 years of Broken marriage, my husband left me with two kids . I felt like my life was about to end i almost committed suicide, i was emotionally down for a very long time. Thanks to a spell caster called Dr Mohammed, which i met online. On one faithful day, as I was browsing through the internet,I came across allot of testimonies about this particular spell caster. Some people testified that he brought their Ex lover back, some testified that he restores womb,cure cancer,and other sickness, some testified that he can cast a spell to stop divorce and so on. i also come across one particular testimony,it was about a woman called Sonia,she testified about how he brought back her Ex lover in less than 2 days, and at the end of her testimony she dropped DR DONKBABA e-mail address. After reading all these,I decided to give it a try. I contacted him via email and explained my problem to him. In just 48hours, my husband came back to me. We solved our issues, and we are even happier than before DR DONKBABA, is really a gifted man and i will not stop publishing him because he is a wonderful man... If you have a problem and you are looking for a real and genuine spell caster to solve all your problems for you.contact him email [email protected] CELL PHONE +2348151795598
With so much in my heart i am here to express myself on how Dr. Ozil saved my marriage from divorce. Myself and my husband were having some misunderstanding and it was tearing our marriage apart to the extend my husband was seeking for a divorce. So i have no option than to go to the internet to seek for solution to my problem it was there i came across Dr.Ozil details and about how he has helped a lot of people by restoring there relationship. I contact Dr.Ozil and in less than 48 hours my husband cancelled the divorce papers. Now myself and my husband live together in peace and harmony all thanks to Dr.Ozil for saving my marriage from breaking up. Dr.Ozil contact information are via email: [email protected] or [email protected] call his phone number: +2348130092712 or view and contact him on his website: http://drozilsolutionhome.webs.com/
I want to testify a great testimony of how Dr.SABU help me,i was heartbroken because i had very ( SMALL PENIS ) not big enough and I couldn't satisfy a real lady in bed and the worse of my problems was that i cum on the sight of a ladies vagina. I battled this problems since i was teen and still couldn't get any good victory. I was online searching for guaranteed results providing products to enlarge the size of my penis when i saw a review of Dr.SABU Penis Enlargement Skills posted by so many poeple that he has help to enlargement penis,i read through the comment and emailed the contact ([email protected]) on it. The doctor called me some few moments later and attended to me more better than i expected. I was convinced and i made purchase of his herbal medicines for Penis Enlargement and Quick Ejaculation. The medicines was sent to me BY UPS DELEVERY and i received it at my home, i applied it as instructed and just few weeks of using the herbal medicines i got a perfect good size of penis. I now measure about 16.0 cm (8.4 inches) when flaccid and 24.3 cm (9.1 inches) when erect. Am so happy and comfortable with the new size of my penis, i feel no discomfort or whatever. Am so happy with this fast results. You too can reach Dr.SABU for same Penis Enlargement medicines on [email protected] or his WHATSAPP/CALLS +2347057624993 or his websites: https://drsabuspelltemple.wixsite.com/dr-sabu
I am a spell caster called Dr Ajayi Ololo, I was the chief priest to this great temple called OLOLO at the age of 18years, and now i am almost 60 years old man. I am telling this to the world to make them know that this temple has been in existence for years now and we have been durable and truthful to all. Our work here is helping anybody to getting there heart desire, Over here in this temple we grant anything you want us to do for you and when you come to this temple you must see result of our work. NOTE: TO THE ONES THAT HAS BEEN DECEIVED BY THOSE ONLINE SCAMMERS, DO NOT COMPARE THIS GREAT TEMPLE WITH ANY OTHER SPELL CASTER THAT HAS RIPPED YOU OFF YOUR MONEY BY CLAMMING TO BE SPELL CASTERS. THEY ARE ALL FAKE AND SCAMMERS ONLINE THERE TO RIP WERE THEY DID NOT SOW, COME TO OLOLO SPELL CASTING TEMPLE AND GET YOUR HEART DESIRE WITHOUT STRESS. In this temple, we work with the spirits in the spiritual realm and they tell us how to solve any problem that you are facing and also i will want you to NOTE: THAT THIS SPELL CASTING DO NOT HAVE ANY SIDE EFFECT ON ANYONE, BUT JUST TO GET YOUR HEART DESIRE AND BE HAPPY. I Dr Ajayi Ololo urge you all that has the following problem 1.} We heal barrenness in women and disturbing menstruation 2.} Get you marriage to the lover of your choice 3.} Guarantee you win the troubling court cases & divorce no matter how what stage 4.} Ensure success in work and business 5.} Mental illness & bewitched 6.} Can?t sleep at night or walking at night 7.} Recover stolen property and whereabouts of people that hurt you. 8.} Bring supernatural luck into 9.} Extreme protection for those doing dangerous jobs like security guards, Bank manager, cash transporters, To mention but a few.11. Get your scam money back 12.} Bring back lost lover, even if lost for a long time 13.} Remove bad spells from homes, business & customer attraction etc. 14.} Get promotion you have desired for a long time at work or in your career.16.} Remove the black spot that keeps on taking your money away 17. Find out why you are not progressing in life and the solution 18.} Eliminate in family fights 19.} Ensure excellent school grades even for children with mental disabilities 20.} Stop your marriage or relationship from breaking apart 21.} I destroy and can send back the Nikolos if requested 22.} . COME HERE AND SOLVE ALL YOUR PROBLEMS HERE AND GET ALL YOUR HEART DESIRES HERE WITHOUT ANY-SIDE EFFECT ON IT. ANY KIND OF PROBLEM YOU ARE FACING IN LIFE SEND YOUR EMAILS TO: [email protected] . to get it solved in the next 5 to 6 hours.
GET YOUR EX BACK WITH THE HELP GREAT SPELL CASTER CALLED DR SAMBO VISIT HIS WEBSITE BELOW; http://divinespellhome.wixsite.com/drsambo
Good dey to you all , my name is moore kathleen i want to share a brief testimony on how Great Dr Sambo helped me re-unite my long gone relationship . I thought I have lost it all until I saw a post abort this great spell caster called Dr Sambo , I contacted him and told him about my story “few years ago , my husband and I were deeply in love I thought life was that easy , because I was with the man of my dream , until his secretary took him from me , he stopped loving me ,we separated because of the incidence , I couldn’t take it anymore , I count live with him know he has someone else out there , but I still loved him . until few week back I contacted Great Dr Sambo who helped me bring back my long gone husband , he apologyzed to me and each day that passes he love me more . if you are out there , because I know a lot of people are experiencing difficulties in their marriage or any form of relationship , just contact Dr Sambo and he will definatily help you as he has done for me here are his details .
I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: [email protected] or you can whatsApp his line:+2348107155060 or see more at http://azaspellcaster.wixsite.com/spell-caster
I was hurt and heart broken when a very big problem occurred in my marriage seven months ago, between me and my husband . so terrible that he took the case to court for a divorce. he said that he never wanted to stay with me again,and that he didn't love me anymore. So he packed out of the house and made me and my children passed through severe pain. I tried all my possible means to get him back,after much begging,but all to no avail.and he confirmed it that he has made his decision,and he never wanted to see me again. So on one evening,as i was coming back from work,i met an old friend of mine who asked of my husband .So i explained every thing to him,so he told me that the only way i can get my husband back,is to visit a spell caster,because it has really worked for him too. So i never believed in spell,but i had no other choice,than to follow his advice. Then he gave me the Email address of the spell caster whom he visited. [email protected]. So the next morning,i sent a mail to the address he gave to me,and the spell caster assured me that i will get my husband back the next day. What an amazing statement!! I never believed,so he spoke with me, and told me everything that i need to do. Then the next morning, So surprisingly, my husband who didn't call me for the past 7 months, gave me a call to inform me that he was coming back. So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and my children. Then from that day,our relationship was now stronger than how it were before,by the help of a spell caster. So, i will advice you out there if you have any problem contact Pristbacasim, i give you 100% guarantee that he will help you.. Email him at: [email protected]
I was hurt and heart broken when a very big problem occurred in my marriage seven months ago, between me and my husband . so terrible that he took the case to court for a divorce. he said that he never wanted to stay with me again,and that he didn't love me anymore. So he packed out of the house and made me and my children passed through severe pain. I tried all my possible means to get him back,after much begging,but all to no avail.and he confirmed it that he has made his decision,and he never wanted to see me again. So on one evening,as i was coming back from work,i met an old friend of mine who asked of my husband .So i explained every thing to him,so he told me that the only way i can get my husband back,is to visit a spell caster,because it has really worked for him too. So i never believed in spell,but i had no other choice,than to follow his advice. Then he gave me the Email address of the spell caster whom he visited. [email protected]. So the next morning,i sent a mail to the address he gave to me,and the spell caster assured me that i will get my husband back the next day. What an amazing statement!! I never believed,so he spoke with me, and told me everything that i need to do. Then the next morning, So surprisingly, my husband who didn't call me for the past 7 months, gave me a call to inform me that he was coming back. So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and my children. Then from that day,our relationship was now stronger than how it were before,by the help of a spell caster. So, i will advice you out there if you have any problem contact Pristbacasim, i give you 100% guarantee that he will help you.. Email him at: [email protected]
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Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
OMG!!! I am so proud and happy to be out here sharing your work Dr. Zaba. I just can't believe this now my ex Husband is really back to me on his knees presenting a ruby rose begging me to take him back and he was feeling regretful and sorry for leaving me and for causing me pains after the divorce which occurred last year. And this whole miracle happened after i ordered an urgent 24hours Dr.Zaba powerful spell which he cast on me and my husband. Sir I am the happiest woman today in this whole wide world. Dr. Zaba you really did it..Yes.. Its a miracle and everlasting pleasure and cheerfulness for me and my family today.. I am so happy now and i dont know how much to convey my thankfulness and appreciation to you sir. And to the whole world, contact him if you need urgent help now because its guaranteed that he will help you. website http://zaba24hoursspell.wixsite.com/zabaspell or email him [email protected] OR [email protected] or call +2347087462033
I HAVE ASKED FOR HELP FROM ALMOST ALL THIS SO CALLED PEOPLE PARADING THEMSELVES TO BE SPELL CASTERS, AT THE END I WAS RIPPED OFF OF MY HARD EARNED MONEY WITHOUT ANY RESULT. TO THE EXTENT THAT I SAW SOME OF THEM REWRITING WHAT I WROTE ON ONLINE SO THANK YOU Dr ALUDA MOST OF THESE PEOPLE ARE TELLING YOU FAKE TESTIMONIES AND FOR YOU TO BEWARE AND BE-WISE TO AVOID BEEN RIPPED OFF, YOU HAVE TO MAKE THE RIGHT CHOICE AT ONCE SO CONTACT THE ONLY REAL AND APPROVED SPELL CASTER IN THE WORLD CALLED Dr ALUDA. AS HE IS THE ONLY MAN YOU CAN ACTUALLY TRUST ON SPELLS I WOULD HAVE BROUGHT HIM RIGHT HERE IN THE STATES IF POSSIBLE. HE DID A WHOLE LOT OF THINGS WHICH I WONT REALLY MENTION HERE BUT THE MOST OF WHAT HE DID FOR ME WAS TO HELP ME GET MY HUSBAND BACK TO ME .OH HE IS GREAT SPELL CASTER AND I MET HIM IN HIS TEMPLE THAT WAS DIRECTED TO ME BY A CHINESE FRIEND OF MINE THAT FOLLOWED ME FOR A CONFERENCE MEETING IN AFRICA. IM STILL VERY MUCH INDEBTED TO Dr ALUDA FOR SAVING MY LIFE AND MY MARRIAGE. I WANT TO LET YOU ALL KNOW THAT YOU DONT HAVE TO TRY ANYBODY ELSE EXCEPT THIS GREAT SPELL CASTER CALLED Dr ALUDA HIS EMAIL CONTACT IS: [email protected] or call or whatsapp him on +2348063930531JASMINE JENNIFERFROM USAREGARDS.
DOCTOR ALUDA IS REAL!!! EVERY OTHER TESTIMONY OR STORY YOU READ HERE IS FAKE.I AM TAKING MY TIME TO SHARE THIS TO MAKE SURE PEOPLE DON'T GET SCAMMED NO MORE.CONTACT DR ALUDA HE IS THE SURE SOLUTION TO YOUR PROBLEM.I NEVER BELIEVED IN LOVE SPELLS UNTIL I MET THIS SPELL CASTER. HE IS REALLY POWERFUL AND CAN HELP CAST SPELLS TO BRING BACK ONE'S GONE,LOST,MISBEHAVING LOVER AND RESTORE ONE'S WOMB OR SPELL FOR A GOOD JOB.MY HUBBY LEFT ME BECAUSE I LOST MY WOMBAND MY LIFE WAS UPSIDE DOWN COS OUR RELATIONSHIP HAS BEEN ON FOR 2YEARS... I REALLY LOVED HIM, BUT HIS MOTHER WAS AGAINST US AND HE HAD NO GOOD PAYING JOB. SO WHEN I MET Dr ALUDA WHICH A FRIEND REFERRED ME TO, I TOLD HIM WHAT HAPPENED AND EXPLAINED THE SITUATION OF THINGS TO HIM.AT FIRST I WAS SKEPTICAL AND DOUBTFUL BECAUSE I HAVE BEEN SCAMMED, BUT I GAVE IT A LAST TRY. AND IN 4 DAYS MY HUBBY CALLED ME APOLOGIZING THAT EVERYTHING HAD BEEN SETTLED WITH HIS MOM AND FAMILY. I WAS VERY HAPPY BUT I LATER ASKED DR ALUDA ABOUT MY WOMB AND HE TOLD ME NOT TO WORRY NO MORE. MY HUSBAND GOT A NEW JOB AND WE GOT MARRIED. HERE I AM NOW WITH MY WOMB INTACT. I SAID I WONT TESTIFY UNTIL I GIVE BIRTH.WELL WE ARE HAPPILY MARRIED NOW AND WE ARE EXPECTING OUR SECOND BABY. IN CASE ANYONE NEEDS THE SPELL CASTER FOR SOME HELP, HIS EMAIL ADDRESS IS; [email protected] OR CALL AND WHATSAPP HIM ON +2348063930531
I read some testimonies about a love spell caster called Dr.OBILUJESU on how he has helped lots of people in bringing back their ex lovers within 24 hours, Sincerely I was just thinking if that was real and if this man could really help bring back my lover whom I love so much. I decided to contact him because I love my boyfriend very much and we have been apart for a couple of months I really missed him so much, I have tried all other means to get him back but I couldn't. I contacted Dr.OBILUJESU and he told me that my ex will come back to me in the next 48 hours,Dr.OBILUJESU released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. My ex is now back to me again.As I`m writing this testimony right now I`m the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up.All thanks goes to Dr.OBILUJESU for the excessive work that he has done for me by helping me to get back with my ex boyfriend. I would like to drop Dr.OBILUJESU mail address and hope you see this testimony and contact him if you have a lover that you really want back so badly, His mail: [email protected] or [email protected] web sit: http://drobilujesuspellca.wixsite.com/drobilujesu OR call him +2348143481000
Hello readers, I USED LOVE SPELL RITUAL TO GET BACK THE MAN I LOVE. It was unfortunate that his ex gf decided to contact him to get him back the moment we planned to build a family together. I was hurt for seeing him confused and indecisive after I sacrificed time, energy, heart, life, money, soul for him only for him to pay me with heart break by saying that he is now unsure of our relationship. He developed an attitude of becoming cold and distant towards me. I know he loves me very much for the past 4 years and I love him more before his ex tries to steal his heart. I was advised by a good friend of mine to fight back both physical and spiritual if I truly love him. Her advised encouraged and lead me to posts and site that talks about love spell from Dr. Wakina https://drwakinalovetemple.webs.com, it was my first attempt but I took the opportunity that came once which Dr. Wakina provided via his email [email protected]. It was said and done by Dr. Wakina, I never knew his ex used evil powers on him. I fought a good fight which I and my lover are very proud of after his freedom and decision to continue with me until we get married, his ex has vanished from his life. I never doubted Dr. Wakina for once, he proved his worth. Thanks for your time…
My name is Bart Cruz am from U.S.A. I want to share a testimony of how Dr Okusu herbal oil saves me from shame and disgrace, my penis was a big problem to me as the size was really so embarrassing,I wasn't able to satisfy my wife any more and she became so tired of my sex life issues, and she was getting angry every time we make love, because i always cum quickly in 10minutes and it was very frustrating,but i never loose hope, i went out to search on a way to make my penis big and lastly in bed. miraculously i saw a testimony of how Dr Okusu has helped my fellow men, on getting help and making their penis bigger and stronger, i wrote to Dr Okusu for his herbal oil and he told me once i used his oil in 5days my penis will be 9inches bigger be able to satisfy my wife longer in bed and also be able to control my ejaculation to cum whenever i want to. to my shocking surprise after using the herbal oil 5days after i was shocked when my penis grow bigger to 9inches exactly as i wanted and now my wife enjoy me on bed and even crave for more. My fellow friends suffers no more, if you need your penis issue to be solved you can contact Dr Okusu today and right now on his email:[email protected] you can as well visit his websites:https://drokususolutiontem.wixsite.com/dr-okusu or call him or what-app him through Mobile:+2349037835860
I was hurt and heart broken when a very big problem occurred in my marriage seven months ago, between me and my husband . so terrible that he took the case to court for a divorce. he said that he never wanted to stay with me again,and that he didn't love me anymore. So he packed out of the house and made me and my children passed through severe pain. I tried all my possible means to get him back,after much begging,but all to no avail.and he confirmed it that he has made his decision,and he never wanted to see me again. So on one evening,as i was coming back from work,i met an old friend of mine who asked of my husband .So i explained every thing to him,so he told me that the only way i can get my husband back,is to visit a spell caster,because it has really worked for him too. So i never believed in spell,but i had no other choice,than to follow his advice. Then he gave me the Email address of the spell caster whom he visited. [email protected]. So the next morning,i sent a mail to the address he gave to me,and the spell caster assured me that i will get my husband back the next day. What an amazing statement!! I never believed,so he spoke with me, and told me everything that i need to do. Then the next morning, So surprisingly, my husband who didn't call me for the past 7 months, gave me a call to inform me that he was coming back. So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and my children. Then from that day,our relationship was now stronger than how it were before,by the help of a spell caster. So, i will advice you out there if you have any problem contact Pristbacasim, i give you 100% guarantee that he will help you.. Email him at: [email protected]
It's unbelievable how fortunate I feel after finding your website. For the past 6 months, I have been so depressed after losing my husband to another woman. My money situation worsened so much that I thought I'd have to file for bankruptcy. I had a huge amount of debt and I didn't know what to do. Out of complete and total desperation, I contacted many of those so-called individuals who promised powerful magic, witchcraft or black magic. None of them worked and none were as wonderful, affectionate and warm as Dr Marvel has been. He is definitely different from the others and I felt immediate hope and strength from hearing about the promises he had to offer. He carries an air of purity and divine strength that is as pure as fresh snow on the ground. I requested Dr Marvel's most powerful spells and I was relieved right away that I had someone to solve my problems for me. His spells worked wonders and I am now back with my husband and my money troubles resolved itself after winning the lottery. Dr Marvel, I have no idea what I would have done without you being there to help me out. you can contact him via email [email protected] or directly on his website www.extremespellhome.webs.com. Telephone +1 669 221 3962 . With best regards, Erin
I am very grateful to Dr. Believe The Mighty spell caster from Africa for helping me in my problem, I was almost losing hope in life when my lover of 4 years ended our relationship without any reason and I consulted an old friend of mine and explain my situation to her and she said she was in same situation 2 weeks back but a wonderful man who was sent to help people helped her out of the problem and I thought it was all a joke and I ask her if it was real and she said if it wasn't real so how did her lover come back to her, then I decided to give Dr. Believe a try because I had nothing to loose in trying and I contacted him on his email address which is [email protected] and he replied to my email very well.. and to cut the long story short Dr. Believe brought back my Lover in 48 hours.. I'm very happy he brought my man back to me and my lover promise that he will never and ever leave me again. Thanks to Dr. Believe contact him now for any kind of via email: [email protected] or [email protected] or view his website: https://authenticspellcast8.wixsite.com/authenticspellcast and call his mobile number: +2348159740630. Sherry Gibbs from Canada.
I NEED MY EX BACK DESPERATELY EMAIL DR. TRUST THROUGH HIS WEBSITE FOR URGENT AND FAST HELP: https://ultimatespellcast3.wixsite.com/dr-trust
HI AM Sharon Poore Sidwell!! from Australia. with this great tears of joy and happiness in my heart today I want to give an unlimited thanks to this man Called Dr. Trust. I am happy today because my husband is now back with full of love, My husband left me for another woman,just because we had a little misunderstanding. he left me and promise he will never come back to me again. i cry all day looking for a genuine spell caster to help me bring him back. Then i saw a testimony share on the internet by miss Karen. i email him and tell him my problems with my ex husband, and he promise to bring him back home within 2 days. indeed he is wonderful he help me to get my husband back within 2days which he promise and my husband return to me and promise to love me forever and beg me for the pain he caused me. I will forever be grateful to you Dr Trust and i will not stop to publish your name on the internet for people to see how truthful you are. And I want to say to everyone on this site that I will always remain thankful to this man Dr. Trust for all he has done for me,My husband is back with love,caring,truthful and he promise to love me forever.I have made a promise that every body I know will never have a relationship problem, via email ([email protected] or [email protected] call him +2348156885231)
I am Amanda john. I want to write a testimony of a spell caster who helped me. I have been a frustrated woman before i met doctor Alabar. I must recommend this spell caster to everyone who needs help because he does his work. I want to appreciate him for the help he gave to me when my daughter was dying sick of epilepsy because she is blood type SS and this makes her get sick easily. When i contacted this spell caster i never knew he was the one to heal my daughter because my problem has been going on since my girl was a baby. I don't know if you reading this can understand my pain but today its a different story. All thanks to Doctor Alabar. This man is easy to contact if you want him to help you with his power, his personal email is [email protected] or contact him on +233263660856
Am so grateful to Dr TWINS for what he did for me , it is over two years when i contacted this deadly disease, which i don’t know how it happened, this disease “HIV Virus” is seriously spreading all over the united state, i did my possible best for me to be cured, i contacted so many doctors both on net and in my residence non could help me most especially the doctors i contacted on net they failed me by stealing my money and nothing change i have used so many drugs but nothing changed, not until a day, a friend of mine introduced me to this great man who cured me with his powerful medicine.i am indeed thankful to you Doc may God continue to empower you with your good work. i know very well that this disease is all over the united state and some nearby countries so let help our self by reaching Dr TWINS on his mail for he is capable of curing this disease. and i asked him if there is a cure for HPV, he immediately gave me a number of a man he cured recently from HPV, that i should call to confirm from her my self and for real he cured her, am so amazebn13. you can mail Dr TWINS just as i was introduce to him on mail. [email protected] OR rather reach him on mobile +2348077544690
Am so grateful to Dr TWINS for what he did for me , it is over two years when i contacted this deadly disease, which i don’t know how it happened, this disease “HIV Virus” is seriously spreading all over the united state, i did my possible best for me to be cured, i contacted so many doctors both on net and in my residence non could help me most especially the doctors i contacted on net they failed me by stealing my money and nothing change i have used so many drugs but nothing changed, not until a day, a friend of mine introduced me to this great man who cured me with his powerful medicine.i am indeed thankful to you Doc may God continue to empower you with your good work. i know very well that this disease is all over the united state and some nearby countries so let help our self by reaching Dr TWINS on his mail for he is capable of curing this disease. and i asked him if there is a cure for HPV, he immediately gave me a number of a man he cured recently from HPV, that i should call to confirm from her my self and for real he cured her, am so amazebn13. you can mail Dr TWINS just as i was introduce to him on mail. [email protected] OR rather reach him on mobile +2348077544690
i am Benita Gift from usa am here to testify about Doctor EKIADOLA the Great spell caster that brought back my husband to me when i thought all hope was lost. Doctor EKIADOLA used his powerful spell to put a smile on my face by bringing back my man with his spell, at first i thought i was dreaming when my husband came back to me on his knees begging me to forgive him and accept him back and even since then he loves me more than i ever expected so i made a vow to my self the i will let the World know about Doctor EKIADOLA he is a God on earth. Do you have problems in your relationship ? have your partner broke up with you and you still love and want him back ? Do you have problem with your finance ? or do you need help of any kind then contact Doctor ekiadola today for i give you 100% guarantee that he will help you just as he helped me. to contact him on this email;[email protected] or call and watsapp +2349057915075
I'm Philps Goodman from US and i quickly want to talk bout the herbal man that was able to cure my Herpes disease with his herbal medicine after spending a whole lot of my hard earned money on drugs from one hospital to the other. I noticed on time that i was was having this virus cos i immediately went to the hospital for a test when i started feeling some unusual pains and encountering some miserable symptoms in my body and i was diagnosed of Herpes simplex 2, i was so perplexed cos i heard there was no cure but i tried to still get rid of it and started to use really strong drugs from the hospital which really cost me a lot and all was to no avail. I had to check on the internet one day bout this Herpes and how i could be helped and i keep seeing a lot of testimonies bout this man called Dr Akpan, so i contacted him via his email and the mobile contact i saw there on the post of one Mrs Kathy Anne cos i needed this cure so bad as it has affected me deeply. I contacted him and was optimistic that he would cure me, and he responded well which made me really calm, I did all that i was i was asked to do by him and which he sent me the herbal medicine with instructions to take it. Just in 14days i started to feel okay and noticed changes in me and later i went to do a test as he told me to, and which i was confirmed negative and Herpes free. The only thing i can say to you all is that hes an awesome healer and hes blessed with good roots and herbs to cure you as i have promised to share to all when i'm cured. Contact him on his email ; [email protected] or via his mobile contact +2348143502763 for call and whatsapp.
It's unbelievable how fortunate I feel after finding your website. For the past 6 months, I have been so depressed after losing my husband to another woman. My money situation worsened so much that I thought I'd have to file for bankruptcy. I had a huge amount of debt and I didn't know what to do. Out of complete and total desperation, I contacted many of those so-called individuals who promised powerful magic, witchcraft or black magic. None of them worked and none were as wonderful, affectionate and warm as Dr Marvel has been. He is definitely different from the others and I felt immediate hope and strength from hearing about the promises he had to offer. He carries an air of purity and divine strength that is as pure as fresh snow on the ground. I requested Dr Marvel's most powerful spells and I was relieved right away that I had someone to solve my problems for me. His spells worked wonders and I am now back with my husband and my money troubles resolved itself after winning the lottery. Dr Marvel, I have no idea what I would have done without you being there to help me out. you can contact him via email [email protected] or directly on his website www.extremespellhome.webs.com. Telephone +1 669 221 3962 . With best regards, Erin
My husband and I have been married for 3 and a half years. In June it will be 7 years that we have been a part of each other lives. Recently he found out about a guy I slept with when we were separated and I did not tell him about. Before we got married we both agreed to let everything out and start fresh. I didn’t tell him. I was embarrassed bc the guy and I never actually had sex…just everything that led up to it. not to mention it was my brothers best friend and a huge mistake. well it came out recently along with some lies. A friend told my husband I cheated on him and I have never cheated on my husband nor would I ever. He is convinced I am a cheater. we have 2 beautiful kids and all I have ever wanted is a happy family. my kids and my husband truly are my everything. I can not see a future where my husband is not in it. I want to do whatever I can to save my marriage but I feel my husband is already set on the divorce. I know I could have been a better wife to him….a wife he deserves but now I fee like I have realized it too late. I look at my kids and I hate myself for allowing myself to break my family apart. I know my husband is not perfect and he has hurt me tremendously these past 5 years we have been off and on but we made it through for a reason. I love him and I want to save my marriage, one day a friend of my introduce me to a spell caster online who lift me up gave me hope and with-in 2 days after his spell my husband who told me he needed a divorce, called me and take me back home to me i must say today we are happily as one big family again all thanks to Dr happy a father and a real spell caster.Contact email address [email protected] Call and Whats-App him +2348133873774 Also contact and know more about his service at http://happyspelltemple.webs.com/
I used to suffer from HERPES SIMPLEX VIRUS for about 3 years. No medications or treatments ever really did anything for me (i tried them all no one work. but I was actually able to completely cure HERPES naturally after countless hours of online research. What worked for me is as follows: I saw Dr ILLUOBE Information online that he normally cure and treat disease and infirmities with his Herbal Medicine, I never really believed the people who where recommending him thou i saw on blog about 6 people where also testifying on how they were cured by Dr ILLUOBE, and still Ellen recommend Dr ILLUOBE who uses herbal medication to cure HERPES SIMPLEX VIRUS and gave me his email, so i mail him He told me all the things I need to do and also give me instructions to take, which I followed properly.Before I knew what is happening after two weeks the HERPES SIMPLEX VIRUS that was in my body got vanished. so if you are also heart broken and also need a help, you can also email him: [email protected]. if you are infected with any disease like HIV,AIDS, CANCER,HERPES SIMPLEX VIRUS, or any other disease you can also be happy like me by contacting.....Name: Dr.ILLUOBEWhat apps Number: +2348109031502Email: [email protected]: http://drilluobetemple.webs.comI“m given you 100% grantee, soon as you have getting in touchwith Dr.ILLUOBE your problems will be solved accurately.Thanks..
I Met a Powerful Spell caster and Psychic from Africa by name Dr Believe, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on [email protected] or [email protected] or check him directly on his website at http://authenticspellcast8.wix.com/authenticspellcast or call +2348159740630 he is the answer to your problem..
I Met a Powerful Spell caster and Psychic from Africa by name Dr Believe, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on [email protected] or [email protected] or check him directly on his website at http://authenticspellcast8.wix.com/authenticspellcast or call +2348159740630 he is the answer to your problem..
I Met a Powerful Spell caster and Psychic from Africa by name Dr Believe, He is Real, Honest and Genuine.. He helped me get my lover back, there is no bad or negative effect.The Psychic uses white Magic and he helped me get my Lover back, it worked in 2 days.. My lover came back to me, apologized for leaving and now we are back together and as happy as ever… I will encourage and recommend anyone to contact the spell caster and ask him for help.. He does all kinds of spells asides love spells. just tell him your situation and he will help you… you can email him directly on [email protected] or [email protected] or check him directly on his website at http://authenticspellcast8.wix.com/authenticspellcast or call +2348159740630 he is the answer to your problem..
What a wonderful and a straight forward spell caster that has brought back joy and happiness into my life after i saw a post on how he helped a lady called Nicole Morgan; i decided to contact him for help, when i told this God sent man Dr BEN on how my lover left me for 2 years without calling nor texting me, When i shared this my sad experience with Dr BEN he said everything would be okay within 3 days i was like am i sure what this man is saying is real, So i decided to give him a try and at first i was thinking he was a scam and i taught he was like other spell casters who come online to add pain to people's life not knowing there feelings but to make money, this great man Dr BEN is never like that because he is for good and to make people happy with the one they love, am just so happy, Even before the 3 days i just got a call from a man who has left me for 2 years saying that he his sorry and that he wants me back to his life i was so happy, He invited me for a dinner which i met with him there and we both talked, he said he wants to prove that he would never leave me for any other lady he engaged me and also made me had access to all his account am so happy all thanks goes to this great man Dr BEN a man who has brought back joy to my life, friends that need help in getting their lover's back i would advice you contact Dr BEN email: [email protected] website http://benherbaltemple.webs.com or call him +1(310)-220-0075
It's unbelievable how fortunate I feel after finding your website. For the past 6 months, I have been so depressed after losing my husband to another woman. My money situation worsened so much that I thought I'd have to file for bankruptcy. I had a huge amount of debt and I didn't know what to do. Out of complete and total desperation, I contacted many of those so-called individuals who promised powerful magic, witchcraft or black magic. None of them worked and none were as wonderful, affectionate and warm as Dr Marvel has been. He is definitely different from the others and I felt immediate hope and strength from hearing about the promises he had to offer. He carries an air of purity and divine strength that is as pure as fresh snow on the ground. I requested Dr Marvel's most powerful spells and I was relieved right away that I had someone to solve my problems for me. His spells worked wonders and I am now back with my husband and my money troubles resolved itself after winning the lottery. Dr Marvel, I have no idea what I would have done without you being there to help me out. you can contact him via email [email protected] or directly on his website www.extremespellhome.webs.com. Telephone +1 669 221 3962 . With best regards, Erin
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hello everyone in his forum i am little bit ashamed to share this marvelous story about a great spell caster who helped me enlarge my small penis size through his herbal mixture cream,i was heartbroken before because i have a very small penis about 3.5 inches which was so annoying and shameful i could not satisfy my wife in bed ,,my marriage was really breaking and i needed help urgently,i have used pills,vaccines,drugs,surgery but none worked...so one faithful day as i was browsing through the internet i saw few comments on a forum about how Mr CARLOS HENRY from USA testified of how he helped him enlarge his penis and also cured his dad diabetes type 2 disease,,i contacted him through his website link http://drharrysolutio.wixsite.com/dr-harry he replied and gave me steps to follow and i did just as he said ,and he sent me the herbal cream to me through UPS and i received it within 4 days and used the herbal cream for just 14 days to my greatest surprise my penis that was 3.5 inches before enlarge to 10 inches long when fully erected ,for just two weeks of using his herbal mixture cream ,,wow my wife loved me more and was so happy that i can satisfy her very well now in bed ,,i even last longer than before thank you so much DR HARRY for making life a better one to live in ..so if you are out there having similar problems please contact him now on his email [email protected] or visit his website http://drharrysolutio.wixsite.com/dr-harry or whatsapp and call him on +2348143240563 thanksnote he also have cure for DIABETES 1/2,HIV/AIDS,HERPES,CANCER,PREMATURE EJACULATION,WEAK ERECTION,ERECTILE DYSFUNCTION ,PREGNANCY HERBAL MEDICINE,ALS,HPV,SPELL OF ALL KINDS ETC
hello everybody i am steller javania, i am here today to gives out testimony on how i and my husband was cure on SHV 1 and 2 me and my husband has be with this virus over since 1 year and 2 months, we have be going true difference medication just to see we are cured of it' but still yet nothing sits to happens we have be buying difference drugs on it but nothing sits to change. we where not comfortable with this virus it make me more sick when ever i see then on me and on my husband body' they has be very suborn most time and it get me so sad and sick when ever it get pi-chins me. it has spreading allover my body and my husband body. we have tried all we can just to see that we have finally getting cure. one day we where advice by our medical doctor to switch to herbal remedy to see if there could be any help and then that was when i search for herbal remedy and how i could get it, and Luckily i saw a great post of silver sandy Philip who was also sick of this same HSV over since 2 years and how it was cure of it by DR.ODUDU HERBAL TRADITIONAL CURE TREATMENT. i was very very happy to see the wonderful publish on the website, i pick up the email and save it then i discussed it with my husband and with our medical doctor, so we all decided to give a try. that was when we contact him and i explained to him, and he reply by asking when we are needing the treatment and explained about the taking of the medication to us, so that was when will switch to his treatment and get it and also go with the taking of the medication according it was being explain to us, and now as i am publishing this post me and my husband has be confirmed cure and very well healthy. DR.ODUDU has be so good to my family. for those who are sick you can also get reach to him and also get cure on your virus. email [email protected] [email protected] massage him on facebook here https://web.facebook.com/DR.ODUDUHERBALHOME/?fref=ts whasaap call here +2348101571054 thanks everybody for your time of reading.
I was in a serious relationship for about 3 years.We had a split up in March due to his Ex making threats to his children about me being no good and not to be with me.he broke up with me. I pleaded (Mistake) then started the NC rule. After almost a month he contacted me via friends phone and asked me if I was mad. he told me he missed me and loved me.We got back together and that lasted almost 2 months.Now in early february a similar situation happened again, but with my children. he broke it off again. I did the pleading again with no response from her. I started NC again.Friends of mine tell me that he’ll be back again. Through a friend he told them that he loved me & missed me, but to have patience, trust and not to worry.so i go in search online if i can get any help to save my relationship with my bf, so luckily i found a web site saying getting your ex back, fix your marriage, and restored broken relationship. so i contacted and gave a try on http://happyspelltemple.webs.com/ he promise and gave me his word that with-in 2 days i will have my relationship restored with my bf which he did as he promise i must say now that me and my bf and living happily together again also we are about to get married now all thanks to Dr happy he is indeed a real spell caster plz if you need any help like my do not hesitate to contact him on [email protected] or add him on whats-app +2348133873774
I celebrated my fourth year Marriage anniversary yesterday, i can remember it was Dr. Todd that helped me. We broke up when we were dating few years back and i tried all ways to get him back to me but nothing work and after trying all means i had to sick for the help of Dr. Todd because i heard and saw so many testimonies about him on a Internet, i had to consult him for help and he brought him back to me in just few days after the counsel, prayers and spell was done in his temple my man was back to me and we got married after 3 months. Yesterday we celebrated our 4th year marriage anniversary , we have been living together as husband and wife after Dr. Todd brought him back to me, we are still living together peacefully till date and we have a son. Here is his email(manifestspellcast@gmail. com) OR manifestspellcast@yahoo. com if you want to contact him concerning your relationship problem or if you want your relationship to be lovely like mine. I’m so very happy and i just want you all to know that he is very powerful and can change your situation.
Am really happy that i and my husband are cured of (HERPES SIMPLEX VIRUS) with the herbal medicine of Dr UDURO, I AM FROM UK, i have been suffering from this disease for the past 2 years without solution until i came across his email of this doctor who have cure so many people with his herbal medicine, i also choose to give him a chance to help me and my husband, he told me what to do and i kindly did it, and he gave us his herbal medicine and direct me on how to use it, i also follows his instruction for use and he ask us to go for a check up after 2 weeks and which i did, to my greatest surprise our result came out as negative, we are really happy that there is someone like this Dr UDURO is ready to help anytime any day. to all the readers and viewers that is doubting this testimony stop doubting it and contact this Dr and see if he will not actually help you. contact him now if you are also in need of help via email [email protected] may god bless you sir for all your good works. or Whats app him on this +2348138634433.He can also cure many disease like HIV,, DIABETES,,CANCER,,DARIA,, IF YOU NEED A BABY IN YOUR FAMILY AND IF YOU NEED YOUR EX LOVER LIKE YOUR EX WIFE,,GIRL FRIEND,,EX HUSBAND,,EX BOY FRIEND.
hello viewers on this site,i am Mrs Kate from USA florida,i am using this opportunity to give thanks to Dr.OBODO, for bringing my husband back to me successful, a man who can bring peace and happiness into family,a man who can bring family together as one,are you in any problem like...1. HIV/AIDS.2. if you always have miscarriage. 3. Breast Cancer.4. herpes cure.5. AlO and Cerebrovascular Disease.6. pregnancy spell.7. You want to be promoted in your office.8. CHRONIC INSOMNIA.9. If you want your wishes to be granted.10.Stop your marriage or relationship from breaking apart...contact him today on his email and your problem we be well taking care of Email:[[email protected]] or you can give him a call : [+2348138829791]or[+2349055820663] am assuring you that you won't find any regret he is a man that can bring family together as one..contact him and you your problem will be successfully cure...
Now that i got my man back it's would be unfair if I don't share my experience to the whole world on how i got him back.That is why I am here this day to let the world know that there are real powers in spell casting as DR OCUSODO has proven to me by bringing joy and happiness to my relationship and my family in just 2days after i got him contacted . I never believed in any of these things until i loosed my boyfriend, I required help until i found [email protected] or [email protected] A great spell caster,who cast a love spell for me, and he assured me that I will get my boyfriend back in two days after the spell has been cast. two days later, my phone rang, and so shockingly, it was my boyfriend who has not called me for past 4 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me.Dr OCUSODO released him up to know how much i loved and wanted him. And opened his eyes to picture how much we have share together. As I'm writing this testimony right now I'm the most happiest girl on earth and me and my boyfriend is living a happy life and our love is now stronger than how it were even before our break up. So that's why I promised to share my testimony all over the universe.All thanks goes to Dr OCUSODO for the excessive work that he has done for me. Below is the email address in situation you are undergoing a heart break, and I assuring the whole universe that as he has done mine for me,he will definitely help anyone that takes time to get him contacted on. [email protected] or [email protected] +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
After being in relationship with markiss for seven years,he broke up with me, I did everything possible to bring him back but all was in vain, I wanted him back so much because of the love I have for him, I begged him with everything, I made promises but he refused. I explained my problem to someone online and she suggested that I should rather contact a spell caster that could help me cast a spell to bring him back but I am the type that never believed in spell, I had no choice than to try it, I mailed the spell caster, and he told me there was no problem that everything will be okay before three days, that my ex will return to me before three days, he cast the spell and surprisingly in the second day, it was around 4pm. My ex called me, I was so surprised, I answered the call and all he said was that he was so sorry for everything that happened, that he wanted me to return to him, that he loves me so much. I was so happy and went to him, that was how we started living together happily again. Since then, I have made promise that anybody I know that have a relationship problem, I would be of help to such person by referring him or her to the only real and powerful spell caster who helped me with my own problem and who is different from all the ones out there. Anybody could need the help of the spell caster, his email: [email protected] you can email him if you need his assistance in your relationship or anything. I CAN NEVER STOP TALKING ABOUT HIM, HIS EMAIL ADDRESS IS: [email protected] CONTCT HIM NOW FOR SOLUTION TO ALL YOUR PROBLEM
Sincerely i was so crushed when my Husband of 8 years left me and moved to Quebec City to be with another woman.The pains was just too much for me to bear that I couldn’t just bear it anymore. So i had to reached out to the Internet for help until i found out that Dr. Alli was the real deal.. I had tried the whole lot I knew, and with your spells, blessings and extraordinary magical powers, you did all the work for me, which you have guaranteed me positive result in 48 hours, my Ex Husband came back to me and he was remorseful for the whole lot he has done And now my life is balanced and i am happy again. Dr. Alli you do a great service to people. Friends in case you need the help of Dr.Alli kindly mail him on [email protected] or +2348100772528 Sir, i will for ever recommend you
I wake up everyday with a smile on my face and singing praises to this powerful spell caster called Dr,UZOYA who has done me a great favor by rescuing my relationship from break up. I used to think that i have a perfect relationship till when my lover started coming home late and everyday he gives me different excuses then i decided to keep a close eye on him then i discovers that he was having an affair with another girl. I was heart broken because i trusted him so much and knowing that he has a secret relationship, hurts me a lot.. But i thank all those who placed the contact information of Dr,UZOYA on the internet because through that i have been able to contact Dr,UZOYA and today my relationship has been restored with love and more romance than ever before. If you are in any kind of relationship or marriage stress i will do you a favor by dropping the contact information's of Dr,UZOYA which are via email: [email protected] Or Call +13019098775
My name is Elizabeth Shirley and I base in USA Florida...My life is back!!! After 2 year of Broken marriage, my husband left me with four kids . I felt like my life was about to end i almost committed suicide, i was emotionally down for a very long time. Thanks to a spell caster called Dr OCUSODO, which i met online. On one faithful day, as I was browsing through the internet,I came across allot of testimonies about this particular spell caster. Some people testified that he brought their Ex lover back, some testified that he restores womb,cure cancer,and other sickness, some testified that he can cast a spell that can help you get pregnant for your husband and so on. i also come across one particular testimony,it was about a woman called Sandra Denilson she testified about how he brought back her husband back in less than 48 days, and at the end of her testimony she dropped Dr OCUSODO e-mail address. After reading all these,I decided to give him a try. I contacted him via email and explained my problem to him. and he give me some instruction i follow it within 48 days, my husband came back to me. We solved our issues, and we are even happier than before Dr OCUSODO, is really a gifted man and i will not stop publishing him because he is a wonderful man... If you have a problem and you are looking for a real and genuine spell caster to solve all your problems for you. you can contact him on his email [email protected] or [email protected] contact number +2349067457724, WEXSITE https://drocusodospellcast.wixsite.com/dr-ocusodo
Hello, i am Jessica Wayne from TX, USA. Life without my husband was a real mess for me and my children I am so happy to get my Ex back through the help of Dr Noble the spell caster . My greatest surprise was that 48 hours after the Doctor prepared the spell for me, my husband who has abandoned me for 4 years suddenly called me unexpectedly and am so happy that we have come to become one again through the help of Dr Noble and am so happy to be with my husband once again. Dr Noble is a very wonderful spell caster, you can contact him if you need his assistant because i know he can also help you. contact him through his email: [email protected] immediately
Hello, i am Jessica Wayne from TX, USA. Life without my husband was a real mess for me and my children I am so happy to get my Ex back through the help of Dr Noble the spell caster . My greatest surprise was that 48 hours after the Doctor prepared the spell for me, my husband who has abandoned me for 4 years suddenly called me unexpectedly and am so happy that we have come to become one again through the help of Dr Noble and am so happy to be with my husband once again. Dr Noble is a very wonderful spell caster, you can contact him if you need his assistant because i know he can also help you. contact him through his email: [email protected] immediately
WOW!!! This is the most wonderful thing i have ever experience and i need to share this great testimony.About how i get my ex back after a breakup.I never believed it, because i never heard nor learn anything about it before. Hey guys my wife and I have been married for 15 years. For over five years I have been dealing with jealousy issues because of flirting and her having emotional type affairs with other men. I felt like my masculinity was in question and if I said the way she was acting bothered me, it seemed to make matters worse or I was accused of being controling. During this period I did state I wanted a divorce if the behavior was not going to end: texting men a night, leaving for the weekend without letting me know where she was going or not responding to messages. We do have a son and basically it's been him and I for the last year on the weekends. She disconnected completely from being a good wife and mother. In April she said she wanted a separation and I said no we need to get into marriage counseling. We did try that, but she was not very responsive and didn't give any effort. Afterward she was adamant about separation and divorce. I continued to say no and that we needed to save our family. She presented a separation agreement and I had to hire a lawyer. Because of the above behavior my lawyer suggested a private investigator. The investigator discovered my wife was having an affair with a close friend of the family who also is married with children. We know the extended families. I feel like I should tell this man's wife about what happened. This adultery has devastated me emotionally, I feel betrayed and I'm physically drained. I know my wife is passionate in terms of her sexuality, and I can't get the thought of them out of my head. It makes me question my own manhood, and I feel very inferior or that he must be a better lover or what ever. The problem is my wife pursued him. She would go to him and she lured him into this adultery. I felt this was coming for some time and could not stop it. She was not only lying to me but also to our son about what she was doing and where she was going. My family is important, my son loves her and as crazy as it sounds so do I. Can you respond with a course of action on how to proceed? I was still have a very huge place in my heart for her. so i searched for help online and I came across a website that suggested that Dr Ahmed can help solve marital problems, restore broken relationships and so on. So I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a spell for me. 28 hours later, my wife came to me and apologized for the wrongs she did and promise never to do it again. Ever since then, everything has returned back to normal. I, my son and my wife are living together happily again.. All thanks to Dr Ahmed. as it is a place to resolve marriage/relationship issues, do you want to be sure if your spouse is being faithful to you or Do you want your Ex to come back to you Contact.: E-mail: [email protected] or call/Whats-app: +2348160153829 save your crumbling home and change of grades its 100% safe. I suggest you contact him. He will not disappoint you.David L. Ollis, 43yrs, UK
Hi,I really appreciate your inputs on my situation Dr happy and what could I do with-out you. Thank you in advance.I was in a relationship with this guy for the past 1 and a half year.. We became close real fast and were very serious about each other… he’s German and was in the country for her internship.. he left after first 6 months and came back after a couple of months.. he said that he had come down for me.. WE started living – in.. We discussed faMILY, KIDS, future, moving to EU together.. everything was going good until of last August when I started to get jealous about his moves, I got controlling and things started to get worse.. we spoke about it a lot of times where he expressed how he felt but I was not able to understand why I was doing that.. During our discussion I would always say I did .. This February he called it off and since then I have been a mess.. I gave it a lot of thought and realised my mistakes and why I was doing it.. I tried talking to him .. begged him.. pleaded him .. but he says that he has lost all his feelings for me and would never get back to me..I was in my knees praying looking for help, them i fine a comment online how a spell caster help to restored relationship get ex lover back, ” so i decided to give a try coz i love my boyfriend so much. so i contacted Dr happy http://happyspelltemple.webs.com/ tell him all my problem and he gave me 100%guarantee that i will have my boyfriend back after the spell so i was gifted and lucky to have contacted him i did every thing he ask of me and to my greatest surprise a day after the spell my boyfriend call me and apology for what he did to me and ask for my forgiveness to come back again for me, we both are happily living together again plz Contact him urgently [email protected] or add him on whats-app +2348133873774Junia Noel from USA
Here priestEka we are here to make your dream of pregnancy and other related human diseases over in your life have a happy home. Wespecialise in helping people with their problems such as.1 bringing back your lost .2 spell to improve your businesses.3 spell to bring back your ex-lover( bring back your husband/wife/boyfriend/girlfriend)4 spell for business protection from colleague.5 spell to prevent witchcraft from your life and that of your family.6 spell to prevent you from accident during your business journey.7. spell to get employment easilyWe also specialise in providing solutions in any of this spiritual gynaecology diseases affect human existence such as:1 fibroid, asthma, ALL STD,2 weakness of man organ3 infections of all kind (yeast infection)4 blockage from the fallopian tube5 cyst. From the ovaries6 unpleasant smell from the virgina7 irregular menstration, menopause8 infertility for easy Conception.9.skin diseases, Toilet infection and bad body odor…….Etc..10.Watering sperm (low sperm count) not able to get woman pregnant.Simply contact the spiritualist PriestEka on ([email protected]) to get his Herbal Medication to cure your disease and put yourself on a motherhood side of life..(No more adoption, with PriestEka your problem will solve and you will have your child with ease.Contact us at ([email protected]) your solution home!
Hello, My name is Dakota meyermi from Canada.. I am here to let you know if you are having issues with your health, The solution is out there. I was cured of HEP B with the use of a herbal medicine from Dr UDURO I was diagnosed with this virus for almost 5 yrs but i was cured within few weeks after using the herbs which Dr UDURO sent to me. Dr UDURO is a herbalist doctor. I know of this doctor through others people testimonies, they were all thanking him and praying for him. He has herbs for the following illness,gain weight, asthma, cancer, Diabetic 1&2, Gonorrhea, Barrenness, Impotence, High Blood Pressure,Epilepsy, Herpes 1&2, Menopause problem, Menstruation problem, Hiv,Diarrhea, Acne, Diseases of the heart, Arthritis,hpv, copd ,lupu,Gain Weight bring your Ex back boy friend,girlfriend,your husband or your wifeback to you less than 72 hours etc. He is also a spell caster, he cast spell to bring love ones together, a spell to stop divorce, a spell to make more profit in the line of business, good luck spell, etc. Email him now via: [email protected] OR [email protected] rather reach him on mobile +2348138634433 He will help you solve all your problems. Once again thanks Dr UDURO for your good deeds.
I think we should be sensitive about the things we read online every day, some are true while some are false. I am sharing this cos I was deceived initially by some fake spell casters who goes around posting some negative reviews about good and genuine casters. They condemn the real ones and secretly provide a referential email to their victims where they can be scammed. It happened to me twice before I became observant.I fell to their trap the first time I needed help getting my Husband back, I was about contacting Dr. Wakina for help before seeing a strange review that said negative things about him which got me discouraged and made me contact the alternative email they provided, and I was scammed. I tried the second guy because of my desperation, but he did the same thing and stopped replying my emails. This situation caused me to become so curious to really know the good and fake caster between Dr. Wakina and them. There was a huge difference between them, Dr. Wakina is a master filled with wisdom and experience, he gave the accurate reading of my life, even one of my deepest secret. Dr. Wakina restored my marriage like a Boss he is at the exact date, time and location. His work was so magical.After the work, I understood why people are so jealous and try to be him. Like I said earlier, there is a huge difference between Dr. Wakina and others, it was confirmed after the spell he did on my Husband. So be careful while searching for a real spell caster. Here is Dr. Wakina contact email: [email protected]
Hey guys, I'm so excited of getting my husband back after he left me and our 3 kids for another woman!!.After 2 years of marriage, me and my husband has been into one quarrel or the other until he finally left me and moved to California to be with another woman. i felt my life was over and my kids thought they would never see their father again. i tried to be strong just for the kids but i could not control the pains that torments my heart, my heart was filled with sorrows and pains because i was really in love with my husband. Every day and night i think of him and always wish he would come back to me, I was really upset and i needed help, so i searched for help online and I came across a website that suggested that Dr Unity can help get ex back fast. So, I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a (Love spell) for me. 28 hours later, my husband really called me and told me that he miss me and the kids so much, So Amazing!! So that was how he came back that same day,with lots of love and joy,and he apologized for his mistake,and for the pain he caused me and the kids. Then from that day,our Marriage was now stronger than how it were before,All thanks to Dr Unity. he is so powerful and i decided to share my story on the internet that Dr Unity real and powerful spell caster who i will always pray to live long to help his children in the time of trouble, if you are here and you need your Ex back or your husband moved to another woman, do not cry anymore, contact this powerful spell caster now. Here’s his contact: Email him at: [email protected] ,you can also call him or add him on Whats-app: +2348071622464 ,his website:http://unityspelltemple.yolasite.com . Jessica, 26 years, Texas, USA.
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mit
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Food label use and its relation to dietary intake among US adults.
Rates of diet-related chronic disease combined with the lack of current data on patterns of food label use by the US population warrant re-examination of the use and potential influence of this public health tool. The purpose of this study was to describe the prevalence of food label use and the association between food label use and nutrient intake in a nationally representative sample of US adults who participated in the 2005-2006 National Health and Nutrition Examination Survey. Data on food label use were collected during the interview portion of the survey, and nutrient intake was estimated using the average of two 24-hour dietary recalls. In this sample, 61.6% of participants reported using the Nutrition Facts panel, 51.6% looked at the list of ingredients, 47.2% looked at serving size, and 43.8% reviewed health claims at least sometimes when deciding to purchase a food product. There were significant differences (P<0.05) in food label use across all demographic characteristics examined. Significant differences (P<0.05) in mean nutrient intake of total energy, total fat, saturated fat, cholesterol, sodium, dietary fiber, and sugars were observed between food label users and non-users with label users reporting healthier nutrient consumption. The greatest differences observed were for total energy and fat and for use of specific nutrient information on the food label. Despite food label use being associated with improved dietary factors, label use alone is not expected to be sufficient in modifying behavior ultimately leading to improved health outcomes.
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---
title: "SQL Server Agent Properties (Service Tab)"
description: Learn about the options on the Service tab in the SQL Server Agent Properties dialog box, such as the binary path, the process ID, and the start mode.
ms.custom: seo-lt-2019
ms.date: "03/14/2017"
ms.prod: sql
ms.prod_service: sql-tools
ms.reviewer: ""
ms.technology: tools-other
ms.topic: conceptual
ms.assetid: 452857fb-be1b-4e1e-851c-dd2216640f35
author: markingmyname
ms.author: maghan
monikerRange: ">=sql-server-2016||=sqlallproducts-allversions"
---
# SQL Server Agent Properties (Service Tab)
[!INCLUDE [SQL Server Windows Only - ASDBMI ](../../includes/applies-to-version/sql-windows-only-asdbmi.md)]
This service is the [!INCLUDE[msCoName](../../includes/msconame-md.md)] [!INCLUDE[ssNoVersion](../../includes/ssnoversion-md.md)] Agent service. The property values in light gray cannot be changed using this application.
## Options
**Binary Path**
Displays the location of the program files used by this service.
**Error Control**
1 indicates `SERVICE_ERROR_NORMAL`. If the service fails to start during computer start up, the startup program logs the error and displays a pop-up message box but continues the startup operation. This value cannot be changed.
**Exit Code**
When an error occurs, the error number appears in this box. Use this number to troubleshoot failures by searching for the number in the [!INCLUDE[msCoName](../../includes/msconame-md.md)] Knowledge Base or provide the number to your technical support staff.
**Host Name**
Displays the name of the computer or cluster running [!INCLUDE[ssNoVersion](../../includes/ssnoversion-md.md)] Agent.
**Name**
Indicates the display name of the service.
**Process ID**
Displays the [!INCLUDE[msCoName](../../includes/msconame-md.md)] Windows process ID.
**SQL Service Type**
Displays the type of service provided to calling processes. [!INCLUDE[ssNoVersion](../../includes/ssnoversion-md.md)] installs several services.
**Start Mode**
Set this service to the following choices:
- Manual: This service does not automatically start when the computer starts. You must start the service using [!INCLUDE[ssNoVersion](../../includes/ssnoversion-md.md)] Configuration Manager, or some other tool.
- Automatic: This service attempts to start when this computer starts.
- Disabled: This service cannot be started.
**State**
Indicates whether this service is running, stopped, or disabled. "**...**" indicates a state change is pending.
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mit
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Q:
Why std::hash seems to be identity function
#include <iostream>
int main() {
std::hash<int> hash_f;
std::cout << hash_f(0) << std::endl;
std::cout << hash_f(1) << std::endl;
std::cout << hash_f(2) << std::endl;
std::cout << hash_f(3) << std::endl;
}
I compile with "g++ main.cpp -std=c++11" and then the result is :
0
1
2
3
Why is this happening? I don't use any library and I don't have a specialized hashing function.
Addendum : I wanted to define the hash for an unordered_set of unordered_set of int with the hash of a set being the sum of its components hashs, but if it's just identity it's not cool because the hash of {2,4} is the same than the hash of {1,5}. The simplest way to avoid that is may be to use the std::hash double function.
A:
It seems its identity, its allowed as its distinct..
From cpp reference
The actual hash functions are implementation-dependent and are not required to fulfill any other quality criteria except those specified above. Notably, some implementations use trivial (identity) hash functions which map an integer to itself. In other words, these hash functions are designed to work with unordered associative containers, but not as cryptographic hashes, for example. ....
A:
It seems completely reasonable for the hash function int→int to be identity, and it's not clear why you are surprised by that. Performing any further computation would be pointless. This is, in fact, a perfect hash in every sense of the term.
Remember, std::hash is supposed to (almost uniquely) identify values, not encrypt them.
It's only when you want to hash types larger than that of the hash itself (say, uint9999999_t) that you need to do some work to "compress" the value into the size of the hash.
A:
The other answers cover the rationale behind the identity function very well. To address your addendum:
I wanted to define the hash of an unordered_set as the sum of its components hashs, but if it's just identity it's not cool because the hash of {2,4} is the same than the hash of {1,5}. The simplest way to avoid that is may be to use the std::hash function.
As you can see, using the + operator to combine hashes is not the best idea. To be more robust, you could use the XOR (^) operator, or take inspiration from the approach taken, e.g., by boost::hash_combine (details in this SO post):
seed ^= hash_value(v) + 0x9e3779b9 + (seed << 6) + (seed >> 2);
As an example, for your two integer pairs (1,5 / 2,4), and a seed of 0, this would work out to
uint32_t seed = 0;
seed ^= 1 + 0x9e3779b9 + (seed << 6) + (seed >> 2);
seed ^= 5 + 0x9e3779b9 + (seed << 6) + (seed >> 2);
// 3449077526
uint32_t seed = 0;
seed ^= 2 + 0x9e3779b9 + (seed << 6) + (seed >> 2);
seed ^= 4 + 0x9e3779b9 + (seed << 6) + (seed >> 2);
// 3449077584
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mit
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Neuro-psychiatric effects of antimalarials.
To study the neuro-psychiatric adverse effects of antimalarial drugs. Persons who visited a Travel Clinic in Rotterdam over a period of 3 months. Prospective cohort study on 394 persons taking mefloquine, 493 persons taking proguanil and 340 persons not taking antimalarial drugs who visited Africa, South America, Asia, or the Middle East. All persons received a structured questionnaire within 14 days of their return to the Netherlands. The questionnaire consisted of questions regarding use of alcohol, smoking, general health, medical history, tropical diseases during the trip, and other medicines, and contained an extensive list of general complaints regarding all body systems at four levels of severity. A modified and validated version of the Profile of Mood States was included. In the study period, 2541 persons visited the Travel Clinic, of whom 1791 (70%) were both eligible and willing to co-operate. Of these 1791, data were obtained from 1501 (84%). Insomnia was most frequently encountered in users of mefloquine and mouth ulcers in proguanil users. After adjustment for gender, age, destination, and alcohol use, the relative risk for insomnia to mefloquine versus non-users of antimalarials was 1.6, and the excess risk was 6 per 100 users over an average period of 2 months. There were no significant differences between groups in depression, anxiety, agitation, and confusion. Stratification by gender demonstrated that insomnia was more common in women on mefloquine, but not in men. Also, women more frequently mentioned palpitations as an adverse event. After adjustment for age, destination, and alcohol use in women, the relative risks for insomnia and palpitations to mefloquine versus non-use of antimalarials were 2.4, and 22.5, respectively. When travellers were specifically asked for the adverse reactions they had experienced, anxiety, vertigo, agitation, and nightmares were significantly more frequently mentioned by mefloquine users. Insomnia was more commonly encountered during use of mefloquine than proguanil or during non-use of antimalarials.
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PyeongChang
Doo Jae Park and Na Ri Shin on October 26, 2018
Athletes from North and South Korea marched together during the opening ceremonies of the 2018 Winter Olympics in PyeongChang, South Korea. (photo by Chang W. Lee/The New York Times)
Sport has important political power in contemporary culture. When North and South Korean athletes marched under a unified flag during the opening ceremonies of the 2018 Winter Olympics in PyeongChang, South Korea, it provided a powerful sign of cooperation between the two nations. Seeing the two Koreas marching together symbolized hope for reunification in the Korean peninsula. The 2018 Olympics, however, were only one chapter in a much longer story about the ways in which South Korea has invested substantial resources in attempts to foster a (global) Koreanness through success in sporting mega-events. In fact, cultural anthropologist Rachael Miyung Joo has argued that South Korea sees transnational sport as the most useful way to demonstrate the potential of a “global Korea.” Sport, in this respect, is used as a cultural apparatus to build a collective identity—what political scientist Benedict Anderson called an “imagined community.” Here, we aim to deconstruct South Korean sporting nationalism by analyzing how sport operates to establish and reinforce nationalism in South Korea.
Danyel Reiche on February 5, 2018
Although it is often framed as a “global” event, the Winter Olympics is quite exclusive and Eurocentric. (Photo by Mike Crane Photography)
While a record number of countries and athletes are expected to participate in the 2018 Olympic Games in PyeongChang, South Korea, the Winter edition of the Olympics remains an exclusive event. South Korea will be just the 12th country that has ever hosted the Winter Olympics, a quadrennial event that was inaugurated in France in 1924. Only a few countries have the geographic and economic conditions to host the event, which accounts for the fact that only 6 percent of the 206 recognized National Olympic Committees have ever done so. Further, a majority of countries still do not participate in the Winter Olympic Games. At the 2014 Sochi Games, 89 countries participated. This number increases to 92 in PyeongChang (plus the “Olympic Athlete from Russia” category), which still leaves 55 percent of countries out of the Games. According to the organizing committee website, 31 nations are participating with just one athlete (18 countries) or two athletes (13 countries). Seven of the eight participating African countries will send only one or two athletes to PyeongChang, while Nigeria has the largest African delegation with three athletes. For comparison, the United States of America is participating with 242 athletes.
Liv Yoon on June 7, 2017
Mount Gariwang, formerly a protected conservation area, is being developed into a skiing venue for the 2018 Winter Olympics. Photo by Liv Yoon.
PyeongChang is a small county in the northeastern province of Gangwon in South Korea (hereafter, ‘Korea’), with an aging population of approximately 43,000 (in which 1 in 4 people are 65 years or older). This mountainous region, known for its quaint charm and small-scale agriculture-based economy, will host the 2018 Winter Olympic and Paralympic Games, and as a result, joins a long list of host cities that have witnessed turmoil due to sport mega-events. An issue surrounding the PyeongChang Games that has gained some attention – but certainly not enough – is the destruction of Mount Gariwang, a former Class 1 Protected Area for Forest Genetic Resource Conservation, now transformed into the official alpine skiing venue. Because this area had long been protected from any kind of development, public or private, a ‘Special Act’ that “may revoke the designation of all, or a part of a conservation zone” was legislated to pave way for the development.
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Q:
How to find nodes with Python string matching functions in Networkx?
Given a dependency parse graph, if I want to find the shortest path length between two fixed nodes, this is how I've coded it:
nx.shortest_path_length (graph, source='cost', target='20.4')
My question here is: What if I want to match for all sentences in the graph or collection a target with any number formatted approximately as a currency? Would I have to first find every node in the graph that is a currency, and then iterate over the set of currency values?
It would be ideal to have:
nx.shortest_path_length (graph, source='cost', target=r'^[$€£]?(\d+([\.,]00)?)$')
Or from @bluepnume ^[$€£]?((([1-5],?)?\d{2,3}|[5-9])(\.\d{2})?)$
A:
You could do it in two steps, without having to loop over.
Step 1: Calculate the shortest distance from your 'cost' node to all reachable nodes.
Step 2: Subset (using regex) just the currency nodes that you are interested in.
Here's an example to illustrate.
import networkx as nx
import matplotlib.pyplot as plt
import re
g = nx.DiGraph()
#create a dummy graph for illustration
g.add_edges_from([('cost','apples'),('cost', 'of'),
('$2', 'pears'),('lemon', '£1.414'),
('apples', '$2'),('lemon', '£1.414'),
('€3.5', 'lemon'),('pears', '€3.5'),
], distance=0.5) # using a list of edge tuples & specifying distance
g.add_edges_from([('€3.5', 'lemon'),('of', '€3.5')],
distance=0.7)
nx.draw(g, with_labels=True)
which produces:
Now, you can calculate the shortest paths to your nodes of interest, subsetting using regex like you wanted to.
paths = nx.single_source_dijkstra_path(g, 'cost')
lengths=nx.single_source_dijkstra_path_length(g,'cost', weight='distance')
currency_nodes = [ n for n in lengths.keys() if re.findall('(\$|€|£)',n)]
[(n,len) for (n,len) in lengths.items() if n in currency_nodes]
produces:
[('$2', 1.0), ('€3.5', 1.2), ('£1.414', 2.4)]
Hope that helps you move forward.
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You don't have to be a Golden State Warriors fan to learn from their NBA championship run.
Entrepreneur contributor Justin Bariso collects a series of examples businesses could learn from the team that hadn’t won a championship in 40 years, led by a rookie coach, but ended up winning the 2015 NBA Championship.
One example is coach Steve Kerr’s willingness to listen to all of his employees, from players to interns. Another is to make decisions that are best for the team, not the individual.
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The Reserve Bank of India (RBI) on Monday said that its central board has accepted the recommendations of Bimal Jalan Committee and approved surplus transfer to the government. The RBI will transfer a sum of Rs 1,76,051 crore to the government which comprises a surplus of Rs 1,23,414 crore for 2018-19.
"The Central Board of the Reserve Bank of India (RBI) today decided to transfer a sum of Rs 1,76,051 crore to the Government of India (Government) comprising of Rs 1,23,414 crore of surplus for the year 2018-19 and Rs 52,637 crore of excess provisions identified as per the revised Economic Capital Framework (ECF) adopted at the meeting of the Central Board today," the RBI said in a statement.
The Bimal Jalan Committee was constituted to suggest appropriate reserves that RBI should maintain and dividends it should pay to the government. The Jalan Committee had submitted its report to the RBI Governor on Friday.
The Reserve Bank follows a July-June financial year and the dividend is usually distributed in August after annual accounts are finalised.
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Jan
24
If cheapskating is going to increase, we might consider whether individual stocks that cater to cheap skates might have inordinate returns. This is the kind of things that my kids might make money with in terms of the category of stock, rather than its financial characteristics. Perhaps. On another front, I believe it is important to be especially cheap after having a good year. I think of Rimm every day with grave loathsomeness.
Art Cooper writes:
It's been a market theme for quite some time to buy stocks like Family Dollar Stores, Dollar General, etc. instead of retail stocks which cater to the middle class. The high-end retail market is a different market, as it responds to different forces.
Jeff Watson writes:
I'm always accused of being a cheap person and try to not be penny wise and pound foolish. I never pay retail for anything and try to buy only stuff that will hold value. Herb Cohen is a person I look up to. He might look a little seedy, but he makes great sense and teaches sound methods of bargaining. His first $19.95 book I ever bought was probably the best investment I ever made, saving at least a million bucks, by bargaining with some of his techniques over a 30 year period. That's a hell of a return and his techniques work…
Pitt T. Maner III writes:
Cheapskating is likely to be an increasingly popular topic as hidden inflation and taxes go up. Perhaps there is an opportunity for a "Global Skinflint"!
"Jeff Yeager, dubbed "The Ultimate Cheapskate" by Matt Lauer on NBC's Today show, is a very cheap guy. He re-cants, as opposed to decants, the wine he proudly serves his dinner guests, funneling cheap box wine into premium-label bottles. He believes you should never spend more than USD 1 per pound on food items. And to save time and energy costs, he soft-boils his morning eggs along with the dirty dishes in the dishwasher."
And then there is the TLC show :
"Be aware of what you're using. Victoria Hunt, who retired from her accounting career at 48 has been tracking her expenses and her income on a spreadsheet since 1989. "Every minute of every day has something to do with how I can make a better decisions financially," she points out."
Rocky Humbert writes:
Mr. Yeager is either wasting money on his super-heated dishwasher or he's stretching the truth about his eggs. Dishwashers (generally) do not heat the water about 140 degrees. See this article on naturalhandyman. To get the egg white solid, it requires about 180 degrees. Even my Miele doesn't get the water to 180 degrees! This does not compute! (That is, he's making his money selling books. Not cooking eggs.) I would suggest that he should instead put his Pop Tarts and morning sausage on his car engine's manifold. By the time he gets to work, he'll have a well-cooked breakfast. (And he can similarly roast hot dogs on his drive home.)
Dr. Johnson writes:
Ballyhoo? Like any good Spec, one must test, and test I did, the claim that an egg can be cooked in a dishwasher during a normal wash/dry cycle.
Equipment- Miele G5775.
Note: Perhaps not the ideal brand for testing a cheapskate's assertion.
Conclusion: Not Ballyhoo! One important consideration for those cheapskates who want to try this method is that egg shells are semipermeable, therefore unless the taste of detergent combined with a menagerie of old food waste is to your liking, sealing the egg in plastic wrap is advisable (also which at +140 F will transmit unwanted substances).
David Hillman writes:
Yes, let us commend Dr. Johnson both on his testing and on his using Phil's Farm Fresh Free Range eggs, the chicken egg of preference at Casa DGH…..cage-free, no chemicals, natural whole grain feed, laid in nests, and certified humane!
That said, even though my Bosch heats water to 160F and air dries at what seems to be 1200K if one opens the door during the 'sanitize' cycle and is met by a blast of superheated air, this whole business of cooking eggs in a dishwasher seems a bit impractical.
One, it seems like using a sledgehammer to place a pushpin in a cork board. Two, while the dishwasher here is run every 2-3 days, typically in the evening, eggs are a daily breakfast staple. What to do on 'accumulation' days? Three, counting time to heat water or a pan, it takes about 10 minutes to fry, poach, baste, scramble or soft boil eggs on the range. Why wait 76 minutes? Four, dishwasher cooking uses a heck of a lot of water and electricity v. range top cooking, multitasking notwithstanding.
Pitt T. Maner III adds:
A few older links, but possibly of interest to those seeking to find ways to ride the money-saving trend and as a possible example of a company that finds quickly (identifying trends) and uses new inventions from private inventors. Khubani the CEO started with ad in National Enquirer.:
1) From 2010: 'A.J. Khubani, the man behind many “As Seen on TV” gadgets such as the PedEgg foot scraper, is making cheapskate gimmicks a priority at his company Telebrands, one of the nation’s top direct-response TV marketing companies.
More than half of Telebrands’ gadgets, sold online and at 90,000 stores, are now focused on helping shoppers be cheap. Khubani, who has been traveling around the country to meet inventors, is speeding up the number of new products he’s launching to every 30 days from every 60 days. “The mood of the country has changed,” said Khubani. “We’ve had tremendous opportunity with this recession.”'
Since 2007, Telebrands’ revenue has doubled to several hundred million dollars, he said.
Read more.
3) From 2012: "For the first time in our company's 29 year history, TeleBrands had 15 products ranked in a single year including our most recent hits like, Slice-O-Matic, Plaque Blast, Slim Away, OrGreenic and Bake Pops," said TeleBrands' CEO/Founder, AJ Khubani. "Each year, we continue to solidify our spot as the largest and most successful marketer of DRTV products aimed at solving everyday problems and reaching mass audiences at affordable prices. In 2011 alone, we rolled-out 12 products — the most in a single year in our company's history."
"The son of Indian immigrants, Khubani started out at 23, spending a few thousand dollars on an ad inNational Enquirer — a move that led to his first big hit. Since then, he's sold hundreds of millions of "As Seen on TV" products, including AmberVision sunglasses, the PedEgg and Doggy Steps. He has bolstered the careers of ubiquitous TV pitchmen, including the late Billy Mays, who enthusiastically hawked products now found on the shelves of more than 100,000 retailers. Today, Khubani is the leader in the $20 billion direct consumer marketing industry, turning out more "low-tech" products than ever before."
"But will anyone care about dust mites? Khubani wasn’t achieving much traction among his Telebrands staff with his bed-spray idea, when along came a proposal for an anti-dust-mite pillow, from a colleague Khubani mysteriously describes only as “a business associate.” It’s hardly a new concept—there are several such pillows already marketed to allergy sufferers and asthmatics. But so far, nobody has had the brilliance to incite a national panic around flesh-eating creatures that feast on human remains—and lurk in the pillow of every man, woman, and child. “The hum you sometimes hear at night?” Khubani asks eerily. “That’s the sound of 2 million dust mites eating your dead skin.” Or perhaps it’s the sound of one man in Fairfield, New Jersey, homing in on your next anxiety. "
Victor Niederhoffer adds:
Of course the main virtue about cheapskating is that it prepares you for such activities in your business. As the oil magnate said, "I am not smart enough to act one way in my personal life and another in my business. My margin is 8%, and if I gave away 8% on everything my 200,000 employees would be out of a job. So I make them pay for their telephone calls." Regrettably, the oil magnate was victimized by old man's disease (the same disease as the sage), and he was locked up in England for 20 years, with his retinue preventing him from going back to us for fear that he might change his will, and he was soporifisized by many nubile girls and other attractive women he would meet at museums.
Funny. More important even then the fine posts with examples and tests of cheapskating is the query I have received from many of the younger hearted on the list. "Where are those museums that the oil magnate frequented?".
Gary Rogan suggests:
Stefan Jovanovich writes:
I hope Gary means the original one in Malibu, the villa whose design Getty himself supervised but never saw. The monstrosity built on top of the landfill by the 405 is absolutely the worst place in LA for the amusements Getty had in mind. If he were alive today and living in SoCal, he would be going to OCMA to appraise the latest generation of lovelies.
Jim Sogi adds:
Eggs can be cooked sous vide at 144 -155 for 20 plus minutes for a wonderfully cooked smooth soft boiled egg with a consistent texture throughout.
Food grade hydrogen peroxide diluted to a 3% solution is an excellent way to sanitize kitchen and utensils and not toxic like chlorine.
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---
abstract: |
The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. This is a paper from 2012.
We first studied provability logics with restricted realizations, and show that for various natural candidates of theory $T$ and restriction set $\Gamma$, where each sentence in $\Gamma$ has a well understood (meta)-mathematical content in $T$, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic ([${\mathrm{PRA}}$]{}), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between [${\mathrm{PRA}}$]{}and $\Sigma_1$.
We then study interpretability logics, obtaining some upper bounds for [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{}, whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\subset {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$.
address:
- Stanford University
- Universidad de Sevilla
author:
- 'Thomas F. Icard'
- 'Joost J. Joosten'
title: Provability and Interpretability Logics with Restricted Realizations
---
Introduction
============
In a recent discussion on a mailing list on the foundations of mathematics[^1] Joe Shipman asked for important theorems that have essentially only one proof. In reply, Giovanni Sambin provided the example of Solovay’s arithmetical completeness theorem of the provability logic **GL** ([@Sol76]).
This paper deals with restricted cases of Solovay’s theorem where alternative proof-methods are available. One of the broad motivations for this paper is the hope of obtaining an alternative proof of Solovay’s Theorem (see Section \[section:Solovay\]). However, the method of provability logics with restricted realizations, we feel, merits interest in its own right, as we shall explain shortly. Let us first briefly restate Solovay’s Completeness Theorem, which is the cornerstone result in the study of provability logics.
Provability Logics
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The propositional modal logic **GL**, known as *Gödel-Löb Logic*, captures exactly the behavior of the standard provability predicate in arithmetic. For a given theory $T$ (e.g. Peano Arithmetic), formulas $\Box A$ are interpreted as, “$A$ is provable in $T$". It is defined by extending the basic modal logic **K** with a schematic formalization of Löb’s Theorem (**L** in the following definition).
[**GL**]{.nodecor} is given by all boolean tautologies, in addition to all instances of the following schemata $$\begin{array}{lll}
\mbox{\textnormal{\textbf{K}}} &:& \Box (A\to B) \to (\Box A \to \Box B);\\
\mbox{\textnormal{\textbf{L}}} &:& \Box (\Box A\to A) \to \Box A.\\
\end{array}$$ The logic is closed under *modus ponens* and necessitation.
**GL** enjoys modal completeness with respect to a simple class of frames, in particular the class of finite, irreflexive, and transitive frames, which we henceforth refer to as **GL**-frames. The logic is linked to formalized provability via *arithmetical realizations*. An arithmetical realization is a function $*$ that maps propositional variables to sentences in the language of (a given) arithmetic, sending $\bot$ to $0=1$. A realization $*$ can be extended uniformly so that we can interpret an arbitrary modal formula as an arithmetical formula by stipulating, $$\begin{array}{l}
(A\to B)^* = A^* \to B^*\\
(\Box A)^* = {\sf Bew}_T(\ulcorner A^* \urcorner).
\end{array}$$ Here $\ulcorner \cdot \urcorner$ is a function that maps a formula $\varphi$ to its code $\ulcorner \varphi \urcorner$ and ${\sf Bew}_T(\cdot)$ is a predicate in the language of $T$ formalizing provability in $T$, so that $T\vdash \varphi$ just in case $\mathbb{N} \vDash {\sf Bew}_T(\ulcorner \varphi \urcorner)$.
We define **PL**($T$), the *provability logic* of a theory $T$, as follows $$\mbox{\textbf{PL}($T$)} := \{ A \mid \forall * T \vdash A^* \}.$$ Since Löb ([@Loeb55]) it is known that **GL** is sound for a large class of theories $T$, that is, **GL** $\subseteq$ **PL**(). The reverse inclusion is Solovay’s completeness result.
\[theorem:solovay\] [**PL**(]{.nodecor}$T$[)]{.nodecor} = [**GL**]{.nodecor} for a wide range of theories $T$.
For soundness, i.e. **GL**$\subseteq$**PL**$(T)$, the theory can be as weak as $\textsf{I}\Delta_0 + \Omega_1$ or equivalently Buss’s ${\sf S^1_2}$ (see [@BerarducciVerbrugge] and [@Buss86]). Arithmetical completeness, i.e. that **PL**${(T)\subseteq}$**GL**, is known to hold for any sound[^2] theory extending $\textsf{I}\Delta_0 + \exp$ (see [@JongJumeletMontagna]).
Solovay proved that whenever **GL** $\nvdash A$, there is a realization $*$ so that $\nvdash A^*$. An outline of the proof runs as follows. First, a modal countermodel $\mathcal{M}$ in the form of a rooted tree is taken that witnesses **GL** $\nvdash A$. Next, a new root is added to this model. A primitive recursive function $f$ on this model is defined in terms of its own provable limit behavior. This definition is made using an arithmetical fixed point. The function $f$ starts in the newly added root and $f(x)$ remains where it is unless $x$ is a proof that the function does not have the node $y$, which is accessible from $x$, as a limit, in which case the function jumps to $y$. If $T$ is a sound theory, the function must stay where it started, in the newly added root. The realization $*$ is defined as a disjunction of the limit-statements $\lambda_y$ of the function $f$, where $\lambda_y$ says “$y$ is the limit of $f$". More specifically $p^*:= \bigvee_{\mathcal{M}, y \Vdash p}\lambda_y$.
Restricted Realizations
-----------------------
This ingenious proof thus gives us the concrete realization $*$. However, the arithmetical content of this realization $*$ is not exactly transparent.[^3] A natural question to ask is whether we can find translations with more clear arithmetical and proof theoretic content. And conversely, given a set of arithmetical sentences with a clear arithmetical content, what modal logics results from restricting realizations to this particular set? These questions motivate the following definition. We shall write, *par abus de langage*, $*\in \Gamma$ to mean that the realization $*$ takes on all its values within the set of sentences $\Gamma$.
[**PL**]{.nodecor}$_{\Gamma}(T) := \{ A : \forall *\in \Gamma , \ T\vdash A^* \}$
Notice that in a strict sense, [**PL**]{.nodecor}$_{\Gamma}(T)$ need not even be a logic as in general it is not closed under substitution. From the definition the following lemma is evident.
\[lemma:upperboundmethod\] If $\Gamma \subseteq \Delta$, then [**PL**]{.nodecor}$_{\Delta}(T) \ \subseteq$ [**PL**]{.nodecor}$_{\Gamma}(T)$.
Clearly, by taking $\Gamma$ to be the set of *all* arithmetical sentences we get **PL**$_{\Delta}(T)=$ **PL**$(T)$. For a large class of theories however, we can improve this to the following theorem.
\[theorem:BoolSigmaPL\] For all those theories $T$ for which Solovay’s Theorem \[theorem:solovay\] can be proved using the original proof we have that $$\textnormal{\textbf{PL}}_{\mathcal{B}(\Sigma_1)}(T) \ \subseteq \textnormal{\textbf{PL}}(T).$$ Here, $\mathcal{B}(\Sigma_1)$ denotes the class of Boolean combinations of $\Sigma_1$ sentences.
By close inspection of the proof of Solovay’s theorem, we see that all substitutions are disjunctions of limit statements. It is clear that for elementary functions $h$, the statement “$h$ has a limit" can be expressed in a $\Sigma_2$ fashion. However, as Albert Visser pointed out to us, the statement “$h$ has limit $i$" which is only actually needed in Solovay’s proof, can be expressed as $\mathcal{B}(\Sigma_1)$: $$\left[ \exists x\; h(x)=i\right] \ \ \wedge \ \ \left[ \forall y,z \; ((y\leq z \wedge hy=i) \to h(z) = i)\right].$$ Taking disjunctions of those sentences will of course not get us out of the class $\mathcal{B}(\Sigma_1)$.
The simple Lemma \[lemma:upperboundmethod\] can be used to establish upperbounds for a provability logic if one is unable to find the full provability logic. For example, it is a long standing open question what the provability logic is of bounded arithmetics such as $\sf S^1_2$.[^4]
Applications and plan of the paper {#subsection:ApplicationsAndPlan}
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One can thus use $\Gamma$ to study the provability logic of $T$. On the other hand, we shall see that **PL**$_{\Gamma}(T)$ can also be used to *characterize* the fragment $\Gamma$. For example, in Theorem \[theorem:mainTheorem\] below we consider the closed fragment $\mathcal{B}$ of provability logic, which consists of boolean combinations of iterated (in)consistency statements. This fragment is given by the following grammar. $$\mathcal{B}\ \ \ := \ \ \ \bot \ \mid \ \mathcal{B}\to \mathcal{B} \ \mid \ \Box \mathcal{B}.$$ We shall see that the modal formulas valid under all realizations from this fragment are exactly the formulas valid on all finite strict linear orders. This can be said to provide yet further evidence that reflection principles and iterated consistency statements are inherently linearly ordered.
Moreover, this fact also gives us information on what kind of arithmetical fixed point constructions are needed in the proof of Theorem \[theorem:solovay\]. By the modal Fixed Point Theorem, independently due to de Jongh and Sambin (see [@Sambin] (de Jongh actually never published his proof)), we know that certain applications of the arithmetical fixed point theorem can be dispensed with. More precisely, if we have a formula $A(x)$ where the $x$ only occurs directly under the scope of a ${\sf Bew}_T$ predicate then applying the fixed point to this formula does not give us new expressive power. That is, if we can prove $B \leftrightarrow A(\ulcorner B \urcorner)$ then $B$ is actually provably equivalent to a formula in the language of provability logic. Thus, these sort of applications of the arithmetical fixed point theorem only yield formulas in $\mathcal{B}$, whence, *pace* Theorem \[theorem:closedFragment\], cannot suffice for a proof of Solovay’s completeness result, Theorem \[theorem:solovay\].
Another example of restricting the substitutions is known in the literature. In [@viss:81] Visser studied the provability logic that arises when restricting substitutions to $\Sigma_1$ sentences.
In Section \[section:glp\] we shall consider a fragment $\mathcal{D}$ which contains infinitely many copies of $\mathcal{B}$ for increasingly strong provability predicates. It turns out that even for this richer fragment we do not move beyond linear frames (cf. Theorem \[theorem:ClosedFragmentGLP\]). However, in Section \[section:NonLinear\] we shall see that there is a natural fragment for whose associated provability logic lies strictly in between the logic of linear frames and **GL**.
In Section \[section:InterpretLogics\] we shall see how restricted realizations can also be applied to interpretability logics.
Fragments and Logics {#fragmentslogics}
====================
In this section we show that certain conditions on a given fragment translate to a semantic characterization of the corresponding restricted provability logics. First, some preliminaries on basic relational semantics for provability logics.
Recall that a *frame* $\mathbb{F}$ for **GL** is an ordered pair $\langle W, R \rangle$, where $W$ is a set of points and $R \subseteq W \times W$ is a finite, irreflexive, and transitive relation. Given a set of propositional variables, a *model* $\mathcal{M}$ *based on $\mathbb{F}$* is a triple $\langle W,R,V \rangle$, where $V: \textsf{Prop} \rightarrow \wp(W)$ is a *valuation function* assigning to each variable the set of the points where it is true. We shall also write $V$ for the straightforward extension of $V$ to arbitrary modal formulas. We then write $\langle W,R,V \rangle , w \vDash A$, just in case $w \in V(A)$. We write $\langle W,R,V\rangle \vDash A$ if $A\in V(w)$ for all $w\in W$. Overloading notation, we also write $\langle W,R \rangle \vDash A$, if $\langle W,R,V \rangle \vDash A$ for all $V$. We say $A$ is *valid* in the model and in the frame, respectively. When dealing with fragments, however, arbitrary variables will not be present. All of the fragments we shall consider in this paper will extend the fragment $\mathcal{B}$ defined above, by adding constants $\sigma _1, \sigma _2, \sigma _3,...$, with some clear arithmetical content. As these constants will be fixed, and as we would like to characterize the sentences in this fragment modally, we shall add constants $s_1, s_2, s_3,...$, to our modal language, and correspondingly extend the definition of a realization to ensure that $(s_i)^* = \sigma _i$. In fact, given this convention, we will be able to define our fragments in a single language and throughout treat each constant simultaneously as a constant in the modal language and as a specified arithmetical formula, disambiguating whenever the distinction is not clear from context. In other words, we will usually not distinguish between $A$ and $A^*$.
On the other hand, as far as the relational semantics is concerned, the constants $s_1, s_2, s_3,...$ are simply treated as variables. Therefore the above notation is extended in the obvious way to this setting.
Suppose we would like to obtain a modal characterization of **PL**$_{\mathcal{F}}(T)$. Under certain circumstances, it suffices to know how $\mathcal{F}$ is characterized according to $T$. To be precise, if we have a model $\mathcal{M}$ based on a frame $\mathbb{F}$, such that for each $A \in \mathcal{F}$, the following condition holds: $$\label{equation:FroedelCondition}
T \vdash A \ \Leftrightarrow \ \mathcal{M} \vDash A,$$ then, as is shown in Theorem \[theorem:mainTheorem\] below, **PL**$_{\mathcal{F}}(T) = \mathcal{L}(\mathbb{F})$. Here, $\mathcal{L}(\mathbb{F})$ is the set of formulas in the basic modal language (with propositional variables) valid on the frame $\mathbb{F}$.
There are two side conditions to our theorem. One of them involves image-finiteness. We call a model *image-finite* if $\{y : xRy\}$ is finite for each $x$. We shall denote the set $\{y : xRy\} \cup \{x\}$ by $x\uparrow$. Our theorem thus reads as follows:
\[theorem:mainTheorem\] Suppose that holds for a model $\mathcal{M}$ based on frame $\mathbb{F}$. Suppose moreover that $\mathcal{M}$ is image-finite and that each point $x\in \mathcal{M}$ is uniquely definable by a formula $D_x \in \mathcal{F}$. Then, we have that [**PL**]{.nodecor}$_{\mathcal{F}}(T) = \mathcal{L}(\mathbb{F})$.
In the light of it suffices to prove that $$\forall \, *\, \in \mathcal{F},\ \mathcal{M}\vDash B^*\ \Leftrightarrow \ \mathbb{F} \vDash B.$$
- Consider some arbitrary $* \in \mathcal{F}$ and define $V_*(p) := \{ i : \mathcal{M},i \vDash p^*\}$. By induction on $A$ we see that for each $i \in \mathbb{F}$ $$\langle \mathbb{F}, V_* \rangle, i \Vdash A \ \Leftrightarrow \ \mathcal{M}, i \Vdash A [p/p^*]$$ and we are done.
- Given some $i \in \mathbb{F}$ and some arbitrary valuation $V$ we define $*$ by $$p^* := \bigvee_{x\in V(p) \cap i\uparrow} D_x.$$ As the frame is image-finite, the disjunction is finite. By an induction[^5] on $C$ we see again that $$\langle \mathbb{F}, V \rangle, i \vDash C \ \Leftrightarrow \ \mathcal{M}, i \vDash C^*.$$ As $i$ was arbitrary, we see that $\mathbb{F} \vDash C$.
As we shall see below, in many occasions we actually will have something stronger than . In particular we shall often find ourselves in a situation where we have, apart from the frame, also a modal logic **L** for which we have $$T \vdash A \ \Leftrightarrow \ {\bf L} \vdash A \ \Leftrightarrow \ \mathcal{M} \vDash A.$$ This logic **L** will facilitate our calculations considerably.
The Closed Fragment {#section:closedfragment}
===================
With Theorem \[theorem:mainTheorem\] we can calculate our first provability logic with restricted substitutions. Recall the definition of the closed fragment $\mathcal{B}$ in Subsection \[subsection:ApplicationsAndPlan\].
[**GL.3**]{.nodecor} is the logic [**GL**]{.nodecor} together with the linearity axiom: $$\Box(\Box A \to B) \vee \Box (\Box^+ B \to A).$$ Here and below, $\Box^+ A$ is short for $A \wedge \Box A$.
\[theorem:closedFragment\] [**PL**$_{\mathcal{B}}$($T$)]{.nodecor}= [**GL.3**]{.nodecor} for a large class[^6] of theories $T$.
It is well known that the truth of a closed formula at a particular point in a model depends solely on the rank of that point. Here, the rank of a point $x$ is defined as the supremum of lengths of paths leading from $x$ to a leaf. See for example Chapter 7 from [@Bool93].
Thus, the linear frame $\langle \omega, > \rangle$ is universal for $\mathcal{B}$ in the sense that if a formula $\varphi \in \mathcal{B}$ is false at some point in some frame, then it is actually false at some point in $\langle \omega, > \rangle$. Thus, by Theorem \[theorem:solovay\], we have $T \vdash A \ \Leftrightarrow \ \langle\omega,>\rangle \vDash A$.
Furthermore, it is known that the logic of the frame $\langle \omega, > \rangle$ is axiomatized by **GL.3**. (See, for example, Chapter 13 of [@Bool93].) Thus, $\langle\omega,>\rangle \vDash A\ \Leftrightarrow \ \mbox{\textbf{GL.3}}\vdash A$ and Condition \[equation:FroedelCondition\] is satisfied for any model based on $\langle\omega,>\rangle$.
Note that $\langle\omega,>\rangle$ is image-finite and that the point $n$ is defined by $\Diamond^n \top \wedge \Box^{n+1}\bot$. Thus, by Theorem \[theorem:mainTheorem\] we have our result.
Substitutions from the Closed Fragment of **GLP** {#section:glp}
=================================================
Japaridze’s Logic **GLP** ([@Jap86]) describes all of the universally valid schemata for reflection principles of restricted logical complexity in arithmetic. It is formulated in a language with infinitely many modalities, where $[n]A$ is read arithmetically as,
*$A$ is provable from $T$ along with all true $\Pi_n$ sentences.*
Arithmetical completeness with respect to this interpretation was proven in [@Ign93], for sound theories containing only a modest amount of arithmetic.
[**GLP**]{.nodecor} is given by the following axiom schemata,
1. All boolean tautologies;
2. $[n]([n]A \rightarrow A) \rightarrow [n]A$, for all $n$;
3. $[m]A \rightarrow [n]A$, for $m \leq n$;
4. $\langle m \rangle A \rightarrow [n]\langle m \rangle A$, for $m<n$;
in addition to the rules of *modus ponens* and necessitation for each $[n]$.
While **GLP** does not admit of any frame semantics, various other models have been given (see, e.g. [@Bek09] and [@BBI09]). In particular, Ignatiev [@Ign93] has defined a *universal frame* for the closed fragment of **GLP**, denoted **GLP**$_0$, which will be of use.[^7]
Define $\mathcal{D}$ to be the fragment given by the following infinite grammar: $$\mathcal{D} \; := \; \bot \; \vert \; \mathcal{D} \rightarrow \mathcal{D} \; \vert \; [0]\mathcal{D} \; \vert \; [1]\mathcal{D} \; \vert \; [2]\mathcal{D} \; \vert \; ...$$ That is, **GLP**$_0$ is simply **GLP** restricted to the fragment $\mathcal{D}$, with no variables.
