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I became the Leadership Editor of Forbes in December 2008, just as the American business world was crashing down and taking the jobs and homes of millions with it. Had I started the job a year or two earlier, I might have found that covering things like how to be a manager, corporate strategy, risk management, governance, and corporate social responsibility was worthy but possibly sometimes a little dull. Now I found that my beat was everything that had gone terribly wrong and was going to have to go very right to get us all back to prosperity. Since then, I've had the pleasure of publishing some of the world's best minds on every aspect of leadership.
Previously I was a senior editor of Forbes magazine, and before that I was for many years the managing editor of American Heritage and the editor of the quarterly Invention & Technology. I've emceed the annual induction ceremony at the National Inventors Hall of Fame, done the play-by-play over the P.A. system on a cruise ship as it passed through the Panama Canal, and written on the history of bourbon whiskey and the making of Steinway pianos, among many, many other things. I prepared for all that by majoring in music in college and writing a senior thesis on the music of Hector Berlioz.
Follow me on Twitter here.
Names You Need to Know in 2011: FiveFingers
Here’s something you need to know about: Could we be seeing the beginning of the end of the modern running shoe? A growing legion of runners and recreational joggers—I am one—backed by a rising number of physiologists, believe that running shoes do more harm than good for millions of people. With their inflexibility, cushioning, and raised heels, they almost force you to crash down on your heel and send the impact of all your weight in every step straight into your knees and hips. That’s not how our bodies were designed to work, and it does terrible harm. (This post is part of an ongoing effort to crowd-source a repeating feature in Forbes magazine entitled Names You Need to Know. We are looking for the people, places, products and ideas that will have significant impact in the near future. Join the ongoing conversation here. )
Millions of years of evolution gave us a much springier way of running, which begins with our coming down on the front of our foot, a little like when you run in place. You necessarily run that way when you run barefoot, because hitting with your heel would be unbearably painful. But who is going to run barefoot in the modern world?
Enter FiveFingers, a line of shoes by the Italian company Vibram that are really gloves for the feet, with minimal protection and a finger for each toe. They look like the feet part of a gorilla costume, and they feel almost like being barefoot. They have grown a cult of devotees over the past few years, and they’re starting to go mainstream. You see them on more and more runners, and the stores that sell them can’t keep them in stock. The running shoe industry has begun trying to figure out how to either beat them or join them, and they and other new minimalist shoes are the subject of a long article in the November 2010 issue of Runner’s World.
I’m a believer. I became one after seeing the wacky-looking things on someone, asking about them, learning more online, and deciding to spend the $85 to try out a pair of FiveFingers KSOs (a model that covers the top of the foot and “Keeps Stuff Out”). I feel freer, faster, lighter on my feet, and my knees and hips don’t hurt anywhere near the way they used to. Pronation and plantar fascitis seem to disappear when you start using your feet the way they were designed to be used.
You have to adapt to barefoot or FiveFingers running very carefully and slowly. Much of the impact that used to shoot into your knees and hips now gets absorbed by your calf muscles. Those muscles will get very sore at first and then very strong. You have to wake up muscles all over your feet that you haven’t been using. You have to toughen the bottoms of your feet. You can get blisters—I wear five-fingered socks to prevent that. But I’m loving running more than in many years. I feel throughout my body that I’m doing something right that I always used to do wrong. If my experience and that of thousands of others is any guide, the FiveFingers phenomenon can only grow and spread. The age of the modern running shoe that began in the 1970s with Phil Knight and Nike could just possibly begin to enter its twilight.
Do you agree with me that FiveFingers is a name to know in 2011? Please comment and let me know. I’ve written this as part of Forbes’s new “Names You Need to Know in 2011″ project. Find out about the project and how you can contribute to it here.
Post Your Comment
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Forbes writers have the ability to call out member comments they find particularly interesting. Called-out comments are highlighted across the Forbes network. You'll be notified if your comment is called out.
I hadn’t seen that. Thanks. I see the government banned the shoes because they “detract from a military appearance,” not because of any perceived physiological shortcoming in them. I guess the Army prefers guerrilla warfare to gorilla warfare.
My name is Michael Renwick. I am the PR Director for www.fiftysense.com. We are an information based site dedicated to educating through common sense ideas for active baby boomers. Our panel of testers, all over the age of 50, have been putting different fivefingers models to the test for over two years now. For a group with sore knees, less padding in their feet and an aversion toward new things–they love them. http://fiftysense.com/reviews/vibram.shtml I thought a first-hand account from testers themselves might be useful. Let me know if fiftysense.com can be of any further use to you.
I’m a college student in Denver Colorado developing a complimentary product for this new emerging market. I bought my first pair of VFF’s in 2009 at a mountaineering and outdoor store in Glennwood Springs, CO and couldn’t believe the fit! (Prior to this I was a staunch shoe and sock wearer and never would consider walking anywhere bare footed.) Nor could I stop laughing for the first week every time I looked down at my feet. I am barely a novice climber and only run for enjoyment or out of necessity, so my perspective is a little different than most of the perspectives I’ve found during my research thus far. I did not follow the suggestions by the company with regards to allowing myself to gradually work into wearing the shoes and I didn’t experience much post exercise soreness or discomfort. However I didn’t over exert my feet either. I certainly would not try running competitively in these shoes, especially right off the bat but over time I think they will provide an edge for all who wear them. I don’t know about making the current shoe designs a thing of the past, but I am confident that these types of shoes will continue to grow in popularity.
I am a non-runner. I will occasionally jog for a minute or two at a time during a warm up or cool down at the gym… However, I wear my VFFs to Zumba class at least 3-4 times per week (along with injinjii toe socks). All my life, I have had issues with blisters. I have never found a sneaker that was comfortable for walking all day or exercising. Every day at a theme park (and I live 2.5 hours from Disney World – so theme parks are just destined to be part of my son’s growing-up years!) resulted in at least 4 or 5 blisters on each foot.
I got my VFF KomodoSports a year or two ago. I was wary of spending $100 on a pair of shoes, considering that I’ve never owned a truly comfortable pair of shoes. Moreover, I have high arches and always had a tendency to get cramps in my arches, and the beginnings of arthritis in my toes, which would have me removing my shoes and massaging my toes after about 20 minutes of a Zumba class. From the very first day of wearing my Vibrams, I have never had discomfort in my feet, and I have had at least a dozen theme park days with zero blisters.
I intend to be a lifelong, faithful customer of Vibram FiveFingers. Who knows, maybe I’ll even become a runner like my husband and son… or perhaps at least less of a non-runner!
I don’t like to be an apostate, but I just discovered Terra Plana’s Vivobarefoot Evo II shoe. It looks more like an ordinary shoe, without articulated toes, but it’s loose around its nice big toe box and holds your foot by the heel and ankle. The result, for me, is even greater comfort and an even fuller feeling of freedom than the Fivefingers. And you can wear it anywhere without drawing as much attention.
I’ve also, incidentally, found that I was making the balls of my feet hurt a little by being too protective of my heels, barely touching them to the ground when I ran. I’ve learned to run almost flat-footed, bringing down the front of the foot slightly before the rest, but all of it down firmly.
The crazy appearance and getting to chit-chat with perfect strangers is half the fun ;-)
For me, having the separated toes is a large part of the comfort, though just wearing the injinjii socks with regular sneakers gives me some improvement compared to regular socks. I just have crazy feet that love these crazy shoes. The FedEx guy will be here with a new pair today and the UPS guy will bring another pair on Friday!
Another thing, I love about the VFFs is how well they hold up in the wash (as they’re machine washable), how quickly they dry (I clip them on a hanger to dry) and how clean they get. I’ve worn my to mow the lawn and had them clean and dry for fitness classes the next day!
|
{
"pile_set_name": "Pile-CC"
}
|
Philistina
Philistina is a genus of beetles belonging to the family Scarabaeidae, typically placed in the tribe Phaedimini .
Species
Philistina aurita (Arrow, 1910)
Philistina benesi Drumont, 1998
Philistina bicoronata (Jordan, 1894)
Philistina campagnei (Bourgoin, 1920)
Philistina fujiokai (Jakl, 2011)
Philistina gestroi (Arrow, 1910)
Philistina inermis (Janson, 1903)
Philistina javana (Krikken, 1979)
Philistina khasiana (Jordan, 1894)
Philistina knirschi (Schürhoff, 1933)
Philistina manai Antoine, 2002
Philistina microphylla (Wood-Mason, 1881)
Philistina minettii (Antoine, 1991)
Philistina nishikawai Sakai, 1992
Philistina pilosa (Mohnike, 1873)
Philistina rhinophylla (Wiedemann, 1823)
Philistina sakaii Alexis & Delpont, 2001
Philistina salvazai (Bourgoin, 1920)
Philistina squamosa Ritsema, 1879
Philistina tibetana (Janson, 1917)
Philistina tonkinensis (Moser, 1903)
Philistina vollenhoveni Mohnike, 1871
Philistina zebuana Kraatz, 1895
Category:Cetoniinae
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Metal hydrides can be used to store hydrogen on-board fuel cell vehicles, but the process of fracture which such materials undergo when exposed to hydrogen makes them poor conductors of the heat generated during hydriding. This fracture process creates particles having irregular faceted shapes...