We can describe Ignatiev’s universal frame for **GLP**$_0$ as follows. Let $\Omega$ consist of the set of $\omega$-sequences of ordinals $(\alpha _0,\alpha _1, \alpha _2,...)$, where each $\alpha _i < \epsilon _0$. Recall $\epsilon _0$ is the least fixed point of the equation $\omega ^{\alpha} = \alpha$. If the Cantor Normal Form of $\alpha$ is $\omega ^{\lambda _n} +...+ \omega ^{\lambda _1}$, then let $e(\alpha) := \lambda _1$ and set $e(0) = 0$.
Ignatiev’s universal frame is defined as $\mathcal{U} := \langle U, \{R_n\}_{n<\omega}\rangle$, with, $$U \; := \; \{\vec{\alpha} \in \Omega: \forall i < \omega, \alpha _{i+1} \leq e(\alpha _i)\};$$ $$\vec{\alpha}R_n\vec{\beta} \; :\Leftrightarrow \; (\forall m < n, \alpha _m = \beta _m \; \& \; \alpha _n > \beta _n).$$
Notice that each point in $U$ can be seen as a finite, strictly decreasing sequence of ordinals less than $\epsilon _0$, as each sequence ends in an infinite tail of zeros. For a visualization of the frame, see Figure \[fig:ignatiev\].
A point of the form $(\alpha,e(\alpha),e(e(\alpha)),...)$, where $\alpha _{i+1} = e(\alpha _i)$ for all $i$, is called a *root point*, and is denoted by $\widehat{\alpha}$ when $\alpha$ is the first coordinate. Thus every coordinate of $\widehat{\alpha}$ is uniquely determined by $\alpha$. The following lemma is then obvious, given the definition of $\mathcal{U}$.
\[compar\] If $\widehat{\alpha}$ and $\widehat{\beta}$ are root points, then either $\widehat{\alpha} R_0 \widehat{\beta}$, $\widehat{\beta}R_0\widehat{\alpha}$, or $\widehat{\alpha} = \widehat{\beta}$.
In addition to the more routine soundness, the following strong completeness theorem has also been proven using several different methods in the works cited above.
\[arithcom\] If [**GLP**]{.nodecor}$_0 \nvdash A$, then there is a root point $\widehat{\alpha} \in U$, such that $\mathcal{U},\widehat{\alpha} \nvDash A$.
With these results we can now show that even with this much richer fragment the resulting provability logic is exactly the same as for the fragment with only the single $\Box$-operator (c.f. Theorem \[theorem:closedFragment\]).
\[theorem:ClosedFragmentGLP\] [**PL**$_{\mathcal{D}}$() = **GL.3**]{.nodecor}.
By Theorem \[theorem:closedFragment\], by Lemma \[lemma:upperboundmethod\] and by observing that $\Box$ is just $[0]$, it is clear that **PL**$_{\mathcal{D}}(\textsf{PRA}) \subseteq \textbf{GL.3}$. For the other inclusion, we must show, under the arithmetical interpretation, $$\textsf{PRA} \vdash \Box(\Box A \rightarrow B) \vee \Box (\Box ^+B \rightarrow A),$$ for any $A,B \in \mathcal{D}$. However, this follows by arithmetical completeness and by the universality of Ignatiev’s frame.
For, suppose $\mathcal{U}, \vec{\alpha} \vDash \Diamond (\Box A \wedge \neg B) \wedge \Diamond (\Box^+B \wedge \neg A)$, for some $\vec{\alpha}$. By Theorem \[arithcom\] there are root points $\widehat{\beta}$ and $\widehat{\gamma}$, such that $\mathcal{U},\widehat{\beta} \vDash \Box A \wedge \neg B$, and $\mathcal{U},\widehat{\gamma} \vDash \Box^+B \wedge \neg A$. By Lemma \[compar\], either $\widehat{\beta} R_0 \widehat{\gamma}$, $\widehat{\gamma}R_0\widehat{\beta}$, or $\widehat{\beta} = \widehat{\gamma}$. All three lead to contradiction.
[Non-Linear **GL**-frames]{}\[section:NonLinear\]
Theorems \[theorem:closedFragment\] and \[theorem:ClosedFragmentGLP\] suggest that it may not be straightforward to define a fragment whose associated restricted provability logic is anything other than **GL.3** or just **GL**. In this section we fill in this gap by giving some sufficient conditions on constants, so that we obtain logics of non-linear **GL**-frames. We will be working with generic fragments $\mathcal{F}_n$, with some finite number $n$ of constants: $$\mathcal{F}_n \; := \; s_1 \; \vert \; s_2 \; \vert \; ... \; \vert \; s_n \; \vert \; \bot \; \vert \; \mathcal{F}_n \rightarrow \mathcal{F}_n \; \vert \; \Box \mathcal{F}_n$$ As before, we will be viewing formulas in $\mathcal{F}_n$ simultaneously as arithmetical formulas, where each $s_i$ is a specified formula in the language of arithmetic and $\Box$ is the standard provability predicate, and as modal formulas, where each $s_i$ is interpreted as a constant and $\Box$ is a normal modal operator.
[Fragments, Logics and Models]{} Let $\vec{{\bf s_i}}$ stand for the sentence $\bigwedge _{j \in J} s_{j+1} \wedge \bigwedge _{k \in K} \neg s_{k+1}$, where $J$ is the set of places in the binary expansion for $i$ with value 1, and $K$ is the complement of $J$ in $\{ 0, \dots,i-1\}$. Then we define the following class of logics.
The logic [**FGL**]{.nodecor}$_n$ is formulated in the language $\mathcal{F}_n$ and thus, contains no propositional variables. The axioms and rules are specified by the axioms and rules of **GL** together with the list of the $2^n$ many axioms below, one axiom for each Boolean combination of the $s_i$. The $B$ in these axioms stands for any formula that is a Boolean combination of formulas of the form $\Box^{\alpha}\bot$, where $\alpha < \omega +1$ and $\Box ^{\omega}\bot := \top$.
- $\Box( \vec{{\bf {s_0}}} \rightarrow B) \rightarrow \Box B$;
- $\vdots$
- $\Box( \vec{{\bf s_{2^{n}-1}}} \rightarrow B) \rightarrow \Box B$.
These logics [**FGL**]{.nodecor}$_n$ come with an associated model, based on the following frames:
The frame $\mathfrak{G}_n := \langle G_n, R_n \rangle$, where $G_n := \{\langle m,i \rangle : m \in \omega, i < 2^n\}$, and $\langle m, i \rangle R_n \langle p,j \rangle$ just in case $p < m$.
The associated model defined on this frame is given *via* the binary expansion, where $J_j$ is given as above, relative to $j$.
$\mathfrak{G}_n^{\bullet}$ is the triple $\langle G_n, R_n, V_n \rangle$, where $V_n(s_j) = \{\langle m,i \rangle : i \in J_j\}$.
For a visualization of $\mathfrak{G}_1^{\bullet}$, see Figure \[figu:pglsemantics\].
\[theorem:frodelcompleteness\] For all formulas $A \in \mathcal{F}_n$, [**FGL**]{.nodecor}$_n \vdash A$, if and only if $\mathfrak{G}_n^{\bullet} \vDash A$.
\[Proof Sketch\] The full proof for the case of $\mathcal{F}_1$ is established in [@Joo04]. Here we give a sketch for the general case. Soundness is routine. For completeness, we use the following two lemmata.
\[lemma:normalforms\] Each $A \in \mathcal{F}_n$ is equivalent in [**FGL**]{.nodecor}$_n$ to a Boolean combination of formulas of the form $s_1$,..., $s_n$, or $\Box ^{\alpha}\bot$. In particular [**FGL**]{.nodecor}$_n \vdash \Box A \leftrightarrow \Box^{\alpha}\bot$ for some $\alpha < \omega +1$.
\[lemmabox\] If [**FGL**]{.nodecor}$_n \vdash \Box A$, then [**FGL**]{.nodecor}$_n \vdash A$.
These lemmata are straightforwardly proven by manipulation of modal normal forms. Completeness is then clear. If **FGL**$_n \nvdash A$, then by Lemma \[lemmabox\], **FGL**$_n \nvDash \Box A$, and by Lemma \[lemma:normalforms\], **FGL**$_n \vdash \Box A \leftrightarrow \Box ^{\alpha} \bot$, for some $\alpha < \omega$ (in particular $\alpha \neq \omega$). By soundness, for any point $\langle m,i \rangle \in G_n$ we know $\mathfrak{G}_n^{\bullet}, \langle m,i \rangle \vDash \Box A \leftrightarrow \Box ^{\alpha} \bot$. Certainly $\mathfrak{G}_n^{\bullet}, \langle \alpha, 0 \rangle \nvDash \Box ^{\alpha} \bot$, so $\mathfrak{G}_n^{\bullet}, \langle \alpha, 0 \rangle \nvDash \Box A$. That, in turn, means for some $\langle \beta, j \rangle$ with $\beta < \alpha$, we have $\mathfrak{G}_n^{\bullet}, \langle \beta , j \rangle \vDash \neg A$. So $A$ is falsified on $\mathfrak{G}_n^{\bullet}$.
[Conditions for completeness]{} \[subsection:conditions\] Suppose we have a given theory $T$ and some fragment $\mathcal{F}_n$, and we would like a characterization of **PL**$_{\mathcal{F}_n}(T)$. In Section \[fragmentslogics\] we showed that if condition (1) holds for some logic **L** and model $\mathcal{M}$, then Theorem \[theorem:mainTheorem\] will follow. Recall Condition (1): $$T \vdash A \ \Leftrightarrow \ \textbf{L} \vdash A \ \Leftrightarrow \ \mathcal{M} \vDash A.$$ To show (1) holds for this case, one merely needs to show arithmetical soundness and completeness of **L** for $T$ . However, given Lemmata \[lemma:normalforms\] and \[lemmabox\], arithmetical completeness of **L** depends only on arithmetical soundness of **L**.
To see this, suppose **FGL**$_n \nvDash A$. Then by Lemma \[lemma:normalforms\], **FGL**$_n \nvDash \Box A$. Since **FGL**$_n \vDash \Box A \leftrightarrow \Box ^{\alpha} \bot$, for $\alpha \neq \omega$, as long as we have soundness of **L**, $T \vdash \Box A \leftrightarrow \Box ^{\alpha} \bot$, under the arithmetical interpretation. Now, if moreover $T$ is a sound theory in the sense that it does not prove any false statements we get $T \nvdash \Box A$, from which it follows $T \nvdash A$.
Consequently, the following is a corollary of Theorem \[theorem:mainTheorem\] and Theorem \[theorem:frodelcompleteness\]. Note that both image finiteness and definability of the states in the model $\mathfrak{G}_n^{\bullet}$ are evident.
\[corollary:sufficient\] [**PL**]{.nodecor}$_{\mathcal{F}_n}(T) = \mathcal{L}(\mathfrak{G}_n)$ whenever $[\textnormal{\textbf{FGL}}_n \vdash A \ \Rightarrow T \vdash A.]$
In Section \[qgl\], we shall see that each of these frames $\mathfrak{G}_n$ has a simple axiomatization. For the rest of this section, we exhibit a suitable constant for the case of $\mathcal{F}_1$.
[A Constant for .]{} Recall is the theory ([@Tarski]) along with induction over $\Sigma _1$ formulas. This theory is finitely axiomatizable, so let $\sigma$ stand for the sentence axiomatizing it. We then define the fragment $\mathcal{Q}$ as a special case of $\mathcal{F}_1$: $$\mathcal{Q} \; := \; \sigma \; \vert \; \bot \; \vert \; \mathcal{Q} \rightarrow \mathcal{Q} \; \vert \; \Box Q$$ Our theory $T$ will be Primitive Recursive Arithmetic (), essentially just with function symbols for all of the primitive recursive functions and induction over $\Delta _0$ formulas. The relationship between and is well studied and understood ([@Min72], [@Par72], [@Bek96]). By Corollary \[corollary:sufficient\], we need to show that **FGL**$_n$ is sound with respect to . It is already well known that **PL**() = **GL**, so certainly all the axioms and rules of **GL** are sound. We need only observe the following also hold:
- $\vdash \Box(\sigma \rightarrow B) \rightarrow \Box B$,
- $\vdash \Box(\neg \sigma \rightarrow B) \rightarrow \Box B$.
In fact, item (i) is a direct consequence of what is known as Parson’s Theorem (named after Charles Parsons, but discovered independently by Grigori Mints and Gaisi Takeuti), which says that $\Sigma _1$ is $\Pi _2$-conservative over . In [@Bek96] it is shown that this theorem is in fact formalizable in , which gives us (i).
\[Parson’s Theorem\] []{.nodecor} $\vdash \forall ^{\Pi _2} B \ (\Box (\sigma \rightarrow B) \rightarrow \Box B)$.
So this certainly holds for $\mathcal{B}(\Sigma _1)$ formulas consisting of Boolean combinations of formulas of the form $\Box ^{\alpha} \bot$. As for (ii), it is shown in [@Joo04] that the negation of the sentence axiomatizing $\Sigma _1$ is $\Pi _3$-conservative over . That is, we have the following lemma:
[]{.nodecor} $\vdash \forall ^{\Pi _3} B \ (\Box (\neg \sigma \rightarrow B) \rightarrow \Box B)$.
Thus, we can state the following corollary:
\[corollary:logicofq\] [**PL**]{.nodecor}$_{\mathcal{Q}}(\textnormal{\textsf{PRA}}) = \mathcal{L}(\mathfrak{G}_1)$.
While the logic **GL.3** of the linear frame $\mathfrak{G}_0$ is well known, that of $\mathfrak{G}_1$ is not. Therefore in the following section we provide a simple axiomatization. Our work can then be generalized to arbitrary $\mathfrak{G}_n$.
[The Logic of $\mathfrak{G}_1$]{} \[qgl\]
[The Modal Logic **GL.4** and its corresponding class of frames]{} We define **GL.4** to be the normal modal logic obtained by adding to **GL** the following two axiom schemata:
- $\Box (\Box A \rightarrow (B \vee C)) \vee \Box (\Box ^+ B \rightarrow (A \vee C)) \vee \Box (\Box ^+ C \rightarrow (A \vee B))$;
- $\Diamond (\Diamond A \wedge \Box B) \rightarrow \Box(\Diamond A \vee B)$.
**GL.4** in fact defines a natural class of frames. We define $\mathcal{C}$ to be the class satisfying the following properties:
- Finite, irreflexive and transitive;
- Non-triple branching: $(xRy \; \& \; xRz \; \& \; xRw) \Rightarrow$ $$(wRy \vee yRw \vee zRw \vee wRz \vee yRz \vee zRy \vee w=y \vee z=y \vee w=z);$$
- Strongly confluent: $(xRy \; \& \; xRz \; \& \; yRw) \Rightarrow (zRw \vee wRz \vee yRz)$.
\[qglcom\] [**GL.4**]{.nodecor} is sound and complete with respect to $\mathcal{C}$.
Soundness is proven as usual by induction on complexity of proofs. As for completeness, we shall appeal to the canonical model of **GL.4** (see Definition 4.18 of [@Bla01]). In particular we use the finite filtration method to transform the canonical model into a model in the class $\mathcal{C}$.
Recall the canonical model $\mathfrak{M}$ of **GL.4** is the triple, $\langle W^{GL.4}, R^{GL.4}, V^{GL.4} \rangle$ with
- $W^{GL.4}$ is the set of maximal **GL.4**-consistent sets;
- For $\Gamma, \Delta \in W^{GL.4}$, define $\Gamma R^{GL.4} \Delta$ if for all $\phi \in \Delta$ we have $\Diamond \phi \in \Gamma$;
- $V(p) = \{\Gamma : p \in \Gamma\}$, for propositional variables $p$.
First, we make some key observations about this model, the verifications of which are straightforward.
\[c2\] C2 holds on $\mathfrak{M}$.
\[c3\] C3 holds on $\mathfrak{M}$.
In fact, these follow by the fact that axiom *Q1* is *canonical* for property *C2*, as is axiom *Q2* for *C3* (see [@Bla01], Definition 4.31). Thus, it remains to show that we can transform the underlying frame of $\mathfrak{M}$ into a finite partial order, while preserving validity of formulas.
\[Proof of Theorem \[qglcom\]\] Suppose that **GL.4** $\nvdash A$, for some formula $A$. We would like to find a maximal consistent set $\Gamma$ such that $(\Box A \wedge \neg A) \in \Gamma$, so that $\Gamma$ is an ‘irreflexive’ point in the canonical model.
By the fact that $A$ is not a theorem, we are guaranteed of some $\Delta \in W^{GL.4}$ such that $A \notin \Delta$. If $\Box A \in \Delta$, then set $\Gamma := \Delta$. Otherwise, since $\neg \Box A \in \Delta$, by the contrapositive form of Löb’s Theorem $\Diamond (\Box A \wedge \neg A) \in \Delta$. Thus by the so-called ‘Existence Lemma’ ([@Bla01], Lemma 4.20) for normal modal logics, $\Delta$ is $R^{GL.4}$-related to some $\Sigma$ for which $(\Box A \wedge \neg A) \in \Sigma$. In that case, set $\Gamma := \Sigma$.
Either way we have some $\Gamma$ with $(\Box A \wedge \neg A) \in \Gamma$. Notice also, if $\Box C$ is a subformula of $A$, and $\Box C \notin \Gamma$, then by the same argument there is some ‘irreflexive’ $\Delta$ such that $\Gamma R^{GL.4} \Delta$ and that $(\Box C \wedge \neg C) \in \Delta$. Moreover, by Lemma \[c2\] there are at most two distinct such $\Delta$.
With these observations in place, our filtrated model $\mathfrak{M'} = \langle W,R,V \rangle$ will be defined as a submodel of $\mathcal{M}$:
- $W := \{\Gamma\} \cup \{\Delta: \Gamma R^{GL.4} \Delta$, and there is $\Box C$ subsentence of $A$, such that\
$(\Box C \wedge \neg C) \in \Delta$ and $\neg \Box C \in \Gamma\}$;
- $R$ is just $R^{GL.4}$ restricted to points in $W$;
- $V(p) := V^{GL.4}(p) \cap W$.
The model $\mathfrak{M'}$ satisfies *C2*, *C3*, and transitivity simply because $\mathfrak{M}$ does. It is clearly finite. And irreflexivity, as hinted above, follows from the fact that each point in $W$ was chosen to contain some formulas $\Box C$ and $\neg C$, ensuring the point is not related to itself. It follows $\mathcal{M'}$ is in $\mathcal{C}$.
The standard ‘Truth Lemma’ is then proven by induction:
If $\Delta \in W$ and $B$ is a subsentence of $A$, then $B \in \Delta$ iff $\mathfrak{M'}, \Delta \vDash B$.
Concluding the proof, since $A \notin \Gamma$, we have that $\mathfrak{M'},\Gamma \nvDash A$.
[The Class $\mathcal{C}$ and the Frame $\mathfrak{G}_1$]{}
We must now show that **GL.4** is the logic of the frame $\mathfrak{G}_1$.
Recall a $p$-morphism from $\mathbb{F} = \langle W,R \rangle$ to $\mathbb{F'} = \langle W',R' \rangle$ is a function $f: W \rightarrow W'$, such that $xRy$ implies $f(x)R'f(y)$; and if $f(x)R'y'$ then there is some $y \in W$ such that $f(y) = y'$ and $xRy$. The following theorem is standard:[^8]
If there is a $p$-morphism from $\mathbb{F}$ to $\mathbb{F'}$, then the existence of a valuation $V'$ and point $w' \in W'$ such that $\langle \mathbb{F'},V' \rangle , w' \nvDash A$, ensures the existence of a valuation $V$ and point $w \in W$, such that $\langle \mathbb{F}, V \rangle , w \nvDash A$.
To demonstrate that **GL.4** is the logic of $\mathfrak{G}_1$, we use the following proposition:
\[pmorph\] For any frame $\mathbb{F} \in \mathcal{C}$ and any point $x$ in $\mathbb{F}$, there is some point $\langle m,i \rangle$ in $\mathfrak{G}_1$, such that there exists a $p$-morphism from the subframe generated by $\langle m,i \rangle$ to the subframe generated by $x$.
In other words, falsifiability is reflected by $p$-morphisms, which gives us the following corollary of Proposition \[pmorph\] and improvement upon Corollary \[corollary:logicofq\].
[**PL**]{.nodecor}$_\mathcal{Q}$[() =]{.nodecor} [**GL.4**]{.nodecor}.
It remains only to verify Proposition \[pmorph\].
\[Proof Sketch of Proposition \[pmorph\]\] The proof proceeds by induction on the number of points in a frame in $\mathcal{C}$. The basic case is obvious. Supposing we have a frame with one point, say $x$, then consider the subframe generated by $\langle 0,0 \rangle$, and the $p$-morphism mapping $\langle 0,0 \rangle$ to $x$.
Supposing we have a frame in $C$ with $n+1$ points, consider the subframe $\mathbb{F} = \langle W,R \rangle$ generated by some point $x\in C$. We would like to use the inductive hypothesis to obtain a $p$-morphism to some subframe of $\mathbb{F}$ containing $\leq n$ points, and extend it to all of $\mathbb{F}$. To do this we consider three cases: (i) $x$ has no successors; (ii) $x$ has one immediate successor (i.e. point $y$ such that $xRy$ and there is no $z$ with $xRzRy$); and (iii) $x$ has two immediate successors. More than 2 immediate successors is ruled out by property *C2*.
Case (i) is trivial. For case (ii), let $\mathbb{F'}$ be $\mathbb{F}$ without the point $x$, and let $y$ be the unique immediate successor of $x$. Then since $\mathbb{F'} \in \mathcal{C}$ and it has $n$ points, we have a $p$-morphism $f$ from the subframe generated by some point $\langle m,i \rangle$ in $\mathfrak{G}_1$ to $\mathbb{F'}$, the subframe generated by $y$. We then consider the subframe generated by $\langle m+1,i \rangle$ instead, and extend the $p$-morphism $f$ so that $f(\langle m+1,i \rangle) = x$ and $f(\langle m,i-1 \rangle) = y$.
Verifying case (iii) is similar, except that instead of removing the point $x$, we must remove the ‘maximal’ points of $\mathbb{F}$. Then the $p$-morphism obtained by inductive hypothesis is extended by shifting each point in the morphism by one. Thus, e.g. if $\langle m,i \rangle$ is mapped to $y$, then in the new mapping $\langle m+1,i \rangle$ is mapped to $y$. And we let $f(\langle 0,0 \rangle) = f(\langle 0,1 \rangle) = x$. The details are straightforward and are left to the reader (or can be found in [@Ica07]).
The methods in this section carry over to the general case of frames $\mathfrak{G}_n$ for arbitrary $n$. By an analogous argument, one can prove the logic is simply *Q2* (strong confluence) and the axiom corresponding to “non-n+2-ary-branching", which is just a generalization[^9] of non-branching and non-triple-branching: $$\bigvee_{i \leq n+1} \Box(\Box^+ A_i \to \bigvee_{i\neq j} A_j).$$
[On the proof of Solovay’s Theorem]{} \[section:Solovay\] In Sections \[section:closedfragment\] and \[section:glp\] we showed that **PL**$_{\mathcal{F}}(T) =$ **GL.3** for a wide range of arithmetical theories $T$ and fragments $\mathcal{F}$. Otherwise put, **PL**$_{\mathcal{F}}(T)$ gives us the logic of non-branching **GL**-frames. *Prima facie*, one might imagine the possibility of strategically adding sentences into the fragment $\mathcal{F}$ (where $\mathcal{F}$ is, e.g. $\mathcal{B}$), so as to obtain the logic of non-triple-branching **GL**-frames, then that of non-quadruple-branching **GL**-frames, and so on. Assuming this could be generalized it would be possible to define an infinite fragment $\mathcal{H}$, for which **PL**$_{\mathcal{H}}(T) =$ **GL**. At that point, to the extent that Solovay’s Theorem is not already assumed in the determination of $\mathcal{H}$, we would have a new proof of the result. After all, any non-theorem of **GL** can be falsified on some finite, and thus finitely branching, frame. So the witnessing realization would make use of some finite subset of the fragment, sufficient to falsify the formula.
What we have shown is that the first step in this process is (almost) possible, *vis-à-vis* Corollary \[corollary:logicofq\]. Adding the constant for and capitalizing on the well studied relationship between that theory and , we are able to obtain the logic of non-triple-branching (and strongly confluent) **GL**-frames. Two important questions remain, however, before taking the next step.
The first and most obvious question is what the further constants will be. The particular case of $\Sigma _1$ and is already well studied. Going beyond that may require some significant arithmetical investigation. In Section \[subsection:conditions\] we isolated what arithmetical facts are sufficient to hold. So on the proposed strategy it would simply be a matter of finding a theory and a fragment that satisfy these requirements.
The second, and more curious, question is how to dispense with property *C3*, strong confluence. We have seen that the logic of the frame **G**$_n$ always contains the formula *Q2*, and so it will clearly remain in the limit. However *Q2* is obviously not a theorem of **GL**. Finding constants whose associated provability logics do not validate *Q2* may prove a challenge. Understanding this situation may shed light on Solovay’s original proof.
Interpretability Logics with Restricted Substitutions {#section:InterpretLogics}
=====================================================
Interpretations are used throughout mathematics and logic. Loosely speaking, an interpretation from a theory $V$ into a theory $U$ is structure preserving map that translates theorems of $V$ to theorems of $U$. The notion of interpretability that we discuss below is *grosso modo* that of [@Tarski] and details can be found in, e.g. [@JapJongh] or in [@Visser97].
Interpretability Logics
-----------------------
Interpretability can be seen as a generalization of provability. By $\alpha \rhd_T \beta$ we denote a natural formalized version of the statement that $T + \beta$ is interpretable in $T+\alpha$.
*Interpretability Logics* are designed to capture the structural behavior of formalized interpretability. The language of these logics is that of provability logic together with a binary modality $\rhd$, orthographically identical to the arithmetical operator, to model formalized interpretability. And indeed, *arithmetical realizations* are extended as expected by imposing that $$(A \rhd B)^* \ =\ A^* \rhd B^*.$$ For a clear distinction, let [${\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}$ ]{}denote the class of modal formulas in language of interpretability logic and [${\sf Form}_{{{\ensuremath{\textup{\textbf{GL}}}}\xspace}}$ ]{}the standard modal language of basic provability logic. In analogy with the definition of **PL**$(T)$ we define **IL**$(T)$, the interpretability logic of a theory $T$ $$\textbf{IL}(T) := \{ A \in {\ensuremath{{\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}} \xspace}\mid \forall * \ T \vdash A^*\} \ \ \ \mbox{and} \ \ \ \textbf{IL}_{\Gamma}(T) := \{ A \in {\ensuremath{{\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}} \xspace}\mid \forall *{\in} \Gamma \ T \vdash A^*\}.$$ By Theorem \[theorem:solovay\] and Footnote \[footnote:ClassOfTheories\] we see that provability logics are the same for all sufficiently strong theories. This is certainly not the case for interpretability logics, which turn out to be more sensitive to differences between theories. One such example is the notion of an *essential reflexive* theory.
A theory is *reflexive* if it proves the consistency of any finite subpart of it. A theory is essentially reflexive whenever any finite extension of it is reflexive. The following theorem is due independently to A. Berarducci and V. Shavrukov. The definition of [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}will follow below.
\[theo:shav\] If $T$ is an essentially reflexive and $\Sigma_1$ sound theory, then ${{\ensuremath {\textup{\textbf{IL}}}({\rm T})}}={{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$.
However, if a theory is finitely axiomatizable we get a different outcome where, again, [[$\textup{\textbf{IL}}{\sf\ensuremath{P}}$]{}]{}is defined below.
\[thm:ilp\] If $T$ is finitely axiomatizable, $\Sigma_1$ sound, and extending $\textsf{I}\Delta_0 + {\sf supexp}$, then ${{\ensuremath {\textup{\textbf{IL}}}({\rm T})}}={{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{P}}}\xspace}}$.
A prominent problem in formalized interpretability is to determine the maximal interpretability logic that is contained in any reasonable arithmetical theory.
\[defi:ilall\] The interpretability logic of all reasonable arithmetical theories, written [[$\textup{\textbf{IL}}$([All]{})]{}]{}, is the set of formulas $\varphi$ such that for all $T$ and $*$, $T\vdash \varphi^*$. Here we let $T$ range over all reasonable[^10] arithmetical theories.
Clearly, [[$\textup{\textbf{IL}}$([All]{})]{}]{} is in the intersection of [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}and [[$\textup{\textbf{IL}}{\sf\ensuremath{P}}$]{}]{}but apparently it possesses a very rich structure (see [@joo:prol00], and [@GorisJoostenNewPrinc]). In this paper, it is only important to know that a certain very weak logic to be defined below is part of [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{}.
\[fact:IlwInILall\] ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{W}}}\xspace}}\subset {{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$
For most theories that do not fall under Theorems \[theo:shav\] and \[thm:ilp\], the interpretability logic is unknown. The theory [${\mathrm{PRA}}$]{}is a notable example: the logic [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{} is still unknown. The most recent results for [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{} are presented in [@BilDickJo].
[${\mathrm{PRA}}$]{}is known to be the same as [[$\mathrm{I}\Sigma^R_{1}$]{}]{} where [[$\mathrm{I}\Sigma^R_{n}$]{}]{} is defined as $\textsf{I}\Delta_0 + \exp$ plus the $\Sigma_n$ induction rule. See for example [@Bek:ReflArith]. In that paper a proof can also be found for the following theorem.
\[theorem:ReflexiveExtensionsOfRules\] [[$\mathrm{I}\Sigma^R_{n}$]{}]{} is reflexive, as is any extension of [[$\mathrm{I}\Sigma^R_{n}$]{}]{} by $\Sigma_{n+1}$ formulas.
The logical complexity of interpretability is $\Sigma_3$ and in [@Shav97] it is shown that it is essentially so. However, by a theorem due to Orey and Hájek we can often reduce the $\Sigma_3$ notion of interpretability to the $\Pi_2$ notion of $\Pi_1$-conservativity. A theory $V$ is $\Pi_1$-conservative over $U$, we write $U\rhd_{\Pi_1}V$, whenever for all $\Pi_1$ sentences $\pi$ we have that \[$V\vdash \pi$ implies $U\vdash \pi$\].
\[theorem:OH\] For reflexive theories $U$ and $V$ we have $$(U\rhd V ) \ \ \ \Leftrightarrow \ \ \ (U \rhd_{\Pi_1}V)$$ and this equivalence is provable in [${\rm{EA}}$]{}.
One advantage of this characterization is evidently that the logical complexity of $\Pi_1$-conservativity is lower than that of interpretability. Another advantage is that the so-called $\Pi_1$-conservativity logic is a relatively stable notion. The $\Pi_1$-conservativity logic of a theory $T$ is just the set of modal formulas in [${\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}$ ]{}that are provable in $T$ under any arithmetical realization where the $\rhd$ modality is mapped to $\rhd_{\Pi_1}$.
\[theorem:PiILM\] For any sound theory $T$ extending $\textnormal{\textsf{I}}\Pi_1^-$ we have that the $\Pi_1$-conservativity logic of $T$ is [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}.
The theorem was first proven by Hájek and Montagna in [@HM90] and [@HM92] to hold for any sound theory containing $\textsf{I}\Sigma_1$. Beklemishev and Visser in [@BeklVisser2005] lowered the threshold to the rather weak theory $\textsf{I}\Pi_1^-$ that allows only induction for parameter free formulas of complexity $\Pi_1$. It is well known that [${\mathrm{PRA}}$]{}extends $\textsf{I}\Pi_1^-$ ([@Bek:ReflArith]).
\[remark:Sigma2Subs\] The proof of Theorem \[theorem:PiILM\] is rather similar to that of Solovay’s original proof and again (see Theorem \[theorem:BoolSigmaPL\]), the substitutions in the completeness proof can be taken[^11] to be $\Sigma_2$.
The logics [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}and [[$\textup{\textbf{IL}}{\sf\ensuremath{P}}$]{}]{}have elegant syntactical presentations. We shall define them in parts. First, we define a logic [[$\textup{\textbf{IL}}$]{}]{}that is present to all interpretability logics studied. Next this logic [[$\textup{\textbf{IL}}$]{}]{}is extended by adding more axiom schemata.
(When we write formulas in [${\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}$ ]{}we adhere to the following binding conventions. We say that $\rhd$ binds stronger than $\to$ but weaker than all other connectives. Using this convention we can save a lot of brackets.)
\[defi:il\] The logic [[$\textup{\textbf{IL}}$]{}]{}is the smallest set of formulas being closed under the rules of Necessitation and of Modus Ponens, that contains all tautological formulas and all instantiations of the following axiom schemata.
1. \[ilax:l1\] $\Box(A\rightarrow B)\rightarrow(\Box A\rightarrow\Box B)$
2. \[ilax:l2\] $\Box A\rightarrow \Box\Box A$
3. \[ilax:l3\] $\Box(\Box A\rightarrow A)\rightarrow\Box A$
4. \[ilax:j1\] $\Box(A\rightarrow B)\rightarrow A\rhd B$
5. \[ilax:j2\] $(A\rhd B)\wedge (B\rhd C)\rightarrow A\rhd C$
6. \[ilax:j3\] $(A\rhd C)\wedge (B\rhd C)\rightarrow A\vee B\rhd C$
7. \[ilax:j4\] $A\rhd B\rightarrow(\Diamond A\rightarrow \Diamond B)$
8. \[ilax:j5\] $\Diamond A\rhd A$
Apart from the axiom schemata enumerated in Definition \[defi:il\] we will need consider other axiom schemata too.
1. $A \rhd B \rightarrow A \wedge \Box C \rhd B \wedge \Box C$
2. $A \rhd B \rightarrow \Box (A \rhd B)$
3. $A \rhd B \rightarrow A \rhd B \wedge \Box \neg A$
If $\sf X$ is a set of axiom schemata we will denote by [$\textup{\textbf{IL}}{\sf\ensuremath{X}}$]{} the logic that arises by adding the axiom schemata in $\sf X$ to [[$\textup{\textbf{IL}}$]{}]{}.
The closed fragment {#the-closed-fragment}
-------------------
Because closed formulas in [[$\textup{\textbf{IL}}{\sf\ensuremath{W}}$]{}]{}can be reduced to those of [[$\textup{\textbf{GL}}$]{}]{}([@haje:norm91]) we can prove that ${{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{B}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$ is again the logic of linear frames.
The logic $\mbox{\bf ILW.3}$ is obtained by adding the linearity axiom schema $\Box (\Box A \rightarrow B) \vee
\Box (\boxdot B \rightarrow A)$ to [[$\textup{\textbf{IL}}{\sf\ensuremath{W}}$]{}]{}.
\[ril\] ${{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{B}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}=\mbox{\bf ILW.3} $
We give a translation from formulas $\varphi$ in [${\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}$ ]{}to formulas $\varphi^{\sf tr}$ in [${\sf Form}_{{{\ensuremath{\textup{\textbf{GL}}}}\xspace}}$ ]{}such that $$\begin{array}{cl}
\mbox{\bf ILW.3}\vdash \varphi \Leftrightarrow \mbox{\bf GL.3} \vdash
\varphi^{\sf tr} & (*)\\
\mbox{ and } & \ \\
\mbox{\bf ILW.3} \vdash \varphi \leftrightarrow \varphi^{\sf tr}. & (**)
\end{array}$$ If we moreover know $(*{*}*): \ \ \mbox{\bf ILW.3} \vdash \varphi \Rightarrow
\forall \, *{\in} \mathcal{B}\; {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash \varphi^*$ we would be done. For then we have by $(**)$ and $(*{*}*)$ that $$\begin{array}{ll}
\forall \, * {\in} {\sf Sub}(\mathcal{B})\; {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash
\varphi^* \leftrightarrow (\varphi^{\sf tr})^*
\end{array}$$
and consequently
$$\begin{array}{ll}
\forall \, * {\in} \mathcal{B}\; {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash \varphi^* & \Leftrightarrow \\
\forall \, * {\in} \mathcal{B}\; {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash {(\varphi^{\sf tr})}^* &
\Leftrightarrow \\
\mbox{\bf GL.3} \vdash \varphi^{\sf tr} & \Leftrightarrow \\ \mbox{\bf ILW.3}\vdash \varphi.
\end{array}$$
We first see that $(*{*}*)$ holds. Certainly, by Fact \[fact:IlwInILall\], we have that ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{W}}}\xspace}}\subseteq {{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{B}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$. Thus it remains to show that ${\ensuremath{{\mathrm{PRA}}}\xspace}\vdash \Box (\Box A^* \rightarrow B^*) \vee \Box (\boxdot B^* \rightarrow A^*)$ for any formulas $A$ and $B$ in [${\sf Form}_{{{\ensuremath{\textup{\textbf{IL}}}}\xspace}}$ ]{}and any $*{\in}\mathcal{B}$. As any formula in the closed fragment of [[$\textup{\textbf{IL}}{\sf\ensuremath{W}}$]{}]{}is equivalent to a formula in the closed fragment of [[$\textup{\textbf{GL}}$]{}]{}(see [@haje:norm91]), Theorem \[theorem:closedFragment\] gives us that indeed the linearity axiom holds for the closed fragment of [[$\textup{\textbf{GL}}$]{}]{}.
Our translation will be the identity translation except for $\rhd$. In that case we define $$(A\rhd B)^{\sf tr} := \Box (A^{\sf tr} \rightarrow (B^{\sf tr}\vee
\Diamond B^{\sf tr})).$$
We first see that we have $(**)$. It is sufficient to show that $\mbox{\bf ILW.3}\vdash p\rhd q \rightarrow \Box (p \rightarrow (q\vee \Diamond q))$. We reason in $\mbox{\bf ILW.3}$. An instantiation of the linearity axiom gives us $\Box (\Box \neg q \rightarrow (\neg p \vee q)) \vee
\Box ((\neg p \vee q) \wedge \Box (\neg p \vee q)\rightarrow \neg q)$. The first disjunct immediately yields $ \Box (p \rightarrow (q\vee \Diamond q))$.
In case of the second disjunct we get by propositional logic $\Box (q \rightarrow \Diamond (p\wedge \neg q))$ and thus also $\Box (q \rightarrow \Diamond p )$. Now we assume $p\rhd q$. By ${\sf W}$ we get $p\rhd q\wedge \Box \neg p$. Together with $\Box (q \rightarrow \Diamond p)$, this gives us $p\rhd \bot$, that is $\Box \neg p$. Consequently we have $ \Box (p \rightarrow (q\vee \Diamond q))$.
We now prove $(*)$. By induction on $\mbox{\bf ILW.3}\vdash \varphi$ we see that $\mbox{\bf GL.3} \vdash \varphi^{\sf tr}$. All the specific interpretability axioms turn out to be provable under our translation in ${{\ensuremath{\textup{\textbf{GL}}}}\xspace}$. The only axioms where the $\Box A \rightarrow \Box \Box A$ axiom scheme is really used is in ${\sf J_2}$ and ${\sf J_4}$. To prove the translation of ${\sf W}$ we also need ${\sf L_3}$.
If $\mbox{\bf GL.3} \vdash \varphi^{\sf tr}$ then certainly $\mbox{\bf ILW.3} \vdash \varphi^{\sf tr}$ and by $(**)$, $\mbox{\bf ILW.3} \vdash \varphi$.
We thus see that $\mbox{\bf ILW.3}$ is an upperbound for ${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$. Using the translation from the proof of Theorem \[ril\], it is not hard to see that both the principles ${\sf P}$ and ${\sf M}$ are provable in $\mbox{\bf ILW.3}$. This tells us that the upperbound is actually not very informative as we know that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\nvdash {{\ensuremath{{\sf {M}}\xspace}}}$. By a straight-forward generalization of Lemma \[lemma:upperboundmethod\] we see that choosing larger $\Gamma$ will generally yield a smaller ${{\ensuremath {\textup{\textbf{IL}}}_{\Gamma}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$ and thus a sharper upperbound. Subsection \[subsection:refutingM\] consists of reflections on just how large the $\Gamma$ should be as to refute [[${\sf {M}}\xspace$]{}]{} in ${{\ensuremath {\textup{\textbf{IL}}}_{\Gamma}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$. First we shall include some observations on a fragment slightly larger than the closed fragment.
The closed fragment with a constant for [[$\mathrm{I}\Sigma_{1}$]{}]{}
----------------------------------------------------------------------
If we consider the proof of Theorem \[theorem:mainTheorem\], we see that it does not make any assumptions on the signature of the modal logic under considerations. In particular, the theorem still holds for interpretability logics. In the theorem below we use this to give a semantic characterization of ${{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{F}_1}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$.
In [@Joo05] it is established that for a certain frame, that we will denote here by $\widetilde{\mathfrak{G}_1^{\bullet}}$, we have the following equivalence. $$\forall A \in \mathcal{F}_1\ [\ \widetilde{\mathfrak{G}_1^{\bullet}}\models A \ \ \Leftrightarrow \ \ {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash A\ ] \ \ \ \ \ \ (\dag)$$ For the purpose of this paper it is not material to know what exactly the frame $\widetilde{\mathfrak{G}_1^{\bullet}}$ looks like and we shall refrain from giving a formal definition. It is only important to know that $\widetilde{\mathfrak{G}_1^{\bullet}}$ is just $\mathfrak{G}_1^{\bullet}$ with some additional accessibility relations to model the $\rhd$ modality. This, together with the mere equivalence $(\dag)$ suffices to obtain the following theorem.
${{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{F}_1}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}} = \mathcal{L}(\widetilde{\mathfrak{G}_1^{\bullet}})$
Image-finiteness and definability of separate points is clear as interpretability logic is an extension of provability logic. Thus, by Theorem \[theorem:mainTheorem\] we obtain the result.
In [@Joo05], also a logic **PIL** is given such that we actually have $$\forall A \in \mathcal{F}_1\ [\ \widetilde{\mathfrak{G}_1^{\bullet}}\models A \ \ \Leftrightarrow \ \ {\ensuremath{{\mathrm{PRA}}}\xspace}\vdash A\ \ \Leftrightarrow \ \ \mathbf{PIL}\vdash A\ ] .\ \ \ $$ This suggests that the following conjecture should not be too hard to prove. In this conjecture, **ILM.4** denotes the logic that arises by joining [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}and **GL.4**.
$\mathcal{L}(\widetilde{\mathfrak{G}_1^{\bullet}})= {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\textbf{.4}$
The inclusion $\mathcal{L}(\widetilde{\mathfrak{G}_1^{\bullet}})\supseteq {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\textbf{.4}$ is actually very easy and follows from a direct verification of the validity of the axioms on $\widetilde{\mathfrak{G}_1^{\bullet}}$. The other direction is harder but not too interesting as we still have ${{\ensuremath{{\sf {M}}\xspace}}} \in {{\ensuremath {\textup{\textbf{IL}}}_{\mathcal{F}_1}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$.
Fragments for refuting [[${\sf {M}}\xspace$]{}]{} in ${{\ensuremath {\textup{\textbf{IL}}}_{\Gamma}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}$ {#subsection:refutingM}
--------------------------------------------------------------------------------------------------------------------------------------------------
In [@Visser97] it is shown that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\nvdash A\rhd \Diamond B \rightarrow \Box (A\rhd \Diamond B)$. It is easy to see that ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\vdash A\rhd \Diamond B \rightarrow \Box (A\rhd \Diamond B)$. This implies that [[${\sf {M}}\xspace$]{}]{} is certainly not derivable in [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{}. We can also find explicit realizations that violate [[${\sf {M}}\xspace$]{}]{}, as the following lemma tells us.
\[lemm:vetgeenM\] For $n\geq 1$, we have that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}\nvdash {{\ensuremath{{\sf {M}}\xspace}}}$.
We define a realization $*$ such that ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} \nvdash (p\rhd q \rightarrow p\wedge \Box r \rhd q \wedge \Box r)^*$.
It is well-known that ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} \subsetneq
{{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}\subsetneq {{\ensuremath{\mathrm{I}\Sigma^R_{n+1}}}\xspace}$ and that, for every $n{\geq} 1$, ${{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}$ is finitely axiomatized. Let $\sigma_n$ be the single sentence axiomatizing ${{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}$. It is also known that (for $n\geq 1$) ${{\ensuremath {\textup{\textbf{IL}}}({\rm {{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}})}}={{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{P}}}\xspace}}$ and that ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{P}}}\xspace}}\nvdash p\rhd q \rightarrow p\wedge \Box r \rhd q \wedge \Box r$. Thus, for any $n{\geq} 1$ we can find $\alpha_n ,\beta_n$ and $\gamma_n$ such that $${{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}\nvdash \alpha_n\rhd\beta_n\rightarrow
\alpha_n\wedge \Box \gamma_n \rhd \beta_n \wedge \Box \gamma_n.$$ Note that $${\ensuremath{{\rm{EA}}}\xspace}\vdash \alpha_n\rhd_{{{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}}\beta_n \leftrightarrow
\sigma_n \wedge \alpha_n\rhd_{{{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}} \sigma_n \wedge \beta_n$$ and $${\ensuremath{{\rm{EA}}}\xspace}\vdash \Box_{{{\ensuremath {\mathrm{I}\Sigma_{n}}}\xspace}}\gamma_n \leftrightarrow
\Box_{{{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}}(\sigma_n \rightarrow \gamma_n).$$ Thus, we have $${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\nvdash
\sigma_n \wedge \alpha_n\rhd \sigma_n \wedge \beta_n \rightarrow
\sigma_n \wedge \alpha_n \wedge \Box (\sigma_n \rightarrow \gamma_n)
\rhd
\sigma_n \wedge \beta_n \wedge \Box (\sigma_n \rightarrow \gamma_n)$$ and we can take $p^*=\sigma_n \wedge \alpha_n$, $q^*=\sigma_n \wedge \beta_n $ and $r^*=\sigma_n \rightarrow \gamma_n$.
We see that the realizations used in the proof of Lemma \[lemm:vetgeenM\] get higher and higher complexities. The complexity is certainly higher than $\Sigma_2$.
By Theorem 1 from [@BilDickJo] (Theorem 12.1.1 from [@Joo04]) we know that for $\alpha,\beta \in \Sigma_2$ we have $${\ensuremath{{\mathrm{PRA}}}\xspace}\vdash (\alpha \rhd \beta) \to ((\alpha \wedge \Box \gamma)\rhd (\beta \wedge \Box \gamma))$$ for any sentence $\gamma$. This translates to ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\vdash {{\ensuremath{{\sf {M}}\xspace}}}$ and indicates that an arithmetical completeness proof for [[$\textup{\textbf{IL}}$([[${\mathrm{PRA}}$]{}]{})]{}]{} can not work with only $\Sigma_2$-realizations.
For ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}$, $n\geq 2$ we know that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}\subset {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$. This follows from the next lemma.
\[restrict\] ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}={{\ensuremath {\textup{\textbf{IL}}}_{\Delta_{n+1}}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}={{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$ whenever $n\geq 2$.
We shall use that the logic of $\Pi_1$-conservativity for theories containing $\textsf{I}\Pi_1^-$ is [[$\textup{\textbf{IL}}{\sf\ensuremath{M}}$]{}]{}as mentioned in Theorem \[theorem:PiILM\].
If, for two classes of sentences we have $X\subseteq Y$, then ${{\ensuremath {\textup{\textbf{IL}}}_{Y}({\rm T})}} \subseteq {{\ensuremath {\textup{\textbf{IL}}}_{X}({\rm T})}}$. We will thus show that ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}\subseteq {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$ and ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\subseteq {{\ensuremath {\textup{\textbf{IL}}}_{\Delta_{n+1}}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}$.
First, we prove by induction on the complexity of a modal formula $A$ that for all $ * {\in} \Delta_{n+1}\ \ {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\vdash A^*_{\Pi_1}
\leftrightarrow A^*_{\rhd}$ and that the logical complexity of $A^*_{\Pi_1}$ is at most $\Delta_{n+1}$. The basis is trivial and the only interesting induction step is whenever $A= (B\rhd C)$. We reason in ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}$:
$$\begin{array}{cl}
(B\rhd C)^*_{\rhd} & \leftrightarrow_{\mbox{def.}}\\
{{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} + B^*_{\rhd} \rhd {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} + C^*_{\rhd}
&\leftrightarrow_{\mbox{i.h.}}\\
{{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} + B^*_{\Pi_1} \rhd {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} + C^*_{\Pi_1}
&\leftrightarrow_{\mbox{Orey-H\'ajek}}\\
{{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} +B^*_{\Pi_1} \rhd_{\Pi_1} {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} + C^*_{\Pi_1}
&\leftrightarrow_{\mbox{def.}}\\
(B\rhd C)^*_{\Pi_1}& \\
\end{array}$$
Note that we have access to the Orey-Hájek characterization as $B^*_{\Pi_1}$ is at most of complexity $\Delta_{n+1}$ and thus ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace} +B^*_{\Pi_1}$ is a reflexive theory by Theorem \[theorem:ReflexiveExtensionsOfRules\]. Also note that $(B\rhd C)^*_{\Pi_1}$ is a $\Pi_2$-sentence and thus certainly $\Delta_{n+1}$ whenever $n\geq 2$.
If now ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\vdash A$ then ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\vdash A^*_{\Pi_1}$ and thus whenever $* \in \Delta_{n+1}$, ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\vdash A^*_{\rhd}$ and ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\subseteq {{\ensuremath {\textup{\textbf{IL}}}_{\Delta_{n+1}}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}$.
If ${{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}\nvdash A$ then by Remark \[remark:Sigma2Subs\] for some $* \in \Sigma_2$ we have ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\nvdash A^*_{\Pi_1}$ whence ${{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}\nvdash A^*_{\rhd}$. We may conclude that ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{n}}}\xspace}})}}\subseteq {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$.
${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\subset {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$
Although the proof of Lemma \[restrict\] does not give us that ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{1}}}\xspace}})}}= {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$, it does give us that ${{\ensuremath {\textup{\textbf{IL}}}_{\Sigma_2}({\rm {{\ensuremath{\mathrm{I}\Sigma^R_{1}}}\xspace}})}}\subseteq {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$. By earlier observations we saw that ${{\ensuremath {\textup{\textbf{IL}}}({\rm {\ensuremath{{\mathrm{PRA}}}\xspace}})}}\neq {{\ensuremath{\textup{\textbf{IL}}{\sf\ensuremath{M}}}\xspace}}$.
Future research
===============
We have seen that adding a constant for [[$\mathrm{I}\Sigma_{1}$]{}]{} to [${\mathrm{PRA}}$]{}is sufficient to obtain a non-trivial provability logic. By a Theorem of Leivant it is known that ${{\ensuremath {\mathrm{I}\Sigma_{1}}}\xspace} \equiv <2>_{{\ensuremath{{\rm{EA}}}\xspace}}\top$. An interesting fragment to consider next for [${\mathrm{PRA}}$]{}would be the closed fragment together with the set of constants $$\{ (<1>_{{\ensuremath{{\rm{EA}}}\xspace}}<2>_{{\ensuremath{{\rm{EA}}}\xspace}})^n\top \mid n\in \omega\}$$ or variants thereof.
Acknowledgements
================
We would like to thank Lev Beklemishev, Dick de Jongh and Albert Visser for fruitful comments and discussions.
[restr]{}
L.D. Beklemishev. Bimodal logics for extensions of arithmetical theories. *Journal of Symbolic Logic*, 61(1):91-124.
L.D. Beklemishev. Reflection principles and provability algebras in formal arithmetic. *Russian Mathematical Surveys* 60(2): 197-268 2005.
L.D. Beklemishev. Kripke semantics for provability logic **GLP**. *Annals of Pure and Applied Logic*, In press.
L.D. Beklemishev, G. Bezhanishvili, and T. Icard. On topological semantics of **GLP**. Forthcoming in R. Schindler, editor, *Ways of Proof Theory*, Ontos Verlag, 2010.
L.D. Beklemishev, J.J. Joosten, and M. Vervoort. A Finitary Treatment of the Closed Fragment of Japaridze’s Provability Logic. *Journal of Logic and Computation*, 15(4):447-463, 2005.
L.D. Beklemishev, A. Visser. On the Limit Existence Principles in Elementary Arithmetic and [$\Sigma^0_n$]{}-consequences of Theories. *Annals of Pure and Applied Logic*, 136(1-2): 56–74, 2005.
A. Berarducci. The interpretability logic of [P]{}eano arithmetic. *Journal of Symbolic Logic*, 55:1059–1089, 1990.
A. Berarducci, and R. Verbrugge. On the provability logic of bounded arithmetic. *Annals of Pure and Applied Logic*, 61(1-2): 75-93, 1993.
M. Bilkova, D. de Jongh, J. J. Joosten. Interpretability in [${\mathrm{PRA}}$]{}. *Annals of Pure and Applied Logic*, 161: 128–138, 2009.
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P. Hájek, and F. Montagna. The logic of $\Pi_1$-conservativity continued. *Archiv für Mathematische Logik und Grundlagenforschung*, 32: 57-63, 1992.
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T. Icard. A topological study of the closed fragment of **GLP**. *Journal of Logic and Computation*, In press.
T. Icard. Towards an Alternative Proof of Solovay’s Arithmetical Completeness Therem. In *Proceedings of the 12$^{\textit{th}}$ ESSLLI Student Session*, Dublin, 2007.
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[^1]: FOM mailing list, 8/19/2009, [http://cs.nyu.edu/pipermail/fom/2009-August/013994.html ]{}
[^2]: $\Sigma_1$-sound is sufficient here.\[footnote:ClassOfTheories\]
[^3]: There is a paper by de Jongh, Jumelet and Montagna [@JongJumeletMontagna] where an alternative proof of Solovay’s theorem is given. In that proof, using the diagonal lemma, one finds some sentences with the required properties rather than defining the sentences and then proving the necessary properties. However, the obtained sentences are essentially the same as the ones defined in Solovay’s original proof.
[^4]: This question has been studied in depth in [@BerarducciVerbrugge]. The question also has important connections to matters in computational complexity. For example, it is shown in [@Buss86] that if $\sf S^1_2$ proves $\Pi_1^b$-completeness with parameters ($\Pi_1^b$ is the set of formulas $(\forall x \leq t)\ \theta$ with $\theta$ sharply bounded), then [NP = coNP]{}.
[^5]: In order to get the inductive step for the $\Box$ operator going we should prove the slightly stronger statement that for all $j\in i\uparrow$ we have $\langle \mathbb{F}, V \rangle, j \vDash C \ \Leftrightarrow \ \mathcal{M}, j \vDash C^*$.
[^6]: See Footnote \[footnote:ClassOfTheories\] on conditions on theories. The current proof of this theorem invokes Solovay’s completeness result, Theorem \[theorem:solovay\], in full. However, in [@Joo03] it is shown how we can substitute the use of Solovay’s completeness result by the proof of Theorem \[theorem:mainTheorem\]. Thus, Theorem \[theorem:closedFragment\] actually holds for a larger class of theories including $\textsf{I}\Delta_0 + \Omega_1$.
[^7]: This frame is studied in detail in [@BJV] and [@Ica09].
[^8]: See, e.g. [@Bla01], Definition 3.13, where $p$-morphisms go under the name *bounded morphism*.
[^9]: It is not hard to see that $\Box(\Box A \to B) \vee \Box(\Box^+ B \to A)$ is equivalent to $\Box(\Box^+ A \to B) \vee \Box(\Box^+ B \to A)$ over **GL**.
[^10]: The boundaries are not exactly determined and will depend a bit on the answer. It is legitimate to think of any theory extending $\textsf{I}\Delta_0 + \exp$.
[^11]: Albert Visser (p.c.) notes that close inspection of the proof actually reveals that the substitutions can be taken to be $\Delta_2(\textsf{I}\Pi_1^-)$. That is, a $\Sigma_2$ sentences that is probably in $\textsf{I}\Pi_1^-$ equivalent to a $\Pi_2$ sentence.
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Enes Kanter Illustration by Tom Bachtell
On a recent off day for the New York Knicks, Enes Kanter—the team’s gifted new center, who moved to the United States from Turkey eight years ago—was at the White Plains Ritz-Carlton trying to decide if it was safe for him to go out for halal kebabs. Kanter is six feet eleven, with manicured facial hair that makes him look older than his twenty-five years—until he smiles, which he does often. The Knicks had just practiced at their training facility in Tarrytown, and Kanter had showered and changed into clean sweats. There was a halal joint called Turkish Cuisine down the block, but Kanter was hesitant. “I don’t know if it’s a my Turkish-people Turkish restaurant or a their Turkish-people Turkish restaurant,” he said, sitting in the lobby. “It could be a problem!”
Kanter is an outspoken critic of Recep Tayyip Erdoğan, Turkey’s increasingly authoritarian President. On Twitter, he has called Erdoğan “the Hitler of our century.” In person, he rattles off stats on Erdoğan’s rollback of civil liberties since a July, 2016, coup attempt against him. “There have been more than sixty thousand people jailed,” he said. “Turkey is No. 1 at putting reporters in jail.” Civilian critics are silenced, too. “If you go to Turkey and you try to check my Twitter, it says ‘BANNED.’ ”
Kanter’s parents and sister are still in Turkey, but he has not talked with them since the coup. “I cannot communicate with my family,” he said, explaining that government censors “listen to everything.” Last summer, Kanter’s father denounced him in the Turkish press, saying, “With a feeling of shame I apologize to our president and the Turkish people for having such a son.” Kanter says the comment was coerced: “My dad has to say a thing like that. If he wouldn’t say it, he’d be in jail right now.”
Last spring, during the Knicks’ off-season, Kanter had plans to hold youth basketball clinics in sixteen countries, a bit of charitable globe-trotting that was to include, he said, stops in “Philippines, Tokyo, Korea—the good one, not the crazy one,” and then a visit to Europe, to see the “Pop . . . Papa?” He grabbed his phone and scrolled through his Twitter feed until he came to @Pontifex. “Pope!”
He smiled and went on with the story. “So we landed in Indonesia; we had our clinic; everything was cool.” At around 2:30 A.M., his manager knocked on his hotel-room door. “He told me the Army and intelligence service had come to the school where we just had the clinic. He said the Turkish government called Indonesia and told them, ‘Enes Kanter? He’s a dangerous man.’ ”
Recalling the night, Kanter arched his eyebrows, still befuddled. He’d made a split-second decision, and told his manager, “We should escape this country right now.” They caught the next flight, a 5:30 A.M. to Singapore, then continued to Romania, hoping to resume the clinics. “I gave my passport to this lady at the Bucharest airport. She said, ‘Your passport is cancelled—you cannot get into this country.’ ” She referred him to a police officer. “So I’m, like, ‘What is the reason?’ The policeman said, ‘It’s a secret.’ ” Kanter decided to head back to New York, via London.
At the gate at Heathrow, Kanter said, “there was this gentleman waiting from U.S. Homeland Security. He asked for my green card, took some notes, and he said, ‘O.K., you’re good to go.’ And I was, like, ‘Man. Now we’re safe.’ ”
Kanter joined the Knicks in September, as part of the trade that sent Carmelo Anthony to the Oklahoma City Thunder. He’d been beloved in Oklahoma, where he and a close friend, the hirsute Kiwi bruiser Steven Adams, became known as the Stache Brothers. Still, Kanter was thrilled about the trade. “Before, I could reach out to, like, a million people,” he said. “In New York, I can reach out to a hundred million. I have an even stronger voice. So I need to use it.”
After the passport incident, Kanter said, his family’s home in Istanbul was raided by the police for electronics—“They wanted to see if I’m still in contact”—and his father was jailed for five days. “I get death threats. I don’t know if I should take them seriously,” he said, sighing. Then the smile came back. “You never know. He”—Erdoğan—“is a crazy man.”
In New York, Kanter said, it’s hard to know how Turkish immigrants will react to him. He is leery of unfamiliar Turkish restaurants. “You could go to a place and find it’s run by a President supporter!” he said. “And they might just kick me out.”
Finally, he headed out to case Turkish Cuisine. He stood across the street from the restaurant, a plain storefront with a giant fork and spoon on the sign. “So this is the place,” he said, squinting. “I don’t wanna get too close. I still don’t know—is it my Turkish or somebody else’s Turkish?”
After a minute, he turned back toward the hotel. “I’ll take you to a Turkish restaurant in the city,” he said. “The one I know, on Fifty-seventh Street? It’s really, really good!” But, reached by telephone later, the owner of Turkish Cuisine, Apo Kilic, who is Kurdish, dismissed Kanter’s apprehensions. “He’s more than welcome to come,” he said. “He can say anything he wants!” ♦
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Background {#Sec1}
==========
Mobile elements have been extremely successful at colonizing human genomes. As much as two thirds of human genomic content is either made up of mobile elements or mobile element relics \[[@CR1],[@CR2]\]. The two most significant classes of human mobile elements are long interspersed elements (LINEs) and short interspersed elements (SINEs), and together, these elements account for at least 38% of genomic content \[[@CR2],[@CR3]\]. In humans, the most significant members of the LINE and SINE families are L1 and Alu, respectively.
L1 elements have been amplifying in mammalian genomes since before the divergence of placental and marsupial mammals 170 million years ago \[[@CR4]\]. Since becoming established in primate genomes (\~40 million years ago), there has been a linear evolution of L1 subfamilies, with each newly active L1 subfamily replacing the one that came before \[[@CR5],[@CR6]\]. Today, the predominant active L1Hs family of L1 retrotransposons is responsible for the majority of known L1 retrotransposition events \[[@CR4],[@CR7]-[@CR10]\]. L1 promoters and other functional sequences vital to L1 retrotransposition accumulate disruptive mutations as a function of time. Hence, it is not surprising that studies have shown the elements most likely to be active are those that have most recently inserted and therefore may not have become fixed in the population \[[@CR11]-[@CR13]\]. These polymorphic L1 elements are often capable of significant levels of L1 retrotransposition \[[@CR11],[@CR14]\].
Active full length L1 is a \~6 kb long element that codes for two open reading frames (ORFs): ORF1 which encodes a protein with nucleic acid chaperone and RNA binding properties, and ORF2 which encodes a protein with endonuclease (EN) and reverse transcriptase (RT) activities \[[@CR15],[@CR16]\]. L1 mobilizes through a process called retrotransposition \[[@CR17],[@CR18]\]. Briefly, full length, intact L1 loci are transcribed by RNA polymerase II, to generate a bicistronic L1 mRNA encoding two proteins, ORF1p and ORF2p \[[@CR19]\]. The L1 mRNA and L1 proteins form a ribonucleoprotein (L1 RNP) which enters the nucleus where the ORF2p-encoded endonuclease (EN) and reverse transcriptase (RT) create a cDNA copy of the L1 mRNA in a new genomic location through a process called target primed reverse transcription (TPRT) \[[@CR17],[@CR18]\]. Through retrotransposition, L1 has been able to amplify itself to its current copy number of \~500,000 copies per genome. However, the majority of these L1 elements are non-functional relics due to severe truncation at their 5' ends caused by aborted TPRT events and accumulation of deleterious mutations either over time or as a result of unfaithful RT activity \[[@CR2],[@CR20]-[@CR23]\].
An estimated 10-30% of new L1 insertions are full length with the potential for further retrotransposition \[[@CR11]\]. Negative evolutionary selection leads to depletion of the full-length elements and there are only about 5000 full-length elements in the human genome \[[@CR2]\]. Most of these are old and mutated, and current estimates of the number of potentially active full length L1s stand at about 80--100 per individual \[[@CR11],[@CR14]\]. Next generation sequencing (NGS) studies have predominantly focused on detecting L1 insertional mutagenesis through detection of the 3' end of L1 elements \[[@CR24]-[@CR27]\]. Through these NGS techniques, various groups have identified the insertional burden of L1 in germline and somatic tissues, including various cancers. Few studies, however, focus on the specific detection of full-length L1 elements in these genomes that could potentially be hot for retrotransposition and responsible for the accumulation of further L1 insertion events. This is an important facet of the retrotranspositional burden in cancers because these full length L1 elements have been shown to be capable of extremely high levels of continued retrotransposition in tumors with newly inserted hot L1 copies continuing to be actively mobile \[[@CR28]-[@CR30]\].
Studies that have looked at full-length L1 elements in the reference build of the human genome have indicated that the bulk of retrotransposition occurs from a handful of hot L1 elements \[[@CR11]\]. Additionally, full length L1 elements that are polymorphic between individuals are significantly more likely to be hot than fixed, reference full length L1s \[[@CR11],[@CR31]\]. Recent reports support the idea that different L1 elements are active in different individuals and that single hot L1 elements can result in incredibly high levels of insertional mutagenesis \[[@CR28],[@CR29]\]. Additionally, hot L1 elements in tumors have recently been shown to mobilize to hundreds of new locations within some tumors, with some of these new insertions also further mobilizing to additional locations \[[@CR30]\]. Despite the extensive mutagenesis caused by hot L1s, few attempts have been made to identify full length L1 elements in human populations that may lead to successive insertion events.
In this report, we describe a NGS-based method to identify full length L1 elements in human genomes that we call Sequencing Identification and Mapping of Primed L1 Elements (SIMPLE). Using SIMPLE we have identified 228 polymorphic L1 elements in seven independent individuals. SIMPLE has 94% sensitivity and is robustly able to identify nearly all full length L1 elements in a human genome, making it possible to identify hot L1s capable of causing high degrees of insertional mutagenesis.
Results {#Sec2}
=======
Description of SIMPLE library generation {#Sec3}
----------------------------------------
Various anchored PCR strategies have been effectively employed to identify mobile element insertions in a whole-genome setting. They all share the use of one primer from within the known sequence of the element, and use different strategies to target a second primer outside of the element. However, there are several significant drawbacks to most of those methods. Those that ligate a linker to a restriction site located randomly outside of the mobile element insertion only create one amplified fragment size from each element. Thus, it is impossible to tell whether duplicate sequences are independent ligations or PCR duplicates. A similar argument can be made for the use of an arbitrary primer to prime outside the mobile element. Furthermore, restriction sites located at different lengths away from the element will amplify with differential efficiencies, as will different arbitrary primers. Thus, these methods require pooling of multiple experiments using different restriction enzymes or primers to come close to saturating the potential insertions. In addition, even those protocols that utilize randomly sheared DNA to apply anchors still use a major ligation step in their preparation. These ligation steps, such as those used in the preparation of Illumina libraries, lead to low levels of chimeric fragment ligation between genomic fragments that can confound data analysis \[[@CR32]\].
Sequencing Identification and Mapping of Primed L1 Elements (SIMPLE) is a unique L1-detection method based on the principles of random shearing combined with t-linker ligation-mediated PCR that takes advantage of the massive parallelization offered by high throughput NGS technology \[[@CR33]\]. A primer specific to the 5' UTR of full length L1 elements is used in a single round of primer extension in the 5' direction of the L1 element, which allows for priming on all of the \~5000 full length L1 elements in the human genome and the generation of a single adenine overhang at the end of those extended fragments (Figure [1](#Fig1){ref-type="fig"}A). The vast majority of the genome will remain single-stranded so that little besides the extended mobile element fragments will have the A overhang necessary for duplex linker ligation via a 3' thymidine overhang. Following linker ligation, PCR amplification using L1 5' UTR and linker-specific primers allows for specific amplification of only those DNA fragments anchored by an L1 extension event (Figure [1](#Fig1){ref-type="fig"}B). Additional Illumina adapter sequences are added in a subsequent PCR reaction and the SIMPLE library is size fractionated on an agarose gel before final library amplification (Figure [1](#Fig1){ref-type="fig"}C). SIMPLE libraries can be directly loaded onto the next generation sequencing platform for sequence analysis. Thus, each sample is generated from a randomly sheared fragmentation, making it easy to differentiate PCR duplicates from authentic detection of the same element multiple times. In addition, we reasoned that the ligation step used in SIMPLE would be less likely to create chimeras than a traditional linker ligation because of the nature of the 3' overhangs generated by primer extension and that most of the fragments will remain single-stranded.Figure 1**Description of SIMPLE. (A)** Randomly sheared genomic DNA (black line) is subjected to a single round of primer extension using a primer specific to the L1 5' UTR (blue arrow). Only DNA fragments containing a full length L1 element (large red arrow) will undergo linear extension from the L1-specific primer. DNA fragments containing no L1 sequence or truncated L1 copies (bottom left and right respectively) will not be extended. Following primer extension, a duplex "t-linker" (orange) is ligated to L1-primer extended ends. **(B)** Following ligation of duplex linkers, adapter sequences needed for Illumina sequencing (purple bars) are added by nested PCR to only those fragments anchored by an L1 extension event. **(C)** Following gel based size fractionation, SIMPLE libraries are run on an agarose gel and analyzed by Agilent BioAnalyzer to confirm size and library quality.
Validation of SIMPLE {#Sec4}
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To assess the ability of SIMPLE to detect full length L1 elements from the human genome, we sought to determine the efficiency of SIMPLE at pulling out known, reference full length L1 elements that are fixed in the human population. These L1 elements represent evolutionarily established L1 elements that inserted in a primate genome before the split of *Homo sapiens*, and thus represent L1 elements universally present in all human genomes at a diploid level. We employed SIMPLE to determine how many known, fixed full-length L1 elements on three randomly selected chromosomes we could detect. We limited our initial analysis to three chromosomes to allow a more exhaustive manual analysis of any variants. Of the 543 fixed full-length L1 elements on three randomly selected chromosomes, SIMPLE successfully detected the vast majority with an average read depth of ten independent (having a different linker location) reads per element. Of these, 511/543 (94%) were detected by at least one SIMPLE read, with 502/543 (92%) detected by more than three independent SIMPLE events (reads) (Figure [2](#Fig2){ref-type="fig"}A and B). The 32/543 (6%) full-length L1 elements not detected by SIMPLE were located in regions of repetitive DNA (nearly all other genomic L1 elements), which confounds the bioinformatics mapping of these elements.Figure 2**Validation of 5'-SIMPLE using known full length L1 elements. (A)** CLC Genomics Workbench window view of a representative fixed full length L1 element detected by SIMPLE. L1PA4 represents a known full length L1 element on chromosome 1. Small green lines at bottom are forward SIMPLE reads, reading into the 5' end of the L1PA4 element. **(B)** Frequency distribution of the number of reference full length L1 elements detected by SIMPLE versus the number of independent SIMPLE hits detected for that element. On average, a reference full length L1 was assayed by nine independent SIMPLE reads, with the vast majority of reference full length L1 elements assayed by at least two independent reads. **(C)** CLC Genomics Workbench window view of a representative non-reference full length L1 element detected by SIMPLE. As in **(A)**, green bars represent forward SIMPLE reads, however in this case, there is no reference full length L1 at the expected site in the genome. **(D)** Frequency distribution of the number of non-reference, known polymorphic full length L1 elements detected by SIMPLE versus the number of independent SIMPLE hits detected for that element. On average, a non-reference full length L1 was assayed by seven independent SIMPLE reads, with the vast majority of reference full length L1 elements assayed by at least two independent reads.
To determine how SIMPLE performed for non-reference full-length L1 elements (ie, those elements either present as a single copy or not present in any given individual) we assessed the detection rate of previously described polymorphic full-length L1 elements on the same three chromosomes \[[@CR11],[@CR25]-[@CR27]\]. While the three chromosomes selected were found to have 152 previously described full-length, polymorphic elements on them, we would only expect a small proportion of these elements to be assayed in our individuals by SIMPLE given the polymorphic nature of these elements. In fact, SIMPLE determined that 23 of these known polymorphic full-length L1 elements were present in our tested population of seven individuals. As with fixed L1 elements, SIMPLE performed well for polymorphic elements, with a median of seven independent SIMPLE reads per polymorphic element (Figure [2](#Fig2){ref-type="fig"}C and D). Additionally, the results of our SIMPLE analysis were compared to a lower-throughput method which identified 68 novel full length L1 elements in the genomes of six individuals of diverse backgrounds \[[@CR31]\]. Using SIMPLE, we confirmed the presence of 29/68 of these non-reference full length L1 elements in at least one of our seven individuals, including the identification of elements with estimated allele frequencies as low as 2%, many of which were known to be hot for retrotransposition (Additional file [1](#MOESM1){ref-type="media"}: Table S3) \[[@CR31]\]. These data indicate SIMPLE is a powerful tool to detect virtually all full length L1 elements in a given individual in a single experiment, allowing for the detection of novel non-reference full length L1s that may contribute to mutagenesis by mobilization in these individuals.
Extent of full length L1 polymorphism {#Sec5}
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To determine the extent of human full length L1 polymorphism between unrelated individuals, we applied SIMPLE to seven individual genomic DNA samples from non-related Caucasian individuals to assess the degree of full-length L1 polymorphism between these individuals. For ease of analysis, SIMPLE reads mapping to known full-length L1 elements were discarded, and only those reads that did not represent reference elements were taken into account for analysis.
Using SIMPLE, we detected 228 non-reference full-length L1 elements in the genomes of seven unrelated individuals. The range of non-reference, full-length L1 elements varied from a low of 73 per individual to a high of 134 per individual with an average of 96 non-reference full length L1 elements per individual (Table [1](#Tab1){ref-type="table"}). Of the non-reference, full-length L1 elements detected by SIMPLE, 160 were elements present in dbRIP or described by previous studies \[[@CR24],[@CR26],[@CR27],[@CR31],[@CR34]\]. The remaining 68 non-reference full length L1 elements detected by SIMPLE in these individuals were novel elements that had not previously been described (Table [1](#Tab1){ref-type="table"}, Additional file [2](#MOESM2){ref-type="media"}: Table S2).Table 1**Summary of polymorphic full length L1 elements in the seven individuals tested in this studyIndividualTotal polymorphicsTotal Unique (New)1**737 (6)**2**758 (8)**3**13428 (14)**4**12013 (6)**5**966 (3)**6**914 (2)**7**803 (2)**Avg.*96**10 (6)***Column two represents the total number of polymorphic full length elements detected in a particular individual. Column three represents the number of non-reference full length L1 elements unique to one individual (and the number of those that represent novel elements).
To get a better understanding of the frequency distribution of non-reference, full-length L1 elements detected by SIMPLE, we performed pair-wise comparisons of the SIMPLE-detected non-reference L1 elements between each of the 7 individuals (Figure [3](#Fig3){ref-type="fig"}). Our data reveal any two individuals shared between 29 and 94 non-reference full length L1 elements with the average number of non-reference L1s shared between any two individuals being 52 (Additional file [3](#MOESM3){ref-type="media"}: Figure S1A). Similar pairwise comparisons revealed any one individual has between 13 and 91 non-reference, full-length L1 elements that are not present in another single individual with an average of 44 non-reference, full-length L1 elements not shared between any two individuals (Figure [3](#Fig3){ref-type="fig"}A). Analysis of novel non-reference, full-length L1 elements (ie, those not previously reported) revealed individuals contained between 9 and 26 novel non-reference full length L1 elements, with an average of 15 novel non-reference elements per individuals (Figure [4](#Fig4){ref-type="fig"}B). Of these, most are unique to a single individual, but as many as 10% are shared between at least two individuals in our sample (Figure [3](#Fig3){ref-type="fig"}B, Figure [4](#Fig4){ref-type="fig"}A, Additional file [3](#MOESM3){ref-type="media"}: Figure S1A). Taken together, these data indicate an individual genome may contain upwards of 90 non-reference full length L1 elements with the potential to be active.Figure 3**Significant variation in the number of non-reference full length L1 elements is observed between individuals. (A)** Two-way matrix showing the total number of polymorphic full length L1 loci present in one individual but not in another. For example, the first row represents the number of polymorphic full length L1 elements present in individual one but not in individuals two through seven (left to right). Total numbers of polymorphic full length L1 elements per individual are included at the end of each row and the bottom of each column for reference. Numbers vary from a low of 13 elements in individual seven that are not found in individual six (red square) to a high of 91 elements that are found in individual three, but are not found in individual two (blue square). **(B)** Similar to **(A)** except showing the subset of only novel (previously unreported) polymorphic full length L1 loci present in one individual but not in another. For example, the first row represents the number of novel polymorphic full length L1 elements present in individual one but not in individuals two through seven (left to right). Total numbers of novel polymorphic full length L1 elements per individual are included at the end of each row and the bottom of each column for reference. Numbers vary from a low of five elements in individuals five and six that are not found in individual seven (red squares) to a high of 24 elements that are found in individual three, but are not found in individual five (blue square). **(C)** Representative gels depicting 5' flank PCR of two randomly selected non-reference full length L1 elements. Lane labels: NTC=no template control, M=1 kb DNA ladder, Arabic numerals=individual number as per Table [1](#Tab1){ref-type="table"}. **(D)** Representative gel depicting 3' flank PCR of non-reference full length L1 element. Lane labels are as in **(C)**.Figure 4**Further SIMPLE validation. (A)** Frequency distribution showing the number of novel (ie, previously undescribed) non-reference SIMPLE-detected full length L1 elements (Y-axis) versus the number of individuals in which they were detected (X-axis). A majority (42/68, 62%) of novel non-reference full length L1 elements were detected in only one individual, however 26/68 (38%) were detected in at least two individuals and 7/68 (10%) were detected in three or more individuals. **(B)** Graph showing the number of novel non-reference full length L1 elements found in each individual. Numbers ranged from three novel non-reference elements in individual seven to 28 in individual three. The average number of novel non-reference full length L1 elements per individual was ten.
Validation of SIMPLE-detected polymorphisms {#Sec6}
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Non-reference, full-length L1 polymorphisms detected by SIMPLE were subjected to PCR validation to confirm the presence of the non-reference elements. We randomly chose 15 non-reference, full-length L1 elements detected by SIMPLE to perform PCR based validation using L1 junction PCR to amplify across the 5' L1 junction. This method allowed us to determine the 5' junction of the non-reference L1 element by Sanger sequencing of PCR product. We successfully validated 15/15 (100%) SIMPLE-detected L1 polymorphisms by 5' junction PCR and confirmed the insertions by Sanger sequencing of the 5' junction (Figure [3](#Fig3){ref-type="fig"}C, Additional file [4](#MOESM4){ref-type="media"}: Table S4). Additionally, a random sample of validated 5'-SIMPLE reads was also validated for 3' junctions. We successfully validated seven full length SIMPLE junctions at the 3' end, and were able to determine polyA tail length, L1 EN cleavage site, and target site duplication size for these polymorphic elements, which showed these elements displayed the normal features expected of insertion by retrotransposition (short target site duplications, polyA tails, and L1 endonuclease consensus cleavage sites) (Additional file [5](#MOESM5){ref-type="media"}: Table S5). Taken together, these data show SIMPLE is a robust method for detecting non-reference full length L1 elements.
Distribution of non-reference full length L1 elements {#Sec7}
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We next analyzed the chromosomal distribution of non-reference, full-length L1 elements detected in our study. SIMPLE successfully detected non-reference L1 elements on all 22 autosomes and X with no significant bias for any chromosome to harbor either a particularly high or particularly low number of non-reference, full-length L1 elements. Novel full-length L1 elements were detected on all chromosomes except 8 and 19 (Additional file [6](#MOESM6){ref-type="media"}: Figure S2). Analysis of the number of individuals in our cohort sharing any given non-reference full length L1 reveals that the majority of novel full-length L1 elements detected by SIMPLE are present in only a single individual, indicating they occur at low allele frequency. However, surprisingly, a sizeable number (38%) of novel full length L1s were detected in at least two independent individuals. In fact, 7/68 (10%) novel full length L1 elements detected by SIMPLE were present in at least three independent individuals (Figure [4](#Fig4){ref-type="fig"}A). Additionally, our data show that any individual contains on average 10 unique non-reference, full-length L1 elements with more than half of those (on average, six) having not been previously described (Figure [4](#Fig4){ref-type="fig"}B and Table [1](#Tab1){ref-type="table"}). However, the vast majority of the novel full length L1s detected by our study represent relatively rare alleles that are limited to only one or two individuals in our sample. To highlight this, we compared the allele frequencies of the 68 novel full length L1s described by this study to the 160 previously described polymorphic full length L1s also detected in our seven individuals. We find that while the previously described polymorphic L1 elements range from rare to common among our seven individuals, the majority of the novel polymorphic full length L1 elements detected by SIMPLE are unique to a single individual (Additional file [7](#MOESM7){ref-type="media"}: Figure S3). Taken together, these data suggest that the extent of low frequency (rare) non-reference, full-length L1 insertion polymorphisms in the population may be greater than expected by previous estimates \[[@CR11],[@CR12],[@CR35]\].
Allele frequency estimates of novel full length L1s {#Sec8}
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Because our study identified numerous novel full-length L1s that we anticipate are present at low allele frequency based on their frequency within our seven-individual population (Additional file [7](#MOESM7){ref-type="media"}: Figure S3), we wanted to determine the allele frequencies of the polymorphic full length L1 elements detected in our study in a larger population of individuals. We conducted a population study to determine the allele frequency of 43 randomly chosen polymorphic elements in 80 individuals from four geographically diverse backgrounds (African American, Asian, German Caucasian, and South American). The vast majority (79%) of full-length polymorphic L1 elements tested existed at an allele frequency less than 50%, with most (58%) existing at less than 25% allele frequency. Additionally, 33% of polymorphic full length L1s were rare in the population, with allele frequencies less than 10% (Figure [5](#Fig5){ref-type="fig"}). These frequencies remained consistent whether we looked at the total population, or within each of the four subpopulations (Additional file [8](#MOESM8){ref-type="media"}: Figure S4). Together, these data suggest that the polymorphic elements detected by SIMPLE represent low frequency polymorphisms in the population that are younger and most likely to be active.Figure 5**Allele frequency determination of full length polymorphic L1s.** The allele frequency of 40 randomly selected full length polymorphic L1s detected by SIMPLE was determined by screening a panel of 80 unrelated individuals from four distinct populations. Most polymorphic L1s tested were found to be low allele frequency elements, with only very few elements existing at allele frequencies above 50%.
Application of SIMPLE to L1 3' ends {#Sec9}
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To determine if SIMPLE could be easily adapted to detect all novel L1 insertions, we sought to modify SIMPLE to target the 3' end of human specific L1 elements (L1Hs) because the vast majority of new L1 insertions are 5' truncated \[[@CR2],[@CR20],[@CR36],[@CR37]\]. We designed primers targeting only the youngest, most active L1 subfamilies and performed SIMPLE targeting L1Hs 3' ends in a single individual. To determine the sensitivity of L1 SIMPLE for L1 3' ends, we performed a similar analysis to that done in Figure [2](#Fig2){ref-type="fig"}B. By mapping 3'-SIMPLE data to the human genome reference build and assaying three random chromosomes for detection of known L1Hs 3' ends, we determined SIMPLE performs similarly for L1 3' ends as it does for 5' ends, detecting 93% of known L1Hs elements on the three chromosomes assayed (Figure [6](#Fig6){ref-type="fig"}A). Like with 5'-SIMPLE, those elements not detected by 3'-SIMPLE are elements located in areas rich in repetitive DNA sequences, compounding mapping. Like 5'-SIMPLE, 3'-SIMPLE is able to detect L1Hs 3' ends with high confidence with an average of seven independent 3'-SIMPLE hits per known L1Hs (Figure [6](#Fig6){ref-type="fig"}B). These results support the use of SIMPLE for detection of *de novo* L1 insertion events, which would prove useful in studying L1 mutagenesis in various cancer types.Figure 6**Adaptation of SIMPLE to detect L1 3' ends. (A)** CLC Genomics Workbench window view of a representative fixed human specific L1 element (L1Hs) detected by 3'-SIMPLE. L1Hs represents a known human specific L1 element on chromosome 1. Small red lines at bottom are reverse 3'-SIMPLE reads, reading into the3 end of the L1Hs element. **(B)** Frequency distribution of the number of reference L1Hs elements detected by 3'-SIMPLE versus the number of independent 3'-SIMPLE hits detected for that element. On average, a reference L1Hs element was assayed by seven independent 3'-SIMPLE reads, with the vast majority of reference L1Hs elements assayed by at least two independent reads.
Discussion {#Sec10}
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Full length polymorphic L1 elements are responsible for virtually all of the disease-causing retroelement insertions and the bulk of retroelement mobilization observed in tumors. We have developed a robust tool to identify full-length polymorphic L1 elements in individual genomes, which also has potential to be applied to the detection of *de novo* somatic L1 insertion events in tumors or other samples. Because DNA fragmentation for SIMPLE occurs by random shearing of genomic DNA by sonication, SIMPLE is able to detect L1 elements that may be located too distally from restriction enzyme cut sites used by other high-throughput sequencing methods \[[@CR25],[@CR26]\]. Additionally, SIMPLE has benefits over other fosmid-based approaches used in the past in that it is extremely high throughput, allowing the detection of essentially all genomic L1 elements in a single SIMPLE reaction \[[@CR31]\]. Those rare elements missed by SIMPLE, which reside within repetitive regions of the genome (ie, other L1 elements) are likely a function of read length. It is possible that future iterations of SIMPLE using longer reads would allow for exact mapping of reads that lie partially within repetitive genomic regions.
Using SIMPLE, we have determined the typical individual contains about 100 non-reference, full-length L1 elements. Based on previous estimates, we expect 63/100 of these elements to be active, with 54/100 exhibiting particularly high levels of activity \[[@CR31]\]. These 100 non-reference, full-length L1 elements are in addition to the active polymorphic L1 elements present in the human genome reference build, indicating the potential burden of active L1s in any one individual exceeds previous estimates of 80--100 elements per individual \[[@CR11]\]. Additionally, our results indicate that the number of non-reference, full-length L1 elements in the population likely exceeds previous estimates given the finding that a number of novel elements detected by SIMPLE were present in multiple individuals. Further, the variation we see between individuals in terms of the number of non-reference, full-length L1 elements implies there are likely individual differences in each person's "retrotransposition potential", with some individuals in a population being more prone to L1 jumping than others, which is consistent with previous hypotheses \[[@CR38]\] as well as with recent data showing some hot L1s are differentially active between different cancer \[[@CR28]-[@CR30]\].
It is interesting to note that although all seven individuals assayed in this study were Caucasian, there is still significant variation between individuals with respect to non-reference, full-length L1 elements. We anticipate, however, that analysis of diverse populations will yield similar results in terms of absolute numbers to those reported here. We would expect an individual from any single population to harbor the average 100 non-reference full length L1 elements with the only difference being in the identity of the particular elements found in that individual. Two individuals from similar backgrounds will share a greater number of non-reference elements than two individuals from diverse backgrounds, but both individuals will have \~100 non-reference full length L1 elements each regardless of their background.
Conclusion {#Sec11}
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SIMPLE represents a useful tool not only for both population and tumor-based studies of L1 insertion polymorphism. The vast majority of L1 retrotransposition is likely to occur due to the approximately 100 polymorphic L1 loci in each individual. Work looking at L1 mobilization in tumors has identified a number of tumor types that appear to support L1 mobilization (colorectal, prostate, lung, ovarian) as well as others that do not (glioblastoma, multiple myeloma) \[[@CR25]-[@CR27],[@CR39],[@CR40]\]. While this preference for particular tumor types can be explained by cell-specific factors affecting L1 mobilization, there is also significant variation in the extent of L1 mobilization within cancers of the same type that could be due to differences in polymorphic, active L1 elements between two individuals \[[@CR27],[@CR40]\]. Using SIMPLE, it may be possible to determine the nature of biases in L1 retrotransposition rates between different individuals or tumors and attribute them to differences in the number or location of active L1 elements between two individuals. Additionally, with the application of SIMPLE to L1Hs 3' ends, it can be used to determine the total levels of L1 retrotransposition in different individuals and tumors.
Methods {#Sec12}
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Cell lines and oligonucleotides {#Sec13}
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Fibroblast cell lines GM01631, GM01632, GM05510, GM05568, GM16094, GM15983, and GM08207 were obtained from the Coriell Institute (Camden, NJ). Cell lines were maintained in EMEM supplemented with non-essential amino acids, sodium pyruvate, and 10% fetal bovine serum. DNA oligonucleotides and duplex linkers were obtained from Integrated DNA Technologies (Coralville, IA). Oligonucleotide sequences used in this study are presented in Additional file [9](#MOESM9){ref-type="media"}: Table S1.
5' SIMPLE library generation {#Sec14}
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Genomic DNA from fibroblast cell lines was extracted using the DNEasy Blood and Tissue Kit from Qiagen (Germantown, MD). DNA was sheared to approximate 750-1200 bp using a Diagenode BioRuptor on High, 30s on/30s off for 12 minutes. 50 ng of sheared gDNA was subject to a primer extension reaction using Taq polymerase and a L1 5' UTR specific primer (L15'UTRP1) which sits \~100 bp from the start of the L1 element. Phosphorylated duplex T-linkers (IDT, Coralville, IA) were ligated using T4 DNA ligase. First round PCR was performed for 20 cycles using primers L15'UTRP1 and LinkerP1. One-million fold dilution of PCR I was performed and 1uL of this dilution was subjected to a nested PCR using primers L15'UTRP2 and LinkerP1 for 25 cycles. PCR products were run on a gel and a gel slice at \~500-700 bp was extracted using the Qiagen Gel Extraction kit. Following extraction, the final library was amplified using Phusion polymerase (Thermo, Waltham, MA) for 12 cycles as per the Illumina library generation protocol, and gel purified to yield the final 500-700 bp library.
Library quantitation and illumina sequencing {#Sec15}
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Illumina sequencing libraries were quantified by qPCR and checked for quality by Agilent BioAnalyzer trace at Elim Biopharmaceuticals (Hayward, CA). 100 bp paired end Illumina sequencing was performed on an Illumina HiSeq 2000 by Elim Biopharmaceuticals (Hayward, CA).
Reference genome masking {#Sec16}
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Reference repetitive element annotation files were downloaded from UCSC genome browser as GTF files and used to annotate the GRCh37 (hg19) reference genome in CLC Genomics Workbench (CLC Bio, Cambridge MA). Two custom GFF annotation files were also generated: 1) representing known polymorphic elements and 2) representing all known full length L1 elements plus 600 bp of upstream genomic flanking sequence.
To identify and annotate 5' L1 regions of the genome, the first 300 base pairs of the L1.3 consensus sequence was aligned to the human reference genome (CRCh37/hg19) via NCBI BLAST (blastn algorithm). Initial search parameters were relaxed to increase sensitivity for older, more mutated elements (word size=7, gap cost=3, gap extension cost=3, match score=2, mismatch penalty=−3, minimum E score of 10). Blast output was generated in tabular format. Because many of the hits obtained from this search consisted of isolated smaller regions (20-30 bp) that were independent of any identifiable L1 element, we further filtered for those hits where alignments were made across \>=250 bp of the 300 bp query. This greatly improved our specificity while maintaining sensitivity to older, more mutated elements. Tabular format blast results were converted to GFF via Perl scripting, and an additional 600 bp of flanking sequence was added upstream and downstream to each matching position.
Annotation of 3' L1 ends and flanking sequenced was performed using a similar strategy as above, except that initial identification of 3' end locations was based on the UCSC Genome Browser hg19 RepeatMasker table track, which allowed for subfamily classification based on all available sequence from the insertion.
5' SIMPLE mapping strategy {#Sec17}
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FASTQ sequencing files were analyzed for quality using FastQC. PCR duplicate reads were removed using custom in-house Perl scripting. Sequence read mappings were performed using CLC Genomics Workbench. Single end genomic flank reads were mapped using a two-pronged mapping strategy. First, reads that either mapped to within 600 bp of known, fixed full length L1 elements or those reads that did not map unambiguously were discarded, as they represented either known elements or unmappable reads. The remaining reads were then mapped uniquely to an annotated GRCh73/hg19 reference genome to identify novel polymorphic or potentially *de novo* full length L1 elements.
Polymorphic full length L1 allele frequency determination {#Sec18}
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### DNA samples {#Sec19}
Confirmed novel L1 elements were genotyped to determine the allele frequency of the insert on a DNA panel of 80 diverse individuals (20 African Americans, 20 Asians, 20 Europeans and 20 South Americans) obtained from the Coriell Institute for Medical Research, Camden, NJ. The efficacy of each primer pair was initially assessed through PCR using human DNA cell line HeLa (ATCC CCL-2) and/or chimpanzee DNA cell line Clint (S006006), confirming the size of the predicted PCR products.
### PCR primer design {#Sec20}
BLAT was used to locate candidate L1 sequences and flanking DNA these sequences in the hg 19 human reference genome, and add 1,000 bp of flanking sequence both upstream and downstream. Sequences were then screened through RepeatMasker confirming the absence of candidate L1 inserts from the hg 19 sequence and localizing unique genomic regions for primer design. For the most part, prospective PCR primers flanking each candidate L1 locus were designed using Primer3 software. Once screened through the BLAT genome browser, primers were selected for PCR if they were predicted to amplify a single locus. In addition, a virtual PCR was performed for each locus using the in silico function of BLAT to compute the expected PCR product size, annealing temperature, and further verify that only one locus would be amplified. For candidate L1 loci directly flanked by substantial stretches (more than 1 kb) of repetitive genomic sequence, primers were designed manually. In such instances, flanking sequences (up to 1 kb) were screened using the BLAT genome browser and aligned with closely reported matches (usually 4--12 loci) in BioEdit (Ibis Biosciences). Point mutations, insertions, and/or deletions specific to the candidate L1 loci were manually positioned toward the 3' end of the primers. In addition, internal primers, purposed to anneal to the 3' end of a full length human specific L1 element, were available \[[@CR41]\].