|
{
"pile_set_name": "Pile-CC"
}
|
unit MouseActionDialog;
{$mode objfpc}{$H+}
interface
uses
Classes, Forms, Controls, ExtCtrls, StdCtrls, ButtonPanel, Spin, CheckLst,
SynEditMouseCmds, LazarusIDEStrConsts, KeyMapping, IDECommands, types;
var
ButtonName: Array [TSynMouseButton] of String;
ClickName: Array [TSynMAClickCount] of String;
ButtonDirName: Array [TSynMAClickDir] of String;
type
{ TMouseaActionDialog }
TMouseaActionDialog = class(TForm)
ActionBox: TComboBox;
ActionLabel: TLabel;
AltCheck: TCheckBox;
BtnDefault: TButton;
BtnLabel: TLabel;
ButtonBox: TComboBox;
ButtonPanel1: TButtonPanel;
CaretCheck: TCheckBox;
chkUpRestrict: TCheckListBox;
ClickBox: TComboBox;
DirCheck: TCheckBox;
PaintBox1: TPaintBox;
PriorLabel: TLabel;
OptBox: TComboBox;
CtrlCheck: TCheckBox;
CapturePanel: TPanel;
OptLabel: TLabel;
Opt2Spin: TSpinEdit;
Opt2Label: TLabel;
ShiftCheck: TCheckBox;
PriorSpin: TSpinEdit;
procedure ActionBoxChange(Sender: TObject);
procedure BtnDefaultClick(Sender: TObject);
procedure ButtonBoxChange(Sender: TObject);
procedure CapturePanelMouseDown(Sender: TObject; Button: TMouseButton; Shift: TShiftState;
{%H-}X, {%H-}Y: Integer);
procedure DirCheckChange(Sender: TObject);
procedure FormCreate(Sender: TObject);
procedure PaintBox1MouseWheel(Sender: TObject; Shift: TShiftState; WheelDelta: Integer;
{%H-}MousePos: TPoint; var {%H-}Handled: Boolean);
private
FKeyMap: TKeyCommandRelationList;
procedure AddMouseCmd(const S: string);
procedure FillListbox;
public
{ public declarations }
Procedure ResetInputs;
Procedure ReadFromAction(MAct: TSynEditMouseAction);
Procedure WriteToAction(MAct: TSynEditMouseAction);
property KeyMap: TKeyCommandRelationList read FKeyMap write FKeyMap;
end;
function KeyMapIndexOfCommand(AKeyMap: TKeyCommandRelationList; ACmd: Word) : Integer;
implementation
uses Math;
{$R *.lfm}
const
BtnToIndex: array [TSynMouseButton] of Integer = (0, 1, 2, 3, 4, 5, 6);
ClickToIndex: array [ccSingle..ccAny] of Integer = (0, 1, 2, 3, 4);
IndexToBtn: array [0..6] of TSynMouseButton = (mbXLeft, mbXRight, mbXMiddle, mbXExtra1, mbXExtra2, mbXWheelUp, mbXWheelDown);
IndexToClick: array [0..4] of TSynMAClickCount = (ccSingle, ccDouble, ccTriple, ccQuad, ccAny);
function KeyMapIndexOfCommand(AKeyMap: TKeyCommandRelationList; ACmd: Word): Integer;
var
i: Integer;
begin
for i := 0 to AKeyMap.RelationCount - 1 do
if AKeyMap.Relations[i].Command = ACmd then
exit(i);
Result := -1;
end;
{ MouseaActionDialog }
procedure TMouseaActionDialog.AddMouseCmd(const S: string);
var
i: Integer;
s2: String;
begin
i:=0;
if IdentToSynMouseCmd(S, i) then begin
s2 := MouseCommandName(i);
if s2 = '' then s2 := s;
ActionBox.Items.AddObject(s2, TObject(ptrint(i)));
end;
end;
procedure TMouseaActionDialog.FillListbox;
const
cCheckSize=35;
var
r: TSynMAUpRestriction;
s: string;
i, Len: integer;
begin
for r := low(TSynMAUpRestriction) to high(TSynMAUpRestriction) do
case r of
crLastDownPos: chkUpRestrict.AddItem(synfMatchActionPosOfMouseDown, nil);
crLastDownPosSameLine: chkUpRestrict.AddItem(synfMatchActionLineOfMouseDown, nil);
crLastDownPosSearchAll: chkUpRestrict.AddItem(synfSearchAllActionOfMouseDown, nil);
crLastDownButton: chkUpRestrict.AddItem(synfMatchActionButtonOfMouseDown, nil);
crLastDownShift: chkUpRestrict.AddItem(synfMatchActionModifiersOfMouseDown, nil);
crAllowFallback: chkUpRestrict.AddItem(synfContinueWithNextMouseUpAction, nil);
else begin
WriteStr(s, r);
chkUpRestrict.AddItem(s, nil);
end;
end;
// update scrollbar
Len := 0;
with chkUpRestrict do
begin
for i := 0 to Items.Count-1 do
Len := Max(Len, Canvas.TextWidth(Items[i])+cCheckSize);
ScrollWidth := Len;
end;
end;
procedure TMouseaActionDialog.FormCreate(Sender: TObject);
var
mb: TSynMouseButton;
cc: TSynMAClickCount;
begin
ButtonName[mbXLeft]:=dlgMouseOptBtnLeft;
ButtonName[mbXRight]:=dlgMouseOptBtnRight;
ButtonName[mbXMiddle]:=dlgMouseOptBtnMiddle;
ButtonName[mbXExtra1]:=dlgMouseOptBtnExtra1;
ButtonName[mbXExtra2]:=dlgMouseOptBtnExtra2;
ButtonName[mbXWheelUp]:=dlgMouseOptBtnWheelUp;
ButtonName[mbXWheelDown]:=dlgMouseOptBtnWheelDown;
ClickName[ccSingle]:=dlgMouseOptBtn1;
ClickName[ccDouble]:=dlgMouseOptBtn2;
ClickName[ccTriple]:=dlgMouseOptBtn3;
ClickName[ccQuad]:=dlgMouseOptBtn4;
ClickName[ccAny]:=dlgMouseOptBtnAny;
FillListbox;
ButtonDirName[cdUp]:=lisUp;
ButtonDirName[cdDown]:=lisDown;
Caption := dlgMouseOptDlgTitle;
CapturePanel.Caption := dlgMouseOptCapture;
CapturePanel.ControlStyle := ControlStyle + [csTripleClicks, csQuadClicks];
CaretCheck.Caption := dlgMouseOptCaretMove;
ActionBox.Clear;
GetEditorMouseCommandValues(@AddMouseCmd);
ButtonBox.Clear;
for mb := low(TSynMouseButton) to high(TSynMouseButton) do
ButtonBox.Items.add(ButtonName[mb]);
ClickBox.Clear;
for cc:= low(TSynMAClickCount) to high(TSynMAClickCount) do
ClickBox.Items.add(ClickName[cc]);
DirCheck.Caption := dlgMouseOptCheckUpDown;
ShiftCheck.Caption := dlgMouseOptModShift;
AltCheck.Caption := dlgMouseOptModAlt;
CtrlCheck.Caption := dlgMouseOptModCtrl;
ActionLabel.Caption := dlgMouseOptDescAction;
BtnLabel.Caption := dlgMouseOptDescButton;
BtnDefault.Caption := dlgMouseOptBtnModDef;
PriorLabel.Caption := dlgMouseOptPriorLabel;
Opt2Label.Caption := dlgMouseOptOpt2Label;
end;
procedure TMouseaActionDialog.ResetInputs;
var
r: TSynMAUpRestriction;
begin
ActionBox.ItemIndex := 0;
ButtonBox.ItemIndex := 0;
ClickBox.ItemIndex := 0;
DirCheck.Checked := False;
ShiftCheck.State := cbGrayed;
AltCheck.State := cbGrayed;
CtrlCheck.State := cbGrayed;
for r := low(TSynMAUpRestriction) to high(TSynMAUpRestriction) do
chkUpRestrict.Checked[ord(r)] := False;
ActionBoxChange(nil);
OptBox.ItemIndex := 0;
end;
procedure TMouseaActionDialog.BtnDefaultClick(Sender: TObject);
begin
ShiftCheck.State := cbGrayed;
AltCheck.State := cbGrayed;
CtrlCheck.State := cbGrayed;
end;
procedure TMouseaActionDialog.ButtonBoxChange(Sender: TObject);
begin
DirCheck.Enabled := not(IndexToBtn[ButtonBox.ItemIndex] in [mbXWheelUp, mbXWheelDown]);
chkUpRestrict.Enabled := DirCheck.Enabled and DirCheck.Checked;
end;
procedure TMouseaActionDialog.ActionBoxChange(Sender: TObject);
var
ACmd: TSynEditorMouseCommand;
i: Integer;
begin
OptBox.Items.Clear;
ACmd := TSynEditorMouseCommand({%H-}PtrUInt(Pointer(ActionBox.items.Objects[ActionBox.ItemIndex])));
if ACmd = emcSynEditCommand then begin
OptBox.Enabled := True;
OptBox.Clear;
for i := 0 to KeyMap.RelationCount - 1 do
if (KeyMap.Relations[i].Category.Scope = IDECmdScopeSrcEdit) or
(KeyMap.Relations[i].Category.Scope = IDECmdScopeSrcEditOnly)
then
OptBox.Items.AddObject(KeyMap.Relations[i].GetLocalizedName,
TObject({%H-}Pointer(PtrUInt(KeyMap.Relations[i].Command))));
OptLabel.Caption := dlgMouseOptionsynCommand;
OptBox.ItemIndex := 0;
end
else
begin
OptBox.Items.CommaText := MouseCommandConfigName(ACmd);
if OptBox.Items.Count > 0 then begin
OptLabel.Caption := OptBox.Items[0];
OptBox.Items.Delete(0);
OptBox.Enabled := True;
OptBox.ItemIndex := 0;
end else begin
OptLabel.Caption := '';
OptBox.Enabled := False
end;
end;
end;
procedure TMouseaActionDialog.CapturePanelMouseDown(Sender: TObject;
Button: TMouseButton; Shift: TShiftState; X, Y: Integer);
begin
ButtonBox.ItemIndex := BtnToIndex[SynMouseButtonMap[Button]];
ClickBox.ItemIndex := 0;
if ssDouble in Shift then ClickBox.ItemIndex := 1;
if ssTriple in Shift then ClickBox.ItemIndex := 2;
if ssQuad in Shift then ClickBox.ItemIndex := 3;
ShiftCheck.Checked := ssShift in Shift;
AltCheck.Checked := ssAlt in Shift;
CtrlCheck.Checked := ssCtrl in Shift;
end;
procedure TMouseaActionDialog.DirCheckChange(Sender: TObject);
begin
chkUpRestrict.Enabled := DirCheck.Checked;
end;
procedure TMouseaActionDialog.PaintBox1MouseWheel(Sender: TObject; Shift: TShiftState;
WheelDelta: Integer; MousePos: TPoint; var Handled: Boolean);
begin
if WheelDelta > 0
then ButtonBox.ItemIndex := BtnToIndex[mbXWheelUp]
else ButtonBox.ItemIndex := BtnToIndex[mbXWheelDown];
ClickBox.ItemIndex := 4;
ShiftCheck.Checked := ssShift in Shift;
AltCheck.Checked := ssAlt in Shift;
CtrlCheck.Checked := ssCtrl in Shift;
end;
procedure TMouseaActionDialog.ReadFromAction(MAct: TSynEditMouseAction);
var
r: TSynMAUpRestriction;
begin
ActionBox.ItemIndex := ActionBox.Items.IndexOfObject(TObject({%H-}Pointer(PtrUInt(MAct.Command))));
ButtonBox.ItemIndex := BtnToIndex[MAct.Button];
ClickBox.ItemIndex := ClickToIndex[MAct.ClickCount];
DirCheck.Checked := MAct.ClickDir = cdUp;
CaretCheck.Checked := MAct.MoveCaret;
ShiftCheck.Checked := (ssShift in MAct.ShiftMask) and (ssShift in MAct.Shift);
if not(ssShift in MAct.ShiftMask) then ShiftCheck.State := cbGrayed;
AltCheck.Checked := (ssAlt in MAct.ShiftMask) and (ssAlt in MAct.Shift);
if not(ssAlt in MAct.ShiftMask) then AltCheck.State := cbGrayed;
CtrlCheck.Checked := (ssCtrl in MAct.ShiftMask) and (ssCtrl in MAct.Shift);
if not(ssCtrl in MAct.ShiftMask) then CtrlCheck.State := cbGrayed;
PriorSpin.Value := MAct.Priority;
Opt2Spin.Value := MAct.Option2;
for r := low(TSynMAUpRestriction) to high(TSynMAUpRestriction) do
chkUpRestrict.Checked[ord(r)] := r in MAct.ButtonUpRestrictions;
ActionBoxChange(nil);
ButtonBoxChange(nil);
if OptBox.Enabled then begin
if MAct.Command = emcSynEditCommand then
OptBox.ItemIndex := OptBox.Items.IndexOfObject(TObject({%H-}Pointer(PtrUInt(MAct.Option))))
else
OptBox.ItemIndex := MAct.Option;
end;
end;
procedure TMouseaActionDialog.WriteToAction(MAct: TSynEditMouseAction);
var
r: TSynMAUpRestriction;
begin
MAct.Command := TSynEditorMouseCommand({%H-}PtrUInt(Pointer(ActionBox.items.Objects[ActionBox.ItemIndex])));
MAct.Button := IndexToBtn[ButtonBox.ItemIndex];
MAct.ClickCount := IndexToClick[ClickBox.ItemIndex];
MAct.MoveCaret := CaretCheck.Checked;
if DirCheck.Checked
then MAct.ClickDir := cdUp
else MAct.ClickDir := cdDown;
MAct.Shift := [];
MAct.ShiftMask := [];
if ShiftCheck.State <> cbGrayed then MAct.ShiftMask := MAct.ShiftMask + [ssShift];
if AltCheck.State <> cbGrayed then MAct.ShiftMask := MAct.ShiftMask + [ssAlt];
if CtrlCheck.State <> cbGrayed then MAct.ShiftMask := MAct.ShiftMask + [ssCtrl];
if ShiftCheck.Checked then MAct.Shift := MAct.Shift + [ssShift];
if AltCheck.Checked then MAct.Shift := MAct.Shift + [ssAlt];
if CtrlCheck.Checked then MAct.Shift := MAct.Shift + [ssCtrl];
MAct.Priority := PriorSpin.Value;
MAct.Option2 := Opt2Spin.Value;
MAct.ButtonUpRestrictions := [];
for r := low(TSynMAUpRestriction) to high(TSynMAUpRestriction) do
if chkUpRestrict.Checked[ord(r)] then
MAct.ButtonUpRestrictions := MAct.ButtonUpRestrictions + [r];
if OptBox.Enabled then begin
if MAct.Command = emcSynEditCommand then begin
MAct.Option := TSynEditorMouseCommandOpt({%H-}PtrUInt(Pointer(OptBox.Items.Objects[OptBox.ItemIndex])));
end
else
MAct.Option := OptBox.ItemIndex;
end
else
MAct.Option := 0;
end;
end.
|
{
"pile_set_name": "Github"
}
|
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in our future tutorial, as soon as the ssl vpn config on sonicwall firewall URL loads up, we can than see the page that was hosted on backend server. This shows that our reverse proxy is correctly configured working. We will learn how to configure the apache reverse proxy as loadbalancer.
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{
"pile_set_name": "Pile-CC"
}
|
We Are Closing
Special Offers on all Stock Items
Epson ink up to 50% Off
With effect from Saturday 24th June our shop at Enterprise 5 will close.
We will still be contactable via email for the next couple of months for existing queries to complete ongoing sales and collect outstanding payments, therefore, if you require copy invoices or other simple requests please contact us on sales@htbl.co.uk
Please be aware that we will no longer be able to carry out Apple or Epson repairs.
For support queries please contact:
Apple – www. apple.com/uk – 0800 107 6285
Leeds Apple Store – 0113 251 1000
Epson www.epson.co.uk – 03439 037766
Large format – Equinox – www.equinox.co.uk – 01684 290000
BenQ Projectors – Equinox – www.equinox.co.uk – 01684 290000
Thanks for all your custom over the last 26 years, we wish all our customers the best for the future.
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{
"pile_set_name": "Pile-CC"
}
|
Q:
How to have opacity on my texture in directx 9?
Been trying hard to find a solution to this, but the results are pretty bad.
Basically I want to draw a texture (it's made up of 2 triangles so it's a quad), and make them have alpha values (0-255, but 0-1 will do too). This is so that I can have that fade in/out effect when I wish.
A:
Found my answer:
Link to Source
DWORD AlphaValue;
AlphaValue = D3DCOLOR_ARGB(100,255,255,255);
mpDevice->SetTextureStageState(0, D3DTSS_COLOROP, D3DTOP_MODULATE);
mpDevice->SetTextureStageState(0, D3DTSS_COLORARG1, D3DTA_TEXTURE);
mpDevice->SetTextureStageState(0, D3DTSS_COLORARG2, D3DTA_DIFFUSE);
mpDevice->SetTextureStageState(0, D3DTSS_ALPHAOP, D3DTOP_MODULATE);
mpDevice->SetTextureStageState(0, D3DTSS_ALPHAARG1, D3DTA_TEXTURE);
mpDevice->SetTextureStageState(0, D3DTSS_CONSTANT, AlphaValue);
mpDevice->SetTextureStageState(0, D3DTSS_ALPHAARG2, D3DTA_CONSTANT);
mpDevice->SetTextureStageState(1, D3DTSS_COLOROP, D3DTOP_DISABLE);
mpDevice->SetTextureStageState(1, D3DTSS_ALPHAOP, D3DTOP_DISABLE);
pMesh->Draw();
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{
"pile_set_name": "StackExchange"
}
|
How many poets does it take to screw in a light bulb?
I have one [PG] fantasy of reading poems in comedy clubs and telling jokes at poetry readings. Why waste a fantasy on it? Why ruin a good comedy night for those unsuspecting patrons? I don’t know. I don’t want to answer those questions. They’re rather aggressive, if you ask me. Sorry, I didn’t mean to offend you. It’s okay. I’d rather explore what that might do, in my mind, to read funny poems, funny poems that are often also quite sad, on stage, against a brick wall, beneath a blinding Klieg or two, alone. The set up sounds like a firing squad.
I’d like for the boundary between what is funny and what is poetry to be torn down or at least be outfitted with a glory hole. I feel there is one (a boundary, geez!). I feel it when I read a funny poem in a terribly lit, modular classroom and am met with unblinking eyes (and no laughs). Or when I read on an elevated stage at a fancy literary festival and hear only the groan of a chair (and no laughs). Maybe it’s because you’re not funny? Get a life. What I’m getting at is there is a set of expectations that surrounds poems and poetry. There is the expectation that the person in front of us is smart(er than us), that poetry is depressing, or worse, poignant, that it is a puzzle and so needs focus lest you miss a vital piece, that it requires silence to be shared. Not super into expectations, especially those. With all of that seriousness seething in the audience it’s a small wonder that I ever get a laugh at all, and I do, okay, so zip it. I feel it when I’m in the audience, too; when my laugh is the only one bashing itself to death against the stark white walls of the auditorium.
For the next three months, I’m going to write about funny poetry and funny poets. Shit, I might even share what I think are Alice Notley’s funniest poems and some of you will be like, WHAT? I had no idea! Yup. She can be hilarious if you let her. (Most things can.) So check back in, with fondness please, because what if really poetry isn’t funny and this is a huge failure.
For the next three months, I will write about how and when poetry is funny, and maybe why, too. When I told a handful of friends the idea of this series, they said, Uh, yeah, but is it? Which is funny. So yes, handful of friends, it is.
Sommer Browning is the author of several books, including the poetry collection, Backup Singers (Birds, LLC), the artist book, The Circle Book (Cuneiform), the joke book, You’re On My Period (Counterpath Press), the collection of comics, Everything But Sex (Low Frequency Press), and a book of dreams, WANT TO HEAR ABOUT THIS DREAM I HAD (Reality Beach). She runs an art space out of her garage called GEORGIA and she’s a librarian in Denver.
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{
"pile_set_name": "Pile-CC"
}
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The isolation of alpha-aminolevulinate synthetase (ALA-S) from livers of induced chick embryos is being continued; with purified enzyme, an antibody will be developed to study the induction mechanism and the transport of the enzyme from the microsome into the mitochondria where it is active. Chlorinated hydrocarbons both induce ALA-S synthesis and block the biosynthetic chain so that uroporphyrin accumulates. The mechanism of the block will be sought. Iron and metabolites of the chlorinated hydrocarborns may be involved. These studies may lead to our understanding of the environmental toxic effects of the chlorinated hydrocarbons including tetrachlorodibenzodioxin. In Friend leukemia cells dimethyl sulfoxide caused their differentiation into erythropoetic cells. The steps in the development of the enzymes of the heme biosynthetic chain will be examined to note whether there is a sequence in time of the developing enzymes that follows the sequence of enzymes of the heme biosynthetic chain. In animals, the synthesis of the first committed product of the heme biosynthetic chain is ALA. It is well establised that ALA is formed by an enzyme that uses glycine plus succinyl CoA as its substrates. In plants there is evidence that ALA may be formed from other substrates; the mechanism of ALA formation will be studied with appropriate labelled substrates and with metabolic inhibitors.