### PCR analysis {#Sec21}
PCR assays were performed in four stages. (1) An empty site PCR analysis was initially conducted using HeLa DNA. External primers flanking each candidate L1 locus were used to amplify PCR products matching the predicted empty site band lengths previously generated using BLAT's in silico PCR feature. This assay was capable of amplifying PCR products no larger than 1,500 bp. As a result, alleles lacking the L1 insertion were exclusively amplified. Neither alleles possessing an L1 insertion nor unsuitable primer pairs amplified PCR products. Thus it was impossible to distinguish suitable primers pairs flanking L1 inserts that were homozygous present in HeLa DNA from an unsuccessful reaction (failure to amplify an existing empty site). Therefore, given the human specific nature of the predicted L1 inserts, chimpanzee DNA was used in a control PCR analysis examining each primer pair that failed to amplify an empty site during the initial PCR. (2) An internal primer test was conducted to verify the presence of novel L1 inserts and verify that they were full length. These PCR analyses were performed using DNA samples from the individual(s) in which the inserts were originally identified, an external primer flanking the predicted 3' end of the L1, and a 3' internal primer \[[@CR41]\]. (3) A long amplification PCR was conducted to further confirm the presence or absence of candidate L1 insertion characterized in the former stages. This assay was capable of amplifying PCR products up to 15 kb. Thus the flanking external primers were used to amplify the entire L1 insertion. (4) An allele frequency PCR analysis was conducted using a DNA panel of 80 individuals (20 African Americans, 20 Asians, 20 Europeans and 20 South Americans). This assay subjected each sample to both an empty site assay (stage 1) and an internal primer test (stage 2).
Stages 1, 2, and 4 were performed in 25 μL reactions containing 25 ng of template DNA, 200 nM of each oligonucleotide primer, 1.5 mM MgCl2, 10× PCR buffer (50 mM KCl, 10 mM TrisHCl; pH 8.4), 0.2 mM deoxyribonucleotide triphosphates and 1 to 2 U Taq DNA polymerase. PCR reactions were performed as follows: initial denaturation at 94°C for 60 seconds, followed by 32 cycles of denaturation at 94°C for 30 seconds, 30 seconds at primer annealing temperature (determined previously with HeLa DNA), and extension at 72°C for 30 seconds. PCR reactions were terminated with a final extension at 72°C for 2 minutes. Fractionation of 20 μL of each PCR product was performed in a horizontal gel chamber on a 2% agarose gel containing 0.2 μg/mL ethidium bromide for 45 minutes at 200 V. UV-fluorescence was used to visualize the amplified DNA fragments.
Stage 3 was performed using Takara LA-Taq (long amplification) DNA polymerase (Clontech Laboratories, Inc., Mountain View, CA). These PCR reactions were performed in 25 μL reactions containing 25 ng of template DNA and LA-Taq PCR reagents according to the manufacturer's suggested protocol: initial denaturation at 94°C for 1 minute and 20 seconds, followed by 32 cycles of denaturation at 94°C for 20 seconds, 20 seconds at the optimized annealing temperature, and a long extension step at 68°C for eight minutes and 30 seconds. These long-amplification reactions were terminated with a final extension at 68°C for 10 minutes. PCR products were size fractionated on a 1% agarose gel for 150 minutes to 180 minutes at 150 V. UV-fluorescence was used to visualize the DNA fragments.
3'-SIMPLE library generation {#Sec22}
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Library generation for 3'-SIMPLE was performed in the same manner as 5'-SIMPLE with the following modifications. Primer extension used L1HsP1 primer. First round PCR was 22 cycles rather than 20 and used primers L1HsP1 and LinkerP1. Second round PCR used primers L1HsP2 and LinkerP1. All remaining steps of 3'-SIMPLE library generation were identical to those for 5'-SIMPLE.
3'-SIMPLE mapping strategy {#Sec23}
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Raw reads were processed for quality and duplicate removal in the same manner as for 5'-SIMPLE. A similar two-pronged mapping strategy was used for 3'-SIMPLE involving mapping reads to known L1Hs elements and then taking unmapped reads and mapping unambiguously to an annotated GRCh37/hg19 genome.
Additional files {#Sec24}
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Additional file 1: Table S3.Detection of previously reported polymorphic full length L1s. This table shows details for 68 previously identified polymorphic L1s \[[@CR31]\], some of which were also detected by this study.Additional file 2: Table S2.Polymorphic full length L1 elements identified in this study. This table shows details for all 228 polymorphic full length L1s identified by this study.Additional file 3: Figure S1.Pairwise comparison of polymorphic L1 elements shared between any two individuals. (A) Two-way matrix showing the total number of polymorphic full length L1 loci shared between any two individuals. For example, the first row represents the number of polymorphic full length L1 elements shared between individual one and each of individuals two through seven (left to right). Total numbers of polymorphic full length L1 elements per individual are included at the end of each row and the bottom of each column for reference. Numbers vary from a low of 29 elements shared between individuals two and seven (red square) to a high of 94 elements shared between individuals three and four (blue square). (B) Similar to (A) except showing the subset of only novel (previously unreported) polymorphic full length L1 loci shared between any two individuals. For example, the first row represents the number of novel polymorphic full length L1 elements shared between individual one and each of individuals two through seven (left to right). Total numbers of novel polymorphic full length L1 elements per individual are included at the end of each row and the bottom of each column for reference. Numbers vary from a low of zero elements shared between individuals one and five and individuals two and five (red squares) to a high of eight elements that are shared between individuals one and two and individuals three and four (blue squares).Additional file 4: Table S4.Validation of SIMPLE-detected full length L1s by 5' flank PCR. This table shows details of the validation of SIMPLE-detected polymorphic L1s using 5' junction PCR.Additional file 5: Table S5.Validation of SIMPLE-detected full length L1s by 3' flank PCR. This table shows details of the validation of SIMPLE-detected polymorphic L1s using 3' junction PCR.Additional file 6: Figure S2.Chromosomal distribution of polymorphic full length L1 elements across seven individuals. Polymorphic full length L1 elements were detected by SIMPLE on all autosomes and chromosome X without any apparent bias for one genomic region over another. Novel full length L1 elements (grey) were also detected across nearly all autosomes and chromosome X without apparent bias.Additional file 7: Figure S3.Allele frequency of known and novel full length L1s among the seven individuals in our population. Graph shows the frequency (as a percentage) of known (blue bars) or novel to this study (red bars) polymorphic full length L1 elements within our population. Previously described polymorphic L1s show an even distribution with some representing rare alleles (present in only one or two individuals) and others representing common alleles (present in all or most individuals). However, the novel polymorphic L1s show a distribution much more skewed to rare alleles, with most novel polymorphic L1s being unique to only a single individual.Additional file 8: Figure S4.Allele frequency of 40 selected polymorphic full length L1s from four diverse populations. Histogram showing the allele frequencies of 40 randomly selected polymorphic full length L1s broken down by population subtype. 20 unrelated individuals from each of four geographically distinct ethnic groups (African American, Asian, German Caucasian, and South American) were tested for the presence or absence of each selected polymorphic full length L1.Additional file 9: Table S1.List of oligonucleotides used in this study. NNNNN represent random barcodes used for multiplexing.
**Competing interests**
The authors declare that they have no competing interests.
**Authors' contributions**
VAS designed and performed most experiments, analyzed data, and wrote the manuscript. VEJ conducted population study experiments. SL and DJH created data analysis tools. MAB analyzed data and revised the manuscript. PLD designed and conceived experiments, analyzed data, and revised the manuscript. All authors read and approved the final manuscript.
The authors thank Dr. Cecily DeFreece and the members of the Deininger lab for critical comments on this manuscript. This work was supported by the National Institutes of health \[R01GM045668, P20RR020152, and P20GM103518 to PLD, R01GM59290 to MAB. Vincent Streva was supported by a Louisiana State Board of Regents Fellowship. Funding for open access charge: National Institutes of Health.
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pNameplates
The primary goal of this nameplate project is to maintain a lightweight footprint while providing as many "quality of life" features as possible. pNameplates is ideal for any player seeking a quality, lightweight nameplate replacement that is more than just the bare minimum.
Sorry am I missing something? Where did my profiles go? I loved this addon before. You could easily change from profile to profile by just simply typing /pp load <profile name>. Now there is no option I see for any profiles and it looks like my previous saved profiles are now gone so I have to recreate them. When I do adjust them in the interface the fonts/sizes revert back to what it was before when toggling on/off.
With the 3.xx update, profiles and slash commands have been temorarily disabled due to various reasons. I do plan to add that functionality back in as soon as possible. I apologize for the inconvenience
As for the font and font size issue, is this occurring every time you attempt to change these options via the configuration interface or is it a more isolated issue, such as these options not retaining their settings upon logging in/out?
Sorry am I missing something? Where did my profiles go? I loved this addon before. You could easily change from profile to profile by just simply typing /pp load <profile name>. Now there is no option I see for any profiles and it looks like my previous saved profiles are now gone so I have to recreate them. When I do adjust them in the interface the fonts/sizes revert back to what it was before when toggling on/off.
i tested the new version, the option panel works, but some nameplates stay on default h/w and the castbar seems adjusted unless its a channeled spell; then it turns to default h/w as you can see on the screeny i made
http://postimg.org/image/n73p10eoj/
might have something to do with saved variables? i tried to edit the lua code so the scale is exactly as i want, it seems that even then it sometimes bugs out. any ideas?
there is no option to adjust castbar (h/w) and pls set minimum height lower or maybe a scale option?
Added these options back into the fold. Plus a couple of extras! (check changelog for details)
i tested the new version, the option panel works, but some nameplates stay on default h/w and the castbar seems adjusted unless its a channeled spell; then it turns to default h/w as you can see on the screeny i made
I would like to ask for 3 features is possible:
1. Option to recolor only the healthbar edge in threat mode.
2. Change the alpha of the non-targeted units' nameplates.
3. Change the name text width or make it equal to the healthbar's width.
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Theory and Experiments on Spatial Competition
Article excerpt
I. INTRODUCTION
Why do convenience stores tend to choose dispersed locations close to consumers, whereas fast food restaurants tend to locate near each other? Why do automobile dealers locate close to one another, yet go to great effort to differentiate their products through advertising? It is clear that firms can compete on many dimensions including price, location and product characteristics. Previous work that has sought to explain competition on many dimensions has been less than successful (see Osborne and Pitchik |1987~).
In our investigation, we allow competition to occur only in the form of spatial product differentiation. In the spirit of Hotelling |1929~, each firm selects a location at some point on a line segment. Identical consumers with downward sloping demand curves are uniformly distributed over the market. Customers incur a transportation cost which varies linearly with their distance from the closer firm. The FOB (free on board) price charged by each firm is the same and exogenous. The price paid by a consumer is the sum of the fixed FOB price and the transportation cost. This gives a negative relationship between a customer's distance from the nearest supplier and the quantity demanded.
A firm has two considerations when selecting a location: market share and proximity to consumers. When the firms locate symmetrically, each at a distance of one-half from the center of the market, they split the market in half. Each firm obtains a 50 percent market share and the most remote consumer is a distance of only one-fourth of the total market space away from one of the firms. Because of this, by locating at the quartiles of the market the firms maximize combined sales. However, in this situation a firm can typically increase it sales by moving toward its rival, thereby increasing its market share as buyers switch firms.
The decision to locate away from the center of the market carries a degree of risk. Any position away from the center leaves "unprotected" market share. The other firm has an opportunity to gain market share (and profit) by locating close by, but a little closer to the center. By locating at the center of the market, a firm assures itself of at least one-half of the market regardless of the other firm's location. Thus market share is secure. In this case, the firms share the market equally, the most remote customer is a distance of one-half from each firm, and combined sales are relatively low compared to a quartile outcome. This corresponds to the notion of minimum differentiation that is attributable to Hotelling. However, if the transportation cost is sufficiently high, this is unlikely to be the outcome.
The theory of location choice has been considered by economists and political scientists. Smithies |1941~, Eaton |1972~ and Eaton and Lipsey |1975~ all consider a uniform distribution of consumers over a linear market. Using verbal arguments Smithies shows that for a duopoly facing linear demand, the type of equilibrium depends on the ratio of the transportation cost to the choke price(1) on the demand curve. Eaton provides the mathematical details. Eaton and Lipsey consider unit demand and extend the analysis to n-firms and a two-dimensional market. In a duopoly on a linear market they conclude that the firms will locate at the center.
Hinich and Ordeshook |1970~ consider a competition for votes between political candidates.(2) They show the candidates will choose the same location in equilibrium if each candidate's objective is to maximize plurality (the excess of its votes over its opponent). However, if each attempts to maximize votes, the candidates may choose different positions in equilibrium, depending on the sensitivity of voter preferences. Much of their analysis is analogous to the case of unit demand and linear transportation cost.
Gabszewicz and Thisse |1986~ maintain unit demand and conclude that due to the harsh competition that such an environment fosters, "the only location equilibrium consists of both firms clustered at the center of the market. …
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Using HealthMap to Analyse Middle East Respiratory Syndrome (MERS) Data.
In this paper, a web-based disease surveillance platform known as HealthMap is used to retrieve and analyze data pertaining to the Middle East Respiratory Syndrome (MERS) within the geographical confinements of the Arab World and North African countries. An account was opened with HealthMap to gain access to data related to MERS for the time-period 9 September 2015 to 7 March 2015. HealthMap accumulates and assesses outbreak data from a range of sources, this includes news media (Google News (Arabic)), validated official alerts (WHO) and expert-curated accounts (ProMED). Search terms "MERS" and "Coronavirus" were used to search HealthMap for relevant alerts pertaining to MERS. The search terms were geographically limited to the Arab World and North African countries. Our results show that the prevalence of MERS still remains to be the highest in Saudi Arabia, however, between 9 September 2015 and 7 March 2015 there has been a slight overall trend in decreasing number of MERS related alerts within Saudi Arabia and the Arab world. All countries other than Saudi Arabia such as, UAE, Tunisia, Qatar, Oman, Lebanon, Kuwait, Jordan and Egypt together amounted for only 25.6% (n=214) of total MERS alerts (n=837). Our findings show that the rise in the aggregated contribution of internet based participatory surveillance systems for tracking non-communicable diseases such as MERS has aided in improving the accuracy, sensitivity and timeliness for monitoring disease outbreaks. Use of resources such as HealthMap can aid in the timely assessment of risk factors, vaccination development initiatives, prevention strategies and measured allocation of healthcare resources while running at a low cost with greater flexibility and increased scalability compared to isolated diseases surveillance systems.
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This invention relates to an axial retention system and components thereof for a bladed rotor, particularly a fan rotor of a gas turbine engine.
A fan rotor of the type used in an aircraft gas turbine engine includes a hub capable of rotating about a rotational axis and an array of blades extending radially from the hub. The hub includes a series of circumferentially distributed peripheral slots. Each slot extends in an axial or predominantly axial direction and has a pair of overhanging lugs, each with an inwardly facing bearing surface. When viewed in the radial direction, each slot may be linear, with the slot centerline oriented either parallel or oblique to the rotational axis, or may have a curved centerline and a corresponding curved shape. Each slot is typically open at either the forward end of the hub, the aft end of the hub, or both to facilitate installation and removal of the blades.
Each blade includes an attachment feature that occupies one of the slots and an airfoil that projects radially beyond the hub periphery. Bearing surfaces on the flanks of the attachment contact the bearing surfaces of the slot lugs to trap the blade radially in the hub. An axial retention system prevents the installed blades from migrating axially out of the slots.
During operation of the engine, the fully assembled bladed rotor rotates about its rotational axis. Each blade is followed by one of its two adjacent neighbors and is led by its other adjacent neighbor in the direction of rotation. Accordingly, each blade in the blade array is said to have a following neighbor and a leading neighbor.
During operation, a blade fragment can separate from the rest of the blade. A separation event usually results from foreign object ingestion or fatigue failure. Because the separated blade fragment can comprise a substantial portion of the entire blade, separation events are potentially hazardous and, although rare, must be safely accounted for in the design of the engine. Engine designers have devised numerous ways to safely tolerate the separation of a single blade. However it has proven inordinately difficult to accommodate the separation of two or more blades without introducing excessive weight, cost or complexity into the engine. Accordingly, it is important that the separation of one blade not provoke the separation of additional blades.
A separated blade can cause the separation of its following neighbor if the initially separated blade contacts the airfoil of the following blade. The following blade urges the initially separated blade aftwardly and, in doing so, experiences a forwardly directed reaction force. The reaction force can overwhelm the axial retention system that normally traps the following blade axially in its hub slot, thereby ejecting the blade from the slot. Accordingly, it is important that the axial retention system be able to withstand such an event.
Another desirable feature of an aircraft engine fan rotor is resistance to windmilling induced wear. Windmilling is a condition that occurs when an aircraft crew shuts down a malfunctioning or damaged engine in flight. The continued forward motion of the aircraft forces ambient air through the fan blade array causing the fan rotor to slowly rotate or xe2x80x9cwindmillxe2x80x9d. Windmilling also occurs when wind blows through the engine of a parked aircraft. Windmilling rotational speeds are too slow to urge the blade attachment flanks centrifugally against the disk slot lugs. As a result, the blade attachments repeatedly chafe against the surfaces of the hub slots causing accelerated wear of the blade attachments and the hub. Since both the hub and blades are extremely expensive, accelerated wear is unacceptable to the engine owner.
Accelerated attachment and hub wear can be mitigated by ensuring a snug fit between the blade attachment and the hub slot. Alternatively, the attachment can be radially undersized relative to the slot with the size difference being taken up by a tightly fitting spacer that occupies the hub slot radially inboard of the blade attachment. Either way, excessive tightness complicates blade installation and removal. Moreover, surfaces that slide relative to each other during blade installation or removal are susceptible to damage from abrasive contaminants that might be present on the surfaces. Excessive tightness exacerbates the risk of damage. Accordingly, it is important not only to ensure a snug fit, but also to minimize the risk of damaging the expensive components during blade installation and removal.
It is, therefore, an object of the invention to provide an improved axial retention system for a bladed rotor, such as a turbine engine fan rotor.
It is an additional object to minimize windmilling induced damage and to ensure that the blades are easily installable and removable without excessive risk of damage
According to the invention, an axial retention system for a bladed rotor includes a hub with bayonet hooks, a bayonet ring with bayonet projections that engage the hooks, and a load transfer element that occupies an annulus defined by the hooks. Ideally, the load transfer element is a substantially circumferentially continuous snap ring. If a separation event or other abnormality exerts an excessive axial load on a blade, the snap ring safely distributes that load to the bayonet hooks to prevent the blade from severing the snap ring and being ejected axially from its slot. The rotor blades themselves feature a chamfered attachment that improves the energy absorption capability of the snap ring. The interface between each blade and its respective slot is tiered. Ideally the interface is a tiered spacer that occupies the hub slot radially inboard of the blade attachment. The spacer ensures a tight fit to resist windmilling induced wear. The tiered character of the spacer reduces the risk of damage during blade installation and removal. The spacer also helps to transmit axial loads to the snap ring during a blade separation event.
The principal advantage of the invention is its ability to prevent the separation of multiple blades. A further advantage is the ability of the tiered spacer to prevent or minimize damage to the hub and blades during windmilling and during blade installation and removal.
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Former first daughter Chelsea Clinton called out Rep. Ilhan Omar (D-MN) on Sunday evening for using an anti-Jewish stereotype when she alleged that members of Congress had been paid to support Israel.
Omar began by tweeting “It’s all about the Benjamins baby” in response to a comment by left-wing Israel critic Glenn Greenwald in which he lamented that Congress in general, and Republicans in particular, supported Israel.
It’s all about the Benjamins baby https://t.co/KatcXJnZLV — Ilhan Omar (@IlhanMN) February 10, 2019
Omar’s response triggered a torrent of criticism — including from left-wing pundits critical of Israel.
Would love to know who @IlhanMN thinks is paying American politicians to be pro-Israel, though I think I can guess. Bad form, Congresswoman. That’s the second anti-Semitic trope you’ve tweeted. https://t.co/FTCaCe7WyG — Batya Ungar-Sargon (@bungarsargon) February 11, 2019
The first-term Minnesota Democrat — whom Speaker of the House Nancy Pelosi (D-CA) promoted to the House Foreign Affairs Committee last month — answered:
Omar’s reference to the American Israel Public Affairs Committee (AIPAC) referred to a common theme in antisemitic commentary on the extreme left and right. AIPAC is a lobby group that does not donate to candidates.
Ungar-Sargon, among others, was not impressed:
Please learn how to talk about Jews in a non-anti-Semitic way. Sincerely, American Jews. https://t.co/qo3BU2Pqgn — Batya Ungar-Sargon (@bungarsargon) February 11, 2019
In the ensuing Twitter melée, Omar retweeted a left-wing journalist who said of her comments: “She might as well call us hook-nosed.” She apparently only noticed the first part of his tweet, where he echoed her criticism of Israel.
Eventually, Chelsea Clinton weighed in:
Co-signed as an American. We should expect all elected officials, regardless of party, and all public figures to not traffic in anti-Semitism. — Chelsea Clinton (@ChelseaClinton) February 11, 2019
That drew a response from others on the left, who continued to slam AIPAC.
Disappointed @ChelseaClinton is pilling on @IlhanMN instead of reaching out to her. She is a new MoC and figuring out how to navigate calling out AIPAC ( which is a terrible organization aligned with Bibi) and learning that a lot of folks will assume the worst. https://t.co/eZMy2jyY7y — melissa byrne (@mcbyrne) February 11, 2019
AIPAC is not “aligned” with any particular Israeli government, but rather supports the U.S.-Israel alliance in general.
Others were even more abusive — and Clinton stood her ground against Omar’s “anti-Semitic language and tropes”:
Hi Alexandr – my children are thankfully asleep and have been for hours. I think we always have to call out anti-Semitic language and tropes. I feel that way particularly as a parent. Hope you have a good night. https://t.co/mp3GNStW82 — Chelsea Clinton (@ChelseaClinton) February 11, 2019
Nevertheless, Chelsea Clinton responded by promising to “reach out” to Omar:
Hi Melissa – I will reach out to her tomorrow. I also think we have to call out anti-Semitic language and tropes on all sides, particularly in our elected officials and particularly now. — Chelsea Clinton (@ChelseaClinton) February 11, 2019
Omar replied to the overturn by suggesting her critics were attacking her for her Muslim faith:
Chelsea – I would be happy to talk. We must call out smears from the GOP and their allies. And I believe we can do that without criticizing people for their faith. I look forward to building an inclusive movement for justice with you. https://t.co/EGA9NQfBCi — Ilhan Omar (@IlhanMN) February 11, 2019
At no point did Omar acknowledge that what she had said was in any way antisemitic or offensive.
Chelsea Clinton’s husband, Marc Mezvinsky, is Jewish.
Joel B. Pollak is Senior Editor-at-Large at Breitbart News. He is a winner of the 2018 Robert Novak Journalism Alumni Fellowship. He is also the co-author of How Trump Won: The Inside Story of a Revolution, which is available from Regnery. Follow him on Twitter at @joelpollak.
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---
abstract: |
We conjecture a formula for the generating function of virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten.
These virtual $\chi_y$-genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki’s formula, the latter can be expressed in terms of Seiberg-Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on ${\mathbb P}^2$ and ${\mathbb P}^1 \times {\mathbb P}^1$. Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases.
In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include $\mu$-classes, thereby interpolating between Vafa-Witten’s formula and Witten’s conjecture for Donaldson invariants.
author:
- Lothar Göttsche and Martijn Kool
title: 'Virtual refinements of the Vafa-Witten formula'
---
*with an appendix by Lothar Göttsche and Hiraku Nakajima*
Introduction
============
Let $S$ be a smooth projective complex surface with $b_1(S) = 0$ and polarization $H$. We denote by $$M:=M_{S}^{H}(r,c_1,c_2)$$ the moduli space of rank $r$ Gieseker $H$-stable torsion free sheaves on $S$ with Chern classes $c_1 \in H^2(S,{\mathbb Z})$, $c_2 \in H^4(S,{\mathbb Z})$. Suppose that no rank $r$ strictly Gieseker $H$-semistable sheaves with Chern classes $c_1, c_2$ exist. Then $M_{S}^{H}(r,c_1,c_2)$ is projective. T. Mochizuki [@Moc] studied a perfect obstruction theory on $M$ with $$\label{Tvir}
T^{\mathrm{vir}}= R\pi_* R{\mathcal{H}{\it{om}}}({\mathbb E},{\mathbb E})_0[1],$$ where ${\mathbb E}$ denotes the universal sheaf on $M \times S$, $\pi : M \times S \rightarrow M$ is projection, and $(\cdot)_0$ denotes the trace-free part.[^1]
This leads to a virtual cycle on $M$ of degree equal to the virtual dimension $$\label{vd}
{\operatorname{vd}}(M)=2rc_2-(r-1)c_1^2-(r^2-1)\chi({\mathcal O}_S).$$ The (algebraic) Donaldson invariants are then obtained by capping certain classes with the virtual cycle $$\label{invgeneral}
\int_{[M]^{{\mathrm{vir}}}} \tau_{\alpha_1}(\sigma_1) \cdots \tau_{\alpha_m}(\sigma_m),$$ where $\sigma_1, \ldots \sigma_m \in H^*(S,{\mathbb Q})$, $\alpha_1, \ldots, \alpha_m \geq 0$ are the descendence degrees, and the insertions $\tau_{\alpha_i}(\sigma_i)$ are defined in Section \[sec1\]. One of the main achievements of [@Moc] is a beautiful formula expressing for $r=2$ in terms of Seiberg-Witten invariants of $S$ and certain explicit integrals over $S^{[n_1]} \times S^{[n_2]}$, where $S^{[n_i]}$ denotes the Hilbert scheme of $n_i$ points on $S$. This formula was used by the first named author, H. Nakajima, and K. Yoshioka to prove the Witten conjecture for algebraic surfaces [@GNY3].
We are interested in the virtual $\chi_y$-genus of $M$ defined in [@FG] $$\chi_{-y}^{{\mathrm{vir}}}(M) := \sum_{p \geq 0} (-y)^p \chi^{\mathrm{vir}}(M,\Omega_{M}^{p,{\mathrm{vir}}}) \in {\mathbb Z}[y],$$ where $\chi^{\mathrm{vir}}(M, \cdot)$ is virtual holomorphic Euler characteristic and $\Omega_{M}^{p, {\mathrm{vir}}} = \Lambda^p (T_{M}^{{\mathrm{vir}}})^{\vee}$.[^2] We will usually use its shifted version $$\overline \chi_{-y}^{{\mathrm{vir}}}(M):=y^{-\frac{{\operatorname{vd}}(M)}{2}}\chi_{-y}^{{\mathrm{vir}}}(M),$$ which is a *symmetric* Laurent polynomial in $y^{\pm \frac{1}{2}}$ [@FG Cor. 4.9]. The virtual Euler characteristic is defined as $$e^{\mathrm{vir}}(M) :=\overline \chi_{-1}^{{\mathrm{vir}}}(M)= \int_{[M]^{\mathrm{vir}}} c_{{\operatorname{vd}}(M)}(T^{\mathrm{vir}}),$$ where the last equality is the virtual Hopf index theorem [@FG Cor. 4.8]. We are interested in the coefficients of the generating function $${\mathsf Z}_{S,H,c_1}(x,y) := \sum_{c_2} \overline{\chi}_{-y}^{\mathrm{vir}}(M_{S}^{H}(2,c_1,c_2)) \, x^{4c_2 - c_{1}^{2} - 3 \chi({\mathcal O}_S)},$$ where the power is ${\operatorname{vd}}(M_{S}^{H}(2,c_1,c_2))$. We will encounter the Dedekind eta function and three of the Jacobi theta functions $$\begin{aligned}
\begin{split} \label{thetas}
\eta(q) &= q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^n), \quad \ \ \theta_1(q,y) = \sum_{n \in {\mathbb Z}} (-1)^n q^{\big(n+\frac{1}{2}\big)^2} y^{n+\frac{1}{2}} \\
\theta_2(q,y) &= \sum_{n \in {\mathbb Z}} q^{\big(n+\frac{1}{2}\big)^2} y^{n+\frac{1}{2}}, \quad \theta_3(q,y) = \sum_{n \in {\mathbb Z}} q^{n^2} y^{n}.
\end{split}\end{aligned}$$ We define $\overline{\eta}(q) := q^{-\frac{1}{24}} \eta(q)$ and write the corresponding “Nullwerte” by $\theta_i(q) = \theta_i(q,1)$.
Let $S$ be a surface with $b_1(S) = 0$ and $p_g(S)>0$. We denote by ${\mathrm{SW}}(a)$ the Seiberg-Witten invariant of $S$ in class $a \in H^2(S,{\mathbb Z})$.[^3] A class $a \in H^2(S,{\mathbb Z})$ is called a Seiberg-Witten basic class when ${\mathrm{SW}}(a) \neq 0$. Many surfaces only have Seiberg-Witten basic classes $0$ and $K_S \neq 0$ with corresponding Seiberg-Witten invariants $1$ and $(-1)^{\chi({\mathcal O}_S)}$. E.g. minimal surfaces of general type with $p_g(S)>0$ and $b_1(S)=0$ [@Mor Thm. 7.4.1].
\[conj\] Let $S$ be a smooth projective surface with $b_1(S) = 0$ and $p_g(S)>0$. Suppose the Seiberg-Witten basic classes of $S$ are $0$ and $K_S \neq 0$. Let $H, c_1,c_2$ be chosen such that there exist no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1,c_2$. Then $\overline{\chi}_{-y}^{{\mathrm{vir}}}(M^H_S(2,c_1,c_2))$ is the coefficient of $x^{4c_2-c_1^2-3\chi({\mathcal O}_S)}$ of $$\begin{aligned}
\psi_S(x,y):=8 \Bigg( \frac{1}{2} \prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{10}(1-x^{2n} y)(1-x^{2n} y^{-1})} \Bigg)^{\chi({\mathcal O}_S)} \Bigg( \frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})} \Bigg)^{K_{S}^2}.\end{aligned}$$
In this conjecture, we can view $\psi_S(x,y)$ as generating function for $\overline{\chi}_{-y}^{{\mathrm{vir}}}(M^H_S(2,c_1,c_2))$ for all $c_1,c_2$. As a consequence of the conjecture, ${\mathsf Z}_{S,H,c_1}(x,y)$ is independent of the choice of polarization $H$, which we often omit from our notation. There is actually a closed formula for ${\mathsf Z}_{S,c_1}(x,y)$, which we give in Proposition \[fixedc1prop\]. Note that the first factor is related to $\theta_1(q,y)$ by the Jacobi triple product formula $$\theta_1(q,y) =q^{\frac{1}{4}}(y^{\frac{1}{2}}-y^{-\frac{1}{2}})\prod_{n=1}^\infty(1-q^{2n})(1-q^{2n} y)(1-q^{2n} y^{-1}).$$ We also present a generalization of this conjecture to *arbitrary* smooth projective surfaces $S$ with $b_1(S) = 0$ and $p_g(S)>0$ (Conjecture \[generalsurfconj\]), but the above conjecture is easier to state.
\[bar\] Replacing $x$ by $x y^{\frac{1}{2}}$, we go from generating functions for $\overline{\chi}_{-y}^{{\mathrm{vir}}}$ to generating functions for $\chi_{-y}^{{\mathrm{vir}}}$. Therefore we also get a conjectural generating function for the non-shifted virtual $\chi_y$-genera of the moduli spaces. Under the same assumptions, Conjecture \[conj\] gives that $\chi^{{\mathrm{vir}}}_{-y}(M^H_S(2,c_1,c_2))$ is the coefficient of $x^{4c_2-c_1^2-3\chi({\mathcal O}_S)}$ of $$\psi_S(xy^{\frac{1}{2}},y)=8 \Bigg( \frac{1}{2} \prod_{n=1}^{\infty} \frac{1}{(1-x^{2n}y^n)^{10}(1-x^{2n} y^{n+1})(1-x^{2n}y^{n-1})} \Bigg)^{\chi({\mathcal O}_S)} \Bigg( \frac{2 \overline{\eta}(x^4y^2)^2}{\theta_3(xy^{\frac{1}{2}},y^{\frac{1}{2}})} \Bigg)^{K_{S}^2}$$ with $$\overline\eta(x^4y^2)=\prod_{n=1}^\infty (1-x^{4n}y^{2n}), \quad \theta_3(xy^{\frac{1}{2}},y^{\frac{1}{2}})=\sum_{n\in {\mathbb Z}} y^{\binom{n+1}{2}}x^{n^2}.$$ This formula also makes it evident that $\chi^{{\mathrm{vir}}}_{-y}(M^H_S(2,c_1,c_2))$ is a polynomial.
Next, denote by $${\mathsf Z}_{S,c_1}(x) := {\mathsf Z}_{S,c_1}(x,1)$$ the generating function of virtual Euler characteristics. When $-K_S H > 0$ or $K_S=0$ (and some other cases including elliptic surfaces), these are just ordinary topological Euler characteristics because the obstructions vanish and the moduli space is smooth. Then ${\mathsf Z}_{S,c_1}(x)$ was studied by many people, e.g. [@Kly; @Got1; @Got2; @GH; @Yos1; @Yos2; @Yos3]. Conjecture \[conj\] implies the following formula which follows by specializing to $y=1$.
\[Proposition \[fixedc1prop\]\] \[cor\] Assume Conjecture \[conj\]. Let $S$ be a smooth projective surface with $b_1(S) = 0$ and $p_g(S)>0$. Suppose the Seiberg-Witten basic classes of $S$ are $0$ and $K_S \neq 0$. Let $H, c_1$ be chosen such that there exist no rank 2 Gieseker strictly $H$-semistable sheaves with first Chern class $c_1$. Then $$\begin{aligned}
{\mathsf Z}_{S,c_1}(x) =2\sum_{k=0}^3 \frac{(i^k)^{c_1^2 - \chi({\mathcal O}_S)}}{(2\overline{\eta}((-1)^kx^2)^{12})^{\chi({\mathcal O}_S)}}\Bigg(\frac{2\overline{\eta}(x^4)^2}{\theta_3(i^kx)} \Bigg) ^{K_S^2},\end{aligned}$$ where $i = \sqrt{-1}$.
From definition , we see that $\theta_3(i^k x)=\theta_3(x^4)+i^k\theta_2(x^4)$. Therefore we can rewrite the formula for ${\mathsf Z}_{S,c_1}(x)$ of Corollary \[cor\] as $$\begin{aligned}
\begin{split} \label{evirgenfunv2}
&\frac{2}{(2\overline{\eta}(x^2)^{12})^{\chi({\mathcal O}_S)}} \Bigg\{ \Bigg( \frac{\theta_3(x^4)+\theta_2(x^4)}{2\overline{\eta}(x^4)^2} \Bigg)^{-K_S^2} + (-1)^{c_1^2 - \chi({\mathcal O}_S)} \Bigg( \frac{\theta_3(x^4)-\theta_2(x^4)}{2\overline{\eta}(x^4)^2} \Bigg)^{-K_S^2} \Bigg\} \\
&+\frac{2 (-i)^{c_1^2 - \chi({\mathcal O}_S)}}{(2\overline{\eta}(-x^2)^{12})^{\chi({\mathcal O}_S)}} \Bigg\{ \Bigg( \frac{\theta_3(x^4)-i\theta_2(x^4)}{2\overline{\eta}(x^4)^2} \Bigg)^{-K_S^2} + (-1)^{c_1^2 - \chi({\mathcal O}_S)} \Bigg( \frac{\theta_3(x^4)+i\theta_2(x^4)}{2\overline{\eta}(x^4)^2} \Bigg)^{-K_S^2} \Bigg\}.
\end{split}\end{aligned}$$ In [@VW], C. Vafa and E. Witten study certain invariants related to $S$-duality. On ${\mathbb P}^2$ their invariants are topological Euler characteristics $e(M)$. For surfaces $S$ with smooth connected canonical divisor, they give a formula [@VW Eqn. (5.38)]. Equation coincides with *part of* their formula, namely all except the first two terms of [@VW Eqn. (5.38)] and up to an overall factor $x^{-\chi({\mathcal O}_S) + K_S^2/3}$ coming from our choice of normalization. Likewise, for $y=1$, the more general Conjecture \[generalsurfconj\] specializes to (part of) a formula from the physics literature due to R. Dijkgraaf, J.-S. Park, and B. Schroers [@DPS].
By work of S. Donaldson, D. Gieseker and J. Li, and others, if we fix $S$ with $b_1(S)=0$ and ample class $H$, then $M:=M^H_S(2,c_1,c_2)$ is irreducible and generically smooth of the expected dimension for sufficiently large ${\operatorname{vd}}=4c_2-c_1^2-3\chi({\mathcal O}_S)$. It is then also normal and lci. See [@HL Chap. 9] for references. In this case [@FG Thm. 4.15] implies that $e^{\mathrm{vir}}(M)$ is the degree of the Fulton Chern class $c_{F}(M)$, which agrees with the Euler characteristic of any smoothening of $M$, and Corollary \[cor\] predicts this number. More generally, all virtual Chern numbers of $M$ coincide with the corresponding Chern numbers of any smoothening [@FG Rem. 4.16]. In particular the virtual $\chi_y$-genus of $M$ equals the $\chi_y$-genus of any smoothening and Conjecture \[conj\] predicts these genera.
Recently, Y. Tanaka and R. P. Thomas [@TT1] defined a symmetric perfect obstruction theory on the moduli space of stable Higgs pairs $(E,\phi)$ on $S$, where $E$ has fixed determinant and $\phi$ is trace-free.[^4] Stable Higgs pairs are related by a Hitchin-Kobayashi correspondence to solutions of the Vafa-Witten equations. There is a ${\mathbb C}^*$-scaling action on the Higgs field and Tanaka-Thomas define “$\mathrm{SU}(r)$ Vafa-Witten invariants” by virtual localization with respect to this action. They show that the contribution to the invariant of the components corresponding to $\phi = 0$ are precisely the virtual Euler characteristics ${\mathsf Z}_{S,c_1}(x)$ that we study (though Tanaka-Thomas’s invariants are defined for any rank). Moreover in the rank 2 case and for $S$ with smooth connected canonical divisor and $b_1(S) = 0$, they conjecture that the contribution of the *other* components of the ${\mathbb C}^*$-fixed locus corresponds to the first two terms of [@VW Eqn. (5.38)]. Recall that these are precisely the two terms that we do not see. They gather evidence for this by computing the contributions of other components for some low orders. Therefore by the calculations of this paper and their conjecture, their invariant indeed appears to be the correct mathematical definition of Vafa-Witten’s invariant [@VW Eqn. (5.38)]. Also recently, A. Gholampour, A. Sheshmani, and S.-T. Yau studied Donaldson-Thomas invariants of local surfaces [@GSY]. Their invariants are closely related to Tanaka-Thomas’s invariants. The virtual Euler characteristics that we calculate are part of their invariants.
We approach Conjecture \[conj\] as follows:
- We use the virtual Hirzebruch-Riemann-Roch formula[^5] to express $\overline{\chi}_{-y}(M)$ in terms of certain descendent Donaldson invariants (Proposition \[chiyinsert\]).
- We apply Mochizuki’s formula to these descendent Donaldson invariants. This expresses ${\mathsf Z}_{S,c_1}(x,y)$ in terms of Seiberg-Witten invariants of $S$ and certain integrals over $S^{[n_1]} \times S^{[n_2]}$. Although Mochizuki’s formula requires $p_g(S)>0$, these integrals make sense for *any* surface $S$.
- We show that the integrals over $S^{[n_1]} \times S^{[n_2]}$ can be expressed in terms of seven universal series $A_1, \ldots, A_7 \in 1+q \, {\mathbb Q}[y](\!(s)\!)[[q]]$ (Proposition \[univ\]).
- The universal series $A_i$ are entirely determined by their values on $S = {\mathbb P}^2$ and ${\mathbb P}^1 \times {\mathbb P}^1$. We calculate $A_i(s,y,q)$ to order $q^{7}$ and $A_i(s,1,q)$ to order $q^{30}$ by Atiyah-Bott localization (Section \[toricsec\] and Appendix \[app1\]).
- We then verify Conjecture \[conj\] up to a certain order in $x$ for examples of the following types: blow-ups ${\operatorname{Bl}}_p K3$, double covers of ${\mathbb P}^2$, double covers of ${\mathbb P}^1 \times {\mathbb P}^1$, double covers of Hirzebruch surfaces ${\mathbb F}_a$, surfaces in ${\mathbb P}^3$, ${\mathbb P}^2 \times {\mathbb P}^1$ and ${\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1$, and complete intersections in ${\mathbb P}^4$ and ${\mathbb P}^5$ (Section \[evidencesec\]).
The reduction to toric surfaces also allows us to relate ${\mathsf Z}_{S,c_1}(x,1)$ to the Nekrasov partition function with one adjoint matter and one fundamental matter (Section \[Nekrasovsec\]). As a consequence, ${\mathsf Z}_{S,c_1}(x,1)$ can be expressed in terms of four universal series $F_0, H, G_1, G_2$ (see Remark \[4univ\] for details). This is not used elsewhere in the paper.
We present two generalizations of Conjecture \[conj\]:
- Conjecture \[numconj\] is a statement purely about intersection numbers on Hilbert schemes of points. Together with a strong form of Mochizuki’s formula (Remark \[strongform\]), it implies Conjecture \[conj\] (see Proposition \[numconjimpliesconj\]). In addition, the strong form of Mochizuki’s formula and Conjecture \[numconj\] imply a version of Conjecture \[conj\] for arbitrary blow-ups of surfaces $S$ with $b_1(S)=0$, $p_g(S)>0$, and Seiberg-Witten basic classes $0$ and $K_S \neq 0$ (Proposition \[nonmingt\]). We test Conjecture \[numconj\] in many cases in Section \[evidencesec\].
- Conjecture \[generalsurfconj\] generalizes Conjecture \[conj\] to *arbitrary* surfaces with $b_1(S) = 0$ and $p_g(S)>0$. This conjecture has two further applications. (1) It implies a blow-up formula for virtual $\chi_y$-genera, which is reminiscent of the blow-up formula of W.-P. Li and Z. Qin (Proposition \[blowupcor\]). (2) It implies a formula for surfaces with canonical divisor with irreducible reduced connected components (Proposition \[disconn\]). For $y=1$, the latter recovers another formula of Vafa-Witten [@VW (5.45)]. We check Conjecture \[generalsurfconj\] up to a certain order in $x$ in the following cases (other than the cases above): $K3$ surfaces, blow-ups ${\operatorname{Bl}}_p {\operatorname{Bl}}_q K3$, and elliptic surfaces $E(n)$ for various $n \geq 4$.
In Appendix \[appB\], the first named author and H. Nakajima conjecture a formula unifying the virtual Euler characteristic specialization of Conjecture \[generalsurfconj\] and Witten’s conjecture for Donaldson invariants.
In [@GK] we extend these results and conjectures to virtual elliptic genera and virtual cobordism classes. Besides Mochizuki’s formula [@Moc], this paper uses ideas from [@GNY1; @GNY2; @GNY3]. The physics approach to the calculation of elliptic genera of instanton moduli spaces was discussed in N. Nekrasov’s PhD thesis [@Nek1] and the papers [@LNS; @BLN]. We refer to [@LLZ] for applications to Gopakumar-Vafa invariants.
We thank Richard Thomas for suggesting to look at Mochizuki’s book in the context of $e^{{\mathrm{vir}}}$ many years ago. The first named author thanks Hiraku Nakajima and K$\bar{\textrm{o}}$ta Yoshioka for collaboration and many useful discussions on instanton counting and Mochizuki’s formula over many years, and in particular to Hiraku Nakajima for very useful discussions and comments. The second named author was supported by Marie Sk[ł]{}odowska-Curie IF 656898.
Mochizuki’s formula {#sec1}
===================
Let $S$ be a smooth projective surface with $b_1(S) = 0$ and polarization $H$. Denote by $M:=M_{S}^{H}(r,c_1,c_2)$ the moduli space of rank $r$ Gieseker $H$-stable torsion free sheaves on $S$ with Chern classes $c_1,c_2$. We assume there are no rank $r$ strictly Gieseker $H$-semistable sheaves with Chern classes $c_1,c_2$. Then $M_{S}^{H}(r,c_1,c_2)$ is projective. In this section, we first show that ${\chi}_{-y}^{{\mathrm{vir}}}(M)$ can be expressed in terms of descendent Donaldson invariants. We then recall Mochizuki’s formula [@Moc] and apply it to our setting.
We start with some notation. Assume there exists a universal sheaf ${\mathbb E}$ on $M \times S$ (in fact, we get rid of this assumption at the end of this section in Remark \[univexists\]). Let $\sigma \in H^*(S,{\mathbb Q})$ and $\alpha \geq 0$, then we define $$\tau_{\alpha}(\sigma) := \pi_{M*} \big( {\operatorname{ch}}_{2+\alpha}({\mathbb E}) \cap \pi_{S}^{*} \, \sigma \big),$$ where $\pi_M : M \times S \rightarrow M$ and $\pi_S : M \times S \rightarrow S$ denote projections. We refer to $\tau_{\alpha}(\sigma)$ as a descendent insertion of descendence degree $\alpha$. The insertions $\tau_0(\sigma)$ are called primary insertions.
We introduce some further notation which will be useful later. Suppose $X$ is any projective ${\mathbb C}$-scheme and $E$ a vector bundle on $X$, then we define $$\Lambda_y E := \sum_{p=0}^{{\operatorname{rk}}(E)} [\Lambda^p E] \, y^p \in K^0(X)[[y]].$$ This element is invertible in $K^0(X)[[y]]$ allowing us to define $\Lambda_y E$ for any element $E$ of $K^0(X)$ by extension. Next, for any element $E$ of $K^0(X)$ we define $$\begin{aligned}
\label{defT}
{\mathsf T}_{y}(E,t) := t^{-{\operatorname{rk}}E} \sum_k \left\{ {\operatorname{ch}}(\Lambda_{y} E^{\vee}) \, {\mathrm{td}}(E) \right\}_k t^{k},\end{aligned}$$ where $\{\cdot\}_k \in A^k(X)_{{\mathbb Q}}$ selects the degree $k$ part in the Chow ring. Since we have $$\Lambda_y (E_1 + E_2) = \Lambda_y E_1 \otimes \Lambda_y E_2,$$ the standard properties of Chern characters and Todd classes give $$\label{Tprod}
{\mathsf T}_y(E_1 + E_2,t) = {\mathsf T}_y(E_1,t) \, {\mathsf T}_y(E_2,t).$$ This multiplicative property will be crucial in Section \[secuniv\]. We also note that for a line bundle $L$ on $X$ with $c_1(L) = x$, we have $$\label{Tlinebd}
{\mathsf T}_y(L,t) = \frac{x (1+y e^{-x t})}{1-e^{-x t}}.$$ The variable $t$ can be exploited for a convenient normalization. Indeed if we take $t = 1+y$, then equations and imply $$\label{trivialshift}
{\mathsf T}_{y}(E - {\mathcal O}_X^{\oplus r},1+y) = {\mathsf T}_{y}(E,1+y)$$ for all $r \geq 0$. This will be used in Section \[secuniv\] as well. Another convenient consequence of the specialization $t = 1+y$ is that ${\mathsf T}_{y}(E,1+y)$ is a *polynomial* in $1+y$. Moreover, its leading coefficient is given by $$\label{specialize}
{\mathsf T}_{y}(E,1+y) \Big|_{y=-1} = c(E),$$ where $c(\cdot)$ denotes total Chern class. This essentially follows from [@FG Thm. 4.5(c)].
We go back to $M:=M_{S}^{H}(r,c_1,c_2)$. The next proposition involves an argument that appears more generally in the context of stable pairs on 3-folds in [@She] (see also [@Pan]).
\[chiyinsert\] For every $S,H,r,c_1,c_2$ as above, there exists a polynomial expression $P({\mathbb E})$ in certain descendent insertions $\tau_{\alpha}(\sigma)$ and $y$ such that $$\chi^{{\mathrm{vir}}}_{-y}(M_{S}^{H}(r,c_1,c_2)) = \int_{[M_{S}^{H}(r,c_1,c_2)]^{{\mathrm{vir}}}} P({\mathbb E}).$$
By the virtual Hirzebruch-Riemann-Roch theorem [@FG Cor. 3.4] $$\begin{aligned}
\chi^{{\mathrm{vir}}}_{-y}(M) &= \int_{[M]^{{\mathrm{vir}}}} {\mathsf T}_{-y}(T^{{\mathrm{vir}}},1-y),\end{aligned}$$ where ${\mathsf T}_{-y}(T^{{\mathrm{vir}}},1-y)$ can be expressed as a ${\mathbb Q}$-linear combination of monomials in $c_i(T^{{\mathrm{vir}}})$ and $y$. By and Grothendieck-Riemann-Roch, each $c_{i}(T^{{\mathrm{vir}}})$ can be expressed as a ${\mathbb Q}$-linear combination of monomials in $$\label{twoterms}
\pi_{M*} \big( {\operatorname{ch}}_\alpha({\mathbb E}) {\operatorname{ch}}_\beta({\mathbb E}) \cap \pi_{S}^{*} \sigma \big),$$ where $\sigma$ is one of the components of ${\mathrm{td}}(S)$. Therefore it suffices to show that every expression of the form is a polynomial in descendent insertions. This will then define the universal polynomial $P({\mathbb E})$.
Let $\pi_{ij}$ and $\pi_i$ be the projections from $M \times S \times S$ to factors $(i,j)$ and $i$ respectively. Then equals $$\label{MxSxS}
\pi_{1*} \big( \pi_{12}^{*} {\operatorname{ch}}_\alpha({\mathbb E}) \ \pi_{13}^{*} {\operatorname{ch}}_\beta({\mathbb E}) \cap \pi_2^* \sigma \ \pi_{23}^{*} \Delta \big),$$ where $\Delta \in H^4(S \times S,{\mathbb Q})$ is (Poincaré dual to) the class of the diagonal. Next we consider the Künneth decomposition $$\Delta = \sum_{i+j=4} \theta_{1}^{(i)} \otimes \theta_{2}^{(j)},$$ where $\theta_{1}^{(i)} \in H^i(S,{\mathbb Q})$ and $\theta_{2}^{(j)} \in H^j(S,{\mathbb Q})$. Substituting this decomposition into , factoring the push-forward as $\pi_1 = \pi_M \circ \pi_{12}$, and applying the projection formula gives $$\pi_{M*} \big( {\operatorname{ch}}_\alpha({\mathbb E}) \, {\operatorname{ch}}_\beta({\mathbb E}) \cap \pi_{S}^{*} \sigma \big) = \sum_{i+j=4} \tau_\alpha(\sigma \theta_{1}^{(i)}) \, \tau_\beta(\theta_{2}^{(j)}). \qedhere$$
Note that in this section (and the next) we use $\chi_{-y}^{{\mathrm{vir}}}$ instead of $\overline{\chi}_{-y}^{{\mathrm{vir}}}$ as in Conjecture \[conj\]. The reason is that the intermediate formulae of this and the next section look slightly easier for $\chi_{-y}^{{\mathrm{vir}}}$ whereas the final formula of Conjecture \[conj\] looks more elegant for $\overline{\chi}_{-y}^{{\mathrm{vir}}}$. One can easily pass from one to the other by Remark \[bar\].
Next we recall Mochizuki’s formula [@Moc Thm. 1.4.6]. His formula is derived by a beautiful argument involving *geometric wall-crossing* for moduli spaces of so-called Bradlow pairs depending on a stability parameter $\alpha \in {\mathbb Q}_{>0}$. For $\alpha \rightarrow \infty$ the moduli spaces are empty and for $\alpha \rightarrow 0^+$ the moduli space is a projective bundle over $M$. On a wall, Mochizuki uses a “master space” with ${\mathbb C}^*$-action, whose fixed locus contains components corresponding to the moduli spaces of Bradlow pairs on either side of the wall. Other components of the fixed locus can be seen as contributions from wall-crossing. They lead to the Seiberg-Witten invariants and integrals over $S^{[n_1]} \times S^{[n_2]}$ described below. These wall-crossing terms are computed by Graber-Pandharipande’s virtual localization on the master space. This geometric wall-crossing is very different from *motivic wall-crossing*, as in [@Joy], which does not work for ample $K_S$ due to non-vanishing ${\operatorname{Ext}}^2$. Among other things, Mochizuki’s formula was used in [@GNY3] to prove Witten’s conjecture for algebraic surfaces.
There are two ingredients for Mochizuki’s formula. The first ingredient is the Seiberg-Witten invariants ${\mathrm{SW}}(a)$ of $S$ in class $a \in H^2(S,{\mathbb Z})$ mentioned in the introduction.
The second ingredient is certain integrals over products of Hilbert schemes of points. On $S^{[n_1]} \times S^{[n_2]} \times S$ we have the pull-backs of the universal ideal sheaves ${\mathcal I}_1$, ${\mathcal I}_2$ from $S^{[n_1]} \times S$, $S^{[n_2]} \times S$. For any line bundle $L \in {\operatorname{Pic}}(S)$ we denote by $L^{[n_i]}$ the tautological vector bundle on $S^{[n_i]}$ defined by $$L^{[n_i]} := p_* q^* L,$$ with $p : {\mathcal Z}_i \rightarrow S^{[n_i]}$, $q : {\mathcal Z}_i \rightarrow S$ projections from the universal subscheme ${\mathcal Z}_i \subset S^{[n_i]} \times S$.
We consider $S^{[n_1]} \times S^{[n_2]}$ to be endowed with a trivial ${\mathbb C}^*$-action and we denote the generator of the character group by ${\mathfrak s}$.[^6] Moreover we write $s$ for the generator of $$H^*(B{\mathbb C}^*,{\mathbb Q}) = H^*_{{\mathbb C}^*}(pt,{\mathbb Q}) \cong {\mathbb Q}[s].$$ Let $P({\mathbb E})$ be any polynomial in descendent insertions $\tau_{\alpha}(\sigma)$, which arises from a polynomial in Chern numbers of $T^{\mathrm{vir}}$ (e.g. such as in Prop. \[chiyinsert\]). For any $a_1, a_2 \in A^1(S)$ and $n_1, n_2 \in {\mathbb Z}_{\geq 0}$, Mochizuki defines $\Psi(a_1,a_2,n_1,n_2)$ as follows $$\label{Psi}
{\mathrm{Coeff}}_{s^0} \Bigg( \frac{P({\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1} \oplus {\mathcal I}_2(a_2) \otimes {\mathfrak s})}{Q({\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1}, {\mathcal I}_2(a_2) \otimes {\mathfrak s})} \frac{{\mathrm{Eu}}({\mathcal O}(a_1)^{[n_1]}) \, {\mathrm{Eu}}({\mathcal O}(a_2)^{[n_2]} \otimes {\mathfrak s}^2)}{(2s)^{n_1+n_2 - \chi({\mathcal O}_S)}} \Bigg).$$ Let us explain this notation. Here ${\mathrm{Eu}}(\cdot)$ denotes ${\mathbb C}^*$-equivariant Euler class and ${\mathrm{Coeff}}_{s^0}$ refers to taking the coefficient of $s^0$.[^7] The notation ${\mathcal I}_i(a_i)$ is short-hand for ${\mathcal I}_i \otimes \pi_S^* {\mathcal O}(a_i)$. Furthermore, for any ${\mathbb C}^*$-equivariant sheaves ${\mathbb E}_1$, ${\mathbb E}_2$ on $S^{[n_1]} \times S^{[n_2]} \times S$ flat over $S^{[n_1]} \times S^{[n_2]}$, Mochizuki defines $$Q({\mathbb E}_1,{\mathbb E}_2) :={\mathrm{Eu}}(- R{\mathcal{H}{\it{om}}}_{\pi}({\mathbb E}_1,{\mathbb E}_2) - R{\mathcal{H}{\it{om}}}_{\pi}({\mathbb E}_2,{\mathbb E}_1)),$$ where $\pi : S^{[n_1]} \times S^{[n_2]} \times S \rightarrow S^{[n_1]} \times S^{[n_2]}$ denotes projection and $$R{\mathcal{H}{\it{om}}}_\pi(\cdot, \cdot) := R\pi_* R{\mathcal{H}{\it{om}}}(\cdot,\cdot).$$ Finally, $P(\cdot)$ is the expression obtained from $P({\mathbb E})$ by formally replacing ${\mathbb E}$ by $\cdot$. We define $$\widetilde{\Psi}(a_1,a_2,n_1,n_2,s)$$ by expression but *without* applying ${\mathrm{Coeff}}_{s^0}$.
Next, let $c_1,c_2$ be a choice of Chern classes and let ${\operatorname{ch}}= (2,c_1,\frac{1}{2}c_1^2 - c_2)$ denote the corresponding Chern character. For any decomposition $c_1 = a_1 + a_2$, Mochizuki defines $$\label{cA}
{\mathcal A}(a_1,a_2,c_2) := \sum_{n_1 + n_2 = c_2 - a_1 a_2} \int_{S^{[n_1]} \times S^{[n_2]}} \Psi(a_1,a_2,n_1,n_2).$$ We denote by $\widetilde{{\mathcal A}}(a_1,a_2,c_2,s)$ the same expression with $\Psi$ replaced by $\widetilde{\Psi}$.
\[mocthm\] Let $S$ be a smooth projective surface with $b_1(S) = 0$ and $p_g(S) >0$. Let $H, c_1,c_2$ be chosen such that there exist no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1,c_2$. Suppose the universal sheaf ${\mathbb E}$ exists on $M_{S}^{H}(2,c_1,c_2) \times S$. Suppose the following conditions hold:
1. $\chi({\operatorname{ch}}) > 0$, where $\chi({\operatorname{ch}}) := \int_S {\operatorname{ch}}\cdot {\mathrm{td}}(S)$ and ${\operatorname{ch}}= (2,c_1,\frac{1}{2}c_1^2 - c_2)$.
2. $p_{{\operatorname{ch}}} > p_{K_S}$, where $p_{{\operatorname{ch}}}$ and $p_{K_S}$ are the reduced Hilbert polynomials associated to ${\operatorname{ch}}$ and $K_S$.
3. For all Seiberg-Witten basic classes $a_1$ satisfying $a_1 H \leq (c_1 -a_1) H$ the inequality is strict.
Let $P({\mathbb E})$ be any polynomial in descendent insertions, which arises from a polynomial in Chern numbers of $T^{\mathrm{vir}}$ (e.g. such as in Prop. \[chiyinsert\]). Then $$\int_{[M_{S}^{H}(2,c_1,c_2)]^{{\mathrm{vir}}}} P({\mathbb E}) = -2^{1-\chi({\operatorname{ch}})} \sum_{{\scriptsize{\begin{array}{c} c_1 = a_1 + a_2 \\ a_1 H < a_2 H \end{array}}}} {\mathrm{SW}}(a_1) \, {\mathcal A}(a_1,a_2,c_2).$$
\[univexists\] The assumption that ${\mathbb E}$ exists on $M \times S$, where $M:=M_{S}^{H}(2,c_1,c_2)$, is unnecessary. As remarked in the introduction, $T^{{\mathrm{vir}}} = -R{\mathcal{H}{\it{om}}}_{\pi}({\mathbb E},{\mathbb E})_0$ always exists globally so the left-hand side of Mochizuki’s formula always makes sense. Moreover, Mochizuki [@Moc] works over the Deligne-Mumford stack of oriented sheaves, which always has a universal sheaf. This can be used to show that global existence of ${\mathbb E}$ on $M \times S$ can be dropped from the assumptions. In fact, when working on the stack, $P$ can be *any* polynomial in descendent insertions defined using the universal sheaf on the stack.
Mochizuki’s formula (Theorem \[mocthm\]) holds without the assumption that there are no strictly $H$-Gieseker semistable sheaves and without assumption (iii). Then $\int_{[M]^{{\mathrm{vir}}}} P({\mathbb E})$ is defined via the moduli space of oriented reduced Bradlow pairs [@Moc Def. 7.3.2]. In this more general setup the definition of ${\mathcal A}(a_1,a_2,c_2)$ has to be modified slightly: when $a_1 H = a_2 H$ the sum in is over all $n_1>n_2$ satisfying $n_1 + n_2 = c_2 - a_1 a_2$.
\[assumpmocthm\] It is conjectured in [@GNY3] that assumptions (ii) and (iii) can be dropped from Theorem \[mocthm\] and the sum can be replaced by a sum over *all* Seiberg-Witten basic classes. We will see in our computations that assumption (i) is necessary.
\[Mochcor\] Let the setup be as in Theorem \[mocthm\]. Then $${\chi}_{-y}^{\mathrm{vir}}(M_{S}^{H}(2,c_1,c_2)) = -2^{1-\chi({\operatorname{ch}})} \sum_{{\scriptsize{\begin{array}{c} c_1 = a_1 + a_2 \\ a_1 H < a_2 H \end{array}}}} {\mathrm{SW}}(a_1) \, {\mathcal A}(a_1,a_2,c_2),$$ where $P({\mathbb E})$ is determined by the following expression $$P({\mathbb E}) := {\mathsf T}_{-y}(-R{\mathcal{H}{\it{om}}}_{\pi}({\mathbb E},{\mathbb E})_0,1-y)$$ with ${\mathbb E}= {\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1} \oplus {\mathcal I}_2(a_2) \otimes {\mathfrak s}$. We note that the rank of $$-R{\mathcal{H}{\it{om}}}_{\pi}({\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1} \oplus {\mathcal I}_2(a_2) \otimes {\mathfrak s},{\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1} \oplus {\mathcal I}_2(a_2) \otimes {\mathfrak s})_0$$ is given by $$\begin{aligned}
&\chi({\mathcal O}_S)+(2n_1+2n_2 - 2\chi({\mathcal O}_S)) + (n_1+n_2-\chi({\mathcal O}(a_2-a_1))) \\
&\qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \! + (n_1+n_2-\chi({\mathcal O}(a_1-a_2))) \\
&= 4c_2-c_1^2-3\chi({\mathcal O}_S),\end{aligned}$$ which equals the rank of $T^{\mathrm{vir}}$ given by .
Universality {#secuniv}
============
In this section $S$ is *any* smooth projective surface. We start with a well-known lemma, which we include for completeness.
\[lem1\] Let $\pi : S^{[n]} \times S \rightarrow S^{[n]}$ denote projection. Then $$-R{\mathcal{H}{\it{om}}}_\pi({\mathcal I},{\mathcal I})_0 \cong {\curly Ext}_{\pi}^{1}({\mathcal I},{\mathcal I})_0 \cong T_{S^{[n]}},$$ where ${\mathcal I}$ denotes the universal ideal sheaf and $T_{S^{[n]}}$ denotes the tangent bundle.
Since ${\curly Ext}_{\pi}^{1}({\mathcal I},{\mathcal I})_0 \cong T_{S^{[n]}}$, it suffices to show that for any ideal sheaf $I = I_Z\subset {\mathcal O}_S$ we have ${\operatorname{Hom}}(I,I)_0 = {\operatorname{Ext}}^2(I,I)_0 = 0$. Clearly ${\operatorname{Hom}}(I,I)_0 = 0$ because $I$ is simple.
Next we consider the trace map $${\operatorname{Ext}}^2(I,I) \rightarrow H^2({\mathcal O}_S).$$ First applying $- \otimes K_S$ and then ${\operatorname{Hom}}(I,\cdot)$ to $$\label{ses}
0 \rightarrow I \rightarrow {\mathcal O}_S \rightarrow {\mathcal O}_Z \rightarrow 0$$ gives a long exact sequence $$0 \rightarrow {\operatorname{Hom}}(I,I \otimes K_S) \rightarrow {\operatorname{Hom}}(I,K_S) \rightarrow {\operatorname{Hom}}(I,K_S|_Z) \rightarrow \cdots.$$ The natural map $H^0(K_S) \rightarrow {\operatorname{Hom}}(I,I \otimes K_S)$ is Serre dual to the trace map, so our goal is to show that this map is an isomorphism. It is enough to show that the restriction map $H^0(K_S) \rightarrow {\operatorname{Hom}}(I,K_S)$, which factors through ${\operatorname{Hom}}(I,I \otimes K_S)$, is an isomorphism. This in turn can be seen by applying ${\operatorname{Hom}}(\cdot, K_S)$ to and using that $Z$ is 0-dimensional.
Our main object of study is the following generating function.
\[defZX\] For any $a$ in the Chow group $A^1(S)$ we abbreviate $\chi(a) := \chi({\mathcal O}(a))$. For any $a_1, c_1 \in A^1(S)$, we define $$\begin{aligned}
&{\mathsf Z}_{S}(a_1,c_1,s,y, q) := \\
&\sum_{n_1, n_2 \geq 0} q^{n_1+n_2} \int_{S^{[n_1]} \times S^{[n_2]}} \frac{{\mathsf T}^{{\mathbb C}^*}_{-y}(E_{n_1,n_2},1-y) \, {\mathrm{Eu}}({\mathcal O}(a_1)^{[n_1]}) \, {\mathrm{Eu}}({\mathcal O}(c_1-a_1)^{[n_2]} \otimes {\mathfrak s}^2)}{{\mathrm{Eu}}(E_{n_1,n_2} - \pi_1^* \, T_{S^{[n_1]}} - \pi_2^* \, T_{S^{[n_2]}})}, \end{aligned}$$ where ${\mathsf T}^{{\mathbb C}^*}$ denotes the ${\mathbb C}^*$-equivariant analog of and $$\begin{aligned}
E_{n_1,n_2} := -R{\mathcal{H}{\it{om}}}_{\pi}({\mathbb E},{\mathbb E})_0+\chi({\mathcal O}_S) \otimes {\mathcal O}+ \chi(c_1-2a_1) \otimes {\mathcal O}\otimes {\mathfrak s}^2+ \chi(2a_1-c_1) \otimes {\mathcal O}\otimes {\mathfrak s}^{-2}
$$ with $${\mathbb E}= {\mathcal I}_1(a_1) \otimes {\mathfrak s}^{-1} \oplus {\mathcal I}_2(a_2) \otimes {\mathfrak s}.$$ This can be rewritten as $$\begin{aligned}
E_{n_1,n_2}
= \pi_1^* \, T_{S^{[n_1]}} + \pi_2^* \, T_{S^{[n_2]}} &+ {\mathcal O}(c_1-2a_1)^{[n_2]} \otimes {\mathfrak s}^2 + {\mathcal O}(2a_1-c_1 + K_S)^{[n_1] *} \otimes {\mathfrak s}^2 \\
&- R{\mathcal{H}{\it{om}}}_{\pi}({\mathcal O}_{{\mathcal Z}_1}(a_1),{\mathcal O}_{{\mathcal Z}_2}(c_1-a_1)) \otimes {\mathfrak s}^2 \\
&+ {\mathcal O}(2a_1-c_1)^{[n_1]} \otimes {\mathfrak s}^{-2} + {\mathcal O}(c_1-2a_1+K_S)^{[n_2] *} \otimes {\mathfrak s}^{-2} \\
&- R{\mathcal{H}{\it{om}}}_{\pi}({\mathcal O}_{{\mathcal Z}_2}(c_1-a_1),{\mathcal O}_{{\mathcal Z}_1}(a_1)) \otimes {\mathfrak s}^{-2}.\end{aligned}$$ Here we used Serre duality and $(\cdot)^*$ denotes the dual vector bundle. The complex $E_{n_1,n_2}$ has rank $4n_1+4n_2$. Note that $$\label{leadingcoeff}
{\mathsf Z}_{S}(a_1,c_1,s,y,q) \in 1 + q \, {\mathbb Q}[y](\!(s)\!)[[q]].$$ For $y=1$, using the ${\mathbb C}^*$-equivariant version of , we obtain $${\mathsf T}^{{\mathbb C}^*}_{-y}(E_{n_1,n_2},1-y) \Big|_{y=1} = c^{{\mathbb C}^*}(E_{n_1,n_2}).$$
Suppose we have a decomposition $c_1 = a_1 + a_2$. Then Corollary \[Mochcor\] implies $$\begin{aligned}
\begin{split} \label{keyexpr}
&\sum_{c_2 \in {\mathbb Z}} \widetilde{{\mathcal A}}(a_1,c_1-a_1,c_2,s) \, q^{c_2} = {\mathsf Z}_{S}\Big(a_1,c_1,s,y,\frac{q}{2s}\Big) \\
&\times (2s)^{\chi({\mathcal O}_S)} \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg)^{\chi(c_1-2a_1)} \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg)^{\chi(2a_1-c_1)} q^{a_1 (c_1-a_1)}.
\end{split}\end{aligned}$$ Besides Corollary \[Mochcor\], this equality uses the following facts: Lemma \[lem1\], equation , equation , and equation . For $y=1$ the second line of simplifies to $$(2s)^{\chi({\mathcal O}_S)} \Bigg( \frac{2s}{1+2s} \Bigg)^{\chi(c_1-2a_1)} \Bigg( \frac{-2s}{1-2s} \Bigg)^{\chi(2a_1-c_1)} q^{a_1 (c_1-a_1)}.$$
\[univ\] There exist universal functions $$A_1(s,y,q), \ldots, A_7(s,y,q) \in 1 + q \, {\mathbb Q}[y](\!(s)\!)[[q]]$$ such that for any smooth projective surface $S$ and $a_1, c_1 \in A^1(S)$ we have $${\mathsf Z}_{S}(a_1,c_1,s,y,q) = A_1^{a_1^2} \, A_2^{a_1 c_1} \, A_3^{c_1^2} \, A_4^{a_1 K_S} \, A_5^{c_1 K_S} \, A_6^{K_S^2} \, A_7^{\chi({\mathcal O}_S)}.$$
**Step 1: Universality.** For any smooth projective surface $S$, define $S_2 := S \sqcup S$ and denote by $S_2^{[n]}$ the Hilbert scheme of $n$ points on $S_2$. Then $$S_2^{[n]} = \bigsqcup_{n_1+n_2=n} S^{[n_1]} \times S^{[n_2]}.$$ We endow $S_2^{[n]}$ with trivial ${\mathbb C}^*$-action. Denote by ${\mathcal I}_1$, ${\mathcal I}_2$ the sheaves on $S_2^{[n]}$ whose restriction to $S^{[n_1]} \times S^{[n_2]}$ are the sheaves ${\mathcal I}_1, {\mathcal I}_2$ we considered earlier. Denote by $L(a_1,c_1)$ the vector bundle whose restriction to $S^{[n_1]} \times S^{[n_2]}$ is $$\pi_1^* {\mathcal O}(a_1)^{[n_1]} \oplus \pi_2^*{\mathcal O}(c_1-a_1)^{[n_2]} \otimes {\mathfrak s}^2.$$ Let $X$ be any rational function in the following list of Chern classes and with coefficients in ${\mathbb Q}[y](\!(s)\!)$ $$\begin{aligned}
&c_{i_1}^{{\mathbb C}^*}(R{\mathcal{H}{\it{om}}}_{\pi}({\mathcal I}_1(a_1),{\mathcal I}_2(c_1-a_1)) \otimes {\mathfrak s}^2), \\
&c_{i_2}^{{\mathbb C}^*}(R{\mathcal{H}{\it{om}}}_{\pi}({\mathcal I}_2(c_1-a_1),{\mathcal I}_1(a_1)) \otimes {\mathfrak s}^{-2}), \\
&c_{i_3}(T_{S^{[n]}_2}), \ c_{i_4}^{{\mathbb C}^*}(L(a_1,c_1)).\end{aligned}$$ Of course we assume that only ${\mathbb C}^*$-moving terms appear in the denominator of $X$. Then there exists a polynomial $Y$ in $a_1^2$, $a_1 c_1$, $c_1^2$, $a_1 K_S$, $c_1 K_S$, $K_S^2$, $\chi({\mathcal O}_S)$ with coefficients in ${\mathbb Q}[y](\!(s)\!)$ such that $$\int_{S_2^{[n]}} X = Y,$$ for any smooth projective surface $S$ and $a_1, c_1 \in A^1(S)$. This is essentially [@GNY1 Lem. 5.5], which in turn is an adaptation of [@EGL].
We conclude that for each $n \geq 0$, $i \in {\mathbb Z}$, there exists a universal polynomial $Y_{n,i}$ in $a_1^2$, $a_1 c_1$, $c_1^2$, $a_1 K_S$, $c_1 K_S$, $K_S^2$, $\chi({\mathcal O}_S)$ such that $${\mathsf Z}_{S}(a_1,c_1,s,y,q) = \sum_{n \geq 0} \sum_{i} Y_{n,i} s^i q^n,$$ for all $S,a_1,c_1$. The coefficient of $q^0$ is 1 by . Hence there exists a universal power series $$G \in {\mathbb Q}[y,x_1, \ldots, x_7](\!(s)\!)[[q]],$$ such that $${\mathsf Z}_{S}(a_1,c_1,s,y,q) = \exp \, G(a_1^2, a_1 c_1, c_1^2, a_1 K_S, c_1 K_S, K_S^2, \chi({\mathcal O}_S))$$ for all $S,a_1,c_1$.\
**Step 2: Multiplicativity.** Let $S = S' \sqcup S''$, where $S', S''$ are not necessarily connected smooth projective surfaces. Let $a_1, c_1 \in A^1(S)$ be such that $$a_1|_{S'} = a_1', \ c_1|_{S'} = c_1', \ a_1|_{S''} = a_1'', \ c_1|_{S''} = c_1''.$$ Then $$S_{2}^{[n]} = \bigsqcup_{n_1+n_2 = n} S_{2}^{\prime [n_1]} \times S_{2}^{\prime \prime [n_2]}.$$ Let ${\mathcal I}_1 = {\mathcal I}_1' \boxplus {\mathcal I}_1''$, ${\mathcal I}_2 = {\mathcal I}_2' \boxplus {\mathcal I}_2''$. As observed in [@GNY1], we then have $$\begin{aligned}
&R {\mathcal{H}{\it{om}}}_\pi ({\mathcal I}_1(a_1), {\mathcal I}_2(c_1-a_1)) = \\
&R {\mathcal{H}{\it{om}}}_\pi ({\mathcal I}_1'(a_1'), {\mathcal I}_2'(c_1'-a_1')) \boxplus R {\mathcal{H}{\it{om}}}_\pi ({\mathcal I}_1''(a_1''), {\mathcal I}_2''(c_1''-a_1''))\end{aligned}$$ and similarly with ${\mathcal I}_1$ and ${\mathcal I}_2$ interchanged. Furthermore $$\begin{aligned}
T_{S_{2}^{[n]}} &= \bigoplus_{n_1+n_2=n} T_{S_{2}^{\prime [n_1]}} \boxplus T_{S_{2}^{\prime \prime [n_2]}} \\
L(a_1,c_1) &= L(a_1',c_1') \boxplus L(a_1'',c_1''), \\
\chi(c_1-2a_1) &= \chi(c_1'-2a_1') + \chi(c_1''-2a_1''), \\
\chi(2a_1-c_1) &= \chi(2a_1'-c_1') + \chi(2a_1''-c_1''), \\
a_1 (c_1-a_1) &= a_1' (c_1' - a_1') + a_1'' (c_1''-a_1'').\end{aligned}$$ Note that ${\mathrm{Eu}}(\cdot)$ and ${\mathsf T}^{{\mathbb C}^*}_{-y}(\cdot,1-y)$ are both group homomorphisms from $(K^{0}_{{\mathbb C}^*}(M),+)$ to $(A^*(M)_{{\mathbb Q}}[y](\!(s)\!),\cdot)$, where “${\mathbb C}^*$” stands for “${\mathbb C}^*$-equivariant”. For ${\mathsf T}^{{\mathbb C}^*}_{-y}(\cdot,1-y)$ this follows from the crucial multiplicative property . Therefore $${\mathsf Z}_{S}(a_1,c_1,s,y,q) = {\mathsf Z}_{S'}(a_1',c_1',s,q) \, {\mathsf Z}_{S''}(a_1'',c_1'',s,y,q).$$ The same reasoning as in [@GNY1] gives the universal functions $A_1, \ldots, A_7$.
For a $7$-tuple $\underline \alpha=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7) \in {\mathbb Z}^7$ we denote $$\begin{aligned}
\begin{split} \label{defsfA}
{\mathsf A}_{\underline \alpha }(s,y,q)
:=&- 2 \Bigg( 2^{-1} \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg)^{2} \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg)^{2} q^{-1} A_1(s,y,q/s) \Bigg)^{\alpha_1} \\
&\times \Bigg( 2 \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg)^{-2} \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg)^{-2} q A_2(s,y,q/s) \Bigg)^{\alpha_2} \\
&\times \Bigg( 2^{-\frac{1}{2}} \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg)^{\frac{1}{2}} \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}}\Bigg)^{\frac{1}{2}} A_3(s,y,q/s) \Bigg)^{\alpha_3} \\
&\times \Bigg( \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg) \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg)^{-1} A_4(s,y,q/s) \Bigg)^{\alpha_4} \\
&\times \Bigg( 2^{\frac{1}{2}} \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg)^{-\frac{1}{2}} \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg)^{\frac{1}{2}} A_5(s,y,q/s) \Bigg)^{\alpha_5} \\
&\times A_6(s,y,q/s) ^{\alpha_6} \\
&\times \Bigg( \frac{s}{2} \Bigg( \frac{1-e^{-2s(1-y)}}{1-y e^{-2s(1-y)}} \Bigg) \Bigg( \frac{1-e^{2s(1-y)}}{1-y e^{2s(1-y)}} \Bigg) A_7(s,y,q/s) \Bigg)^{\alpha_7}.
\end{split}\end{aligned}$$
\[univcor\] Suppose $S$ satisfies $b_1(S)=0$ and $p_g(S) > 0$. Let $H, c_1, c_2$ be chosen such that there exist no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1, c_2$. Assume furthermore that:
- $c_2 < \frac{1}{2} c_{1}(c_1-K_S) + 2\chi({\mathcal O}_S)$.
- $p_{{\operatorname{ch}}} > p_{K_S}$, where $p_{{\operatorname{ch}}}$ and $p_{K_S}$ are the reduced Hilbert polynomials associated to ${\operatorname{ch}}= (2,c_1,\frac{1}{2}c_1^2-c_2)$ and $K_S$.
- For all SW basic classes $a_1$ satisfying $a_1 H \leq (c_1 -a_1) H$ the inequality is strict.
Then $$\chi_{-y}^{{\mathrm{vir}}}(M_{S}^{H}(2,c_1,c_2))={\mathrm{Coeff}}_{s^0 q^{c_2}}
\Big[ \sum_{{\scriptsize{\begin{array}{c} a_1 \in H^2(S,{\mathbb Z}) \\ a_1 H < (c_1-a_1) H \end{array}}}} {\mathrm{SW}}(a_1) \,
{\mathsf A}_{(a_1^2,a_1 c_1,c_1^2,a_1 K_S, c_1 K_S, K_S^2, \chi({\mathcal O}_S))}(s,y,q)\Big].$$
This follows from Remark \[Mochcor\], equation , and Proposition \[univ\].
\[strongform\] By Remark \[assumpmocthm\], we conjecture that this corollary holds without assuming (ii) and (iii) and that the sum can be replaced by a sum over *all* Seiberg-Witten basic classes. We refer to this as “the strong form of Mochizuki’s formula”.
Toric calculation {#toricsec}
=================
The universal functions $A_1, \ldots, A_7$ are entirely determined by the generating function ${\mathsf Z}_S(a_1,c_1,s,y,q)$ in the following cases $$\begin{aligned}
(S,a_1,c_1) = \, &({\mathbb P}^2,{\mathcal O},{\mathcal O}), ({\mathbb P}^2,{\mathcal O}(1),{\mathcal O}(1)), ({\mathbb P}^2,{\mathcal O},{\mathcal O}(1)), ({\mathbb P}^2,{\mathcal O}(1),{\mathcal O}(2)), \\
&({\mathbb P}^1 \times {\mathbb P}^1, {\mathcal O},{\mathcal O}), ({\mathbb P}^1 \times {\mathbb P}^1, {\mathcal O}(1,0),{\mathcal O}(1,0)), ({\mathbb P}^1 \times {\mathbb P}^1, {\mathcal O},{\mathcal O}(1,0)).\end{aligned}$$ For these choices the $7 \times 7$ matrix with rows $$(a_1^2,a_1 c_1,c_1^2,a_1 K_S, c_1 K_S, K_S^2, \chi({\mathcal O}_S))$$ has full rank. In each of these cases $S$ is a toric surface.
Suppose $S$ is a toric surface with torus $T = {\mathbb C}^{*2}$. Let $\{U_\sigma\}_{\sigma = 1, \ldots, e(S)}$ be the cover of maximal $T$-invariant affine open subsets of $S$. On $U_\sigma$ we use coordinates $x_\sigma, y_\sigma$ such that $T$ acts with characters of weight $v_\sigma, w_\sigma \in {\mathbb Z}^2$ $$t \cdot (x_\sigma,y_\sigma) = (\chi(v_\sigma)(t) \, x_\sigma, \chi(w_\sigma)(t) \, y_\sigma).$$ Here $\chi(m) : T \rightarrow {\mathbb C}^*$ denotes the character of weight $m \in {\mathbb Z}^2$. Consider the integral over $S^{[n_1]} \times S^{[n_2]}$ of Definition \[defZX\] $$\label{HilbxHilb}
\int_{S^{[n_1]} \times S^{[n_2]}} \frac{{\mathsf T}_{-y}^{{\mathbb C}^*}(E_{n_1,n_2},1-y) \, {\mathrm{Eu}}({\mathcal O}(a_1)^{[n_1]}) \, {\mathrm{Eu}}({\mathcal O}(c_1-a_1)^{[n_2]} \otimes {\mathfrak s}^2)}{{\mathrm{Eu}}\big(E_{n_1,n_2}-\pi_1^*T_{S^{[n_1]}}-\pi_2^*T_{S^{[n_2]}}\big)}.$$ Define $\widetilde{T} := T \times {\mathbb C}^*$, where ${\mathbb C}^*$ denotes the trivial torus factor of Section \[sec1\]. The $T$-fixed locus of $S^{[n_1]} \times S^{[n_2]}$ is indexed by pairs $({\boldsymbol \lambda},{\boldsymbol \mu})$ with $${\boldsymbol \lambda}= \{\lambda^{(\sigma)}\}_{\sigma=1, \ldots, e(S)}, \ {\boldsymbol \mu}= \{\mu^{(\sigma)}\}_{\sigma=1, \ldots, e(S)},$$ where $\lambda^{(\sigma)}$, $\mu^{(\sigma)}$ are partitions such that $$\label{size}
\sum_\sigma |\lambda^{(\sigma)}| = n_1, \ \sum_\sigma |\mu^{(\sigma)}| = n_2.$$ Here $|\lambda|$ denotes the size of $\lambda$. A partition $\lambda = (\lambda_1 \geq \cdots \geq \lambda_\ell)$ corresponds to a monomial ideal of ${\mathbb C}[x,y]$ $$I_{Z_\lambda} := (y^{\lambda_1}, xy^{\lambda_2}, \ldots, x^{\ell-1}y^{\lambda_\ell}, x^{\ell}),$$ where $\ell(\lambda) = \ell$ denotes the length of $\lambda$. For $\lambda^{(\sigma)}$ we denote the subscheme defined by the corresponding monomial ideal in variables $x_\sigma,y_\sigma$ by $Z_{\lambda^{(\sigma)}}$ and similarly for $\mu^{(\sigma)}$.
In order to apply localization, we make a choice of $T$-equivariant structure on the line bundles ${\mathcal O}(a_1)$, ${\mathcal O}(c_1-a_1)$. For any $T$-equivariant divisor $a$, the restriction ${\mathcal O}(a)|_{U_{\sigma}}$ is trivial with $T$-equivariant structure determined by some character of weight $a_\sigma \in {\mathbb Z}^2$. By Atiyah-Bott localization, the integral equals $$\begin{aligned}
&\sum_{({\boldsymbol \lambda},{\boldsymbol \mu})} \prod_\sigma \frac{{\mathrm{Eu}}(H^0({\mathcal O}(a_1)|_{Z_{\lambda^{(\sigma)}}}))}{{\mathrm{Eu}}(T_{Z_{\lambda^{(\sigma)}}})} \frac{{\mathrm{Eu}}(H^0({\mathcal O}(c_1-a_1)|_{Z_{\mu^{(\sigma)}}}) \otimes {\mathfrak s}^2)}{{\mathrm{Eu}}(T_{Z_{\mu^{(\sigma)}}})} \\
&\times \frac{{\mathsf T}^{\widetilde{T}}_{-y}(E_{n_1,n_2}|_{(Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})},1-y)}{{\mathrm{Eu}}(E_{n_1,n_2}|_{(Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})} - T_{Z_{\lambda^{(\sigma)}}} - T_{Z_{\mu^{(\sigma)}}})}.\end{aligned}$$ Here ${\mathrm{Eu}}(\cdot)$ denotes $\widetilde{T}$-equivariant Euler class, ${\mathsf T}^{\widetilde{T}}$ is the $\widetilde{T}$-equivariant version of , and the sum is over all $({\boldsymbol \lambda},{\boldsymbol \mu})$ satisfying . Moreover, $T_{Z_{\lambda}}$ denotes the $T$-equivariant Zariski tangent space of $({\mathbb C}^{2})^{[n]}$ at $Z_\lambda$ where $n = |\lambda|$. The calculation of the above product reduces to the computation of the following elements of the $T$-equivariant $K$-group $K_0^{T}(pt)$ $$\begin{aligned}
&H^0({\mathcal O}(a)|_{Z_{\lambda^{(\sigma)}}}), \\
&R{\operatorname{Hom}}_S({\mathcal O}_{Z_{\lambda^{(\sigma)}}}, {\mathcal O}_{Z_{\lambda^{(\sigma)}}}), \\
&R{\operatorname{Hom}}_S({\mathcal O}_{Z_{\lambda^{(\sigma)}}}, {\mathcal O}_{Z_{\mu^{(\sigma)}}}(a)),\end{aligned}$$ for various $T$-equivariant divisors $a$. By definition we have $$Z_{\lambda^{(\sigma)}} = \sum_{i=0}^{\ell(\lambda^{(\sigma)}) - 1} \sum_{j=0}^{\lambda^{(\sigma)}_{i+1}-1} \chi(v_\sigma)^i \, \chi(w_\sigma)^j$$ and similarly for $Z_{\mu^{(\sigma)}}$. Multiplying by $\chi(a_\sigma)$ gives $H^0({\mathcal O}(a)|_{Z_{\lambda^{(\sigma)}}})$. Define $$\overline{\chi(m)} := \chi(-m) = \frac{1}{\chi(m)},$$ for any $m \in {\mathbb Z}^2$. This defines an involution on $K_0^{T}(pt)$ by ${\mathbb Z}$-linear extension.
Let $W,Z \subset S$ be 0-dimensional $T$-invariant subschemes supported on a chart $U_\sigma \subset S$ and let $a$ be a $T$-equivariant divisor on $S$ corresponding to a character of weight $a_\sigma \in {\mathbb Z}^2$ on $U_\sigma$. Then we have the following equality in $K_0^{T}(pt)$ $$R{\operatorname{Hom}}_S({\mathcal O}_W, {\mathcal O}_Z(a)) = \chi(a_\sigma) \, \overline{W} Z \frac{(1-\chi(v_\sigma)) (1- \chi(w_\sigma))}{\chi(v_\sigma) \chi(w_\sigma)}.$$
The proof we present is similar to the calculation in [@MNOP Sec. 4.7]. Let $v := v_\sigma$, $w := w_\sigma$, and $a := a_\sigma$. Write $U_\sigma = {\operatorname{Spec}\,}R$ with $R = {\mathbb C}[x_\sigma,y_\sigma]$. Then $$R{\operatorname{Hom}}_S({\mathcal O}_W, {\mathcal O}_Z(a)) = R {\operatorname{Hom}}_{U_\sigma}({\mathcal O}_{W}, {\mathcal O}_{Z}(a)),$$ because $W,Z$ are supported on $U_\sigma$. We claim $$\begin{aligned}
&\Gamma(U_\sigma, {\mathcal O}(a)) - R {\operatorname{Hom}}_{U_\sigma}(I_W,I_Z(a)) = \chi(a) \Big( Z + \frac{\overline{W}}{\chi(v) \chi(w)} - \overline{W} Z \frac{(1-\chi(v)) (1- \chi(w))}{\chi(v) \chi(w)} \Big).\end{aligned}$$ The result of the proposition follows from this using $I_Z = {\mathcal O}_{U_\sigma} - {\mathcal O}_{Z}$, $I_W = {\mathcal O}_{U_\sigma} - {\mathcal O}_{W}$, because the first term on the right-hand side is $H^0(U_\sigma,{\mathcal O}_Z(a))$ and the second term is $$\begin{aligned}
R{\operatorname{Hom}}_{U_\sigma}({\mathcal O}_W,{\mathcal O}_{U_\sigma}(a)) &= H^0(U_\sigma,{\mathcal O}_W(-a) \otimes K_{U_\sigma})^* \\
&= H^0(U_\sigma,{\mathcal O}_W)^* \otimes \frac{\chi(a)}{\chi(v)\chi(w)}\end{aligned}$$ by $T$-equivariant Serre duality. In order to prove the claim, choose $T$-equivariant graded free resolutions $$\begin{aligned}
0 \rightarrow E_r &\rightarrow \cdots \rightarrow E_0 \rightarrow I_{W} \rightarrow 0, \\
0 \rightarrow F_s &\rightarrow \cdots \rightarrow F_0 \rightarrow I_{Z} \rightarrow 0,\end{aligned}$$ where $$\begin{aligned}
E_i = \bigoplus_j R(d_{ij}), \ F_i = \bigoplus_j R(e_{ij}).\end{aligned}$$ Then we have Poincaré polynomials $$P_W = \sum_{i,j} (-1)^i \chi(d_{ij}), \ P_Z = \sum_{i,j} (-1)^i \chi(e_{ij}),$$ which are independent of the choice of resolution. Moreover $$\begin{aligned}
\begin{split} \label{WZ}
W &= {\mathcal O}_{U_\sigma} - I_W \\
&= \frac{1-P_W}{(1-\chi(v)) (1-\chi(w))}, \\
Z &= {\mathcal O}_{U_\sigma} - I_Z \\
&= \frac{1-P_Z}{(1-\chi(v)) (1-\chi(w))}.
\end{split}\end{aligned}$$ Furthermore $$\begin{aligned}
R {\operatorname{Hom}}_{U_\sigma}(I_W,I_Z(a)) &= \sum_{i,j,k,l} (-1)^{i+k} {\operatorname{Hom}}(R(d_{ij}),R(a+e_{kl})) \\
&= \sum_{i,j,k,l} (-1)^{i+k} R(a+e_{kl} - d_{ij}) \\
&= \frac{\chi(a) \overline{P}_W P_Z}{(1-\chi(v))(1-\chi(w))}.\end{aligned}$$ Eliminating $P_W$, $P_Z$ using gives the desired result.
We implemented the calculation of into a PARI/GP program (and some parts into Maple as well). This allows us to compute $A_1(s,y,q), \ldots, A_7(s,y,q)$ up to order $q^7$, where we calculated the coefficient of $q^i$ up to order $s^{29-3i}$ . We also calculated $A_1(s,1,q), \ldots, A_7(s,1,q)$ up to order $q^{30}$ and any order in $s$. The latter are listed up to order $q^4$ in Appendix \[app1\].
Nekrasov partition function {#Nekrasovsec}
===========================
In Section \[toricsec\], we considered a toric surface $S$ and we reduced ${\mathsf Z}_S(a_1,c_1,s,y,q)$ to a purely combinatorial expression on each toric patch $U_\sigma \cong {\mathbb C}^2$. We now study these local contributions on the toric patches in terms of the Nekrasov partition function. The content of this section is not used elsewhere in this paper. For simplicity we restrict attention to the case of virtual Euler characteristics, i.e. $y=1$, although very similar arguments work for virtual $\chi_y$-genus. In particular, we will see that $${\mathsf Z}_S(a_1,c_1,s,q):={\mathsf Z}_S(a_1,c_1,s,1,q)$$ can be expressed in terms of four universal functions. We use the notation from *Lectures on instanton counting* by Nakajima-Yoshioka [@NY1] (see also [@GNY1]).
Let $M(n)$ be the framed moduli space of pairs $(E,\Phi)$. Here $E$ is a rank 2 torsion free sheaf on ${\mathbb P}^2$ with $c_2(E)=n$ and locally free in a neighbourhood of the “line at infinity” $\ell_\infty \subset {\mathbb P}^2$. Furthermore $$\Phi : E|_{\ell_\infty} \stackrel{\cong}{\rightarrow} {\mathcal O}_{\ell_\infty}^{\oplus 2}$$ denotes the framing. The moduli space $M(n)$ is a fine moduli space and is a smooth quasi-projective variety of dimension $4n$. Let $T = {\mathbb C}^{*2}$ acting on ${\mathbb C}^2 = {\mathbb P}^2 \setminus \ell_\infty$ by $$(t_1,t_2)\cdot (x,y) = (t_1 x, t_2 y).$$ This action lifts to $M(n)$, which has an additional ${\mathbb C}^*$ action by scaling the framing $$(s_1,s_2) \in {\mathcal O}_{\ell_\infty}^{\oplus 2} \mapsto (e^{-1}s_1,es_2) \in {\mathcal O}_{\ell_\infty}^{\oplus 2}.$$ We denote the corresponding equivariant parameters of these actions by $\epsilon_1, \epsilon_2, a$. Following the conventions of [@GNY1], we write their characters as $e^{\epsilon_1}, e^{\epsilon_2}, e^{a}$. The $\Gamma = T \times {\mathbb C}^*$ fixed locus of $M(n)$ is given by ${\operatorname{Hilb}}^n({\mathbb C}^2 \sqcup {\mathbb C}^2)^T$, where ${\mathbb C}^2 = {\mathbb P}^2 \setminus \ell_{\infty}$ as described in [@NY1]. In particular, the fixed locus consists of finitely many isolated reduced points indexed by pairs of partitions $(\lambda,\mu)$ satisfying $|\lambda|+|\mu|=n$ as in Section \[toricsec\]. Concretely, the pair $(\lambda,\mu)$ corresponds to the direct sum of ideal sheaves $I_{Z_\lambda} \oplus I_{Z_\mu}$.