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{
"pile_set_name": "NIH ExPorter"
}
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Growth inhibition of Microcystis aeruginosa by white-rot fungus Lopharia spadicea.
Harmful cyanobacterial blooms cause water deterioration and threaten human health. It is necessary to remove harmful cyanobacteria with useful methods. A bio-treatment may be one of the best ways to do this. A strain of specific white-rot fungus, Lopharia spadicea, with algicidal ability was isolated. Its algicidal ability on algae under various conditions was determined using three main influence factors: initial chlorophyll-a content, initial pH, and algal cell mixture. The result showed that the chlorophyll-a content of Microcystis aeruginosa FACHB-912, Oocystis borgei FACHB-1108, and Microcystis flos-aquae FACHB-1028 decreased from 798+/-13, 756+/-40, and 773+/-24 microg/L to 0 within 39 h. L. spadicea could also remove more than 95% chlorophyll-a when initial chlorophyll-a content increased from 397+/-13 to 2,132+/-4 microg/L. Moreover, the strain has great removal ability under a broad initial pH range of 5.5 to 9.5. The chlorophyll-a content of the three algal strain mixtures decreased from about 672+/-23 microg/L to 0 within 45 h. After superoxide dismutase (SOD) and malondialdehyde (MAD) were assessed in a co-culture of L. spadicea, it was observed that an increase in MAD content was correlated with the decrease in chlorophyll-a content of M. aeruginosa FACHB-912. This result suggested that the algae was not only greatly inhibited but also severely damaged by the fungus.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Q:
GREP numeric range
I have log file in following format
14:15
14:16
14:17
14:30
14:31
14:41
I want to grep based on 15 minute time interval.
So in one case I want to grep time b/w 14:0-14:15
in another case I want to grep time b/w 14:15 14:30
Is there a way to do that in grep ?
A:
$ awk -v start=14:15 -v end=14:30 'start<=$1 && $1<=end' log.txt
14:15
14:16
14:17
14:30
|
{
"pile_set_name": "StackExchange"
}
|
JERUSALEM — The publication late last week of eyewitness accounts by Israeli soldiers alleging acute mistreatment of Palestinian civilians in the recent Gaza fighting highlights a debate here about the rules of war. But it also exposes something else: the clash between secular liberals and religious nationalists for control over the army and society.
Several of the testimonies, published by an institute that runs a premilitary course and is affiliated with the left-leaning secular kibbutz movement, showed a distinct impatience with religious soldiers, portraying them as self-appointed holy warriors.
A soldier, identified by the pseudonym Ram, is quoted as saying that in Gaza, “the rabbinate brought in a lot of booklets and articles and their message was very clear: We are the Jewish people, we came to this land by a miracle, God brought us back to this land and now we need to fight to expel the non-Jews who are interfering with our conquest of this holy land. This was the main message, and the whole sense many soldiers had in this operation was of a religious war.”
Dany Zamir, the director of the one-year premilitary course who solicited the testimonies and then leaked them, leading to a promise by the military to investigate, is quoted in the transcripts as expressing anguish over the growing religious nationalist elements of the military.
|
{
"pile_set_name": "OpenWebText2"
}
|
Failure to demonstrate a role for line Ib tumor-associated surface antigen in the etiology of age-dependent polioencephalomyelitis.
In the preceding paper [Morris et al. (1982), Molec. Immun. 19, 973-982] we demonstrate an associative interaction between the line Ib tumor-associated surface antigen (Ib-TASA) and the Dk/Kk regions of the major histocompatibility complex, i.e. 'altered-self' antigen. We originally hypothesized that age-dependent polioencephalomyelitis (ADPE) occurred as the result of immune recognition of a 'self'-determinant on the 'altered-self' antigen. In this report we used the non-ionic detergent, NP-40, to solubilize Ib cell surface antigens. Although immunization of immunocompetent C58 mice with the soluble NP-40 Ib cell extract afforded protection to lethal tumor challenge, the extract failed to induce ADPE in immunosuppressed mice. Data presented here demonstrate that Ib-TASA is not involved in the etiology of ADPE. The evidence suggests that lactic dehydrogenase virus, which is a silent virus passaged with line Ib leukemia, is the causative agent of the paralytic disease.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
1 F.3d 1231NOTICE: First Circuit Local Rule 36.2(b)6 states unpublished opinions may be cited only in related cases.
Danny M. KELLY, Plaintiff, Appellant,v.Niles L. NORDBERG, et al., Defendants, Appellees.
No. 93-1138.
United States Court of Appeals,First Circuit.
August 17, 1993
APPEAL FROM THE UNITED STATES DISTRICT COURT FOR THE DISTRICT OF MASSACHUSETTS
Danny M. Kelly on brief pro se.
Scott Harshbarger, Attorney General, and Steve Berenson, Assistant Attorney General, on Memorandum in Support of Appellee's Motion for Summary Affirmance, for appellee.
D. Mass.
VACATED AND REMANDED.
Before Breyer, Chief Judge, Selya and Stahl, Circuit Judges.
Per Curiam.
1
The narrow question before us is whether plaintiff was required to exhaust state administrative remedies before bringing this suit. Plaintiff appears pro se seeking unspecified damages, injunctive and declaratory relief against the Massachusetts Department of Employment and Training's ("DET's") practice of disqualifying for unemployment benefits those persons who travel outside of the State for the dual purpose of seeking work and engaging in other activities. The district court granted to defendant a judgment on the pleadings. We vacate and remand without prejudice to consideration of any other issue in the case.
2
A grant of judgment on the pleadings is subject to plenary review. International Paper Co. v. Jay, 928 F.2d 480, 482 (1st Cir. 1991). We accept as true all of the non-movant's factual allegations and draw all reasonable inferences in his favor. Santiago de Castro v. Morales Medina, 943 F.2d 129, 130 (1st Cir. 1991). We are aided here by the parties' apparent agreement as to the administrative posture of plaintiff's claim.
3
According to the complaint, plaintiff was qualified to receive unemployment benefits of $282 per week beginning in September, 1991, after he lost his job as a software engineer at Wang Laboratories. He sought new employment locally and in the midwest. When he filed a required periodic claim for benefits in December, 1991, he certified that he would be in Chicago, Illinois from December 23, 1991 until January 5, 1992. He alleged that his reason for travel was to look for work and to visit family and friends. He also certified that while he was there he actively sought work, and was "available" for employment.1
4
The DET denied plaintiff any benefits for the two weeks he was in Chicago because of the dual purpose of his trip. According to both parties' pleadings, the agency's rule, as reflected in its "Service Representative Handbook," is that a claimant who travels or stays outside of the registration area must do so "for the SOLE purpose of seeking new employment or reporting for a pre-arranged job or job interview" in order to qualify for benefits.2 Answer Exh. D., Complaint p 7. Based on plaintiff's written answers to questions about his trip, a DET adjudicator decided that plaintiff's trip "did not meet the requirements of the law because ... looking for employment was not the sole purpose of the trip." Answer p 7, Exh. D.
5
Under M.G.L. c. 151A, Sec. 39(b), a claimant may seek reconsideration of the DET's initial determination by requesting a de novo hearing before a review examiner. In the absence of such a request, the initial determination is final. The parties agree that plaintiff did not request agency review, but the DET spontaneously treated plaintiff's correspondence as a notice of appeal, advising plaintiff of a hearing date. Plaintiff did not appear at the scheduled hearing and did not respond to a further notice from the DET offering to consider any justifications for his failure to appear. DET dismissed the appeal.
6
Plaintiff instead filed this complaint pro se alleging that DET's travel rule unconstitutionally infringed on his right to travel and to enjoy the same benefits as lifelong residents of Massachusetts.3 Defendant answered and moved for judgment on the pleadings on the ground that plaintiff had failed to exhaust his administrative remedies and failed to state a claim. At the hearing on the motion, the judge inquired whether DET was still willing to afford a hearing on plaintiff's claim and gave DET two weeks to respond to the question.4 DET answered with an affidavit stating that it would reschedule a hearing if the plaintiff showed satisfactory reasons for his initial failure to appear. The judge then dismissed the instant action for failure to exhaust administrative remedies, "in view of the Commissioner's willingness to afford what appears to be a meaningful hearing on the merits."
7
We sense in the district court's decision an attempt to fashion an equitable solution to a practical dilemma. The DET procedure strikes us as affording to a pro se plaintiff the benefit of a fast, streamlined, and certainly less expensive procedure for litigating the issue he urges upon the federal courts.5 Moreover, requiring exhaustion of administrative remedies normally "serves the interests of accuracy, efficiency, agency autonomy and judicial economy." Ezratty v. Puerto Rico, 648 F.2d 770, 774 (1st Cir. 1981). While common sense would seem to dictate that plaintiff ought to avail himself of the benefits of the state forum, plaintiff here adamantly insists, as he did in his memorandum below, that he has deliberately chosen to bypass the state's procedure in favor of a federal forum.
8
The court cannot insist on exhaustion of state remedies as a prerequisite to a federal suit, however, where Congress has left that choice to the plaintiff. Reading plaintiff's complaint liberally, especially in light of his pro se status, it appears to assert a claim under 42 U.S.C. Sec. 1983, in that plaintiff alleges that the state defendant adopted a policy which violates his right to equal protection of the laws, and impedes his constitutional right to interstate travel.6 It may also be read as attempting to state a claim for violation by state officials of Title III of the Social Security Act, 42 U.S.C. Sec. 503(a)(1), which requires states receiving federal funds to provide for "methods of administration ... that are ... reasonably calculated to assure full payment of unemployment compensation when due." The courts have consistently recognized a private right of action for equitable relief to enforce this provision.7
9
"It is now firmly settled that exhaustion or resort to state remedies is not a prerequisite to a Sec. 1983 claim." Miller v. Hull, 878 F.2d 523 (1st Cir.) (citing Patsy v. Board of Regents, 457 U.S. 496 (1982)), cert. denied, 493 U.S. 976 (1989). A section 1983 claimant who alleges that he has been injured by an unconstitutional practice need not pursue state administrative remedies "but may proceed directly to federal court" in order to press his claims. Kercado-Melendez v. Aponte-Roque, 829 F.2d 255, 260 (1st Cir. 1987) (while abstention may be warranted where a civil rights plaintiff seeks to use the federal courts to nullify an ongoing coercive state proceeding, where the plaintiff is given the option to initiate a state proceeding, the Patsy rule prevails), cert. denied, 486 U.S. 1044 (1988). Cf. Darby v. Cisneros, 1993 U.S. LEXIS 4246 at (June 21, 1993) (in suit under the APA federal courts do not have the authority to require a plaintiff to exhaust administrative remedies where neither statute nor rules mandate administrative appeals in order to render the agency action final, citing Patsy with approval).