The instanton part of the Nekrasov partition function with one adjoint matter $M$ and one fundamental matter $m$ is defined as follows $${\mathsf Z}^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q) := \sum_{n=0}^{\infty} q^n \int_{M(n)} {\mathrm{Eu}}(T_{M(n)} \otimes e^M) \, {\mathrm{Eu}}(\mathcal{V} \otimes e^m).$$ Since $M(n)$ is non-compact, the above integral is defined by the $\Gamma$-localization formula. Here ${\mathrm{Eu}}(\cdot)$ is the equivariant Euler class with respect to two trivial torus actions with equivariant parameters $m,M$ (and, after localization, it also becomes equivariant with respect to $\Gamma$). Furthemore $T_{M(n)}$ denotes the tangent bundle and $\mathcal{V}$ denotes the rank $n$ vector bundle defined by $$\mathcal{V} := R^1 q_{2*} \Big( \mathcal{E} \otimes q_1^* {\mathcal O}(-\ell{_\infty}) \Big),$$ where $\mathcal{E}$ is the universal sheaf on ${\mathbb P}^2 \times M(n)$ and $q_i$ is projection to the $i$th factor. We note that the term in ${\mathsf Z}^{\mathrm{inst}}$ corresponding to $n=0$ is equal to 1.
In the previous section, we encountered the following expression $$\begin{aligned}
\begin{split} \label{Zcomb}
{\mathsf Z}(\epsilon_1,\epsilon_2,a_1,c_1,s,q) := \, &\sum_{(\lambda,\mu)} q^{|\lambda|+|\mu|} \frac{{\mathrm{Eu}}(H^0({\mathcal O}(a_1)|_{Z_{\lambda}}))}{{\mathrm{Eu}}(T_{Z_{\lambda}})} \frac{{\mathrm{Eu}}(H^0({\mathcal O}(c_1-a_1)|_{Z_{\mu}}) \otimes {\mathfrak s}^2)}{{\mathrm{Eu}}(T_{Z_{\mu}})} \\
&\times \frac{c(E_{n_1,n_2}|_{(Z_{\lambda},Z_{\mu})})}{{\mathrm{Eu}}(E_{n_1,n_2}|_{(Z_{\lambda},Z_{\mu})} - T_{Z_{\lambda}} - T_{Z_{\mu}})},
\end{split}\end{aligned}$$ where the sum is over all pairs of partitions $(\lambda, \mu)$. Moreover, we view $a_1, c_1$ as equivariant parameters (of trivial torus actions) by replacing ${\mathcal O}(c_1-a_1)$ by $e^{c_1-a_1}$ etc. In this section, all Chern classes and Euler classes are equivariant with respect to all tori.
$${\mathsf Z}(\epsilon_1,\epsilon_2,a_1,c_1,s,q) = {\mathsf Z}^{\mathrm{inst}}\Big(\epsilon_1,\epsilon_2,s+\frac{c_1-2a_1}{2},s+\frac{c_1}{2},0,q \Big).$$
The $\Gamma$-representation of $T_{(E,\Phi)} M(n) = {\operatorname{Ext}}^1_{{\mathbb P}^2}(E,E(-\ell_\infty))$ at a $\Gamma$-fixed point is described in [@NY1 Thm. 3.2]. After a bit of rewriting, much like in [@GNY1], it becomes $$\begin{aligned}
-R{\operatorname{Hom}}_{{\mathbb C}^2}(I_{Z_\lambda}, I_{Z_\lambda})_0 -R{\operatorname{Hom}}_{{\mathbb C}^2}(I_{Z_\mu}, I_{Z_\mu})_0 &+ R\Gamma({\mathcal O}_{{\mathbb C}^2} \otimes e^{2a}) - R{\operatorname{Hom}}_{{\mathbb C}^2}(I_{Z_\lambda}, I_{Z_\mu} \otimes e^{2a}) \\
&+ R\Gamma({\mathcal O}_{{\mathbb C}^2} \otimes e^{-2a}) - R{\operatorname{Hom}}_{{\mathbb C}^2}(I_{Z_\mu}, I_{Z_\lambda} \otimes e^{-2a}).\end{aligned}$$ Referring to Definition \[defZX\], we conclude that we want to specialize $a = s + \frac{c_1-2a_1}{2}$. The specialization $m = s+\frac{c_1}{2}$ can be deduced from the fact that the fibre of $\mathcal{V}$ over $Z_\lambda \oplus Z_\mu \in {\operatorname{Hilb}}^n({\mathbb C}^2 \sqcup {\mathbb C}^2)^T$ is given by $$H^1(I_{Z_\lambda}(-\ell_\infty)) \oplus H^1(I_{Z_\mu}(-\ell_\infty)) \cong H^0({\mathcal O}_{Z_\lambda}) \oplus H^0({\mathcal O}_{Z_\mu}),$$ where the first factor has weight $e^{-a}$ and the second $e^a$ with respect to the framing action. The specialization $M=0$ comes from the fact that we are interested in total Chern class, i.e. the virtual Euler characteristic specialization.
Similar to [@NY1], we define $$F^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q) := \log {\mathsf Z}^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q).$$ The Nekrasov conjecture, originally formulated in [@Nek2] and studied in various contexts e.g. in [@NY2; @NO; @BE], states that $F^{\mathrm{inst}}$ (in its original setting) is regular at $(\epsilon_1,\epsilon_2) = (0,0)$ and identifies its value at $(0,0)$ with the corresponding Seiberg-Witten prepotential, an expression in terms of the periods of the corresponding Seiberg-Witten curve, which is typically a family of elliptic curves. In the case of the partition function with one adjoint and one fundamental matter however, the Seiberg-Witten curve, and thus the Seiberg-Witten prepotential, are not available. Nevertheless, it is natural to conjecture[^8] $$\label{weakNEK}
\epsilon_1 \epsilon_2 F^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q) \textrm{ is regular at } (\epsilon_1,\epsilon_2) = (0,0).$$ Since $F^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q)$ is symmetric under $\epsilon_1 \leftrightarrow \epsilon_2$, allows us to write $$\begin{aligned}
\begin{split} \label{leadnekr}
\epsilon_1 \epsilon_2 F^{\mathrm{inst}}(\epsilon_1,\epsilon_2,a,m,M,q) = \, &F_0(a,m,M,q) + (\epsilon_1+\epsilon_2) H(a,m,M,q) + \\
&(\epsilon_1+\epsilon_2)^2 G_1(a,m,M,q) + \epsilon_1 \epsilon_2 G_2(a,m,M,q) + \cdots,
\end{split}\end{aligned}$$ where $\cdots$ stands for terms of order $\epsilon_1^i \epsilon_2^j$ with $i+j \geq 3$.
\[4univ\] For any toric surface $S$ and $a_1, c_1 \in A^1_{T}(S)$, we have (\[defZX\], ) $$\label{prodcharts}
{\mathsf Z}_S(a_1,c_1,s,1,q) = \prod_{\sigma=1}^{e(S)} {\mathsf Z}(v_\sigma,w_\sigma,(a_1)_\sigma,(c_1)_\sigma,s,q).$$ Assume holds. Combining , with localization on $S$ gives $$\begin{aligned}
\begin{split} \label{logZ}
\log {\mathsf Z}_S(a_1,c_1,s,q) =\, &\frac{1}{8}(c_1-2a_1)^2 \, \frac{\partial^2 F_0}{\partial a^2} + \frac{1}{4}(c_1-2a_1)c_1 \, \frac{\partial^2 F_0}{\partial a \partial m} + \frac{1}{8} c_1^2 \, \frac{\partial^2 F_0}{\partial m^2} \\
&+\frac{1}{2}(c_1-2a_1) K_S \, \frac{\partial H}{\partial a} +\frac{1}{2} c_1 K_S \, \frac{\partial H}{\partial m} +K_S^2 \, G_1 + c_2(S) \, G_2 .
\end{split}\end{aligned}$$ Here the right hand side is evaluated at $(a,m,M,q) = (s,s,0,q)$. See [@GNY1 Proof of Thm. 4.2] for a similar, but more complicated, calculation.
Assume holds. From Proposition \[univ\] and the fact that the $A_i$ are determined on ${\mathbb P}^2$, ${\mathbb P}^1 \times {\mathbb P}^1$, we deduce that holds for *any* smooth projective surface $S$ and $a_1, c_1 \in A^1(S)$. Therefore $ {\mathsf Z}_S(a_1,c_1,s,1,q)$ is determined by four universal functions $F_0,H,G_1,G_2$. We do not know the statement of the Nekrasov conjecture in this context, which would be an explicit conjectural formula for $F_0,H,G_1,G_2$, possibly in terms of periods of a family of elliptic curves, as studied in many cases e.g. in [@NY2; @NO; @BE; @GNY3]. One possibility to approach the $y=1$ specialization of Conjecture \[conj\] (and Conjectures \[numconj\], \[generalsurfconj\] below), would be via first finding a formulation of the Nekrasov conjecture in this context, and then a solution, employing strategies somehow related to the ones used in [@NY2; @NO; @BE; @GNY3]. However, this seems to be very difficult, because the corresponding Seiberg-Witten curve is not available.
Two more conjectures and consequences
=====================================
In Section \[toricsec\], we have given a toric procedure to calculate the universal functions $A_1(s,y,q), \ldots, A_7(s,y,q)$ and therefore also ${\mathsf A}_{\underline{\alpha}}(s,y,q)$ defined by . Consequently, we could now go ahead and provide checks of Conjecture \[conj\].
Instead we first present two generalizations of Conjecture \[conj\]. The first conjecture is a statement about intersection numbers on Hilbert schemes of points. It implies a formula for arbitrary blow-ups of surfaces $S$ with $b_1(S)=0$, $p_g(S)>0$, and Seiberg-Witten basic classes $0$ and $K_S \neq 0$. The second conjecture generalizes Conjecture \[conj\] to *arbitrary* surfaces $S$ with $b_1(S)=0$ and $p_g(S)>0$. It implies a blow-up formula, which is reminiscent of the blow-up formula of W.-P. Li and Z. Qin [@LQ1; @LQ2]. It also implies a formula for surfaces with canonical divisor with irreducible reduced connected components. The latter refines a result from the physics literature due to Vafa-Witten [@VW Eqn. (5.45)].
Numerical conjecture
--------------------
Suppose $S$ is a surface with $b_1(S) = 0$, $p_g(S)>0$, and Seiberg-Witten basic classes are $0$ and $K_S \neq 0$. Then Conjecture \[conj\] applies to any choice of $$\underline{\beta} = (\beta_1, \beta_2,\beta_3,\beta_4) = (c_1^2, c_1 K_S, K_S^2, \chi({\mathcal O}_S)),$$ provided we choose a polarization $H$ for which there are no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1, c_2$. Moreover, as long as the assumptions of Corollary \[univcor\] are satisfied, the coefficients of ${\mathsf Z}_{S,c_1}(x,y)$ are calculated by the universal functions $A_1, \ldots, A_7$. This raises the expectation that for *any* choice of the 4-tuple $\underline{\beta} \in {\mathbb Z}^4$ the formula of Corollary \[univcor\] is determined by the coefficients of the modular form of Conjecture \[conj\]. This turns out to be false. Computer calculations show that we need to impose[^9] $$\beta_3 \geq \beta_4 - 3.$$ Indeed let $\underline{\beta} \in {\mathbb Z}^4$ with $\beta_1 \equiv \beta_2 \mod 2$ and $\beta_3 \geq \beta_4 - 3$. Let $n < \frac{1}{2}(\beta_1-\beta_2)+2\beta_4$. We conjecture that $$\begin{aligned}
{\mathrm{Coeff}}_{s^0x^{4n-\beta_1-3\beta_4}} \Big[&(x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(0,0,\beta_1,0,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \\
&+ (-1)^{\beta_4} (x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(\beta_3,\beta_2,\beta_1,\beta_3,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \Big]\end{aligned}$$ equals the coefficient of $x^{4n-\beta_1-3\beta_4}$ of $$8 \Bigg( \frac{1}{2} \prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{10}(1-x^{2n} y)(1-x^{2n} y^{-1})} \Bigg)^{\beta_4} \Bigg( \frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})} \Bigg)^{\beta_3}.$$ In fact, we have a stronger conjecture, which arose by attempts to generalize Conjecture \[conj\] to blow-ups.
\[numconj\] Let $\underline{\beta} \in {\mathbb Z}^4$ be such that $\beta_1 \equiv \beta_2 \mod 2$ and $\beta_3 \geq \beta_4 - 3$. Let $n < \frac{1}{2}(\beta_1-\beta_2)+2\beta_4$. Let $(\gamma_1,\gamma_2)\in {\mathbb Z}^2$. Then $$\begin{aligned}
{\mathrm{Coeff}}_{s^0x^{4n-\beta_1-3\beta_4}} \Big[&(x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(\gamma_1,\gamma_2,\beta_1,\gamma_1,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \\
&+ (-1)^{\beta_4} (x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(\beta_3-\gamma_1,\beta_2-\gamma_2,\beta_1,\beta_3-\gamma_1,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \Big]\end{aligned}$$ equals the coefficient of $x^{4n-\beta_1-3\beta_4}$ of $$\begin{aligned}
\psi_{\gamma_1, \gamma_2, \beta_3, \beta_4}(x,y) := 8 (-1)^{\gamma_2}\Bigg( \frac{\phi(x,y)}{2}\Bigg)^{\beta_4} \Bigg( \frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})} \Bigg)^{\beta_3}\Bigg(\frac{\theta_3(x,y^{\frac{1}{2}})}{\theta_3(-x,y^{\frac{1}{2}})} \Bigg)^{\gamma_1},\end{aligned}$$ where $$\phi(x,y) :=\prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{10}(1-x^{2n} y)(1-x^{2n} y^{-1})}.$$
The evidence for this conjecture is presented in Section \[numconjevidence\]. We now discuss its consequences.
Let $\beta_1\in {\mathbb Z}$ be even, $\beta_4\in {\mathbb Z}_{\le 3}$, and $n < \frac{1}{2}\beta_1+2\beta_4$. Then we conjecture $$\begin{aligned}
\label{LHSK3}
{\mathrm{Coeff}}_{s^0x^{4n-\beta_1-3\beta_4}} \Big[& (x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(0,0,\beta_1,0,0,0,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \Big]\end{aligned}$$ equals the coefficient of $x^{4n-\beta_1-3\beta_4}$ of $$\begin{aligned}
\label{RHSK3}
4 \Bigg( \frac{\phi(x,y}{2} \Bigg)^{\beta_4}.\end{aligned}$$ In the case $\beta_4$ is even this follows from Conjecture \[numconj\] by taking $\gamma_1=\gamma_2=\beta_2=\beta_3 = 0$ because then the two summands on the left-hand side of the conjecture are equal. In the case $\beta_4$ is odd this says that is $0$, because only contains even powers of $x$.
\[numconjimpliesconj\] Conjecture \[numconj\] and the strong form of Mochizuki’s formula (Remark \[strongform\]) imply Conjecture \[conj\].
We only use Conjecture \[numconj\] for $\gamma_1=\gamma_2 = 0$. Suppose $S$ is a surface with $b_1(S) = 0$, $p_g(S)>0$, and Seiberg-Witten basic classes $0$ and $K_S \neq 0$. Choose a polarization $H$ and $c_1, c_2$ such that there are no rank 2 strictly Gieseker $H$-semistable sheaves with these Chern classes. By taking $$(\beta_1, \beta_2,\beta_3,\beta_4) = (c_1^2, c_1K_S, K_S^2, \chi({\mathcal O}_S))$$ we automatically satisfy $\beta_1 \equiv \beta_2 \mod 2$. The fact that $\beta_3 \geq \beta_4 - 3$ can be seen as follows. If $S$ is not minimal then it is the blow-up of a $K3$ in one point and the inequality is trivial. If $S$ is minimal then it is minimal properly elliptic or minimal general type because $p_g(S)>0$. For minimal general type the inequality follows from Noether’s inequality $K_S^2 \geq 2(\chi({\mathcal O}_S) - 3)$. When $\pi : S \rightarrow B$ is minimal properly elliptic, we have $B \cong {\mathbb P}^1$ because $b_1(S) = 0$. The canonical bundle $K_S$ satisfies $K_S^2 = 0$ and can be represented by an effective divisor containing $\pi^* D$, where $D \subset {\mathbb P}^1$ is some effective divisor of degree $$\chi({\mathcal O}_S)-2 \geq 0.$$ Indeed $\chi({\mathcal O}_S) \leq 3$, because otherwise at least $0, F, K_S$ are distinct Seiberg-Witten basic classes [@FM]. By the strong form of Mochizuki’s formula (Corollary \[univcor\] and Remark \[strongform\]), Conjecture \[conj\] follows for all $${\operatorname{vd}}< c_1^2 - 2 c_1 K_S + 5 \chi({\mathcal O}_S).$$ If this inequality is not satisfied, then we replace $c_1$ by $c_1 + t H$ for some $t>0$. Since $${\mathsf Z}_{S,c_1}(x,y) = {\mathsf Z}_{S,c_1 + 2 t H}(x,y)$$ we can compute the coefficients of this generating function for all $${\operatorname{vd}}< c_1^2 - 2 c_1 K_S + 5 \chi({\mathcal O}_S) + 4 t^2 H^2 + 4 t H (c_1 - K_S).$$ By choosing $t \gg 0$ the bound becomes arbitrarily high.
Fixed first Chern class {#fixedc1sec}
-----------------------
\[fixedc1propnum\] Assume Conjecture \[numconj\]. Let $\underline{\beta} \in {\mathbb Z}^4$ be such that $\beta_1 \equiv \beta_2 \mod 2$ and $\beta_3 \geq \beta_4 - 3$ and let $(\gamma_1,\gamma_2)\in {\mathbb Z}^2$. Then $$\begin{aligned}
&{\mathrm{Coeff}}_{s^0} \Big[ (x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(\gamma_1,\gamma_2,\beta_1,\gamma_1,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \\
&\qquad\quad \ \, + (-1)^{\beta_4} (x y^{-\frac{1}{2}})^{-\beta_1-3\beta_4} {\mathsf A}_{(\beta_3-\gamma_1,\beta_2-\gamma_2,\beta_1,\beta_3-\gamma_1,\beta_2,\beta_3,\beta_4)}(s,y,(x y^{-\frac{1}{2}})^4) \Big]\\
&=2(-1)^{\gamma_2}\sum_{k=0}^3 (i^k)^{\beta_1-\beta_4} \Bigg(\frac{\phi(i^k x,y)}{2}\Bigg)^{\beta_4}\Bigg(\frac{2\overline{\eta}(x^4)^2}{\theta_3(i^k x,y^{\frac{1}{2}})}\Bigg)^{\beta_3}\Bigg(\frac{\theta_3(i^k x,y^{\frac{1}{2}})}{\theta_3(-i^k x,y^{\frac{1}{2}})}\Bigg)^{\gamma_1} \\
&\quad +O(x^{\beta_1-2\beta_2+5\beta_4}),\end{aligned}$$ where $i = \sqrt{-1}$ and $$\phi(x,y) :=\prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{10}(1-x^{2n} y)(1-x^{2n} y^{-1})}.$$
Recall the formula for $\psi(x,y) := \psi_{\gamma_1,\gamma_2,\beta_3,\beta_4}(x,y)$ of Conjecture \[numconj\]. We see that the term in the sum on the right-hand side corresponding to $k=0$ equals $\psi(x,y)/4$. Define coefficients $f_n(y)$ by $$\psi(x,y) = \sum_{n=0}^{\infty} f_n(y) \, x^n.$$ Then the right-hand side of the formula of the proposition equals $$\begin{aligned}
\sum_{k=0}^{3} \frac{(i^k)^{\beta_1 - \beta_4}}{4} \psi(i^k x,y) &=\sum_{k=0}^{3} \sum_{n=1}^{\infty} \frac{(i^k)^{\beta_1 - \beta_4 + n}}{4} f_n(y) \, x^n \\
&=\sum_{n=1}^{\infty} \Bigg( \frac{1}{4}\sum_{k=0}^{3} \big(i^{\beta_1 - \beta_4 + n}\big)^k\Bigg) f_n(y) \, x^n \\
&= \sum_{n \equiv -\beta_1 + \beta_4 \mod 4} f_n(y) \, x^n.\end{aligned}$$ Therefore we conclude that the right-hand side of the formula of the proposition is obtained from $\psi(x,y)$ by extracting all terms $x^n$ for which $n \equiv -\beta_1 - 3 \beta_4 \mod 4$ and up to order $O(x^{\beta_1-2\beta_2+5\beta_4})$. The result follows from Conjecture \[numconj\].
The same type of proof applied to Conjecture \[conj\] implies the following.
\[fixedc1prop\] Assume Conjecture \[conj\]. Let $S$ be a smooth projective surface with $b_1(S) = 0$ and $p_g(S)>0$. Suppose the Seiberg-Witten basic classes of $S$ are $0$ and $K_S \neq 0$. Let $H, c_1$ be chosen such that there are no rank 2 strictly Gieseker $H$-semistable sheaves with first Chern class $c_1$. Then $$\begin{aligned}
{\mathsf Z}_{S,c_1}(x,y) = 2\sum_{k=0}^3 (i^k)^{c_1^2 - \chi({\mathcal O}_S)} \Bigg( \frac{\phi(i^k x,y)}{2} \Bigg)^{\chi({\mathcal O}_S)} \Bigg(\frac{2\overline{\eta}(x^4)^2}{\theta_3(i^k x,y^{\frac{1}{2}})} \Bigg)^{K_S^2}.\end{aligned}$$ In particular, Corollary \[cor\] in the introduction follows.
Using Conjecture \[numconj\] we can do better:
\[nonmingt\] Assume Conjecture \[numconj\] and the strong form of Mochizuki’s formula (Remark \[strongform\]). Let $S_0$ be a smooth projective surface with $b_1(S_0) = 0$ and $p_g(S_0)>0$. Suppose the Seiberg-Witten basic classes of $S_0$ are $0$ and $K_{S_0} \neq 0$. Suppose $S$ is obtained from an iterated blow-up (possibly at infinitely near points) of $S_0$ and let $E_1,\cdots, E_m$ denote the total transforms of the exceptional divisors. Suppose furthermore that $K_S^2\ge \chi({\mathcal O}_S)-3$. Let $H, c_1$ be chosen such that there exist no rank 2 strictly Gieseker $H$-semistable sheaves on $S$ with first Chern class $c_1$. Then $$\begin{aligned}
&{\mathsf Z}_{S,c_1}(x,y) = \\
&2\sum_{k=0}^3 (i^k)^{c_1^2-\chi({\mathcal O}_S)}\Bigg(\frac{\phi(i^k x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2\overline{\eta}(x^4)^2}{\theta_3(i^k x,y^{\frac{1}{2}})}\Bigg)^{K_{S}^2} \prod_{j=1}^m\Bigg(1+(-1)^{c_1E_j} \frac{\theta_3(-i^kx,y^{\frac{1}{2}})}{\theta_3(i^kx,y^{\frac{1}{2}})}\Bigg).\end{aligned}$$
The surface $S$ is obtained from $S_0$ by an iterated blow-up in (possibly infinitely near) points $\pi : S \rightarrow S_0$. We denote by $E_1, \ldots, E_m$ the total transforms on the blow-ups. Write $M:=\{1,\ldots,m\}$, and for a subset $I\subset M$ write $E_I=\sum_{i\in I} E_i$. Then $K_S=K_{S_0}+E_M$. Moreover the Seiberg-Witten basic classes are the $E_I$ (with Seiberg-Witten invariant $1$) and the $K_{S_0}+E_I=K_S-E_{M-I}$ (with SW-invariant $(-1)^{\chi({\mathcal O}_S)}$) for all $I\subset M$. E.g. see [@Mor Thm. 7.4.6] for Seiberg-Witten invariants of blow-ups. We denote by $|I|$ the number of elements of $I$. Now note that $$K_S(K_S-E_I)=(K_S-E_I)^2=K_S^2-E_I^2.$$ By the strong form of Mochizuki’s formula (Corollary \[univcor\] and Remark \[strongform\]) and Proposition \[fixedc1propnum\], we obtain the following equation modulo $x^{c_1^2-2c_1K_S+5\chi({\mathcal O}_S)}$ $$\begin{aligned}
&{\mathsf Z}_{S,c_1}(x,y) =\sum_{I\subset M}{\mathrm{Coeff}}_{s^0}\Big[\Big((x y^{-\frac{1}{2}})^{-c_1^2-3\chi({\mathcal O}_S)}{\mathsf A}_{(E_I^2,E_Ic_1,c_1^2,E_IK_S,c_1K_S,K_S^2,\chi({\mathcal O}_S))}(s,y,(x y^{-\frac{1}{2}})^4)\\&\quad+(-1)^{\chi({\mathcal O}_S)}(x y^{-\frac{1}{2}})^{-c_1^2-3\chi({\mathcal O}_S)}
{\mathsf A}_{(K_{S}^2-E_I^2,c_1K_{S}-c_1E_I,c_1^2,K_S^2-E_I^2,c_1K_S,K_S^2,\chi({\mathcal O}_S))}(s,y,(x y^{-\frac{1}{2}})^4)\Big]\\
&=2\sum_{I\subset M}\sum_{k=1}^3 (i^k)^{c_1^2-\chi({\mathcal O}_S)}\Bigg(\frac{\phi(i^k x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2\overline{\eta}(x^4)^2}{\theta_3(i^k x,y^{\frac{1}{2}})}\Bigg)^{K_S^2} (-1)^{c_1E_I} \Bigg(\frac{\theta_3(-i^k x,y^{\frac{1}{2}})}{\theta_3(i^kx,y^{\frac{1}{2}})}\Bigg)^{|I|},
\end{aligned}$$ where we replaced $E_{M-I}$ by $E_I$ for all terms with Seiberg-Witten invariant $(-1)^{\chi({\mathcal O}_S)}$. After interchanging the sums we get the formula of the proposition.
Arbitrary surfaces with holomorphic 2-form
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We present the following conjecture about virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on *arbitrary* smooth projective surfaces $S$ with $b_1(S)=0$ and $p_g(S)>0$. Although this conjecture is strictly stronger than Conjecture \[conj\], the latter is a little easier to state and was therefore the focus of the introduction. We provide some evidence for this conjecture in Sections \[K3sec\], \[ellipticevidence\], and \[blowupevidence\].
\[generalsurfconj\] Let $S$ be a smooth projective surface with $b_1(S) = 0$ and $p_g(S)>0$. Let $H,c_1,c_2$ be chosen such that there are no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1,c_2$ and let $M:=M_S^H(2,c_1,c_2)$. Then $\overline{\chi}_{-y}^{{\mathrm{vir}}}(M)$ equals the coefficient of $x^{{\operatorname{vd}}(M)}$ of $$\begin{aligned}
\psi_{S,c_1}(x,y) &:= 4 \Bigg(\frac{\phi(x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})}\Bigg)^{K_{S}^2} \sum_{a \in H^2(S,{\mathbb Z})} {\mathrm{SW}}(a)(-1)^{c_1 a} \Bigg(\frac{\theta_3(x,y^{\frac{1}{2}})}{\theta_3(-x,y^{\frac{1}{2}})}\Bigg)^{a K_S},
$$ where $$\phi(x,y) :=\prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{10}(1-x^{2n} y)(1-x^{2n} y^{-1})}.$$
Assuming this conjecture and when there are no strictly Gieseker $H$-semistable sheaves with first Chern class $c_1$, the same calculation as in Section \[fixedc1sec\] gives $$\label{Zpsi}
{\mathsf Z}_{S,c_1}(x,y) = \frac{1}{2} \psi_{S,c_1}(x,y) + \frac{1}{2} i^{c_1^2 - \chi({\mathcal O}_S)} \psi_{S,c_1}(ix,y),$$ where $i = \sqrt{-1}$. Specializing to $y=1$ gives (part of) a formula from the physics literature due to Dijkgraaf-Park-Schroers, namely terms two and three of [@DPS Eqn. (6.1)].[^10] This involves a bit of rewriting using $$\label{rewritetheta}
\theta_3(i^k x)=\theta_3(x^4)+i^k\theta_2(x^4),$$ ${\mathrm{SW}}(a) = (-1)^{\chi({\mathcal O}_S)} {\mathrm{SW}}(K_S-a)$ [@Mor Cor. 6.8.4], and $a^2=aK_S$ for Seiberg-Witten basic classes [@Moc Prop. 6.3.1].
A straight-forward calculation shows that this conjecture implies both Proposition \[fixedc1prop\] (without assuming Conjecture \[conj\]) and Proposition \[nonmingt\] (without assuming Conjecture \[numconj\] and without assuming $K_S^2 \geq \chi({\mathcal O}_S)-3$). In fact, this conjecture implies Conjecture \[conj\].
The first application of Conjecture \[generalsurfconj\] is the following blow-up formula.
\[blowupcor\] Assume Conjecture \[generalsurfconj\] holds. Let $\pi : \widetilde{S} \rightarrow S$ be the blow-up in a point of a smooth projective surface $S$ with $b_1(S) = 0$, $p_g(S)>0$. Suppose $H, c_1$ are chosen such that there are no rank 2 strictly Gieseker $H$-semistable sheaves with first Chern class $c_1$. Let $\widetilde{c}_1 = \pi^* c_1 - \epsilon E$ with $\epsilon=0,1$ and suppose $\widetilde{H}$ is a polarization on $\widetilde{S}$ such that there are no rank 2 strictly Gieseker $\widetilde{H}$-semistable sheaves on $\widetilde{S}$ with first Chern class $\widetilde{c}_1$. Then $$\begin{aligned}
{\mathsf Z}_{\widetilde{S}, \widetilde{c}_1}(x,y) = \begin{cases} \frac{\theta_3(x^4,y)}{\overline{\eta}(x^4)^2} \, {\mathsf Z}_{S, c_1}(x,y), & \epsilon=0 \\ \frac{\theta_2(x^4,y)}{\overline{\eta}(x^4)^2} \, {\mathsf Z}_{S, c_1}(x,y), & \epsilon=1. \end{cases}\end{aligned}$$
The Seiberg-Witten basic classes of $\widetilde{S}$ are $\pi^*a$, $\pi^*a+E$, where $a$ is a Seiberg-Witten basic class of $S$ and the corresponding Seiberg-Witten invariants are [@Mor Thm. 7.4.6] $$\begin{aligned}
{\mathrm{SW}}(\pi^*a) = {\mathrm{SW}}(\pi^*a+E) = {\mathrm{SW}}(a). \end{aligned}$$ Using $\chi({\mathcal O}_{\widetilde{S}}) = \chi({\mathcal O}_S)$, $K_{\widetilde{S}} = \pi^* K_{S}+E$, $K_{\widetilde{S}}^2 = K_S^2-1$, Conjecture \[generalsurfconj\] implies $$\psi_{\widetilde{S},\widetilde{c}_1}(x,y) = \frac{1}{2} \Bigg( \frac{\theta_3(x,y^{\frac{1}{2}})}{\overline{\eta}(x^4)^2} + (-1)^{\epsilon} \frac{\theta_3(-x,y^{\frac{1}{2}})}{\overline{\eta}(x^4)^2} \Bigg) \psi_{S,c_1}(x,y).$$ Specializing to $\epsilon=0,1$ and using the result follows.
For $y=1$ this blow-up formula appears in a physics context in [@VW Sect. 4.3].
Let $\pi : \widetilde{S} \rightarrow S$ be the blow-up in a point of a simply connected smooth projective surface $S$. Let $H, c_1$ be chosen on $S$ such that $c_1 H$ is odd. Let $\widetilde{H} = r \pi^* H - E$ for $r \gg 0$ and $\widetilde{c}_1 = \pi^* c_1 - \epsilon E$ for $\epsilon=0,1$ such that $\widetilde{c}_1 \widetilde{H}$ is odd as well. One can show that the moduli spaces $M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2)$ do not depend on the choice of such $\widetilde{H}$ [@LQ1]. In this setting Li-Qin [@LQ1; @LQ2] derived a blow-up formula for the virtual Hodge polynomials of these moduli spaces[^11] $$h(M_S^H(2,c_1,c_2), x_1, x_2), \, h(M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2), x_1, x_2).$$ Normalize the virtual Hodge polynomials as follows $$\begin{aligned}
\overline{h}(M_S^H(2,c_1,c_2), x_1, x_2) &= (x_1 x_2)^{ - {\operatorname{vd}}(M_S^H(2,c_1,c_2))/2} h(M_S^H(2,c_1,c_2), x_1, x_2), \\
\overline{h}(M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2), x_1, x_2) &= (x_1 x_2)^{ - {\operatorname{vd}}(M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2)/2} h(M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2), x_1, x_2).\end{aligned}$$ Then Li-Qin’s formula reads (see also [@Got2 Rem. 3.2]) $$\sum_{c_2} \overline{h}(M_{\widetilde{S}}^{\widetilde{H}}(2,\widetilde{c}_1,c_2), x_1, x_2) x^{4c_2 - \widetilde{c}_{1}^2} = \begin{cases} \frac{\theta_3(x^4,x_1x_2)}{\overline{\eta}(x^4)^2} \sum_{c_2} \overline{h}(M_S^H(2,c_1,c_2), x_1, x_2) x^{4c_2 - c_{1}^2}, & \epsilon=0 \\ \frac{\theta_2(x^4,x_1x_2)}{\overline{\eta}(x^4)^2} \sum_{c_2} \overline{h}(M_S^H(2,c_1,c_2), x_1, x_2) x^{4c_2 - c_{1}^2}, & \epsilon=1. \end{cases}$$ When specializing to $x_1 = y$ and $x_2=1$, this gives the ratios of Proposition \[blowupcor\]. Hence the blow-up formula for virtual $\chi_y$-genera (virtual in the sense of virtual classes) and the blow-up formula for $\chi_y$-genera (defined via virtual Hodge polynomials) coincide. In particular, the blow-up formula for virtual Euler characteristics and classical Euler characteristics involve the same ratio as well.
The second application of Conjecture \[generalsurfconj\] is to surfaces with canonical divisor with irreducible reduced connected components.
\[disconn\] Assume Conjecture \[generalsurfconj\] holds. Let $S$ be a smooth projective surface with $b_1(S) = 0$, $p_g(S)>0$, and suppose $C_1+ \cdots +C_m \in |K_S|$ is a sum of irreducible reduced curves. Suppose $H,c_1$ are chosen such that there are no rank 2 strictly Gieseker $H$-semistable sheaves with first Chern class $c_1$. Then $$\begin{aligned}
&{\mathsf Z}_{S,c_1}(x,y) = 2 \Bigg(\frac{\phi(x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)} \prod_{j=1}^{m} \Bigg\{ \Bigg( \frac{2\overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})} \Bigg)^{C_j^2} + (-1)^{c_1 C_j+h^0(N_{C_j/S})} \Bigg( \frac{2\overline{\eta}(x^4)^2}{\theta_3(-x,y^{\frac{1}{2}})} \Bigg)^{C_j^2} \Bigg\} \\
&+ 2(-i)^{c_1^2 - \chi({\mathcal O}_S)} \Bigg(\frac{\phi(-ix,y)}{2}\Bigg)^{\chi({\mathcal O}_S)} \prod_{j=1}^{m} \Bigg\{ \Bigg( \frac{2\overline{\eta}(x^4)^2}{\theta_3(-i x,y^{\frac{1}{2}})} \Bigg)^{C_j^2} + (-1)^{c_1 C_j+h^0(N_{C_j/S)}} \Bigg( \frac{2\overline{\eta}(x^4)^2}{\theta_3(i x,y^{\frac{1}{2}})} \Bigg)^{C_j^2} \Bigg\}\end{aligned}$$ where $i = \sqrt{-1}$ and $N_{C_j/S}$ denotes the normal bundle of $C_j \subset S$.
Specializing to $y=1$ and using $\theta_3(i^k x)=\theta_3(x^4)+i^k\theta_2(x^4)$ gives a more explicit version of [@VW Eqn. (5.45)].[^12] Before we prove this proposition, we need three lemmas about disconnected curves and their Seiberg-Witten invariants.
\[lemm1\] Let $C, D$ be irreducible reduced mutually disjoint curves on a smooth projective surface $S$ with $b_1(S)=0$. Then precisely one of the following is true:
- $C$ or $D$ is rigid, i.e. $|C|$ or $|D|$ is 0-dimensional.
- $|C| = |D| \cong {\mathbb P}^1$ is a pencil.
Suppose neither of $C,D$ is rigid. Then their linear systems sweep out $S$. Therefore $|C|$ contains an element $F$ which intersects $D$. Note that $F$ is connected because $C$ is irreducible reduced. The intersection cannot be only in dimension 0, because $CD=0$. Therefore $F = D + \sum_{i \in I} n_i F_i$, where $I$ is a finite index set, $n_i>0$, and $D, \{F_i\}_{i \in I}$ are all mutually distinct prime divisors. Suppose $|I|>0$. Then $$0 = CD = FD = D^2 + \sum_{i \in I} n_i F_i D > D^2.$$ Hence $D^2<0$, so $H^0(N_{D/S}) = 0$ contradicting the assumption that $D$ is not rigid. Therefore $I = \varnothing$ and $D \in |C|$. Furthermore, $|C|$ is base-point free and $C^{\prime} C^{\prime \prime} = C D = 0$ for all $C^{\prime}, C^{\prime \prime} \in |L|$ so $|L| \cong {\mathbb P}^1$.
Suppose $C_1, \ldots, C_m$ are irreducible reduced mutually disconnected curves on a smooth projective surface $S$ with $b_1(S) = 0$ and let $M:=\{1, \ldots, m\}$. Then for any $I = \{i_1, \ldots, i_k\} \subset M$, we define $$C_I := \sum_{i \in I} C_i.$$ For $I, J \subset M$ we write $I \sim J$ whenever $C_I \sim_{\mathrm{lin}} C_J$. This defines an equivalence relation. We denote the equivalence class corresponding to $I$ by $[I]$ and denote its number of elements by $|[I]|$.
\[lemm2\] For any $I \subset M$, we have $|[I]| = \binom{\dim |C_M|}{ \dim |C_I|}$.
Since $b_1(S) = 0$, any effective divisor $D \subset S$ satisfies $\dim |D| = h^0(N_{D/S})$. In particular we have $$\label{dimform}
\dim|C_I| = h^0(N_{C_I/S}) = \sum_{i \in I} h^0(N_{C_i/S}) = \sum_{i \in I} \dim |C_i|.$$ Suppose, possibly after relabeling, that $C_1, \ldots, C_a$ are the rigid curves (i.e. their linear systems are 0-dimensional). Then $C_{a+1}, \ldots, C_{m}$ are all linearly equivalent (Lemma \[lemm1\]). Moreover, if $m>a+1$, then their linear systems are pencils (Lemma \[lemm1\]). There are three cases:\
**Case 1:** $m=a$. Then all curves $C_i$ are rigid, so $I \sim J$ if and only if $I=J$ and the statement follows from .\
**Case 2:** $m=a+1$. Then only $C_m$ is not rigid and it is again easy to see that $I \sim J$ if and only if $I=J$. By , we have that $\dim|C_I| = \dim |C_m|$ if $m \in I$ and zero otherwise. The statement follows.\
**Case 3:** $m>a+1$. Then $C_1, \ldots, C_a$ are rigid and $|C_{a+1}| = \cdots = |C_m| \cong {\mathbb P}^1$. Let $A:=\{1, \ldots, a\}$ and $B:=\{a+1, \ldots, m\}$. Then $I \sim J$ if and only if $A \cap I = A \cap J$ and $|B \cap I| = |B \cap J|$. Therefore $|[I]| = \binom{|B|}{|B \cap I|}$. The result follows from equation as follows $$\dim|C_I| = \sum_{i \in A \cap I} \dim|C_i| + \sum_{i \in B \cap I} \dim|C_i| = |B \cap I|. \qedhere$$
\[lemm3\] Let $S$ be a smooth projective surface with $b_1(S) = 0$ and suppose $C_1+ \cdots +C_m \in |K_S|$, where $C_1, \ldots, C_m$ are mutually disjoint irreducible reduced curves. Then the Seiberg-Witten basic classes of $S$ are $\{C_I\}_{I \subset M}$ and $${\mathrm{SW}}(C_I) = |[I]| \prod_{i \in I} (-1)^{h^0(N_{C_i/S})}.$$
The proof combines Lemma \[lemm2\] and the proof of [@Moc Prop. 6.3.1]. We first note that all Seiberg-Witten basic classes must be of the form $\{C_I\}_{I \subset M}$ (this can be seen most easily from the cosection localization of Chang-Kiem [@CK Lem. 3.2]). Let $\varnothing \neq I \subset M$. Then Mochizuki shows that $${\mathrm{SW}}(C_I) = c_{\mathrm{top}}({\operatorname{Ob}}) \cap |C_I|,$$ where ${\operatorname{Ob}}$ is a rank $h^1(N_{C_I/S})$ vector bundle with total Chern class $$(1+h)^{h^1(N_{C_I/S}) - p_g(S)}$$ where $h$ denotes the hyperplane class on $|C_I|$. Hence $${\mathrm{SW}}(C_I) = \binom{h^1(N_{C_I/S})-p_g(S)}{h^1(N_{C_I/S})} = (-1)^{h^1(N_{C_I/S})} \binom{p_g(S)-1}{h^1(N_{C_I/S})}.$$ By Serre duality and adjunction $K_{C_i} = (C_M+C_i)|_{C_i} = 2C_i|_{C_i}$ $$h^1(N_{C_I/S}) = \sum_{i \in I} h^1({\mathcal O}_{C_i}(C_i)) = \sum_{i \in I} h^0({\mathcal O}_{C_i}(C_i)) = h^0(N_{C_I/S}).$$ Therefore Lemma \[lemm2\] implies $$\binom{p_g(S)-1}{h^1(N_{C_I/S})} = \binom{\dim|C_M|}{\dim|C_I|} = |[I]|. \qedhere$$
Combining Conjecture \[generalsurfconj\] and Lemma \[lemm3\] gives $$\begin{aligned}
\psi_{S,c_1}(x,y) &= 4 \Bigg(\frac{\phi(x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})}\Bigg)^{K_{S}^2} \sum_{[I]} |[I]| (-1)^{h^0(N_{C_I/S})+c_1C_I} \Bigg(\frac{\theta_3(x,y^{\frac{1}{2}})}{\theta_3(-x,y^{\frac{1}{2}})}\Bigg)^{C_I^2} \\
&= 4 \Bigg(\frac{\phi(x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})}\Bigg)^{K_{S}^2} \sum_{I} (-1)^{h^0(N_{C_I/S})+c_1C_I} \Bigg(\frac{\theta_3(x,y^{\frac{1}{2}})}{\theta_3(-x,y^{\frac{1}{2}})}\Bigg)^{C_I^2} \\
&= 4 \Bigg(\frac{\phi(x,y)}{2}\Bigg)^{\chi({\mathcal O}_S)}\Bigg(\frac{2 \overline{\eta}(x^4)^2}{\theta_3(x,y^{\frac{1}{2}})}\Bigg)^{K_{S}^2} \prod_{i=1}^{m}\Bigg(1+(-1)^{h^0(N_{C_i/S})+C_i c_1} \Bigg(\frac{\theta_3(x,y^{\frac{1}{2}})}{\theta_3(-x,y^{\frac{1}{2}})}\Bigg)^{C_i^2} \Bigg).\end{aligned}$$ The formula for ${\mathsf Z}_{S,c_1}(x,y)$ follows from after re-organizing the terms.
Verification of the conjectures in examples {#evidencesec}
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In this section we check Conjectures \[conj\], \[numconj\], and \[generalsurfconj\] in many cases. We recall that we calculated $A_1(s,y,q), \ldots, A_7(s,y,q)$ up to order $q^7$, where we calculated the coefficient of $q^i$ up to order $s^{29-3i}$. We computed $A_1(s,1,q), \ldots, A_7(s,1,q)$ up to order $q^{30}$ and any order in $s$ (see Section \[toricsec\]). The latter are listed up to order $q^4$ in Appendix \[app1\].
K3 surfaces {#K3sec}
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Let $S$ be a $K3$ surface. The canonical class is trivial and $b_1(S) = 0$ so we are in the setting of Conjecture \[generalsurfconj\], which states $$\label{K3gen}
{\mathsf Z}_{S,c_1}(x,y)=\frac{1}{2}\big(\phi(x,y)^2-i^{c_1^2} \phi(ix,y)^2\big).$$ This can be restated as saying that $\overline\chi_{-y}^{\mathrm{vir}}(M^H_S(2,c_1,c_2))$ is the coefficient of $4c_2-c_1^2-6$ of $$\phi(x,y)^2=
\prod_{n=1}^{\infty} \frac{1}{(1-x^{2n})^{20}(1-x^{2n} y)^2(1-x^{2n} y^{-1})^2},$$ when there are no rank 2 strictly $H$-semistable sheaves with Chern classes $c_1,c_2$. This looks very similar to Conjecture \[conj\], which does not apply, and which would be off by a factor 2.
In the absence of strictly Gieseker $H$-semistable sheaves, the moduli space $M_{S}^{H}(2,c_1,c_2)$ is smooth of expected dimension. If in addition $c_1$ is primitive, big, and nef, then in [@GH] it is shown that $$\chi_{-y}(M^S_H(2,c_1,c_2))=\chi_{-y}(S^{[2c_2 - c_1^2/2 - 3]}).$$ More generally, in [@Huy Cor. 4.8], when $c_1$ is primitive, and in [@Yos4 Thm. 0.1] in general, it is show that in the absence of strictly Gieseker $H$-semistable sheaves $M_{S}^{H}(r,c_1,c_2)$ is deformation equivalent to the Hilbert scheme $S^{[{\operatorname{vd}}/2]}$. In particular they have the same Hodge numbers. In turn, $\chi_y$-genera of Hilbert schemes of points on $K3$ surfaces were computed by the first named author and W. Soergel in [@GS]. Combining this formula with the result above implies that holds for $K3$ surfaces.
The only Seiberg-Witten basic class of $S$ is $0$ with corresponding Seiberg-Witten invariant 1. Suppose $c_1$ is a choice of first Chern class such that $c_1^2 = c$, then $c \in 2 {\mathbb Z}$. Then we see that also follows from Conjecture \[numconj\]. We put $(\gamma_1,\gamma_2)=(0,0)$ and $(\beta_1,\beta_2,\beta_3,\beta_4)=(c,0,0,2)$. Then the two summands in ${\mathrm{Coeff}}_{s^0x^n}[\cdot]$ in Conjecture \[numconj\] are equal, and each equals the contribution of the Seiberg-Witten basic class $0=K_S$. Therefore, if $c_2$ satisfies the assumptions of Conjecture \[numconj\], it gives that $\overline\chi_{-y}^{\mathrm{vir}}(M^H_S(2,c_1,c_2))$ is the coefficient of $x^{4c_2-c_1^2-6}$ of $$\frac{1}{2}\psi_{0,0,0,2}(x,y)=\phi(x,y)^2.$$ The same argument as in the proof of Proposition \[numconjimpliesconj\] shows that follows from Conjecture \[numconj\] and the strong form of Mochizuki’s formula.
All the same, we want to calculate terms of directly by applying Corollary \[univcor\] and our explicit knowledge of the universal functions $A_i$ as described in Section \[toricsec\]. We use the conjectured strong form of Mochizuki’s formula (Remark \[strongform\]), so our calculations can also be viewed as an additional test of Remark \[strongform\]. The easiest way to satisfy all assumptions of Corollary \[univcor\] is by choosing $c_1$ and $H$ such that $c_1 H > 0$ is odd (though this is unnecessarily strong). According to Theorem \[mocthm\] (i), we can calculate modulo $$\mod x^{c + 10}.$$ In addition we have a bound given by the accuracy to which we calculated the universal functions $A_i(s,y,q)$ and $A_i(s,1,q)$. For $A_i(s,1,q)$ this bound is $q^{30}$, which for ${\mathsf Z}_{S,c_1}(x,1)$ means we can calculate modulo $$\mod x^{-c+118}.$$ For $y=1$, i.e. the case of virtual Euler characteristics, we tested Conjecture \[generalsurfconj\] for all even $c \in \{-6, \ldots, 116\}$ up to the above accuracies. We summarize this in a table, whose format we use in subsequent sections too:
For virtual $\chi_y$-genus, we tested Conjecture \[generalsurfconj\] for:
Elliptic surfaces {#ellipticevidence}
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Let $S \rightarrow {\mathbb P}^1$ be a non-trivial elliptic surface with section, $12n>0$ rational nodal fibres, and no other singular fibres, i.e. an elliptic surface of type $E(n)$. The canonical class is given by $K_S = (n-2) F$, where $F$ denotes the class of the fibre. Note that $\chi({\mathcal O}_S) = n$. Moreover, choose a section $B \subset S$, then its class satisfies $B^2 = -n$.
We assume $n \geq 2$, then $E(n)$ has a smooth canonical divisor which has $m=n-2$ connected components $F_j$; each a smooth elliptic fibre of $S$. The surface $E(3)$ can be used to test Conjecture \[conj\] and the surfaces $E(n)$ with $n \geq 2$ can be used to test Conjecture \[generalsurfconj\]. Note that $E(2)$ is $K3$, which was discussed in Section \[K3sec\], so we take $n \geq 3$.
Let $c_1 = \epsilon B + d F$, for $\epsilon, d \in {\mathbb Z}$ and $B$ the class of the section. Since $F_j^2=0$ and $c_1 F_j=\epsilon$, Conjecture \[generalsurfconj\] gives $$\label{ellgen}
{\mathsf Z}_{S,c_1}(x,y)=\begin{cases}\frac{1}{2}\big(\phi(x,y)^n+(-i)^{c_1^2 - n}\phi(-ix,y)^n\big), & \epsilon \hbox{ odd} \\0, & \epsilon \hbox{ even.}\end{cases}$$ In the case $n=3$, this can be restated as saying that $\overline\chi_{-y}^{\mathrm{vir}}(M^H_S(2,c_1,c_2))$ is the coefficient of $4c_2-c_1^2-9$ of $
\phi(x,y)^3$, which is the statement of Conjecture \[conj\].
In [@Yos3], Yoshioka fixes $c_1, c_2$ such that $c_1 F$ is odd and an ample divisor $H = B + \beta F$ with $\beta \gg c_2$. By choosing $\beta$ of the right parity $c_1H$ is odd, so there are no rank 2 strictly $\mu$-semistable sheaves with Chern classes $c_1,c_2$. Then $M:=M_{S}^{H}(2,c_1,c_2)$ is smooth irreducible of expected dimension and independent of $H$. In this setting, the Betti numbers and Euler characteristics of $M:=M_{S}^{H}(2,c_1,c_2)$ were calculated in [@Yos3] and are indeed given by the specialization $y=1$ of (\[ellgen\]).
Next we want to calculate terms of directly by applying Corollary \[univcor\] and our explicit knowledge of the universal functions $A_i$ as described in Section \[toricsec\]. There are numerous choices for the polarization $H$ for which the conditions of Corollary \[univcor\] are satisfied. Specifically, let $H = \alpha B + \beta F$ with $\alpha>0$ and $\beta > \alpha n$ be a polarization. When $$\begin{aligned}
c_1 H &= (\beta - \alpha n) \epsilon + \alpha d > 2\alpha (n-2) = 2 K_S H, \\
c_1 H &= (\beta - \alpha n) \epsilon + \alpha d \qquad \mathrm{odd},\end{aligned}$$ all conditions are satisfied. For fixed $\epsilon > 0$ and $d \in {\mathbb Z}$ not both even, there are many solutions $\alpha>0$ and $\beta > \alpha n$ to these equations. By [@FM], the Seiberg-Witten basic classes are $0, F \ldots, (n-2)F$ and $${\mathrm{SW}}(pF) = (-1)^p \binom{n-2}{p}.$$ For $y=1$, we verified Conjecture \[generalsurfconj\] in the following cases:
For virtual $\chi_y$-genus, we tested Conjecture \[generalsurfconj\] for:
Blow-ups {#blowupevidence}
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Let $\pi : \widetilde{S} \rightarrow S$ be the blow-up of a $K3$ surface $S$ in a point and denote the exceptional divisor by $E$. Then $K_{\widetilde{S}} = E$ is a smooth canonical divisor. We want to gather evidence for Conjecture \[conj\] by applying Corollary \[univcor\] and our explicit knowledge of the universal functions $A_i$ as described in Section \[toricsec\]. Let $\widetilde{c}_1 = \pi^* c_1 + \epsilon E$ and $c_1^2=c \in 2 {\mathbb Z}$. The easiest way to satisfy the conditions of Corollary \[univcor\] is to take a polarization $H$ on $S$ such that $c_1 H > 0$ is odd and taking $\widetilde{H} = r H - E$ with $r \gg 0$ and $r + \epsilon$ odd, but more general choices are possible. The blow-up formula for Seiberg-Witten invariants implies that $\widetilde{S}$ has Seiberg-Witten basic classes $0,E$ and ${\mathrm{SW}}(0)={\mathrm{SW}}(E) = 1$ [@Mor Thm. 7.4.6]. For $y=1$, we verified Conjecture \[conj\] in the following cases:
For virtual $\chi_y$-genus, we tested Conjecture \[conj\] for:
Next we consider the blow-up of a $K3$ surface $S$ in two distinct points $$\pi : \widetilde{S} \rightarrow S$$ and we denote the exceptional divisors by $E_1, E_2$. We gather evidence for Conjecture \[generalsurfconj\] by applying Corollary \[univcor\] and our explicit knowledge of the universal functions $A_i$ as described in Section \[toricsec\]. By the blow-up formula for Seiberg-Witten invariants, the Seiberg-Witten basic classes are $$0, \, E_1, \, E_2, \, E_1+E_2$$ and the invariant is 1 in each case. We consider classes $$\widetilde{c}_1 = \pi^* c_1 + \epsilon_1 E_1 + \epsilon_2 E_2.$$ As before the easiest way to satisfy the conditions of Corollary \[univcor\] is to take a polarization $H$ on $S$ such that $c_1 H > 0$ is odd and taking $\widetilde{H} = r H - E_1 - E_2$ with $r \gg 0$ and parity such that $r + \epsilon_1 + \epsilon_2$ is odd, though other choices are possible. For $y=1$, we verified Conjecture \[generalsurfconj\] in the following cases:
For virtual $\chi_y$-genus, we tested Conjecture \[generalsurfconj\] for:
Double covers
-------------
Next, we consider double covers $$\pi : S_d \rightarrow {\mathbb P}^2$$ branched over a smooth curve $C$ of degree $2d$. Then $K_{S_d} = (d-3) L$, where $L$ is the pull-back of the class of a line on ${\mathbb P}^2$. These surfaces satisfy $b_1(S_d) = 0$. It is easy to calculate $$\begin{aligned}
K_{S_d}^2 &= 2(d-3)^2, \\
\chi({\mathcal O}_{S_d}) &=\frac{1}{2}d(d-3)+2.\end{aligned}$$ Since $\alpha L$ is base-point free for all $\alpha > 0$, the canonical linear system $|K_{S_d}|$ contains smooth connected canonical divisors when $d \geq 4$, which we assume from now on. Let $c_1 = \epsilon L$. As before, we want to gather evidence for Conjecture \[conj\] by applying Corollary \[univcor\] and our explicit knowledge of the universal functions $A_i$ as described in Section \[toricsec\]. The Seiberg-Witten basic classes are $0, K_S \neq 0$ with Seiberg-Witten invariants $$1, \, (-1)^{\chi({\mathcal O}_{S_d})}.$$ We first take $H=L$ as the polarization on $S$. Then conditions (ii), (iii) of Corollary \[univcor\] require $$\begin{aligned}
c_1 H = 2 \epsilon > 4(d-3) = 2K_S H,\end{aligned}$$ i.e. $\epsilon > 2(d-3)$. In the case $\epsilon=2k$ is even, we can ensure the absence of rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1, c_2$ by choosing $c_2$ such that $$\frac{1}{2} c_1(c_1 - K_S) - c_2 = \epsilon(\epsilon - (d-3)) - c_2$$ is odd [@HL Rem. 4.6.8].
Now assume that $\epsilon=2k+1$ is odd. If $L$ generates the Picard group of $S_d$, then there are no rank 2 strictly $\mu$-semistable sheaves with Chern classes $\epsilon L$ and $c_2$. In general the Picard group of $S_d$ can have more generators, but $L$ is still ample and primitive. In this case we take the polarization $H$ general and sufficiently close to $L$ (i.e. of the form $nL+B$ for $n$ sufficiently large), so that conditions (ii) and (iii) of Corollary \[univcor\] still hold when $\epsilon>2(d-3)$, and so that there are no rank 2 strictly $\mu$-semistable sheaves with Chern classes $\epsilon L$ and $c_2$.
Recall that conjecturally conditions (ii), (iii) of Corollary \[univcor\] are not necessary (see Remark \[strongform\]). Therefore we will also test Conjecture \[conj\] for $\epsilon, d$ which do not satisfy $\epsilon > 2(d-3)$.
Note that $\mu$-stability is invariant under tensorizing by a line bundle and that Gieseker stability with respect to $H$ is invariant under tensorizing by a power of $H$. Therefore we know that $M^H_{S_d}(2,L,c_2)$ together with its virtual structure is isomorphic to $M^H_{S_d}(2,(2k+1)L,c_2+2(k^2+k))$, and when $c_2$ is odd, the same holds for $M^L_{S_d}(2,0,c_2)$ and $M^L_{S_d}(2,2kL,c_2+2k^2)$.
For $y=1$, we verified Conjecture \[conj\] in the following cases:
For virtual $\chi_y$-genus, we tested Conjecture \[conj\] for:
In order to give an idea of the complexity of the numbers involved, we give some examples. First we compute the virtual Euler numbers. We only show cases where $\epsilon > 2(d-3)$, so that we do not need to use the strong form of Mochizuki’s formula, and thus we have unconditionally proved these formulas.
The formula of Corollary \[cor\] is proved for $(d=4,\epsilon=5)$, modulo $x^{50}$, for $(d=4,\epsilon =6)$ modulo $x^{40}$, for$(d=5,\epsilon=5)$ modulo $x^{45}$ for $(d=5,\epsilon=6)$ modulo $x^{31}$. When $\epsilon$ is even, let ${\mathsf Z}^{\mathrm{odd}}_{S_d,H,0}(x,1)$ be the part of the generating function corresponding to $c_2$ odd, so the moduli spaces only consist of Gieseker stable sheaves. Using the invariance under tensoring by $H$ or $L$ respectively we have $$\begin{aligned}
{\mathsf Z}_{S_4,H,L}(x,1)=\,&120x^2 + 80800x^6 + 18764544x^{10} + 2352907648x^{14} + 192977504816x^{18}\\
&+ 11510039013632x^{22} + 533695300891136x^{26} +O(x^{30}),\\
{\mathsf Z}^{\mathrm{odd}}_{S_4,L,0}(x,1)=\,&2 + 1354148x^8 + 22293779698x^{16} + 80622742217604x^{24} \\
&+ 115687108304998636x^{32} + O(x^{40}),\\
{\mathsf Z}_{S_5,H,L}(x,1)=\,&-256x - 2622464x^5 - 4399076864x^9 - 3005594355712x^{13}\\
&- 1137273257362688x^{17} - 278765441520823296x^{21} +O(x^{25}),\\
{\mathsf Z}^{\mathrm{odd}}_{S_5,L,0}(x,1)= \,&- 123928576x^7 - 62207974965248x^{15}- 3825621677917863936x^{23} + O(x^{31}).\end{aligned}$$ Now we give some examples of the virtual $\chi_{y}$-genus. In this case we need to use the strong form of Mochizuki’s formula. For $d=4$, $c_1=0,L$ we get the virtual refinements $$\begin{aligned}
&{\mathsf Z}_{S_4,H,L}(xy^{\frac{1}{2}},y)=(14y^2 + 92y + 14)x^2 + (154y^6 + 2540y^5 + 16398y^4 + 42616y^3 + \ldots)x^6 \\
&\quad+ (756y^{10} + 17360y^9 + 185020y^8 + 1145700y^7 + 4174352y^6 + 7718168y^5 +\ldots)x^{10} + O(x^{14}),\\
&{\mathsf Z}^{\mathrm{odd}}_{S_4,L,0}(xy^{\frac{1}{2}},y)=2 + (364y^8 + 7300y^7 + 64090y^6 + 293556y^5 + 623528y^4 + \ldots)x^8 + O(x^{16}),\end{aligned}$$ where the missing terms are determined by the symmetry of the virtual $\chi_y$-genus.
Next let ${\mathbb F}_a = {\mathbb P}({\mathcal O}_{{\mathbb P}^1} \oplus {\mathcal O}_{{\mathbb P}^1}(a))$ denote the $a$th Hirzebruch surface, where $a \geq 0$. Suppose $B$ is the section corresponding to the surjection ${\mathcal O}_{{\mathbb P}^1} \oplus {\mathcal O}_{{\mathbb P}^1}(a) \twoheadrightarrow {\mathcal O}_{{\mathbb P}^1}$ onto the first factor. Then $B^2 = -a$. We denote the class of the fibre of ${\mathbb F}_a \rightarrow {\mathbb P}^1$ by $F$. Let $d_1>0$ and $d_2>ad_1$. A rich source of examples are double covers $$\pi : S_{a,d_1,d_2} \rightarrow {\mathbb F}_a$$ branched over a smooth connected curve in $|{\mathcal O}(2d_1 B + 2d_2 F)|$. Let $\widetilde{B} := \pi^* B$ and $\widetilde{F} = \pi^* F$. We choose $d_1 > 2$ and $d_2 > a(d_1-1)+2$, so the canonical divisor $$K_{S_{a,d_1,d_2}}=(d_1-2)\widetilde{B}+ (d_2-(a + 2))\widetilde{F}$$ is the pull-back of a very ample divisor and its linear system contains smooth connected curves. These surfaces satisfy $b_1(S_{a,d_1,d_2}) = 0$. We have $$\begin{aligned}
K_{S_{a,d_1,d_2}}^2 &=4(d_1-2)\Big(d_2-\frac{d_1a}{2}-2\Big), \\
\chi({\mathcal O}_{S_{a,d_1,d_2}}) &=(d_1-1)\Big(d_2-\frac{d_1a}{2}-1\Big)+1.\end{aligned}$$ We first note that it is enough to determine ${\mathsf Z}_{S_{a,d_1,d_2},\epsilon_1\widetilde{B}+\epsilon_2\widetilde{F}}(x,y)$ for $a=0,1$.
Assume the strong form of Mochizuki’s formula holds (Remark \[strongform\]). Let $a \geq 0$, $d_1 > 2$, $d_2 > (a+1)(d_1-1) + 1$, and $d_2 > (a+1)d_1$. Let $H$ be a polarization on $S_{a,d_1,d_2}$ such that there exist no rank 2 strictly Gieseker $H$-semistable sheaves with Chern classes $c_1 := \epsilon_1 \widetilde{B} + \epsilon_2 \widetilde{F}$ and $c_2$. Let $H'$ be a polarization on $S_{a+2,d_1,d_1+d_2}$ such that there exist no rank 2 strictly Gieseker $H'$-semistable sheaves with Chern classes $c_1' := \epsilon_1 \widetilde{B} + (\epsilon_1+\epsilon_2) \widetilde{F}$ and $c_2$. Then $$\chi_{-y}^{{\mathrm{vir}}}(M_{S_{a,d_1,d_2}}^{H}(2,\epsilon_1 \widetilde{B} + \epsilon_2 \widetilde{F},c_2)) = \chi_{-y}^{{\mathrm{vir}}}(M_{S_{a+2,d_1,d_1+d_2}}^{H'}(2,\epsilon_1 \widetilde{B} + (\epsilon_1+\epsilon_2) \widetilde{F},c_2)),$$ for all $$\begin{aligned}
c_2 &< 2\chi({\mathcal O}_{S_{a,d_1,d_2}}) + \frac{1}{2} c_1(c_1-K_{S_{a,d_1,d_2}}) \\
&= 2\chi({\mathcal O}_{S_{a+2,d_1,d_1+d_2}}) + \frac{1}{2} c_1^{\prime}(c_1^\prime - K_{S_{a+2,d_1,d_1+d_2}}).\end{aligned}$$
Write $S:=S_{a,d_1,d_2}$ and $S':=S_{a+2,d_1,d_1+d_2}$. Fix $c_1:=\epsilon_1\widetilde{B}+\epsilon_2\widetilde{F}$ on $S$ and $c_1':=\epsilon_1\widetilde{B}+(\epsilon_1+\epsilon_2)\widetilde{F}$ on $S'$. The map $$\phi : H^2(S,{\mathbb Z})\to H^2(S',{\mathbb Z}), \ \widetilde{B}\mapsto \widetilde{B}+\widetilde{F}, \ \widetilde{F}\mapsto \widetilde{F}$$ is an isomorphism between the subgroups generated by $\widetilde{B}, \widetilde{F}$ on $S$, $S'$ and it preserves the intersection forms on these subgroups. It sends $K_{S}$ to $K_{S'}$, and thus it sends the Seiberg-Witten basic classes $0$, $K_ {S}$ of $S$ to the corresponding Seiberg-Witten basic classes of $S'$. We also see that $c_1$ is send to $c_1'$. Moreover $\chi({\mathcal O}_{S})=\chi({\mathcal O}_{S'})$. We apply the strong form of Mochizuki’s formula to $S$ with $H,c_1,c_2$ and $S'$ with $H',c_1',c_2$. Since $$(a_1^2, a_1 c_1, c_1^2, a_1 K_S, c_1 K_S, K_S^2,\chi({\mathcal O}_S)) = (a_1'^2, a_1' c_1', c_1^{ \prime 2}, a_1' K_{S'}, c_1' K_{S'}, K_{S'}^2,\chi({\mathcal O}_{S'}))$$ for $(a_1,a_1') = (0,0)$ and $(a_1,a_1') = (K_S,K_{S'})$, the assertion follows from Corollary \[univcor\] and Remark \[strongform\].
For $y=1$, we checked Conjecture \[conj\] in the following cases:
where $$\begin{aligned}
&M := -\frac{5}{2} a d_1^2+\frac{5}{2} a d_1+ 5 d_1 d_2-5d_1-5d_2 +10+4 a \epsilon_1 d_1 - 4 a \epsilon_1-4 \epsilon_1 d_2+8 \epsilon_1-4 \epsilon_2 d_1\\
&\qquad\ \, +8 \epsilon_2-2 a \epsilon_1^2+4 \epsilon_1 \epsilon_2, \\
&N := 2a \epsilon_1^2 - 4 \epsilon_1 \epsilon_2 -3 (d_1-1)(d_2-\frac{d_1 a}{2} - 1) - 3 \\
&\qquad \ \, + 8\min\Big\{0,a(d_1-2)(\epsilon_1 - (d_1-2)) + (d_1-2)(\epsilon_2-(d_2-(a+2))) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\ \ \ \, + (d_2 - (a+2))(\epsilon_1 - (d_1-2)) \Big\}.\end{aligned}$$
For virtual $\chi_{y}$-genus, we checked Conjecture \[conj\] in the following cases:
Some examples of these tables are the following: $$\begin{aligned}
{\mathsf Z}^{\mathrm{odd}}_{S_{0,3,3},0}(x,1)&=- 105472x^5 - 17699188736x^{13} - 311478544324608x^{21} +O(x^{29}),\\
{\mathsf Z}_{S_{0,3,4}, \widetilde{B}}(x,1)&= -36864 x^{3}-123928576 x^{7}-125523673088 x^{11} +O( x^{15} ),\\
{\mathsf Z}_{S_{1,3,5}, \widetilde{F}}(x,1) &= 1248 x^{2}+3740160 x^{6}+3011270208 x^{10}+1143966397440 x^{14} +O(x^{18}) ,\\
{\mathsf Z}_{S_{1,3,5}, \widetilde{F}+\widetilde B}(x,1) &=8 + 85920x^4 + 119219712x^8 + 63288183168x^{12}+O(x^{16}).\end{aligned}$$ Here $c_1,c_2$ always have the property that there are no rank 2 strictly Gieseker $H$-semistable sheaves with these Chern classes. The virtual refinement of ${\mathsf Z}_{S_{1,3,5}, \widetilde{F}}(x,1)$ is given by $$\begin{aligned}
{\mathsf Z}_{S_{1,3,5}, \widetilde{F}}(xy^{\frac{1}{2}},y)=\,&(216y^2 + 816y + 216)x^2 + (13720y^6 + 184128y^5\\
&+ 876264y^4 + 1591936y^3 + 876264y^2 + 184128y + 13720)x^6+O(x^{10}).\end{aligned}$$
Divisors in products of projective spaces
-----------------------------------------
Let $S_d$ be a smooth surface of degree $d$ in ${\mathbb P}^3$, then $$K_{S_d} = (d-4)L,$$ where $L$ is the hyperplane section. Moreover $S_d$ is simply connected by the Lefschetz hyperplane theorem. We take $d \geq 5$, so $S_d$ is of general type. One readily calculates $$\begin{aligned}
K_{S_d}^2 &= d(d-4)^2, \\
\chi({\mathcal O}_{S_d}) &= \frac{1}{6} d (d^2+11) - d^2.\end{aligned}$$ The hyperplane section $L$ on $S_d$ is very ample so $|K_S|$ contains smooth connected canonical divisors. We test Conjecture \[conj\] using Corollary \[univcor\].
We take as polarization $H = L$ and $c_1 = \epsilon L$. If $S_d$ is very general, i.e. in the complement of countably many closed subvarieties in the projective space of hypersurfaces of degree $d$, then ${\operatorname{Pic}}(S_d)={\mathbb Z}L$ by the Noether-Lefschetz Theorem. For $$\begin{aligned}
c_1 H &= d \epsilon> 2 d (d-4) = 2 K_{S_d} H, \\
c_1 H &= d \epsilon \quad \hbox{odd, or } S_d \hbox{ very general and } \epsilon \ \mathrm{ odd}\end{aligned}$$ there are no rank 2 strictly $\mu$-semistable sheaves with first Chern class $c_1$ and conditions (ii), (iii) of Corollary \[univcor\] are satisfied.
We also perform computations when these conditions are not satisfied (recall Remark \[strongform\]). We verified Conjecture \[conj\] in the following cases:
For virtual $\chi_y$-genus, we tested Conjecture \[conj\] for:
We list a few examples. First we deal with the specialization to the virtual Euler number and in this case we only consider the case that $\epsilon>2(d-4)$, so that we do not use the strong from of Mochizuki’s formula and the formulas are proved unconditionally. For $d=5$ and $\epsilon=3,4$ we obtain $$\begin{aligned}
{\mathsf Z}_{S_5,L,L}(x,1)=\,&8 + 52720x^4 + 48754480x^8 + 17856390560x^{12} + 3626761297400x^{16} \\
&+ 482220775619120x^{20}+ 46283858505022160x^{24} +O(x^{28}),\\
{\mathsf Z}^{\mathrm{odd}}_{S_5,L,0}(x,1)=\,&- 316096x^5 - 70399202880x^{13} - 1550539821466560x^{21} + O(x^{29}).\end{aligned}$$ For $d=5$, $\epsilon=1$ and using the strong form of Mochizuki’s formula, we obtain the refinement $$\begin{aligned}
&{\mathsf Z}_{S_5,L,L}(xy^{\frac{1}{2}},y)=8 + (1280y^4 + 11440y^3+ 27280y^2+ 11440y+1280)x^4 \\
&\quad +(25520y^8+ 448640y^7+ 3228960y^6 + 11405320y^5+18537600y^4+\ldots)x^8 + O(x^{12}).\end{aligned}$$
Next we move on to smooth surfaces $S_{d_1,d_2}$ of bidegree $(d_1,d_2)$ in ${\mathbb P}^2 \times {\mathbb P}^1$. Again $S_{d_1,d_2}$ is simply connected and one readily calculates $$\begin{aligned}
K_{S_{d_1,d_2}}^2 &= (d_1-3)^2 d_2 + 2d_1(d_1-3)(d_2-2), \\
\chi({\mathcal O}_{S_{d_1,d_2}}) &= \frac{1}{2}d_1(d_1d_2-d_1-3d_2+3)+d_2.\end{aligned}$$ We take $d_1 \geq 4$ and $d_2 \geq 3$ so $K_{S_{d_1,d_2}}$ is very ample and its linear system contains a smooth connected canonical divisor.
Let $L_i$ be the restriction of $\pi_i^{*} L$ to $S$, where $\pi_1 : {\mathbb P}^2 \times {\mathbb P}^1 \rightarrow {\mathbb P}^2$ and $\pi_2 : {\mathbb P}^2 \times {\mathbb P}^1 \rightarrow {\mathbb P}^1$ denote projections and $L$ is the hyperplane class on each factor. We tested Conjecture \[conj\] using Corollary \[univ\] and the strong version of Mochizuki’s formula (Remark \[strongform\]) in the following cases:
where $$\begin{aligned}
M :=\, &\frac{5}{2}d_1(d_1d_2-3d_2-d_1+3)+5d_2 + \epsilon_1^2 d_2 + 2 \epsilon_1 \epsilon_2 d_1 - 2 \epsilon_1 (d_1-3) d_2 - 2 \epsilon_2 (d_1-3) d_1 \\
&- 2\epsilon_1 (d_2-2) d_1, \\
N :=\, &- \epsilon_1^2 d_2 - 2 \epsilon_1 \epsilon_2 d_1- \frac{3}{2}d_1(d_1d_2-3d_2-d_1+3)-3d_2 \\
&+ 4\min\Big\{0,(d_1-3)(\epsilon_1-(d_1-3))d_2 + (d_1-3)(\epsilon_2 - (d_2-2))d_1 \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \!\! + (d_2-2)(\epsilon_1 - (d_1-3))d_1\Big\}.\end{aligned}$$
For general $y$, we checked Conjecture \[conj\] in the following cases:
Examples contained in these tables with $(d_1,d_2) = (4,3)$ are $$\begin{aligned}
{\mathsf Z}_{S_{4,3},3 L_1 }(x,1)&=128 + 5350656x^4 + 18196176128x^8 + 20761214894592x^{12} +O(x^{16}),\\
{\mathsf Z}_{S_{4,3},3 L_2 }^{\mathrm{odd}}(x,1) &= - 2704756736 x^{7} +O(x^{15}),\end{aligned}$$ where we choose a polarization $H$ such that $L_1 H$ is odd in the first case. A virtual refinement of the second formula is given by $$\begin{aligned}
{\mathsf Z}_{S_{4,3},3 L_2 }^{\mathrm{odd}}(xy^{\frac{1}{2}},y) =\, &- \big(6323328y^{7} +81371136y^{6} +394518784y^{5} + 870165120(y^4+y^3) \\
&+394518784y^{2}+81371136y+6323328\big)x^7 + O(x^{15}).\end{aligned}$$
Finally we consider smooth surfaces $S_{d_1,d_2,d_3}$ of tridegree $(d_1,d_2,d_3)$ in ${\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1$. Then $S_{d_1,d_2,d_3}$ is simply connected and one can compute $$\begin{aligned}
K_{S_{d_1,d_2,d_3}}^2 &= 2 \prod_{{\scriptsize{\begin{array}{c} (i,j,k) \\ i,j,k \, \mathrm{distinct} \end{array}}}} d_i (d_j - 2)(d_k-2) , \\
\chi({\mathcal O}_{S_{d_1,d_2,d_3}}) &= \frac{1}{2} d_1d_2d_3-\frac{1}{6}d_1d_2-\frac{1}{6}d_1d_3-\frac{1}{6} d_2 d_3-\frac{1}{3}d_1-\frac{1}{3}d_2-\frac{1}{3}d_3+2.\end{aligned}$$ Taking $d_1,d_2,d_3 \geq 3$, the canonical linear system is very ample and contains smooth connected curves. Denote by $L_i$ the restriction of $\pi_i^{*} L$ to $S$, where $\pi_i : {\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1 \rightarrow {\mathbb P}^1$ are the projections and $L$ is the class of a point on ${\mathbb P}^1$. We tested Conjecture \[conj\] using Corollary \[univ\] and the strong version of Mochizuki’s formula (Remark \[strongform\]) in the following cases:
where $$\begin{aligned}
M := \, &\frac{5}{2} d_1d_2d_3-\frac{5}{6}d_1d_2-\frac{5}{6}d_1d_3-\frac{5}{6} d_2 d_3-\frac{5}{3}d_1-\frac{5}{3}d_2-\frac{5}{3}d_3+10 \\
&- 2 \prod_{{\scriptsize{\begin{array}{c} (i,j,k) \\ i,j,k \, \mathrm{distinct} \end{array}}}} (d_i-2) d_j e_k + 2 \prod_{{\scriptsize{\begin{array}{c} (i,j,k) \\ i,j,k \, \mathrm{distinct} \end{array}}}} e_i e_j d_k, \\
N := \, &-\frac{3}{2} d_1d_2d_3+\frac{1}{2}d_1d_2+\frac{1}{2}d_1d_3+\frac{1}{2} d_2 d_3+d_1+d_2+d_3-6 \\
&-\prod_{{\scriptsize{\begin{array}{c} (i,j,k) \\ i,j,k \, \mathrm{distinct} \end{array}}}} e_i e_j d_k +4\min\Big\{0,\prod (d_i-2)(e_j - (d_j-2)) d_k \Big\}.\end{aligned}$$
For arbitrary $y$, we checked Conjecture \[conj\] in the following cases:
An example covered by these tables with $(d_1,d_2,d_3) = (3,3,3)$ is the following $${\mathsf Z}_{S, -L_1+2L_2+2L_3}(x,1) =-147456 x -8358985728 x^5 +O( x^{9}),$$ where we choose a polarization $H$ such that $L_1 H$ is odd. Its virtual refinement is $${\mathsf Z}_{S,3L_1}(xy^{\frac{1}{2}},y)={\mathsf Z}_{S, -L_1+2L_2+2L_3}(xy^{\frac{1}{2}},y)= (-73728y- 73728)x+O(x^5).$$
Complete intersections in projective spaces
-------------------------------------------
For $d_1,d_2\in {\mathbb Z}_{\ge 2}$, with $d_1+d_2\ge 6$, let $S_{d_1,d_2}$ be a smooth complete intersection of bidegree $(d_1,d_2)$ in ${\mathbb P}^4$. Then $S_{d_1,d_2}$ is simply connected. Let $L$ be the restriction of the hyperplane class on ${\mathbb P}^4$ to $S_{d_1,d_2}$. If $S_{d_1,d_2}$ is very general, then by the Noether-Lefschetz theorem for complete intersections (see e.g. [@Kim]) the Picard group of $S_{d_1,d_2}$ is generated by $L$. Putting $d:=d_1d_2$, $D:=d_1+d_2$, we have $K_{S_{d_1,d_2}}=(D-5)L$ is very ample and $$\begin{aligned}
K_{S_{d_1,d_2}}^2&=d(D-5)^2, \\
\chi({\mathcal O}_{S_{d_1,d_2}})&=\frac{d}{12}(2D^2-15D-d+35).\end{aligned}$$ If the Picard group of $S_{d_1,d_2}$ is ${\mathbb Z}L$ and $H=L$, then there are no rank $2$ strictly $\mu$-semistable sheaves on $S_{d_1,d_2}$ with $c_1=\epsilon L$ and $\epsilon$ odd.
For $y=1$, we checked Conjecture \[conj\] in the following cases:
Here $$\begin{aligned}
M&:=\epsilon^2d-2\epsilon d(D-5)+\frac{5d}{12}(2D^2-15D-d+35),\\
N&:=-\epsilon^2d-\frac{3d}{12}(2D^2-15D-d+35)+4\min\Big\{0,d(D-5)(\epsilon-(D-5))\Big\}.\end{aligned}$$ For $y$ general, we checked Conjecture \[conj\] in the following cases:
For $\epsilon=3$ we determined ${\mathsf Z}_{S_{3,3},L,L}(x,1)$, and ${\mathsf Z}_{S_{2,4},L,L}(x,1)$ modulo $x^{25}$ and $x^{34}$, where the second is only written modulo $x^{26}$. $$\begin{aligned}
{\mathsf Z}_{S_{3,3},L,L}(x,1)=\,&-1152x - 11784960x^5 - 18762235136x^9 - 11903890079232x^{13} \\
&- 4135344957021312x^{17} - 924519456314916096x^{21} + O(x^{25}),\\
{\mathsf Z}_{S_{2,4},L,L}(x,1)=\,&6912x^2 + 30124032x^6 + 31867565056x^{10} + 15237098061824x^{14} \\
&+ 4243875728564736x^{18} + 789670403161694208x^{22} +O(x^{26}).\end{aligned}$$ For $\epsilon=1$ and using the strong form of Mochizuki’s formula (Remark \[strongform\]), we get $$\begin{aligned}
\chi_{-y}^{\mathrm{vir}}(M^L_{S_{3,3}}(2,L,7))&=-576y - 576, \\
\chi_{-y}^{\mathrm{vir}}(M^L_{S_{2,4}}(2,L,7))&=1344y^2 + 4224y + 1344.\end{aligned}$$
For $d_1,d_2,d_3\in {\mathbb Z}_{\ge 2}$, with $d_1+d_2+d_3\ge 7$, let $S_{d_1,d_2,d_3}$ be a smooth complete intersection of tridegree $(d_1,d_2,d_3)$ in ${\mathbb P}^5$. Then $S_{d_1,d_2,d_3}$ is simply connected. When $S_{d_1,d_2,d_3}$ is very general, the Picard group of $S_{d_1,d_2,d_3}$ is generated by the restriction $L$ of a hyperplane class on ${\mathbb P}^5$. Putting $d:=d_1d_2d_3$, $D:=d_1+d_2+d_3$, we have $K_{S_{d_1,d_2,d_3}}=(D-6)L$ is very ample and $$\begin{aligned}
K_{S_{d_1,d_2,d_3}}^2&=d(D-6)^2,\\
\chi({\mathcal O}_{S_{d_1,d_2,d_3}})&=\frac{d}{12}(2D^2-18D+51-d_1d_2-d_1d_3-d_2d_3).\end{aligned}$$ For $y=1$, we checked Conjecture \[conj\] in the following cases:
As an example we get $$\begin{aligned}
{\mathsf Z}_{S_{2,2,3},L,L}(x,1)=&-1261568x^3 - 7379091456x^7 - 11717181702144x^{11}\\
&- 8585117244063744x^{15} - 3662336916158939136x^{19} + O(x^{23}).\end{aligned}$$
Verifications of Conjecture \[numconj\] {#numconjevidence}
---------------------------------------
Let $\underline{\beta} \in {\mathbb Z}^4$ and $(\gamma_1,\gamma_2)\in {\mathbb Z}^2$ such that $\beta_1 \equiv \beta_2 \mod 2$ and $\beta_3 \geq \beta_4 - 3$. Put $$\begin{aligned}
M_1&:=\min\Big\{2\beta_1-2\beta_2+8\beta_4, 124-4\gamma_1+4\gamma_2,124+4\beta_2-4\beta_3+4\gamma_1-4\gamma_2\Big\}-\beta_1-3\beta_4,\\
M_2&:= \min\Big\{2\beta_1-2\beta_2+8\beta_4, 32-4\gamma_1+4\gamma_2,32+4\beta_2-4\beta_3+4\gamma_1-4\gamma_2\Big\}-\beta_1-3\beta_4.\end{aligned}$$ For $y=1$, we verified Conjecture \[numconj\] up to order $x^{M_1-1}$ in the following cases:
- $\gamma_1=\gamma_2=0$ and any $|\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| \le 16$,
- any $|\gamma_1|, |\gamma_2|, |\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| \le 4$.
For arbitrary $y$, we verified Conjecture \[numconj\] up to order $x^{M_2-1}$ in the following cases:
- $\gamma_1=\gamma_2=0$ and any $|\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| \le 3$,
- any $|\gamma_1|, |\gamma_2|, |\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| \le 2$.
We have examples showing that the condition $\beta_3 \geq \beta_4 - 3$ is necessary.
Any choice of $\underline{\beta} \in {\mathbb Z}^4$ with $\beta_1 \equiv \beta_2 \mod 2$ satisfying $$\frac{1}{12} \beta_3 \leq \beta_4 < \frac{1}{9} \beta_3$$ does *not* arise geometrically and automatically implies $\beta_3 \geq \beta_4-3$. These inequalities correspond to “$e(S) \geq 0$ but violation of the Bogomolov-Miyaoka-Yau inequality”, so in this regime $\underline{\beta}$ cannot be realized by a smooth projective algebraic surface. With these restrictions, we verified Conjecture \[numconj\] for $y=1$ up to order $x^{M_1-1}$ for all $$\begin{aligned}
\gamma_1=\gamma_2=0 \hbox{ and } |\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| &\le 25,\end{aligned}$$ and for arbitrary $y$ up to order $x^{M_2-1}$ for all $$\begin{aligned}
\gamma_1=\gamma_2=0 \hbox{ and } |\beta_1|, |\beta_2|, |\beta_3|, |\beta_4| &\le 27.\end{aligned}$$
$A_1, \ldots, A_7$ {#app1}
==================
We calculated $A_1(s,1,q), \ldots, A_7(s,1,q)$ up to order $q^{30}$ and list them up to order $q^4$.
\
Vafa-Witten formula with $\mu$-classes\
*by Lothar Göttsche and Hiraku Nakajima* {#appB}
========================================
Let $S$ be a projective algebraic surface with $b_1(S)=0$ and $p_g(S)>0$. For an ample line bundle $H$ on $S$, we denote again $M=M^H_S(c_1,c_2)$ the moduli space of $H$-semistable rank $2$ sheaves on $S$ with Chern classes $c_1\in H^2(S,{\mathbb Z})$ and $c_2\in H^4(S,{\mathbb Z})={\mathbb Z}$. We assume that there are no rank $2$ strictly Gieseker $H$-semistable sheaves with first Chern class $c_1$. Let ${\mathbb E}$ be a universal sheaf over $M$. For $\beta\in H_k(S,{\mathbb Q})$ we denote $$\mu(\beta):=\pi_{M*}\big((c_2({\mathbb E})-c_1({\mathbb E})^2/4)\cap \pi_S^*\beta\big)\in H^{4-k}(M,{\mathbb Q}).$$ Formally we can write $(c_2({\mathbb E})-c_1({\mathbb E})^2/4)=-{\operatorname{ch}}_2({\mathbb E}\otimes \det({\mathbb E})^{-1/2})$.
The Witten conjecture for $S$ expresses the generating function of the Donaldson invariants of $S$ in terms of the Seiberg-Witten invariants of $S$. On the other hand the version of the Vafa-Witten conjecture of this paper (Conjecture \[generalsurfconj\] with $y=1$) expresses the generating function of the virtual Euler numbers of the $M^H_S(c_1,c_2)$ in terms the Seiberg-Witten invariants of $S$. We want to give a common conjectural generalization of both formulas. For $\alpha\in H_2(S,{\mathbb Q})$ and $p\in H_0(S,{\mathbb Z})$ the class of a point, the Donaldson generating function is $$\begin{aligned}
{\mathsf D}^S_{c_1}(\alpha z+p u, x):=&\,\sum_{n} \int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}\exp\big(\mu(\alpha z+p u)\big) x^{{\operatorname{vd}}(S,c_1,n)}\\=&
\sum_{n}\sum_{l+2m={\operatorname{vd}}(S,c_1,n)}\int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}\mu(\alpha)^l\mu(p)^m\frac{z^l}{l!}\frac{u^m}{m!}x^{l+2m}.\end{aligned}$$ Here again ${\operatorname{vd}}(S,c_1,n)=4n-c_1^2-3\chi({\mathcal O}_S)$ is the expected dimension of the moduli space $M^H_S(c_1,n)$. The Vafa-Witten partition function is $${\mathsf Z}^S_{c_1}(x):=\sum_{n} \int_{[M^H_S(c_1,n)]^{\mathrm{vir}}} c_{{\operatorname{vd}}(S,c_1,n)} (T^{\mathrm{vir}}_{M^H_S(c_1,n)} )x^{{\operatorname{vd}}(S,c_1,n)}.$$ We consider the following new generating function $${\mathsf Z}^S_{c_1}(\alpha z+p u,t,x):=\sum_{n} \int_{[M^H_S(c_1,n)]^{\mathrm{vir}}} c^{{\mathbb C}^*}_{{\operatorname{vd}}(S,c_1,n)} (T^{\mathrm{vir}}_{M^H_S(c_1,n)}\otimes \frak{t}) \exp\big(\mu(\alpha z+p u)\big)x^{{\operatorname{vd}}(S,c_1,n)}.$$ Here $\frak{t}$ is the trivial line bundle with equivariant first Chern class $t$. Note that $$\begin{aligned}
{\mathsf Z}^S_{c_1}(\alpha z+p u,t,x)&=\sum_{n,k}\int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}c _k(T^{\mathrm{vir}}_{M^H_S(c_1,n)})t^{{\operatorname{vd}}(S,c_1,n)-k} \exp\big(\mu(\alpha z+p u)\big)x^{{\operatorname{vd}}(S,c_1,n)}\\
&=\sum_{n} \int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}c (T^{\mathrm{vir}}_{M^H_S(c_1,n)}) \exp\big(\mu(\alpha zt+p ut^2)\big)x^{{\operatorname{vd}}(S,c_1,n)}\\&={\mathsf Z}^S_{c_1}(\alpha zt+p ut^2,1,x),\end{aligned}$$ thus the variable $t$ is redundant, but it serves to interpolate between ${\mathsf Z}^S_{c_1}(x)$ and ${\mathsf D}^S_{c_1}(\alpha z+p u, x)$. In fact it is obvious from the above that $$\begin{aligned}
{\mathsf Z}^S_{c_1}(\alpha z+ p u,t,x)|_{t=0}&={\mathsf Z}^S_{c_1}(x),\\
{\mathrm{Coeff}}_{t^{k}}\big[ {\mathsf Z}^S_{c_1}(\alpha z+ p u,t^{-1},xt)\big]&=\sum_{n}\int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}c _k(T^{\mathrm{vir}}_{M^H_S(c_1,n)}) \exp\big(\mu(\alpha z+p u)\big)x^{{\operatorname{vd}}(S,c_1,n)}.\end{aligned}$$ In particular ${\mathsf Z}^S_{c_1}(\alpha z+ p u,t^{-1},xt)|_{t=0}={\mathsf D}_{c_1}^S(\alpha z+pu,x).$
Let $$\overline G_2(x)=G_2(x)+\frac{1}{24}=\sum_{n>0}\sigma_1(n) x^n$$ be the Eisenstein series of weight $2$, write $D:=x\frac{d}{dx}$. Let $$\begin{aligned}
G_p(x)&:=2\overline G_2(x^2)\quad
G_Q(x)=\frac{1}{2}(D G_2)(x^2),\quad
G_{S}(x):=(G_2(x)-G_2(-x))/2.
$$
We denote $Q(\alpha)=\int_S \alpha^2$, the quadratic form and for $C\in H^2(S,{\mathbb Z})$ let ${\langle}C,\alpha{\rangle}$ be the intersection product. Let $a_1,\ldots,a_s$ be the Seiberg-Witten basic classes of $S$ (in the Mochizuki notation, the corresponding characteristic cohomology classes are $\widetilde a_i=K_S-2a_i$). Denote $$\begin{aligned}
\widetilde {\mathsf Z}^S_{c_1}(\alpha z+pu,t,x)=&\,4 \left(\frac{1}{2\overline{\eta}(x^2)^{12}}\right)^{\chi({\mathcal O}_S)}\left( \frac{2 \overline{\eta}(x^4)^2}{\theta_3(x)} \right)^{K_{S}^2}\exp\big(G_Q(x)Q(\alpha)z^2t^2+G_p(x)(ut^2-{\langle}K_S,\alpha{\rangle}zt)\big)\\
&\cdot\sum_{i=1}^s {\mathrm{SW}}(a_i)(-1)^{{\langle}c_1,a_i{\rangle}} \Big(\frac{\theta_3(x)}{\theta_3(-x)}\Big)^{{\langle}K_S,a_i{\rangle}}\exp\big(G_S(x) {\langle}K_S-2 a_i,\alpha{\rangle}zt\big).\end{aligned}$$ Then we have the conjecture
\[WVW\] $${\mathsf Z}^S_{c_1}(\alpha z+pu,t,x)=\frac{1}{2}\big(\widetilde {\mathsf Z}^S_{c_1}(\alpha z+pu,t,x)+(\sqrt{-1})^{c_1^2-\chi({\mathcal O}_S)}\widetilde {\mathsf Z}^S_{c_1}(\alpha z+pu,t,\sqrt{-1}x)\big).$$
By definition it is obvious that at $t=0$, Conjecture \[WVW\] specializes to Conjecture \[generalsurfconj\]. Now we look at the specialization at $t^{-1}=0$.
Conjecture \[WVW\] is true for ${\mathsf Z}^S_{c_1}(\alpha z+ p u,t^{-1},xt)$ modulo $t^2$.