10
And the cases recognizing a private right of action to enforce 42 U.S.C. Sec. 503 leave little doubt that state administrative exhaustion cannot be required where the challenge is to a state rule that allegedly conflicts with the "payment ... when due" provision. See, e.g., Java, 402 U.S. at 121 (where private plaintiffs brought class action challenging state practice of suspending unemployment benefits pending appeal, suit commenced before conclusion of administrative hearings allowed, without discussion); Wheeler v. Vermont, 335 F. Supp. 856, 860 (D. Vt. 1971) (exhaustion of state administrative remedies not required where claimant challenges agency's initial redetermination practice and terminates benefits before a hearing); cf. International Union, UAW v. Brock, 477 U.S. 274 (1986) (citing cases decided under 42 U.S.C. Sec. 503 for holding that Eleventh Amendment does not bar suits challenging application of federal guidelines to benefit claims, even though individual eligibility for benefits may be confined to state processes); Shaw v. Valdez, 819 F.2d 965, 966 n.2 (10th Cir. 1987) (availability of state judicial remedies does not bar private suit challenging state's notice provisions under Sec. 503(a)(3) where deprivation is allegedly caused by established state procedure, rather than random or unauthorized act).
11
Exhaustion is not required in cases challenging system wide errors at the initial benefits determination stage because of the economic aims of the statute. Prompt replacement of wages is vital to effectuate "[b]oth the humane (or redistributive) objectives of unemployment insurance and its macroeconomic objective (dampening the business cycle by keeping up the purchasing power of people laid off in a recession) ... " Jenkins, 691 F.2d at 1229 (Posner, J.); see also Java, 402 U.S. at 131-32 (Congress' intention in enacting Sec. 503(a)(1) was to assure both purposes by making payments available at the earliest stage that is administratively feasible). While individual administrative appeals may effectively correct errors in individual cases, the process may not result in speedy correction of systemic errors at the initial determination stage. Cf. Schoolcraft v. Sullivan, 971 F.2d 81, 87 (8th Cir. 1992) (under statute allowing discretionary waiver of exhaustion requirements, applying similar reasoning to waive requirement).
12
In conclusion we decide here only the exhaustion of remedies issue presented to us. We express no opinion on any other question of justiciability, including standing, ripeness, mootness, or the like. And as our footnotes repeatedly emphasize we express no opinion on the merits of plaintiff's claims, the desirability of the relief sought, nor the ability of these claims to withstand a proper motion for summary judgment or other dismissal on the merits.
13
Vacated and remanded.
1
Under Massachusetts' Employment Security Law, to be eligible for unemployment compensation during any week a claimant must provide evidence to the employment office that he is available for and actively engaged in a systematic and sustained effort to obtain work. M.G.L. c. 151A, Secs. 24, 30
2
The record does not explain DET's rulemaking practices, but we note a suggestion in the case law that agency rules relating to eligibility are frequently incorporated into circulars, rather than the Code of Massachusetts Regulations ("CMR"). See Grand v. Director of the Div. of Employment Sec., 393 Mass. 477, 480-81, 472 N.E.2d 250, 252 (1984) (rejecting claimant's argument that review examiner acted without standards when he held claimant's job search to be inadequate, because agency's circularized notice "constitutes a guideline or standard set forth by the division"). We note, too, rules in the CMR for interstate claims subject to plans approved by the Interstate Conference of Employment Security Agencies. See 430 C.M.R. Secs. 4.02, 4.05; see also M.G.L. c. 151A, Sec. 66. There are insufficient facts in this record to determine the relevancy, if any, of the codified rules
3
Claimants who remained in the State were allowed benefits if they actively sought work "at least three days a week and made at least four job contacts/week," according to a 1984 Supreme Judicial Court opinion. Grand, 393 Mass. at 481, 472 N.E. at 252. The record before us offers no facts as to DET's current eligibility rules for those who remain in the state while seeking work, facts against which any claim of unequal treatment necessarily must be measured. Without a full record we imply no opinion as to the ability of the instant claim to withstand a motion to dismiss on the merits
4
Since the parties have not provided a transcript of the hearing, our understanding of the proceedings below is limited to the judge's abbreviated written orders
5
Plaintiff maintains that an agency factual hearing would be futile since the examiner would have no power to change the DET's admitted policy, only to award benefits. However, state law also provides a subsequent discretionary appeal to the Board of Review, which is expressly empowered to search the record for errors of law as well as fact. M.G.L. c. 151A, Secs. 40, 41. And claimants are further afforded a streamlined method for appeal to the state's district courts where jurisdiction includes any constitutional errors, errors of law or procedure. M.G.L. c. 151A, Sec. 42; M.G.L. c. 30, Sec. 14(7)
6
We emphasize again that the record is too slim to assess the ability of these claims to withstand a proper motion to dismiss on the merits. We have before us no information on basic issues like the actual burden, if any, on interstate travel or commerce and the state's legitimate interest or need for the rule. Moreover the factual basis for plaintiff's unequal treatment claim is not clear, see supra n.3. See generally Hooper v. Bernalillo County Assessor, 472 U.S. 612, 624 (1985); Zobel v. Williams, 457 U.S. 55, 58-65 (1982); Jones v. Helms, 452 U.S. 412, 417-22 (1981); Shapiro v. Thompson, 394 U.S. 618 (1969) (overruled in part on another and by Edelman v. Jordan, 416 U.S. 1000 (1974)); Edwards v. California, 314 U.S. 160 (1941); Crandall v. Nevada, 73 U.S. (6 Wall.) 35 (1868). We observe only that general federal question jurisdiction is sufficiently pleaded under 28 U.S.C. Sec. 1331. See Charles A. Wright et. al., 5 Federal Practice and Procedure Sec. 1209 (2d Ed. Supp. 1993)
7
See California Dep't of Human Resources Dev. v. Java, 402 U.S. 121 (1971); Ohio Bureau of Employment Servs. v. Hodory, 431 U.S. 471 (1977). Though the statute contains no language allowing a private action, to assure state compliance, the result makes "practical sense." Jenkins v. Bowling, 691 F.2d 1225, 1228 (7th Cir. 1982); see also Shaw v. Valdez, 819 F.2d 965 (10th Cir. 1987); Wilkinson v. Abrams, 627 F.2d 650 (3d Cir. 1980); Pennington v. Ward, 1989 U.S. Dist. LEXIS 7651, at (N.D. Ill.) (citing Maine v. Thiboutot, 448 U.S. 1 (1980) for point that Sec. 1983 embraces claims that state defendants violated rights secured by statute); Brewer v. Cantrell, 622 F. Supp. 1320 (W.D. Va. 1985), aff'd without op., 796 F.2d 472 (4th Cir. 1986)
Payment "when due" is interpreted by the federal regulations to mean with "the greatest promptness that is administratively feasible," 20 C.F.R. Sec. 640.3(a). We have not been offered a direct explanation of DET's procedure for handling travel claims, but its brief suggests that the travel rule is an initial administrative "rule of thumb." DET states that despite the "sole purpose" language in the rule and the dual purpose of plaintiff's trip, plaintiff's benefits could have been reinstated at a factual hearing. A review examiner, we are told, could have weighed evidence of the comparative time plaintiff devoted to seeking work versus the time he spent on personal matters to arrive at a result different from that mandated by the rule. We read this as implying that DET initially denies benefits to claimants who travel for a dual purpose as an administrative "rule of thumb" subject to change on appeal in individual cases. Whether this procedure is one sufficiently calculated to result in payment "when due" within the meaning of 42 U.S.C. Sec. 503(a)(1), is a fact specific issue which we cannot meaningfully assess on the rudimentary record before us. See Fusari v. Steinberg, 419 U.S. 379, 387, 389 (1975). Nor can we determine the relevancy, if any, of the federal statute encouraging certain interstate payments and procedures on behalf of unemployed workers who relocate while seeking employment. See 26 U.S.C. Sec. 3304(B)(9); see also M.G.L. c. 151A, Sec. 66.
|
{
"pile_set_name": "FreeLaw"
}
|
A New York school district received the clear to fire a teacher accused of abusing special education students, officials announced Monday.
Long Beach Middle School teacher Lisa Weitzman, 37, allegedly dug a high heel into a student’s foot, threatened to restrain a student’s hands and ankles with zip ties, sprayed Lysol on students and sent them to the bathroom as a time-out, Newsday reported Monday.
Weitzman faced eight allegations, including incidents involving five former special needs students beginning in the 2012-2013 academic year. She denied the allegations, however, according to Newsday.
Long Beach Public Schools (LBPS) placed Weitzman on paid leave beginning in November 2014 and reportedly paid her over $540,000 by the middle of August 2018, according to Newsday.
“The Board of Education will meet on Tuesday, April 2, to immediately take action on Ms. Weitzman’s termination,” the LBPS Board of Education said in a statement Monday. “The safety and well-being of our students is the district’s top priority, and we are grateful that this matter has been concluded for the benefit of our children.”
Weitzman’s disciplinary hearing began in March 2016 and ended in May 2017. (RELATED: Former Yale Soccer Coach Expected To Plead Guilty Over Involvement In College Admissions Scandal)
Weitzman was a tenured teacher, and therefore could only be punished or fired for a “just cause,” New York State Education Department officials told The Daily Caller News Foundation. Tenured teachers are entitled to a hearing, full pay and benefits during the disciplinary process. The state, however, does not get to determine the tenured teacher’s hearing decision.
LBPS and Weitzman’s attorney Debra Wabnik did not immediately respond to TheDCNF’s request for comment.
Follow Neetu on Twitter
Send tips to: neetu@dailycallernewsfoundation.org
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 licensing@dailycallernewsfoundation.org.