Specializing $t^{-1}=0$ in ${\mathsf Z}^S_{c_1}(\alpha z+ p u,t^{-1},xt)$, Conjecture \[WVW\] says that $${\mathsf D}^S_{c_1}(\alpha z+pu,x)=\frac{1}{2}\big(\widetilde {\mathsf D}^S_{c_1}(\alpha z+pu,x)+(\sqrt{-1})^{c_1^2-\chi({\mathcal O}_S)}\widetilde {\mathsf D}^S_{c_1}(\alpha z+pu,\sqrt{-1}x)\big),$$ with $$\begin{aligned}
&\widetilde {\mathsf D}^S_{c_1}(\alpha z+pu,x):=\widetilde {\mathsf Z}^S_{c_1}(\alpha z+pu,t^{-1},xt)|_{t=0}\\&=2^{2+K_S^2-\chi({\mathcal O}_S)}\exp\Big(\frac{Q(\alpha)}{2}x^2z^2+2x^2u)\Big)
\sum_{i=1}^s {\mathrm{SW}}(a_i)(-1)^{{\langle}c_1,a_i{\rangle}}\exp\big({\langle}K_S-2 a_i,\alpha{\rangle}xz\big),\end{aligned}$$ which is a reformulation of the Witten conjecture for Donaldson invariants of algebraic surfaces, proved in [@GNY3].
By [@HL Prop. 8.3.1], we have $$c _1(T^{\mathrm{vir}}_{M^H_S(c_1,n)})=-2\mu(K_S)$$ in $H^2(M^H_S(c_1,n),{\mathbb Q})$. Thus after the substitution $t\to t^{-1}$, $x\to xt$, and using the result for the coefficient of $t^0$, the coefficient of $t^1$ of the right hand side of Conjecture \[WVW\] is $$\begin{aligned}
&\sum_{n}\int_{[M^H_S(c_1,n)]^{\mathrm{vir}}}c _1(T^{\mathrm{vir}}_{M^H_S(c_1,n)}) \exp\big(\mu(\alpha z+p u)\big)x^{{\operatorname{vd}}(S,c_1,n)}=z^{-1} \frac{\partial}{\partial w}{\mathsf D}^S_{c_1}(\alpha z-2K_Swz+pu,x) \Big|_{w=0}\\&=\frac{1}{2}\big(F_1(x)+(\sqrt{-1})^{c_1^2-\chi({\mathcal O}_S)}F_1(\sqrt{-1}x)\big)
,\end{aligned}$$ with $$\begin{aligned}
F_1(x)=&\,z^{-1} \frac{\partial}{\partial w}\widetilde {\mathsf D}^S_{c_1}(\alpha z-2K_Swz+pu,x) \Big|_{w=0}\\=&\,2^{2+K_S^2-\chi({\mathcal O}_S)}\exp\Big(\frac{Q(\alpha)}{2}x^2z^2+2x^2u)\Big)\\&\cdot
\sum_{i=1}^s \big({\langle}-2K_S,\alpha{\rangle}x^2z +{\langle}K_S-2a_i,-2K_S{\rangle}x\big){\mathrm{SW}}(a_i)(-1)^{{\langle}c_1,a_i{\rangle}}\exp\big({\langle}K_S-2 a_i,\alpha{\rangle}xz\big).\end{aligned}$$ And for the left hand side of Conjecture \[WVW\] we get $$\frac{\partial}{\partial t}{\mathsf Z}^S_{c_1}(\alpha z+pu,t^{-1},xt)|_{t=0}=\frac{1}{2}\big(F_2(x)+(\sqrt{-1})^{c_1^2-\chi({\mathcal O}_S)}F_2(\sqrt{-1}x)\big)$$ with $$\begin{aligned}
F_2(x)=\,&\frac{\partial}{\partial t}\widetilde {\mathsf Z}^S_{c_1}(\alpha z+pu,t^{-1},xt)|_{t=0}=2^{2+K_S^2-\chi({\mathcal O}_S)}\exp\Big(\frac{Q(\alpha)}{2}x^2z^2+2x^2u)\Big)\\&\cdot
\sum_{i=1}^s \big(-2K_S^2x -{\langle}2K_S,\alpha{\rangle}x^2z+4{\langle}a_i,K_S{\rangle}x\big){\mathrm{SW}}(a_i)(-1)^{{\langle}c_1,a_i{\rangle}}\exp\big({\langle}K_S-2 a_i,\alpha{\rangle}xz\big).\end{aligned}$$
Let $\pi$ and $q$ be the projections from $S^{[n_1]}\times S^{[n_2]}\times S$ to $S^{[n_1]}\times S^{[n_2]}$ and $S$. For $\beta\in H_*(S,{\mathbb Q})$ $a_1,c_1 \in A^1(S)$, put $$\nu(\beta):=\pi_{*}^{{\mathbb C}^*}\big(-{\operatorname{ch}}_2^{{\mathbb C}^*}\big({\mathcal I}_1(a_1-c_1/2)\otimes {\mathfrak s}\oplus {\mathcal I}_2(c_1/2-a_1)\otimes {\mathfrak s}^{-1}\big)\cap q^*(\beta)\big).$$
For $\alpha\in H_2(S,{\mathbb Q})$ and $p\in H_0(S,{\mathbb Z})$ the class of a point, we put $$\begin{aligned}
&{\mathsf Z}_S(a_1,c_1,s,\alpha z+pu,q):=\\
&\sum_{n_1, n_2 \geq 0} q^{n_1+n_2} \int_{S^{[n_1]} \times S^{[n_2]}} \frac{c^{{\mathbb C}^*}(E_{n_1,n_2})\exp\big(\nu(\alpha)zt+\nu(p)ut^2\big) \, {\mathrm{Eu}}({\mathcal O}(a_1)^{[n_1]}) \, {\mathrm{Eu}}({\mathcal O}(c_1-a_1)^{[n_2]} \otimes {\mathfrak s}^2)}{{\mathrm{Eu}}(E_{n_1,n_2} - \pi_1^* \, T_{S^{[n_1]}} - \pi_2^* \, T_{S^{[n_2]}})}.\end{aligned}$$
We denote by $\widetilde{\mathcal A}(a_1,c_1-a_1,c_2,s)$ the expression from , in the case $$P({\mathbb E})=c(T_M^{\mathrm{vir}})\exp(\mu(\alpha) zt +\mu(p)u t^2).$$ As in we find that $$\begin{split}
\sum_{c_2\in {\mathbb Z}}\widetilde{\mathcal A}(a_1,c_1-a_1,c_2,s)q^{c_2}=\,&{\mathsf Z}_S(a_1,c_1,s,\alpha z+pu,q) \\
&\cdot (2s)^{\chi({\mathcal O}_S)}\left(\frac{2s}{1+2s}\right)^{\chi(c_1-2a_1)}\left(\frac{-2s}{1-2s}\right)^{\chi(2a_1-c_1)}q^{a_1(c_1-a_1)}.
\end{split}$$ Now we have the following analogue of Proposition \[univ\].
For $i=1,\ldots,7$ denote $A_i:=A_i(s,1,q)$ for $A_i(s,y,q)$ the universal functions from Proposition \[univ\]. There exist universal functions $A_8(s,q), \ldots, A_{12}(s,q)\in 1+q{\mathbb Q}(\!(s)\!)[[q]]$, such that for any smooth projective surface $S$ and any $a_1,c_1 \in A^1(S)$, $\alpha\in H_2(S,{\mathbb Q})$, we have $${\mathsf Z}_S(a_1,c_1,s,\alpha z+pu,q)=A_1^{a_1^2} A_2^{a_1c_1}A_3^{c_1^2}A_4^{a_1K_S}A_5^{c_1K_S}A_6^{K_S^2}A_7^{\chi({\mathcal O}_S)}A_8^{\alpha^2 z^2 t^2} A_9^{a_1\alpha z t}A_{10}^{c_1\alpha z t}
A_{11}^{K_S\alpha z t} A_{12}^{ut^2}.$$
The proof is a simple adaptation of the proof of Proposition \[univ\] (which is an adaptation of [@GNY1]), note that already in [@GNY1] it was shown how to deal with the classes $\nu(\beta)$. Write $M=M^H_S(c_1,c_2)$. The same proof as that of Corollary \[univcor\] shows that under the assumptions of Corollary \[univcor\] we have $$\begin{aligned}
&\int_{[M^{\mathrm{vir}}]}
c_{{\operatorname{vd}}(M)}^{{\mathbb C}^*}(T^{{\mathrm{vir}}}_M\otimes {\frak t})\exp\big(\mu(\alpha)z+\mu(p)u\big)=\int_{[M^{\mathrm{vir}}]}
c(T^{{\mathrm{vir}}}_M)\exp\big(\mu(\alpha)zt+\mu(p)ut^2\big)\\&={\mathrm{Coeff}}_{s^0q^{c_2}}
\Bigg[\sum_{{a_1\in H^2(S,{\mathbb Z})}\atop a_1H<{(c_1-a_1)H}}
{\mathrm{SW}}(a_1){\mathsf A}_{(a_1^2,a_1c_1,c_1^2,a_1K_S,c_1K_S,K_S^2,\chi({\mathcal O}_S))}A_8^{\alpha^2 z^2 t^2} A_9^{a_1\alpha z t}A_{10}^{c_1\alpha z t}
A_{11}^{K_S\alpha z t} A_{12}^{ut^2}\Bigg].\end{aligned}$$ Let $F$, $G$ be the classes of the fibres of the two projections of ${\mathbb P}^1\times {\mathbb P}^1$. The functions $A_8,\ldots,A_{12}$ are determined by $A_1,\ldots, A_7$ and ${\mathsf Z}_{{\mathbb P}^1\times {\mathbb P}^1}(a_1,c_1,s,\alpha z,q)$, for $$(a_1,c_1,\alpha)=(0,0,G-F), \ (0,0,-F),\ (G,G,F-G),\ \ (0,G,F-G),$$ and by ${\mathsf Z}_{{\mathbb P}^1\times {\mathbb P}^1}(0,0,s,pu ,q)$.
In these cases $S$ is a toric surface with torus $T={\mathbb C}^{*2}$. We use the notations and conventions from Section 4. Thus for $\sigma=1,\ldots, e(S)$ let $p_\sigma$ be a fixpoint of the $T$-action on $S$, let $U_\sigma$ be a maximal $T$-invariant open affine neighbourhood of $p_\sigma$, with coordinates $x_\sigma,y_\sigma$ such that $T$ acts by $$t\cdot (x_\sigma,y_\sigma)=(\chi(v_\sigma)x_\sigma,\chi(w_\sigma)y_\sigma),$$ where $\chi(v_\sigma)$ is a character with weight $v_\sigma$. Then in the notations of Section 4 the $T$-fixed points on $S^{[n_1]}\times S^{[n_2]}$ are parametrized by pairs $({\boldsymbol \lambda}, {\boldsymbol \mu})$ with ${\boldsymbol \lambda}=\{\lambda^{(\sigma)}\}_{\sigma=1,\ldots,e(S)},$ ${\boldsymbol \mu}=\{\mu^{(\sigma)}\}_{\sigma=1,\ldots,e(S)}$ tuples of partitions. Writing $\lambda^{(\sigma)}=(\lambda_1,\ldots,\lambda_\ell)$, here $\lambda^{(\sigma)}$ corresponds to the subscheme $Z_{\lambda^{(\sigma)}}$ supported in $p_\sigma$ with ideal $$I_{\lambda^{(\sigma)}}=(y_\sigma^{\lambda_1},x_\sigma y_\sigma^{\lambda_2},\ldots,x_\sigma^{\ell-1}y_\sigma^{\lambda_\ell},x_\sigma^\ell),$$ and similar for the $\mu^{(\sigma)}$. The fixpoint of $S^{[n_1]}\times S^{[n_2]}$ corresponding to $({\boldsymbol \lambda}, {\boldsymbol \mu})$ is the pair $(Z_{{\boldsymbol \lambda}},Z_{{\boldsymbol \mu}})$ with $Z_{{\boldsymbol \lambda}}$ the disjoint union of the $Z_{\lambda^{(\sigma)}}$ and similar for $Z_{{\boldsymbol \mu}}$. A $T$-equivariant divisor $a$ is again on $U_\sigma$ given by a character of weight $a_\sigma$. In the $T$-equivariant $K$-group $K_0^T(pt)$, the fibre of ${\mathcal I}_1(a)\oplus {\mathcal I}_2(-a)$ at the point $(p_\sigma,Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})$ is $$\chi(a_\sigma) \big(1-(1-\chi(v_\sigma))(1-\chi(w_\sigma))Z_{\lambda^{(\sigma)}}\big)+\chi(-a_\sigma) \big(1-(1-\chi(v_\sigma))(1-\chi(w_\sigma))Z_{\mu^{(\sigma)}}\big).$$ In particular, denoting $i^*_{(p_\sigma,Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})}$, the equivariant pullback to a point via the embedding of the fixpoint $(p_\sigma,Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})$, we get $$i^*_{(p_\sigma,Z_{\lambda^{(\sigma)}},Z_{\mu^{(\sigma)}})}{\operatorname{ch}}^T_2({\mathcal I}_1(a)\oplus {\mathcal I}_2(-a))=a_\sigma^2-v_\sigma w_\sigma (|\lambda^{(\sigma)}|+|\mu^{(\sigma)}|),$$ and thus for $b$ an equivariant divisor on $S$, we get $$i^*_{(Z_{{\boldsymbol \lambda}},Z_{{\boldsymbol \mu}})}\pi_{S^{[n_1]}\times S^{[n_2]} *} \big({\operatorname{ch}}_2^T({\mathcal I}_1(a)\oplus {\mathcal I}_2(-a)) \pi_S^* c_1(b)\big)=\sum_{\sigma=1}^{e(S)} \Big(\frac{b_\sigma a_\sigma^2}{v_\sigma w_\sigma}-b_\sigma (|\lambda^{(\sigma)}|+|\mu^{(\sigma)}|\Big).$$ On ${\mathbb P}^1\times {\mathbb P}^1$ we can represent the class of a point by $c_2({\mathbb P}^1\times {\mathbb P}^1)/4$, which gives $$i^*_{(Z_{{\boldsymbol \lambda}},Z_{{\boldsymbol \mu}})}\pi_{S^{[n_1]}\times S^{[n_2]} *} \big({\operatorname{ch}}_2^T({\mathcal I}_1(a)\oplus {\mathcal I}_2(-a)) \pi_S^*(pt)\big)=\sum_{\sigma=1}^{e(S)} \frac{1}{4}\big({a_\sigma^2}-v_\sigma w_\sigma (|\lambda^{(\sigma)}|+|\mu^{(\sigma)}|)\big).$$ Combining this with the formulas of Section 4 we can compute in the above cases ${\mathsf Z}_{{\mathbb P}^1\times {\mathbb P}^1}(a_1,c_1,s,\alpha z,q)$ by localization.
This has been implemented as as a PARI/GP program, which computed $A_8(s,q)$, $\ldots$, $A_{12}(s,q)$ up to order $q^{10}$ and any order in $s$. We then checked Conjecture \[WVW\] in a number of cases, in all these cases $\alpha$ is an arbitrary class in $H_2(S,{\mathbb Q})$. In all these cases the conjecture has been verified with the bounds of Section 7 adapted accordingly.
1. $K3$ surfaces, for $c_1^2=0,\ldots,30$.
2. Blowup $\widehat S$ of a $K3$ surface $S$ in a point with exceptional divisor $E$. Here we take $c_1=L+kE$ with $L$ the pullback of a line bundle on $S$ with $L^2=0,\ldots,30$ and $k=0,1$.
3. Elliptic surfaces $S\to {\mathbb P}^1$ of type $E(n)$, as in Section 7.2, for $n=3,\ldots,7$ with $c_1=kF$ or $B+kF$, with $B$ the class of a section with $B^2=-n$ and $F$ the class of a fibre and $k=0,\ldots 7$.
4. Double cover of ${\mathbb P}^2$, branched along a curve of degree $2d$ for $d=4,5$, with $c_1$ the pullback of $k$-times with hyperplane class for $k=-1\ldots,4$.
5. General quintic in ${\mathbb P}^3$, with $c_1$ the restriction of $k$ times the hyperplane class with $k=0,1,2,3$.
[MNOP]{} A. Braverman and P. Etingof, *Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential*, Studies in Lie theory, Progr. Math. 243, 61–78, Birkhäuser, Boston (2006). L. Baulieu, A. Losev, and N. Nekrasov, *Chern-Simons and twisted supersymmetry in various dimensions*, Nuclear Phys. B 522 (1998) 82–104. I. Ciocan-Fontanine and M. Kapranov, *Virtual fundamental classes via dg-manifolds*, Geom. Topol. 13 (2009) 1779–1804. H.-l. Chang and Y.-H. Kiem, *Poincaré invariants are Seiberg-Witten invariants*, Geom. and Topol. 17 (2013) 1149–1163. R. Dijkgraaf, J.-. Park, and B. J. Schroers, *$N=4$ supersymemtric Yang-Mills theory on a Kähler surface*, hep-th/9801066 ITFA-97-09. G. Ellingsrud, L. Göttsche, and M. Lehn, *On the cobordism class of the Hilbert scheme of a surface*, Jour. Alg. Geom. 10 (2001) 81–100. B. Fantechi and L. Göttsche, *Riemann-Roch theorems and elliptic genus for virtually smooth schemes*, Geom. Topol. 14 (2010) 83–115. R. Friedman and J. W. Morgan, *Obstruction bundles, semiregularity and Seiberg-Witten invariants*, Comm. Anal. Geom. 7 (1999) 451–495. L. Göttsche and D. Huybrechts, *Hodge numbers of moduli spaces of stable bundles on K3 surfaces*, Int. J. Math. 7 (1996) 359–372. L. Göttsche and M. Kool, *A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula*, in preparation. L. Göttsche, H. Nakajima, and K. Yoshioka, *Instanton counting and Donaldson invariants*, J. Diff. Geom. 80 (2008) 343–390. L. Göttsche, H. Nakajima, and K. Yoshioka, *K-theoretic Donaldson invariants via instanton counting*, Pure Appl. Math. Q. 5 (2009) 1029–1111. L. Göttsche, H. Nakajima, and K. Yoshioka, *Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting*, Publ. Res. Inst. Math. Sci. 47 (2011) 307–359. L. Göttsche, *Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces*, Math. Z. 223 (1996) 247–260. L. Göttsche, *Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces*, Comm. Math. Phys. 206 (1999) 105–136. L. Göttsche, W. Soergel, *Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces*, Math. Ann. 296 (1993) 235–245. A. Gholampour, A. Sheshmani, and S.-Y. Yau, *Localized Donaldson-Thomas theory of surfaces*, arXiv:1701.08902. D. Huybrechts, M. Lehn, *The geometry of moduli spaces of sheaves*, Cambridge University Press (2010). D. Huybrechts, *Compact hyper-Kähler manifolds: basic results*, Invent. Math. 135 (1999) 63–113. D. Joyce, *Configurations in abelian categories. IV. Invariants and changing stability conditions*, Adv. Math. 217 (2008) 125–204. S.-O. Kim, *Noether-Lefschetz locus for surfaces*, Trans. Amer. Math. Soc. 324 (1991) 369–384. A. A. Klyachko, *Vector bundles and torsion free sheaves on the projective plane*, preprint Max Planck Institut für Mathematik (1991). J. Li, K. Liu, and J. Zhou, *Topological string partition functions as equivariant indices*, Asian J. Math. 10 (2006) 81–114. A. Losev, N. Nekrasov, and S. Shatashvili, *Issues in topological gauge theory*, Nuclear Phys. B 534 (1998) 549–611. W.-P. Li and Z. Qin, *On blowup formulae for the $S$-duality conjecture of Vafa and Witten*, Invent. Math. 136 (1999) 451–482. W.-P. Li and Z. Qin, *On blowup formulae for the $S$-duality conjecture of Vafa and Witten II: the universal functions*, Math. Res. Lett. 5 (1998) 439–453. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, *Gromov-[W]{}itten theory and [D]{}onaldson-[T]{}homas theory, [I]{}*, Compos. Math. 142 (2006) 1263–1285. T. Mochizuki, *Donaldson type invariants for algebraic surfaces*, Lecture Notes in Math. 1972, Springer-Verlag, Berlin (2009). J. W. Morgan, *The Seiberg-Witten equations and applications to the topology of smooth four-manifolds*, Math. Notes 44, Princeton Univ. Press, Princeton NJ (1996). N. Nekrasov, *Four dimensional holomorphic theories*, PhD Thesis, Princeton (1996). N. Nekrasov, *Seiberg-Witten prepotential from instanton counting*, Adv. Theor. Math. Phys. 7 (2003) 831–864. N. Nekrasov and A. Okounkov, *Seiberg-Witten prepotential and random partitions*, The unity of mathematics, 525–596, Progr. Math. 244, Birkhäuser, Boston (2006). H. Nakajima and K. Yoshioka, *Lectures on instanton counting*, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes vol. 38, AMS Providence RI, 31–101 (2004). H. Nakajima and K. Yoshioka, *Instanton counting on blowup. I. 4-dimensional pure gauge theory*, Invent. Math. 162 (2005) 313–355. A. Okounkov, *Lectures on $K$-theoretic computations in enumerative geometry*, arXiv:1512.07363. R. Pandharipande, *Descendents for stable pairs on 3-folds*, arXiv:1703.01747. J. Shen, *Cobordism invariants of the moduli space of stable pairs*, J. London Math. Soc. 94 (2016) 427–446. Y. Tanaka and R. P. Thomas, *Vafa-Witten invariants for projective surfaces I: stable case*, arXiv:1702.08487. Y. Tanaka and R. P. Thomas, *Vafa-Witten invariants for projective surfaces II: semistable case*, arXiv:1702.08488. C. Vafa and E. Witten, *A strong coupling test of $S$-duality*, Nucl. Phys. B 431 (1994) 3–77. K. Yoshioka, *The Betti numbers of the moduli space of stable sheaves of rank 2 on ${\mathbb P}^2$*, J. Reine Angew. Math. 453 (1994) 193–220. K. Yoshioka, *The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface*, Math. Ann. 302 (1995) 519–540. K. Yoshioka, *Number of ${\mathbb F}_q$-rational points of the moduli of stable sheaves on elliptic surfaces*, Moduli of vector bundles, editor: M. Maruyama, Lect. Notes in Pure and Appl. Math. 179, Marcel Dekker, New York (1996). K. Yoshioka, *Some examples of Mukai’s reflections on K3 surfaces*, J. Reine Angew. Math. 515 (1999) 97–123.
[^1]: Although ${\mathbb E}$ may only exist étale locally, $R\pi_* R{\mathcal{H}{\it{om}}}({\mathbb E},{\mathbb E})_0$ exists globally [@HL Sect. 10.2].
[^2]: We denote the $K$-group generated by locally free sheaves on $M$ by $K^0(M)$. Virtual holomorphic Euler characteristic and the definition of $\Lambda^p E \in K^0(M)$ for an arbitrary element $E \in K^0(M)$ are explained in [@FG]. The notation $(\cdot)^\vee$ is for derived dual.
[^3]: We use Mochizuki’s convention, where ${\mathrm{SW}}(a)$ stands for $\widetilde{{\mathrm{SW}}}(2a-K_S)$ with $\widetilde{{\mathrm{SW}}}(b)$ the usual Seiberg-Witten invariant in class $b \in H^2(S,{\mathbb Z})$.
[^4]: In [@TT1], Tanaka-Thomas consider the case where semistability and stability of Higgs pairs coincide. They treat the semistable case in a separate paper [@TT2].
[^5]: See [@FG Cor. 3.4], or [@CFK] in the context of $[0,1]$-manifolds.
[^6]: This action originates from the ${\mathbb C}^*$-action on the master space mentioned above.
[^7]: This differs slightly from Mochizuki who uses $p_g(S)$ instead of $\chi({\mathcal O}_S)$ and takes a residue. Consequently our definition differs by a factor $2$ from Mochizuki’s. The difference is accounted for by the fact that Mochizuki works on the moduli stack of oriented sheaves which maps to $M$ via a degree $\frac{1}{2}:1$ étale morphism [@GNY3 Rem. 4.2].
[^8]: We verified this using the toric calculations of Section \[toricsec\] for (1) $M=0$ and order $q^{\leq 2}$ and (2) $M=0$, $\epsilon_1 = -\epsilon_2$, and order $q^3$.
[^9]: The inequality $K_S^2 \geq \chi({\mathcal O}_S) - 3$ is reminiscent of the (stronger) Noether’s inequality $K_S^2 \geq 2(\chi({\mathcal O}_S)-3)$ for minimal surfaces of general type.
[^10]: Up to an overall factor $x^{-\chi({\mathcal O}_S) + K_S^2/3}$ coming from our choice of normalization.
[^11]: Here the adjective “virtual” does not refer to virtual cycles. The definition of virtual Hodge polynomials involves Deligne’s weight filtration. They coincide with ordinary Hodge polynomials for smooth projective varieties.
[^12]: Up to an overall factor $x^{-\chi({\mathcal O}_S) + K_S^2/3}$ coming from our choice of normalization.
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Q:
403 Forbidden error when downloading paid OpenMapTiles from website
Every time there's a new release of OpenMapTiles tiles, I get a 403 forbidden error from maptiler.download. If I wait a day or two, the download works, but I'm paying good money for these weekly updates, so how do I download them on time?
A:
Looks like they fixed it by moving to a different CDN provider.
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Gels and gel pastes containing a silicone elastomer or a silicone organic elastomer have recently been described in PCT/US07/006833, PCT/US07/006894, and PCT/US07/006936; which are assigned to the same assignee of the present application. The present disclosure relates to aqueous emulsion compositions of these gels and gel pastes containing a silicone elastomer or a silicone organic elastomer.
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The downturn in the struggling manufacturing sector showed no sign of easing during March as businesses failed to beat tough competition from overseas rivals for new contracts.
While the 600 industrial companies questioned in a survey by Markit and the Chartered Institute of Purchasing & Supply gave a more positive reading of 48.3 on current trading and outlook, compared with February’s 47.9, it remained firmly below the crucial 50 mark that separates growth and contraction.
Companies were hit hard by a sharp drop in new export orders amid weak demand from Europe and strong competition in the United States and
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Light and electron microscopy of the ovarian interstitial tissue in the senile rat: normal aspect and response to HCG of "deficiency cells' and 'epithelial cords'.
The two components of the interstitial tissue in the senile rat's ovary--deficiency cells and epithelial cellular cords--both contain steroidogenic organelles at a very low level of activity, which can be notably raised by HCG. Deficiency cells and epithelial cords are not the expression of primary ovarian aging, but the consequence of senile gonadotropic hypofunctioning. For that same reason, these ovarian structures become permanent and their possible cellular aging can thus be studied. No ultrastructural signs of senescence have as yet been observed.
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Q:
How to use -v awk option for filtering out lines from a file
I have a text file with the format
key01=value
key02=value
...
keyN=value
And I would like to retrieve the value for a specific key number. I made it work using a literal, but it failed when trying to use the $SEARCHKEY variable as input for search. For example:
$echo $SEARCHKEY
KEY02
$awk -F= -v "searchKey=$SEARCHKEY" '/KEY02/ {print $2 " => " searchKey}' \
./parameterValue.lst
Output:
BBBB => KEY02
However, I cannot make it work when I change the /KEY02/ literal by searchKey:
awk -F= -v "searchKey=$SEARCHKEY" '/searchKey/ {print $2 " => " searchKey}' \
./parameterValue.lst
Returns nothing. What am I doing wrong?
A:
Try this:
awk -F= -v "searchKey=$SEARCHKEY" '$0 ~ searchKey {print $2 " => " searchKey}' input
variable are passed as strings into awk when using the -v-flag.
/searchKey/ hence searches for the string "searchKey" To prevent this, use the regex matching operator ~ instead.
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EU Nations Increasingly Divided over UN Migration Pact
BRUSSELS - Just days before scores of countries sign up to a landmark U.N. migration pact, a number of European Union nations have begun joining the list of those not willing to endorse the agreement.
The 34-page U.N. Global Compact for Safe, Orderly and Regular Migration is to be formally approved in Marrakech, Morocco, on Dec. 10-11.
The drafting process was launched after all 193 U.N. member states, including the United States under President Barack Obama, adopted in 2016 a declaration saying no country can manage international migration on its own and agreed to work on a global compact
But the United States, under President Donald Trump, pulled out a year ago, claiming that numerous provisions in the pact were 'inconsistent with U.S. immigration and refugee policies.''
Despite its non-binding nature, Bulgaria signaled this week that it will not sign the pact, as did Slovakia, whose foreign minister resigned in protest at his government's stance. Meanwhile, Belgium's government was teetering on the brink of collapse, riven by coalition differences over the pact.
'It's way too pro-migration. It doesn't have the nuance that it needs to have to also comfort European citizens,'' Belgium's migration minister, Theo Francken, said Thursday.
'It's not legally binding, but it's not without legal risks,'' he said, adding that rights laws are being interpreted widely in EU courts and those rulings are tying the hands of migration policy-makers.
Francken said his right-wing N-VA party wants 'nothing to do with it.''
But Belgian Prime Minister Charles Michel took the migration agreement to parliament Thursday, where it was approved against the wishes of the N-VA, the biggest party in his governing coalition.
The arrival in Europe in 2015 of well over 1 million migrants - most fleeing conflict in Syria or Iraq - plunged the EU into a deep political crisis over migration, as countries bickered over how to manage the challenge and how much help to provide those countries hardest hit by the influx. Their inability to agree helped fuel support for anti-migrant parties across Europe.
Experts say the pact is an easy target. Leaving it can play well with anti-migrant domestic audiences and pulling out has no obvious negative impacts on governments.
'The ones who opposed the global compact, have they read it? It is only a framework of cooperation with all countries,'' EU Migration Commissioner Dimitris Avramopoulos said Thursday. 'It is not binding. It doesn't put in question national sovereignty.''
Other EU countries to turn their back on the document are Hungary and Poland, which have opposed refugee quotas aimed at sharing the burden of Mediterranean countries like Italy, Greece and more recently Spain, where most migrants are arriving.
But the withdrawal of Austria - holder of the EU's presidency until the end of the year - has been of high symbolic importance.
Conservative Austrian Chancellor Sebastian Kurz, in a coalition with the nationalist, anti-migration Freedom Party, announced Austria's departure from the pact in October, highlighting 'some points that we view critically and where we fear a danger to our national sovereignty.''
Francken said that never before had the head negotiator for the European states, Austria, 'pulled out the plug. That gave a lot of political shock effect in all countries.''
It remains to be seen whether North African countries - and others like Turkey, which the EU has outsourced its migrant challenge to - see this as a new sign that migration management can only be done on Europe's terms.
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Q:
Combining 2 codes in Python
I have generated these two codes.
The first generates a country according to the probability given by the number of customers in each country:
import random
selectp = random.choices(
population=["A", "B", "C", "D", "E"],
weights=[0.2307, 0.0769, 0.3846, 0.1923, 0.1153],
k=1
)
print(selectp)
And the second generates a random number of 4 characters by filling in the numbers on the left with zero where appropriate:
import random
num = random.randrange(1, 9999)
# using string's zfill
num_with_zeros = str(num).zfill(4)
print (num_with_zeros)
What I need to do is to combine these codes to generate a result of the following type, for example, in a random way:
A-0213, B-2345, A-0001...
The added complication is that for each letter there are a limited number of combinations, which are:
For A, the numbers will go from A-0000 to A-0300
For B, the numbers will go from B-0000 to B-0100
For C, the numbers will go from C-0000 to C-0500
For D, the numbers will go from D-0000 to D-0250
For E, the numbers will go from E-0000 to E-0150
Any idea how to do it?
Edit:
So what I need is a code that randomly generates a combination of the type:
A-0001, B-0035...
keeping in mind that the probability distribution for the letters is:
A = 0.2307
B = 0.0769
C = 0.3846
D = 0.1923
E = 0.1153
And that the number must be between the following numbers:
For A, the numbers will go from A-0000 to A-0300
For B, the numbers will go from B-0000 to B-0100
For C, the numbers will go from C-0000 to C-0500
For D, the numbers will go from D-0000 to D-0250
For E, the numbers will go from E-0000 to E-0150
A:
You will need a dictionary to store what the allowed numbers are, and then you can simply add to strings with +
import random
max_vals = {"A": 300, "B": 100, "C": 500, "D": 250, "E": 150}
def random_code():
letter = random.choices(list(max_vals.keys()),
weights=[0.2307, 0.0769, 0.3846, 0.1923, 0.1153],
k=1)[0]
num = random.randrange(0, max_vals[letter]) # You allow 0000 in the examples in the question
# using string's zfill
num_with_zeros = str(num).zfill(4)
return letter + "-" + num_with_zeros
for _ in range(10):
print(random_code())
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Harvey Weinstein should go sign up for the dating website OkCupid and make sure he gets that pink ribbon on his profile, branding him a feminist and supporter of Planned Parenthood. The media mogul is the ultimate bro-choicer, an adoring donor to the Democratic Party, defended by some of the left’s most powerful women.
Weinstein has been accused in a lengthy New York Times story of doing some truly horrendous things to women, totally unsolicited − also known as sexual harassment − over a period of nearly three decades. His alleged victims were talented young actresses and other women who had the potential to build successful careers on their own without some ugly, creepy man coercing them into giving him a massage or worse.
Gross.
But this is Hollywood, and the feminists are the women who defend creeps like Weinstein.
Anita Dunn, a close associate of President Obama who became his communications director for a time at the White House, has been reported to have counseled Weinstein after the allegations broke in the media.
Even more troubling is that celebrity women’s advocate attorney Lisa Bloom is representing Weinstein. This is a woman who has represented victims in high-profile sexual harassment cases. Throwing that legacy away, she agreed to represent Weinstein, whom she said she has counseled that “times have changed” and he “needs to evolve into a higher standard.” Really?
Bloom’s reasoning regarding Weinstein gets better, though: “He has acknowledged mistakes he has made. He is reading books and going to therapy. He is an old dinosaur learning new ways. He wants to reach out to any of the women who may have issues with him to talk to them in a respectful, peaceful way, with me present if that is acceptable to them.”
I hope Bloom has a better answer for the women who Weinstein is accused of sexually harassing and forcing to look at him half-clothed or with no clothes at all, with tears in their eyes, as he emotionally blackmailed them for sex.
But this is Hollywood. This is Hugh Hefner’s world. This is the liberal paradise of promoting equal rights and justice for women, yet defending the men who use them for their own satisfaction, discarding them for younger versions every year, and then complaining about it years later when the women finally have the courage to say “enough!”
This is not true feminism. Supporting men who tear down women for their own use, supporting free access to taxpayer-funded abortion so men can cover up their own conduct, and mistreating women for decades is nothing for women to get behind.
These powerful women, these self-proclaimed feminists, are only hurting women when they stand up for and promote men who proudly use their own power to harass and demean women.
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Chaz and AJ Podcasts - 6/27/13
Today, Attorney Norm Pattis talked about the George Zimmerman trial and Aaron Hernandez's arrest, is "Chaz" a bully's name? Why did RC Smith pick on Phil's mom? Why is it so enjoyable to listen to Magilla get zapped with a shock collar?
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News
Congratulations to Kassandra Hordal on her gold medal at the Royal Canadian Legion Youth National Championships in the U-16 category. Kassandra successfully cleared 2.90m to win the Gold, and she continued jumping to clear 3.20m. Great Job Kassandra!
On the boys side, Corey Woolnough had an amazing warm up, but unexpectedly needed even bigger poles. We attempted to borrow a pole from another athlete, but unfortunately, it wasn't to be. Corey had an injury during the warm-ups and was unable to compete. Keep your head up Corey, great things are coming.
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Network Working Group P. Sarolahti
Request for Comments: 4138 Nokia Research Center
Category: Experimental M. Kojo
University of Helsinki
August 2005
Forward RTO-Recovery (F-RTO): An Algorithm for Detecting
Spurious Retransmission Timeouts with TCP and the
Stream Control Transmission Protocol (SCTP)
Status of This Memo
This memo defines an Experimental Protocol for the Internet
community. It does not specify an Internet standard of any kind.
Discussion and suggestions for improvement are requested.
Distribution of this memo is unlimited.
Copyright Notice
Copyright (C) The Internet Society (2005).
Abstract
Spurious retransmission timeouts cause suboptimal TCP performance
because they often result in unnecessary retransmission of the last
window of data. This document describes the F-RTO detection
algorithm for detecting spurious TCP retransmission timeouts. F-RTO
is a TCP sender-only algorithm that does not require any TCP options
to operate. After retransmitting the first unacknowledged segment
triggered by a timeout, the F-RTO algorithm of the TCP sender
monitors the incoming acknowledgments to determine whether the
timeout was spurious. It then decides whether to send new segments
or retransmit unacknowledged segments. The algorithm effectively
helps to avoid additional unnecessary retransmissions and thereby
improves TCP performance in the case of a spurious timeout. The
F-RTO algorithm can also be applied to the Stream Control
Transmission Protocol (SCTP).
Sarolahti & Kojo Experimental [Page 1]
RFC 4138 Forward RTO-Recovery August 2005
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Terminology . . . . . . . . . . . . . . . . . . . . 4
2. F-RTO Algorithm . . . . . . . . . . . . . . . . . . . . . 4
2.1. The Algorithm . . . . . . . . . . . . . . . . . . . 5
2.2. Discussion . . . . . . . . . . . . . . . . . . . . 6
3. SACK-Enhanced Version of the F-RTO Algorithm . . . . . . 8
4. Taking Actions after Detecting Spurious RTO . . . . . . . 10
5. SCTP Considerations . . . . . . . . . . . . . . . . . . . 10
6. Security Considerations . . . . . . . . . . . . . . . . . 11
7. Acknowledgements . . . . . . . . . . . . . . . . . . . . 12
8. References . . . . . . . . . . . . . . . . . . . . . . . 12
8.1. Normative References. . . . . . . . . . . . . . . . 12
8.2. Informative References. . . . . . . . . . . . . . . 13
Appendix A: Scenarios . . . . . . . . . . . . . . . . . . . . 15
Appendix B: SACK-Enhanced F-RTO and Fast Recovery . . . . . . 20
Appendix C: Discussion of Window-Limited Cases . . . . . . . 21
1. Introduction
The Transmission Control Protocol (TCP) [Pos81] has two methods for
triggering retransmissions. First, the TCP sender relies on incoming
duplicate ACKs, which indicate that the receiver is missing some of
the data. After a required number of successive duplicate ACKs have
arrived at the sender, it retransmits the first unacknowledged
segment [APS99] and continues with a loss recovery algorithm such as
NewReno [FHG04] or SACK-based loss recovery [BAFW03]. Second, the
TCP sender maintains a retransmission timer which triggers
retransmission of segments, if they have not been acknowledged before
the retransmission timeout (RTO) expires. When the retransmission
timeout occurs, the TCP sender enters the RTO recovery where the
congestion window is initialized to one segment and unacknowledged
segments are retransmitted using the slow-start algorithm. The
retransmission timer is adjusted dynamically, based on the measured
round-trip times [PA00].
It has been pointed out that the retransmission timer can expire
spuriously and cause unnecessary retransmissions when no segments
have been lost [LK00, GL02, LM03]. After a spurious retransmission
timeout, the late acknowledgments of the original segments arrive at
the sender, usually triggering unnecessary retransmissions of a whole
window of segments during the RTO recovery. Furthermore, after a
spurious retransmission timeout, a conventional TCP sender increases
the congestion window on each late acknowledgment in slow start.
This injects a large number of data segments into the network within
one round-trip time, thus violating the packet conservation principle
[Jac88].
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RFC 4138 Forward RTO-Recovery August 2005
There are a number of potential reasons for spurious retransmission
timeouts. First, some mobile networking technologies involve sudden
delay spikes on transmission because of actions taken during a
hand-off. Second, given a low-bandwidth link or some other change in
available bandwidth, arrival of competing traffic (possibly with
higher priority) can cause a sudden increase of round-trip time.
This may trigger a spurious retransmission timeout. A persistently
reliable link layer can also cause a sudden delay when a data frame
and several retransmissions of it are lost for some reason. This
document does not distinguish between the different causes of such a
delay spike. Rather, it discusses the spurious retransmission
timeouts caused by a delay spike in general.
This document describes the F-RTO detection algorithm. It is based
on the detection mechanism of the "Forward RTO-Recovery" (F-RTO)
algorithm [SKR03] that is used for detecting spurious retransmission
timeouts and thus avoids unnecessary retransmissions following the
retransmission timeout. When the timeout is not spurious, the F-RTO
algorithm reverts back to the conventional RTO recovery algorithm,
and therefore has similar behavior and performance. In contrast to
alternative algorithms proposed for detecting unnecessary
retransmissions (Eifel [LK00], [LM03] and DSACK-based algorithms
[BA04]), F-RTO does not require any TCP options for its operation,
and it can be implemented by modifying only the TCP sender. The
Eifel algorithm uses TCP timestamps [BBJ92] for detecting a spurious
timeout upon arrival of the first acknowledgment after the
retransmission. The DSACK-based algorithms require that the TCP
Selective Acknowledgment Option [MMFR96], with the DSACK extension
[FMMP00], is in use. With DSACK, the TCP receiver can report if it
has received a duplicate segment, enabling the sender to detect
afterwards whether it has retransmitted segments unnecessarily. The
F-RTO algorithm only attempts to detect and avoid unnecessary
retransmissions after an RTO. Eifel and DSACK can also be used for
detecting unnecessary retransmissions caused by other events, such as
packet reordering.
When an RTO expires, the F-RTO sender retransmits the first
unacknowledged segment as usual [APS99]. Deviating from the normal
operation after a timeout, it then tries to transmit new, previously
unsent data, for the first acknowledgment that arrives after the
timeout, given that the acknowledgment advances the window. If the
second acknowledgment that arrives after the timeout advances the
window (i.e., acknowledges data that was not retransmitted), the F-
RTO sender declares the timeout spurious and exits the RTO recovery.
However, if either of these two acknowledgments is a duplicate ACK,
there will not be sufficient evidence of a spurious timeout.
Therefore, the F-RTO sender retransmits the unacknowledged segments
in slow start similarly to the traditional algorithm. With a
Sarolahti & Kojo Experimental [Page 3]
RFC 4138 Forward RTO-Recovery August 2005
SACK-enhanced version of the F-RTO algorithm, spurious timeouts may
be detected even if duplicate ACKs arrive after an RTO
retransmission.
The F-RTO algorithm can also be applied to the Stream Control
Transmission Protocol (SCTP) [Ste00], because SCTP has acknowledgment
and packet retransmission concepts similar to TCP. For convenience,
this document mostly refers to TCP, but the algorithms and other
discussion are valid for SCTP as well.
This document is organized as follows. Section 2 describes the basic
F-RTO algorithm. Section 3 outlines an optional enhancement to the
F-RTO algorithm that takes advantage of the TCP SACK option. Section
4 discusses the possible actions to be taken after detecting a
spurious RTO. Section 5 gives considerations on applying F-RTO with
SCTP, and Section 6 discusses the security considerations.
1.1. Terminology
The keywords MUST, MUST NOT, REQUIRED, SHALL, SHALL NOT, SHOULD,
SHOULD NOT, RECOMMENDED, MAY, and OPTIONAL, when they appear in this
document, are to be interpreted as described in [RFC2119].
2. F-RTO Algorithm
A timeout is considered spurious if it would have been avoided had
the sender waited longer for an acknowledgment to arrive [LM03].
F-RTO affects the TCP sender behavior only after a retransmission
timeout. Otherwise, the TCP behavior remains the same. When the RTO
expires, the F-RTO algorithm monitors incoming acknowledgments and if
the TCP sender gets an acknowledgment for a segment that was not
retransmitted due to timeout, the F-RTO algorithm declares a timeout
spurious. The actions taken in response to a spurious timeout are
not specified in this document, but we discuss some alternatives in
Section 4. This section introduces the algorithm and then discusses
the different steps of the algorithm in more detail.
Following the practice used with the Eifel Detection algorithm
[LM03], we use the "SpuriousRecovery" variable to indicate whether
the retransmission is declared spurious by the sender. This variable
can be used as an input for a corresponding response algorithm. With
F-RTO, the value of SpuriousRecovery can be either SPUR_TO
(indicating a spurious retransmission timeout) or FALSE (indicating
that the timeout is not declared spurious), and the TCP sender should
follow the conventional RTO recovery algorithm.
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2.1. The Algorithm
A TCP sender MAY implement the basic F-RTO algorithm. If it chooses
to apply the algorithm, the following steps MUST be taken after the
retransmission timer expires. If the sender implements some loss
recovery algorithm other than Reno or NewReno [FHG04], the F-RTO
algorithm SHOULD NOT be entered when earlier fast recovery is
underway.
1) When RTO expires, retransmit the first unacknowledged segment and
set SpuriousRecovery to FALSE. Also, store the highest sequence
number transmitted so far in variable "recover".
2) When the first acknowledgment after the RTO retransmission arrives
at the sender, the sender chooses one of the following actions,
depending on whether the ACK advances the window or whether it is
a duplicate ACK.
a) If the acknowledgment is a duplicate ACK OR it acknowledges a
sequence number equal to the value of "recover" OR it does not
acknowledge all of the data that was retransmitted in step 1,
revert to the conventional RTO recovery and continue by
retransmitting unacknowledged data in slow start. Do not enter
step 3 of this algorithm. The SpuriousRecovery variable
remains as FALSE.
b) Else, if the acknowledgment advances the window AND it is below
the value of "recover", transmit up to two new (previously
unsent) segments and enter step 3 of this algorithm. If the
TCP sender does not have enough unsent data, it can send only
one segment. In addition, the TCP sender MAY override the
Nagle algorithm [Nag84] and immediately send a segment if
needed. Note that sending two segments in this step is allowed
by TCP congestion control requirements [APS99]: An F-RTO TCP
sender simply chooses different segments to transmit.
If the TCP sender does not have any new data to send, or the
advertised window prohibits new transmissions, the recommended
action is to skip step 3 of this algorithm and continue with
slow start retransmissions, following the conventional RTO
recovery algorithm. However, alternative ways of handling the
window-limited cases that could result in better performance
are discussed in Appendix C.
3) When the second acknowledgment after the RTO retransmission
arrives at the sender, the TCP sender either declares the timeout
spurious, or starts retransmitting the unacknowledged segments.
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a) If the acknowledgment is a duplicate ACK, set the congestion
window to no more than 3 * MSS, and continue with the slow
start algorithm retransmitting unacknowledged segments. The
congestion window can be set to 3 * MSS, because two round-trip
times have elapsed since the RTO, and a conventional TCP sender
would have increased cwnd to 3 during the same time. Leave
SpuriousRecovery set to FALSE.
b) If the acknowledgment advances the window (i.e., if it
acknowledges data that was not retransmitted after the
timeout), declare the timeout spurious, set SpuriousRecovery to
SPUR_TO, and set the value of the "recover" variable to SND.UNA
(the oldest unacknowledged sequence number [Pos81]).
2.2. Discussion
The F-RTO sender takes cautious actions when it receives duplicate
acknowledgments after a retransmission timeout. Because duplicate
ACKs may indicate that segments have been lost, reliably detecting a
spurious timeout is difficult due to the lack of additional
information. Therefore, it is prudent to follow the conventional TCP
recovery in those cases.
If the first acknowledgment after the RTO retransmission covers the
"recover" point at algorithm step (2a), there is not enough evidence
that a non-retransmitted segment has arrived at the receiver after
the timeout. This is a common case when a fast retransmission is
lost and has been retransmitted again after an RTO, while the rest of
the unacknowledged segments were successfully delivered to the TCP
receiver before the retransmission timeout. Therefore, the timeout
cannot be declared spurious in this case.
If the first acknowledgment after the RTO retransmission does not
acknowledge all of the data that was retransmitted in step 1, the TCP
sender reverts to the conventional RTO recovery. Otherwise, a
malicious receiver acknowledging partial segments could cause the
sender to declare the timeout spurious in a case where data was lost.
The TCP sender is allowed to send two new segments in algorithm
branch (2b) because the conventional TCP sender would transmit two
segments when the first new ACK arrives after the RTO retransmission.
If sending new data is not possible in algorithm branch (2b), or if
the receiver window limits the transmission, the TCP sender has to
send something in order to prevent the TCP transfer from stalling.
If no segments were sent, the pipe between sender and receiver might
run out of segments, and no further acknowledgments would arrive.
Therefore, in the window-limited case, the recommendation is to
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revert to the conventional RTO recovery with slow start
retransmissions. Appendix C discusses some alternative solutions for
window-limited situations.
If the retransmission timeout is declared spurious, the TCP sender
sets the value of the "recover" variable to SND.UNA in order to allow
fast retransmit [FHG04]. The "recover" variable was proposed for
avoiding unnecessary, multiple fast retransmits when RTO expires
during fast recovery with NewReno TCP. Because the sender
retransmits only the segment that triggered the timeout, the problem
of unnecessary multiple fast retransmits [FHG04] cannot occur.
Therefore, if three duplicate ACKs arrive at the sender after the
timeout, they probably indicate a packet loss, and thus fast
retransmit should be used to allow efficient recovery. If there are
not enough duplicate ACKs arriving at the sender after a packet loss,
the retransmission timer expires again and the sender enters step 1
of this algorithm.
When the timeout is declared spurious, the TCP sender cannot detect
whether the unnecessary RTO retransmission was lost. In principle,
the loss of the RTO retransmission should be taken as a congestion
signal. Thus, there is a small possibility that the F-RTO sender
will violate the congestion control rules, if it chooses to fully
revert congestion control parameters after detecting a spurious
timeout. The Eifel detection algorithm has a similar property, while
the DSACK option can be used to detect whether the retransmitted
segment was successfully delivered to the receiver.
The F-RTO algorithm has a side-effect on the TCP round-trip time
measurement. Because the TCP sender can avoid most of the
unnecessary retransmissions after detecting a spurious timeout, the
sender is able to take round-trip time samples on the delayed
segments. If the regular RTO recovery was used without TCP
timestamps, this would not be possible due to the retransmission
ambiguity. As a result, the RTO is likely to have more accurate and
larger values with F-RTO than with the regular TCP after a spurious
timeout that was triggered due to delayed segments. We believe this
is an advantage in the networks that are prone to delay spikes.
There are some situations where the F-RTO algorithm may not avoid
unnecessary retransmissions after a spurious timeout. If packet
reordering or packet duplication occurs on the segment that triggered
the spurious timeout, the F-RTO algorithm may not detect the spurious
timeout due to incoming duplicate ACKs. Additionally, if a spurious
timeout occurs during fast recovery, the F-RTO algorithm often cannot
detect the spurious timeout because the segments that were
transmitted before the fast recovery trigger duplicate ACKs.
However, we consider these cases rare, and note that in cases where
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F-RTO fails to detect the spurious timeout, it retransmits the
unacknowledged segments in slow start, and thus performs similarly to
the regular RTO recovery.
3. SACK-Enhanced Version of the F-RTO Algorithm
This section describes an alternative version of the F-RTO algorithm
that uses the TCP Selective Acknowledgment Option [MMFR96]. By using
the SACK option, the TCP sender detects spurious timeouts in most of
the cases when packet reordering or packet duplication is present.
If the SACK blocks acknowledge new data that was not transmitted
after the RTO retransmission, the sender may declare the timeout
spurious, even when duplicate ACKs follow the RTO.
Given that the TCP Selective Acknowledgment Option [MMFR96] is
enabled for a TCP connection, a TCP sender MAY implement the
SACK-enhanced F-RTO algorithm. If the sender applies the
SACK-enhanced F-RTO algorithm, it MUST follow the steps below. This
algorithm SHOULD NOT be applied if the TCP sender is already in SACK
loss recovery when retransmission timeout occurs. However, when
retransmission timeout occurs during existing loss recovery, it
should be possible to apply the principle of F-RTO within certain
limitations. This is a topic for further research. Appendix B
briefly discusses the related issues.
The steps of the SACK-enhanced version of the F-RTO algorithm are as
follows.
1) When the RTO expires, retransmit the first unacknowledged segment
and set SpuriousRecovery to FALSE. Set variable "recover" to
indicate the highest segment transmitted so far. Following the
recommendation in SACK specification [MMFR96], reset the SACK
scoreboard.
2) Wait until the acknowledgment of the data retransmitted due to the
timeout arrives at the sender. If duplicate ACKs arrive before
the cumulative acknowledgment for retransmitted data, adjust the
scoreboard according to the incoming SACK information. Stay in
step 2 and wait for the next new acknowledgment. If RTO expires
again, go to step 1 of the algorithm.
a) if a cumulative ACK acknowledges a sequence number equal to
"recover", revert to the conventional RTO recovery and set the
congestion window to no more than 2 * MSS, like a regular TCP
would do. Do not enter step 3 of this algorithm.
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b) else, if a cumulative ACK acknowledges a sequence number
(smaller than "recover", but larger than SND.UNA) transmit up
to two new (previously unsent) segments and proceed to step 3.
If the TCP sender is not able to transmit any previously unsent
data -- either due to receiver window limitation, or because it
does not have any new data to send -- the recommended action is
to refrain from entering step 3 of this algorithm. Rather,
continue with slow start retransmissions following the
conventional RTO recovery algorithm.
It is also possible to apply some of the alternatives for
handling window-limited cases discussed in Appendix C. In this
case, the TCP sender should follow the recommendations
concerning acknowledgments of retransmitted segments given in
Appendix B.
3) The next acknowledgment arrives at the sender. Either a duplicate
ACK or a new cumulative ACK (advancing the window) applies in this
step.
a) if the ACK acknowledges a sequence number above "recover",
either in SACK blocks or as a cumulative ACK, set the
congestion window to no more than 3 * MSS and proceed with the
conventional RTO recovery, retransmitting unacknowledged
segments. Take this branch also when the acknowledgment is a
duplicate ACK and it does not acknowledge any new, previously
unacknowledged data below "recover" in the SACK blocks. Leave
SpuriousRecovery set to FALSE.
b) if the ACK does not acknowledge sequence numbers above
"recover" AND it acknowledges data that was not acknowledged
earlier (either with cumulative acknowledgment or using SACK
blocks), declare the timeout spurious and set SpuriousRecovery
to SPUR_TO. The retransmission timeout can be declared
spurious, because the segment acknowledged with this ACK was
transmitted before the timeout.
If there are unacknowledged holes between the received SACK blocks,
those segments are retransmitted similarly to the conventional SACK
recovery algorithm [BAFW03]. If the algorithm exits with
SpuriousRecovery set to SPUR_TO, "recover" is set to SND.UNA, thus
allowing fast recovery on incoming duplicate acknowledgments.
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4. Taking Actions after Detecting Spurious RTO
Upon retransmission timeout, a conventional TCP sender assumes that
outstanding segments are lost and starts retransmitting the
unacknowledged segments. When the retransmission timeout is detected
to be spurious, the TCP sender should not continue retransmitting
based on the timeout. For example, if the sender was in congestion
avoidance phase transmitting new, previously unsent segments, it
should continue transmitting previously unsent segments after
detecting a spurious RTO. This document does not describe the
response to spurious timeouts, but a response algorithm is described
in RFC 4015 [LG04].
Additionally, different response variants to spurious retransmission
timeout have been discussed in various research papers [SKR03, GL03,
Sar03] and IETF documents [SL03]. The different response
alternatives vary in whether the spurious retransmission timeout
should be taken as a congestion signal, thus causing the congestion
window or slow start threshold to be reduced at the sender, or
whether the congestion control state should be fully reverted to the
state valid prior to the retransmission timeout.
5. SCTP Considerations
SCTP has similar retransmission algorithms and congestion control to
TCP. The SCTP T3-rtx timer for one destination address is maintained
in the same way as the TCP retransmission timer, and after a T3-rtx
expires, an SCTP sender retransmits unacknowledged data chunks in
slow start like TCP does. Therefore, SCTP is vulnerable to the
negative effects of the spurious retransmission timeouts similarly to
TCP. Due to similar RTO recovery algorithms, F-RTO algorithm logic
can be applied also to SCTP. Since SCTP uses selective
acknowledgments, the SACK-based variant of the algorithm is
recommended, although the basic version can also be applied to SCTP.
However, SCTP contains features that are not present with TCP that
need to be discussed when applying the F-RTO algorithm.
SCTP associations can be multi-homed. The current retransmission
policy states that retransmissions should go to alternative
addresses. If the retransmission was due to spurious timeout caused
by a delay spike, it is possible that the acknowledgment for the
retransmission arrives back at the sender before the acknowledgments
of the original transmissions arrive. If this happens, a possible
loss of the original transmission of the data chunk that was
retransmitted due to the spurious timeout may remain undetected when
applying the F-RTO algorithm. Because the timeout was caused by a
delay spike, and it was spurious in that respect, a suitable response
is to continue by sending new data. However, if the original
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transmission was lost, fully reverting the congestion control
parameters is too aggressive. Therefore, taking conservative actions
on congestion control is recommended, if the SCTP association is
multi-homed and retransmissions go to alternative addresses. The
information in duplicate TSNs can be then used for reverting
congestion control, if desired [BA04].
Note that the forward transmissions made in F-RTO algorithm step (2b)
should be destined to the primary address, since they are not
retransmissions.
When making a retransmission, an SCTP sender can bundle a number of
unacknowledged data chunks and include them in the same packet. This
needs to be considered when implementing F-RTO for SCTP. The basic
principle of F-RTO still holds: in order to declare the timeout
spurious, the sender must get an acknowledgment for a data chunk that
was not retransmitted after the retransmission timeout. In other
words, acknowledgments of data chunks that were bundled in RTO
retransmission must not be used for declaring the timeout spurious.
6. Security Considerations
The main security threat regarding F-RTO is the possibility that a
receiver could mislead the sender into setting too large a congestion
window after an RTO. There are two possible ways a malicious
receiver could trigger a wrong output from the F-RTO algorithm.
First, the receiver can acknowledge data that it has not received.
Second, it can delay acknowledgment of a segment it has received
earlier, and acknowledge the segment after the TCP sender has been
deluded to enter algorithm step 3.
If the receiver acknowledges a segment it has not really received,
the sender can be led to declare spurious timeout in the F-RTO
algorithm, step 3. However, because the sender will have an
incorrect state, it cannot retransmit the segment that has never
reached the receiver. Therefore, this attack is unlikely to be
useful for the receiver to maliciously gain a larger congestion
window.
A common case for a retransmission timeout is that a fast
retransmission of a segment is lost. If all other segments have been
received, the RTO retransmission causes the whole window to be
acknowledged at once. This case is recognized in F-RTO algorithm
branch (2a). However, if the receiver only acknowledges one segment
after receiving the RTO retransmission, and then the rest of the
segments, it could cause the timeout to be declared spurious when it
is not. Therefore, it is suggested that, when an RTO expires during
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fast recovery phase, the sender would not fully revert the congestion
window even if the timeout was declared spurious. Instead, the
sender would reduce the congestion window to 1.
If there is more than one segment missing at the time of a
retransmission timeout, the receiver does not benefit from misleading
the sender to declare a spurious timeout because the sender would
have to go through another recovery period to retransmit the missing
segments, usually after an RTO has elapsed.
7. Acknowledgements
We are grateful to Reiner Ludwig, Andrei Gurtov, Josh Blanton, Mark
Allman, Sally Floyd, Yogesh Swami, Mika Liljeberg, Ivan Arias
Rodriguez, Sourabh Ladha, Martin Duke, Motoharu Miyake, Ted Faber,
Samu Kontinen, and Kostas Pentikousis for the discussion and feedback
contributed to this text.
8. References
8.1. Normative References
[APS99] Allman, M., Paxson, V., and W. Stevens, "TCP Congestion
Control", RFC 2581, April 1999.
[BAFW03] Blanton, E., Allman, M., Fall, K., and L. Wang, "A
Conservative Selective Acknowledgment (SACK)-based Loss
Recovery Algorithm for TCP", RFC 3517, April 2003.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[FHG04] Floyd, S., Henderson, T., and A. Gurtov, "The NewReno
Modification to TCP's Fast Recovery Algorithm", RFC 3782,
April 2004.
[MMFR96] Mathis, M., Mahdavi, J., Floyd, S., and A. Romanow, "TCP
Selective Acknowledgement Options", RFC 2018, October 1996.
[PA00] Paxson, V. and M. Allman, "Computing TCP's Retransmission
Timer", RFC 2988, November 2000.
[Pos81] Postel, J., "Transmission Control Protocol", STD 7, RFC
793, September 1981.
Sarolahti & Kojo Experimental [Page 12]
RFC 4138 Forward RTO-Recovery August 2005
[Ste00] Stewart, R., Xie, Q., Morneault, K., Sharp, C.,
Schwarzbauer, H., Taylor, T., Rytina, I., Kalla, M., Zhang,
L., and V. Paxson, "Stream Control Transmission Protocol",
RFC 2960, October 2000.
8.2. Informative References
[ABF01] Allman, M., Balakrishnan, H., and S. Floyd, "Enhancing
TCP's Loss Recovery Using Limited Transmit", RFC 3042,
January 2001.
[BA04] Blanton, E. and M. Allman, "Using TCP Duplicate Selective
Acknowledgement (DSACKs) and Stream Control Transmission
Protocol (SCTP) Duplicate Transmission Sequence Numbers
(TSNs) to Detect Spurious Retransmissions", RFC 3708,
February 2004.
[BBJ92] Jacobson, V., Braden, R., and D. Borman, "TCP Extensions
for High Performance", RFC 1323, May 1992.
[FMMP00] Floyd, S., Mahdavi, J., Mathis, M., and M. Podolsky, "An
Extension to the Selective Acknowledgement (SACK) Option
for TCP", RFC 2883, July 2000.
[GL02] A. Gurtov and R. Ludwig. Evaluating the Eifel Algorithm
for TCP in a GPRS Network. In Proc. of European Wireless,
Florence, Italy, February 2002.
[GL03] A. Gurtov and R. Ludwig, Responding to Spurious Timeouts in
TCP. In Proceedings of IEEE INFOCOM 03, San Francisco, CA,
USA, March 2003.
[Jac88] V. Jacobson. Congestion Avoidance and Control. In
Proceedings of ACM SIGCOMM 88.
[LG04] Ludwig, R. and A. Gurtov, "The Eifel Response Algorithm for
TCP", RFC 4015, February 2005.
[LK00] R. Ludwig and R.H. Katz. The Eifel Algorithm: Making TCP
Robust Against Spurious Retransmissions. ACM SIGCOMM
Computer Communication Review, 30(1), January 2000.
[LM03] Ludwig, R. and M. Meyer, "The Eifel Detection Algorithm for
TCP", RFC 3522, April 2003.
[Nag84] Nagle, J., "Congestion Control in IP/TCP Internetworks",
RFC 896, January 1984.
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RFC 4138 Forward RTO-Recovery August 2005
[SKR03] P. Sarolahti, M. Kojo, and K. Raatikainen. F-RTO: An
Enhanced Recovery Algorithm for TCP Retransmission
Timeouts. ACM SIGCOMM Computer Communication Review,
33(2), April 2003.
[Sar03] P. Sarolahti. Congestion Control on Spurious TCP
Retransmission Timeouts. In Proceedings of IEEE Globecom
2003, San Francisco, CA, USA. December 2003.
[SL03] Y. Swami and K. Le, "DCLOR: De-correlated Loss Recovery
using SACK Option for Spurious Timeouts", work in progress,
September 2003.
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Appendix A: Scenarios
This section discusses different scenarios where RTOs occur and how
the basic F-RTO algorithm performs in those scenarios. The
interesting scenarios are: a sudden delay triggering retransmission
timeout, loss of a retransmitted packet during fast recovery, link
outage causing the loss of several packets, and packet reordering. A
performance evaluation with a more thorough analysis on a real
implementation of F-RTO is given in [SKR03].
A.1. Sudden Delay
The main motivation behind the F-RTO algorithm is to improve TCP
performance when a delay spike triggers a spurious retransmission
timeout. The example below illustrates the segments and
acknowledgments transmitted by the TCP end hosts when a spurious
timeout occurs, but no packets are lost. For simplicity, delayed
acknowledgments are not used in the example. The example below
applies the Eifel Response Algorithm [LG04] after detecting a
spurious timeout.
...
(cwnd = 6, ssthresh < 6, FlightSize = 6)
1. <---------------------------- ACK 5
2. SEND 10 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
3. <---------------------------- ACK 6
4. SEND 11 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
5. |
[delay]
|
[RTO]
[F-RTO step (1)]
6. SEND 6 ---------------------------->
(cwnd = 6, ssthresh = 3, FlightSize = 6)
<earlier xmitted SEG 6> --->
7. <---------------------------- ACK 7
[F-RTO step (2b)]
8. SEND 12 ---------------------------->
9. SEND 13 ---------------------------->
(cwnd = 7, ssthresh = 3, FlightSize = 7)
<earlier xmitted SEG 7> --->
10. <---------------------------- ACK 8
[F-RTO step (3b)]
[SpuriousRecovery <- SPUR_TO]
(cwnd = 7, ssthresh = 6, FlightSize = 6)
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11. SEND 14 ---------------------------->
(cwnd = 7, ssthresh = 6, FlightSize = 7)
12. <---------------------------- ACK 9
13. SEND 15 ---------------------------->
(cwnd = 7, ssthresh = 6, FlightSize = 7)
14. <---------------------------- ACK 10
15. SEND 16 ---------------------------->
(cwnd = 7, ssthresh = 6, FlightSize = 7)
...
When a sudden delay (long enough to trigger timeout) occurs at step
5, the TCP sender retransmits the first unacknowledged segment (step
6). The next ACK covers the RTO retransmission because the
originally transmitted segment 6 arrived at the receiver, and the TCP
sender continues by sending two new data segments (steps 8, 9). Note
that on F-RTO steps (1) and (2b), congestion window and FlightSize
are not yet reset because in the case of spurious timeout, the
segments sent before the timeout are still in the network. However,
the sender should still be equally aggressive toward conventional
TCP. Because the second acknowledgment arriving after the RTO
retransmission acknowledges data that was not retransmitted due to
timeout (step 10), the TCP sender declares the timeout to be spurious
and continues by sending new data on the next acknowledgments. Also,
the congestion control state is reversed, as required by the Eifel
Response Algorithm.
A.2. Loss of a Retransmission
If a retransmitted segment is lost, the only way to retransmit it is
to wait for the timeout to trigger the retransmission. Once the
segment is successfully received, the receiver usually acknowledges
several segments at once, because other segments in the same window
have been successfully delivered before the retransmission arrives at
the receiver. The example below shows a scenario where
retransmission (of segment 6) is lost, as well as a later segment
(segment 9) in the same window. The limited transmit [ABF01] or SACK
TCP [MMFR96] enhancements are not in use in this example.
...
(cwnd = 6, ssthresh < 6, FlightSize = 6)
<segment 6 lost>
<segment 9 lost>
1. <---------------------------- ACK 5
2. SEND 10 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
3. <---------------------------- ACK 6
4. SEND 11 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
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5. <---------------------------- ACK 6
6. <---------------------------- ACK 6
7. <---------------------------- ACK 6
8. SEND 6 --------------X
(cwnd = 6, ssthresh = 3, FlightSize = 6)
<segment 6 lost>
9. <---------------------------- ACK 6
10. SEND 12 ---------------------------->
(cwnd = 7, ssthresh = 3, FlightSize = 7)
11. <---------------------------- ACK 6
12. SEND 13 ---------------------------->
(cwnd = 8, ssthresh = 3, FlightSize = 8)
[RTO]
13. SEND 6 ---------------------------->
(cwnd = 8, ssthresh = 2, FlightSize = 8)
14. <---------------------------- ACK 9
[F-RTO step (2b)]
15. SEND 14 ---------------------------->
16. SEND 15 ---------------------------->
(cwnd = 7, ssthresh = 2, FlightSize = 7)
17. <---------------------------- ACK 9
[F-RTO step (3a)]
[SpuriousRecovery <- FALSE]
(cwnd = 3, ssthresh = 2, FlightSize = 7)
18. SEND 9 ---------------------------->
19. SEND 10 ---------------------------->
20. SEND 11 ---------------------------->
...
In the example above, segment 6 is lost and the sender retransmits it
after three duplicate ACKs in step 8. However, the retransmission is
also lost, and the sender has to wait for the RTO to expire before
retransmitting it again. Because the first ACK following the RTO
retransmission acknowledges the RTO retransmission (step 14), the
sender transmits two new segments. The second ACK in step 17 does
not acknowledge any previously unacknowledged data. Therefore, the
F-RTO sender enters the slow start and sets cwnd to 3 * MSS. The
congestion window can be set to three segments, because two round-
trips have elapsed after the retransmission timeout. Finally, the
receiver acknowledges all segments transmitted prior to entering
recovery and the sender can continue transmitting new data in
congestion avoidance.
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A.3. Link Outage
The example below illustrates the F-RTO behavior when 4 consecutive
packets are lost in the network causing the TCP sender to fall back
to RTO recovery. Limited transmit and SACK are not used in this
example.
...
(cwnd = 6, ssthresh < 6, FlightSize = 6)
<segments 6-9 lost>
1. <---------------------------- ACK 5
2. SEND 10 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
3. <---------------------------- ACK 6
4. SEND 11 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
5. <---------------------------- ACK 6
|
|
[RTO]
6. SEND 6 ---------------------------->
(cwnd = 6, ssthresh = 3, FlightSize = 6)
7. <---------------------------- ACK 7
[F-RTO step (2b)]
8. SEND 12 ---------------------------->
9. SEND 13 ---------------------------->
(cwnd = 7, ssthresh = 3, FlightSize = 7)
10. <---------------------------- ACK 7
[F-RTO step (3a)]
[SpuriousRecovery <- FALSE]
(cwnd = 3, ssthresh = 3, FlightSize = 7)
11. SEND 7 ---------------------------->
12. SEND 8 ---------------------------->
13. SEND 9 ---------------------------->
Again, F-RTO sender transmits two new segments (steps 8 and 9) after
the RTO retransmission is acknowledged. Because the next ACK does
not acknowledge any data that was not retransmitted after the
retransmission timeout (step 10), the F-RTO sender proceeds with
conventional recovery and slow start retransmissions.
A.4. Packet Reordering
Because F-RTO modifies the TCP sender behavior only after a
retransmission timeout and it is intended to avoid unnecessary
retransmissions only after spurious timeout, we limit the discussion
on the effects of packet reordering on F-RTO behavior to the cases
where it occurs immediately after the retransmission timeout. When
Sarolahti & Kojo Experimental [Page 18]
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the TCP receiver gets an out-of-order segment, it generates a
duplicate ACK. If the TCP sender implements the basic F-RTO
algorithm, this may prevent the sender from detecting a spurious
timeout.
However, if the TCP sender applies the SACK-enhanced F-RTO, it is
possible to detect a spurious timeout when packet reordering occurs.
Below, we illustrate the behavior of SACK-enhanced F-RTO when segment
8 arrives before segments 6 and 7, and segments starting from segment
6 are delayed in the network. In this example the TCP sender reduces
the congestion window and slow start threshold in response to
spurious timeout.
...
(cwnd = 6, ssthresh < 6, FlightSize = 6)
1. <---------------------------- ACK 5
2. SEND 10 ---------------------------->
(cwnd = 6, ssthresh < 6, FlightSize = 6)
3. <---------------------------- ACK 6
4. SEND 11 ---------------------------->
5. |
[delay]
|
[RTO]
6. SEND 6 ---------------------------->
(cwnd = 6, ssthresh = 3, FlightSize = 6)
<earlier xmitted SEG 8> --->
7. <---------------------------- ACK 6
[SACK 8]
[SACK F-RTO stays in step 2]
8. <earlier xmitted SEG 6> --->
9. <---------------------------- ACK 7
[SACK 8]
[SACK F-RTO step (2b)]
10. SEND 12 ---------------------------->
11. SEND 13 ---------------------------->
(cwnd = 7, ssthresh = 3, FlightSize = 7)
12. <earlier xmitted SEG 7> --->
13. <---------------------------- ACK 9
[SACK F-RTO step (3b)]
[SpuriousRecovery <- SPUR_TO]
(cwnd = 7, ssthresh = 6, FlightSize = 6)
14. SEND 14 ---------------------------->
(cwnd = 7, ssthresh = 6, FlightSize = 7)
15. <---------------------------- ACK 10
16. SEND 15 ---------------------------->
...
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After RTO expires and the sender retransmits segment 6 (step 6), the
receiver gets segment 8 and generates duplicate ACK with SACK for
segment 8. In response to the acknowledgment, the TCP sender does
not send anything but stays in F-RTO step 2. Because the next
acknowledgment advances the cumulative ACK point (step 9), the sender
can transmit two new segments according to SACK-enhanced F-RTO. The
next segment acknowledges new data between 7 and 11 that was not
acknowledged earlier (segment 7), so the F-RTO sender declares the
timeout spurious.
Appendix B: SACK-enhanced F-RTO and Fast Recovery
We believe that a slightly modified, SACK-enhanced F-RTO algorithm
can be used to detect spurious timeouts also when RTO expires while
an earlier loss recovery is underway. However, there are issues that
need to be considered if F-RTO is applied in this case.
In step 3, the original SACK-based F-RTO algorithm requires that an
ACK acknowledges previously unacknowledged non-retransmitted data
between SND.UNA and send_high. If RTO expires during earlier
(SACK-based) loss recovery, the F-RTO sender must use only
acknowledgments for non-retransmitted segments transmitted before the
SACK-based loss recovery started. This means that in order to
declare timeout spurious, the TCP sender must receive an
acknowledgment for non-retransmitted segment between SND.UNA and
RecoveryPoint in algorithm step 3. RecoveryPoint is defined in
conservative SACK-recovery algorithm [BAFW03], and it is set to
indicate the highest segment transmitted so far when SACK-based loss
recovery begins. In other words, if the TCP sender receives
acknowledgment for a segment that was transmitted more than one RTO
ago, it can declare the timeout spurious. Defining an efficient
algorithm for checking these conditions remains a future work item.
When spurious timeout is detected according to the rules given above,
it may be possible that the response algorithm needs to consider this
case separately, for example, in terms of which segments to
retransmit after RTO expires, and whether it is safe to revert the
congestion control parameters. This is considered a topic for future
research.
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Appendix C: Discussion of Window-Limited Cases
When the advertised window limits the transmission of two new
previously unsent segments, or there are no new data to send, it is
recommended in F-RTO algorithm step (2b) that the TCP sender continue
with the conventional RTO recovery algorithm. The disadvantage is
that the sender may continue unnecessary retransmissions due to
possible spurious timeout. This section briefly discusses the
options that can potentially improve performance when transmitting
previously unsent data is not possible.
- The TCP sender could reserve an unused space of a size of one or
two segments in the advertised window to ensure the use of
algorithms such as F-RTO or Limited Transmit [ABF01] in window-
limited situations. On the other hand, while doing this, the TCP
sender should ensure that the window of outstanding segments is
large enough for proper utilization of the available pipe.
- Use additional information if available, e.g., TCP timestamps with
the Eifel Detection algorithm, for detecting a spurious timeout.
However, Eifel detection may yield different results from F-RTO
when ACK losses and an RTO occur within the same round-trip time
[SKR03].
- Retransmit data from the tail of the retransmission queue and
continue with step 3 of the F-RTO algorithm. It is possible that
the retransmission will be made unnecessarily. Thus, this option
is not encouraged, except for hosts that are known to operate in an
environment that is prone to spurious timeouts. On the other hand,
with this method it is possible to limit unnecessary
retransmissions due to spurious timeout to one retransmission.
- Send a zero-sized segment below SND.UNA, similar to TCP Keep-Alive
probe, and continue with step 3 of the F-RTO algorithm. Because
the receiver replies with a duplicate ACK, the sender is able to
detect whether the timeout was spurious from the incoming
acknowledgment. This method does not send data unnecessarily, but
it delays the recovery by one round-trip time in cases where the
timeout was not spurious. Therefore, this method is not
encouraged.
- In receiver-limited cases, send one octet of new data, regardless
of the advertised window limit, and continue with step 3 of the
F-RTO algorithm. It is possible that the receiver will have free
buffer space to receive the data by the time the segment has
propagated through the network, in which case no harm is done. If
the receiver is not capable of receiving the segment, it rejects
the segment and sends a duplicate ACK.
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Authors' Addresses
Pasi Sarolahti
Nokia Research Center
P.O. Box 407
FIN-00045 NOKIA GROUP
Finland
Phone: +358 50 4876607
EMail: [email protected]
http://www.cs.helsinki.fi/u/sarolaht/
Markku Kojo
University of Helsinki
Department of Computer Science
P.O. Box 68
FIN-00014 UNIVERSITY OF HELSINKI
Finland
Phone: +358 9 191 51305
EMail: [email protected]
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Full Copyright Statement
Copyright (C) The Internet Society (2005).
This document is subject to the rights, licenses and restrictions
contained in BCP 78, and except as set forth therein, the authors
retain all their rights.
This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET
ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE
INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.
Intellectual Property
The IETF takes no position regarding the validity or scope of any
Intellectual Property Rights or other rights that might be claimed to
pertain to the implementation or use of the technology described in
this document or the extent to which any license under such rights
might or might not be available; nor does it represent that it has
made any independent effort to identify any such rights. Information
on the procedures with respect to rights in RFC documents can be
found in BCP 78 and BCP 79.
Copies of IPR disclosures made to the IETF Secretariat and any
assurances of licenses to be made available, or the result of an
attempt made to obtain a general license or permission for the use of
such proprietary rights by implementers or users of this
specification can be obtained from the IETF on-line IPR repository at
http://www.ietf.org/ipr.
The IETF invites any interested party to bring to its attention any
copyrights, patents or patent applications, or other proprietary
rights that may cover technology that may be required to implement
this standard. Please address the information to the IETF at ietf-
[email protected].
Acknowledgement
Funding for the RFC Editor function is currently provided by the
Internet Society.
Sarolahti & Kojo Experimental [Page 23]
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Constraint-Induced Aphasia Therapy in the Acute Stage: What Is the Key Factor for Efficacy? A Randomized Controlled Study.
Constraint-induced aphasia therapy (CIAT) has proven effective in patients with subacute and chronic forms of aphasia. It has remained unclear, however, whether intensity of therapy or constraint is the relevant factor. Data about intensive speech and language therapy (SLT) are conflicting. To identify the effective component of CIAT and assess the feasibility of SLT in the acute stage after stroke. A total of 60 patients with aphasia (68.2 ± 11.7 years) were enrolled 18.9 days after first-ever stroke. They were randomly distributed into 3 groups: (1) CIAT group receiving therapy for 3 hours per day (10 workdays, total 30 hours); (2) conventional communication treatment group, with same intensity without constraints; and (3) control group receiving individual therapy twice a day as well as group therapy (total 14 hours). Patients were assessed pretreatment and posttreatment using the Aachener Aphasia Test (primary end point: token test) and the Communicative Activity Log (CAL). Pretreatment, there were no between-group differences. Posttreatment, all groups showed significant improvements without between-group differences. It was found that 14 hours of aphasia therapy administered within 2 weeks as individual therapy, focusing on individual deficits, combined with group sessions has proven to be most efficient. This approach yielded the same outcome as 30 hours of group therapy, either in the form of CIAT or group therapy without constraints. SLT in an intensive treatment schedule is feasible and was well tolerated in the acute stage after stroke.
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The present disclosure relates to compositions composed of grafted diene polymer and processes for producing the same.
Known are maleic anhydride (MAH) grafted olefin-based polymer resins (MAH-g-PO). Such resins are commonly used as tie layer in multilayer films for food packaging and specialty packaging. The MAH-g-PO tie layer is typically used to bind a polyolefin layer to other layers containing a polar substrate, such as nylon, for example. Currently, MAH-g-PO resin is produced via free radical grafting of maleic anhydride onto polyolefin in a melt blend process.
The art recognizes the benefits of MAH grafted ethylene/α-olefin/diene terpolymer. However, free radical grafting of an ethylene/α-olefin/diene terpolymer is problematic because the resultant MAH-g-ethylene/α-olefin/diene terpolymer has insufficient flow at elevated temperature (i.e., temperature above ambient) and low tensile strength. In addition, when the ethylene/α-olefin/diene terpolymer contains propylene comonomer, undesired chain scission side reactions (resulting in lower molecular weight and higher melt flow rate) occur during free radical grafting.
Desirable would be an ethylene/α-olefin/diene terpolymer with MAH functionalization that is flowable at elevated temperature and is not subject to crosslinking and/or chain scission during the production thereof.
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8 Things to do in Taupo for an Adventurous Weekend
The best things to do in Taupo – from the Tongariro Crossing to natural hot springs, street art, and skydiving.
With a population of just over 20,000, there are a surprisingly large number of things to do in Taupo, a small mountain town surrounded by stunning landscapes in the middle of the North Island of New Zealand.
Living on the Pacific Coast and in a very flat part of California for most of my life, I’m still getting used to the variety of terrain to be found around New Zealand.
I think the only thing missing in Taupo is a proper beach (which isn’t surprising, being that it’s smack dab in the middle of the North Island), but since I was visiting during the winter anyway, I was content to drool over the picturesque snowcapped mountains and bathe in the natural hot springs.
I’ve already expressed my fondness over Taupo in past posts, so it’s no wonder that after spending a few days in the place, I realized just how many things there are to do and see around town.
These are my top recommendations for what to do if you find yourself in Taupo, New Zealand one day.
8 Things to do in Taupo
Go to McDonald’s
You may think it’s laughable that I, as an American, would recommend first and foremost to head to McDonald’s when you get into Taupo, but believe me, you won’t regret it.
Although I didn’t eat at this particular “Mackas”, it is seriously the coolest McDonalds you will come across in New Zealand, if not worldwide. There’s a decommissioned red and silver DC3 plane attached to the restaurant, and rumor has it you can even eat in the pilot seat!
Apparently, the plane was bought in 1985 by the mayor of Taupo at the time to be used as his office, because obviously, those are the kind of offices that appealed to New Zealand politicians in the ’80s…I mean, why not? It later became an icon when attached to the McDonalds in town, because again, why not?
It’s so random, and therefore very Kiwi, but a good place to stop by when you get into town for a photo op, or just to check out the infamous Mackas that has an actual plane nestled into it.
If you’re not after fast food and looking for a really good place to eat after checking out the exterior of McDonald’s, stop by the Crafty Trout. It has awesome locally brewed craft beer and wood-fired pizzas that you can enjoy while cozying up next to the fire.
Try out the Hole in One Challenge
Just a short walk from the McDonalds, there’s the famous Hole in One Challenge to be found along the waterfront, where people passing by can test their luck (or random skill) at making a hole in one from the waterfront into a tiny hole on a small platform floating out on the great Lake Taupo. As you can imagine, it’s a lot harder when there is any kind of wind.
It’s $1 per golf ball, $15 for 18 golf balls, or $20 for 25 golf balls. If you manage to make a hole in one into the red flag, 102 meters away, you can win NZD$10,000!! There are an array of other prizes as well if you make a ball into another one of the flags on the floating platform.
Although we didn’t see any winners during the time we were spectating, there’s usually about one hole in one made every 2 weeks.
Take a dip in the natural hot springs
Probably one of my favorite activities to do in Taupo is discovering one of the many natural hot springs around town. Although some of the hot springs cost money, if you care about facilities like toilets and showers, you can find free ones as well.
We went to the Taupo info center to find out where the closest free hot springs were located and walked to the Spa Thermal Park within 30 minutes from town.
There’s also a lake right beside and connected to the hot springs, so if you feel yourself getting too hot, just swim towards the lake and you’ll find refreshing ice cold water or a happy medium between the two.
It felt wonderful swimming in the hot springs. As dusk started to fall in the brisk Taupo setting, the mini waterfall beat at our backs and the minerals soaked in around our skin. It was a great way to relax from the long bus ride and enjoy our first night in Taupo before hiking the Tongariro Crossing the next morning.
Tackle the Tongariro Alpine Crossing
I recently wrote about hiking the Tongariro Crossing and how it’s considered the best one day hike in New Zealand (and in the top 10 worldwide!), and I cannot stress enough how much of an amazing experience it was. It’s by far one of my favorite things to do in Taupo.
Whatever season you’re in when you visit Taupo, the Tongariro Crossing is a doable activity, heck, I did it in winter and ended up having a beautiful day up on the snowy mountains. As long as there’s not a major snowstorm coming through the crossing you should be just fine, although it’s highly recommended to have a guide in the winter, unless you’re some kind of snow ninja.
Photo cred: Mats Blaauboer
The hike is not for the faint of heart, there are a lot of steep inclines and kilometers to be walked, and you have to be a reasonably fit person to finish the hike in the course of a day. With that said, DO IT! Go do it now, it is the hardest hike I’ve been on and I don’t regret taking on that challenge for a moment.
Visit Huka Falls
Huka Falls is a boisterous example of the pure strength of hydro power. More than 220,000 liters of water flows over the falls per second, say what!? Needless to say, you would not want to try your luck floating down these seemingly short falls, in boat, floatie, or person – you will be crushed.
The incredibly swift water comes from the fact that the Waikato River passes through a narrow ravine of hard volcanic rock before shooting over the falls. There’s even a footbridge directly over the river, just before the falls, so you can feel the full impact of how fast that water is moving beneath you.
Huka is the Maori word for foam, which is fitting if you see the falls up close, it looks like one foamy whirlpool below the drop.
There are a few different trails to hike alongside the falls as well and even a jet boat ride that takes you to the base of the falls, if you want to experience them to the fullest.
Wander around Craters of the Moon
Craters of the Moon is a bit of a tourist trap, but with the admission price at only $8, I was willing to bite and check it out anyway. If you’re really into geothermal activity, this place is the right place for you. If you just like seeing steam come up from the earth and hot mud pools spit up bubbles, then it’s also pretty cool.
It’s a fairly large park, actually the largest geothermal field in New Zealand, and you’re able to follow the steam vents around a circular path and back out to the entrance. Although not as exciting as some of the other activities Taupo has to offer, it was still an interesting and unusual place to check out and spend a good hour or so.
Go skydiving or bungy jumping
Taupo is the skydiving capital of THE WORLD. It’s no surprise, with the epic landscape, glistening mountains in the distance, and beautiful blue lake down below. I knew if I wanted to skydive, I had to do it here. I finally got up the nerve my last day in Taupo, and it was such a scary and freeing experience, I wrote a whole post about it here.
I still can’t get over the ethereal feeling I had under the canopy after free fall, and the giddiness I felt in jumping out of a plane.
Bungy jumping is also a huge activity in Taupo, although not as popular as it is in the adventure capital of Queenstown. As we were walking to the hot springs on our first day in Taupo, we stopped by to watch the bungy jumpers on the way.
The Taupo bungy is over a gorgeous cliff out on a beautiful lake, not a bad place to throw yourself off a platform, if I do say so myself.
Check out the street art around town
I was pleasantly surprised with the amount of street art I found around Taupo. For some reason, I don’t usually expect small towns to be a hot bed of artistic graffiti, but Taupo blew my socks off with the talented art I saw in back alleys and around town.
For the best pieces of art I found around the city, check out my post on a glimpse at Taupo’s street art here.
I had a feeling I would enjoy my time in Taupo, and the town exceeded any expectations I had coming in. Taupo is cozy, mountainous, clean, and a hell of a lot of fun when you realize all there is to do.
Have you ever been to Taupo? Have I missed anything that you would add to the list of things to do in Taupo?
——————–
Looking for more North Island travel inspiration? Check out these related posts below!
Mimi founded The Atlas Heart to create a community of travelers inspired to see the world. The Atlas Heart is a space where you'll find anecdotes on slow travel, craft beer, outdoor adventures, and all the eccentric bits in between that this world has to offer.
About The Atlas Heart
Thanks for stopping by and welcome to The Atlas Heart! My name is Mimi, I’m an adventurous traveler who loves getting outside and getting to know local cultures through slow travel. My goal is to find a daily dose of beauty everywhere I go, to learn with an open mind, and to live with an Atlas Heart.
After living abroad for five years in Australia, New Zealand, Thailand, Greece, and Portugal I have finally found a home in San Francisco, California. Join me on my journey around the world!
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Here’s what the distributor says about their film: “From Academy Award-nominated screenwriter JOHN LOGAN (Gladiator, The Aviator, Hugo, Skyfall) and acclaimed, Tony Award-winning director MICHAEL GRANDAGE in his feature film debut, comes ‘Genius,’ a stirring drama about the complex friendship and transformative professional relationship between the world-renowned book editor Maxwell Perkins (who discovered F. Scott Fitzgerald and Ernest Hemingway) and the larger-than-life literary giant Thomas Wolfe.
Based on the inspirational biography Max Perkins: Editor of Geniusby Pulitzer Prize winner A. SCOTT BERG, Genius stars COLIN FIRTH as Perkins, JUDE LAW as Wolfe, NICOLE KIDMAN as Aline Bernstein, a costume designer sharing a tumultuous relationship with Wolfe, Laura Linney as Louise Perkins, Max’s wife and a talented playwright, GUY PEARCE as F. Scott Fitzgerald and DOMINIC WEST as Ernest Hemingway.
Finding fame and critical success at a young age, Wolfe was a blazing talent with a larger-than-life personality to match. Perkins was one of the most respected and well-known literary editors of all time.”
Positive—I’ve wanted to see this true life story for some time now, and I wasn’t disappointed. The sepia tones set the scene nicely for NYC in the 1920s when authors are desperate to see their literary babies brought to life in a publisher’s hands, not least Thomas Wolfe, F. Scott Fitzgerald (The Great Gatsby) and Ernest Hemingway. It’s a fascinating look inside the writer’s mind (Jude Law as Wolfe) and how the editing process (Colin Firth as Max Perkins) takes place.
The effect writing a book can have on loved ones, where obsession becomes the watchword, is powerfully portrayed and well acted.
Only one brief scene—is really unnecessary to the plot, insinuating that a steamy sexual act is about to take place, though you see next to nothing. Just as editors edit, so the director edits scenes and this one implied the writer is ecstatic about his new found success after many rejections, and he celebrates by drinking a lot and having sex with his long suffering mistress, Nicole Kidman.
There is a lot of smoking in the movie, but, like the acting, the script is strong, and the story is interesting and made me want to write something, though not 5,000 words a day with a pencil.My Ratings: Moral rating: Better than Average / Moviemaking quality: 5
…Made up largely of conversations—including way too many in which Perkins discusses ways for Wolfe to cut down his prose – the story indulges in long scenes focused on dialogue. When Grandage decides scenes are running on too long, he just changes the location, so exchanges jump from the office to stations, to trains and to the street, but the waffling continues. …
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Q:
Switch from left click to right click to make character walk in the Top Down Example, in UE4?
When creating a new project in the Unreal Engine 4, you can choose to create a project already with some added features and things (Top Down Example):
This project has a character (which I changed its appearance) and it moves around the map when I click the left mouse button:
I looked at the blueprints of the project, but I did not identify where the command is that makes the character walk, so I'd like it to walk when I right-click.
Alls blueprints of project:
M_Cursor_Decal:
TopDownCharacter:
TopDownController:
TopDownGameMode:
The circled blocks of reds are the ones I suspected could be where I should make the modification, but no information on them gave me the certainty that that was where I was supposed to change.
I still do not create projects in C ++ programming mode, because I do not have Visual Studio 2017.
Some links from websites where I spent before coming here and that can help you to help me:
https://docs.unrealengine.com/en-us/Engine/Blueprints
https://www.youtube.com/watch?v=EFXMW_UEDco
The video is 2015, so although it's good, I did not take it so well, because I know that with all this time that has already happened, a lot is different.
Sorry for any translation errors, I'm not English speaker.
A:
Go to Edit => ProjectSettings => Engine => Input => Action Mappings:
1 = Project settings
2 = Input
3 = Action Mappings
4 = Right mouse button (That's where the modification is made)
Link that helped me:
https://docs.unrealengine.com/en-US/Gameplay/Input
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All relevant data are within the manuscript and its Supporting Information files.
Introduction {#sec001}
============
Marine environments, and coastal regions in particular, are heavily exposed to anthropogenic stressors. These fragile habitats are affected by vigorous development, in the form of ports and factories, as well as by tourism and leisure enterprises. Other factors of disturbance result from land-based contaminants reaching the sea via rivers and floods \[[@pone.0215005.ref001]--[@pone.0215005.ref003]\]. There are various tools currently in use by which to monitor the health of marine ecosystems. Routine chemical assessment is a common practice worldwide; however, it provides a limited 'snapshot' of the specific time point of measurement. In addition, chemicals are not always identified nor are they concentrated at a level that enables detection. Thus, sporadic events, such as sewage spills, may easily be missed. Furthermore, chemical analysis does not provide information concerning the effect on living biota \[[@pone.0215005.ref004],[@pone.0215005.ref005]\]. Another tool used for quality assessment is biocenotic indexes, i.e. the presence or absence of indicator species within a community, and their tolerance to pollutants. However, this tool is limited to use for well-studied and defined communities for which there are sufficient data available. It is not able to provide an early warning of disturbance \[[@pone.0215005.ref004],[@pone.0215005.ref006]\].
Proteins are the immediate effectors in biological processes and provide an accurate account of the current state of a cell, tissue, or the entire organism. In many cases the response to a change in environmental conditions will be reflected in the response of a protein network, independent of gene expression or translation of new mRNA transcripts. Thus, proteins serve as excellent biomarkers directly linking impact and physiological response \[[@pone.0215005.ref004],[@pone.0215005.ref007]\]. Proteomics-based approaches are capable of illuminating protein differential expression or alterations without previous knowledge of toxicity mechanisms, although a profound knowledge of the mechanism of action of an enzyme array, for instance, may be necessary \[[@pone.0215005.ref004],[@pone.0215005.ref008],[@pone.0215005.ref009]\]. Applying 'shotgun' proteomics (also termed 'bottom-up' proteomics), a method in which peptide fragments are correlated to protein sequences in the database \[[@pone.0215005.ref010]\], enables the identification of proteins or their homologues, pointing at possible expression pathways as a result of a changed environment, and potential new protein biomarkers. This emerging field of environmental proteomics further allows for the integration of non-model organisms that comprise the majority of the environment and are the most ecologically relevant \[[@pone.0215005.ref009],[@pone.0215005.ref011],[@pone.0215005.ref012]\]. Shotgun proteomics has been applied in studies focusing on the response of mussels to pharmaceuticals \[[@pone.0215005.ref013]\], crustaceans to heavy metals \[[@pone.0215005.ref014]\] and oysters to ocean acidification \[[@pone.0215005.ref015]\].