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{
"pile_set_name": "OpenWebText2"
}
|
To answer your question - in the past, I created with leather & components that would be called African. As you know, currently I'm into silver & turquoise. One thing I know for sure is that I lean to minimalist designs - both in creating & in what I wear...and I do love that Basic Black. I need a lesson in how you wrapped the ends of the leather. ;)
Love your wire wrapped earrings! My favorites to work with are natural beads, both gemstones and pearls and basic stringing. I am still new to the art of using other stringing materials, such as leather or silk.
So hard to pick a favorite, they are all so wonderful! I LOVE the scarlet necklace, the deep color of the silk is just scrumptious! I would wear the Tucson earrings anyday, love them and all of the pearls and leather are fab!!
Want to be in on the latest?
The Pearl Geek Behind the Blog,...
Hello, I'm Michelle (Shel) and Pearls are my favorite gem! I'm a CPAA Certified Pearl Specialist and I hold a Graduate Pearls Diploma from GIA. I'd love for you to join me here as I share with you my love of pearls and jewelry and possibly some macro shots, a little 'Door Porn' now and then and other fun things!
~ "Peace, Love and Pearls, Baby" ~
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{
"pile_set_name": "Pile-CC"
}
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Android is Winning - speg
http://techcrunch.com/2012/08/14/android-is-winning/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+Techcrunch+%28TechCrunch%29
======
factorialboy
Of course Android wins as a platform. Apple still makes good money for share-
holders. Win-win?
|
{
"pile_set_name": "HackerNews"
}
|
Boko Haram remote control bomb kills two Niger soldiers
NIAMEY (Reuters) - A bomb planted and remotely detonated by Boko Haram militants near the southeastern Niger town of Diffa has killed two soldiers and wounded a third, Niger military sources said on Wednesday.
"We had two soldiers killed on Wednesday in a remote controlled explosion. We took up the chase and killed the two militants responsible for the attack," near a bridge over the Kamadougou river, an officer said.
It is the first time the army has said it was attacked using a remotely detonated bomb since it launched a campaign in conjunction with Chad, Cameroon and Nigeria this year against the Islamist militant group.
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{
"pile_set_name": "Pile-CC"
}
|
using System;
using Org.BouncyCastle.Asn1;
namespace Org.BouncyCastle.Asn1.Misc
{
public class IdeaCbcPar
: Asn1Encodable
{
internal Asn1OctetString iv;
public static IdeaCbcPar GetInstance(
object o)
{
if (o is IdeaCbcPar)
{
return (IdeaCbcPar) o;
}
if (o is Asn1Sequence)
{
return new IdeaCbcPar((Asn1Sequence) o);
}
throw new ArgumentException("unknown object in IDEACBCPar factory");
}
public IdeaCbcPar(
byte[] iv)
{
this.iv = new DerOctetString(iv);
}
private IdeaCbcPar(
Asn1Sequence seq)
{
if (seq.Count == 1)
{
iv = (Asn1OctetString) seq[0];
}
}
public byte[] GetIV()
{
return iv == null ? null : iv.GetOctets();
}
/**
* Produce an object suitable for an Asn1OutputStream.
* <pre>
* IDEA-CBCPar ::= Sequence {
* iv OCTET STRING OPTIONAL -- exactly 8 octets
* }
* </pre>
*/
public override Asn1Object ToAsn1Object()
{
Asn1EncodableVector v = new Asn1EncodableVector();
if (iv != null)
{
v.Add(iv);
}
return new DerSequence(v);
}
}
}
|
{
"pile_set_name": "Github"
}
|
Kenkaku: My favorite system against 4. e3, the Classical, and the Leningrad as black is to meet them with 4...c5. Are there any other proponents of this system on here? It's fairly universal in the Nimzo-Indian, and strong as well.
Benjamin Lau: Against 4. e3, ...c5 is very strong, perhaps the strongest move (I think ...b6 is another good candidate). Black can usually equalize with it. Against the classical 4. Qc2, 4... c5?! is considered a little suspect at top levels these days, but has not been refuted and likely never will be. At lower levels, I think that it might be classified as ...c5 rather than c5?!. The point is that it often leaves black with a very difficult to defend backward d pawn, worst of all, it's on an open file (as in the classical Pirc variation 4. Qc2 c5 5. dxc5). ...c5 is a very good move in the Leningrad. Statistics show that it at least equalizes, but more often brings black a slight advantage with best play from both sides.
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This forum is for this specific opening and nothing else. If you want to discuss chess in general, or
this site, you might try the Kibitzer's Café.
Messages
posted by Chessgames members do not necessarily represent the views of Chessgames.com, its employees, or sponsors.
|
{
"pile_set_name": "Pile-CC"
}
|
#!/bin/sh
if [ -z "$TESTSUITE" ]; then
CMDLINE=/proc/cmdline
ALIASES=/etc/preseed_aliases
else
CMDLINE=user-params.in
ALIASES=user-params.aliases
fi
# sed out multi-word quoted value settings
for item in $(sed -e 's/[^ =]*="[^"]*[ ][^"]*"//g' \
-e "s/[^ =]*='[^']*[ ][^']*'//g" $CMDLINE); do
var="${item%=*}"
# Remove trailing '?' for debconf variables set with '?='
var="${var%\?}"
if [ "$item" = "--" ] || [ "$item" = "---" ]; then
inuser=1
collect=""
elif [ "$inuser" ]; then
# BOOT_IMAGE is added by syslinux
if [ "$var" = "BOOT_IMAGE" ]; then
continue
fi
# init is not generally useful to pass on
if [ "$var" = init ]; then
continue
fi
# suppress installer-specific parameters
if [ "$var" = BOOT_DEBUG ] || [ "$var" = DEBIAN_FRONTEND ] || \
[ "$var" = INSTALL_MEDIA_DEV ] || [ "$var" = lowmem ] || \
[ "$var" = noshell ]; then
continue
fi
# brltty settings shouldn't be passed since
# they are already recorded in /etc/brltty.conf
if [ "$var" = brltty ]; then
continue
fi
# ks is only useful to kickseed in the first stage.
if [ "$var" = ks ]; then
continue
fi
# We don't believe that vga= is needed to display a console
# any more now that we've switched to 640x400 by default,
# and it breaks suspend/resume. People can always type it in
# again at the installed boot loader if need be.
if [ "$var" = vga ]; then
continue
fi
# Sometimes used on the live CD for debugging initramfs-tools.
if [ "$var" = break ]; then
continue
fi
# Skip live-CD-specific crud
if [ "${var%-ubiquity}" != "$var" ] || \
[ "$var" = noninteractive ]; then
continue
fi
# Skip debconf variables
varnoslash="${var##*/*}"
if [ "$varnoslash" = "" ]; then
continue
fi
# Skip module-specific variables
# varnodot="${var##*.*}"
# if [ "$varnodot" = "" ]; then
# continue
# fi
# Skip preseed aliases
if [ -e "$ALIASES" ] && \
grep -q "^$var[[:space:]]" "$ALIASES"; then
continue
fi
if [ -z "$collect" ]; then
collect="$item"
else
collect="$collect $item"
fi
fi
done
if [ -z "$TESTSUITE" ]; then
# Include default parameters
RET=`debconf-get debian-installer/add-kernel-opts || true`
if [ "$RET" ]; then
collect="$collect $RET"
fi
fi
for word in $collect; do
echo "$word"
done
|
{
"pile_set_name": "Github"
}
|
No matter what you think about the Essential Phone , one area that it dominates all other Android OEMs at (aside from Google, of course) is with its software updates. Less than a month after launching a beta for Android 8.1 Oreo, Essential's now pushing the software to all of its users.
This marks the first public Oreo update for the Essential Phone as Essential chose to skip 8.0 after discovering some bugs towards the end of its beta, but even so, Essential's still beating Samsung, LG, and other big players in these regards.
If you didn't jump on the Oreo beta for the Essential Phone, you'll find a lot of new toys to play with, including picture-in-picture, Android's new emoji design, a dark theme for Quick Settings that changes based on your wallpaper, Google's Autofill API for faster password entry, and way more than I have time to list here.
Essential says to "check your phone now" for the 8.1 Oreo update, and if you don't see it yet, you should be able to download and install it over the next couple of days.
If you've got an Essential Phone, what are you looking forward to the most with Android 8.1?
Essential Phone users can now join the Oreo beta with a simple OTA update
|
{
"pile_set_name": "OpenWebText2"
}
|
Category Archives: Journals
The FDA has approved alemtuzumab (Lemtrada) as a treatment for patients with multiple sclerosis. The FDA was careful to cite that use of alemtuzumab was a high-risk option since the drug may cause an increased risk for malignancies, including thyroid cancer, melanoma, lymphoproliferative disorders, neuropathy, which in some instances proved fatal. The medication is also marketed as Campath, which […]
Recent dramatic study highlights the fact that some neuropathies can be caused by viruses. In this case the virus is known as Dinocampus coccinellae paralysis virus (DcPV), which is a single-stranded, positive-sense RNA virus unique to insects. The virus creates a “zombie ladybug” by inducing a neuropathy that spreads to the ladybug’s entire nervous system and paralyzing […]
European scientists at the Pain Evaluation and Treatment Centre of Hôpital Ambroise Paré, evaluated several medications for neuropathic pain in an article partially titled “Pharmacotherapy for neuropathic pain in adults“. While research reports and findings are peer-reviewed for scientific correctness, the director of the center, Dr.Nadine Attal, finds certain biases creep into the picture. For […]
|
{
"pile_set_name": "Pile-CC"
}
|
Available Options
Known for his cryptic, often unreadable (and more often rendered in pure black) aesthetic, the patchwork pattern adorning this Yohji Yamamoto shirt includes youthful images of the designer and founder himself. Emulating an old school comic book print, the sleeves are detailed with squared-off boxes filled with text enclosed in cartoon-like speech bubbles. Set atop a longline poplin profile, keep pairings to a minimal – and opt for the label’s signature hue.
|
{
"pile_set_name": "Pile-CC"
}
|
Blog dedicated to Street Fighter and Marvel vs. Capcom
Menu
Ed officially revealed for Street Fighter 5
Capcom has officially announced the newest Season 2 DLC character for Street Fighter 5, Ed.
Click to see the trailer and additional info.
Here is a more in-depth look on Ed’s V-Skill, V-Trigger, and Critical Art.
V-Skill: Psycho Snatcher
Ed releases Psycho Power from his palm, pulling himself or opponents depending on how long he charges it up.
V-Trigger: Psycho Cannon
With a full V-meter, Ed releases an enormous amount of Psycho Power. He can follow alongside it, allowing him to pepper in shots when it connects with the opponent.
Critical Art: Psycho Barrage
Ed lunges forward with a series of punches. If at any point he connects, he’ll continue weaving around the opponent, slipping in quick combinations of Psycho-punches ending with a powerful uppercut.
Here are some stills of the story and premium costumes that are coming with Ed.
I gotta say, I had my doubts about the look, but his fighting style looks pretty cool. Like a mash up of two of my favorite Street Fighter characters, M.Bison and Balrog. He looks like he will play a lot like Balrog, but will also weaves in his version of Psycho Power and kicks. I am really digging how he appears to play.
Aesthetically, I’m not a huge fan of the one shoulder pad look, but his story and battle costume look pretty cool. Also, I could have done without another blonde in Street Fighter.
At about 0:25 seconds in to the trailer, we get a look at Ed’s V-Reversal. It appears to be like Rashid’s where he switches sides.
Ed’s V-Skill looks interesting, providing two different ways to close gaps.
It looks like Ed’s V-Trigger is a lot like Decapre’s from Ultra Street Fighter IV.
|
{
"pile_set_name": "Pile-CC"
}
|
NAB Statement in Response to AT&T STELAR Blog
WASHINGTON, D.C. -- In response to a blog post by AT&T on the upcoming expiration of the Satellite Television Extension and Localism Act Reauthorization, the following statement can be attributed to NAB Executive Vice President of Communications Dennis Wharton:
“AT&T is being remarkably disingenuous in suggesting that broadcasters are responsible for rising pay TV prices. Pay-TV prices have increased independent of programming costs for decades. Further, as the attached chart from S&P Global notes, cable network fees are the primary driver of high programming costs. AT&T’s own CNN, TBS and TNT Networks are among the cable networks charging the highest fees, despite ratings that are paltry in comparison to eyeballs delivered by local TV stations.
“There is no reason for ‘screens to go dark’ as a result of the STELAR bill, as claimed by AT&T. Instead, AT&T's DirecTV has the opportunity as a result of STELAR to fulfill its decade-long pledge to deliver local TV to tens of thousands of viewers in 12 ‘unserved’ rural markets. We are hopeful that AT&T delivers on that promise and supports the wishes of Congress in delivering local television signals to rural viewers.”
About
NAB
The National Association of Broadcasters is the premier advocacy association for America's broadcasters. NAB advances radio and television interests in legislative, regulatory and public affairs. Through advocacy, education and innovation, NAB enables broadcasters to best serve their communities, strengthen their businesses and seize new opportunities in the digital age. Learn more at www.nab.org.
|
{
"pile_set_name": "Pile-CC"
}
|
A wide range of medical procedures involve placing objects, such as sensors, tubes, catheters, dispensing devices and implants, within the body. Position sensing systems have been developed for tracking such objects. Magnetic position sensing is one of the methods known in the art. In magnetic position sensing, magnetic field generators are typically placed at known positions external to the patient. One or more magnetic field sensors within the distal end of a probe generate electrical signals in response to these magnetic fields, which are processed in order to determine the position coordinates of the distal end of the probe. These methods and systems are described in U.S. Pat. Nos. 5,391,199, 6,690,963, 6,484,118, 6,239,724, 6,618,612 and 6,332,089, in PCT International Publication WO 1996/005768, and in U.S. Patent Application Publications 2002/0065455 A1, 2003/0120150 A1 and 2004/0068178 A1, whose disclosures are all incorporated herein by reference.
U.S. Pat. No. 6,370,411, whose disclosure is incorporated herein by reference, describes a probe having two parts: a catheter of minimal complexity which is inserted into a patient's body, and a connection cable that connects between the proximal end of the catheter and the console. The catheter comprises a microcircuit that carries substantially only information specific to the catheter, which is not in common with other catheters of the same model. The cable comprises an access circuit which receives the information from the catheter and passes it in a suitable form to the console. In some embodiments, the cable operates with all catheters of a specific model or type, and therefore when a catheter is replaced, there is no need to replace the cable. Catheters that are planned for one-time use do not require replacement of the cable, which does not come in contact with patients.
U.S. Patent Application Publication 2006/0074289 A1, whose disclosure is incorporated herein by reference, discusses an endoscopic probe, whose handle has an orientation sensor that generates signals indicative of the orientation of the handle in an external frame of reference. The output of the orientation sensor may be used to sense movement of the handle relative to its initial position and orientation at the beginning of the endoscopic procedure.
|
{
"pile_set_name": "USPTO Backgrounds"
}
|
Hold Up the Sky: And Other Native American Tales from Texas and the Southern Plains
Any used item that originally included an accessory such as an access code, one time use worksheet, cd or dvd, or other one time use accessories may not be guaranteed to be included or valid. By purchasing this item you acknowledge the above statement.
Description
Nearly all that remains of some Indian tribes of Texas and the Southern Plains are their stories. Here twenty-six tales are brought together from fourteen tribes and at least five different cultures. They are stories of humor, guidance, and adventure that have been passed down through the generations.
From the Tejas story that explains how the universe began, to the Lipan Apache tale in which a small lizard smartly outwits a hungry coyote, these stories are sure to delight young readers. Additional information about each tribe is included in the "About the Storytellers" section.
Once again Jane Louise Curry has skillfully retold traditional tales of Native Americans. Hold Up the Sky is in keeping with the style of her previous, highly acclaimed collections of Native American stories, Back in the Beforetime, The Wonderful Sky Boat, and Turtle Island. This, too, is a collection to be treasured.
|
{
"pile_set_name": "Pile-CC"
}
|
Pages
Tuesday, 1 May 2012
Good Things
As a deeply rooted fan of Eve Lom's Rescue Mask, having never discovered anything even close, I was pleasantly surprise to stumble across a close contender for my 'favourite face mask ever' category. And, at £5.99 rather than £35, it is hard to argue.
The product in question is Good Things Five Minute Facial Mask, which contains avocado and goji berry extract, and is amazing for a cheaper product. It is designed for younger skins, prone to blemishes, and gently soothes and reduces redness and inflammation without drying the skin at all. In fact, it softened my skin, while soothing irritated areas and getting rid of pesky spots. It also smells really good, softly fruity (not in the tutti frutti sense, don't panic), and doesn't set hard, which is a selling point for me.
I'm always a bit cautious of anything designed to get rid of spots, as I think they are overly harsh and drying, and for me, can make the problem much worse. Good Things is based on the concept of gentle skincare, utilising natural ingredients to address skin problems. It's creator, Alice Hart-Davis, is a beauty editor, and has a refreshing approach to skincare. Advocating a consistent cleansing regime and the use of products that don't strip the skin, I was sold on her range instantly.
The price bracket is the best part, nothing costs over £8.99, with the majority of the range under £5. You can't really go wrong!
♥ Good Things Five Minute Facial Mask, available from Boots and Superdrug for £5.99
|
{
"pile_set_name": "Pile-CC"
}
|
List of mayors of Salt Lake City
This is a list of mayors of Salt Lake City, Utah, USA. Salt Lake City was incorporated on January 6, 1851. The mayor of Salt Lake City is a non-partisan position.
References
Harold Schindler, (November 10, 1991) "Mayoral History Awaits Corradini Chapter: Colorful Mayoral History Awaits Unprecedented Corradini Chapter". The Salt Lake Tribune, p. A1.
Salt Lake City
*
Category:1851 establishments in Utah Territory
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Main navigation
Adam Rovner
Adam Rovner is Associate Professor of English and Jewish Literature at the University of Denver. His articles, essays, translations and interviews have appeared in numerous scholarly journals and general interest publications. Rovner’s short documentary on Jewish territorialism, No Land Without Heaven, has been screened at exhibitions in New York, Paris, and Tel Aviv.
|
{
"pile_set_name": "Pile-CC"
}
|
'use strict'
import React from 'react'
import Sortable from 'sortablejs'
class Favorite extends React.PureComponent {
constructor() {
super()
this.state = {
activeKey: null
}
this._updateSortableKey()
}
_updateSortableKey() {
this.sortableKey = `sortable-${Math.round(Math.random() * 10000)}`
}
_bindSortable() {
const {reorderFavorites} = this.props
this.sortable = Sortable.create(this.refs.sortable, {
animation: 100,
onStart: evt => {
this.nextSibling = evt.item.nextElementSibling
},
onAdd: () => {
this._updateSortableKey()
},
onUpdate: evt => {
this._updateSortableKey()
reorderFavorites({from: evt.oldIndex, to: evt.newIndex})
}
})
}
componentDidMount() {
this._bindSortable()
}
componentDidUpdate() {
this._bindSortable()
}
onClick(index, evt) {
evt.preventDefault()
this.selectIndex(index)
}
onDoubleClick(index, evt) {
evt.preventDefault()
this.selectIndex(index, true)
}
selectIndex(index, connect) {
this.select(index === -1 ? null : this.props.favorites.get(index), connect)
}
select(favorite, connect) {
const activeKey = favorite ? favorite.get('key') : null
this.setState({activeKey})
if (connect) {
this.props.onRequireConnecting(activeKey)
} else {
this.props.onSelect(activeKey)
}
}
render() {
return (<div style={{flex: 1, display: 'flex', flexDirection: 'column', overflowY: 'hidden'}}>
<nav className="nav-group">
<h5 className="nav-group-title"/>
<a
className={'nav-group-item' + (this.state.activeKey ? '' : ' active')}
onClick={this.onClick.bind(this, -1)}
onDoubleClick={this.onDoubleClick.bind(this, -1)}
>
<span className="icon icon-flash"/>
QUICK CONNECT
</a>
<h5 className="nav-group-title">FAVORITES</h5>
<div ref="sortable" key={this.sortableKey}>{
this.props.favorites.map((favorite, index) => {
return (<a
key={favorite.get('key')}
className={'nav-group-item' + (favorite.get('key') === this.state.activeKey ? ' active' : '')}
onClick={this.onClick.bind(this, index)}
onDoubleClick={this.onDoubleClick.bind(this, index)}
>
<span className="icon icon-home"/>
<span>{favorite.get('name')}</span>
</a>)
})
}</div>
</nav>
<footer className="toolbar toolbar-footer">
<button
onClick={() => {
this.props.createFavorite()
// TODO: auto select
// this.select(favorite);
}}
>+</button>
<button
onClick={
() => {
const key = this.state.activeKey
if (!key) {
return
}
showModal({
title: 'Delete the bookmark?',
button: 'Delete',
content: 'Are you sure you want to delete the selected bookmark? This action cannot be undone.'