Ascidians (Chordata, Ascidiacea) are sessile filter-feeding organisms closely related to vertebrates \[[@pone.0215005.ref016]\], able to filter even minute particulate matter \[[@pone.0215005.ref017],[@pone.0215005.ref018]\]. They are well known for their ability to accumulate heavy metals \[[@pone.0215005.ref019],[@pone.0215005.ref020]\] and to thrive in eutrophic (nutrient-rich), and polluted environments \[[@pone.0215005.ref021]\]. There are nearly 3000 species of ascidians inhabiting all marine habitats from shallow water to the deep sea, with some species, in particular invasive species, creating large aggregations on artificial structures \[[@pone.0215005.ref021],[@pone.0215005.ref022]\]. Their ability to bioaccumulate a variety of contaminants and tolerate large fluctuations in salinity, temperature, and even pollution \[[@pone.0215005.ref023]\], make them ideal candidates for bio-monitoring a wide variety of marine habitats on a global scale, and for investigating the pathological effects of a variety of stressors. In addition, annotated genome sequences are available for key ascidian species such as *Ciona intestinalis* \[[@pone.0215005.ref024]\] and *Botryllus schlosseri* \[[@pone.0215005.ref025]\], further facilitating the use of changes in gene expression in environmental toxicology. The current study focuses on the identification of proteins that could serve as biomarkers of a stress response, with emphasis on the differentially-expressed proteins that may play a role in cellular stress-response mechanisms. Our analysis enabled us to identify a large number of significantly differentiated proteins expressed in the studied species, some of which are known to be stress-related, that could be further established as biomarkers and used for assessing the health of coastal marine environments.
Materials and methods {#sec002}
=====================
Ethics statement {#sec003}
----------------
This study was carried out in strict accordance with the Israeli law. The Israeli Nature and National Parks Protection Authority provided permission to collect the animals and work at all field locations (permit 2017/41626).
Ascidians and sampling sites {#sec004}
----------------------------
*Microcosmus exasperatus* Heller 1878 (Stolidobranchia; Pyuridae) ([Fig 1A](#pone.0215005.g001){ref-type="fig"}) has a wide global distribution and is very common in tropical and sub-tropical waters. It is considered a non-indigenous species to the Eastern Mediterranean, commonly found along the Israeli Mediterranean coast on both artificial and natural substrates \[[@pone.0215005.ref026],[@pone.0215005.ref027]\], known to reproduce in this area year-round, except for the winter months (January-February) \[[@pone.0215005.ref028]\]. This species was sampled from five sites along the Mediterranean coast of Israel ([Fig 2A](#pone.0215005.g002){ref-type="fig"}), representing different levels of disturbance: The two northern sites, Akko and Haifa, are known as the most perturbed sites due to years of exposure to various pollutants from multiple sources and due to their proximity to Haifa Port. Dor-Habonim is considered as the least disturbed site and was previously selected as a control site in the National Monitoring Program of Israel's Mediterranean waters \[[@pone.0215005.ref029]\]. The two southern sites, Palmachim and Ashqelon, are also known as disturbed sites due to being exposed to episodic pollution events from wastewater and sewage; the Dan Region Wastewater Treatment Plant (Shafdan), Soreq River estuary, and sewage spills from Gaza. Ascidians in Palmachim and Ashqelon were collected from natural reefs and in Akko, Haifa, and Dor-Habonim from patches of rocks scattered on the seabed. Due to technical limitations, sampling of *M*. *exasperatus* from Haifa site was executed a month later than from other sites (June vs. July) and as a consequence the temperature in Haifa at time of sampling was higher ([Table 1](#pone.0215005.t001){ref-type="table"}).
{#pone.0215005.g001}
{ref-type="table"}.](pone.0215005.g002){#pone.0215005.g002}
10.1371/journal.pone.0215005.t001
###### Sampling data of *M*. *exasperatus* and *P*. *mytiligera* collected for protein profiling.
n = 15 (per site). SW = seawater.
{#pone.0215005.t001g}
Species and location Date Site Coordinates Depth (m) SW temperature (°C) [\*](#t001fn001){ref-type="table-fn"}
---------------------------------------- ------------------- -------------------------------- -------------------------------- ----------- -----------------------------------------------------------
*M*. *exasperatus* (Mediterranean Sea) 01/06/2017 Akko 32°55\'15.0\"N, 35° 4\'27.0\"E 1--3 22.8
18/07/2017 Haifa 32°49\'58.0\"N, 34°59\'26.0\"E 2--3 29.8
02/06/2017 Dor-Habonim 32°38\'40.0\"N, 34°55\'26.5\"E 1--3 23.3
08/06/2017 Palmachim 31°55\'39.0\"N, 34°41\'38.0\"E 4--7 23.6
08/06/2017 Ashqelon 31°39\'57.0\"N, 34°32\'31.0\"E 5--8 23.6
*P*. *mytiligera* (Red Sea) 20/6/2017 The Princess Beach 29°29\'43.3\"N, 34°54\'28.1\"E 5 24.8
20/6/2017 Tamar Reef 29°30\'44.2\"N, 34°55\'34.0\"E 6--7 24.8
19/6/2017 The Coral Nursery 29°32\'27.0\"N, 34°58\'24.0\"E 13--15 24.9
19/6/2017 Kisoski 29°32\'50.5\"N, 34°57\'15.4\"N 13--15 24.9
\* For the Mediterranean sampling sites, seawater temperature data were derived from the Hadera IOLR (Israel Oceanographic and Limnological Research) monitoring station (sensor at 12 m depth, mean of maximum and minimum recorded temperatures), while for The Red Sea sampling sites temperature data were derived from the IUI (The Inter-University Institute for Marine Sciences) monitoring station (sensor at 1 m depth, temperature recorded at 12 pm).
*Polycarpa mytiligera* (Stolidobranchia; Styelidae) ([Fig 1B](#pone.0215005.g001){ref-type="fig"}) is an abundant solitary ascidian native to the Red Sea \[[@pone.0215005.ref030],[@pone.0215005.ref031]\]. Similar to *M*. *exasperatus*, it reproduces nearly year-round, excluding the coldest winter months (January-February) \[[@pone.0215005.ref032]\] It was collected from four different sites along the Israeli coastal stretch of the Red Sea, near the city of Eilat ([Fig 2B](#pone.0215005.g002){ref-type="fig"}). Each of these sites is characterized by different water conditions, current regime, and proximity to terrestrial runoff. The two northern sites in the Red Sea, the Nursery and Kisoski, are devoid of a natural coral reef and are located in close proximity to the city of Eilat; whereas the southern sites, Tamar Reef and Princess Beach, are part of a coral-reef marine reserve, with Tamar Reef being an artificial underwater construction. Unlike the other three sites, the Nursery site is a floating structure in the water column, 10 meters above the sea floor. In addition, the northern coasts and sites of Eilat have been exposed to multiple eutrophication events over the years \[[@pone.0215005.ref033],[@pone.0215005.ref034]\]. Located at the site of the former fish cages, which had a major impact on nutrient enrichment \[[@pone.0215005.ref034],[@pone.0215005.ref035]\], the Nursery site is exposed to freshwater runoff, especially at times of flooding, and is located in proximity to an open canal (the "Kinet" canal) that carries brackish water and terrestrial solvents into the sea \[[@pone.0215005.ref033]\].
At each site, 15 animals were sampled by SCUBA or snorkeling at depths ranging from 1 to 15 m. Sampling data are given in [Table 1](#pone.0215005.t001){ref-type="table"} and [Fig 2](#pone.0215005.g002){ref-type="fig"}.
Both species were detached from the substrate with utmost care by hand or, when the ascidian was attached to a crevice, using a knife. However, whereas detaching *M*. *exasperatus* from the substrate was fairly easy to perform, detaching *P*. *mytiligera* was more difficult.
Upon removal from the water, all individuals were transferred within a few minutes to liquid nitrogen, in order to minimize a biased protein expression profile. As a general rule, batches of three individuals were wrapped together in aluminum foil prior to freezing, since direct flash-freeze had been shown to result in shattered specimens. Samples were transferred frozen to the laboratory at Tel Aviv University, either on dry-ice or in liquid nitrogen, where they were kept at -80 °C until processed. At the lab, the tunic of each animal was removed on ice for minimal thawing, and the inner bodies of every three ascidians were pooled into one sample. Removing the tunics was essential since the tunics of both species were covered with epibionts which could have affected the protein expression profile. When specimens were found shattered, making it difficult to distinguish between individuals, their fragments were jointly weighed and divided into equal weight samples. In total, each site produced five replicas (i.e. N = 5) of three individuals.
Protein extraction {#sec005}
------------------
All protein-related procedures were performed at the Smoler Proteomics Center, Technion, Haifa. Tissue was homogenized at a ratio of 1 g tissue: 3 mL buffer, composed of: 8 M Urea, 100 mM Ammonium bicarbonate, 10 mM DTT. Prior to digestion, 2 mL homogenates were sonicated (to break up any remaining tissue chunks) and briefly centrifuged to pellet insoluble debris. Protein amount was estimated using Bradford protein assay. 20 g of protein from each sample were reduced with 10 mM DTT (60 °C for 30 min.) and modified with 8.8 mM iodoacetamide in 400 mM ammonium bicarbonate (in the dark, at room temperature for 30 min.), followed by digestion overnight at 37 °C in 2 M Urea, 25 mM ammonium bicarbonate with modified trypsin (Promega) at a 1:50 enzyme-to-substrate ratio. A second digestion with trypsin was done for 4 hours at 37 °C. Protein extraction and trypsinization of the different samples were done at the same day to reduce technical variability.
Protein analysis by LC-MS/MS {#sec006}
----------------------------
The tryptic peptides were desalted using C18 tips (Harvard), dried and then re-suspended in 0.1% Formic acid. Two μg of the resulted peptides were resolved by reverse-phase chroμatography on 0.075 X 180-mm fused silica capillaries (J&W) packed with Reprosil reversed phase material (Dr Maisch GmbH, Germany). The peptides were eluted with a linear gradient of acetonitrile in 0.1% formic acid in water, as detailed: 5 to 28% for 180 min., 28 to 95% for 15 min., and 95% acetonitrile for 25 min. Flow rate was 0.15 μL/min. Mass spectrometry was performed by Q Exactive plus mass spectrometer (Thermo Scientific) in a positive mode (m/z 350--1800, resolution 70,000) using repetitively full MS scan followed by collision induces dissociation (HCD) of the ten most dominant ions (\> 1 charges) selected from the first MS scan. A dynamic exclusion list was enabled with exclusion duration of 20 sec. LC-MS/MS analyses of the different samples were done as one set on the same week with washes and empty runs between them to avoid contaminations.
The MS raw data from all the biological repeats were analyzed using the MaxQuant software (version 1.5.2.8) for peak picking and quantitation, followed by identification using the Andromeda search engine vs. the Ascidiacea database- (Taxon ID 7713, 56, 748 proteins) and *Ciona intestinalis* subsets of the uniprot database, and against Stolidobranchia subset of NCBI database, with mass tolerance of 20 ppm for the precursor masses and fragment ions. Methionine oxidation was set as variable post-translational modifications, and carbamidomethyl on cysteine as a static one. Minimal peptide length was set to six amino acids, and a maximum of two miscleavages was allowed. To eliminate identifications from the reverse database and common contaminants, peptide- and protein-level false discovery rates (FDRs) were filtered to 1% using the target--decoy strategy. Data were quantified by label free analysis using the same MaxQuant software, based on extracted ion currents (XICs) of peptides from each LC/MS run, enabling quantitation of all the peptides identified in any experiment. Normalization was done using the LFQ algorithm base on similar protein profile \[[@pone.0215005.ref036]\]. Since several organisms were included in the database, homolog proteins were clustered as one protein group, and only proteins that were identified with at least two peptides were listed. As the data need to be in a linear scale, in order to apply the statistical tests, they were transformed to log2 intensities.
Statistical analysis {#sec007}
--------------------
Significant differences in protein expressions between sites of each sea were tested via two-sample *t*-tests with Permutation-based FDR (with 250 randomizations, threshold value = 0.05).
Additionally, hierarchical clustering-based heatmaps were produced in order to easily identify up or down-regulated proteins. Both analyses were performed with Perseus 1.6.0.7 software (Mathias Mann's group). Only proteins that met a minimum of 60% valid value criteria (in our study three out of five repeats were needed to provide an intensity reading) were used for the analyses.
In order to reveal any grouping of sites, a principal component analysis (PCA) of significantly expressed protein profiles was undertaken, followed by analysis of similarity (ANOSIM). A similarity percentages analysis (SIMPER) was used to indicate percentage of similarity/dissimilarity between sites and to identify the proteins whose expression contributed the most to the observed dissimilarities. PCA, ANOSIM and SIMPER were performed with the PRIMER-E software, version 6. Significant differences were accepted when *p-*values were \< 0.05.
Results and discussion {#sec008}
======================
Identified proteins and biological functions {#sec009}
--------------------------------------------
Overall, 431 proteins were identified from *M*. *exasperatus* sampled at the Mediterranean sites, of which 403 had met the 60% minimum valid value criteria and were used for analyses. Of these, 193 proteins showed significant differential expression between sampling sites (*p* \< 0.05, two-sample *t*-test) ([Fig 3A](#pone.0215005.g003){ref-type="fig"}; for a complete list of significant proteins see [S1 Table](#pone.0215005.s001){ref-type="supplementary-material"} and [S1 File](#pone.0215005.s004){ref-type="supplementary-material"}) and were assigned to several main functional categories: protein synthesis, cytoskeleton, energy metabolism, response to stress, metabolic-related and proteolysis-related ([Fig 4A](#pone.0215005.g004){ref-type="fig"}). Over 60% of the significant proteins were associated with protein synthesis (39%) and cytoskeleton (22%). Since data on the proteome of ascidians in the field are scarce, this finding (i.e. the plethora of proteins with which to experiment) is of high importance. In addition to proteins such as peroxiredoxin, hsp70 and 14-3-3, all of which are stress-related and were previously suggested to serve as universal indicators for cellular stress \[[@pone.0215005.ref037],[@pone.0215005.ref038]\], other proteins associated with various biological functions may also qualify as biomarkers. For example, the synthesis of ribosomal proteins, which comprised most of the proteome in our study, is affected by the cell's environment, repressed in response to stress, and induced under favorable conditions \[[@pone.0215005.ref039]\]. In addition to their role in ribosome biogenesis and cellular development \[[@pone.0215005.ref040]\], many ribosomal proteins have critical roles in various cellular functions, such as DNA repair \[[@pone.0215005.ref041]\], apoptosis \[[@pone.0215005.ref042]\], and cellular differentiation \[[@pone.0215005.ref043]\]. Due to their relatively high abundance in the proteome of the sampled *M*. *exasperatus*, correlating changes in ribosomal proteins expression to various environmental factors/stressors could establish them as valuable biomarkers.
{ref-type="supplementary-material"} and [S2](#pone.0215005.s002){ref-type="supplementary-material"} Tables. For (A): Ak = Akko, Ha = Haifa, DH = Dor-Habonim, Pa = Palmachim, As = Ashqelon. For (B): PB = Princess Beach, TR = Tamar Reef, NSY = The Nursery, KSI = Kisoski. For both (A) and (B) n = 5, ±SE.](pone.0215005.g003){#pone.0215005.g003}
{#pone.0215005.g004}
In contrast to the large number of identified proteins from *M*. *exasperatus*, only 126 were identified from *P*. *mytiligera* individuals sampled at the Red Sea. Of the 70 proteins that met the 60% minimum valid value criteria, 13 proteins significantly differentiated exclusively between the two southern sites (Tamar artificial reef and the natural reef of Princess Beach) and the northern unique site (Nursery) in the northern zone (*p* \< 0.05, two-sample *t*-test) ([Fig 3B](#pone.0215005.g003){ref-type="fig"}, [S2 Table](#pone.0215005.s002){ref-type="supplementary-material"} and [S2 File](#pone.0215005.s005){ref-type="supplementary-material"}). These proteins were assigned to four main functional categories (response to stress, protein synthesis, cytoskeleton, and metabolic-related), with most proteins (77%) associated with the former two categories, five proteins in each ([Fig 4B](#pone.0215005.g004){ref-type="fig"}). Although these species belong to different families (Pyuridae and Styelidae), they are of the same order (Stolidobranchia) and the same database was used for identification (Ascidiacea). The obtained results are therefore surprising, since fewer proteins were identified for *P*. *mytiligera*. However, it should be noted that because the animals were sampled from different habitats and under different conditions, we do not expect to obtain the same results. Moreover, the additional effort required to detach *P*. *mytiligera* from its substrate may also account for a different pattern of protein expression. The list of significant proteins nonetheless includes potential biomarkers for various stressors such as actin \[[@pone.0215005.ref044]\], 14-3-3 proteins \[[@pone.0215005.ref037]\], and Hsc71 \[[@pone.0215005.ref045]\].
Distinction between sites/zones {#sec010}
-------------------------------
PCA ([Fig 5A](#pone.0215005.g005){ref-type="fig"}), and hierarchical clustering-based heatmap ([Fig 6](#pone.0215005.g006){ref-type="fig"}) of the significant proteins showed a clear separation between northern (Akko, Haifa and Dor-Habonim) and southern sites (Palmachim and Ashqelon) of the Mediterranean.
{#pone.0215005.g005}
{#pone.0215005.g006}
Most apparent were clusters of upregulated proteins, with 68 upregulated proteins in the northern sites and 80 in the southern sites. The majority of proteins in the northern sites were cytoskeletal proteins (26, 38%) and energy metabolism related (22, 32%), whereas most proteins in the southern sites were assigned to protein synthesis (53, 66%). ANOSIM for the two zones (i.e. north vs. south), confirmed that they were significantly dissimilar (R = 0.868, *p* = 0.001). ANOSIM for sites (global R = 0.731, *p* \< 0.01) showed that the most dissimilar sites were Akko and Ashqelon (pairwise test R = 1, *p* = 0.008). Although less apparent from the PCA, ANOSIM revealed significant dissimilarities among northern sites too, with Haifa being most dissimilar to the other two sites; Akko and Haifa (R = 0.836, *p* = 0.008), Haifa and Dor-Habonim (R = 0.904, *p* = 0.008) and Akko and Dor-Habonim (R = 0.616, *p* = 0.008). This difference among northern sites was more noticeable in the heatmap, where a cluster of 31 upregulated proteins distinguished Haifa from Akko and Dor-Habonim ([S1 Fig](#pone.0215005.s003){ref-type="supplementary-material"}). A distinction between zones/sites was demonstrated for *P*. *mytiligera* too, with noticeable differences between the southern sites and the Nursery site in the north ([Fig 5B](#pone.0215005.g005){ref-type="fig"}). This can be seen also in the hierarchical clustering-based heatmap ([Fig 7](#pone.0215005.g007){ref-type="fig"}), in which roughly two clusters of proteins differentially up-regulated between the northern Nursery site and the two southern sites. Although *t*-test results indicate that expression of proteins at Kisoski was not significant, ANOSIM (global R = 0.527, *p* \< 0.01) suggests that protein profiles of organisms sampled at this site were slightly dissimilar to those sampled at Princess Beach and Tamar Reef (southern sites, R = 0.376, *p* = 0.016 and R = 0.62, *p* = 0.008, respectively).
{#pone.0215005.g007}
Seven proteins were up-regulated at the Nursery site vs the two southern sites, belonging to the following categories: Cytoskeleton (1, 14%), response to stress (3, 43%), and protein synthesis (3, 43%). SIMPER analysis shows that the proteins which contributed most to the observed differences between the southern sites and the Nursery were catalase (17.8%) and cytochrome c oxidase subunit 2 (13.2%), both response to stress proteins.
The observed dissimilarities between sites and zones in both seas are most intriguing. Although Dor-Habonim is considered a relatively clean site and was previously selected as a control site in the National Monitoring Program of Israel's Mediterranean waters \[[@pone.0215005.ref029]\], the PCA indicated that protein profiles of *M*. *exasperatus* sampled at this site are closely related to those of Akko and Haifa. Since these sites are located in Haifa Bay, a region which has been polluted by a variety of contaminants for many years, greater dissimilarity between clean and polluted sites would be expected. The present finding, however, indicates that other factors may be involved: e.g. food availability, as well as specific combinations of hydrological conditions and various pollutants \[[@pone.0215005.ref046],[@pone.0215005.ref047]\], could have masked the effect of contaminants on organisms from Haifa Bay and have resulted in what seems to be a mismatch. Further investigation of the complex of conditions at each site is required in order to better elucidate these results. Nevertheless, although subtle, significant differences were observed among the northern sites, with Haifa being most dissimilar to the others. A possible explanation for the dissimilarity between Haifa and the two other northern sites can be found in the large proportion of stress-related proteins within the cluster of 31 upregulated proteins (14, 45%), including hsp90s alpha and beta, hsp70 and endoplasmin-like. Heat-shock proteins, which are known to be very important to the survival of cells and organisms in response to various stressors \[[@pone.0215005.ref048]\], may suggest that organisms of this habitat may be exposed to different or additional stressors than those at Akko and Dor-Habonim. Interestingly, since sediments in both Akko and Haifa have been reported as intermediately to heavily polluted by various contaminants such as toxic organic substances (e.g. PCB's) and heavy metals \[[@pone.0215005.ref049]\], it should be expected that sedentary organisms residing in both habitats and exposed to similar pollutants would exhibit similar protein profiles. The observed differences between sites could be the result of expansion of the Haifa Port project which has been operating for the last couple of years, and the extensive dredging works involved. Due to the proximity of the Haifa site to the port, organisms in this habitat may be exposed to an increased suspended sediment load, and hence to higher concentrations of contaminants. A possible bias in our results may be due to the temperature difference between Haifa and the other sites: since seawater temperature at Haifa was 6 °C higher than that at the other sites at the time of sampling ([Table 1](#pone.0215005.t001){ref-type="table"}), expression levels of some proteins, such as hsp70, could have been affected by the temperature difference. However, since not all proteins are temperature-dependent and not all members of the hsp family are affected by natural temperature fluctuations (e.g. hsp90 \[[@pone.0215005.ref050]\]), we suggest that the observed differences were probably associated with the heavy pollution and other perturbations to which organisms at this site have been continuously exposed.
With respect to the clear differences between protein profiles of ascidians from the northern and southern sites in both the Mediterranean and the Red Sea, several possible explanations exist: first and foremost, the special feature of the Nursery site in the Red Sea, floating in the water column high above the seabed. This is a probable cause of the observed differences between the proteome of *P*. *mytiligera* from this site and those from Princess Beach and Tamar Reef. Environmental factors, including mechanical disturbance, have been found to impact cell metabolism and also the expression of stress proteins such as hsp70 \[[@pone.0215005.ref051],[@pone.0215005.ref052]\]. The investment of less energy by ascidians in removing suspended sediment matter at the Nursery site could have resulted in the observed differences between sites. This may also explain the differences between the protein profiles of Dor-Habonim and the two southern sites of the Mediterranean: whereas *M*. *exasperatus* in Dor-Habonim was collected from substrate in close proximity to the sediment, the ascidians in Palmachim and Ashqelon were sampled from natural reefs, considerably higher above the sediment.
The lack of a natural coral reef at the northern sites of the Red Sea, unlike the southern sites, which are located within a coral-reef marine reserve, is reflected also in their sediment: in the northern sites, sediment particles are smaller and of terrestrial origin, while sediment at the southern sites is composed of larger particles of marine biogenic origin \[[@pone.0215005.ref033]\]. Although the northern sites have experienced nutrient enrichment events during past years, the findings of the national monitoring program did not reveal any contaminants as such (e.g. terrestrial runoff) in the year preceding our sampling \[[@pone.0215005.ref033]\]. Thus, the observed differences in protein expression between the northern and southern sites of the Red Sea could not be attributed to such events.
In regard to the Mediterranean sites, although both northern (Akko and Haifa) and southern (Palmachim and Ashqelon) sites are subjected to various pollutants, protein profiles were significantly different. It is reasonable to assume that this finding reflects the differences between the substances the ascidians are exposed to in the two regions, as well as the pattern of exposure. Whereas *M*. *exasperatus* in the Bay of Haifa (i.e. Akko and Haifa sites) are chronically exposed to a large number of toxicants, as this area is in close proximity to the largest petrochemical plants in Israel, as well as to one of its largest ports, ascidians in Palmachim and Ashqelon are exposed intermittently to pollutants from other sources, mostly from wastewater and sewage.
Although significant, SIMPER analysis indicated that the observed dissimilarities were not greater than 1.1% in the Mediterranean and 3.2% in the Red Sea. We suggest that this is due to our processing and analyzing whole organisms rather than selected tissues. As was previously shown, the expression profile and magnitude of specific proteins in response to known contaminants may differ between tissues. Lopez et al.'s \[[@pone.0215005.ref053]\] study on the proteomic response to elevated temperatures in ovaries of the solitary ascidian *Ciona intestinalis*, revealed that many of the identified proteins were stress-related. Chora et al. \[[@pone.0215005.ref054]\] showed that actin, an important cytoskeletal protein, was expressed differently in the gills and digestive glands of the clam *Ruditapes decussatus* when exposed to cadmium. It is probable that if we had analyzed selected tissues of the studied ascidians, the relative expression of specific proteins would have been different and dissimilarities would have been more pronounced.
Discussing the obtained results would not be complete without considering possible confounding factors that could have affected the presented protein profiles. In mussels, for instance, factors such as age, reproduction stage, and food availability were shown to induce various biological responses that could be misinterpreted as an exposure effect \[[@pone.0215005.ref055],[@pone.0215005.ref056],[@pone.0215005.ref057]\]. Similarly, in the present study, the obtained site-specific differences in protein expression levels could not be linked to a specific stressor, endogenic or exogenic. Nevertheless, considering the differences in pollution levels that the sampling sites are exposed to and the major sources of the pollutants (i.e. anthropogenic sources), these findings demonstrate the potential of ascidians to function as bioindicators, possibly of anthropogenic stressors.
That being said, since the combined effect of various stressors on ascidians in their natural environment has not been investigated to date, the subject merits further study. Isolated populations (e.g. due to geographical distance) could also have led to biased conclusions, with site-specific selection pressures possibly yielding differences in protein profiles. Nevertheless, both species used in this study are solitary ascidians that spawn their gametes into the water column. As the gametes may spend a period of hours to several days in the water until fertilization, and the developing embryos also drift in the water for several hours, the distance from release of gametes to settlement of the larvae may reach hundreds of kilometers. A similar pattern of dispersal was previously demonstrated for *Pyura gibbosa gibbosa* \[[@pone.0215005.ref058]\], a solitary ascidian of the same family as *M*. *exasperatus* (Pyuridae), for which five populations of *P*. *gibbosa gibbosa* located in a distance of up to 215 kilometers apart from each other were found to support a single panmictic population. Considering the limited geographical distance between the sampling sites in our study, 147 km and 17 km, Mediterranean and Red Sea respectively, and a dominant south-to-north current in the Mediterranean (similar to the unidirectional current that exists in Ayre et al.'s (1997) study region, which was assumed to link between sampling sites), it is unlikely that either of the species has established isolated populations within the limited geographic distribution. A future population genetic study should clarify whether these populations are panmictic or reproductively isolated.
Conclusions {#sec011}
===========
In the current study, we demonstrate how ascidians from different localities exhibited significantly differentiated protein profiles, which could reflect the conditions of the environments they inhabit. The obtained proteomes distinguished between sites, providing essential information regarding the current environmental health status at these areas, thus providing us with a more accurate estimation of the health of a specific site. Our results show evidence for differentially-expressed proteins, which are stress-related or previously suggested to serve as universal biomarkers in other studies. We find the large number of identified differentiated proteins which we obtained advantageous for an additional reason: Due to the complexity of pollutants in the environment and the wide spectrum of proteomic responses these may induce in organisms \[[@pone.0215005.ref007]\], a combination of well-studied proteomic biomarkers, rather than a single one, could be more efficient in identifying the stressors involved and would improve the environmental hazard assessment (reviewed by Trapp et al. \[[@pone.0215005.ref004]\]).
These findings corroborate the potential of the proteomic approach for monitoring coastal marine environment health, with ascidians as bioindicators. Furthermore, the study also presents the first application of ascidians as bioindicators in coral reefs, in regard to the southern sites of the Red Sea. Additional controlled exposure experiments and field studies will contribute to our understanding of the stress-response pathways involved, and will enable validating the potential of proteins of interest to serve as qualified biomarkers.
Supporting information {#sec012}
======================
###### A complete list of significantly differentiated proteins expressed in *M*. *exasperatus*, sampled from the Mediterranean.
When possible, molecular function was specified for uncharacterized proteins for which a name was not specified in UniProt. Abbreviations of the proteins which appear in the column chart ([Fig 3A](#pone.0215005.g003){ref-type="fig"}) are specified in parentheses, bold font, following their full name. na = not available.
(DOCX)
######
Click here for additional data file.
###### All significantly differentiated proteins expressed in *P*. *mytiligera*, sampled from the Red Sea.
Abbreviations of the proteins which appear in the column chart ([Fig 3B](#pone.0215005.g003){ref-type="fig"}) are specified in parentheses, bold font, following their full name.
(DOCX)
######
Click here for additional data file.
###### Heatmap showing a cluster of 31 upregulated proteins in *M*. *exasperatus* from Haifa.
Protein names are listed in [S1 Table](#pone.0215005.s001){ref-type="supplementary-material"}. na = not available, assigned to proteins for which data on their biological functions could not be obtained.
(TIF)
######
Click here for additional data file.
###### The analysis data of all significantly differentiated proteins expressed in *M*. *exasperatus*.
Intensity values (LFQ) were produced from raw MS data using MaxQuant software, and significant differences between sites were tested via two-sample *t*-tests, using Perseus software.
(XLSX)
######
Click here for additional data file.
###### The analysis data of all significantly differentiated proteins expressed in *P*. *mytiligera*.
Intensity values (LFQ) were produced from raw MS data using MaxQuant software, and significant differences between sites were tested via two-sample *t*-tests, using Perseus software.
(XLSX)
######
Click here for additional data file.
We would like to thank N. Paz for the editorial assistance. We are grateful for the generous help of M. Harel of the T. Geiger Lab at the Sackler Faculty of Medicine at Tel Aviv University, for her guidance and good advices with the analyses. We thank S. Ben-Lulu of the Smoler Protein Research Center at the Technion for her guidance and patience. Also, we would like to thank the Inter-University-Institute in Eilat staff and scientists, and the Steinhardt Museum of Natural History.
[^1]: **Competing Interests:**The authors have declared that no competing interests exist.
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minipile
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NaturalLanguage
|
mit
| null |
Q:
Can I say that domain doesn't exist if Socket.Connect fails?
I want to check if some email domain exist.
When Dns.GetHostEntry(domain) throw an exception I know for sure that the domain doesn't exist.
Can I say that if Dns.GetHostEntry(domain) succeeded then the domain does exists or even if Dns.GetHostEntry(domain) succeeded that doesn't mean (yet) that domain exists?
Can I say the same when s.Connect fails to connect? I mean if connect throw an exception can I say that such domain doesn't exist?
If (1) is true, so in order to check if domain exists (1) will be enough, right?
public static bool Lookup(string domain)
{
if (domain == null) throw new ArgumentNullException("domain");
try {
IPHostEntry ipHost = Dns.GetHostEntry(domain);
var endPoint = new IPEndPoint(ipHost.AddressList[0], _dnsPort);
return Transfer(endPoint);
}
catch (SocketException ex)
{
++attempts;
}
return false;
}
public static bool Transfer(IPEndPoint endPoint)
{
int attempts = 0;
while(attempts <= _attempts)
{
try
{
var s = new Socket(endPoint.AddressFamily, SocketType.Stream, ProtocolType.Tcp);
s.Connect(endPoint);
}
catch (SocketException ex)
{
++attempts;
}
finally
{
s.Close();
}
}
}
A:
When trying to connect with a socket you're saying a few things:
There is a network path from my machine to the target machine
The network path is clear of firewall and other restrictions
The target machine is hosting a service on the target port
The target machine is accepting my connections on that target service
All of these occur after the hostname has been resolved to an IP.
So the answer to your question is no. If Socket.Connect fails it could be that the domain doesn't exist, or any of the above reasons (and perhaps more).
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In the old days, inside the Green Mountain, there lived a giant serpent who acted as if it were the master of the place, rixxo. The serpent was so old that its skin was as rough as the bark of a tree. Its aspect and size were the cause of much fear among the villagers.
They said that it would emerge to sunbathe during every rainy season, twisting itself around the green mountain, through the hills, ravines and plains of the Ngigua region. The animals and vegetation were abundant there because of the frequent rain that came down during the entire year. These downpours lasted for five or six days at a time, and the rivers would rise a lot. People did not come out of their huts when these downpours came because of the fear they had of the serpent of the Green Mountain.
When the rains began, the villagers knew that it meant difficult and dangerous times because the great serpent would emerge to warm itself up after the downpours. This is what happened, sometimes two or three times a year. These were fearful and dangerous days, as much for the people as for the animals; no one dared to even get close to the hills to care for his animals or to gather wood because, those who did not take the proper precautions, ran the risk of becoming very frightened, or even devoured by the enormous serpent.
After the heavy rains, the Green Mountain was enveloped in a thick fog that prevented anyone from seeing anything all around. As the fog cleared, little by little, people were astonished and afraid of what they could see from a distance, the giant head of the serpent whose body was twisted three or four times around the highest part of the hill. Its tongue was out and its eyes were shining like two balls of fire; they only got close when the animal was sleeping, or when the day became cloudier. All of its body was shining in the sun, as if it were something beautiful, but only when it went back in did people and their animals feel safe.
The years passed in this way. Many people came to love and respect the serpent because, thanks to it, they did not suffer through droughts; their land was fertile and produced good crops. On all sides, there were springs with running water. Still, there were others who were thinking about killing the serpent because it had eaten their animals or had frightened them terribly when they were on the hill.
On the side where the sun rose, the Green Mountain had an entrance that looked like a door. This was supposedly how the serpent came and left.
Time went on, and after the downpours people noticed that some of their animals were missing. They blamed the serpent for eating them. A group of people who were very angry at this decided to finish off the serpent. During the last season, the beast had come out very aggressively and scared everybody. It was after this that many men got together and started collecting firewood and stacking it up at the entrance of the mountain. Then they set it on fire so that the serpent would not be able to come out again. But the serpent, who possessed special powers, did not die. They repeated their actions over and over, but each time, the serpent came out to warm itself on the hill.
The serpent of Green Mountain, rixxo, became angry because the people of the hill would not leave it in peace. They were not aware of how good it really was. Thanks to the serpent, they did not suffer from droughts as they did later. But because they bothered it so much, it decided to leave the place, punishing the village that didn't want it. It came out on the other side of the mountain, and from there went to the coast. Wherever it slithered, the earth turned into stones. In some places, the mark can still be seen.
By going to other places, it took everything good that had existed in the village, Ngiguas: fruits, vegetables and, especially, rain. The elders say that everything changed from then on. The rains didn't come for many days, the rivers did not rise, the vegetation and the animals were getting scarcer. For lack of rain, the land was becoming barren and many places were developing ravines. The lands were shrinking because of erosion!
Nothing was ever heard again about the serpent of the Green Mountain.
When some downpours occur, and the year looks promising, Green Mountain is covered with fog again. Then the elders hope to see again, coiled around the hill, the giant serpent that their ancestors had known when it would rain as often as it was needed. But they fall into sadness when they realize that this is not possible, even when the white fog comes down as far as Coixtlahuaca. They know that nothing will ever be the same as it was when the serpent of Green Mountain was there.
Earl Shorris was a prominent social critic and author. His works include Ofay; The Boots of the Virgin; A Novel of Pancho Villa; The Death of the Great Spirit; The Oppressed Middle: Scenes From Corporate Life; Latinos: A Biography of the People; and New American Blues: A Journey Through Poverty to Democracy among others. He was the coeditor of In the Language of Kings: An Anthology of Mesoamerican Literature, Pre-Columbian to the Present; The Life and Times of Mexico; and While Someone Else Is Eating: Poets and Novelists on Reaganism. He was a contributing editor to Harper’s, and his essays and articles appeared in the Nation, the Atlantic Monthly, the New York Times, the Los Angeles Times, American Educator, the Antioch Review, and many more publications. He founded and chaired the advisory board of The Clemente Course in the Humanities; and cofounded—with Howard Meredith and members of the Kiowa, Cherokee, Chickashaw, Maya, Nahua, Lakota, CYup'ik, and other tribes and nations—the Pan-American Indian Humanities Center at the University of Science and Arts of Oklahoma. He died in May of 2012.
Sylvia Sasson Shorris is the author of Talking Pictures: With the People Who Made Them and co-editor of While Someone Else Is Eating: Poets and Novelists on Reaganism and In the Language of Kings: An Anthology of Mesoamerican Literature Pre-Columbian to the Present. She has published articles in The Nation, Chicago Tribune, Fork Roads, and Review (a publication of the Center for Inter-American Relations), and has been a translator in Mexico for Luis Montes Film Distribution, and in New York for 20th Century-Fox Film Corporation.
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Words without Borders opens doors to international exchange through translation, publication, and promotion of the best international literature. Every month we publish select prose and poetry on our site. In addition we develop print anthologies, work with educators to bring literature in translation into classrooms, host events with foreign authors, and maintain an extensive archive of global writing.
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Toronto FC announced Wednesday that the club has re-signed midfielder Victor Vázquez to a multi-year contract extension.
Vázquez, 31, was named to the Major League Soccer Best XI for the 2017 season, after finishing second in the league in assists (16), and leading MLS with six game-winning assists. Vázquez made 39 combined appearances, registering 10 goals and 20 assists across all competitions (MLS league, MLS Cup Playoffs and Canadian Championship). He was named MLS Player of the Week four times (Weeks 13, 25, 27 & 28).
“Following the conclusion of the 2016 season, teams within MLS started to figure out they could sit back on us. It was critical for our group to find a top-level talent that could help unlock teams and put us over the top,” said Toronto FC Sr. Vice-President, Soccer Operations & General Manager Tim Bezbatchenko. “Victor ultimately would become that piece helping us in our success last season in winning the treble. We are pleased to get this deal done so that Victor can help us defend our titles this season and beyond.”
Vázquez has made five appearances across MLS league play and the Concacaf Champions League this season for Toronto FC.
TRANSACTION: Toronto FC re-signs midfielder Victor Vázquez
VICTOR VÁZQUEZ #7
Position: Midfielder
Height: 5’9
Weight: 167 lbs
Birthdate: January 20, 1987 (Age – 31)
Birth Place: Barcelona, Spain
Nationality: Spanish
Last Club: Toronto FC
Pronunciation: “VAS-KEZ”
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Q:
How to use AJAX to get data from DB and display on page
I'm working in a project and I'm stuck here, I don't know why I can't get the list from my database\
Here is my JAVASCRIPT
$(document).ready(function(){
$.ajax({
url:'datos.php?accion=ac',
success:function(datos){
for(x = 0;x<datos.length;x++){
//$("#PAIS").append("<option value='"+datos[x].id_pais+"'>"+datos[x].pais+"</option>");
$("#PAIS").append(new Option( datos[x].pais, datos[x].id_pais));
}
}
})
$("#PAIS").change(function(){
//var felix=$('#PAIS option:selected').val();
//alert(felix);
$.ajax({
url:'datos.php?accion=ad',
alert('hola22');
success:function(datos1){
console.log("hola");
for(x = 0;x<=datos1.length;x++){
//$("#PAIS").append("<option value='"+datos[x].id_pais+"'>"+datos[x].pais+"</option>");
$("#REGION").append(new Option( datos1[x].region, datos1[x].id_region));
}
}
})
});
})
And my functions.php:
<?php
$server="localhost";
$usr="root";
$passwd="";
$data="combo";
$db=mysqli_connect($server,$usr,$passwd,$data) or die ("Error en la conexion1");
$Accion = $_GET['accion'];
if($Accion=="ac"){
header('Content-Type: application/json');
$paises = array();
$Consulta = mysqli_query($db,"SELECT * FROM paises")or die ("Error en la conexion7");
while($Fila=mysqli_fetch_assoc($Consulta)){
$paises[] = $Fila;
}
echo json_encode($paises);
}
if($Accion=="ad"){
header('Content-Type: application/json');
$regiones = array();
$Consulta1 = mysqli_query($db,"SELECT * FROM regiones WHERE id_pais=4");//.$_REQUEST['id_pais']);
while($Fila=mysqli_fetch_assoc($Consulta1)){
$regiones[] = $Fila;
//echo json_encode($Fila);
}
echo json_encode($regiones);
}
?>
Well, my problem it's that I really don't know how the first really works :D, but when I'm calling url:datos.php=ad this block doesn't work :/
A:
First, you will find it helpful to review these simple examples about AJAX. Do not just read them, copy them to your server and make them work. Change a few names or values -- see how they work.
AJAX request callback using jQuery
Next, here is a post that gives an overview to how PHP / web page / AJAX all work together. Take a few minutes and read it carefully. See if you can follow the logic. I bet the lightbulb will come on for you.
PHP - Secure member-only pages with a login system
Make your code as standard as possible. Don't take any shortcuts. Use the full $.ajax() structure, not the shortcuts of $.post() or $.get() (these are both shortcut forms of $.ajax(). Don't skip anything. As you get better, you can start to take some shortcuts. But for now, make sure your AJAX code block looks like this:
var var_value = $('#someElement').val();
$.ajax({
type: 'post',
url: 'your_ajax_processor.php',
data: 'post_var_name=' +var_value,
success: function(dataz){
if (dataz.length) alert(dataz);
$('body').append(dataz);
}
});
In your PHP, you will receive the value you posted in the $_POST array variable. If, in AJAX, you named your variable post_var_name (as we did in the example above), then that is how you access the contents:
$myVar = $_POST['post_var_name'];
When you are having trouble, a great idea is to put in some tests. (1) On the PHP side, comment out everything and at the top put in an echo command, like:
<?php
echo 'I got here';
die();
Back on the web page, in the AJAX success function, just alert what you get:
success: function(d){
alert(d);
}
At that point, you know two things:
Your AJAX -to- PHP communications are working, and
You see what value got passed over to PHP.
Then, you might do something like this
js/jQuery:
var var_value = $('#someElement').val();
$.ajax({
type: 'post',
url: 'your_ajax_processor.php',
data: 'post_var_name=' +var_value,
success: function(dataz){
if (dataz.length) alert(dataz);
$('body').append(dataz);
}
});
PHP:
<?php
$myVar = $_POST['post_var_name'];
//Now you can do something with variable `$myVar`, such as:
$out = '
<div class="red-background"> '.$myVar.' </div>
';
//This content is received in the AJAX code block using the variable name you specified: "dataz"
echo $out;
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Q:
HTTP Error 503. The service is unavailable when port 80 is already free
Before you go and suggest it as a duplicate of other IIS HTTP Error 503 questions, I have already tried many solutions available on SO and also on Google but the problem is not resolved. Below are the most popular solutions that I have found:
IIS - HTTP Error 503. The service is unavailable
Solve HTTP Error 503. The service is unavailable in IIS
In my case:
IIS is not working while the port 80 is open and free to use.
As far as I know the error is caused if the application pool is not enabled (started) for the website in IIS.
I have already closed all other applications like Skype, Google Disk etc who can possibly use port 80 and checks if the port is in use or not.
Then I looked into the Application Pools of IIS and found DefaultAppPool status stopped.
After starting the DefaultAppPool I recheck my web and gets the same error.
And now the situation is like, DefaultAppPool status shows started until I don't hit any URL. Whenever I request any URL for my local websites or localhost, the browsers returns me the Service Unavailable error message and the DefaultAppPool status change to stopped automatically.
I have already tried the iisreset command but after resetting IIS the problem is still there.
A:
Most probably identity under which you are running your app pool i.e. administrator has wrong password or disabled. Try changing this identity to something like network service or local service or any other valid identity.
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`vh` and `vw`, `vmin` and `vmax`
Hide the scrollbar in webkit browser
Use `border-radius` to style rounded corners of an element
Highlight input forms using `:focus` pseudo-class
How to flip an image
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When the Dead Return
When the Dead Return is a one-reel 1911 American motion picture produced by Kalem Company and directed by Sidney Olcott with Gene Gauntier, Jack J. Clark and JP McGowan in the leading roles.
Cast
Jack J. Clark as Antoine
Gene Gauntier as Marcele
JP McGowan as Jacques
External links
When the Dead Return website dedicated to Sidney Olcott
Category:1911 films
Category:American films
Category:American drama films
Category:American silent short films
Category:Films directed by Sidney Olcott
Category:1910s short films
Category:1910s drama films
Category:American black-and-white films
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---
abstract: |
Superconductivity in the topological non-trivial Dirac semimetal PdTe$_2$ was recently shown to be type-I. We here report measurements of the relative magnetic penetration depth, $ \Delta \lambda$, on several single crystals using a high precision tunnel diode oscillator technique. The temperature variation $\Delta \lambda (T)$ follows an exponential function for $T/T_c < 0.4$, consistent with a fully-gapped superconducting state and weak or moderately coupling superconductivity. By fitting the data we extract a $\lambda (0)$-value of $\sim 500$ nm. The normalized superfluid density is in good agreement with the computed curve for a type-I superconductor with nonlocal electrodynamics. Small steps are observed in $\Delta \lambda (T)$, which possibly relates to a locally lower $T_c$ due to defects in the single crystalline sample.
[Keywords]{}: Superconductivity, Dirac Semimetal, Penetration depth
address:
- '$^1$ Van der Waals - Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands'
- '$^2$ Institut Néel, CNRS & Université Grenoble Alpes, BP 166, 38042 Grenoble Cedex 9, France'
author:
- 'M. V. Salis$^1$, P. Rodière$^2$, H. Leng$^1$, Y. K. Huang$^1$, A. de Visser$^1$'
bibliography:
- 'References\_PdTe2.bib'
title: 'Penetration depth study of the type-I superconductor PdTe$_2$'
---
Introduction
============
Transition metal dichalcogenides display a wide variety of interesting electronic properties. Recently, these layered materials received considerable attention because they offer a fruitful playground for the realization of topological non-trivial electronic band structures [@Soluyanov2015; @Huang2016; @Yan2017; @Bahramy2018]. Notably, it has been proposed that transition metal dichalcogenides exhibit a generic coexistence of type-I and type-II three-dimensional Dirac fermion states [@Bahramy2018]. A prominent example is PdTe$_2$ that was reported to be a type-II Dirac semimetal, based on *ab-initio* electronic structure calculations combined with angle-resolved photoemission spectroscopy (ARPES) [@Liu2015a; @Fei2017; @Noh2017; @Bahramy2018; @Clark2017]. In a type-II Dirac semimetal the Dirac cone is tilted and Lorentz invariance is broken [@Soluyanov2015]. The Dirac point then forms the touching point of electron and hole pockets. PdTe$_2$ is of special interest because it is a superconductor as well ($T_c = 1.6$ K [@Guggenheim1961]). Especially, it has been proposed to be an improved platform for topological superconductivity [@Fei2017]. Topological non-trivial superconductors attract a strong interest since they are predicted to host protected Majorana zero modes at their surface (for recent reviews see references ). This offers a unique design route for devices based on topological quantum computation.\
Recently, the superconducting phase of PdTe$_2$ was characterized in detail by magnetic and transport experiments by Leng *et al.* [@Leng2017]. Surprisingly, dc-magnetization measurements on high-quality single crystals show that PdTe$_2$ presents a rare case of a binary compound that is a type-I superconductor. The crystals also show the intermediate state as demonstrated by the differential paramagnetic effect that is observed in low frequency ac-susceptibility measurements in applied dc-fields. The magnetization data reveal superconductivity is a bulk property with $T_c = 1.64$ K and a critical field $H_c (T)$ that follows a quadratic temperature variation with $\mu_0 H_c (0) = 13.6$ mT. This provides strong evidence that PdTe$_2$ is a conventional type-I Bardeen-Cooper-Schrieffer (BCS) superconductor. Interestingly, in applied fields $H_a > H_c$ large diamagnetic screening signals persist in the ac-susceptibility measured for small driving fields [@Leng2017]. This is attributed to superconductivity of the surface sheath. However, the critical field for surface superconductivity $H_c ^S$ does not obey the standard Saint-James - de Gennes relation with critical field $H_{c3} = 2.39 \times \kappa H_c$ , where $\kappa$ is the Ginzburg-Landau parameter. Therefore, it was proposed [@Leng2017] superconductivity of the surface layer has a topological nature and originates from the topological surface states that were detected by ARPES [@Liu2015a; @Noh2017].\
The superconducting properties of PdTe$_2$ were further investigated by heat capacity measurements [@Amit2018], as well as by scanning tunneling microscopy (STM) and spectroscopy (STS) [@Das2018; @Clark2017]. The specific heat data confirm conventional BCS superconductivity with a ratio $\Delta C/ \gamma T_c \approx 1.52$, which is close to the weak-coupling value 1.43. Here $\Delta C$ is the size of the step in the specific heat at $T_c$ and $\gamma$ the Sommerfeld coefficient. The zero-field STM/STS measurements point to a fully-gapped superconducting state. The absence of in-gap states seems to rule out topological superconductivity at the surface [@Das2018; @Clark2017]. However, the spectra in applied fields differ in important details. The conductance images taken by Das *et al.* [@Das2018] are consistent with type-I superconductivity with a critical field of 25 mT, but at certain points on the surface much larger critical fields (up to 4 T) were observed. This was attributed to the presence of randomly distributed impurities/defects. We remark these higher fields are reminescent of the transport critical field $\mu_0 H_c ^R (0) =0.3$ T $ > \mu_0 H_c ^S (0)$ reported by Leng *et al*. [@Leng2017]. On the other hand, Clark *et al*. [@Clark2017] claim to observe a vortex, indicating the superconductivity they probe is type-II. We remark, the presence of the type-I intermediate state that arises due to the sizeable non-zero demagnetization factor, $N$, has not been considered in the interpretation of the tunneling experiments. We conclude these conflicting results demand for a further examination of superconductivity in the bulk and on the surface of PdTe$_2$.\
We here present measurements of the relative magnetic penetration depth, $\Delta \lambda$, of PdTe$_2$ single crystals using the tunnel diode oscillator technique. The $\Delta \lambda (T)$-variation shows an exponential temperature dependence for $T/T_c < 0.4$ consistent with a fully-gapped superconducting state. The $\lambda (0)$-value extracted by fitting $\Delta \lambda (T)$, measured for several crystals, to the standard BCS expression amounts to 500 nm. The normalized superfluid density is in good agreement with the computed curve for a type-I superconductor with nonlocal electrodynamics and $\lambda (0) = 121$ nm.
Experiment
==========
PdTe$_2$ crystallizes in the trigonal CdI$_2$ structure (space group P$\bar{3}$m1) [@Thomassen1929]. The crystals used in this work were cut from the crystal grown by a modified Bridgman technique [@Lyons1976] as reported and characterized in reference [@Leng2017]. Scanning Electron Microscopy (SEM) with Energy Dispersive X-ray (EDX) spectroscopy showed the proper 1:2 stoichiometry within the experimental resolution of 0.5 %. Powder X-ray diffraction confirmed the CdI$_2$ structure. From transport measurements a residual resistance ratio $R$(300K)/$R$(2K) = 30 was deduced. Magnetic measurements show $T_c = 1.64$ K and $\mu _0 H_c(0) = 13.6$ mT [@Leng2017] (for an applied field directed in the plane of the layers, $H_a \parallel a$-axis). Three crystals (labeled s1, s2 and s3) with approximate size $1.0 \times 1.0 \times 0.1$ mm$^3$ were cut from the bulk crystal using a scalpel blade. The thin direction is taken along the $c$-axis. After the measurements the thickness of crystal s3 was reduced to $\sim 7~\mu$m by the Scotch tape technique. This crystal is labeled s4. The dimensions of the crystals are listed in table \[tab:table1\].\
Magnetic penetration depth measurements were carried out with the tunnel diode oscillator (TDO) technique [@Degrift1975]. Changes in the resonant frequency $f$ of a high stability $LC$ circuit are measured with a tunnel diode as gain operating at $\sim$ 13.4 MHz [@Diener2008]. The ac-excitation field, $H_{ac}$, is below 1 $\mu$T and the earth magnetic field is shielded by mu-metal. The sample is fixed with vacuum grease at the end of a sapphire rod attached to the cold finger of a home-built helium-3 refrigerator, which allows for measurements down to 0.38 K [@Diener2008]. During measurements the sample is positioned in the center of the excitation coil with inductance $L$. The crystal can be extracted *in-situ* from the excitation coil for calibration purposes. Changes in the penetration depth with respect to temperature are reflected in the resonant frequency via the mutual inductance of the sample and the coil. The frequency shift $\Delta f$ is measured relative to the frequency at the base temperature of the refrigerator. Measurements were made on crystals s1-s4 mounted vertically, *i.e.* with the field applied in the plane of the layers, $H_{ac} \perp c$, such that the screening currents flow in the plane and along the $c$-axis. In this case the frequency shift $\Delta f$ can be converted to a relative shift of $\lambda$ according to [@Fletcher2007]:
$$\label{eq:freqtolambda}
\frac{\Delta f}{\Delta f_0} = \frac{2}{t} \bigg(\Delta\lambda_{a} + \frac{t}{w} \Delta\lambda_c \bigg).$$
Here $\Delta \lambda _a$ and $\Delta \lambda _c$ are the relative magnetic penetration depths along the $a$- and $c$-axis, respectively, and $t$ and $w$ are the thickness and width of the crystal, respectively. $\Delta f_0$ is the frequency shift obtained by extracting the sample from the coil and accounts for demagnetization effects and the coil calibration factor. For thin square sample geometries the latter term in the brackets can be neglected since $w \gg t$. Crystal s1 was also measured in the horizontal configuration, with $H_{ac} \parallel c$, and the screening currents in the $aa^*$-plane. For a thin square geometry, with the magnetic field applied perpendicular to the broad surface, $\Delta f$ can be converted to the in-plane relative magnetic penetration depth $\Delta \lambda _a$ according to [@Prozorov2000]: $$\Delta \lambda _a = -R \frac{\Delta f}{\Delta f_0},
\label{eq:hcfreqtolambda}$$ where $R$ is a characteristic geometrical factor given by $$R = \frac{w/2}{
2 \bigg[ 1 + \bigg[ 1 + \bigg( \frac{2t}{w} \bigg)^2 \bigg] \textrm{arctan} \bigg( \frac{w}{2t} \bigg) - \frac{2t}{w} \bigg] },$$ with $t$ the sample thickness and $w$ the mean of the in-plane dimensions.
--------- ------------ ----------- --------------- -------------------
Crystal length width thickness $T_c^{onset}$
(mm) (mm) (mm) (K)
s1 0.88 0.84 0.097 1.67
s2 1.20 1.15 0.096 1.65
s3 1.48 0.96 0.060 1.63
s4 1.18 0.88 $\sim 0.007$ 1.76
--------- ------------ ----------- --------------- -------------------
: Dimensions (length, width and thickness) and onset superconducting transition temperature, $T_c^{onset}$, of the PdTe$_2$ crystals investigated.\
\[tab:table1\]
Results and analysis
====================
![Temperature dependence of $\Delta \lambda _a$ of four PdTe$_2$ crystals labeled s1-s4 with $H_{ac} \perp c$. $\Delta \lambda _a$ is computed from the frequency shift $\Delta f$ according to equation \[eq:freqtolambda\]. Dashed lines for crystal s1 show how $T_c ^{onset} = 1.67$ K is determined. In labeling crystal s1 we added the symbol $\perp$ since this crystal was measured in two configurations (see text). []{data-label="fig:figure1"}](figure1.eps){width="10cm"}
The overall temperature variation of the in-plane relative magnetic penetration depth of all four crystals is reported in figure \[fig:figure1\]. In this experiment the crystals were mounted vertically and $\Delta \lambda_a (T)$ was extracted using equation \[eq:freqtolambda\]. For crystals s1-s3 the superconducting transition is sharp, with $T_c ^{onset}$-values in the range 1.63-1.67 K, while for the thin crystal (s4) the transition is broad with $T_c ^{onset} = 1.76$ K. Above $T_c$ the tunnel diode oscillator technique probes the skin depth $\delta$, which is temperature independent in this case since the electrical resistivity, $\rho(T)$, attains a constant value near $T_c$ [@Leng2017; @Hardy1993]. In the measurements on crystals s1, s2 and s4 tiny steps in $\Delta \lambda_a$ are observed as shown in figure \[fig:figure2\]. The steps have different sizes up to 3 nm and appear both upon cooling and warming and are reproducible (see inset \[fig:figure2\]). Smoothed data sets for crystal s1 and s2 are readily obtained after removing the steps as shown in figure \[fig:figure3\]. In the following paragraphs we analyze $\Delta \lambda_a (T)$ as measured and with the steps removed. We remark the results differ in details only, and the presence or absence of steps does not affect any of our main conclusions.\
![Zoom-in of the data presented in figure \[fig:figure1\] for 0.4 K $ < T < $ 1 K. Small steps are visible in $\Delta \lambda _a$ (indicated by arrows), notably for crystals s2 and s4. The inset shows data taken on heating (red symbols) and cooling (blue symbols) at a rate of 0.02 K/min, which demonstrates that the steps are reproducible with little hysteresis.[]{data-label="fig:figure2"}](figure2.eps){width="10cm"}
The gap symmetry of the superconducting state can be elucidated by examining the temperature dependence of $\lambda _a$, or $\Delta \lambda _a$. For a superconductor with an isotropic superconducting gap, $\Delta \lambda (T)$ at low temperatures follows the relation [@Poole2007]
$$\label{eq:bcsfit}
\frac{\Delta \lambda(T)}{\lambda(0)} = \sqrt{\frac {\pi \Delta(0)}{2 k_B T}} e^{- \frac{\Delta(0)}{k_B T}}.$$
Here $\lambda(0)$ and $\Delta(0)$ are the penetration depth and superconducting gap for $T \rightarrow 0$, respectively, and $k_B$ is the Boltzmann constant. For a weak-coupling BCS superconductor $\Delta(0) = 1.76k_B T_c$. We have fitted $\Delta \lambda_a$ to equation \[eq:bcsfit\] with $\lambda_a(0)$ and $\Delta(0)$ as fit parameters, where a small offset has been added as a fit parameter to $\Delta \lambda_a$ to account for the exponential variation between $\lambda_a(0)$ and $\lambda_a(0.38$K$)$. It appears good fits can be obtained when we restrict the temperature range to $T/T_c < 0.4$ as shown in figure \[fig:figure3\] for crystals s1-s3. In table \[tab:table2\] we summarize all fit parameters obtained by fitting the as-measured and smoothed data for the four crystals. The value for $\lambda_a(0)$ ranges between 468 and 586 nm, with an outlier of 745 nm for the as-measured crystal s2. The latter large value is attributed to the ubiquitous presence of steps in the fitted range. The ratio $\Delta(0) /k_B T_c$ ranges from 1.77 to 1.86, again with the value 1.97 as an outlier for the as-measured crystal s2. These values are close to the weak-coupling BCS value 1.76. The error bar on the fit parameters is of the order of 10 %, a value obtained by inspecting the stability of the fit when slightly changing the reduced temperature range, $0.35 < T/T_c < 0.45$. The values of the fit parameters listed in table \[tab:table2\] show smoothing the data does not bring about major changes, the main reason being that the steps in the fitted temperature range are very small (except for crystal s2).
--------------- ----------- ------------------------- -------------- --
Crystal range $\Delta(0)/k_BT_c$ $\lambda(0)$
measured data ($T/T_c$) (nm)
s1$\perp$ 0.4 1.77 471
s2 0.5 1.97 745
s3 0.4 1.78 553
s4 0.5 1.86 586
smoothed data
s1$\perp$ 0.4 1.77 468
s1$\parallel$ 0.4 1.74 377
s2 0.4 1.83 482
s3 - - -
s4 0.5 1.83 471
--------------- ----------- ------------------------- -------------- --
: Fit parameters $\Delta(0)/k_BT_c$ and $\lambda(0)$ obtained by fitting equation \[eq:bcsfit\] to the as-measured PdTe$_2$ data (upper block) and data with steps removed (lower block). Entries for crystals s2-s4 are for $H_{ac} \perp c$-axis and for crystal s1 with $H_{ac} \perp c$-axis and $\parallel c$-axis.\
\[tab:table2\]
![$\Delta \lambda_a$ as a function of the reduced temperature $T/T_c$ for PdTe$_2$ crystals s1-s3 measured for $H_{ac} \perp c$. Open circles: experimental data, with small steps removed for crystals s1 and s2 (see text). Solid lines: BCS fits using equation \[eq:bcsfit\] up to $T/T_c = 0.4$. []{data-label="fig:figure3"}](figure3.eps){width="10cm"}
In figure \[fig:figure4\] we report $\Delta \lambda_a (T)$ of crystal s1 with the excitation field applied parallel to the $c$-axis. Here $\Delta \lambda_a (T)$ is calculated by using equation \[eq:hcfreqtolambda\]. In this configuration a few tiny steps appeared in the as-measured data as well, but less frequently than for the same crystal in the perpendicular configuration. In figure \[fig:figure4\] these steps have been removed. A fit to equation \[eq:bcsfit\] up to $T/T_c = 0.4$ yields $\lambda_a(0) = 377 $ nm and $\Delta(0)/k_B T_c = 1.74$. These fit parameters are listed in table \[tab:table2\] as well. For comparison we show in figure \[fig:figure4\] $\Delta \lambda_a (T)$ measured for $H_{ac} \perp c$ on the same crystal. Overall, the data are in good agreement, as well as the extracted values of $\lambda_a(0)$ given a 10% error margin. We remark that the calculated value of $\lambda_a(0)$ for $H_{ac} \perp c$ would be slightly reduced if we do not neglect the second term in equation \[eq:freqtolambda\].
![Temperature dependence of $\Delta\lambda_a$ for crystal s1 with the driving field perpendicular (black circles) and parallel to the $c$-axis (red circles), where small steps have been removed for both data sets. The solid lines represent fits to equation \[eq:bcsfit\] up to $T/T_c = 0.4$. Inset: Temperature dependence of $\Delta \lambda_a$ in a larger temperature range.[]{data-label="fig:figure4"}](figure4.eps){width="10cm"}
Superfluid density
==================
To further elucidate the nature of the superconducting phase of PdTe$_2$, especially the type-I or type-II character, we analyze $\Delta \lambda (T)$ (from now on we drop the subscript a) in terms of the normalized superfluid density $\rho_s(T)$ [@Poole2007]
$$\label{eq:nrho}
\rho_s (T) = \frac{n_s(T)}{n_s(0)} = \frac{\lambda (0)^2}{\lambda (T)^2} =\Bigg( \frac{\lambda(0)}{\lambda(0) + \Delta \lambda(T)}\Bigg)^2.$$
We first trace in figure \[fig:figure5\]a the normalized superfluid density of crystal s1 ($H_{ac} \perp c$) calculated from the temperature variation of $\Delta \lambda (T) $, where we used $\lambda (0) = 468$ nm (see table \[tab:table2\]). Here the reduced temperature $T/T_c$ is calculated with $T_c = 1.63$ K, which locates the midpoint of the superconducting transition, rather than with $T_c^{onset} = 1.67$ K. Next we compare the experimental curve to theoretical expressions for $\rho _s$ evaluated with (type-I) and without (type-II) nonlocal electrodynamics.
For a type-I superconductor nonlocal electrodynamics has to be taken into account, because the field penetration depends on the ratio of the BCS coherence length $\xi_0$ and the electron mean free path $\ell$. This results in an enhancement of the ratio $\lambda(T) / \lambda_L(T)$, where $\lambda_L(T)$ is the London penetration depth and $\lambda(T)$ is the magnetic penetration depth due to nonlocal electrodynamics [@Miller1959]. In the limit $T \rightarrow 0$, the London penetration depth is given by $\lambda_L (0) = (m^*/\mu_0 n_s e^2)^{1/2}$, where $m^*$ is the effective mass, $n_s$ the superfluid density, $\mu_0$ the vacuum permeability and $e$ the elementary charge. With a carrier density $n=5.5 \times 10^{27}$ m$^{-3}$ [@Leng2017] and $ m^* \approx 0.3m_e$ [@Dunsworth1975] (here we use an average value $ m^*$ and $m_e$ is the free electron mass) a value of $\lambda _L(0)$ of 39 nm had been derived [@Leng2017]. Considering Pippard’s nonlocal electrodynamics [@Tinkham1996] the penetration depth is enhanced by the ratio $\lambda(0) / \lambda_L(0)= (1+\xi_0 / \ell)^{1/2}$ in the limit $T \rightarrow 0$. The BCS coherence length is given by $\xi_0 = \hbar v_F / \pi \Delta(0) = 0.18 \hbar^2 k_F /(k_B T _c m^*)$ with $v_F$ the Fermi velocity and $k_F$ the Fermi wave number. With $k_F = (3 \pi ^2 n) ^{1/3} = 5.5 \times 10^9$ m$^{-1}$ we calculate $\xi_0 = 1.8$ $\mu$m. We remark $\xi_0$ is larger than the coherence length $\xi = 439$ nm derived using Ginzburg-Landau relations[^1]. An estimate for $\ell$ can be taken from the Drude transport model $\ell = 3 \pi^2 \hbar/\rho_0 e^2 k_F ^2$. With the residual resistivity $\rho_0 = 0.76~ \mu \Omega$cm [@Leng2017] we calculate $\ell = 531$ nm. Consequently $\xi_0 / \ell \approx 3.4$ and $\lambda(0) / \lambda_L(0) = 2.1$.
A full fledged function $\lambda(T) / \lambda_L(T)$ in the case of nonlocal electrodynamics can be computed following Miller [@Miller1959], where
$$\label{eq:miller}
\lambda = \pi \Bigg[ \int_0^{\infty} \textrm{ln}\bigg[ 1 + q^{-2} K(q) \bigg]dq \Bigg]^{-1}.$$
Here the response function $K(q)$ can be calculated with help of equations 5.26 and 5.3 in reference [@Nam1967]. Using the parameters $\lambda_L(0) = 39$ nm, $\xi_0 = 1.8$ $\mu$m and $\ell = 531$ nm derived above, we have computed $\rho_s (T)$ and the result is shown as the solid red line in figure \[fig:figure5\]a. In the inset we show the calculated ratio $\lambda(T) / \lambda_L(T)$. We remark that for temperatures below $T/T_c = 0.4$ the ratio is constant within 1 %, indicating that at low temperatures the local and nonlocal cases are indistinguishable, apart from the absolute value of the ratio. At higher temperatures the ratio is no longer constant, which allows one to discriminate between the local and nonlocal case via the superfluid density. In the limit $T \rightarrow 0$, $\lambda (0)/ \lambda_L(0) = 3.1$. This value can easily be verified by the corresponding entry in table \[tab:table1\] in reference [@Miller1959]. Consequently, $\lambda(0)= 121$ nm. The model curve is in good agreement with the experimental one calculated with $\lambda = 468$ nm (open black circles). We comment on the different values of $\lambda (0)$ in the Discussion section.
![(a) Normalized superfluid density $\rho_s(T)$ for PdTe$_2$. Open circles: calculated values for crystal s1 using equation \[eq:nrho\] with $\lambda (0) = 468$ nm. Red solid line: computed $\rho_s$ for the nonlocal case with help of equations \[eq:nrho\] and \[eq:miller\] with parameters $\lambda_L(0) = 39$ nm, $\xi(0)=$ 1.8 $\mu$m and $\ell= 531$ nm (see text). Black solid line: computed $\rho_s$ in the London limit with help of 8 and parameter $\Delta(0)/k_BT_c = 1.77$. Inset: Computed temperature variation of the ratio $\lambda(T) / \lambda_L(T)$. (b) and (c) The same analysis for crystals s2 and s3 with $\lambda (0) = 482$ nm and 553 nm, respectively. []{data-label="fig:figure5"}](figure5.eps){width="12cm"}
The theoretical expression of $\rho_s(T)$ for a conventional type-II superconductor (in the London limit) is given by [@Poole2007; @Mitra2017]
$$\rho_s = 1 + \int^{\infty}_{-\infty} \frac{\partial f}{\partial E} d\epsilon,$$
with the Fermi function $f = 1/ (1 + \exp(E/k_B T))$ and the Bogoliubov quasiparticle energy $E = (\Delta(T)^2 + \epsilon^2)^{1/2}$. $\Delta(T)$ can be obtained from the gap interpolation formula given by [@Gross1986]
$$\label{eq:inter}
\Delta(T) = \Delta(0) \textrm{tanh}\bigg[ \frac{\pi k_B T_c}{\Delta(0)} \sqrt{a\Big(\frac{\Delta C}{\gamma T_c} \Big)\Big(\frac{T_c}{T} -1 \Big)} \bigg],$$
where $a = 2/3$. We computed $\rho_s(T)$ for crystal s1 (with steps removed and $H_{ac} \perp c$) with help of equations \[eq:nrho\] and \[eq:inter\], where we inserted $\Delta(0)/k_BT_c = 1.77$ (see table \[tab:table2\]), and the corresponding $\Delta C / \gamma T_c$-value of 1.45 [@Carbotte1990]. The computed theoretical curve is shown as the black solid line in figure \[fig:figure5\]a. The model curve for type-II superconductivity considerably deviates from the experimental one for crystal s1 (black open circles), which is to be expected if superconductivity in PdTe$_2$ is type-I. On the other hand, a reasonable match between the computed curve and the experimental one can be obtained up to $T/T_c \sim 0.5$ when we reduce $\lambda(0)$ to about 350 nm in equation 5. However, in this case the mismatch at higher temperatures is large. The analysis of the superfluid density of crystals s2 and s3 yields very similar results compared to crystal s1, as demonstrated by the computed $\rho_s$-curves shown in figure \[fig:figure5\]b,c.
Discussion
==========
The BCS fits of $\Delta \lambda_a (T)$, presented in figures \[fig:figure3\] and \[fig:figure4\], are consistent with PdTe$_2$ being a conventional $s$-wave superconductor. The ratio $\Delta(0)$/$k_BT_c$ = 1.77 to 1.97 (table \[tab:table2\] points to weak- or moderately-coupling superconductivity, in agreement with specific heat data [@Amit2018]. The value of the penetration depth $\lambda_a (0)$ extracted from these fits is equal to 468 nm and 377 nm for crystal s1 measured in the perpendicular and parallel configuration, respectively, and largely exceeds the London penetration depth $\lambda_L (0) = 39$ nm. This difference can partly be accounted for by considering non-local electrodynamics, which results in $\lambda(0) = 121$ nm as calculated for a mean free path $\ell= 531$ nm and BCS coherence length $\xi_0=$1.8 $\mu$m. The values $\lambda_L (0) = 39$ nm and $\lambda (0) = 121$ nm fall well within the range of $\lambda$-values of type-I superconductors [@Poole2007]. A small value of $\lambda$ is also in line with a Ginzburg-Landau parameter $\kappa = \lambda / \xi_0 < 1/\surd2$. With the above value $\lambda (0)=468$ nm and $\xi_0$ we calculate $\kappa \approx 0.29$. Type-I superconductivity is furthermore supported by the computed superfluid density in the non-local scenario, which matches the experimental data better than the type-II London model (see figure \[fig:figure5\]).
At the moment we cannot reconcile the different values of $\lambda (0)$. A possible source of error in the calculation of $\lambda (0)$ via $\ell$ and $\xi_0$ is the assumption of a spherical Fermi surface, which is obviously an oversimplification [@Dunsworth1975]. Another uncertainty is the value of the carrier density, $n$, which was measured on a bulk crystal. Possibly, $n$ varies in different pieces cut from the crystal because of the semimetallic nature of PdTe$_2$ and sample inhomogeneities. A highly relevant experimental technique to determine the absolute value of $\lambda (0)$ in another way is transverse field muon spin rotation ($\mu$SR) [@Blundell1999]. However, this method is suitable for type-II superconductors only, because in a type-I superconductor the magnetic induction is zero (Meissner state) and muon spin precession is absent. While preparing our manuscript other measurements of the magnetic penetration depth of PdTe$_2$ by the TDO technique were reported [@Teknowijoyo2018]. In this work two crystals were measured in the parallel configuration ($H_{ac} \parallel c-$axis). By fitting the BCS expression to $\Delta \lambda$, $\lambda(0)$-values of 220 and 240 nm were derived. These values are about a factor two lower than the values we reported in table \[tab:table2\]. We remark in reference [@Teknowijoyo2018], $\Delta \lambda (0)$ was fitted to a power law temperature variation, $T^n$, to compare with predictions for an unconventional superconducting gap function. The authors report good fits with $n=4.2-4.3$, but also remark the data follow the exponential BCS expression (equation \[eq:bcsfit\]) equally well.
The appearance of small steps in $\Delta \lambda_a$ as shown in figure \[fig:figure2\] is striking. They are predominantly observed in crystals s1, s2 and s4, but not in s3. Note that crystal s4 was prepared by the Scotch technique by thinning crystal s3. They appear throughout the whole temperature range below $T_c$ and are reproducible. The steps have different sizes up to 3 nm, which is several times the $c$-axis lattice parameter, 0.513 nm [@Lyons1976]. A few larger steps appeared as well. We made a histogram of the step sizes, but did not find a clear periodicity with respect to the thickness of one PdTe$_2$ layer. Possibly microfractures, dislocations and local deformations in the crystal cutting or thinning process lead to a locally lower $T_c$. This in turn might cause “delayed” superconducting transitions of (parts of) multiple PdTe$_2$ layers. Crystal s1 was measured for $H_{ac} \parallel c$ as well. In this configuration the steps are weaker and occur less frequently. We remark steps are absent in measurements on other materials made with the same experimental set-up and are not an artefact of the measurement method [@Diener2008]. Also no steps were reported in the measurements on PdTe$_2$ reported in reference [@Teknowijoyo2018].
Besides bulk superconductivity below $T_c = 1.64$ K, the magnetic measurements revealed superconductivity of the surface layer in applied fields below $T_c^S = 1.33$ K [@Leng2017]. We have carefully inspected $\lambda (T)$ in the vicinity of $T_c^S$ but its variation is smooth and no change is detected when cooling the crystals below $T_c^S$. Measurements of $\lambda (T)$ in applied fields may be conducted to shed light on this unusual type of surface superconductivity.
Summary
=======
We have reported measurements of the relative magnetic penetration depth, $\Delta \lambda (T)$, in the superconducting state ($T_c =1.6$ K) of the Dirac semimetal PdTe$_2$ using the tunnel diode oscillator technique. The experiments have been performed on several crystals and for driving fields $H_{ac} \parallel c$ and $\perp c$. In all cases the temperature variation $\Delta \lambda (T)$ follows an exponential function for $T/T_c < 0.4$, which is consistent with a fully-gapped superconducting state and weak or moderately coupling superconductivity. By fitting the data we extract a magnetic penetration depth $\lambda (0)$ of $\sim 500$ nm for $T \rightarrow 0$. We compare the experimental normalized superfluid density $\rho _s$ to the computed curve for type-I superconductivity with nonlocal electrodynamics and the theoretical expression for type-II superconductivity in the London limit. In the nonlocal electrodynamics model we obtain $\lambda(0) = 121$ nm, with as input parameters for our crystal the London penetration depth $\lambda _L (0) =39$ nm, a mean free path $\ell= 531$ nm and the BCS coherence length $\xi_0=~1.8~\mu$m. We conclude the nonlocal model of type-I superconductivity gives a better match to the experimental superfluid density. At the moment we cannot reconcile the different values of $\lambda (0)$ extracted from the experimental data and the model. Part of the differences might be attributed to simplifications, such as a spherical Fermi surface, in the determination of the relevant parameters, and to variations in the carrier concentration $n$. Small steps are observed in $\Delta \lambda (T)$, which possibly relates to a locally lower $T_c$ due to crystal defects in the single crystalline sample. A next step in the research is to investigate the magnetic penetration depth in small dc-fields, notably in view of the unusual superconductivity of the surface sheath that was detected in applied fields below $T_c = 1.33$ K [@Leng2017].
Acknowledgements - H. Leng acknowledges the Chinese Scholarship Council for grant 201604910855. This work was part of the research program on Topological Insulators funded by FOM (Dutch Foundation for Fundamental Research on Matter).
References {#references .unnumbered}
==========
[^1]: In reference [@Leng2017] the calculation of the coherence length $\xi$ contains an error. The correct value is 439 nm
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She's used to getting carded at bars because of her youthful appearance, but when it comes to making people laugh, Lisa Corrao is 100 percent woman. "I don't mind looking young," says Corrao, who will perform Saturday during the Girls Night Out comedy show at the Funky Buddha Lounge and Brewery in Boca Raton. "But when people say you look like a prepubescent teen, that sucks."
The pint-size comic, known as the "Festival Flea Market girl" for her commercial work, chatted via phone about her upcoming projects and unorthodox sleeping habits.
I just finished filming a new episode of [the web sitcom] "Labor Days" this past weekend, and I'm producing a charity standup show with my "Labor Days" co-stars John Wynn and Forrest Shaw. The show takes place Nov. 7 at the Miami Improv, and all the proceeds will benefit the Miami-Dade Animal Shelter. The idea came about because we all have rescued pets. I'm also going to be featured on an upcoming Nick Jr. standup comedy show on Nickelodeon.
What's your standup about?
I make fun of myself a lot. Like, the fact that I'm sleeping in a bunk bed in a room with my daughter. It's cool when you're 10 but not so much now.
You really sleep in a bunk bed?
Yeah, I had an apartment, but it became too expensive, and I gave it up because I wanted to do comedy full-time. So I'm homeless right now. Actually, I live with my parents in Fort Lauderdale to save money. My plan is to move to Los Angeles or New York, because as a comedian, that's where the work is.
You seem really confident.It’s funny, I have big dreams for being this tiny, little Italian girl. You know, there’s no exact way or path to become a successful standup comic. It’s all trial and error. Sometimes, you’re really broke, and every day is an adventure. But I’m happy, and you can’t put a price on happy.
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principles of freedom chapter 14
CHAPTER XIV
MILITARISM
I
To defend or recover freedom men must be always ready for the appeal to arms. Here is a principle that has been vindicated through all history and needs vindication now. But in our time the question of rightful war has been crossed by the evil of militarism, and in our assertion of the principle, that in the last resort freemen must have recourse to the sword, we find ourselves crossed by the anti-militarist campaign. We must dispose of this confusing element before we can come to the ethics of war. Of the evil of militarism there can be no question, but a careful study of some anti-militaristic literature discloses very different motives for the campaign. I propose to lay some of the motives bare and let the reader judge whether there may not be an insidious plot on foot to make a deal between the big nations to crush the little ones. For this purpose I will consider two books on the question, one by Mr. Norman Angell, "The Great Illusion," and one by M. Jacques Novikow, "War and Its Alleged Benefits." In the work of Mr. Angell the reader will find the suggestion of the deal, while in the work of M. Novikow is given a clear and honest statement of the anti-militarist position, with which we can all heartily agree. Those of us who would assert our freedom should understand the right anti-militarist position, because in its exponents we shall find allies at many points. But with Mr. Angell's book it is otherwise. These points emerge: the basis of morality is self-interest; the Great Powers have nothing to gain by destroying one another, they should agree to police and exploit the territory of the "backward races"; if the statesmen take a different view from the financiers, the financiers can bring pressure to bear on the statesmen by their international organisation; the capitalist has no country. Well, our comment is, the patriot has a country, and when he wakens to the new danger, he may spoil the capitalist dream, and this book of Mr. Angell's may in a sense other than that the author intended be appropriately named "The Great Illusion."
II
The limits of this essay do not admit of detailed examination of the book named. What I propose to do is make characteristic extracts sufficiently full to let the reader form judgment. As we are only concerned for the present with the danger I mention, I take particular notice of Mr. Angell's book, and I refer the reader for further study to the original. But the charge of taking an accidental line from its context cannot be made here, as the extracts are numerous, the tendency of all alike, and more of the same nature can be found. I divide the extracts into three groups, which I name:
1. The Ethics of the Case.
2. The Power of Money.
3. The Deal.
Where italics are used they are mine.
1. THE ETHICS OF THE CASE."The real basis of Social Morality is self-interest." ("The Great Illusion," 3rd Ed., p. 66.) "Have we not abundant evidence, indeed, that the passion of patriotism, as divorced from material interest, is being modified by the pressure of material interest?" (p. 167.) "Piracy was magnificent, doubtless, but it was not business." (Speaking of the old Vikings, p. 245.) "The pacifist propaganda has failed largely because it has not put (and proven) the plea of interest as distinct from the moral plea." (p. 321.)
2. THE POWER OF MONEY."The complexity of modern finance makes New York dependent on London, London upon Paris, Paris upon Berlin, to a greater degree> than has ever yet been the case in history." (p. 47.)
"It would be a miracle if already at this point the whole influence of British Finance were not thrown against the action of the British Government." (On the assumed British capture of Hamburg, p. 53).
"The most absolute despots cannot command money." (p. 226.)
"With reference to capital, it may almost be said that it is organised so naturally internationally that formal organisation is not necessary." (p. 269.)
3. THE DEAL."France has benefited by the conquest of Algeria, England by that of India, because in each case the arms were employed not, properly speaking, for conquest at all, but for police purposes." (p. 115.)
"While even the wildest Pan-German has never cast his eyes in the direction of Canada, he has cast them, and does cast them, in the direction of Asia Minor.... Germany may need to police Asia Minor." (pp. 117, 118.)
"It is much more to our interest to have an orderly and organised Asia Minor under German tutelage than to have an unorganised and disorderly one which should be independent." (p. 120.)
"Sir Harry Johnston, in the 'Nineteenth Century' for December, 1910, comes a great deal nearer to touching the real kernel of the problem.... He adds that the best informed Germans used this language to him: 'You know that we ought to make common cause in our dealings with backward races of the world!'"
The quotations speak for themselves. Note the policing of the "backward races." The Colonies are not in favour. Mr. Angell writes: "What in the name of common sense is the advantage of conquering them if the only policy is to let them do as they like?" (p. 92.) South Africa occasions bitter reflections: "The present Government of the Transvaal is in the hands of the Boer Party." (p. 95.) And he warns Germany, that, supposing she wishes to conquer South Africa, "she would learn that the policy that Great Britain has adopted was not adopted by philanthropy, but in the hard school of bitter experience." (p. 104.) We believe him, and we may have to teach a lesson or two in the same school. It may be noted in passing Mr. Angell gives Ireland the honour of a reference. In reply to a critic of the Morning Post, who wrote thus: "It is the sublime quality of human nature that every great nation has produced citizens ready to sacrifice themselves rather than submit to external force attempting to dictate to them a conception other than their own of what is right." (p. 254.) Mr. Angell replied: "One is, of course, surprised to see the foregoing in the Morning Post; the concluding phrase would justify the present agitation in India, or in Egypt, or in Ireland against British, rule." (p. 254.) Comment is needless. The reading and re-reading of this book forces the conclusion as to its sinister design. Once that design is exposed its danger recedes. There is one at least of the "backward races" that may not be sufficiently alive to self-interest, but may for all that upset the capitalist table and scatter the deal by what Ruskin described in another context as "the inconvenience of the reappearance of a soul."
III
We must not fail to distinguish the worth of the best type of anti-militarist and to value the truth of his statement. It is curious to find Mr. Angell writing an introduction to M. Novikow's book, for M. Novikow's position is, in our point of view, quite different. He does not draw the fine distinction of policing the "backward races." Rather, he defends the Bengalis. Suppose their rights had never been violated, he says: "They would have held their heads higher; they would have been proud and dignified, and perhaps might have taken for their motto, Dieu et mon droit." ("War and Its Alleged Benefits," p. 12.) He can be ironical and he can be warm. Later, he writes; "The French (and all other people) should vindicate their rights with their last drop of blood; so what I write does not refer to those who defend their rights, but to those who violate the rights of others." (Note p. 70.) He does not put by the moral plea, but says: "Political servitude develops the greatest defects in the subjugated peoples." (p. 79.) And he pays his tribute to those who die for a noble cause: "My warmest sympathy goes out to those noble victims who preferred death to disgrace." (p. 82.) This is the true attitude and one to admire; and any writer worthy of esteem who writes for peace never fails to take the same stand. Emerson, in his essay on "War," makes a fine appeal for peace, but he writes: "If peace is sought to be defended or preserved for the safety of the luxurious or the timid, it is a sham and the peace will be base. War is better, and the peace will be broken." And elsewhere on "Politics," he writes: "A nation of men unanimously bent on freedom or conquest can easily confound the arithmetic of the statists and achieve extravagant actions out of all proportions to their means." Yes, and by our unanimity for freedom we mean to prove it true.
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Shop the Look: Nicole’s Floral Keyhole Dress in People Style Watch
Who: Nicole RichieWhat: Charlotte Ronson Floral Keyhole Dress with Belt in Russian BlueThe Way She Wore: Nicole chose the “Anastasia” dress—often nick-named the “Nicole” dress, since she looked so stunning in it— in the royal blue shade. She paired the silk chiffon floral-print with a tan belt and black pumps, finishing off her look with red lips and a sleek top-knot for a cool and subtly elegant look.
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Opioid antagonist impedes exposure.
Exposure is a rapid and effective treatment for simple phobias. This study tested the assumption that endorphin release may be involved in exposure to a feared situation. Thirty spider-phobic Ss underwent exposure to 17 phobic-related, graded performance tasks. Half the Ss were randomly assigned to naltrexone, an opioid antagonist, and half to a placebo. Measures of heart rate, blood pressure, self-efficacy, anxiety, and cognitions were obtained during treatment. Six of the 15 Ss in the naltrexone group dropped out after the 10th step in the treatment compared with 1 of the 15 Ss in the placebo group, chi 2(1, N = 30) = 4.7, p = .03. The naltrexone group took significantly longer to complete the first 10 steps (the last step that included all Ss) compared with the placebo group, F(9, 252) = 2.17, p = .024. Maximum heart rate and anxiety were significantly greater at Step 10 in the naltrexone group, but no differences were found for self-efficacy or cognitions. The study provides further evidence that the endogenous opioid system may be involved in the process of exposure.
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Controversial British plan to ‘divide and rule’ Afghanistan
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A Tory MP proposes hacking up Afghanistan into separate “kingdoms”, each ruled by a foreign power, and to include members of the Taliban.
The plan is the brainchild of conservative MP and Foreign Office aide Tobias Ellwood, a former army captain in the Royal Green Jackets, and is already under discussion in London and Washington, according to a report which was seen by the British newspaper The Independent on Sunday.The report puts forward a regionalized state under a powerful new prime minster and would attempt to deal with weak government, corruption and tribal disputes, which have plagued Afghanistan. The blueprint – which has been labeled Plan C – is to split Afghanistan up into eight zones based around the economic hubs of Kabul, Kandahar, Herat, Mazar-i-Sharif, Kunduz, Jalalabad, Khost and Bayman. The areas would then be governed by a council representing different ethnic groups and overseen by one or more foreign countries; in the same way that Berlin was run after the Second World War. Other options being considered include Plan A, to turn Afghanistan into a smoothly functioning democracy and Plan B, to hand the war over to the Afghan security forces, which would be overseen by American military advisors. Ellwood warned that Afghanistan faces a bleak future once the International Security Force (ISAF) withdraws at the end of 2014. He also said that “The Taliban will not enter into a meaningful dialogue if there is no feasible political strategy within which they can participate.”But experts on Afghanistan were critical of the plan, which they view as a colonial attempt to impose a democratic system. Instead, Western powers should think about a military exit strategy that would enable them to withdraw their forces by the 2014 deadline.Thomas Ruttig, co-director of the Afghanistan Analysis Network said: “Splitting the country into such regions will result in the empowerment of what we have started calling “local or regional powerbrokers,” what were previously known as “warlords”, whose misrule between 1992 and 1996 caused the rise of the Taliban in the first place.”While Wazma Frogh, executive director of Afghanistan’s Research Institute for Woman Peace and Security, was more scathing.“Who is this British MP sitting in London and deciding for Afghanistan? It should be us the people of this country, deciding if we want to divide into states and collapse as a nation,” she said.The current plan, when ISAF troops finally pull out, is to leave several thousand American trainers in the country in the hope that the Afghans will be largely able to police themselves; similar to the situation set up in Iraq after the withdrawal of Coalition forces in 2010.
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Benefits of photosynthesis for insects in galls.
Insect-induced plant galls are predominantly reputed to act as strong carbon sinks, although many types of galls contain chlorophyll and have the potential to photosynthesize. We investigated whether the photosynthetic capacity of bud galls induced by a Pteromalid wasp, Trichilogaster acaciaelongifoliae, in Acacia longifolia subsidises carbon budgets or provides O(2) to the larvae while concurrently consuming CO(2) in the dense gall tissue, thereby maintaining (O(2)) and (CO(2)) within the range of larval tolerance. Low (O(2)) (<5 % v/v) were found within the internal tissues of galls, and these concentrations responded only marginally to light, suggesting that the photosynthetic activity within the gall is inconsequential in the provision of O(2) to the larvae. The metabolic response of larvae to reduced (O(2)) and elevated (CO(2)) indicated that larvae were tolerant of hypoxia/hypercarbia and also capable of reducing their respiratory rates to cope with hypercarbia. The low mortality of larvae in galls shaded with Al-foil for 20 days showed that photosynthesis was not vital for the survival of the larvae, although growth of shaded galls was substantially reduced. Gas exchange measurements confirmed that, while photosynthesis never fully compensated for the respiratory costs of galls, it contributed substantially to the maintenance and growth, especially of young galls, reducing their impact as carbon sinks on the host. We conclude that, although photosynthesis may contribute to O(2) provision, its main role is to reduce the dependence of the insect-induced gall on the host plant for photosynthates, thereby reducing intra-plant, inter-gall competition and enhancing the probability that each gall will reach maturity.
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Effects of verapamil on coronary vascular resistance in rabbits: measurement with pulsed Doppler velocimetry.
Verapamil is an effective vasodilator. The purpose of this study was to investigate the in vivo effect of verapamil on coronary blood flow velocity and vascular resistance in anesthetized, open-chest rabbits. Twenty-one male New Zealand white rabbits were anesthetized, and a 3-mm suction-type pulsed Doppler velocimeter probe was applied to the proximal part of the left anterior descending coronary artery after median sternotomy. The rabbits received intravenous bolus infusion of 4 different doses of verapamil (0.01 mg/kg, n = 5; 0.1 mg/kg, n = 5; 1 mg/kg, n = 5, and 10 mg/kg, n = 6). The percent changes in coronary blood flow velocity and coronary vascular resistance were examined. There was 10.0+/-1.6% increase in coronary blood flow (CBF) and 12.5+/-1.9% reduction in coronary vascular resistance (CVR) after infusion of 0.01 mg/kg of verapamil. The CBF increased 23.0+/-9.5% and CVR decreased 24.2+/-5.2% after infusion of 0.1 mg/kg of verapamil. Infusion of 1 mg/kg of verapamil induced 34.8+/-10.5% increase in CBF and 32.6+/-2.5% reduction in CVR. The CBF increased 41.1+/-14.8% and CVR decreased 45.1+/-5.4% after infusion of 10 mg/kg of verapamil. Compared with baseline condition, all doses of verapamil increased coronary blood flow velocity and decreased coronary vascular resistance significantly in anesthetized, open-chest rabbits.
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VASCULARITY TARGETED PERCUTANEOUS ETHANOL INJECTION OF TOXIC ADENOMAS: OUTCOMES OF A FEASIBILITY STUDY PERFORMED IN THE USA.
Abstract Objective: The recommended treatment options for toxic adenomas (TA) in the USA are radioactive iodine ablation and surgical resection, and continued observation for pre-toxic adenomas (PTA). Percutaneous ethanol ablation (PEI) has proven efficacy in the treatment of TA and is widely available in Europe, but not so in the USA. Methods: Retrospective analysis of all patients who underwent PEI for TA/PTA at the University of Utah, from January 2010 to 2018 was performed. Ultrasound guided PEI, with injections targeting power Doppler mapped blood vessels within the adenomas was conducted. Functionality was confirmed using thyroid scintigraphy prior to PEI. Results: Eighteen adults (15 female) underwent PEI. Mean age was 41 ± 13.7 years. Baseline TSH was suppressed (0.06 ± 0.09 mU/L) with normal FT4 1.43 ± 0.39 ng/dL. Median nodule volume was 5.7 cm3 (IQR 4.8-7.7 cm3). 78% (n=14) underwent 2 or less PEI sessions. Median volume of ethanol used was 0.46 ml/ml nodule volume (IQR 0.3-0.6 ml). There was a significant increase in TSH concentrations within the first 3 months after PEI (TSH 0.06 ± 0.09 mU/L vs 1.22 ± 1.88 mU/L; p=0.02) with a concomitant significant decrease in FT4 concentrations (FT4 1.43 ± 0.39 vs 1.13 ± 0.25 ng/d; p<0.01). Significant nodular volume reduction was observed following PEI [median 5.7 cm3 (IQR 4.8-7.7 cm3) vs 2.5 cm3 (IQR 2.0-7.8 cm3); p<0.01]. Conclusion: Vascularity targeted PEI is safe and effective for treating PTA and TA. This unique approach required lower injected alcohol volume and fewer injections for therapeutic success.
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A transoesophageal echocardiographic image acquisition protocol for wide-view fusion of three-dimensional datasets to support atrial fibrillation catheter ablation.
The aim of this paper is to propose a transoesophageal echocardiography (TOE) image acquisition protocol which provides a systematic manner of acquiring a minimal number of overlapping 3D TOE datasets allowing the reconstruction of a wide 3D view of the left atrium (LA) with anatomical landmarks that are important for atrial fibrillation catheter ablation. In eight cardiac surgical patients, 3D TOE datasets were acquired with a six-step protocol. In the protocol, step 1 aims to acquire the central view of the mitral valve (MV), aortic valve (AV) and left atrial appendage (LAA). Step 2 was developed to acquire the left pulmonary veins (PVs) and step 3 to acquire the right PVs. Steps 4, 5 and 6 were developed to create a sufficient overlap between different datasets. 3D TOE datasets were registered and fused manually in end diastole. The image acquisition protocol was feasible in all patients. In the fused 3D dataset, a wide 3D view of the LA is shown, and left and right PVs could be seen simultaneously. The LAA, MV, AV and fossa ovalis (FO) were visualised clearly in the 3D TOE datasets. The PV ostia, which are located at the edges of the 3D datasets, suffered more from the artefact of echo loss. The volume overlaps between neighbouring TOE datasets were 50-75 %. The major part of the LA anatomy incorporating the PVs, LAA, MV, AV and FO as important anatomical landmarks can be reconstructed by registering and fusing 3D datasets acquired with the six-step TOE image acquisition protocol.
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ECFC FA Cup Rosette
A red and white rosette depicting the FA Cup and labelled Exeter City to commemorate City's 3rd round tie against the Manchester United of Best, Law and Charlton in January 1969. United won the game 3-1, Alan Banks scoring for City.
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Pages
Wednesday, August 6, 2008
2am Musings
**My dog snores.. Not puppy snores, oh no not my dog. She sounds like a full sized human male sleeping on the couch with a beer in one hand & remote in the other. (note- Daddy Chaos doesn't drink but it doesn't sound as good as saying a pop in one hand.. lol)
**No matter how much I wiggle and bounce on the bed (a waterbed at that) Daddy Chaos sleeps right thru the middle of the night feeding/fussings of Baby Bug. Not that I try to wake him up, that would be mean right?
**Do you know how hard it is to hold a hungry baby & measure the correct amount of formula into a bottle all with your eyes only half open? - note to self, make bottles before attempting sleep. It'll be easier, believe me.
** I miss my bed.. I wonder when Baby Bug will start being more accepting of his cradle? Tomorrow would be great, I'd say tonite but come on, I've already wasted half of it on the couch.. lol
2 comments:
I am so sorry that baby bug is not sleeping on his own yet. that has got to be hard! I remember when we got placed with our bug and it was so hard. It was all new and we were totally not prepaired for an infant, but I am glad we did it none the less. And yes, after about a month of trying to make the formula with a crying hungry baby in the middle of the night was a JOKE! We quickly learned to make up two bottles before heading to bed and just warming them up when needed worked wonders! YEA!
My little fellah is 15 months old and still wakes frequently. The first waking after I've gone to bed (he's already woken once or twice before then) I give my hubby a good night squeeze and baby and I curl up in the spare bed in his room, with a (pre made-up) heated bottle of milk, and that's where we stay till morning. He's returning home in a few months but as his family is Asian and tend to co-sleep too, they'll hopefully continue with night wakings in the same vein as me.
Search This Blog
About Me
I am a 30 something SAHM to 6 kiddos. I am a writer (of blogs) , a reader (of mainly romance) and a ruler (of the Kingdom of Chaos). I like to keep busy by reviewing books and now proofreading as well.. I am also a gestational surrogate. I wear many hats; my head is getting heavy.
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Is your computer running slow? Are you experiencing computer crashes or windows error messages? Do you want to improve your system performance and stability? If you are answering yes, then you very likely need to clean up your Windows registry.
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The auto-start feature of RegDoctor helps to support its regular use. By using RegDoctor regularly to clean and repair your registry, your system should not only be more stable but it will also help Windows and your software run faster. Click
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Easy to use interface. Lots of search parameters. Shows you exactly what it found, unlike some others than just show you. Also it does delete what it says it finds. It beat out Registry Cleaner 2006(32), WinAso Registry Optimizer (20) and Registry Mechanic (22) in my own personal heads up testing phase. It found registry entries for Norton (which I deleted more than 6 months ago) that the others didn't. The total 92 errors found, were also totally different except for maybe like 10 entries that were exactly alike with all 4 programs. It seems the others just tend to clean the surface and this goes very in-depth. Anyway this by far is my new favorite!
Your
Windows
registry
is a
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more
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incorrect
removal
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drivers
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startup
programs
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Thanks
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understanding.
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Smith wasn’t the only one; the vandal left their gold markings all along Haskins Drive in Suffolk. In all, three cars, three mail boxes, a speed limit sign and manhole cover were all tagged with gold spray paint.
“This is very upsetting because we work hard for what we have and for someone to come and destroy it, it’s pretty upsetting,” Smith says.
Smith says he was lucky; he was able to use paint thinner to clean his vehicle. He also took action to help his neighbors.
“I felt bad for the rest of the neighbors so I went down and helped get all the paint off their cars,” he says.
You can hardly tell his truck was once tagged, but Smith is making sure if there’s a next time the vandal will be caught on video.
“Because everyone wants to protect their valuable stuff,” he says.
Smith installed three home surveillance cameras around his house.
He has a message for the criminal.
“I hope you get caught,” Smith says. “I really do. I hope you get caught and I hope you get what’s coming to you.”
If you have any information for police, you can take action against this crime and call the Crime Line at 1-888-LOCK-U-UP.
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Olympian Matthew Mitcham Apologizes for Friend’s Homophobic Remark
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From his Facebook– “To those asking for my opinion (inc. media): I know a lot of ppl are upset by what Steph Rice said. I agree, it was offensive & very thoughtless, but being friends […]
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Q:
Kotlin, How to use the collection function to simplify this code
I am using Kotlin for a project, I write this code can complete the requirement:
val rewards = ArrayList<Map<String, Int>>()
rangeExams
.forEach { examAnswer ->
var reward = hashMapOf("Score" to examAnswer.answerScore)
var questionIds = examAnswer
.answers
.map { it.id }
reward.put("NewQuestion", questionIds.size)
rewards.add(reward)
}
"rangeExams" is a list of collection.
I would like to combinate Kotlin Functions of Collection,
to put elements of rangeExams into a map
and put this map to a new list,
how can I simplify this code by Kotlin ?
ExamAnswer is a pojo:
class ExamAnswer (val id: String, val answerScore: Int, val answers:List<Answer>)
Thank you for your reply
A:
Since you add an item to the rewards for each element of rangeExams, the .forEach { ... } call can be transformed to .map { ... }.
Also, you only use the result of examAnswer.answers.map { it.id } to get its size, so you can remove .map { it.id } and use the size of the original collection.
If you don't need to mutate the maps afterwards, you can replace hashMapOf(...) with mapOf(...).
val rewards = rangeExams.map {
mapOf(
"Score" to it.answerScore,
"NewQuestion" to it.answers.size)
}
If you need to mutate the rewards list after it's created, add .toMutableList() in the end.
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Sleaford Rural District
Sleaford was a rural district in Lincolnshire, Parts of Kesteven, England, from 1894 to 1931.
It was created by the Local Government Act 1894 based on the Sleaford rural sanitary district. In 1931, under a County Review Order, it was abolished, with a small part going to West Kesteven Rural District, the parishes of Leadenham, Welbourn and Wellingore going to North Kesteven Rural District, and the larger part going to the East Kesteven Rural District.
References
https://web.archive.org/web/20071001030328/http://www.visionofbritain.org.uk/relationships.jsp?u_id=10025345
Category:Districts of England created by the Local Government Act 1894
Category:Local government in Lincolnshire
Category:Rural districts of England
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Q:
Why total internal reflection is called total?
We know light rays incident on a surface face three types of events.
proportion of it is absorbed
some of it is reflected &
remaining part of it gets refracted.
In case of Total Internal Reflection, we know, not only does normal reflection occur (2), but also its refractive part (2) get reflected.
What happens to the (1) part? Is it also reflected or just get absorbed by the surface? If part (1) is absorbed like all other cases, why do we call it 'total'?
A:
The word "total" in "total internal reflection" is used in the following sense: all of the light that could possibly propagate away from this surface is reflected, and none is refracted. The absorbed portion is still absorbed, as usual, but since it doesn't propagate away from the surface, it's still consistent with our usage of "total" above.
As a side note, though total internal reflection means that no light will refract through the surface, that doesn't mean that there is no electromagnetic field on the other side of the surface. In reality, beyond the surface, there exist exponentially-decaying evanescent waves which do not propagate or carry energy away from the surface, but can still be detected and used to, for example, trap small molecules: https://en.wikipedia.org/wiki/Evanescent_field
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1. Technical Field
The present invention relates to a technology for controlling electro-optic elements for use in an electro-optic device or the like.
2. Related Art
Recently, display devices, each employing electro-optic elements, such as organic EL elements, which emit light with a luminance level in accordance with an amount of electric current flowing therethrough, have been in practical use. Such a display device, generally, uses pixel circuits each controlling an electric current supplied to the electro-optic element in accordance with an electric potential of the gate of a transistor (hereinafter, referred to as a “driving transistor”). The electric-current characteristic of the driving transistor varies depending on temperature. Thus, a variation of temperature results in a variation of the brightness of displayed images. In this regard, in JP-A-2011-221127, there has been disclosed a technology for detecting a temperature and correcting the luminance of electro-optic element on the basis of the detected temperature.
In conventional temperature correction methods, nevertheless, a correction for temperature has been made on the basis of a detected temperature on a real time basis. Thus, any correction error immediately results in a variation of the luminance of currently displayed images. In particular, in the case where the pixel circuit is formed on a silicon substrate, a sub-threshold region of the driving transistor is sometimes used as an operation area of the driving transistor. Since, in the sub-threshold region of the driving transistor, the electric-current characteristic thereof sensitively and largely varies, any real time process of measuring a temperature and controlling the luminance of a currently displayed image based on the measured temperature causes frequent and large variations of the luminance of the currently displayed image. Thus, there has been a problem that this frequent and large variations of the luminance of an image on a display screen result in the occurrence of flickering of the image.
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Introduction
============
To navigate in the natural environment, an animal must estimate its relative distance to obstacles along its path in order to avoid collision and reach its goal. Previous research in flying animals, such as honeybees ([@B39], [@B38]; [@B3]), budgerigars ([@B6]) and *Drosophila* ([@B43]) has provided evidence that these visually guided species rely on optic flow, the angular velocity of image motion across the animal's eyes, to guide locomotion ([@B39]). Optic flow provides continuous feedback to an animal about its relative velocity and distance to objects in its environment, and experimental manipulations of optic flow cues produce changes in animal flight trajectory and speed ([@B39], [@B38]; [@B3], [@B2]; [@B8]; [@B6]; [@B29]; [@B18]).
In the dark, optic flow cues become unreliable, and flying animals may need to rely on other strategies or sensory cues to move through a complex environment. For example, some bats have evolved an active sensing system -- echolocation -- to exploit echo-acoustic information to guide movements in the dark. Echolocating bats produce acoustic signals in the ultrasonic range and extract features of the environment from information carried by echo returns from surrounding objects. Some bat species use constant frequency (CF) signals, which are well suited to measure relative velocity ([@B27]; [@B22]), while other species rely exclusively on frequency modulated (FM) signals, which are well suited for spatial localization ([@B20]; [@B37]). Bats compute the distance to objects from the time delay between sonar emissions and echo returns ([@B31]) and the angular offset of objects from inter-aural difference cues ([@B17]; [@B36]; [@B20]). Populations of neurons in the bat auditory system show selective responses to 3D spatial acoustic information, i.e., distance (pulse-echo delay), azimuth and elevation, providing the neuro-computational substrate for dynamic sonar scene representation (reviewed in [@B40]; [@B7]; [@B44]). A recent neurophysiological study in the FM bat, *Phyllostomus discolor*, reports changes in echo-delay tuned neural responses to playbacks of pulses and echoes meant to simulate patterns of echo flow that a bat might receive as it flies past an obstacle ([@B4]). At the behavioral level, changes in echo delay, inter-aural differences and Doppler cues must be integrated across time from successive vocalizations to render updated echo scenes. The intermittent sampling of spatial information through echolocation occurs at intervals spanning tens to hundreds of milliseconds and contrasts with the nearly continuous sampling of information through vision.
The big brown bat, *Eptesicus fuscus*, is a nocturnal mammal that flies at speeds between 2 and 6 m/s ([@B9]), while probing its environment with short, downward-sweeping FM biosonar sounds, which contain several harmonics in the range of 25--130 kHz (FM1: ∼65--25 kHz, FM2: ∼130--50 kHz) and last between 0.5 and 15 ms ([@B12]; [@B33]; [@B10]; [@B42]). This bat dynamically decreases the duration and interval between sonar emissions as it approaches and intercepts prey ([@B33]), and must rapidly process changing echo information to guide appropriate adjustments in flight and echolocation behavior ([@B42]; [@B32]; [@B21]). Previous studies have shown that bats adapt echolocation call features in response to information extracted from echoes, which in turn influences the subsequent biosonar call design. For example, bats that produce CF signals adjust the frequency of their echolocation calls to compensate for Doppler shifts introduced by their own flight velocity ([@B26], [@B27]; [@B23]). Both CF and FM insectivorous bats adapt the emission rate, duration, and bandwidth of their sonar signals, depending on changing distance to prey in both open and cluttered environments ([@B33], [@B32]; [@B42]; [@B28]; [@B41]). While performing obstacle avoidance and prey capture tasks that demand high spatial localization accuracy, FM bats adjust their call rate and produce sonar sound groups, clusters of echolocation calls with short intervals, flanked by calls at longer pulse intervals ([@B19]; [@B24]; [@B16]; [@B25]). Therefore, echoes from objects in the bat's environment directly affect the bat's call timing, which in turn, directly influences the temporal patterning of echo returns. Laboratory research further demonstrates dynamic adjustments in flight speed, which are coordinated with adaptive echolocation behaviors ([@B24]; [@B9]). For example, [@B24] show that the big brown bat decreases its call rate and flight speed with increasing clutter density, and they report an increase of sonar sound groups in high clutter conditions. [@B9] further report shorter call durations in cluttered compared to open flight spaces. Overall, these results suggest that the big brown bat dynamically adapts its flight speed and biosonar call parameters when flying in cluttered environments.
Adjustments in bat flight and echolocation behaviors can be related to the echo-acoustic scenes the animal perceives via echolocation. Following each vocalization, a bat receives a cascade of echoes from objects ensonified by its sonar beam (**Figure [1A](#F1){ref-type="fig"}**). The horizontal beam of the big brown bat is spatially broad and spans approximately ±40-70°, depending on sound frequency (-6 dB beam width, [@B13]). Therefore, when the bat moves through its environment, the broad sonar beam will often ensonify not only prey, but also other objects. The bat's analysis of its echo scene thus involves the integration and segregation of cascades of echoes arriving from different objects at different spatial locations and distances ([@B21]). This is a particularly challenging task, as echoes from closely spaced objects may overlap in time, creating complex interference patterns ([@B34]; [@B5]).
{ref-type="fig"}**). The onset of call 2 (call at *t*~2~) occurs when echoes from call 1 (call at *t*~1~) are still arriving (dashed black arrow). The distance that the bat has flown in between two calls in schematically plotted on the (*y*-axis). Calls (blue) and echoes (orange/green/gray) are matched between panels **(A)** and **(B)**. Note that head and ear movements are not displayed and may further complicate echo patterns. **(C)** Schematic of waveforms across time (*x*-axis) as the bat flies down the corridor and emits successive vocalizations (blue). Pulse-echo overlap may occur (e.g., call 2, 4, and 6), complicating the spectral and temporal echo structures. The wall echo (gray) will increase in intensity and move closer to the time of vocalization as distance to the end of the corridor decreases. *Y*-axis plot distance flown between successive calls. Note that waveforms in **(B)** and **(C)** are cartoons illustrating the complex merging of the echoes returning to the bat; they do not represent how the bat perceives the flow of echoes.](fnbeh-10-00081-g001){#F1}
Past studies of echolocation behavior in complex environments challenged the bat to maneuver around obstacles, which introduces uncontrolled echo-acoustic variables that are difficult to quantify across individual animals (e.g., [@B9]). In this study, we attempt to address this limitation by investigating the bat's echolocation and flight behavior in experimentally controlled corridors, which constrain the animal's flight trajectory and allow echo-acoustic information to be systematically varied. As the bat flies in the controlled corridor, it experiences differential changes of returning echo patterns over a series of echolocation calls. These echo cascades arrive at the ears of the moving bat and vary with the animal's velocity, its head aim, and distance to objects in the environment (**Figure [1](#F1){ref-type="fig"}**). We here refer to these changes in echo patterns as "echo flow." While the construction of our bat flight corridor was inspired by animal studies of visually guided flight, it is important to note that fundamental differences between vision and echolocation preclude direct comparisons between optic flow and echo flow.
In the present study, we investigated the effect of echo flow patterns on bat flight and biosonar behaviors. Echolocating big brown bats flew through a corridor with walls constructed from individual poles, mimicking rows of trees that a bat might encounter in its natural environment. This setup allowed us to manipulate the echo patterns of each corridor wall that each bat would receive as it flew. We hypothesized that the density of poles comprising each corridor wall would influence bat flight trajectory and echolocation behavior. Specifically, we predicted that bats would fly along the midline of the flight corridor with balanced pole-spacing on opposite walls, and that they would show wall-following behavior in conditions with imbalanced pole-spacing. We further hypothesized that the bats would adapt their call duration and use of sonar sound groups with the density of poles comprising the corridor walls. Specifically, we predicted that bats would shorten call duration and increase sound group production when they flew through corridors with walls comprised of more densely spaced poles.
Materials and Methods {#s1}
=====================
Animals
-------
Seven wild-caught big brown bats, *E. fuscus* (three males and four females) were individually trained in an empty flight room to fly through a hole cut into a custom-built foam wall (**Figure [2A](#F2){ref-type="fig"}**, inset; see below). Bats were fed with mealworms (*Tenebrio molitor*) daily to maintain their individual weights between 13 and 16 g for the period of training and testing. The animals were housed in two group cages under reversed 12-h light/dark-cycle in a colony room kept at 24 to 28° C at 40 to 50% relative humidity. The experimental procedures were approved by the Johns Hopkins University Institutional Animal Care and Use Committee.
{ref-type="supplementary-material"}. Codes: s, single call/PI; d, doublet call/PI; t, triplet call/PI; q, quadruplet call/PI; p, pre-/post-sound group PI.](fnbeh-10-00081-g002){#F2}
Experimental Paradigm
---------------------
Experiments were carried out in a custom-built carpeted flight room (6 m × 6 m × 2.5 m), with the walls and ceiling lined with acoustic foam (Sonex Classic, Sonex Acoustics, San Jose, USA) and shielded from outside electrical noise. Along the diagonal of the flight room a corridor was built from individually moveable wooden poles, 2.5 cm in diameter each that spanned from floor to ceiling (**Figure [2A](#F2){ref-type="fig"}**, gray circles). The corridor measured 6.2 m in length and 1.2 m in width. Each pole was individually set up and straightened by referencing to a Bosch self-leveling Line Laser (Robert Bosch Tool Corporation LLC., Michigan, USA). The left and right walls of the corridor were constructed with experimentally controlled spacing of the poles (**Figure [2](#F2){ref-type="fig"}**). Four conditions with different configurations of pole spacing on the left and right sides of the corridor were tested in the experiment (**Figure [2B](#F2){ref-type="fig"}**). At the opening to the corridor, a black felt curtain with an elliptic hole (31 cm × 38 cm; Rainbow Felt Black, Fabric.com, USA), attached to a large frame (251 cm × 120 cm) of acoustic foam prevented the bat from gaining information about the wall configurations until the animal began its flight through the corridor on each trial (see **Figure [2A](#F2){ref-type="fig"}**, black line labeled "felt curtain," and gray bar and inset labeled "frontal view").
Prior to each experiment, the designated bat was removed from its cage and allowed to fly for several minutes in an adjacent flight chamber. Once the bat was actively flying, water-soluble glue (Grimas Mastix Water Soluble, Heemstede, Holland) was used to attach a custom-built head marker to the bat's head (**Figure [2C](#F2){ref-type="fig"}**). The head marker was 3.5 cm in length and 5 mm in width. Two reflective spheres (diameter: 8 mm) were glued to each end of the marker, so that one sphere was positioned between the bat's ears (**Figure [2C](#F2){ref-type="fig"}**, m2), and the other was positioned on, but not glued to, the bat's back (**Figure [2C](#F2){ref-type="fig"}**, m1). The total weight of the marker was 0.9 g, which corresponds to about 5--7% of body weight of individual bats. When the glue had dried and all recording systems had been configured, the experiment started. After collecting data over at least 10 trials, the head and body markers were carefully removed, and the animal was returned to its cage. Every day the seven different bats were tested in the same order and at the same time. For each trial three experimenters were present: the first experimenter released the bat from behind the curtain and otherwise stayed in that location. A second experimenter was responsible for catching the bat after a trial had ended and safely return it to the curtain-enclosed space. The "catcher" was otherwise waiting between the corner of the felt curtain and the flight room wall (**Figure [2](#F2){ref-type="fig"}**). The third experimenter recorded notes on every trial and triggered recording system for trial capture (see below).
Each bat navigated four different corridor configurations (**Figure [2B](#F2){ref-type="fig"}**) over a minimum of 10 trials. To study the bat's flight and echolocation behavior in response to the density of clutter and the flow of echo information, we tested each animal in combinations of dense and sparse spacing of wooden poles. Due to the duration it took to prepare each corridor setup, a given condition was tested on a single day. For all conditions, dense spacing refers to a 12 cm gap between two poles, and sparse spacing refers to a 36 cm gap between two poles (see **Figure [2B](#F2){ref-type="fig"}**). Bats were released at ca. 1.5 to 1 m distance from the acoustic foam curtain, and they entered the corridor by flying through the entrance hole. Each of the seven bats navigated through a corridor whose two walls were comprised of densely spaced (**Figure [2B](#F2){ref-type="fig"}**, 1) or sparsely spaced (**Figure [2B](#F2){ref-type="fig"}**, 2) poles. These corridor wall configurations served as baseline conditions. To test the effect of imbalanced left/right echo patterns on behavior, each animal also flew through a corridor in which one wall was comprised of sparsely spaced poles (**Figure [2B](#F2){ref-type="fig"}**, 3R; 4L) and the other one of densely spaced poles (**Figure [2B](#F2){ref-type="fig"}**, 3L; 4R).
To restrict bats from using visual cues ([@B14]), all data collection was done in a dark room that was solely illuminated with dim infrared light for motion-tracking detection of the two reflective markers on the bat. Measurements of the light levels in the flight room at the beginning, middle and end of the corridor each revealed a light intensity of \<10^-2^ lux. Measurements were done using a spectrophotometer (GS-1500, Gamma Scientific, San Diego, CA, USA) at experimental conditions.
Data Recording
--------------
For each trial, synchronized audio and motion-tracking data of the flying bat were captured. Audio data were recorded using 4 ultrasonic microphones (D500X external microphone, Pettersson Elektronik Uppsala, Sweden) bandpassed between 10 and 100 kHz, mounted at the end of the corridor (**Figure [2](#F2){ref-type="fig"}**, m). Audio data were sampled at 250 kHz (NI PXI board 6143). The bat's flight trajectory was acquired through 13 high-speed IR motion-capture cameras (Nexus, Vicon, Vicon Motion Systems Ltd., UK) mounted on the ceiling within the corridor. The motion-tracking system tracked the two reflective spheres attached to each bat at 300 frames per second. After all trials for the day were collected, the motion-tracking program also collected data on the position of the microphones, the location of the entrance hole, and the poles that made up the corridor walls. Every trial was manually triggered by an investigator after the bat had traversed the corridor at full length. Data acquired within 6 s prior to the trigger were stored for analysis.
Data from the motion-tracking system were processed offline, and custom-written MATLAB programs (Mathworks, Natick, MA, USA) were used to digitally analyze the audio data and 3D flight tracks of the bat.
Data Processing and Analysis
----------------------------
Motion-capture data were processed with custom-written MATLAB code to reconstruct 3D tracks of each bat's navigational patterns on a given day. In subsequent processing, we used the 2D projection of the bat's flight path onto the horizontal plane (floor) to compute its deviation from the midline of the corridor (**Figure [2A](#F2){ref-type="fig"}**, gray dashed line). Positive numbers indicate a deviation to the right side of the corridor, negative numbers indicate a deviation to the left side (see Results). Data points are calculated as distance from the end of the corridor, which has been defined as the plane created by the last poles on the left and right sides (**Figure [2A](#F2){ref-type="fig"}**, gray circles closest to 'm').
Echolocation calls produced by bats flying in the corridor were manually processed using an open-source MATLAB package that is archived on the GitHub software repository at: [https://github.](https://github.com/leewujung/call_marking_gui)[com/leewujung/call_marking_gui](https://github.com/leewujung/call_marking_gui). We extracted the start and end times of each call that was emitted during the portion of the flight path that had been previously reconstructed. With these parameters we then calculated the call rate, pulse interval, and duration of each call.
For all variables of interest, data points at distances smaller than 1 m from the end of the corridor and greater than 6 m from the end of the corridor (the "start" of the corridor) were excluded. Only data collected from the middle portion (a total of 5 m) of the corridor were analyzed.
We used JMP to perform analyses with a mixed effects model on the relationship between experimental condition and parameters of interest (deviation from the midline, flight speed, call rate, call duration, pulse interval). A Tukey's range test (HSD) was used for *post hoc* testing. For analysis of flight speed and sound parameters, we collapsed the two conditions that create an imbalanced flow of echoes (LS-RD, LD-RS) into one condition (S/D), as these conditions are essentially identical with regard to those variables. Unless otherwise noted, the mixed effects model analysis used condition (*N* = 4 for track data, *N* = 3 for all other data) as fixed effects and bat (*N* = 7) as random effects. *Z*-tests to test the deviation from the midline (zero-point) were done across condition and trials. Only trials in which the bat traversed at least AAA of the corridor were used for analysis: a total of 312 trials (LD-RD: *N* = 67, LD-RS: *N* = 64, LS-RD: *N* = 68, LS-RS: *N* = 62, LDS-RD: *N* = 51 trials) were analyzed.
Results
=======
Here we report how the spacing of poles along the corridor walls influenced the bat's flight and echolocation behaviors. All flight data are plotted according to the bat's deviation from the midline. Thus, negative numbers represent deviations to the left of the midline, whereas positive numbers represent deviations to right of the midline.
Flight Tracks
-------------
**Figure [3](#F3){ref-type="fig"}** shows the bats' average deviation from the midline (dashed black line) in the corridor across different conditions. When the corridor consisted of densely spaced poles on both sides (**Figure [3](#F3){ref-type="fig"}**, LD-RD), the bats centered their flight path and on average (black triangle) deviated 0.0075 m from the midline toward the right side. Similarly, when the corridor was comprised of sparsely spaced poles on both sides (**Figure [3](#F3){ref-type="fig"}**, LS-RS), the bats stayed close to the midline, deviating on average (black triangle) 0.0013 m from the midline. In both conditions, where the spacing of poles on either side of the corridor was imbalanced (**Figure [3](#F3){ref-type="fig"}**, LS-RD, LD-RS) the bats steered away from the wall with more densely spaced poles. Statistical analyses confirms a difference in the distributions of deviation from the midline across conditions. Using a Tukey's HSD test we report that flight paths in LS-RD and LD-RS differ from baseline conditions (LD-RD, LS-RS) and from one another (*F*~3,18~ = 21.597, *p* \< 0.0001; LD-RD: *M* = 0.0075 m, *SE* = 0.0004, LS-RS: *M* = 0.0013 m, *SE* = 0.0006, LD-RS: *M* = 0.055 m, *SE* = 0.0006, LS-RD: *M* = -0.0607, *SE* = 0.0005). The baseline conditions did not differ from one another. Importantly, flight paths in LS-RD and LD-RS also differ from zero (i.e., the midline of the corridor; *z*-test, LS-RD: z = -2.25, *p* = 0.012; LD-RS: *z* = 1.94, *p* = 0.026). In contrast, neither LD-RD nor LS-RS differ from zero (i.e., the midline of the corridor; *z*-test, LD-RD: *z* = 0.226, *p* = 0.82, LS-RS: z = -0.26, *p* = 0.79).
{#F3}
**Figure [4](#F4){ref-type="fig"}** plots the raw flight tracks (gray) and their mean (red) of each bat along the entire corridor for each condition.
{#F4}
Flight Speed
------------
**Figure [5A](#F5){ref-type="fig"}** compares the bats' flight speed (circles and solid line) and their call rate (squares and dashed line, see below) across flight corridor configurations. On average, bats navigated all conditions at around 3.8 m/s. Flight speed did not differ across conditions (*F*~2,12~ = 1.77, *p* = 0.21; LD-RD: *M* = 3.76 m/s, *SE* = 0.0054, LS-RS: *M* = 3.74 m/s, *SE* = 0.006, S/D: *M* = 3.83 m/s, *SE* = 0.004).
{#F5}
Echolocation Sampling
---------------------
Previous research has shown that a change of flight speed in the echolocating bat is often accompanied by a reciprocal change in call rate ([@B24]; [@B9]). We did not find a change in flight speed and also found that the call rates remained constant across baseline conditions, but showed a slight decrease in acoustically imbalanced conditions (S/D) (*F*~2,12~ = 5.7531, *p* = 0.017; LD-RD: *M* = 22.33 calls/s, *SE* = 0.077, LS-RS: *M* = 22.15 calls/s, *SE* = 0.096, S/D: *M* = 20.67 m/s, *SE* = 0.061, **Figure [5A](#F5){ref-type="fig"}**).
Overall, calls are shorter in the LD-RD condition (*M* = 2.89 ms, *SE* = 0.32) compared to the LS-RS condition (*M* = 3.5 ms, *SE* = 0.32) and S/D (3.36 ms, *SE* = 0.32) (*F*~2,12~ = 26.35, *p* \< 0.001, **Figure [5B](#F5){ref-type="fig"}**).
Of the 5,854 calls analyzed across conditions, 70.3% were part of a sound group (see **Figure [6A](#F6){ref-type="fig"}**, doublet, triplet and quadruplet), with calls creating a doublet call group making up the largest proportion (∼62%). The distribution of pulse intervals (PIs) was split into doublet sound groups (a single interval between 2 calls, **Figure [2D](#F2){ref-type="fig"}**, "d"), triplet sound groups (two intervals between three calls, **Figure [2D](#F2){ref-type="fig"}**, "t"), quadruplet sound groups (three intervals between four calls, **Figure [2D](#F2){ref-type="fig"}**, "q") and single sound groups (intervals between single calls that did *not* immediately precede or follow another sound group (**Figure [2D](#F2){ref-type="fig"}**, red "s", Supplementary Figure [S2](#SM3){ref-type="supplementary-material"}, red). Single pulse intervals that immediately preceded or followed a sound group (∼76%, **Figure [2D](#F2){ref-type="fig"}**, "p," Supplementary Figure [S2](#SM3){ref-type="supplementary-material"}, black) were excluded from this analysis. The overall sound group distribution was analyzed using a random effects model with condition (*N* = 3) and sound group (*N* = 4) added as fixed effects, and bat (*N* = 7) added as random effects. **Figure [6B](#F6){ref-type="fig"}** illustrates the mean pulse intervals used within each type of sound group. We did not find an effect of condition (*F*~2,58~ = 0.2831, *p* = 0.75, LD-RD: *M* = 46.48 ms, *SE* = 3.78, LS-RS: *M* = 44.80 ms, *SE* = 3.69, S/D: *M* = 47.65 ms, *SE* = 3.78), but report that pulse intervals differ significantly across sound group types: pulse intervals between single sounds (red "s" in **Figure [2D](#F2){ref-type="fig"}**) are longer (*M* = 61.25 ms, *SE* = 3.74) compared to intervals between sounds that make up a sound group (*F*~3,59.49~ = 12.60, *p* \< 0.0001, doublet: *M* = 39.68 ms, *SE* = 3.74, triplet: *M* = 40.20 ms, *SE* = 3.74, quadruplet: *M* = 44.10 ms, *SE* = 5.86).
{ref-type="fig"}**) are significantly longer, around 60 ms. By contrast, doublets, triplets and quadruplets share a shorter PI around 40 ms. Data are pooled across conditions, as no difference across conditions (LD-RD, LS-RS, and S/D) was found. Asterisk indicates significance at *p* = 0.05 level.](fnbeh-10-00081-g006){#F6}
Discussion
==========
Animals exploit a rich array of sensory cues to find their way in the environment, relying on vision, hearing, olfaction, somatosensation, infrared and magnetic sensing ([@B30]). Early work has suggested that animals relying primarily on vision for navigation, guide their movements through optic flow in order to measure their locomotion with respect to the objects in their surroundings ([@B11]; [@B15]). Studies investigating the effect of optic flow on animal flight have provided evidence that several species (bees, budgerigars, and fish, among others) adapt their movement trajectories and speed in response to experimentally controlled optic flow cues ([@B39], [@B38]; [@B3], [@B2]; [@B6]; [@B29]; [@B18]).
Here we investigated how an acoustically guided animal, the echolocating big brown bat, adapts its flight and sonar behavior in a restricted environment that returns dynamically changing patterns of echo flow. Inspired by paradigms used in optic flow research, we asked how the bat would change its flight trajectory, speed and biosonar behavior in a corridor with walls of differing echoic structure. In this corridor, each sonar emission returns a cascade of echoes from objects at different distances, some overlapping in time, which creates complex echo flow patterns (see **Figure [1](#F1){ref-type="fig"}**). These patterns change dynamically as the bat emits successive sounds in flight. It is noteworthy that acoustic flow in the corridor contrasts with optic flow along several dimensions. First, optic flow depends on relative movement between the animal and its environment, and provides a cue for relative distance. Echolocating bats, on the other hand, compute distance to objects directly from echo delay ([@B31]). Second, a visual animal's movement through the environment induces optic flow from light patterns present in the environment. In the case of echo flow, the dynamic features of echo cascades depend on the bat's active and discrete sampling of sensory information from the environment. Third, optic flow is independent of the contrast or intensity of visual patterns ([@B39]). By contrast, the structure and spatial configuration of corridor walls in our experiment directly influence the echo flow patterns the bat receives as it flies. Fourth, past research has shown that optic flow information can be experimentally manipulated or eliminated (e.g., [@B39]; [@B29]). Echo flow information, however, cannot be removed from our flight corridor, whose walls return echoes that vary with the bat's distance and directional aim of the sonar beam. These differences therefore limit direct comparisons between animal studies of optic flow and echo flow.
In accordance with our experimental predictions, the bats flew closer to the side with wider pole spacing when the corridor walls had different inter-pole spacing on the left and right sides (**Figure [3](#F3){ref-type="fig"}**, LS-RD and LD-RS). In conditions where the spacing was identical on both corridor walls, bats centered their flight paths, though they showed more variability in their flight trajectories along the corridor with wider inter-pole spacing (**Figure [3](#F3){ref-type="fig"}**, LS-RS vs. LD-RD), suggesting that reduced acoustic reflections from the walls influenced flight path planning. Our results show that bats deviate from the midline, away from the wall with dense pole spacing and toward the wall with sparse pole spacing. By contrast, bees continue to fly along the midline of a tunnel that displays different spatial frequencies of black and white vertical stripes on either side ([@B39]). It is noteworthy that this independence of spatial frequency on the bee's centering behavior holds only over a limited range of mismatch in the spatial frequency of patterns on opposite walls of the corridor. When the spatial frequency mismatch of vertical gratings on opposite walls is large, the bumblebees' flight paths deviate toward the wall displaying higher spatial frequency grating patterns ([@B8]). This finding differs from the flight path adaptations observed in our experiment, in which the echolocating bat veers away from the wall with higher density pole spacing. Note, however, that the spatial frequency mismatch of stimuli presented on opposite corridor walls in the study by [@B8] cannot be directly compared to the differences in wall pole spacing in our study, as we manipulated the distance between poles, and not the width of poles on each side.
When bees and budgerigars are presented with corridor walls that display patterns of vertical stripes on one side and horizontal stripes on the other, animals adapt their flight paths to show wall-following behavior, as optic flow cues are absent from the wall displaying horizontal stripes ([@B39]; [@B6]; [@B29]). As noted above, it is not possible to remove echo information from a structured environment, as all physical objects, including walls, return echoes, which are influenced by the bat's distance to those objects as well as the directional aim of their sonar beam. In a previous study, bats abolished centering behavior in a flight corridor with horizontal and vertical poles on opposite walls (Supplementary Figure [S1A](#SM2){ref-type="supplementary-material"}), flying closer to the wall with horizontal poles (Supplementary Figure [S1B](#SM2){ref-type="supplementary-material"}) that return fewer echoes (Supplementary Figure [S1C](#SM2){ref-type="supplementary-material"}), LH-RV, LV-RH; [@B45]). While the overall bat flight paths in this earlier experiment mirror the flight adjustments observed in visually guided animals flying along corridors with vertical and horizontal stripes on opposite walls (e.g., [@B39]; [@B6]; [@B29]), the findings of both research fields must be interpreted independently, as echo flow is not eliminated in horizontal pole condition.
Overall, bats produced shorter duration echolocation calls when flying through corridors with dense pole spacing (**Figure [5B](#F5){ref-type="fig"}**), which indicates that clutter influences sonar behavior. Specifically, shorter duration calls in highly cluttered conditions suggest a strategy that the bat might employ to reduce pulse-echo overlap from multiple, closely spaced reflecting surfaces (**Figures [1A,B](#F1){ref-type="fig"}**) ([@B24]). Contrary to our experimental predictions, flight speed and echolocation call rate in baseline conditions did not vary with differences in the corridor wall pole spacing (**Figure [5A](#F5){ref-type="fig"}**) though on average, bats emitted fewer calls in corridors comprised of imbalanced echoic walls. The observed constant velocity suggests that the bat can adapt its flight behavior (deviation from the midline to steer away from dense pole spacing) without altering its velocity; this is consistent with studies on zebra fish swimming through manipulations of optic flow corridors ([@B29]).
Contrary to our prediction, the bats' call rate did not increase from sparse (LS-RS) to dense (LD-RD) pole spacing conditions, and the use of sound groups (reciprocal of call rate), was consistently prevalent (**Figure [6A](#F6){ref-type="fig"}**, Supplementary Figure [S2](#SM3){ref-type="supplementary-material"}). In fact, bats emitted very similar proportions of sonar sound groups in all trials across conditions, producing doublets ∼62% of the time while flying through the corridor (**Figure [6A](#F6){ref-type="fig"}**, Supplementary Figure [S2](#SM3){ref-type="supplementary-material"}). The interval between single calls was significantly longer than the pulse intervals between clusters of sounds (**Figure [6B](#F6){ref-type="fig"}** single vs. doublet, triplet, and quadruplet). Past research suggests that sonar call groups may help the bat sharpen its representation of a complex environment ([@B16]), and the consistent prevalence of sonar sound groups recorded from bats flying in the corridor (Supplementary Figure [S2](#SM3){ref-type="supplementary-material"}) suggests that this environment presents the bat with a challenging echo scene that requires frequent sonar probing. Supplementary Figure [S2](#SM3){ref-type="supplementary-material"} illustrates typical and individual pulse interval patterns that each bat used when probing the corridor environment across conditions. It is noteworthy, that despite using different temporal patterning, the bats' flight behavior is very similar across conditions (**Figure [4](#F4){ref-type="fig"}**).
Bats can use information carried by echoes to navigate narrow spaces, forage in cluttered environments, avoid objects in their path, and forage on the wing ([@B12]; [@B27]; [@B35]). Similarly, visually guided animals use optic cues to maneuver in complex environments (e.g., [@B39]; [@B1]; [@B18]). The flight adaptations to echo flow patterns and optic flow cues suggest that common principles may underlie guidance of animal movement planning, even though the physics of light and sound render the stimuli distinctly different. While research has demonstrated that visual animals adapt their flight trajectories to balance optic flow, it is yet unclear whether bats adjust their flight paths to balance echo flow, or to simply steer away from the more echoic corridor wall. More detailed investigations into these phenomena may help elaborate on the computations involved in sensory-guided movement.
Author Contributions
====================
MW designed the experiment, collected, processed, and analyzed the data, and wrote the manuscript. W-JL wrote audio processing code and contributed to the writing of the manuscript. AK contributed to audio processing and the writing of the manuscript. CM contributed to the design of the experiment, discussion of data analysis and the writing of the manuscript.
Conflict of Interest Statement
==============================
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
**Funding.** This research was supported by grants from the Human Frontiers Science Program, the National Science Foundation, ONR and AFOSR, and a grant by the Studienstiftung Deutschland.
We thank Jonathan Engelberg, Victoria Perez, and Alycia Skarzinski for help during data collection, Ninad Kothari for MATLAB code, and James W. Garmon for extensive help in devising and building the experimental setup.
Supplementary Material
======================
The Supplementary Material for this article can be found online at: <http://journal.frontiersin.org/article/10.3389/fnbeh.2016.00081>
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[^1]: Edited by: *Miriam Liedvogel, Max-Planck-Gesellschaft, Germany*
[^2]: Reviewed by: *Emily Baird, Lund University, Sweden; Brock Fenton, University of Western Ontario, Canada*
[^3]: ^†^Present address: *Wu-Jung Lee, Applied Physics Laboratory, University of Washington, Seattle, WA, USA*
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Occurrence of rabies in a wolf population in northeastern Alaska.
Nine Alaskan wolves (Canis lupus) were found dead during spring and summer 1985; five of seven animals tested for rabies virus were positive. The 1985 epizootic altered annual den use patterns by wolves in northeastern Alaska, but did not appear to affect population size. We propose that rabies in arctic wolves may be more common than previously thought.
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Leishmania infantum induces expression of the negative regulatory checkpoint, CTLA-4, by human naïve CD8+ T cells.
CD8+ T cells are important in mediating protective responses to intracellular pathogens. However, an uncontrolled response may lead to pathology. The role of CD8+ T cells in different clinical manifestations of human leishmaniasis is controversial and poorly understood. We aim to study the response of CD8+ T cells to the first exposure to different strains of Leishmania, seeking to correlate these findings with clinical manifestations of disease. We have evaluated the expression of granzyme A, inflammatory and anti-inflammatory cytokines, as well as CTLA-4 by human naïve CD8+ T cells exposed to Leishmania braziliensis and two different strains of Leishmania infantum in vitro. We observed that while exposure to L braziliensis induced an inflammatory profile, as measured by the expression of granzyme A, IFN-gamma and IL-17, as well as a higher IFN/IL-10 ratio, exposure to L infantum led to a regulatory profile, as measured by lower IFN/IL-10 ratio and higher expression of CTLA-4. These results may help explain why patients with the visceral clinical form present a weaker cellular response and, consequently, a worse outcome of the disease. The use of CTLA-4 checkpoint inhibitors may emerge as a potential immunotherapy to ameliorate the immune response in visceral leishmaniasis patients.
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The Governments of the Member States of Asean and the Government of the People’s Republic of China,
REAFFIRMING their determination to consolidate and develop the friendship and cooperation existing between their people and governments with the view to promoting a 21st century-oriented partnership of good neighborliness and mutual trust;
COGNIZANT of the need to promote a peaceful, friendly and harmonious environment in the South China Sea (West Philippine Sea) between ASEAN and China for the enhancement of peace, stability, economic growth and prosperity in the region;
COMMITTED to enhancing the principles and objectives of the 1997 Joint Statement of the Meeting of the Heads of State/Government of the Member States of ASEAN and President of the People’s Republic of China;
DESIRING to enhance favorable conditions for a peaceful and durable solution of differences and disputes among countries concerned;
HEREBY DECLARE the following:
1. The Parties reaffirm their commitment to the purposes and principles of the Charter of the United Nations, the 1982 UN Convention on the Law of the Sea, the Treaty of Amity and Cooperation in Southeast Asia, the Five Principles of Peaceful Coexistence, and other universally recognized principles of international law which shall serve as the basic norms governing state-to-state relations;
2. The Parties are committed to exploring ways for building trust and confidence in accordance with the above-mentioned principles and on the basis of equality and mutual respect;
3. The Parties reaffirm their respect for and commitment to the freedom of navigation in and overflight above the South China Sea as provided for by the universally recognized principles of international law, including the 1982 UN Convention on the Law of the Sea;
4. The Parties concerned undertake to resolve their territorial and jurisdictional disputes by peaceful means, without resorting to the threat or use of force, through friendly consultations and negotiations by sovereign states directly concerned, in accordance with universally recognized principles of international law, including the 1982 UN Convention on the Law of the Sea;
5. The Parties undertake to exercise self-restraint in the conduct of activities that would complicate or escalate disputes and affect peace and stability including, among others, refraining from action of inhabiting on the presently uninhabited islands, reefs, shoals, cays, and other features and to handle their differences in a constructive manner.
Pending the peaceful settlement of territorial and jurisdictional disputes, the Parties concerned undertake to intensify efforts to seek ways, in the spirit of cooperation and understanding, to build trust and confidence between and among them, including:
a. holding dialogues and exchange of views as appropriate between their defense and military officials;
b. ensuring just and humane treatment of all persons who are either in danger or in distress;
c. notifying, on a voluntary basis, other Parties concerned of any impending joint/combined military exercise; and
d. exchanging, on a voluntary basis, relevant information.
6. Pending a comprehensive and durable settlement of the disputes, the Parties concerned may explore or undertake cooperative activities. These may include the following:
a. marine environmental protection;
b. marine scientific research;
c. safety of navigation and communication at sea;
d. search and rescue operation; and
e. combating transnational crime, including but not limited to trafficking in illicit drugs, piracy and armed robbery at sea, and illegal traffic in arms.
The modalities, scope and locations, in respect of bilateral and multilateral cooperation should be agreed upon by the Parties concerned prior to their actual implementation.
7. The Parties concerned stand ready to continue their consultations and dialogues concerning relevant issues, through modalities to be agreed by them, including regular consultations on the observance of this Declaration, for the purpose of promoting good neighbourliness and transparency, establishing harmony, mutual understanding and cooperation, and facilitating peaceful resolution of disputes among them;
8. The Parties undertake to respect the provisions of this Declaration and take actions consistent therewith;
9. The Parties encourage other countries to respect the principles contained in this Declaration;
10. The Parties concerned reaffirm that the adoption of a code of conduct in the South China Sea would further promote peace and stability in the region and agree to work, on the basis of consensus, towards the eventual attainment of this objective.
Done on the Fourth Day of November in the Year Two Thousand and Two in Phnom Penh, the Kingdom of Cambodia.
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Friday, July 31, 2009
Tonight marks the first time in his professional career that Cliff Lee will pitch for a team not named the Cleveland Indians. That's good news for him, as he goes from one of the most putrid teams in baseball to the World Fucking Champions. Good deal for him and us.
Of course, we all know the the reasons why Cliff Lee is awesome, but here's another one for you: The man has two first names. So in honor of this fact, here's a list right off the top of my head of famously awesome Cliffs and Lees, just to prove how awesome his name really is.
Cliff Huxtable
How could I not start with Dr. Huxtable? Portrayed by Philadelphia's own Bill Cosby, Cliff Huxtable, the star of the Cosby Show, was the coolest Dr./father on the planet. Sure, his son was dyslexic and his daughter a whore who ran off with a sailor, but there wasn't a smoother, funnier father figure on TV. The Cosby Show was fantastic, and I still watch reruns of it from time to time.
Bruce Lee
Kung-Fu is awesome, no matter how cheesy the movies may be, and Bruce Lee is the original Kung-Fu gangsta. Just watch one of his movies and try not to enjoy it. That man kicked some serious ass. And he even fought Kareem Abdul-Jabbar.
Cliff Clavin
Cheers is easily one of the best shows of all time, and a big reason is everybody's favorite mailman. Cliff Clavin played one of the funniest supporting roles ever and was easily my favorite character on the show. While Norm got all the love, Cliff made me almost piss my pants on many occasions. The highlight, without question, was his appearance on Jeopardy.
Stan Lee
The godfather of comic books, Stan Lee is responsible for many of the greats: Spiderman, the Fantastic Four, X-Men, Iron Man, the Incredible Hulk just to name a few. If you're a fan of comic books or comic book movies, you can thank Stan Lee. He also made a cameo in Mallrats, one of my favorite movies.
Clifford the Big Red Dog
I love dogs, and there's not many more famous ones than Clifford the Big Red Dog (OK, there is, like Snoopy and Lassie and Rin Tin Tin and Hooch and Beethoven but cut me a break). I'm pretty sure every kid in my generation grew up reading Clifford the Big Red Dog, either voluntarily or having it read in class by teachers or librarians. And frankly, that shit was entertaining. Speaking of shit, how big a load do you think Clifford took when he had to go number 2? And did his owners clean it up? How? With a forklift? A bulldozer? These are the type of things I think about. I may have some mental problems.
Lee Harvey Oswald
Perhaps the most famous Lee to ever live, the man charged with killing JFK. While conspiracies abound, Lee Harvey Oswald is the only known shooter in the only Catholic president's assassination, but no one believes this to be true. Either way, the guy shoots people. Moral of the story: Don't mess with a guy named Lee (unless he's handcuffed by police — I hear that's a good time to shoot a guy named Lee).
Man, these games out west are killing me. I'm an old man. I need my rest. And last night, it all finally caught up to me. After watching every inning of all three games against the Diamondbacks, I simply couldn't 'make it last night, which was probably for the better. Once the bottom of the fourth inning rolled around, I was out cold.
Now, the Phillies did lose their second straight game last night, and for the second straight night, they couldn't get runners home in situations where you simply have to get runners home. Take, for instance, the third inning, when Jimmy Rollins led off with a double and stole third, placing him on third with no one out only to see Chase Utley strike out, Jayson Werth pop out and Ryan Howard end the inning without plating Rollins by grounding out. It was reminiscent of Wednesday night's 7th inning, where the Phils had runners on second and third with no one out after Ryan Howard led off with single and Raul Ibanez followed with a double. Then they failed to bring them home.
Normally, that would be cause for a rant or two, but honestly, it's just a little hard to jump off the cliff (no pun intended) at the moment or fly off the handle. After all, the Phils have been white hot, they just got the reigning AL Cy Young award winner who will make his Phillies debut tonight, and it was the first time they've lost two games in a row since the first two days of the month. Right now, it's a good time to be a Phillies fan.
While I'm weary from these road games against the NL West, you can bet your ass I'm going to watch every damn pitch tonight. It's all set up pretty perfectly for Cliff Lee to ease into his role with his new team. The Phils are on the road, so there isn't nearly as much pressure to perform from the fans. He's playing in a pitcher's ballpark against a lineup that, even with the additions of Ryan Garko and Freddy Sanchez, doesn't strike a lot of fear. And the Phils will undoubtedly have a boost from seeing Lee on the mound in their colors for the first time.
Frankly, I'm excited. And if Lee pitches well, and gets down a sacrifice bunt, you know it'll be a great weekend.
Wednesday, July 29, 2009
Well ladies and gentleman, the Phillies now have four starting lefthanders, as evidently, the Phils have traded for Cliff Lee, last year's American League Cy Young winner.
The Phils are shipping Carlos Carrasco, Jason Donald, Jason Knapp and Lou Marson to Cleveland for Lee and Ben Francisco, a 27-year-old righthanded outfielder who is hitting .250 with 10 home runs, 33 RBI and 13 steals.
Welcome to Philadelphia, Cliff Lee. I hope your career is much, much better than Travis Lee's here in Philadelphia.
With reports swirling that the Phillies are about to acquire Cliff Lee, I had a thought so crazy (and incredibly improbable) that it would be the most awesome thing ever … what if the Phillies did the unthinkable and traded for both Cliff Lee and Roy Halladay?
First of all, everyone's head would explode. Second of all, the farm system would be bare thin. But think how awesome it would be to have Halladay, Lee, Hamels and Blanton as your top four starters. Unreal.
Conceivably, if reports are true for the asking prices, the Phillies could do it if they really wanted to. Trade Lou Marson, Jason Donald, Carlos Carrasco and Jason Knapp for Cliff Lee and then swap Kyle Drabek, J.A. Happ and Dominic Brown to Toronto for Roy Halladay. That would be the greatest thing in the history of things.
I'd be thrilled with getting Lee, but only if Toronto doesn't trade Halladay. All indications are the Phils could get him if they're willing to meet the steep price. Halladay > Lee. But Lee is all sorts of good.
Anyway, if the Phils do hold on to Drabek and it turns out the Phils' reluctance to deal him was the only reason they couldn't get Halladay, well, I hope for Drabek's sake he's a perennial Cy Young candidate. Because if he's not, he'll get buried, always remembered as the guy the Phils wouldn't give up to get the Hall-of-Fame-bound Halladay. That's a lot of pressure to put on a kid.
Unless of course the Phils repeat as World Fucking Champions. Then everyone is off the hook, no matter what happens.
Wanna know just how stupid I am? All day yesterday, I thought it was Wednesday. I'm not making this up. I was wondering where the KSK This Week In F—k You was and everything.
Turns out, it was Tuesday. Apparently, today is Wednesday. That sucks. And I'm an idiot. Here's some Wednesday links …
-I'm not entirely sure how I feel about Inquirer columnist John Gonzalez. Sometimes, I enjoy reading him. Other times, I think his columns are complete garbage. One thing I do know, his column today is one of the most insightful and well-written columns I've ever read. Do yourself a favor and read it. He makes the best points you will read about Saturday's tragic murder outside McFadden's.
-Naturally, the Philadelphia newspapers are all over Jim Johnson's passing.
Bowden is clearly obsessed with the number, but not the other parts of what will add up to his legacy as a football coach. Paterno, in contrast, could care less, and said the same thing when he sat behind Bowden in the count just a few years ago. His name is on the library at Penn State. His team has been the case study for the "Grand Experiment," Paterno's attempt to fuse academics and athletics that has Penn State's football players consistently ranked at the top of the Big Ten in graduation rates. (Northwestern's usually first, but they're also Northwestern.) His contributions to the university extend far beyond the football field …
Even if the NCAA decides to recognize the wins, what Joe Paterno has in the win column on and off the field puts him ahead of Bowden in ways a win column can't ever quantify, and that is a score Paterno likely cares about more than any simple count of wins and losses.
Yesterday, I got home from work and performed my customary routine: changed out of my work clothes and tuned in to Daily News Live. A short while later, after discussing the impressive display rookie tight end Cornelius Ingram showcased at that day's morning practice with Ray Didinger, the show went to commercial. When it came back, there was Neil Hartman conveying the sad news: Jim Johnson, the Eagles defensive coordinator for the entire 2000s, had lost his battle with skin cancer and passed away.
Of course, I was saddened, but I seemed to take it in stride at first. We all knew Johnson was not doing well, and I think we all knew deep down he'd never be with the Eagles again. But then, as Les Bowen and Paul Domowitch and Ike Reese and Sal Paolantonio relayed the tales of how Jim Johnson affected their lives, it began to hit me. Jim Johnson, easily the best coordinator the Eagles have had in my lifetime, was gone. The architect of that aggressive, blitz-happy defense gone.
I never met Jim Johnson. I don't really know a damn thing about him other than he was a damn fine coach. But I do know a hell of a lot about the defenses he constructed, and man, they have been devastating. Quarterbacks cringed when they had to face the Eagles, because no matter what, they knew they were going to get hit. Conventional wisdom be damned, Jim Johnson was bringing the blitz and attacking your weakness no matter what. And he did, and did it better than anyone.
It's hard to put things in perspective as to just what Jim Johnson has meant to the Andy Reid-era Eagles. Emphasis always seemed to fall at the feet of Andy's playcalling, never Jim's. But in this city, it was a fascinating dynamic that the fans had with the defensive coordinator. While Andy received the brunt of criticism for the team routinely coming up just short, Johnson has been praised and cherished by this hard-nosed fanbase. Perhaps it's because, no matter what, the Eagles were going to reflect the city on the defensive side of the ball — hard-nosed, aggressive, never backing down, consequences be damned.
Johnson coached some damn fine players: Hugh Douglas, Troy Vincent, Brian Dawkins, Jeremiah Trotter, Sheldon Brown, just to name a few. All of them credit him for their success. All of them.
No one knows what effect his loss will mean to the team. Sean McDermott is a Johnson disciple, a protege groomed to take the helm. If the success of Jim Harbaugh, Steve Spagnuolo and company are any indications, the Eagles should be just fine. Jim Johnson has proven himself an excellent teacher, to both players and coaches. But it's safe to say things will never quite be the same. How can they be? The team, the city just lost an icon.
LeClair, of St. Albans, Vt., became the first American-born NHL player to post three consecutive 50-goal seasons -- all with the Philadelphia Flyers. He played for the Flyers for 10 seasons, totaling 333 goals, and finished his 17-season NHL career with 406 goals in 967 games. He represented the United States in the 1998 Olympics, '96 World Cup and the '88 and '89 World Junior Championships.
LeClair finished second in scoring among all players in the '96 World Cup, helping the Americans to a memorable performance with 6 goals and 10 points in seven games. He was also part of the "Divine Line" at the '02 Winter Olympics, skating alongside Brett Hull and Mike Modano -- the highest-scoring trio (9 goals, 21 points in 6 games) in the tournament. He was drafted 33rd by the Canadiens in 1987 and would win a Stanley Cup in Montreal in 1993. LeClair spent his final two seasons in Pittsburgh.
Congrats, John LeClair. You earned it. You'll always be welcomed in Philadelphia for your monster years as a Flyer.
You may remember him as one of the three Nittany Lions suspended last season for a little incident involving marijuana. Well, apparently, Abe Koroma has been suspended for a year, meaning his junior campaign is over before it even started. Well done, Abe. Whatever you did (on weed), it was worthy of a year-long suspension, your second suspension in as many years, and quite possibly ended your football career. You're almost as stupid as Maurice Evans.
With basketball and hockey on hiatus for the summer and football still a month away from real games that count, there's nothing happening except the dog days of summer. So I'll be publishing my own personal lists of my five favorite players of the four major Philadelphia sports franchises.
View the first installment on the Phillies here, the second installment on the Sixers here, the third installment on the Flyers here and the fourth installment on the Eagles defense here.
I'm Roy Halladay-ed out. The Phillies should get him. If they don't I'm not going to be happy. I hope Rubes reads this today and just pulls the damn trigger. That's all I have left to say about that.
Now let's shift focus to my favorite Eagles on the offensive side of the football in my lifetime.
1. Donovan McNabb
I know half of Philadelphia just threw up in their mouths at placing Donovan McNabb in my top spot. He's everyone's favorite whipping boy, some criticism extremely deserved, some completely absurd. But I'm as big a McNabb as fan as there is. He is, hands down, the most prolific quarterback in Eagles history, and most certainly the best modern-day QB ever to wear an Eagles uniform. Arguing this just sounds uninformed and frankly, ignorant.
Check the numbers: He's the franchise leader in completions (2,534), passing yards (29,320) and passing touchdowns (194). He's posted a career record of 82-45-1 in the regular season, gone to to playoffs seven times in 10 seasons, gone to five Pro Bowls, five NFC Championship games and a Super Bowl. Add in his 3,109 rushing yards and 26 rushing touchdowns, and you have yourself a great player.
Yes, he didn't win the big one. Yes, he's had subpar performances in many championship games. And yes, he threw up in the Super Bowl. He also led the Eagles to all those opportunities, oftentimes without superior talent at the receiver position. He's battled through injuries, controversies and disappointment, all the while establishing himself as the face of the Eagles.
Yes, he can drive me crazy when he buries a pass at someone's feet or fails to take the open field in front of him with his legs, but he also helped bring a lifeless franchise back from the dead, and he's still here some 10 years later. Each and every season with McNabb at the helm, the Eagles are among the favorites in the NFC. As much as those final losses hurt, it sure as hell beats the Ty Detmer/Rodney Peete/Bubby Brister eras of no hope. A major reason is Number 5.
2. Brian Westbrook
Simply put, Brian Westbrook is the most explosion weapon I've ever seen in an Eagles uniform. Beginning as a punt returner who saw spot duty when he was drafted, Westbrook displayed his dynamic game-changing ability from the start, scaring opposing punt coverage teams to death.
His talent could not be ignored, forcing his way onto the field despite concerns about his size and durability. While injuries have nagged him over his career, you'd be hard-pressed to find a more complete back (LT excluded) in his prime during the years Brian Westbrook has been in Philadelphia. His speed and explosiveness made him a threat to take it to the house on any play. He could break it to the outside, but as he has shown, his strong frame made him just as effective between the tackles. Catching the ball out of the backfield was perhaps his biggest strength, creating all kinds of matchup problems (no linebacker alive can cover him) and despite his stature, he turned into a devastating blocker.
In 2006 and 07, he was as good as back as there was in the league. In 06, he rushed for 1,217 yards, seven touchdowns and averaged 5.1 yards per carry, all while catching 77 passes for for 699 yards and four TDs. He followed that up with a remarkable 2007, rushing for 1,333 yards, seven TDs, averaging 4.8 ypc and nabbing 90 receptions for 771 yards and five TDs, leading all the land with 2,104 yards from scrimmage.
And even last year, playing through injury, he put up 936 yards rushing, 9 TDs, and added 54 catches and five more receiving touchdowns. And of course, he made huge plays in the playoffs, especially against the Vikings. It's damn near impossible to not love Brian Westbrook if you're an Eagles fan.
3. Duce Staley
When the Eagles let Duce Staley go, I was admittedly bummed out. Turns out the Birds were right, as Staley had nothing left in the tank, but regardless, I was not pleased. During his seven seasons in Philadelphia, Duce was nothing short of a workhorse, putting up some phenomenal seasons, specifically in 1998, 99 and 2002, when he rushed for over 1,000 yards and carried the ball at least 258 times (325 in 99) each of those seasons.
When a small contingent of moronic Eagles fans booed the selection of Donovan McNabb in 1999 because they wanted Ricky Williams, I applauded the move of picking a potential franchise QB because the Eagles already had a damn good running back in Staley. Sure, Duce wasn't a speed burner with game-changing speed the way Westbrook has been in his career, but he was a dependable, tough workhorse who also could block and catch the ball out of the backfield well. Not to mention that his effort could never be questioned, giving it his all every down.
In fact, in the 2004 NFC Championship game, while the rest of his teammates looked flat and, frankly, disinterested against the Carolina Panthers, Duce played his tail off. Honestly, to me, he looked like the only Eagle, on either side of the ball, who looked like he gave a shit in that game.
And of course, who can forget his incredible performance in the "Pickle Juice Game" in 2000, when he rushed for 201 yards against the stinking Cowboys in Cowboy Stadium?
Huge Duce fan.
4. Irving Fryar
Irving Fryar played 17 seasons in the NFL. Only three of them were with the Eagles, but two of those three were two of his finest seasons of his career. In 1996, he caught 88 balls for 1,195 yards and 11 touchdowns, earning a Pro Bowl bid in his first season with the team.
Fryar followed that up with another Pro Bowl year in 1997, catching 86 passes for 1,316 yards and six TDs. He tapered off in 98, as age finally caught up with him, but a guy who had been considered a bust and troublesome character for the Patriots turned into a very fine player and upstanding citizen as an Eagle. He also made one of the most incredible catches I've ever seen, leaping up the air, full extension and plucking the ball out of mid-air with one hand, reaching back behind him, and coming down with it as he slammed on the Vet's horrific turf and getting hit. It was amazing.
5. Terrell Owens
I know, Terrell Owens was only an Eagle for one and half seasons, effectively dividing a locker room and causing headaches and distractions beyond belief in his second season. He's an asshole. An attention-whore. A certified lunatic who doesn't give a damn about anyone but himself. He also was a part of the most fun professional football season in my lifetime, and easily the most dominant wide receiver the Eagles have ever had in one season.
The arrival of T.O. in 2004 brought an excitement and fun to the Eagles that had been lacking. No longer was Donovan forced to choose between Todd Pinkston and James Thrash. No, he had his man, finally, and McNabb and Owens instantly became the most lethal QB-WR combo in the league: 77 receptions, 1,200 yards, 14 touchdowns. The only thing that could slow them down was Roy Williams and his dirty fucking tackling.
Owens was electric in that one season, even returning from that injury in the Super Bowl, only to be the best player on the field. Big, fast, talented. He made the offense better than it had ever been under Andy Reid. And as painful as the 05 campaign was, I'd do it all over again if I had the chance, no questions asked.
With Owens in the fold, it wasn't if the Eagles were going to win, it was by how much. They cruised through the season. It was just so damn fun. And Owens was everything we'd hoped he'd be and then some.
The man was a beast. He's tired and played out now, but in 2004, shit, that man could do no wrong in this city. And it was easily my favorite season ever for an Eagles wide receiver.
I expect a ton of people to question my decisions here. No Randall Cunningham, you ask? No, no Randall. He never won a playoff game, and I typically find that the same people who lambast McNabb are ardent Cunningham fans. That annoys the shit out of me to no end, because McNabb is a million times better at playing quarterback than Randall ever was. Not that I blame Randall. I was a fan of his, for sure. Fred Barnett and Calvin Williams almost made the list, because that duo was badass. But that's the list. Take it for what you will.
DeSean may be coming in the future. If he remembers to cross the goal line before dropping the ball.
Monday, July 27, 2009
As I'm sure you're all aware of by now, I am a Penn State graduate who has season tickets to Penn State football. Of course, I was a season ticket holder as a student all four years, but last season was my first as a graduate. It was a great year to get on board, what with the Nittany Lions going undefeated at home and winning the Big Ten. That demolition of Michigan was especially gratifying.
Well, this year, the home schedule sucks big, fat donkey balls. The first three games are garbage, as Penn State hosts Akron, Syracuse and then Temple Sept. 5, 12 and 19. To add insult to injury, all three of those shitfests start at noon o'clock. That's some bullshit right there.
The fourth game, an 8:05 night contest with Iowa, sounds pretty good on paper, but without Shonn Greene, the Hawkeyes don't scare me one bit. It is a revenge game, though, so a thorough thumping on Iowa would be sweet.
Beyond that, there isn't even a respectable game at Beaver Stadium until the pinnacle, Ohio State on Nov. 7. In between, I get the "privilege" of taking in Eastern Illinois (Oct. 10) and Minnesota (Oct. 17). Then Penn State wraps up the home schedule with Indiana on Nov. 14. That means Arkansas Fred and I are doling out $880 to see the likes of Akron, Syracuse, Temple, Eastern Illinois, Minnesota and Indiana. What a crock of shit. Give me more than one or two actual good home games to spend this type of dough on, you jerks.
Do me a favor: Read this and this. Those are two of the best things I've read about the Phillies and Roy Halladay.
This is a plea to Ruben Amaro Jr. Go get Roy Halladay. Just do it. Listen, I understand the hesitation in giving up both Kyle Drabek and J.A. Happ. Drabek is your No. 1 pitching prospect, a guy who's storming through the minor leagues and a guy who has the pedigree on his side. Happ is a 6'6" lefty who currently sits at 7-1 this season with a sub-3.00 ERA — the best numbers of any Phillies starter. I get that.
But let's be reasonable here, Ruben. A year ago, Carlos Carrasco was supposedly the next great Phillies pitcher, the untouchable. Now the Phils can't give him away, just one season later. That is to say, Kyle Drabek looks like a future star, but who knows what he'll look like next year? The answer: Your guess is as good as mine.
J.A. Happ, meanwhile, was a late-season call-up in 2008 and performed extremely well. He has done nothing but improve on that performance in 2009. But Happ is still more unknown than you think. For starters, he was sort of like Kyle Kendrick from two years ago. The Phils called him up with little expectations and no real buzz surrounding him. He did a fine job in spot duty and relief, competed for a starting spot this season and ultimately earned it a few months in.
Happ has been really, really good. A true revelation. But he hasn't exactly performed against any world beaters so far. He's beaten bad teams. When he finally faced a good team, Friday against the Cards, he didn't look so hot. Yes, it was a bad matchup with St. Louis's righthand-heavy lineup, but he picked up his first loss and didn't look good doing it.
And what some fans seem to fail to realize is Happ has yet to be seen a second time by a team. The first go-round, the pitcher always has the advantage. Opposing hitters don't know what the guy has, how his pitches look, what his delivery is coming at them in the box. The second time around, it's not so easy for the pitcher. Batters know what's coming, how it's coming and can adjust. Happ has been good, seems to have excellent control, but his 88-92 fastball may not sneak up on guys so easily the second time around.
That is all to say, basically, Happ and Drabek are not Roy Halladay, a known commodity. And it's not just the fact that Halladay is a known commodity. Those are easy to come by. We all know what a handful of veterans are. But Halladay isn't your average, run-of-the-mill available starter. He is a Cy Young winner. An all-star. One of the greatest pitchers to ever walk the face of the earth. He can give you 7, 8, 9 innings every time out. EVERY time out. Not sometimes. Every damn time. That weary Phillies bullpen sure would like that.
The thing is, as Jayson Stark pointed out, the Phils are leery to give up a pair of pitchers who could potentially be very, very good. But as he states, neither will ever be as good as Roy Halladay. At least, the chances of that are razor thin. Halladay is a once-in-a-generation workhorse. The type of pitcher that really doesn't seem to exist anymore. He would not just make the rotation better; he'd make the bullpen better, resting their arms. He'd make the team better. Even if it means trading away Happ and Drabek.
The future is now for the Phillies. Three, four, five years from now, when Kyle Drabek is in the majors, the Phillies will be drastically different. There's no telling if they'll be contenders in that time, whether they gut the farm system or not. But today, there's no question the Phillies have a chance to repeat. Bring in Halladay, and those chances multiply greatly. Keep him in the fold and you have a core that can win for two, three years. World Series are hard to come by. If you have a real chance to put yourself in position to compete for one, you have to do it.
Now, I'm not saying that Ruben should go in there and tell the Blue Jays: "Whatever you want." Try to keep Happ or Drabek. But when push comes to shove, when the deadline is here, if you have to give them both up to make this happen, well, shit, give them both up. It makes your team better, and isn't that the bottom line?
In five years, maybe Kyle Drabek and J.A. Happ are all-star pitchers. Maybe they aren't. But one thing is for certain: Right now, today, Roy Halladay is better — vastly better — than J.A. Happ and Kyle Drabek combined. He give the Phillies a better shot at winning the World Series. He completes a team that is ready to win now.
The story of the weekend should have been the fact that the red-hot World Fucking Champions just took two of three from the NL Central-leading Cardinals, keeping Albert Pujols in check while winning their seventh straight series. Instead, it's about murder and alcohol and laser pointers. Ugh.
I have to admit, I laughed when I saw the laser pointer schtick directed at Juio Lugo on Saturday. The guy was 3-for-3 in the game, after all, so anything you can to get him out. As Tim McCarver was waxing poetic about how disgraceful it is and how the other fans would do themselves proud by turning the perpetrator in, I sat back and laughed. But really, there is no need for something so childish to take place. It's definitely an unfair distraction to the Cardinals, to Lugo and Albert Pujols. Once was enough, people. Let's leave the laser pointers at home.
To make matters worse, on the same day that Philadelphia was getting another black eye thanks to the laser pointer delay on national television, tragedy took place outside the ballpark. A 22-year-old male, part of a bachelor party celebrating at the ballpark, was beaten to death. Talk about a double-whammy. There's certainly nothing funny about that, especially since it hit home. I went to the Phillies game yesterday for a bachelor party, tailgating early and going all out. Thankfully, our group was simply out for fun. No fights, no yelling matches. Just a group of guys getting drunk and having a blast at the Phils game. The way it should be.
The sad thing about all of this is that when it all boils down, Philadelphia comes out looking bad yet again. Do other cities boo? Of course. Do other cities have drunken fights and fans acting a fool? Absolutely. But honestly, we aren't doing ourselves any favors in that regard. We're all tired of the stereotype, the boarish fans who look for trouble, the ones who throw batteries and snowballs and garbage. It doesn't represent us well, and honestly, it doesn't even represent the majority of Philadelphia fans accurately.
Yet here we are, branded again as savages … and how can we even argue? We interfered with a game. More importantly, more horrifying, we created an atmosphere where a fan, seemingly just out to enjoy some beers, a bachelor party and a ball game, winds up dead. For no good reason other than booze and male pride. Certainly, when you mix alcohol and testosterone, things are going to happen. But how could anyone let it get this far? It's embarrassing. I'm embarrassed.
You'll hear plenty about how ridiculous we are from all over the place. As pathetic as it is that just a few bad seeds can ruin the reputation of a whole, it's true. And as long as these things keep happening in our stadiums with our fans, however few, it will keep up that reputation we all get tired of hearing about. Sports are supposed to be fun. To be an escape. To be enjoyed. Yet here we are, in the midst of another pennant race, scorching hot and in pursuit of one of the greatest pitchers of our era, and the headlines read murder, booze and laser pointers.
I'm as tired as the rest of you when we're branded as low class neanderthals. But when things like the transgressions from this weekend occur, it certainly shines light on why so many outside the city look down on us in disgust.
Friday, July 24, 2009
So you know how the Sixers are in need of a veteran point guard to mentor Jrue Holiday? Andre Miller seems all but gone, and none of the names out there really jump out at you. Sure, Jamaal Tinsley can ball, but the Pacers distance themselves from him for a reason. Bobby Jackson is intriguing, especially to mentor the defensive-minded Holiday, but injuries have ravaged him a lot in his career.
But now a name has arisen that has my juices flowing at the possibility of seeing him a Sixers uniform. Yes, ladies and gentleman, the one true Jason Williams, White Chocolate, is available and no longer retired, via The Big Lead.
Now, at first glance, it may seem like an odd fit. After all, J-Will was never known for his defensive prowess, and as a youngster, he was quite the risk-taker. The man never met a pull-up three he didn't like, and the insanely high-risk/high-reward dimes he'd attempt were both frustrating and tantalizing. None of that seems like it bodes well for the learning curve of one Jrue Holiday, whose game is nothing like that of Williams'.
But, Jason Williams was a young guy who matured into a very good leader and floor general, under control and more careful with the ball, under Hubie Brown in Memphis. He was a solid veteran contributor to the championship Miami team, a stabilizing force making big plays in key moments quite often. And shit, it's Jason Williams. How fucking cool would it be to see him as a Sixer? The guy freakin passed the ball off his elbow for chrissakes!
I mean, honestly, the only way I'd be opposed to the Sixers signing Williams was if they decided to go all sorts of crazy and bring back Allen Iverson. This team isn't winning shit in the foreseeable future anyway.
Make my dream happen, Eddie. Bring in J-Will. Make him a Sixer. That's all I'm asking.
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This Halloween (Saturday, October 31) from 10 pm – midnight on 89.3 KPCC, L.A. Theatre Works will air a sci-fi double bill:
The War of the Worlds by H.G. Wells was adapted by Howard Koch and stars John de Lancie, Meagan Fay, Jerry Hardin, Gates McFadden, Leonard Nimoy, Daryl Schultz, Armin Shimerman, Brent Spiner, Tom Virtue and Wil Wheaton. Originally performed by Orson Welles and his Mercury Theatre of the Air, War of the Worlds is truly the mother of all space invasions, offering a rare combination of chills, thrills and great literature. Join actors from Star Trek and Star Trek:The Next Generation as they recreate this classic radio thriller. The breathless pace and convincing details make it clear why the 1938 broadcast of an “eyewitness report” of an invasion from Mars caused a nationwide panic. Stay tuned after the show for a post-performance interview with Leonard Nimoy.
The Lost World by Sir Arthur Conan Doyle, adapted by John DeLancie, stars Josh Clark, Kyle Colerider-Krugh, Peter Paige, Kirsten Potter, Kate Steele, Tom Virtue and Kenneth Alan Williams. The Lost World is a rollicking adventure that follows a scientific expedition deep into the Amazon jungle right back into the time of dinosaurs and cavemen. Before Jurassic Park, before Indiana Jones – there was Sir Arhur Conan Doyle’s The Lost World! Also: Learn about L.A.’s very own prehistoric explorers with a sound-rich tour of the Dinosaur Lab at the Natural History Museum of Los Angeles County.
The broadcast can be heard locally in Southern California on Saturday from 10 pm to midnight on KPCC 89.3 FM, and can also be streamed on demand at http://www.latw.org/. Both productions are directed by Star Trek: The Next Generation’s John de Lancie.
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PROPERTY-BASED INTEGRITY MONITORING OF OPERATING SYSTEM KERNELS
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Abstract
As the foundation of the trusted computing base, the operating system
kernel is a valuable target for attackers of a computer system
seeking maximum control and privilege. Furthermore, because the
majority of modern security solutions rely on the correctness of at
least some portion of the operating system kernel, skilled attackers
who successfully infiltrate kernel memory can remain undetected
indefinitely.
In this dissertation, we present an approach for detecting
attacks against the kernel's integrity (commonly referred to as
"rootkits"). Our approach, which we call
<em>property-based integrity monitoring</em>, works by monitoring and
analyzing the kernel's state at runtime. Unlike traditional security
solutions, our monitor operates in isolation of, and
independently from, the protected operating system and has direct
access to the kernel's runtime state.
The basic strategy behind property-based monitoring is to identify a
set of <em>properties</em> that are practical
to check, yet are effective at detecting the types of changes
an attacker might make - both known and yet-to-be-discovered. In
this work, we describe a practical and effective property for detecting
persistent control-flow modifications in running kernels, called
state-based control-flow integrity (SBCFI). Furthermore, to
address those data-only attacks that do not violate the kernel's
control-flow, we introduce a high-level policy language system for
enforcing <em>semantic integrity</em> constraints in runtime kernel data.
To evaluate the feasibility and effectiveness of our system, we have
implemented two property-based integrity monitors for the Linux
kernel - one using a virtual machine monitor and the other using a
PCI-based coprocessor. We demonstrate that property-based monitoring
is capable of detecting all publicly-available kernel integrity
threats while imposing less than 1% overhead on
the protected system. We conclude that property-based kernel integrity
monitoring can be both practical and effective.
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The photo of Northrop XF-15 A Reporter is on the next page dedicated to the F-15 A, as it indicates the serial number 43-8335 well seen on the tail of aircraft. The photo of the Northrop F-15 A Reporter s/n 45-59335 comes from AeroFiles site.
Greetings from Poland
Bogdan 04/25/2004 @ 10:26 [ref: 7288]
John H. Tinkle Ridgeville, IN
The F-15A reporter was assigned as the primary aircraft for the 8th Photo Recon Squadron, Johnson AB,Japan. All the Specs and information are available and should be included in your presentation. 02/04/2001 @ 10:24 [ref: 1536]
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Targeting human CD2 by the monoclonal antibody CB.219 reduces intestinal inflammation in a humanized transfer colitis model.
The cell adhesion molecule CD2 facilitates antigen-independent T-cell activation and CD2 deficiency or blockade reduces intestinal inflammation in murine models. We here aimed to evaluate the therapeutic potential of monoclonal antibodies (mAb) specific for human CD2 in colitis treatment. Transfer colitis induced by naïve CD4(+) T cells expressing human CD2 was treated with anti-human CD2 mAb. The mAb CB.219 protected from severe colitis in a preventive treatment regimen, while therapeutic treatment ameliorated intestinal inflammation. Diminished intestinal tissue damage was paralleled by a profound suppression of lamina propria lymphocytes to produce pro-inflammatory cytokines and tumor necrosis factor α as well as the neutrophil chemoattractant CXC motif ligand 1 and the CC chemokine ligand 3. Furthermore, infiltration with macrophages and T cells was low. Thus, reduced intestinal inflammation in our humanized colitis model by targeting CD2 on T cells with the mAb CB.219 suggests a novel approach for colitis treatment.
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HARI SREENIVASAN: Astrophysicists and astronomers
all over the world are celebrating a golden
moment this week, the announcement of a scientific
finding that has Nobel Prize written all over
it.
They witnessed the collision of two incredibly
dense neutron stars and found a scientific
Holy Grail in the process.
It provides further proof that Albert Einstein
was a genius, relatively speaking.
The findings help us understand the universe
better, and, as a result, we now know where
all the gold and silver and platinum in the
world comes from.
It's the focus of our Leading Edge segment
this week.
And science correspondent Miles O'Brien joins
me now.
Miles, tell me, why are they so excited about
this?
MILES O'BRIEN: It's a textbook-changer, Hari.
It happened in August, and it began with two
observations, one of gravitational waves,
ripples in spacetime, if you will, followed
right on its heels by the recording of a gamma
ray burst.
This set off this amazing scientific full-court
press that led to this discovery.
The focus of all this, Hari, are neutron stars
and the collision of two of them.
Neutron stars are what is left over after
a supernova.
A star burns out.
These things are the densest things we know
of in the universe.
These, at the focus of this story, were about
the size of Boston.
And yet they have a mass that is 50 percent
greater than our sun.
They're relatively rare to have two of them
collide.
And it happens once about every 100,000 years
in our galaxy, the Milky Way, Hari.
HARI SREENIVASAN: So, did we just get lucky?
Did all these people just get lucky when these
-- kind of all their beepers and bells and
whistles started going off that something
was afoot?
MILES O'BRIEN: Well, luck favors the prepared
scientist, I guess, in this case.
It began with the LIGO instrument.
This team just recently won the Nobel Prize
for a discovery in 2015 of these gravitational
waves, wrinkles in spacetime, that proved
out Einstein's theory of relativity.
It did that by detecting the collision of
black holes.
Now, in our business of television, we prefer
our science illustrated.
So, when they discovered that there might
possibly be a collision of neutron stars,
that includes an explosion and some light,
and that made people feel a little more excited.
In August, the LIGO instrument detected one
of these gravitational waves, but it was ever
so slightly different.
It happened a little longer, because these
neutron stars move a little slower than black
holes.
Another instrument, subsequently, the Fermi,
which is an orbiting instrument, detected
a gamma ray burst.
Scientists thought they were hot on the trail
of one of these elusive neutron star collisions,
and so they started scrambling.
Edo Berger is part of the team.
He's at Harvard University.
EDO BERGER, Harvard University: As soon as
we received an alert from the LIGO instruments
telling us that they detected a gravitational
wave source, we started calling up observatories
all over the world where we have programs
that are ready to go for that purpose.
We gave them the coordinates of the source
in the sky where they would have to point
the telescope, and as soon as they pointed
a telescope in that direction, we could look
at the images coming in.
MILES O'BRIEN: Working together, the gravitational
wave astronomers and the light wave astronomers
were able to kind of pinpoint this location
very quickly, sort of triangulate in on the
galaxy where it was happening, a galaxy that
is 130 million light years away.
And it turns out it was much more than a light
show.
Once they were able to find it and they watched
this explosion unfold, they were able to really
record the entire electromagnetic spectrum.
And, in it, they were able to see the distinct
signatures of all kinds of elements, including
these heavy elements, gold, silver, and platinum,
proving that is what those -- that furnace,
that explosion is what creates those particular
elements.
HARI SREENIVASAN: So now that we know where
some of these heavy elements come from, what
do scientists do with that information?
MILES O'BRIEN: Well, it would be nice to go
out and get the gold, wouldn't it, Hari?
It's 130 million light years away.
It's a little bit of a problem.
Somebody actually calculated how much gold
would have been created by this particular
collision.
Just so you know, it comes out to about 10
octillion dollars' worth.
That's one followed by 27 zeros.
So we could quit our day jobs, if we can get
out there, Hari.
But, obviously, scientists are not as focused
on the gold itself.
For them, knowledge is gold.
Duncan Brown is a physics professor at Syracuse
University.
DUNCAN BROWN, Syracuse University: This really
is a new type of astronomy.
We're now bringing together all the tools
that humans have to bear on observing the
universe.
We can feel ripples in spacetime.
We can see the light from things colliding
out there in the universe and exploding and
the light from stars.
And bringing all these tools together is going
to allow us to learn so much more about the
universe.
HARI SREENIVASAN: You know, give me a sense
of this collaboration.
Right now, in the United States, we can't
get two parties to agree on something, but
you're talking about different teams from
all over the world responding at the drop
of a hat.
MILES O'BRIEN: Yes.
And there are cases where astronomers will
line up observatories to look at events, and
they collaborate on these things.
But as best we can tell, this is unprecedented
in its scope and its speed of response.
It was really lightning fast once the word
got out.
About 70 observatories, ultimately, were pointed
at this unprecedented event.
HARI SREENIVASAN: Now, all this happened just
last August, but, really, it happened when
the dinosaurs were walking around.
MILES O'BRIEN: Year, it's a little bit of
a mind-bender.
When you look at this event 130 million light
years away, it took that long for the light
to reach us.
So, really, it happened 130 million years
ago.
And that gives you an idea of how old the
universe, and it also gives you an idea of
how these particles, which are created so
far away, ultimately have really great meaning
to us.
We end up wearing them.
It's bling.
(LAUGHTER)
HARI SREENIVASAN: All right, Miles O'Brien,
thanks so much.
MILES O'BRIEN: You're welcome, Hari.
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Jabhat al-Nusra's Goals Extend Beyond Syria
At the end of December, in an unprecedented move, Abu Muhammad al-Julani, the leader of Jabhat al-Nusra in Syria, addressed "the people of Syria and the mujahedeen."
Summary⎙ Print As the Syrian war intensifies, Hussein Jemmo examines the reasons behind the rise of al-Qaeda-affiliated Jabhat al-Nusra, and argues that the battle for Syria is only one step in a wider regional strategy for this group.
The speech suggested that the militant front has become the main force in the fight against the Syrian regime, with no mention of the Free Syrian Army (FSA). Instead, the Islamist group's leader replaced the term "FSA" with "the brigades and militant groups."
The speech came in response to the United States designation of Jabhat al-Nusra as a foreign terrorist organization. But it seemed obvious that Julani was speaking from a strong position, warning that "those who sowed the seeds of the revolution will be the ones to reap its fruits." He also warned his supporters and the people of Syria against attempts to replace the Syrian regime with a Western one.
The speech indicates that the FSA is being subsumed. After having been the leading military entity in the Syrian revolution, the FSA has been pushed to the sidelines compared to Jabhat al-Nusra. The militant praised those parties that have condemned the US decision to designate it as a terrorist group.
During my visit to Aleppo, I got the impression that Jabhat al-Nusra and Islamist factions were still on the sidelines of the conflict in Aleppo's countryside. However, inside the city the situation was different, as expected. Jihadist organizations, mainly Jabhat al-Nusra, were well informed and aware of what was taking place at the international and regional levels.
To begin to understand how Jabhat al-Nusra managed to establish itself in this environment, another question must be asked: Is this militant faction merely seasonal, and will it cease to exist when the regime falls — or will it persist to implement a plan that goes far beyond Syria?
War strategy
In Aleppo's countryside, a member of Jabhat al-Nusra showed me a booklet entitled "Regional War Strategy in Syria." The booklet represents a serious vision by an al-Qaeda analyst. It is available on the internet and helps explain the carefully planned beginnings of jihadism in Syria. According to the study, "The title of the next battle of Damascus will be ‘survival of the smartest,'" and explains how the jihadist environment began to emerge in Syria. Jabhat al-Nusra bases its work in Syria on three pillars.
Resilience
The first pillar is to show resilience: "Syria is a vast and far-reaching battlefield, which necessitates a remarkably resilient force that is able to both move and remain stationary."
Jabhat al-Nusra has shown a reasonable level of intelligence in implementing this strategy, outshining more amateur FSA planners. The group began by sending a few people to the Syrian territories on a reconnaissance mission in the rural areas of Aleppo and west of Idlib.
A limited number of militants were then sent to the battlefield, positioned on the most hostile lines of confrontation in the city of Aleppo. They earned the respect of FSA fighters, who have been more than happy to let them take over this deadly position.
When more Arabs, foreign fighters and a limited number of Syrians with past jihadist leanings began to pour into the Syrian territories, Jabhat al-Nusra began to move its combat team to the countryside. It carefully chose areas prone to embracing the jihadist ideology and willing to fight for it. First, the militant group took over the vicinity of fortified military bases in the west of the countryside, such as the Regiment 46 and Sheikh Suleiman base. With the help of allied battalions, Jabhat al-Nusra also took control of the rural areas in the east of Aleppo's countryside.
Jabhat al-Nusra didn't station its militants in civilian neighborhoods that would be easily besieged. The Islamist group has also not been entirely comfortable with FSA military leaders, "who could strike a deal with the West to wipe them out," according to one member of the jihadist organization.
Because of its strategy, Jabhat al-Nusra has been able to increase its control over the supply lines and logistics between Turkey and Aleppo. By attacking large military barracks, jihadist militants have managed to seize large quantities of ammunition and military equipment without the need for external funding, in contrast to the FSA, which until recently continued to receive Kalashnikov bullets through intermediaries in Turkey.
Jabhat al-Nusra has managed to attract battalions with jihadist inclinations to its ranks. For instance, Sheikh Tawfiq Shahabuddin, leader of the Nur al-Din Zanki Brigades, declared the brigades’ defection from the FSA-affiliated al-Tawhid brigade on Dec. 6. Although the sheikh did not join the ranks of Jabhat al-Nusra after his defection, he deserted the FSA right after he participated in an attack on the Sheik Suleiman base in collaboration with the jihadist organization.
The jihadist group has taken another tactical step: reducing the large size of brigades, such as the al-Tawhid brigade, by encouraging marginalized factions within these larger brigades to become self-sufficient.
Interestingly, al-Tawid launched a sizable attack on the infantry school in northern Aleppo without involving Jabhat al-Nusra, which hasn't sought to expand in the northern countryside which has a weak supply line compared to the western countryside of Aleppo. This is also true of the eastern countryside, whose importance derives from the fact that it overlooks the governorates of al-Raqqa, al-Hasaka and Deir al-Zour.
These points suggest that Jabhat al-Nusra has succeeded in adopting a flexible strategy without making significant mistakes. Therefore, when the US listed it as a terrorist group, no military group on the ground dared to support the US move, because the supply lines have come under the control of the jihadists.
Functional role
The second pillar, according to the vision of the "al-Qaeda analyst," is to develop a specific "functional role" for jihadists — whose role has been limited to performing military activity — because such a war in Syria "would return us to the laws of the jungle and survival of the fittest."
According to the same analyst, the intended result is for "the faction that holds power on the ground to eventually hold other types of power." Consequently, the group would temporarily distance itself from administrative and service matters to avoid any divisions of power, as happened in Iraq when al-Qaeda declared the Islamic State of Iraq during the war, established ministries and appointed administrators.
Also, the continuation of the war — which al-Qaeda sees as a prize for jihad — will put the rural or civil communities under military force, unlike less heated and shorter wars, where civil forces begin to express themselves and to demand power and the return of soldiers and fighters to their barracks at their conclusion, as happened in Libya.
How successful is Jabhat al-Nusra in achieving this second strategy?
Clearly, this al-Qaeda-affiliated group in Syria is very flexible in reorganizing its priorities. It is also known for its ability to change its tactics smoothly to achieve its strategic goal of total control under a popular cover that would give it the ability to later turn itself into a local Syrian faction.
It seems that this goal is a priority in the short term, to distance the group from any resemblance to foreign forces in Syrian society.
Jabhat al-Nusra realizes that the process of adhering to the limitations of society requires abandoning the tactics used in the "Islamic State of Iraq" and the complex administrative frameworks that inevitably led to its demise. Moreover, the group's leaders are aware that a clash is inevitable with FSA leaders and civilian figures, with a broad spectrum of Syrian Islamists who do not like the foreign Islamic nature of al-Qaeda. They all promptly endorsed the establishment of local popular bodies to run the liberated neighborhoods in the city of Aleppo with the support of local clerics. They think that the infiltration of Jabhat al-Nusra into civil departments or through loyalists is something of the past, and that all doors have been shut before them except for that of military jihad.
This is true, while taking into account that this is what Jabhat al-Nusra actually wants: not to be involved in administrative affairs in Syria. "Self-protection" is more essential and requires hard power, unlike in Yemen, where al-Qaeda is involved administratively because it is an arena for "self-assertion" requiring soft power.
The third pillar of the al-Qaeda strategy in Syria is the principle of "prolonging the conflict." This is based primarily on rejecting any consensus over a political solution, such as that put forward by UN Envoy Lakhdar Brahimi. Accordingly, a series of enemies are being promoted, including the "Alawite regime" and the "Zionist enemy," in addition to whatever form of rule might replace the Assad regime.
Continuation of the conflict
Interestingly, al-Qaeda is very confident that Washington will remain reluctant to intervene. Jabhat al-Nusra leader Julani accused the US and the international community of "prolonging the reign of the Syrian regime," although this is exactly what the group wants. The scene has become multi-faceted. Just as the Americans rely on the Russian rejection of a UN resolution that would authorize military intervention, Jabhat al-Nusra also feeds on US hesitation to supply weapons to the Syrians. Washington is speaking morally on the back of Moscow, and al-Qaeda is expanding popularly on the back of Washington.
During my meeting with one of the leaders of Jabhat al-Nusra in an Aleppo suburb, we discussed the FSA's lack of arms and munitions. This leader thought that this is the best thing currently happening, because it will put Syria back into the hands of God. But this will not happen before the rebels discover that the West has deceived them for over a year.
Today, a large group of Syrians, who were calling on the US to intervene to save them, are accusing opposition forces of being affiliated with the American agenda! This was even clearer during a protest under the slogan, "There is No Terrorism except Assad's Terrorism."
These three pillars have been working smoothly and successfully until now. Due to the moral chaos surrounding some FSA brigades, and the fact that some of these brigades committed theft, the presence of Jabhat al-Nusra has gained support among the industrial middle class. Many factories have been looted, and the hopes of industry owners who escaped this wave of looting were relying on the movement's entrance into industrial regions "because they do not steal," according to one industry owner. The movement's access to Aleppo's society is not only religious, but also economic. It did not declare the establishment of courts to try civilians for violations they committed. Deep-seated corruption in the delicately improvised judiciary formed by militias was exposed. These indications paved the way for a call for help from the people to Jabhat al-Nusra to manage their daily affairs. In Aleppo, the protesters reiterate daily the following slogan: "The Free Syrian Army are thieves…We want the Islamic army."
The success or failure of Jabhat al-Nusra in Syria is mainly linked to both Aleppo and Idlib, and other cities constitute a future, long-term strategy. For example, in Aleppo, Jabhat al-Nusra is adopting a strategy of direct confrontation with the regime's forces, while the Iraqi model prevails in Damascus and Hama suburb (involving car bombs and suicide bombings).
During the first half of 2012, members of Jabhat al-Nusra — who had come from different countries — lived as guests in Aleppo suburbs, and fought without having permanent residence in the city. Today, a small community is expanding to reach the scale of "al-Qaeda," or at least what al-Qaeda's leaders aspire to. They see Syria as merely a single step in a journey taking them from Aleppo and Idlib to the Alawite mountains in Latakia, and eventually extending to regions beyond Syria.
This article was translated by Sahar Ghoussoub, Naria Tanoukhi and Pascale Menassa.
Al-Monitor brings together top journalists from across the Middle East. Including Egypt, Iran, Iraq, Israel, Lebanon, Palestine, Syria and Turkey. Plus, we have daily translations from 20 major news organizations in the region.
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The Phillies shopping list this Holiday season includes more than a legitimate middle of the order bat like a Bryce Harper or Manny Machado.
The club has needs at the back end of the bullpen and also aims to strengthen the starting rotation behind Aaron Nola and Jake Arrieta.
Jon Morosi of MLB.com reported on Monday the Phillies have been linked to veteran lefty-reliever Andrew Miller. The 33-year-old is coming off an injury-plagued campaign in 2018 but posted outstanding numbers during the four years prior and, if healthy, would be a noticeable upgrade in the setup role for the bullpen.
Another potential option for the setup role is righty Adam Ottavino who posted a career-best 2.43 ERA last season while pitching in Colorado. His 112 strikeouts set a franchise season high for a reliever.
Zach Britton, whom the Phillies have long fancied, is also a potential target. The lefty was traded from Baltimore to New York last year and could potentially slide into the role of closer if needed. The two-time All-Star has a career ERA+ of 153.
As for the starting rotation, multiple options have been linked to the Phillies over the last two weeks - including lefty Patrick Corbin. The Yankees were believed to be the favorites to land the 29-year-old entering the offseason, but things may have changed considering the club recently acquired James Paxton from the Mariners. Corbin tossed 200 innings last year, posting a 3.15 ERA.
Another potential option is righty Nathan Eovaldi. The 28-year-old is coming off a fantastic postseason with Boston and posted a 3.81 ERA in 21 starts last season. He may not make as much of a splash as Corbin, but he would be a welcome addition to the rotation.
The Winter Meetings open in Las Vegas in less than two weeks.
If you think the rumors are flying fast now, you haven't seen anything yet.
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This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. The subproject and investigator (PI) may have received primary funding from another NIH source, and thus could be represented in other CRISP entries. The institution listed is for the Center, which is not necessarily the institution for the investigator. This project investigates the potential for cytokines promoting cell mediated immune responses for restoring/augmenting antiviral immune responses following SIV infection by administration of IL-12 alone or in combination with IL-2 or Il-15 and a structured antiviral chemotherapy interruption protocol using PMPA. The past year has been devoted primarily to the monitoring of the rhesus infected with SIVmac251 and treated with STI and IL-12. Surviving monkeys with variable levels of immune mediated control of the SIV viremia have been subjected to chronic administration of IL-15 followed by IL-7 to document the effect of these T cell directed cytokines on the viral loads and antiviral immune responses in vivo. In addition, parallel studies were initiated in sooty mangabeys in attempt to delineate T cell dysfunction in the pathogenic rhesus vs non pathogenic mangabey monkey. The role of regulatory T cells was investigated longitudinally in both models as well as the potential role of PD-1 expression on CD4 and CD8+ T cells. In addition, a preliminary evaluation of the in vivo administration of inhibitors of potassium channels specific for central memory and effector memory T cells was performed in conjunction with an immunization with influenza.
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;;; fringe-helper.el --- helper functions for fringe bitmaps
;;
;; Copyright (C) 2008 Nikolaj Schumacher
;;
;; Author: Nikolaj Schumacher <bugs * nschum de>
;; Version: 0.1.1
;; Keywords: lisp
;; URL: http://nschum.de/src/emacs/fringe-helper/
;; Compatibility: GNU Emacs 22.x
;;
;; This file is NOT part of GNU Emacs.
;;
;; This program is free software; you can redistribute it and/or
;; modify it under the terms of the GNU General Public License
;; as published by the Free Software Foundation; either version 2
;; of the License, or (at your option) any later version.
;;
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this program. If not, see <http://www.gnu.org/licenses/>.
;;
;;; Commentary:
;;
;; fringe-helper contains helper functions for fringe bitmaps.
;;
;; `fringe-helper-define' allows you to to define fringe bitmaps using a visual
;; string replesentation. For example:
;;
;; (fringe-helper-define 'test-bitmap '(top repeat)
;; "XX......"
;; "..XX...."
;; "....XX.."
;; "......XX")
;;
;; You can also generate arguments for `define-fringe-bitmap' yourself, by
;; using `fringe-helper-convert'.
;;
;; fringe-helper also provides a few stock bitmaps. They are loaded on demand
;; by `fringe-lib-load' and adapt to the current fringe size to a certain
;; extend.
;;
;; `fringe-helper-insert' inserts a fringe bitmap at point and
;; `fringe-helper-insert-region' inserts a fringe bitmap along a region.
;; `fringe-helper-remove' removes both kinds.
;;
;;
;; Here's an example for enhancing `flymake-mode' with fringe bitmaps:
;;
;; (require 'fringe-helper)
;; (require 'flymake)
;;
;; (defvar flymake-fringe-overlays nil)
;; (make-variable-buffer-local 'flymake-fringe-overlays)
;;
;; (defadvice flymake-make-overlay (after add-to-fringe first
;; (beg end tooltip-text face mouse-face)
;; activate compile)
;; (push (fringe-helper-insert-region
;; beg end
;; (fringe-lib-load (if (eq face 'flymake-errline)
;; fringe-lib-exclamation-mark
;; fringe-lib-question-mark))
;; 'left-fringe 'font-lock-warning-face)
;; flymake-fringe-overlays))
;;
;; (defadvice flymake-delete-own-overlays (after remove-from-fringe activate
;; compile)
;; (mapc 'fringe-helper-remove flymake-fringe-overlays)
;; (setq flymake-fringe-overlays nil))
;;
;;
;;; Change Log:
;;
;; 2008-06-04 (0.1.1)
;; Fixed bug where `fringe-helper-remove' missed overlays at the end.
;; Fixed `fringe-lib-load' to work when already loaded.
;;
;; 2008-04-25 (0.1)
;; Initial release.
;;
;;; Code:
(eval-when-compile (require 'cl))
(defun fringe-helper-convert (&rest strings)
"Convert STRINGS into a vector usable for `define-fringe-bitmap'.
Each string in STRINGS represents a line of the fringe bitmap.
Periods (.) are background-colored pixel; Xs are foreground-colored. The
fringe bitmap always is aligned to the right. If the fringe has half
width, only the left 4 pixels of an 8 pixel bitmap will be shown.
For example, the following code defines a diagonal line.
\(fringe-helper-convert
\"XX......\"
\"..XX....\"
\"....XX..\"
\"......XX\"\)"
(unless (cdr strings)
;; only one string, probably with newlines
(setq strings (split-string (car strings) "\n")))
(apply 'vector
(mapcar (lambda (str)
(let ((num 0))
(dolist (c (string-to-list str))
(setq num (+ (* num 2) (if (eq c ?.) 0 1))))
num))
strings)))
(defmacro fringe-helper-define (name alignment &rest strings)
"Define a fringe bitmap from a visual representation.
Parameters NAME and ALIGNMENT are the same as `define-fringe-bitmap'.
Each string in STRINGS represents a line of the fringe bitmap as in
`fringe-helper-convert'."
(declare (indent defun))
`(define-fringe-bitmap ,name
(eval-when-compile (fringe-helper-convert ,@strings))
nil nil ,alignment))
(defun fringe-helper-insert (bitmap pos &optional side face)
"Insert a fringe bitmap at POS.
BITMAP is the name of a bitmap defined with `define-fringe-bitmap' or
`fringe-helper-define'. SIDE defaults to 'left-fringe and can also be
'right-fringe. FACE is used to determine the bitmap's color.
The function returns an object suitable for passing to
`fringe-helper-remove'."
(let* ((display-string `(,(or side 'left-fringe) ,bitmap .
,(when face (cons face nil))))
(before-string (propertize "!" 'display display-string))
(ov (make-overlay pos pos)))
(overlay-put ov 'before-string before-string)
(overlay-put ov 'fringe-helper t)
ov))
(defun fringe-helper-insert-region (beg end bitmap side &optional face)
"Insert fringe bitmaps between BEG and END.
BITMAP is the name of a bitmap defined with `define-fringe-bitmap' or
`fringe-helper-define'. SIDE defaults to 'left-fringe and can also be
'right-fringe. FACE is used to determine the bitmap's color. The
function returns an overlay covering the entire region, which is suitable
for passing to `fringe-helper-remove'. The region grows and shrinks with
input automatically."
(let* ((display-string `(,(or side 'left-fringe) ,bitmap .
,(when face (cons face nil))))
(before-string (propertize "!" 'display display-string))
(parent (make-overlay beg end))
ov)
(save-excursion
(goto-char beg)
(goto-char (point-at-bol 2))
;; can't use <= here, or we'll get an infinity loop at buffer end
(while (and (<= (point) end) (< (point) (point-max)))
(setq ov (make-overlay (point) (point)))
(overlay-put ov 'before-string before-string)
(overlay-put ov 'fringe-helper-parent parent)
(goto-char (point-at-bol 2))))
(overlay-put parent 'fringe-helper t)
(overlay-put parent 'before-string before-string)
(overlay-put parent 'insert-in-front-hooks
'(fringe-helper-modification-func))
(overlay-put parent 'modification-hooks
'(fringe-helper-modification-func))
parent))
(defun fringe-helper-modification-func (ov after-p beg end &optional len)
(if after-p
(if (eq beg end)
;; evaporate overlay
(when (= (overlay-start ov) (overlay-end ov))
(delete-overlay ov))
;; if new lines are inserted, add new bitmaps
(let ((before-string (overlay-get ov 'before-string))
fringe-ov)
(save-excursion
(goto-char beg)
(while (search-forward "\n" end t)
(setq fringe-ov (make-overlay (point) (point)))
(overlay-put fringe-ov 'before-string before-string)
(overlay-put fringe-ov 'fringe-helper-parent ov)))))
;; if a \n is removed, remove the fringe overlay
(unless (= beg end)
(setq beg (max beg (overlay-start ov)))
(setq end (min end (overlay-end ov)))
(save-excursion
(goto-char beg)
(while (search-forward "\n" end t)
(let ((overlays (overlays-in (point) (1+ (point)))))
(while overlays
(when (eq (overlay-get (car overlays) 'fringe-helper-parent) ov)
(delete-overlay (car overlays))
(setq overlays nil))
(pop overlays))))))))
(defun fringe-helper-remove (fringe-bitmap-reference)
"Remove a fringe bitmap."
(unless (or (not (overlay-buffer fringe-bitmap-reference))
(overlay-get fringe-bitmap-reference 'fringe-helper-parent))
;; region
(dolist (ov (overlays-in (overlay-start fringe-bitmap-reference)
(1+ (overlay-end fringe-bitmap-reference))))
(when (eq (overlay-get ov 'fringe-helper-parent) fringe-bitmap-reference)
(delete-overlay ov)))
(delete-overlay fringe-bitmap-reference)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun fringe-lib-load (pattern &optional side)
"Load a stock bitmap.
It returns the symbol name of the loaded bitmap, which is suitable for passing
to `fringe-helper-insert'. The actual work of defining the bitmap is only done once.
PATTERN can be one of the following:
`fringe-lib-exclamation-mark': an exclamation mark
`fringe-lib-question-mark': a question mark
`fringe-lib-zig-zag': a zig-zag pattern
`fringe-lib-wave': a wavy-line pattern
`fringe-lib-stipple': a stipple pattern
`fringe-lib-full': a solid color
SIDE should be either 'left-fringe or 'right-fringe and defaults to the former."
(let ((fringe-width (frame-parameter (selected-frame)
(or side 'left-fringe)))
(alignment (when (eq (car pattern) 'repeat)
(setq pattern (cdr pattern))
'(top t))))
(while (> (caar pattern) fringe-width)
(pop pattern))
(setq pattern (cdar pattern))
(or (car (memq (car pattern) fringe-bitmaps))
(define-fringe-bitmap (car pattern) (cdr pattern) nil nil alignment))))
(defconst fringe-lib-exclamation-mark
`((5 fringe-lib-exclamation-mark-5 .
,(eval-when-compile
(fringe-helper-convert "...XX..."
"..XXXX.."
"..XXXX.."
"...XX..."
"...XX..."
"........"
"........"
"...XX..."
"...XX...")))
(0 fringe-lib-exclamation-mark-0 .
,(eval-when-compile
(fringe-helper-convert ".XX....."
".XX....."
".XX....."
".XX....."
".XX....."
"........"
"........"
".XX....."
".XX.....")))))
(defconst fringe-lib-question-mark
`((5 fringe-lib-question-mark-5 .
,(eval-when-compile
(fringe-helper-convert "...XX..."
"..XXXX.."
"..X..X.."
"....XX.."
"...XX..."
"...XX..."
"........"
"...XX..."
"...XX...")))
(0 fringe-lib-question-mark-0 .
,(eval-when-compile
(fringe-helper-convert ".XX....."
"XXXX...."
"X..X...."
"..XX...."
".XX....."
".XX....."
"........"
".XX....."
".XX.....")))))
(defconst fringe-lib-zig-zag
`(repeat
(0 fringe-lib-zig-zag-0 .
,(eval-when-compile
(fringe-helper-convert "X......."
"X......."
".X......"
".X......"
"..X....."
"..X....."
".X......"
".X......")))))
(defconst fringe-lib-wave
`(repeat
(0 fringe-lib-wave-0 .
,(eval-when-compile
(fringe-helper-convert "X......."
".X......"
"..X....."
"..X....."
"..X....."
".X......"
"X......."
"X.......")))))
(defconst fringe-lib-stipple
`(repeat
(0 fringe-lib-stipple-0 .
,(eval-when-compile
(fringe-helper-convert "XXXXXXXX"
"XXXXXXXX"
"XXXXXXXX"
"........"
"........"
"........")))))
(defconst fringe-lib-full
`(repeat
(0 fringe-lib-full-0 .
,(eval-when-compile
(fringe-helper-convert "XXXXXXXX")))))
(provide 'fringe-helper)
;;; fringe-helper.el ends here
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by Gilad Atzmon
Israeli media outlet Ynet reported last weekend that five staunch supporters of Israel, including Alan Dershowitz, sent a letter to Netanyahu “imploring him to reconsider (a) plan that will see illegal African migrants sent to Uganda or Rwanda, warning of the 'incalculable damage' it could have on Israel and Jews' 'moral standing.'
“We fear,” the letter states, “that a mass expulsion could cause incalculable damage to the moral standing of Israel and of Jews around the world.” If for one minute you thought that Dershowitz & Co care about the thousands of Black refugees and their human rights you were sadly mistaken. The letter writers clearly explain that their concern is for Jews and their reputation.
The five advocates for Israel counselled PM Netanyahu that the expulsion of African refugees will reflect badly on the Jews.
One of the authors of the cautionary letter was Rabbi Marvin Hier, who accepted an invitation by US President Donald Trump to offer a prayer at his presidential inauguration. Heir was also one of the founders of the Simon Wiesenthal Centre and its Museum of Tolerance. Heir probably realised that crying about the Holocaust while being associated with another one isn’t such a clever idea.
I am left wondering, if Dershowitz, Heir, Abe Foxman and the others can see that the treatment of Africans in the Jewish State is ‘bad for the Jews,’ how is it that they fail to see that Israel’s brutal occupation and racist policies against Palestinians have a similar impact on ‘Jewish moral standing’?
I probably expect too much ethical thinking.
If they want to burn it , you want to read it..
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A top oil industry official delivered a clear warning to President Obama Wednesday: approve the Keystone XL pipeline or face “huge political consequences.”
American Petroleum Institute President Jack Gerard urged Obama to quickly approve the pipeline, which would carry oil sands crude from Alberta, Canada, to refineries along the Gulf Coast.
A payroll tax cut package signed into law by Obama last month includes a GOP-backed provision requiring the president to make a final decision on the pipeline within 60 days.“I think it would be a huge mistake on the part of the president of the United States to deny the construction of the Keystone XL pipeline,” Gerard said during the powerful oil industry trade association’s annual “State of American Energy” event Wednesday.
“Clearly, the Keystone XL pipeline is in the national interest. A determination to decide anything less than that I believe will have huge political consequences.”
When I read, "Clearly, the Keystone XL pipeline is in the national interest...." I remember the radio guests and articles that tell a different story. In essence, the U.S. oil industry imports more oil than it uses in America, so the Keystone XL pipeline is just another profit grab for oil executives and has nothing to do with American's interests.
Remember, neocons add a provision in the last last-minute legislative battle, another poison pill for America, just like killing the USPS.
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All in all, Miliband's team will likely be pleased — there aren't
any excruciating moments or major slip-ups, and he seems to win
Brand over by the end of the meeting— but those who criticised
him for trivialising the election by doing the interview in the
first place are unlikely to become sudden converts.
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Handy Sony map shows where all UK Vitas are
PlayStation has put together a map to show consumers where they can play Vita.The new handheld goes on sale on February 22nd, and is available to play in locations up and down the UK, including all GAME and GameStation stores. You can check out the full map here.It is part of the platform holder's aggressive bid to achieve mass awareness for the new portable. The firm is still in the process of taking its Vita rooms tour across the UK, while ads are starting to appear across the country and on TV.
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Contentions
Yes, we’ve all heard this line before. But Obama’s poll numbers are plummeting with Latino voters, and he has Rep. Luis Gutierrez breathing down his neck, so the president has suddenly remembered the urgency of tackling the immigration debate. Or at the very least, he’s remembered the urgency of paying lip service to it:
The guest list alone is enough to give you an idea of the seriousness of the meeting. It included Al Sharpton, Center for American Progress CEO John Podesta, AFL-CIO head Richard Trumka, COO of Facebook Sheryl Sandberg, NYC Mayor Michael Bloomberg, a bishop, a handful of east- and west-coast lawmakers, former Homeland Security Secretary Michael Chertoff and NYC police commissioner Ray Kelly (conspicuously snubbed: anyone from the state of Arizona).
Also note that the meeting only lasted an hour, so Obama’s not even doing a good job pretending to care about this issue.
However, the administration promised afterward that the immigration debate will become a major priority for the president. “The president noted that he will continue to work to forge bipartisan consensus and will intensify efforts to lead a civil debate on this issue in the coming weeks and months,” read a White House statement.
Since Obama hasn’t announced any sort of immigration legislation, and Republicans in Congress would likely slap it down immediately if he did, it’s unclear exactly what “efforts” the president will be “intensifying” in the coming months. More likely Obama will continue to half-heartedly and intermittently bring up the issue until after the election.
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Utilization of healthcare by very-low-birthweight infants during their first year of life.
New knowledge in perinatal medicine has resulted in increased survival of very-low-birthweight (VLBW) infants. After leaving hospital, the child is seen at regular medical check-ups, but there is often a persistent worry about the child which affects the family as a whole. This can lead to an increased utilization of healthcare. Our objective was therefore to describe the utilization of healthcare by VLBW infants during their first year of life and its relation to high-risk diagnoses in the neonatal period. The study group comprised 36 infants born at gestational ages of < or = 31 wk and with a birthweight of < or = 1500 g, and was compared with a control group of 36 full-term infants. Utilization of healthcare by the VLBW infants was higher than that by the control group in paediatric and ophthalmic outpatient clinics. The total number of contacts with healthcare was on average 38.7 versus 17.4. High-risk diagnoses in the neonatal period did not correlate with utilization of care, except for visits to the paediatric outpatient clinic, especially planned visits. Further studies focusing on how to support these families after leaving hospital are therefore needed.
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For too long in our country, politicians have weaponized religion for their own selfish gain, fomenting bigotry, fears and suspicions based on the faith, religion or spiritual practices of their political opponents.
Whether we think of ourselves as Christians, Hindus, Muslims, Sikh, Buddhists, Jews, atheists, agnostics, or anything else, it is imperative that we stand united in our commitment to protect religious freedom and the right to worship or not worship, safely and without the fear of retribution.
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We must stand together, and with one voice condemn those who seek to incite bigotry based on religion. We cannot allow those who are anxious to exploit our differences to drive a wedge between us. We cannot and will not tolerate prejudicial treatment of those with whom we disagree, any more than we would tolerate such treatment of those with whom we agree.
Standing up for freedom of religion for all people is as critical now as it’s ever been--hatred and bigotry are casting a dark shadow over our political system and threatening the very fabric of our country. The heartbreaking atrocity in Pittsburgh, where 11 people were murdered while worshiping at their synagogue, is but the latest reminder of the horrible potential consequence of prejudice and bigotry. The shooting at the synagogue in Pittsburgh, the shooting at the Sikh temple in Oak Creek, Wis., the shooting at the mosque in New York and too many others to list serve as devastating and atrocious reminders of what this kind of hatred and this kind of bigotry can lead to.
While I oppose the nomination of Brian Buescher to the U.S. District Court in Nebraska, I stand strongly against those who are fomenting religious bigotry, citing as disqualifiers Buescher’s Catholicism and his affiliation with the Knights of Columbus. If Buescher is “unqualified” because of his Catholicism and affiliation with the Knights of Columbus, then President John F. Kennedy, and the 'liberal lion of the Senate' Ted Kennedy would have been “unqualified” for the same reasons.
Article 6 of the U.S. Constitution clearly states that there "shall be no religious test" for any seeking to serve in public office.
No American should be told that his or her public service is unwelcome because “the dogma lives loudly within you” as Sen. Dianne Feinstein Dianne Emiel FeinsteinSenators offer disaster tax relief bill Democrats back away from quick reversal of Trump tax cuts Congress must save the Postal Service from collapse — our economy depends on it MORE (D-Calif.) said to Amy Coney Barrett during her confirmation hearings in 2017 to serve as U.S. Circuit Court judge in the 7th Circuit.
While I absolutely believe in the separation of church and state as a necessity to the health of our nation, no American should be asked to renounce his or her faith or membership in a faith-based, service organization in order to hold public office.
The party that worked so hard to convince people that Catholics and Knights of Columbus like Al Smith and John F. Kennedy could be both good Catholics and good public servants shows an alarming disregard of its own history in making such attacks today.
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We must call this out for what it is – religious bigotry. This is true not just when such prejudice is anti-Catholic, but also when it is anti-Semitic, anti-Muslim, anti-Hindu, or anti-Protestant, or any other religion.
In politics or at home, we Americans can disagree with and oppose people if we are concerned about their views, opinions, or their commitment to uphold their constitutional duties. However, we must not claim or imply that an individual is not qualified because of their religion or their membership in a particular religious organization, or their belief in the tenets of their faith.
While deployed in the Middle East, I saw first-hand the suffering and violence that is inherent to religious bigotry. In Iraq in 2005, I saw candidates for political office labelled according to their sectarian religion, with supporters of “Shia candidates” threatening attacks to the polling centers where supporters of “Sunni candidates” were casting their votes, and vice versa. These insane sectarian divides and prejudices gave rise to ISIS and other terrorist organizations and are the cause of immense human suffering and wars in the Middle East.
If we can all agree that we do not want prejudice and bigotry to rule our nation, then we must stand united to denounce it whenever it raises its ugly head.
Elected leaders engaging in religion-baiting are playing with fire. They are sacrificing the well-being, peace and harmony of our country to satisfy their own political ambitions for partisan political interests.
We must stand together, call out and reject religious bigotry no matter where it comes from, and fight to protect the freedoms and principles that bind us together as Americans.
Gabbard represents Hawaii’s 2nd District.
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“Ask Brianna” is a column from NerdWallet for 20-somethings or anyone else starting out. I’m here to help you manage your money, find a job and pay off student loans — all the real-world stuff no one taught us how to do in college. Send your questions about postgrad life to [email protected].
Q: I’m going to college for the first time in my late 20s after working for several years. How do I pay for it while dealing with potentially less income in the meantime?
A: Making a big change like this requires weighing the benefits against the cost. While only you can decide whether college is the right move, in most cases, a postsecondary degree is a straight path to more money and more job security. Median after-tax earnings of bachelor’s degree holders were 61 percent higher than high school graduates’ in 2015, according to a report by the College Board; median earnings of associate’s degree recipients were 23 percent higher.
But expecting that you’ll make more money someday won’t totally ameliorate the sticker shock now. Make college attainable, without amassing a mountain of debt, by maximizing financial aid and re-evaluating your budget. Here’s how.
GET HELP FROM LESSER-KNOWN SOURCES
As an older student going back to school, you have access to money for college that younger students may not. If you’re in a labor union, ask about scholarships open to members. If you plan to keep working — generally the most cost-effective option for students juggling other financial obligations — many companies offer tuition reimbursement up to a certain amount each year. Chances are you don’t even know it’s available, says Ted Beck, president and CEO of the National Endowment for Financial Education.
“I’m surprised how often as a manager I’ve had to remind people that we have it,” Beck says.
Are you eager to change careers or learn a new skill that’s in high demand in your region? Look into nearby community colleges’ workforce development programs, Beck says. Local companies in need of specially trained workers may help pay for your schooling and consider you for a job afterward.
Siemens, for example, offers subsidized apprenticeship programs that can lead to full-time jobs in fields like manufacturing technology; participants are paid while they attend classes and work for the company. Contact your nearest community college or state apprenticeship office , listed on the U.S. Department of Labor’s website, to learn about programs available near you.
MAXIMIZE GRANTS
The Free Application for Federal Student Aid, known as the FAFSA, isn’t just for 18-year-olds. Everyone interested in higher education should fill it out to qualify for federal, state and school-based financial aid. On the FAFSA, students age 24 or older are considered independent students, so only your income and assets — not those of your parents — will be used to determine eligibility for aid.
One of the most valuable undergraduate financial aid options is the federal Pell Grant , which doesn’t have to be paid back. For 2017-18, the maximum award is $5,920, but the amount you’ll receive depends on financial need and your course load. Use the government’s FAFSA4Caster to determine if you qualify for a Pell Grant. They can be used for up to 12 semesters, including summers. Nearly half of Pell Grant recipients in 2014-15 were 24 or older, according to the College Board.
If a Pell Grant won’t cover your college costs, prioritize taking out federal student loans before private loans, as the federal ones come with more flexible repayment options. Keep loan payments affordable by borrowing no more than you expect to earn the year after you graduate.
SPEND AND SAVE WITH PURPOSE
While in school — or if you’re saving up for it — you’ll likely need to re-evaluate spending to address any lost income or to cover out-of-pocket education costs. Put money into an emergency fund, too, so unexpected expenses don’t derail your plans.
Avoid thinking of it as budgeting if that sounds restrictive; frame it as a spending plan that aligns your purchases to your values, says Megan Lathrop, lead money coach at Capital One in San Francisco. Consider cutting in half the amount you spend on eating out, she says, and scrutinize the cost of attending weddings or other pricey obligations. Remember your end goal: more earning potential and a better chance you’ll be able to ride out a tough labor market.
This article was provided to The Associated Press by the personal finance website NerdWallet . Brianna McGurran is a staff writer at NerdWallet. Email: [email protected]. Twitter: @briannamcscribe.
RELATED LINKS
NerdWallet - How to Get Free Pell Grant Money to Pay for College: https://nerd.me/pell-grant
U.S. Department of Labor - ApprenticeshipUSA State Contact List: https://www.doleta.gov/OA/contactlist.cfm
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Reappraisal of serum alpha-foetoprotein as a surveillance test for hepatocellular carcinoma during entecavir treatment.
The aim of this study was to re-evaluate the diagnostic performance of alpha-foetoprotein (AFP) as a surveillance test for hepatocellular carcinoma (HCC) in patients with hepatitis B virus-related chronic liver disease who were treated with entecavir (ETV). Between January 2007 and August 2012, we analysed 373 treatment-naïve patients with HBV-related chronic hepatitis (n = 229) or cirrhosis (n = 144) who were candidates for surveillance test, and were treated with ETV (0.5 mg/day) for longer than 12 months. To minimize the effect of AFP elevation caused by hepatitis activity, serum AFP levels were measured 12 months after the initiation of ETV treatment. Hepatocellular carcinoma developed in 28 patients (7.5%) during a median follow-up period of 48.0 months (IQR = 40.5-57.3 months). The area under the receiver operating characteristic curve for AFP was 0.71 (95% CI = 0.59-0.84). The optimal AFP cut-off value was 13 ng/ml, leading to a sensitivity of 50.0%, specificity of 98.8%, positive predictive value of 77.8% and negative predictive value of 96.1%. In multivariate Cox analysis, an older age, the presence of cirrhosis and AFP levels of ≥20 ng/ml at 12 months after treatment were found to be significantly associated with an increased incidence of HCC. The role of serum AFP as a surveillance test should be re-evaluated in patients with HBV-related chronic liver diseases who were treated with antiviral therapy.
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Talent is a mask we put on perseverance and dedication because we don’t want to see that greatness is within our reach -if we are willing to work for it.
We want to believe that top athletes and performers got to their level of mastery mostly because of “natural abilities”, and that we could do the same if we only had those traits.
It’s easy to dismiss the hard work and commitment behind greatness as mere talent, but we don’t realize how much is behind becoming a top performer. The resilience, the mental toughness, the endless hours of practice…
The truth is that we are not willing to do what it takes, we make excuses and blame other factors to protect our ego. If we were conscious about how hard masters work on their craft we would feel ashamed for ever thinking that talent is the reason for their success.
When people called Michelangelo’s “Pieta” a work of genius, he replied “If people knew how hard I had worked to gain my mastery, it wouldn’t seem so wonderful at all”
While it’s true that some people can learn faster than others, or have physical traits that make their craft easier for them, they still need a great deal of practice and dedication to become a master.
Everyone that has made it to the top in any field had to spend endless hours learning and refining their skill. No exception. None. Even Leonardo Da Vinci or Mozart, who have been wrongly popularized as geniuses that simply “got it”, worked extensively on their craft from an early age.
In the example of Mozart, we are amazed by stories that he could hear a piece of music once and later replicate it without error. While the stories could be exaggerated, if almost everything you have ever done since age 3 from the moment you woke up to the moment you fell asleep was playing and composing music it would be just as surprising if you weren’t able to do such a feat. Mozart worked very hard on his music, no amount of talent without his passion and commitment would have taken him to such level of mastery.
If you want to read more on what allowed Mozart to memorize music so easily download the free report below
Another example is Michael Jordan, one of the greatest sportsman of our time. His work ethic is legendary, he demanded the very best of himself and everyone around him. Most of the time he would be the first one to show up for practice and the last one to leave.
Jordan never settled for the level he had reached, he kept looking for ways to become an even better athlete and player. It would be an insult to call “talent” the lifetime of dedication to the improvement of his skill.
“The mental toughness and the heart are a lot stronger than some of the physical advantages you might have” – Michael Jordan
In the book “Mindset” -one of the most influential works on personal development- Carol Dweck discusses how the belief that we are born with certain traits that can’t be altered limits our thinking and our progress. But embracing the “growth” mindset, the realization that we can change and improve regardless of the traits we have, will help us improve faster and motivate us to take on bigger challenges. The idea that “you either have it or you don’t” is only true if you want to believe it is.
In that sense, taking responsibility for our development is the first step towards mastery. We have to admit -no matter how painful might be- that whatever level of skill we have or don’t have has been a result of our own making. Though factors such as genetics, special circumstances, and overall luck play a part, our progress in learning and mastering skills is largely under our control.
“People always said I had a natural Swing. They thought I wasn’t a hard worker. But when I was young, I’d play and practice all day, then practice more at night by my car’s headlights. My hands bled. Nobody worked harder at golf than I did” -Sam Snead
It saddens me to think of passionate people who want to pursue arts, sports, or any craft but quit before even starting because they were told -sometimes after short assessments- that they were not talented or didn’t “have it”. How many potential great minds, artists, and sportsmen have we repressed? You might be one of them.
Never again let your perceived talent or lack of it steer you away from your passion and your goals. In the search of mastery natural abilities can be an advantage but they are not a requirement.
“Nothing in the world can take the place of Persistence. Talent will not; nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not; the world is full of educated derelicts. Persistence and determination alone are omnipotent. The slogan ‘Press On’ has solved and always will solve the problems of the human race.” -Calvin Coolidge
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Role of caspases in neuronal damage.
There is growing evidence that neuronal apoptosis, or cell suicide, is likely to underlie a number of neurodegenerative disorders. The caspases are a growing family of cysteine proteases that execute the death program by proteolytic cleavage of a number of specific cellular substrates. Recent evidence suggests that they are also involved in processing of neurodegeneration-associated proteins such as amyloid precursor protein and huntingtin. Caspases could thus be attractive therapeutic targets in a number of neural pathologies. The challenge lies in understanding how and which caspases are important in neuronal apoptosis and processing of neurodegeneration-associated proteins and how they are normally regulated. Here we review the evidence for the involvement of caspases in neuronal damage and highlight various studies where caspase inhibition was effective in combating neuronal death.
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New City Council Speaker Corey Johnson pledged at his inauguration Sunday to tackle the city’s “affordability crisis” and its impact on working-class residents, the homeless and local businesses.
Johnson, a Democrat representing Manhattan’s West Side from the southern tip of Central Park down to Canal Street, committed to working with state lawmakers within the coming months to extend rent protections for more than one million city apartments and close loopholes in the rent stabilization law that Johnson said has allowed landlords to deregulate units.
“The affordability crisis that grips our city threatens the very existence of our neighborhoods. New Yorkers who have lived in the same community their entire lives now find themselves priced out, unable to afford their rent or even their groceries,” Johnson told the crowd at the ceremony held in an auditorium of the Fashion Institute of Technology. “Many working families are living literally paycheck to paycheck, one missed shift or one medical expense away from eviction or bankruptcy.”
Johnson was sworn in by Senate Minority Leader Chuck Schumer and was joined by dozens of his colleagues in the Council and other high ranking officials, including Mayor Bill de Blasio, who also spoke at the event.
The new leader of the city’s legislative body related the city’s soaring rents to his own experience moving to New York City from his small Massachusetts hometown, shortly after he came out as a gay teenager in 1999. It set off a “chain reaction” that helped him deal with his mental struggles and launch his career in politics, he said.
“When I came out to my family; when I came out at school, I was pretty much before that literally suicidal,” said Johnson, who illegally lived in a New York University dorm when he moved to the city. “That experience of coming here in 2001 as a 19 year old on a wing and on a prayer and on a dream with two suitcases and knowing two people — that’s becoming harder and harder in 2018. I don’t know how many 19-year-olds can come here who don’t come from a wealthy family and have that experience.”
Johnson, who is also the only openly HIV-positive elected official in the state, will take the reins after a crowded and contentious race for the speakership, which included tense moments between otherwise like-minded Democrats. Many of those losing candidates were in attendance. Though Council members Ydanis Rodriguez and Inez Barron — who cast the only dissenting vote for the speakership, a vote for herself — were no shows.
A spokeswoman for Rodriguez said the councilman missed the inauguration because the ceremony conflicted with his youngest daughter’s birthday party. Barron’s office did not respond to request for comment.
Johnson described the race as “emotionally challenging” but said he was still able to call the candidates his friends. The speaker and de Blasio certainly seemed to be on good terms as well. The mayor praised Johnson as a politician with a “burning desire for justice.”
“We send a message to the whole country that an HIV-positive man is one of the great leaders of our city,” de Blasio said.
The speaker was complimented as someone who could keep the mayor in check, however.
Both Schumer and City Comptroller Scott Stringer referenced Johnson’s announcement earlier this month of the creation of a new Council investigations unit to probe city agencies.
“Corey Johnson has made it clear that this Council is going to be a Council of independence, of issue-focus,” Stringer said, “and when you talk to the Council members, his colleagues, there’s really a moment here that Corey Johnson has become the person we need to lead us for the next four years.”
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Q:
WSO2 Identity Server 5.0.0 Sharepoint 2013 SSO Passive STS failing
I followed the SSO for Microsoft Sharepoint Web Applications with WSO2 Identity Server tutorial, but I am receiving a SOAP error in the wresult on the form that POSTs back to SharePoint.
The soapenv:Reason contains the following text:
Error in creating a SAMLToken using Opensaml library
I've enabled DEBUG logging globally in WSO2 Identity Server, and I can see the error response, but I can't figure out why it is happening:
TID: [0] [IS] [2015-10-29 15:39:18,921] DEBUG {org.wso2.carbon.identity.sts.passive.PassiveSTSService} - Retrieving wreply url for : Portal in tenant : carbon.super {org.wso2.carbon.identity.sts.passive.PassiveSTSService}
TID: [0] [IS] [2015-10-29 15:39:18,921] DEBUG {org.wso2.carbon.identity.sts.passive.PassiveSTSService} - Setting ReplyTo URL : hxxp://portal.domain/_trust for Realm : Portal {org.wso2.carbon.identity.sts.passive.PassiveSTSService}
TID: [0] [IS] [2015-10-29 15:39:18,937] DEBUG {org.apache.rahas.client.STSClient} - Creating request with request type: hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512/Issue and applies to: Portal {org.apache.rahas.client.STSClient}
TID: [0] [IS] [2015-10-29 15:39:18,937] DEBUG {org.apache.rahas.client.STSClient} - Using RSTTemplate: <sp:RequestSecurityTokenTemplate xmlns:sp="hxxp://schemas.xmlsoap.org/ws/2005/07/securitypolicy"><wst:TokenType xmlns:wst="hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512">hxxp://docs.oasis-open.org/wss/oasis-wss-saml-token-profile-1.1#SAMLV1.1</wst:TokenType><wst:KeyType xmlns:wst="hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512">hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512/SymmetricKey</wst:KeyType><wst:KeySize xmlns:wst="hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512">256</wst:KeySize><wst:Claims xmlns:wst="hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512" xmlns:wsp="hxxp://docs.oasis-open.org/ws-sx/ws-trust/200512" wsp:Dialect="hxxp://wso2.org/claims"><wsid:ClaimType xmlns:wsid="hxxp://schemas.xmlsoap.org/ws/2005/05/identity" Uri="{WindowsAccountName|123456789}"></wsid:ClaimType></wst:Claims></sp:RequestSecurityTokenTemplate> {org.apache.rahas.client.STSClient}
TID: [0] [IS] [2015-10-29 15:39:18,937] DEBUG {org.apache.rahas.client.STSClient} - Extracting key size from the RSTTemplate: {org.apache.rahas.client.STSClient}
TID: [0] [IS] [2015-10-29 15:39:18,937] DEBUG {org.apache.rahas.client.STSClient} - Key size from RSTTemplate: 256 {org.apache.rahas.client.STSClient}
TID: [0] [IS] [2015-10-29 15:39:18,952] DEBUG {org.wso2.carbon.identity.sts.passive.processors.RequestProcessor} - STSTimeToLive read from carbon.xml in passive STS 1800000 {org.wso2.carbon.identity.sts.passive.processors.RequestProcessor}
TID: [0] [IS] [2015-10-29 15:39:18,999] DEBUG {org.apache.ws.security.components.crypto.CryptoFactory} - Using Crypto Engine [org.wso2.carbon.security.util.ServerCrypto] {org.apache.ws.security.components.crypto.CryptoFactory}
TID: [0] [IS] [2015-10-29 15:39:19,046] DEBUG {org.apache.xml.security.Init} - Registering default algorithms {org.apache.xml.security.Init}
TID: [0] [IS] [2015-10-29 15:39:19,140] DEBUG {org.wso2.carbon.identity.provider.AttributeCallbackHandler} - Loading claims {org.wso2.carbon.identity.provider.AttributeCallbackHandler}
TID: [0] [IS] [2015-10-29 15:39:19,140] DEBUG {org.wso2.carbon.identity.core.IdentityClaimManager} - IdentityClaimManager singleton instance created successfully {org.wso2.carbon.identity.core.IdentityClaimManager}
TID: [0] [IS] [2015-10-29 15:39:19,140] DEBUG {org.wso2.carbon.user.core.claim.ClaimInvalidationCache} - My Hash code of Claim cache is : 1 {org.wso2.carbon.user.core.claim.ClaimInvalidationCache}
TID: [0] [IS] [2015-10-29 15:39:19,156] DEBUG {org.wso2.carbon.user.core.claim.ClaimInvalidationCache} - Shared Hash code of Claim cache is : 1 {org.wso2.carbon.user.core.claim.ClaimInvalidationCache}
TID: [0] [IS] [2015-10-29 15:39:19,156] DEBUG {org.wso2.carbon.identity.provider.AttributeCallbackHandler} - Processing claim data {org.wso2.carbon.identity.provider.AttributeCallbackHandler}
TID: [0] [IS] [2015-10-29 15:39:19,156] DEBUG {org.wso2.carbon.identity.provider.AttributeCallbackHandler} - Populating claim values {org.wso2.carbon.identity.provider.AttributeCallbackHandler}
TID: [0] [IS] [2015-10-29 15:39:19,187] DEBUG {org.apache.axiom.om.impl.builder.StAXOMBuilder} - WARNING: The current state of the parser is not equal to the state just received from the parser. The current state in the paser is END_ELEMENT the state just received is END_DOCUMENT {org.apache.axiom.om.impl.builder.StAXOMBuilder}
TID: [0] [IS] [2015-10-29 15:39:19,187] DEBUG {org.apache.axis2.handlers.addressing.AddressingOutHandler} - includeOptionalHeaders=false {org.apache.axis2.handlers.addressing.AddressingOutHandler}
TID: [0] [IS] [2015-10-29 15:39:19,187] DEBUG {org.apache.axis2.handlers.addressing.AddressingOutHandler} - WSAHeaderWriter: isFinal=true addMU=false replace=false includeOptional=false role=null {org.apache.axis2.handlers.addressing.AddressingOutHandler}
TID: [0] [IS] [2015-10-29 15:39:19,187] DEBUG {org.apache.axis2.client.Options} - getAction (urn:getResponseResponse) from org.apache.axis2.client.Options@4cdb77b9 {org.apache.axis2.client.Options}
TID: [0] [IS] [2015-10-29 15:39:19,187] DEBUG {org.apache.axis2.transport.local.LocalResponder} - Response - <?xml version="1.0" encoding="utf-8"?><soapenv:Envelope xmlns:soapenv="hxxp://www.w3.org/2003/05/soap-envelope"><soapenv:Header xmlns:wsa="hxxp://www.w3.org/2005/08/addressing"><wsa:Action>urn:getResponseResponse</wsa:Action><wsa:RelatesTo>urn:uuid:83ee0d80-39ff-428f-92b9-bad675cdb820</wsa:RelatesTo></soapenv:Header><soapenv:Body><ns:getResponseResponse xmlns:ns="hxxp://org.apache.axis2/xsd"><ns:return xmlns:xsi="hxxp://www.w3.org/2001/XMLSchema-instance" xmlns:ax2364="hxxp://passive.sts.identity.carbon.wso2.org/xsd" xsi:type="ax2364:ResponseToken"><ax2364:authenticated>true</ax2364:authenticated><ax2364:context>hxxp://portal.domain/_layouts/15/Authenticate.aspx?Source=%2F</ax2364:context><ax2364:replyTo>hxxp://portal.domain/_trust</ax2364:replyTo><ax2364:responsePointer xsi:nil="true"></ax2364:responsePointer><ax2364:results><soapenv:Fault xmlns:soapenv="hxxp://www.w3.org/2003/05/soap-envelope"><soapenv:Code><soapenv:Value>Sender</soapenv:Value><soapenv:Subcode><soapenv:Value xmlns:sts="hxxp://wso2.org/passivests">sts:InvalidRequest</soapenv:Value></soapenv:Subcode></soapenv:Code><soapenv:Reason><soapenv:Text>Error in creating a SAMLToken using Opensaml library</soapenv:Text></soapenv:Reason><soapenv:Detail>none</soapenv:Detail></soapenv:Fault></ax2364:results></ns:return></ns:getResponseResponse></soapenv:Body></soapenv:Envelope> {org.apache.axis2.transport.local.LocalResponder}
http URIs changed to hxxp so that I can submit this question.
A:
I figured it out. The catalina config was set to use the wrong alias in the keystore for encrypting private communications. In my case, the specified alias didn't exist at all. Changed it to a valid alias and now it is working as expected.
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Background {#Sec1}
==========
Allied health services (such as physiotherapy, occupational therapy, speech pathology, social work and dietetics) are now commonly provided on the weekend in hospitals internationally. A survey of tertiary-care hospitals in Canada reported that 97% of facilities provided weekend physiotherapy services \[[@CR1]\]. This was at a lower intensity than during the week and there was high variability in the scope of services provided between hospitals. In an Australian study, 61% of hospitals provided physiotherapy on Saturdays, and 45% on Sundays \[[@CR2]\]. There is little published information examining weekend services amongst other allied health disciplines. One survey of Australian public hospital emergency departments found that 6 out of 21 responding hospitals had rostered occupational therapy weekend services \[[@CR3]\].
There is a body of indirect evidence indicating that provision of earlier and higher intensities of particular allied health services improves health outcomes for a range of hospital patient populations \[[@CR4]-[@CR8]\]. However, increasing service provision on weekends may not have the same effect as increasing services during the week. Staff who are employed in weekend roles may not have the same level of organisational knowledge/expertise/connection as staff who work during the week, which may affect patient health outcomes, discharge planning and subsequent organisational flow outcomes. Also, community support services that allied health services may refer patients to may not operate over the weekend, reducing their ability to facilitate discharges at this time.
There is scant evidence from well-designed studies that directly supports the effectiveness of allied health services provided over the weekend. A systematic review of experimental, quasi-experimental and observational studies concluded that research to date did not provide strong evidence that physiotherapy services provided on the weekend reduced length of stay, improved patient discharge mobility status or discharge destination \[[@CR9]\]. Subsequent research has focused on provision of physiotherapy ± occupational therapy on weekends in rehabilitation wards and has generated some support for service effectiveness in this context \[[@CR10]-[@CR13]\]. However, patients on rehabilitation wards are distinct from those in other hospital wards given their longer length of stay and higher need for rehabilitation therapy. Hence, it is difficult to extrapolate the findings of research conducted in rehabilitation settings to other hospital areas such as acute medical/surgical wards.
An additional factor that should influence whether allied health services are provided on weekends is that of economic efficiency. There are two key factors that may limit the cost-effectiveness of allied health services provided over the weekend, being: the higher cost per hour of employing staff over weekends compared to during the week, \[[@CR14]\] and the possibility of diminishing marginal returns \[[@CR15]\]. This latter principle suggests that the amount of additional benefit gained for each additional unit of service provision will decrease as the overall level of service provision increases, meaning that the cost-effectiveness ratio of these services will decrease with increasing levels of service provision (that is, there may be less benefit achieved from increasing allied health services from 5 to 7 days service per week than when increasing from 3 to 5 days of service per week). There is some evidence from an observation dose-response study indicating this principle applies to physiotherapy rehabilitation services \[[@CR16]\].
Direct evidence of the economic efficiency of allied health services delivered on weekends is scant and inconsistent in its support of weekend allied health services. A partial economic evaluation (cost evaluation) of a weekend physiotherapy service provided to rheumatology patients in the United Kingdom found increased costs of service provision with no reduction in length of stay (indeed, a non-significant increase of 0.5 days) associated with this programme \[[@CR17]\]. A quasi-experimental, historical control group study found weekend physiotherapy provided to patients who had undergone a total hip or knee arthroplasty generated a cost saving to the health fund driven by a reduction in length of stay from 12.28 days to 10.84 days \[[@CR18]\]. An observational study of a 7-day a week, 24-hour, on-call social work service provided in a hospital's emergency department suggested that the programme was operated at little cost to the hospital \[[@CR19]\]. However, this study was purely an accounting exercise based on the estimated accounts paid and actual cost of the on-call services. An economic evaluation arising from a randomised controlled trial investigating the cost-effectiveness of additional Saturday physiotherapy and occupational therapy services on rehabilitation inpatients compared to those receiving usual Monday to Friday services had an incremental cost utility ratio of AUD 41,825 (95% confidence interval (CI) −2,817 to 74,620) per quality-adjusted life year (QALY) gained for the intervention group indicating this approach in this setting is likely to be cost-saving \[[@CR20]\].
Clearly, there is uncertainty as to whether providing weekend allied health services is effective or cost-effective, particularly when provided on acute medical or surgical wards. Current widespread provision of these services complicates conduct of a traditional randomised trial as the intervention is already being provided as a part of usual care (the default control condition in pragmatic research). Our research team has recently devised a novel disinvestment research design that can be applied in the context where a health technology is being applied as a part of routine care, yet there is uncertainty as to the effectiveness, cost-effectiveness or safety of this health technology \[[@CR21]\]. In this research, we will use this novel research approach to evaluate the effectiveness, cost-effectiveness and safety of two models of weekend allied health service being provided on acute medical and surgical wards compared to having no weekend allied health service.
Methods/Design {#Sec2}
==============
Design {#Sec3}
------
This research comprises two studies. Study 1 will consist of two hospital sites undertaking a novel, stepped wedge, roll-in, cluster randomised disinvestment trial whereby the current model of weekend allied health service delivery will be 'rolled back in' (withdrawn). Study 2 will consist of these same two hospital sites undertaking a conventional stepped wedge, roll-out, cluster randomised trial design in which a new stakeholder-driven model of weekend allied health service delivery will be rolled out to the same wards. Study 2 will commence immediately following completion of study 1.
Participants, therapists, centres {#Sec4}
---------------------------------
This research will take place across six acute medical or surgical wards from Dandenong Hospital and Western Hospital (Footscray), in Victoria, Australia. Both of these hospitals are major tertiary, metropolitan hospitals. The wards (Table [1](#Tab1){ref-type="table"}) were selected by project investigators in consultation with managers and clinicians based at each site on the basis of currently having a weekend allied health service, the patient types being treated on that ward (medical or surgical patients, not rehabilitation), and not being anticipated to undergo major structural change (for example, substantive change of patient casemix or refurbishment requiring ward shut-down) during the study period which would confound the trial design. High-risk wards such as intensive and coronary care units, emergency departments and paediatric wards were excluded. Each hospital will commence this research at different time points (Dandenong Hospital commencing February 2014 and Western Hospital (Footscray) commencing April 2014.Table 1**Caseload in participating wardsHospitalWardDescription**Dandenong HospitalSW3/W2 acuteOrthopaedic surgerySW4Stroke unitWest 3Thoracic, Vascular, General surgical and medical unitsWest 4General medicineNorth WardHead and neck, PlasticsNorth 3SurgicalWestern Hospital2BMedical2CMedical2DInfectious diseases; Respiratory medicine2 W^a^Plastics; Head and neck surgery; ENT surgery3 WGeneral surgery; Colorectal; Breast; Endocrine; Urology3EGeneral surgery; Vascular surgery; Thoracic; Upper gastrointestinal^a^Study 1 only.
Data in this study will be collected from three groups of participants. Each group of participants is now described in detail.
### Group 1 {#Sec5}
'All patients on participating wards.' This group will consist of patients admitted to acute medical or surgical wards involved in the trial who are over the age of 18 years. Paediatric patients will be excluded from this study as they are not routinely seen on the selected wards (none are paediatric units or have paediatric beds) and have different provisions for service delivery. It is anticipated, based on previous patient flow data, that each ward will have an average of 174 patient admissions per month during the trial. This means that 6 wards at 1 site will have 7,308 patient admissions during the 7 months of study 1 and 7,308 during the 7 months of study 2. This will lead to an anticipated total of 29,232 patient admissions during the overall study period.
### Group 2 {#Sec6}
'Randomly selected subgroup of patients from participating wards'. A randomly selected subgroup of approximately 600 patients from participating wards will be recruited for additional data collection. Study data collectors will use a random number generator to randomly select wards on specific days during the study period that they will attend and approach all patients who are planned for discharge within the next 24 hours from that ward to consent to participate in this component of the study. These participants will be recruited to contribute data to secondary outcomes for this study.
### Group 3 {#Sec7}
'Health professionals.' A volunteer subgroup of medical, nursing and allied health staff (four to twelve per participating ward) will be sought to participate in qualitative data collection approaches (group interviews and key informant interviews) being used as a part of the project process evaluation and planning for the intervention model to be used in study 2. This will include staff who work on the participating wards during the week and on the weekend. Three waves of data collection will take place with these health professionals (pre-study 1, between studies 1 and 2, and post-study 2).
Intervention/Control {#Sec8}
--------------------
We have provided a summary of our intervention conditions described according to the TIDieR guidelines \[[@CR22]\] in Table [2](#Tab2){ref-type="table"}. Further elaboration of the intervention conditions is now provided according to study.Table 2**Intervention conditions according to TIDieR criteriaTIDieR criteriaStudy 1 interventionStudy 2intervention**Item 1. Brief name: provide the name or a phrase that describes the interventionUsual care weekend allied health serviceStakeholder-driven weekend allied health serviceItem 2. Why: describe any rationale, theory, or goal of the elements essential to the interventionUsual care is the prevailing model of care in the research location that in a pragmatic research design serves as an appropriate reference standard. This model of care has developed incrementally over time and has largely been driven by decisions of individual allied health managers in an '*ad hoc*' mannerA new model of weekend allied health service will be developed where managers and staff of participating wards are engaged to identify the most important tasks that require completion on weekends that could be undertaken by allied health staff. It is anticipated that by directly engaging with these key stakeholders, a new model of care that better meets the needs of individual wards will be developedItem 3. What (materials)?: describe any physical or informational materials used in the intervention, including those provided to participants or used in intervention delivery or in training of intervention providersThere are no specific materials used beyond those materials ordinarily used by allied health professionals during the week. It is left to the discretion of individual staff what materials they use in their clinical practiceThere are no specific materials used beyond those materials ordinarily used by allied health professionals during the week. If the people involved in developing this model of care determine that additional materials are required, these will be identified and described at a later dateItem 4. What (procedures)?: describe each of the procedures, activities, and/or processes used in the intervention, including any enabling or support activitiesAllied health services may include services provided by physiotherapy, occupational therapy, social work, dietetics, speech pathology professionals and allied health assistants. Services delivered are the same as those performed on weekdays, although the intensity of weekend services is lower (fewer hours per ward) than weekday services. Services commonly include mobilisation, chest physiotherapy, discharge planning, assessment and prescription of aids and equipment, swallowing assessment, dietary analysis and prescription, and counsellingServices provided are likely to be similar to that of the usual care weekend allied health service. However, the people involved in developing this model will be able to inform the practitioners involved of the relative priority of the different tasks that they may be asked to perform on each wardItem 5. Who provided?: for each category of intervention provider (for example, psychologist, nursing assistant), describe their expertise, background and any specific training givenAll allied health professionals will have entry-level allied health degrees as a minimum. Orientation of new staff members to the health care organisation and wards that they work on is provided as a part of standard human resources procedures. Allied health assistants do not require formal qualification but most have a certificate III or IV \[[@CR44]\] and all operate under the direction of an allied health professionalServices providers are likely to be similar to that of the usual care weekend allied health service. However, the people involved in developing this model will be able to decide which service providers are best positioned to undertake the tasks that require completion on the weekend, and will also decide if transdisciplinary training is required by individual practitionersItem 6. How?: describe the modes of delivery (such as face to face or by some other mechanism, such as Internet or telephone) of the intervention and whether it was provided individually or in a groupFace to face individual patient interactionFace to face individual patient interactionItem 7. Where: describe the type(s) of location(s) where the intervention occurred, including any necessary infrastructure or relevant featuresHospital acute medical/surgical ward environmentHospital acute medical/surgical ward environmentItem 8. When and how much?: describe the number of times the intervention was delivered and over what period of time including the number of sessions, their schedule, and their duration, intensity or doseIndividual patients will receive variable amounts of weekend allied health service delivery. The intensity of services provided is at the discretion of the allied health professional. The number of hours of weekend allied health service delivered per day will vary between wards within each site and between sitesIndividual patients will receive variable amounts of weekend allied health service delivery. The intensity of services provided is at the discretion of the allied health professional. The number of hours of weekend allied health service delivered per day will vary between wards within each site and between sitesItem 9. Tailoring: if the intervention was planned to be personalised, titrated or adapted, then describe what, why, when, and howAll weekend allied health services will be tailored to the needs of the patients being treated. This will be at the discretion of the treating allied health professional based upon their clinical judgement.All weekend allied health services will be tailored to the needs of the patients being treated. This will be at the discretion of the treating allied health professional based upon their clinical judgement.Item 10. Modifications: if the intervention was modified during the course of the study, describe the changes (what, why, when, and how)Not applicable for protocolNot applicable for protocolItem 11. How well (planned)?: if intervention adherence or fidelity was assessed, describe how and by whom, and if any strategies were used to maintain or improve fidelity, describe themResearch assistants will be present daily on study wards to both promote and monitor intervention fidelity in terms of which wards should be receiving weekend allied health services and which should not. Patient contact statistics are recorded by allied health professionals and are recorded in hospital administrative datasets. These datasets will be used by investigators to measure time spent by weekend allied health personnel with patients on each ward.Research assistants will be present daily on study wards to both promote and monitor intervention fidelity in terms of which wards should be receiving weekend allied health services and which should not. Patient contact statistics are recorded by allied health professionals and are recorded in hospital administrative datasets. These datasets will be used by investigators to measure time spent by weekend allied health personnel with patients on each ward.Item 12. How well (actual)?: If intervention adherence or fidelity was assessed, describe the extent to which the intervention was delivered as plannedNot applicable for protocolNot applicable for protocol
### Study 1 intervention condition - current weekend allied health services {#Sec9}
Current physiotherapy, occupational therapy, social work, dietetics, speech pathology and allied health assistant weekend service delivery on the participating wards will be the intervention for study 1 of this trial. During this study, the weekend allied health services of each participating ward (cluster) will be 'rolled in' sequentially using the stepped wedge design (Figure [1](#Fig1){ref-type="fig"} and Additional file [1](#MOESM1){ref-type="media"}). The current model of weekend allied health service delivery in the hospitals participating in this study has not been developed systematically, rather in an '*ad hoc*' manner driven by decisions made by individual allied health managers. Thus, there is potential that this model is not currently tailored to the needs of individual wards and is, therefore, not delivering optimal outcomes. However, it is the prevailing model of care that in a pragmatic research design serves as an appropriate reference standard \[[@CR23]\].Figure 1**Design of study 1 and study 2 at Dandenong Hospital.**
Design at Western Hospital (Footscray) is similar but with 5 wards involved in study 2 and a 13-month total study period. Black = current weekend allied health service. White = no weekend allied health service. Grey = new, stakeholder-driven weekend allied health service.Twelve months of pre-trial data collection at participating sites indicated that weekend allied health services at each site varied in terms of the number of disciplines involved, the amount of time allocated per ward and the total budgetary allocation to this activity. Steps will be taken during study 1 to ensure intervention fidelity relative to the pre-trial data collection. Allied health staff providing weekend services will be updated as to the amount of time spent on each ward per month and will have pre-trial data used as monthly targets to ensure that service provision levels within study 1 intervention periods are consistent with pre-trial data. Staff will be advised to continue performing tasks and prioritising patients in the same manner with which they had been doing so prior to trial commencement.
### Study 1 - control condition - no weekend allied health services {#Sec10}
The control condition will be characterised by the provision of no weekend allied health services unless a patient meets a 'clinical exception' criteria. A 'clinical exception' refers to circumstances where the trial treatment protocol is permitted to be violated due to the risk of harm to an individual or organisation. This is also intended to mitigate against possible protocol violations by staff and to ease their concerns regarding the health and well-being of patients given the novel research approach being used in this study \[[@CR21]\]. Patients who meet one of these criteria will be permitted to see a weekend allied health staff member despite them being on a ward during a 'control' period. Local nursing or medical staff will identify if the patient meets the pre-determined set of criteria, which allows for them to be seen by an allied health professional over a weekend. This will be escalated to the site allied health director or project site liaison who will approve provision of the service following a final check against the 'clinical exception' criteria. These criteria will be suggested by ward-level staff during pre-trial group interviews, and managers through key-informant and group interviews and will vary from site to site. Staff will be asked to provide evidence to support their request for a clinical exception criteria (for example, research papers, local incident reports, hospital policy), and project investigators will determine whether the strength of evidence presented warrants formation of a clinical exception criterion.
The frequency of clinical exceptions under each criterion will be recorded to monitor control condition fidelity. Project data collectors at each site will be present on each ward each day of this project and will facilitate ongoing communication with ward staff regarding local clinical exception criteria.
### Study 2 - control condition - no weekend allied health services {#Sec11}
This will be identical to the study 1 control condition.
### Study 2 - intervention condition - stakeholder-driven model of weekend allied health services {#Sec12}
A criticism of using the current model of weekend allied health service provision as the intervention in study 1 is that it is possible that the sites involved may have constructed their prevailing model on an *ad hoc* basis, which may not reflect higher standards of service set in other hospitals. This is consistent with evidence already discussed that there is high variability in the amount and focus of allied health services already being delivered on weekends internationally \[[@CR1]\]. However, there is no existent 'gold standard' for how a weekend allied health service delivery model should be structured (which services, how much, what activities) given the paucity of evidence regarding the comparative efficacy of different weekend allied health service delivery models. Developing such a standard across all medical and surgical wards may also be an impossibility, as different types of wards will have different patient casemix and different allied health service requirements.
Rather than conform to a non-existent 'standard' for what a weekend allied health service delivery model should look like, we have sought to standardise a process by which an 'optimal' site-specific service could be developed. Thus, the model of allied health service delivery used as the intervention in study 2 of this research is a complex intervention \[[@CR24]\].
Our process for developing this site-specific weekend allied health service delivery model begins with extensive consultation with relevant stakeholders (medical, nursing and allied health staff and managers) on participating wards. These staff will be interviewed (group and individual, key informant) to drive development of this model; hence, we refer to this model as a stakeholder-driven model. Investigators experienced in conducting qualitative and participatory action research (LO, FM) will facilitate these interviews. These staff will not be asked to say which professional discipline they want to be employed on the weekends, rather, to identify and prioritise the tasks that they believe to be most important for allied health to perform on weekends both in terms of improving patient health outcomes, improving patient flow, and reducing readmissions. They will also be asked to reflect on the strengths and limitations of the current model of care, suggest areas for improvement and to examine patient incident and clinical exception data gathered during the first study of the trial to inform their decisions. Allied health managers will be provided with this list of tasks and other feedback gathered, so that they can propose what the new, stakeholder-driven model of weekend allied health service delivery will be. Transdisciplinary, interdisciplinary and multidisciplinary models of care are all possible candidates for the model of care that will arise from this process. These managers will then be engaged in a Delphi process \[[@CR25]\] to select the preferred model of care from the proposals generated.
Each site will hold constant the budgetary amount allocated to weekend allied health services between studies 1 and 2. During the second 'step' in the stepped wedge design of study 2, the first ward to resume provision of weekend allied health services using the stakeholder-driven model will be provided with a budgetary allocation that is the average per ward of study 1. During the third step, when a second ward resumes provision of weekend allied health services, the total budgetary allocation to be shared between the 2 wards that have resumed provision of the weekend service will be twice the average per ward of study 1. However, provisions within the stakeholder-driven model will be made to allow weekend allied health to distribute their time unequally between the two wards based on the task prioritisation framework provided to them by the stakeholders who designed this model. This provision recognises that an optimal service would target patients who are likely to receive the greatest benefit from this service, regardless of the ward they are physically being treated on.
Outcome measures {#Sec13}
----------------
Current United Kingdom Medical Research Council guidance on the evaluation of complex interventions indicates that a single primary outcome may not make best use of data in these evaluations, rather, that a range of outcome measures will be needed including possible unintended consequences \[[@CR23]\]. In this research, we plan to examine three domains of primary outcome with one domain being analysed in two ways, and another being measured as a composite outcome.
### Primary outcome measure: 1) length of stay {#Sec14}
Mean overall length of stay per patient who is treated on one of the targeted wards will be used as the primary outcome. This measure is often used as an indicator of hospital efficiency \[[@CR26]\]. Length of stay is the key overall driver of inpatient costs and is the most readily available outcome to enable stopping rules to be checked during study 1. However, overall mean length of stay is limited as an outcome for this project as: i) these data can be skewed by highly influential outliers, ii) length of stay of longer-stay patients will not be known until they are discharged (thus interim analyses for non-inferiority to check stopping rules may underestimate mean length of stay) and iii) patient admission casemix profile may vary over the study period (for example, a study ward may start admitting more patients who typically stay longer). To account for these limitations, we will also examine the proportion of patients who stay longer than their Australian Refined Diagnosis-Related Group average 'inlier' length of stay according to data published from the previous year \[[@CR27]\].
### Primary outcome measure: 2) rate of unplanned hospital readmission within 28 days {#Sec15}
This routinely collected measure will be used as a marker of treatment effectiveness, discharge planning and patient readiness for discharge \[[@CR28]\].
### Primary outcome measure: 3) rate of adverse events {#Sec16}
Patient adverse events will be collected using a range of clinical data collection systems. The events quantified will be in-hospital falls, Code Blue/Medical Emergency Team calls, pulmonary embolus, deep vein thrombosis, death, hospital acquired pressure area, and intensive care unit admission from the ward. Rate of adverse events (events per person-time) will be considered as a single, composite end-point when considered as a primary outcome. A composite end point for this primary outcome domain was selected as a number of different clinical events may indicate a clinical failure, whereas the selection of only one type of clinical event as the end point may not present a comprehensive clinical picture \[[@CR29]\].
### Secondary outcome measure: 1) number of complaints {#Sec17}
Complaints emanating from targeted wards (total and allied health specific) will be captured through the hospital administrative databases and local departmental datasets.
### Secondary outcome measure: 2) number of compliments {#Sec18}
Compliments emanating from targeted wards will be captured similarly to the number of complaints.
### Secondary outcome measure: 3) patient discharge destination {#Sec19}
Patient discharge destination will be classified into categories of: i) discharged to the community, ii) transferred to another acute ward, iii) transferred to intensive care, iv) transferred to rehabilitation, and v) discharged to residential aged care.
### Secondary outcome measure: 4) patient satisfaction {#Sec20}
Patient satisfaction with overall care will be measured using data from the 'overall hospital experience' domain of the Victorian Patient Satisfaction Survey \[[@CR30]\]. These data will only be extracted from participant group 2 (randomly selected subgroup of patients from participating wards). These data will not be collected from patients with a diagnosis of dementia or cognitive impairment documented in their medical history.
### Secondary outcome measure: 5) patient functional independence at discharge {#Sec21}
Patient independence at discharge will be measured using participant self-report on the Modified Barthel Index \[[@CR31]\]. These data will be collected from the same participant subgroup as secondary outcome measure: 4) patient satisfaction.
### Secondary outcome measure: 6) patient health-related quality of life at discharge {#Sec22}
Health-related quality of life at discharge from hospital will be measured using the European Quality of Life, 5 dimensions (EQ-5D-5 L) instrument \[[@CR32]\]. These data will be collected from the same participant subgroup as secondary outcome measure: 4) patient satisfaction in study 2.
### Secondary outcome measure: 7) staff absenteeism {#Sec23}
Staff absenteeism (medical, nursing, allied health staff) will be collected through routine finance reporting. As allied health staff are not allocated to specific wards, but may work across a combination of wards with mixed exposure to intervention and control conditions, we will collect these data at only 3 time points: the first month of study 1, the final month of study 1 and the final month of study 2.
### Process measure: 1) allied health hours of service {#Sec24}
Number of occasions of allied health service provision to each patient on weekends and weekdays will be collected.
### Process measure: 2) clinical exceptions {#Sec25}
The frequency and reason for clinical exceptions taking place will be recorded by project research personnel.
### Process measure: 3) proportion of patients discharged on a Saturday or Sunday {#Sec26}
This outcome is a potential indicator of patient flow during the study. It is important to consider patient flow as a process measure to ensure that bed-block is not occurring on the weekend.
### Economic outcome measure: 1) cost of inpatient treatment per patient {#Sec27}
Hospital clinical costing data will be used to measure the costs attributed to each patient. If administrative data for a particular patient is unable to be extracted, costing based on the most recent National Weighted Activity Unit calculators \[[@CR33]\] will be used. This will be based on the overall patient admission, not just their time spent on a ward under an intervention or control condition.
### Qualitative outcomes {#Sec28}
Group and individual interviews will be conducted with staff members from each ward and weekend allied health staff at 3 time points (pre-study 1, between study 1 and study 2, and post-study 2) to explore their satisfaction with and experiences of the different weekend allied health service delivery models being examined.
Procedure {#Sec29}
---------
Approval to conduct this study has been obtained from the Monash Health Human Research Ethics Committee (EC00383 - representing Dandenong Hospital) and the Melbourne Health Human Research Ethics Committee (2013.283 - representing Western Hospital (Footscray)) and this trial has been registered with the Australian New Zealand Clinical Trials Registry ([ACTRN12613001231730](http://www.anzctr.org.au/ACTRN12613001231730) (study 1) and [ACTRN12613001361796](http://www.anzctr.org.au/ACTRN12613001361796) (study 2)). It is anticipated that data collection will commence in February 2014 and be completed in June 2015 with reports and publications finalised by June 2016. A multi-site project executive committee has been developed consisting of all project investigators. This executive committee has developed a project communications plan in consultation with representatives from the Victorian Government, Department of Health. Further pre-trial communications with relevant health services unions, local site clinicians, managers and executives will also be undertaken prior to trial commencement. Extensive staff engagement has been undertaken for this project at both study sites prior to trial commencement.
### Randomisation and allocation concealment {#Sec30}
The order in which weekend allied health services will be rolled in and out from each participating ward will be determined at random by an investigator blinded to ward identity. One investigator will develop pseudonyms for each ward while another investigator blinded to the true identity of each pseudonym will use a random number generator in Microsoft Excel (Microsoft Corporation, Redmond, WA, USA) to allocate ward number locations to each pseudonym. The first investigator will then be able to reveal which pseudonym represented which ward.
### Masking {#Sec31}
Research assistants collecting and entering data will not be blinded to the allocation of wards within the stepped wedge research designs. The randomly selected subgroup of patients from participating wards will not be informed of the purpose of the interview as being for the evaluation of a randomised controlled trial. However, we cannot say that all of these participants will be blinded as to the intervention they were exposed to. Masking will be applied to the trial data analyst. Six mock codes representing different orders in which the wards may have progressed through the stepped wedge design (using a Latin Square approach) will be used to blind the statistician conducting the final quantitative analysis from the true identity of each ward and the time sequence in which each ward was randomised.
### Trial safety {#Sec32}
The process of disinvestment in allied health services has the potential to be detrimental to patient safety and/or organisational outcomes. The checking of stopping rules and development of clinical exceptions will, therefore, be used to ensure safety. A stopping rule allows data to be monitored and the trial to be stopped if there is evidence of a lack of safety or efficacy associated with the intervention \[[@CR34]\]. During study 1, interim analyses will be conducted monthly to ensure that patient outcomes have not dropped below a pre-specified non-inferiority margin. This margin represents the maximum amount of gain anticipated if the amount of resources being saved were to be reallocated to another purpose. Discussion groups with clinicians and project investigators initially determined that the non-inferiority margin should be 0.8 multiplied by the standard deviation of the targeted outcome. This is equivalent to 'Cohen's large effect size' \[[@CR35]\]. However, review of the standard deviations for the outcomes that these stopping rules would be applied to revealed that these margins were too wide to be acceptable to the management of participating hospitals. As a result of these consultations, the following stopping rules were developed:The 95% CI of the effect (difference between means) of having no weekend allied health services exceeding a 1-day increase in mean length of stay.The 95% CI of the effect (difference between proportions) of having no weekend allied health services exceeding an absolute change of 0.02 (2%) in the outcomes of the proportion of patients who stay longer than their average inlier Australian Refined Diagnosis-Related Group (ARD-RG), the proportion of patients who experience one or more of the previously specified adverse events, or the proportion of patients who are unexpectedly readmitted within 28 days.
Management at each site also reserved the right to review other study data (for example, the proportion of patients discharged on the weekend) in determining whether the study should continue at each site. Interim analysis will be conducted during each month of study 1 and reported back to the project executive committee. The project executive committee will then forward their recommendations along with blinded data to an independent data-monitoring committee at each site to make a determination as to whether the project should be ceased on the basis of the analysis conducted.
Data will be collected in an identifiable form to allow data linkage with other project datasets but will then be de-identified for storage and analysis once linkages have been made.
### Recruitment {#Sec33}
A waiver of consent was requested and granted for researchers to access routinely collected, hospital administrative data (for example, length of stay, hospital readmission) covering these forms of data for participants in Group 1. Written consent will be obtained from all participants (where the waiver of consent does not apply). Written consent will be obtained from participants in Group 2 by a project team member (MS, MG) or research assistant. They will be identified as being potentially eligible for consent to participate by screening of ward handover sheets and communication with ward staff. Written consent will be obtained from participants in Group 3 by a project team member (ES, SP, LS, DM, LOB). They will be notified of the possibility of participating in this project by Email and verbal communication.
Data analysis {#Sec34}
-------------
### Sample size estimation {#Sec35}
The sample size in this study is governed by the patient throughput on the participating wards over the trial period. Non-inferiority trials do not preclude testing of superiority and can be done so without statistical penalty \[[@CR36]\]; hence, we undertook power calculations for studies 1 and 2 from a superiority analysis perspective. Current data from study wards indicates there will be 7,308 patient admissions in total per study per site. We used the approach for conducting power analyses for stepped wedge trials advocated by Hussey and Hughes \[[@CR37]\] based upon the Wald statistic. We applied this approach to 3 of our primary outcomes and demonstrated \> 90% power in each case for study 1 and study 2 (Table [3](#Tab3){ref-type="table"}). We reiterate that the actual sample size to be used in the trial has been determined primarily by practical considerations, particularly the availability of suitable wards at the participating sites.Table 3**Outcome of power analysis for three outcomes for each study at each site (assuming six wards per site), and for both sites combinedOutcomeProportion 1 (weekend allied health services)** ^**a**^**Proportion 2 (no weekend allied health services)Single site or both sites combinedAssumed coefficient of variationAssumed number of patients per studyPower**Proportion of patients who stay longer than their AR-DRG average inlier length of stay0.400.42Single0.4n = 7,000^b^0.65Proportion of patients who are readmitted within 28 days0.100.12Single0.4n = 7,000^b^0.96Proportion of patients who experience at least one of the adverse events listed0.100.12Single0.4n = 7,000^b^0.96Proportion of patients who stay longer than their AR-DRG average inlier length of stay0.400.42Both0.4n = 14,000^b^0.99Proportion of patients who are readmitted within 28 days0.100.12Both0.4n = 14,000^b^\>0.99Proportion of patients who experience at least one of the adverse events listed0.100.12Both0.4n = 14,000^b^\>0.99AR-DRG, Australian Refined Diagnosis-Related Group.^a^Baseline proportions based on data drawn from administrative datasets at participating sites covering a 12-month period.^b^Note that for the single site analyses we assume there will be 7,308 per study, but use only 7,000 in the power analysis to allow for loss of patients during the transition phases between intervention and control conditions and for transfers between study wards. Correspondingly, we use 14,000 when considering the power of both sites combined.
### Statistical analysis {#Sec36}
#### Study 1 {#FPar1}
Both non-inferiority and superiority analyses will be conducted. For non-inferiority analyses, if the non-inferiority null hypothesis is to be rejected, the upper limit of the CI around the observed difference should be less than the non-inferiority margin. Multi-level, mixed-effects generalised linear model analyses will be used to construct the 95% CIs that compare effectiveness and safety outcomes between groups. These models will nest patient admissions within wards, treating both as random factors, to account for the clustered nature of these data. Weekend allied health service delivery model provided will be treated as a fixed factor. The distribution of the length of stay outcome will be examined (± transformations as indicated) as a part of model building as these data are commonly skewed. Data from patients who are exposed to both control and intervention conditions within each study will be excluded from the analyses. Analyses will be adjusted for index study ward (first study ward the patient was admitted to during that inpatient episode of care), calendar time using each step in the wedge design as a categorical covariate and for historical data collected for the outcome being examined from each site over the previous 2 years mapped against these steps to account for potential local seasonal variations in the outcomes collected.
Analyses will be conducted at a site-level with pre-planned meta-analyses across both sites using patient-level data from phase 1 and separately for phase 2. These meta-analyses will examine the main effect of the usual care weekend allied health service as opposed to the control condition across sites. A site-by-intervention interaction effect will also be examined to determine if there is heterogeneity in results between sites. This is plausible as a dose-response relationship may exist between the amount of resource allocated to the weekend allied health service and the study outcomes. In this study, local data collected prior to the study indicated that Dandenong Hospital allocated approximately four times as much resource to the acute medical and surgical wards participating in this research. Meta-regression will be employed using ward-level data, treating each ward over time as a pre-post intervention study to identify if the labour costs consumed on each ward for provision of the weekend allied health service explains variation in the change in each outcome within ward.
Patients who are exposed to both intervention and control conditions will be excluded from analyses to minimise potential contamination that is made possible by having the transition periods within the stepped wedge design.
#### Study 2 {#FPar2}
These analyses will be equivalent to the analyses conducted for study 1; however, non-inferiority analyses will not be conducted for this study.
A grand meta-analysis is planned, pooling results between studies 1 and 2 to examine whether there is a significant study-by-intervention interaction effect. This analysis will examine whether the effect of the stakeholder-driven model of weekend allied health service was significantly different to that of the usual care weekend allied health service delivery model.
Sensitivity analyses will be conducted if there are unit closures (for example, surgical unit closures over the Christmas holiday period) that are likely to affect the caseload on related wards at these times. Data within these cells of the stepped wedge design will be treated as missing in the sensitivity analyses.
#### Economic analysis {#FPar3}
The primary economic analysis will be a cost-effectiveness analysis from the hospital perspective that examines the incremental cost per patient admitted. These costs will include the 'total cost' of patients during their admission plus additional costs from an unplanned admission within 28 days. Data from patients who are transferred between wards in different study periods (that is one without access to allied health services, one with access to allied health services) will be excluded from the analyses as will data from patients who remain within the same ward over two or more weekends if they are allowed to access weekend allied health services during one weekend, and not allowed to access allied health services on another. Secondary economic evaluations will examine the incremental cost per difference in clinical outcome for the measures of functional independence (studies 1 and 2) and health-related quality of life (study 2).
#### Qualitative data analysis {#FPar4}
Both content and thematic analysis will be used for the analysis of qualitative data collected as a part of this study \[[@CR38]\].
#### Analysis of workforce absenteeism outcome {#FPar5}
It was initially planned to analyse these data in a similar manner to other outcomes; however, it has become apparent that for some professional groups, staff operate over multiple wards that may be in the intervention or control periods during the study. Hence, we will only compare data collected from the first and last months of each study when all wards within the study are either completely in control or intervention periods.
Deviation from registered trial protocol {#Sec37}
----------------------------------------
There have been two changes to the registered trial protocol. The first change was the withdrawal from both study 1 and study 2 of what was planned to be a third study site. Originally, a third site with five participating wards was planned to be involved and the appropriate ethics approval was gained. The ethics committee at this site retracted this approval due to subsequent opposition from senior medical staff. Project investigators met with these senior medical staff to discuss their concerns; however, agreement could not be reached regarding the issues raised to enable the trial to continue at this site.
The second change was that the broader health network, within which the Western Health (Footscray) site operates, decided to transfer some services from that site to another site within their network. This has resulted in only 5 wards being available to participate in study 2 at this site. This is likely to result in some changes to casemix at the Western Health (Footscray) site during study 2. These changes are planned to take effect in early 2015 when the intervention phase of study 2 is in progress, thus it was decided to make the changes to the study wards involved from the commencement of phase 2.
Discussion {#Sec38}
==========
Disinvestment from health services is becoming a growing priority for health care service providers due to increasing health care expenditure, budget cuts, an increasing burden of chronic disease and ageing in developed nations, and the ongoing introduction of new health technologies that render older technologies obsolete \[[@CR39]-[@CR41]\]. Paradigms for undertaking disinvestment are emerging but current processes rely upon the availability of evidence of effectiveness and cost-effectiveness \[[@CR39],[@CR42],[@CR43]\]. This study uses a novel research design to demonstrate how health services can generate evidence regarding effectiveness, cost-effectiveness and safety during the disinvestment process \[[@CR21]\]. The withdrawal of an intended study site due to opposition from senior medical staff despite prior ethics approval has demonstrated a clear barrier to conducting disinvestment research.
Trial status {#Sec39}
============
Recruitment ongoing, anticipated completion June 2015.
Additional file {#Sec40}
===============
Additional file 1:**SPIRIT 2013 Checklist: recommended items to address in a clinical trial protocol and related documents\*.**
ARD-RG
: Australian Refined Diagnosis-Related Group
EQ-5D-5 L
: European Quality of Life, 5 dimensions
QALY
: quality-adjusted life year
TIDieR
: Template for Intervention Description and Replication
**Competing interests**
The authors declare that they have no competing interests.
**Authors' contributions**
TPH: led development of research design and analysis plan and overall project management. LOB: assisted development of research design, led development of qualitative data collection methods and collection of qualitative data. DMi: site liaison for Monash Health, assisted with overall project management, led development of qualitative data collection methods and collector of qualitative data across all sites. KAB: led development of quantitative data collection methods, assisted with overall project management and collection of quantitative data across all sites. RH: development of local data collection methods and data collection at Monash Health. DMa: assisted development of overall research design, assisted overall project management and conduct of project at Dandenong Hospital. SP: site liaison for Melbourne Health, assisted with overall project management, assisted development of data collection plan at Melbourne Health. TC: assisted development of overall research design, assisted with overall project management, and conduct of project at Western Health. KM: assisted development of overall research design, assisted with overall project management, and conduct of project at Dandenong Hospital. KP: assisted with overall development of research design and assisted with overall project management. DL: contribution to project management, administration and communications strategy. FM: assisted with development of qualitative data collection approach and assistance with qualitative data collection. MS: assisted with development of quantitative data collection approach at Monash Health and data collection at this site. MG: assisted with development of quantitative data collection approach at Western Health and data collection at this site. LS: site liaison for Melbourne Health, assisted with development of quantitative data approach at Melbourne Health. GJ: assisted with overall project management. ES: assisted with development of overall research design, site liaison for Western Health, assisted with development of quantitative data approach at Melbourne Health, assisted with data collection, assisted with overall project management. All authors have read and approved the trial manuscript and given approval for this version to be published.
Funding for this trial has been received from a partnership grant from the National Health and Medical Research Council (NHMRC) Australia (ID 1060696) and from the Department of Health, Victoria. TH is supported by a Career Development Fellowship from the NHMRC (ID 1069758).
Access to trial data {#FPar6}
====================
TPH will have access to the final, de-identified trial dataset.
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After years of praising their virtues, governments across the world are belatedly waking up to the problems posed by big tech. From India and Australia to France and America – and now the UK, with its report from the Digital Competition Expert Panel – politicians have been reckoning with how to mitigate the harms of the world’s largest technology platforms. And they all seem to arrive at the same answer: competition is the magic mechanism that will somehow tame the giants, unleash innovation and fix our digital world.
But what if competition is the problem rather than the solution? After all, it’s competition – not size – that demands more data, more attention, more engagement and more profits at all costs. It’s competition that demands the tech giants expand. It’s competition for ad dollars that drives Google, Facebook and Amazon to ignore privacy concerns and expand data collection. It’s the struggle to dominate voice interfaces and smart-home data that leads Amazon and Google to aggressively push their surveillance machines into our homes. It’s competition for attention that leads apps and platforms to make their products as addictive as possible. It’s competition for users and engagement that makes Twitter, Facebook and others turn a blind eye to abuse, fake news and far-right radicalisation. And it’s competition over who will be the dominant AI provider that leads the tech giants to constantly colonise new sources of data. The government’s efforts to increase competition risk simply aggravating these problems.
We could go even further and socialise the platforms instead of leaving them in private hands
So if increasing competition won’t hold big tech in check, what will? Too often, discussions about this start and end with government and consumers, and pay little heed to workers. That’s a mistake: larger and stronger tech unions can be an important counterbalance to the power of big tech. One area where the lack of competition really matters is in terms of wages: new research suggests that the bigger and more concentrated companies are in a particular industry, the more wages are suppressed.
The right solution is to support efforts to organise tech workers into unions. While still in their early stages, unions of tech workers have already made significant progress in terms of improving working conditions, blocking tech’s cooperation with the surveillance state and bringing to light the tech industry’s latent sexism and racism. The Independent Workers Union of Great Britain, for example, has made major strides in getting basic rights for gig economy workers.
A second step would be to regulate social media platforms like the public utilities they are. In an earlier era, governments didn’t hesitate to regulate natural monopolies where services provided were essential public goods. It didn’t make sense, for instance, to lay down two sets of railroad tracks next to each other in a wasteful and desperate effort to maintain competition. In cases like this, monopolies were recognised as inevitable, but were subject to strict regulations that ensured they worked for the common good and not just private profit.
Today, these regulations could apply to some of the services provided by big tech, and might include rules around who can use the platforms, limits on how the platforms can prioritise their own products on them, restrictions on how these companies can collect and use personal data, and requirements to open up intellectual property or licence patents cheaply. US Senator Elizabeth Warren has, for instance, suggested this recently in her proposals to tackle big tech.
Or we could go even further and socialise the platforms instead of leaving them in private hands, shifting ownership and control to workers, users and citizens. For example, one of the major problems posed by private ride-sharing apps such as Uber is that they produce negative externalities such as congestion and air pollution. A publicly owned platform, by contrast, would have a much wider mandate than simply adding more vehicles on to the road. Transport for London (TfL) could build and run a ride-sharing app that enabled low-emission movement across London. And the data from that platform could then be freely shared with other public bodies, ensuring better planning for housing and transportation.
Big tech has too much monopoly power – it's right to take it on | Kenneth Rogoff Read more
At a different scale, we could decisively move away from Amazon, Google and Microsoft’s control over cloud computing and imagine computing as a basic 21st-century utility. Initiatives such as the European Union’s Open Science Cloud demonstrate one model for providing a publicly funded and operated cloud infrastructure. There’s no reason why governments shouldn’t provide citizens with free access to a public cloud that ensures privacy, security, energy efficiency and equal access for all. Our digital infrastructures should not be left in the hands of profit-seeking monopolies, but instead run democratically for the common good.
Something needs to be done to rein in the increasing economic and political power of big tech, and fast. But the instinctual liberal belief in the value of competition will exacerbate rather than solve the problems. We need to think bigger if we are to take back control of our digital lives.
• Nick Srnicek is a lecturer in digital economy at King’s College London and the author of Platform Capitalism
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“Lennie” Gerber and Pearl Berlin have been together for 48 years.
Think of what it would mean for someone who has been with their partner for decades to confront losing a spouse, while the state they live in insists they’re not really married. That’s exactly what many same-sex couples face in North Carolina.
The American Civil Liberties Union and the ACLU of North Carolina recently filed a new lawsuit asking a court to grant relief for three North Carolina families in desperate situations by overturning North Carolina’s ban on marriage recognition for same-sex couples.
Our clients — three lesbian couples in long-term, loving, and committed relationships — have long been pillars of their community yet North Carolina law treats them as second-class citizens, denying them respect of their marriages.
Because of their age and medical conditions, these couples have a substantial fear that one of them could pass away before their marriage is recognized by North Carolina, depriving them forever of the dignity state-recognition of their marriage would bring.
Ellen “Lennie” Gerber and Pearl Berlin have been together for 48 years and were married in Maine in 2013, though they have lived in North Carolina for more than 40 years. Pearl is 89 years old and was hospitalized three times over the past two years, most recently for having suffered a fall where she hit her head, incurred internal bleeding, and broke three ribs.
“The idea of Pearl having to go through any sort of emergency alone, or have another person make decisions for her is devastating to me,” Lennie says. They worry that Pearl may die before their marriage is recognized in North Carolina. “It would only make things worse at a time of tremendous grief and loss if I were not listed on her death certificate as her spouse,” Lennie says, “which would demean the relationship we’ve built over 48 years and be an insult to her memory.”
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Q:
Can WKWebView instance show webpage while it is loading?
I'm using WKWebView to browse the specific website.
WKWebView instance loads initial page of the website by URL with method load(_ request: URLRequest) -> WKNavigation? Until load request isn't completed I see white screen. Can WKWebView show already loaded parts of the webpage while the rest of the webpage is loading?
Can UIWebView do this trick?
A:
if white screen is your issue, i recommend you to display an custom progress indicator over the webView or some Animation view over your webView.
Once the loading is completed hide the progress indicator.
Start your Animation at:
optional func webView(_ webView: WKWebView,
didStartProvisionalNavigation navigation: WKNavigation!)
{
//Start Progress indicator animation
}
End/Hide your Animation/Progress View at:
optional func webView(_ webView: WKWebView,
didFinish navigation: WKNavigation!)
{
//Stop Progress indicator animation
}
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Google and Amazon are opposing an Illinois bill that would effectively prohibit big tech companies from remotely accessing users’ microphones.
The Internet Association and other trade organizations that represent the three major tech companies — Facebook, Google, and Amazon — criticized SB 1719 in a press statement Wednesday, arguing that the bill is vague and unnecessary given that federal law already prevents such action. The bill prevents third-party actors from gaining remote access to people’s devices.
“Provides that no private entity may turn on or enable, cause to be turned on or enabled, or otherwise use a digital device’s microphone to listen for or collect information, including spoken words or other audible or inaudible sounds, unless a user first agrees to a written policy meeting specified criteria,” the bill notes.
The so-called Keep Internet Devices Safe Act is vague in its description, according to the Internet Association’s statement. “The definition of ‘user’ does not make it clear who should receive notice. Microphones cannot distinguish voices and thus cannot and thus cannot determine who is the user and who is the owner of the device.” the statement reads.
Here are the Amazon/Google talking points against the @CastroforIL22 bill barring tech platforms from secretly recording you in your home. Talking points are laundered through the Illinois Chamber of Commerce and tech trade associations. pic.twitter.com/ZWWa2zDgL9 — Matt Stoller (@matthewstoller) April 10, 2019
The Internet Association has not yet responded to The Daily Caller News Foundation’s request for comment. Google and Facebook have been criticized in recent years for their supposedly haphazard handling of private information. (RELATED: Google Dings AI Ethics Panel After Employees Throw A Fit Over Conservative Member)
Recent reports have discussed how Facebook reportedly spent years forging various partnership agreements with digital media to gobble up troves of users’ private data even when customers are not perusing the company’s platform. The Silicon Valley giant began forming partnerships during the company’s early years with tech businesses, including online retailers, entertainment websites, along with automakers and media organizations, reports show.
CEO Mark Zuckerberg sought to weave the company’s services into virtually every website across the network, inoculating it from competition, The New York Times noted in December 2018. Facebook also reportedly used contact lists from partners like Spotify and Netflix to gain deeper insight into people’s relationships — one partner was Huawei, a Chinese-based company American officials believe is a security threat because of its connections to China.
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Content created by The Daily Caller News Foundation is available without charge to any eligible news publisher that can provide a large audience. For licensing opportunities of our original content, please contact [email protected].
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Cephalosporins having a ureido substituted acyl sidechain and various groups in the 3-position are disclosed by Erickson in U.S. Pat. No. 3,673,183, Welch et al. in U.S. Pat. No. 3,708,479, Dolfini in U.S. Pat. No. 3,833,568, Breuer in U.S. Pat. No. 3,860,591, and Breuer et al. in U.S. Pat. Nos. 3,996,217, 3,996,218, and 4,024,135, and Belgian Pat. No. 833,640.
Various 7.alpha.-methoxy cephalosporins having a ureido substituted acyl sidechain and various groups in the 3-position are disclosed by Dolfini in U.S. Pat. Nos. 3,978,051, 3,989,693, 3,989,697, 4,000,134, and 4,061,852.
Ochiai et al. in U.S. Pat. No. 4,098,888 disclose cephalosporins having an iminothiazolyl acetamido acyl sidechain with an amino, hydroxy, or imino group on the .alpha.-carbon atom of the acyl sidechain.
Cephalosporins having an iminothiazolyl or substituted iminothiazolyl acetamido acyl sidechain and various groups in the 3-position are disclosed by Numata et al. in German Offenlegungsschrift No. 2,461,478. Similarly, desmethoxy cephalosporins having an alkoxyimino or hydroxyimino substituted iminothiazolyl acetamido acyl sidechain and various groups in the 3-position are disclosed in Belgian Pat. Nos. 850,662; 852,860; 852,427; 853,545; and 856,045.
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General: What records should be added onto SSNAP?
Patients with a primary diagnosis of stroke coded as I61, I63 or I64 should be added onto SSNAP. Additionally, any patient who follows the stroke care pathway should be included on SSNAP. If a patient has a serious co-morbidity that would prevent them from following the stroke care pathway, these patients should be excluded from the SSNAP dataset.
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