}).then(() => {
const index = this.props.favorites.findIndex(favorite => key === favorite.get('key'))
this.props.removeFavorite(key)
this.selectIndex(index - 1)
})
}
}
>-</button>
</footer>
</div>)
}
componentWillUnmount() {
this.sortable.destroy()
}
}
export default Favorite
|
{
"pile_set_name": "Github"
}
|
Crash Bandicoot N.Sane Trilogy was a massive hit and so Activision aren’t stopping there. The publisher have revealed they have more remaster releases planned for later this year.
An oldie but a goodie, check out our list of the best old games on PC.
In Activision’s annual investors report they explain their plans for upcoming releases in late 2018, such as World of Warcraft: Battle for Azeroth and the latest Call of Duty game. However, the publishers also state they have plans for “releases of remastered versions of titles from our library of IP.”
This is the first time the publishers have solidly confirmed they plan on releasing remastered titles this year.
Activision have quite the back catalogue of games to choose from, but fans – myself included – are hoping for a remastered Spyro the Dragon Trilogy. Kotaku have previously said this is on the way, according to multiple sources, and if their leak is valid then the Spyro remaster will be announced in March, with the trilogy releasing in Q3 2018, fitting with the investor’s report.
|
{
"pile_set_name": "OpenWebText2"
}
|
Q:
IIIF: Get list of ids for all pages in book given one page in book
I'm trying to download all page images from a book, and those images are hosted on a IIIF server. Here's the page:
https://iiif.bodleian.ox.ac.uk/iiif/image/c444f7e2-ca30-48ae-87b5-54f93d6ed046/full/full/0/default.jpg
(I got the id for that page from a GUI page)
I thought I could get the metadata for that page with the /info.json route:
https://iiif.bodleian.ox.ac.uk/iiif/image/c444f7e2-ca30-48ae-87b5-54f93d6ed046/info.json
But this doesn't include all pages in the book. Does anyone know how One can obtain all pages from a book using the IIIF spec/API? Any help would be appreciated!
A:
Hmm, googling for https://iiif.bodleian.ox.ac.uk/iiif/mirador/c444f7e2-ca30-48ae-87b5-54f93d6ed046 today I saw two results: this post and https://iiif.bodleian.ox.ac.uk/iiif/mirador/c444f7e2-ca30-48ae-87b5-54f93d6ed046.
The latter contains all pages in the book, so I suppose the lesson is to search to see if the given IIIF server has a mirador viewer set up, and if so, to ping that...
Also, the manifest route contains all links expressed cleanly: https://iiif.bodleian.ox.ac.uk/iiif/manifest/db2fade4-61ee-4a11-a894-19361c551eed.json
|
{
"pile_set_name": "StackExchange"
}
|
Q:
How to detach a process from terminal in unix?
When I start a process in background in a terminal and some how if terminal gets closed then we can not interact that process any more. I am not sure but I think process also get killed. Can any one please tell me how can I detach that process from my terminal. So even if I close terminal then I can interact with same process in new terminal ?
I am new to unix so your extra information will help me.
A:
The command you're looking for is disown.
disown <processid>
This is as close as you can get to a nohup. It detaches the process from the current login and allows it to continue running. Thanks David Korn!
http://www2.research.att.com/~gsf/man/man1/disown.html
and I just found reptyr which lets you reparent a disowned process.
https://github.com/nelhage/reptyr
It's already in the packages for ubuntu.
BUT if you haven't started the process yet and you're planning on doing this in the future then the way to go is screen and tmux. I prefer screen.
A:
You might also consider the screen command. It has the "restore my session" functionality. Admittedly I have never used it, and forgot about it.
Starting the process as a daemon, or with nohup might not do everything you want, in terms of re-capturing stdout/stdin.
There's a bunch of examples on the web. On google try, "unix screen command" and "unix screen tutorial":
http://www.thegeekstuff.com/2010/07/screen-command-examples/
GNU Screen: an introduction and beginner's tutorial
A:
First google result for "UNIX demonizing a process":
See the daemon(3) manpage for a short overview. The main thing of daemonizing
is going into the background without quiting or holding anything up. A list of
things a process can do to achieve this:
fork()
setsid()
close/redirect stdin/stdout/stderr to /dev/null, and/or ignore SIGHUP/SIGPIPE.
chdir() to /.
If started as a root process, you also want to do the things you need to be root
for first, and then drop privileges. That is, change effective user to the "daemon"
user or "nobody" with setuid()/setgid(). If you can't drop all privileges and need
root access sometimes, use seteuid() to temporary drop it when not needed.
If you're forking a daemon then also setup child handlers and, if calling exec,
set the close on exec flags on all file descriptors your children won't need.
And here's a HOWTO on creating Unix daemons: http://www.netzmafia.de/skripten/unix/linux-daemon-howto.html
|
{
"pile_set_name": "StackExchange"
}
|
#!/bin/bash
# $Id: //depot/cloud/rpms/nflx-webadmin-gcviz/root/apps/apache/htdocs/AdminGCViz/index#2 $
# $DateTime: 2013/05/15 18:34:23 $
# $Author: mooreb $
# $Change: 1838706 $
cd `dirname $0`
prog=`basename $0`
NFENV=/etc/profile.d/netflix_environment.sh
if [ -f ${NFENV} ]; then
. ${NFENV}
NETFLIX_VMS_EVENTS=checked
else
NFENV=""
fi
cat <<EndOfHeader
Content-Type: text/html
<html>
<head>
<title>AdminGCViz</title>
</head>
<body>
EndOfHeader
cat <<EndOfGenerate
Generate a GC visualization report now
<form action="generate" method="POST">
<input type="checkbox" name="jmap_histo_live" value="on" checked> jmap -histo:live<br>
<input type="checkbox" name="vms_refresh_events" value="on" ${NETFLIX_VMS_EVENTS}> parse catalina logs looking for netflix VMS events<br>
<input type="submit" value="Generate GCViz report">
</form>
EndOfGenerate
echo Look at previous reports
echo "<ul>"
for f in `find /mnt/logs/gc-reports -type f -name "*.png" | LANG=C sort -rn`;
do
u=`echo ${f} | sed 's|/mnt/logs/gc-reports/||'`
echo "<li> <a href=\"/AdminGCVizImages/${u}\">${f}</a>"
done
echo "</ul>"
cat <<EndOfFooter
</body>
</html>
EndOfFooter
|
{
"pile_set_name": "Github"
}
|
Lymphocytes prime activation is required for nicotine-induced calcium waves.
Lymphocytes are reported to express nicotinic acetylcholine receptors (nAChR). However, no data are available on the expression of these nAChR on activated lymphocyte relatively to resting lymphocytes. In this study, we examined nAChR subunits expression in PHA-stimulated versus un-stimulated lymphocytes, and four leukemic cell lines. Cell stimulation with nicotine triggered calcium responses only in some experiments conducted with PHA-stimulated lymphocytes. Likewise, only the Jurkat and HL-60 cell lines displayed calcium waves upon nicotine stimulation, whereas the Raji and CCRF-CEM did not. All responding cells displayed an active form of the nicotinic a-7 nAChR. Indeed, use of 2 different sets of primers for the corresponding mRNA showed that expression of the full-length a-7 subunit mRNA was only present in PHA-stimulated lymphocytes for which calcium waves had been evidenced. Microscopy analysis of lymphocytes structure showed a direct relationship between their size, their a-7 nAChR expression, and calcium release upon nicotine stimulation. Then, this relationship suggested that lymphocytes need a prime activation to express the a-7 nAChR, and therefore to release calcium in response to nicotine.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
1. Field
Example embodiments of inventive concepts described herein relate to user devices supporting defragmentation and/or zone-based defragmentation methods thereof.
2. Description of Conventional Art
Unlike a hard disk, a flash memory may not support an overwrite operation. For this reason, an erase operation may be performed before a write operation. The flash memory may perform an erase operation by a block unit, and a time taken to perform the erase operation may be long. The flash memory characteristics may make it difficult to apply a file system for hard disk to the flash memory without modification. To solve such problems, a flash translation layer FTL may be used as middleware between the file system for hard disk and the flash memory. The FTL may enable the flash memory to be freely read and written like a conventional hard disk.
Defragmentation or defrag may mean a work of reducing the amount of fragmented files of a file system on a hard disk. The defragmentation may be made to speed up by reorganizing, concatenating and/or compacting fragmented files within a hard disk physically. Periodic defragmentation may help maintain the optimal performance of memory systems and constituent data media by reducing the file system overhead and data search time caused by excessive fragmentation.
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{
"pile_set_name": "USPTO Backgrounds"
}
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I use my "influence" on reddit and a few other sites to get traffic to a cute college students youtube vids. As a thank you she masturbates for me on skype and mails me her used panties.
164 shares
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{
"pile_set_name": "OpenWebText2"
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In this article
Owned Entity Types
In this article
Note
This feature is new in EF Core 2.0.
EF Core allows you to model entity types that can only ever appear on navigation properties of other entity types. These are called owned entity types. The entity containing an owned entity type is its owner.
Explicit configuration
Owned entity types are never included by EF Core in the model by convention. You can use the OwnsOne method in OnModelCreating or annotate the type with OwnedAttribute (new in EF Core 2.1) to configure the type as an owned type.
In this example, StreetAddress is a type with no identity property. It is used as a property of the Order type to specify the shipping address for a particular order.
We can use the OwnedAttribute to treat it as an owned entity when referenced from another entity type:
Implicit keys
Owned types configured with OwnsOne or discovered through a reference navigation always have a one-to-one relationship with the owner, therefore they don't need their own key values as the foreign key values are unique. In the previous example, the StreetAddress type does not need to define a key property.
In order to understand how EF Core tracks these objects, it is useful to think that a primary key is created as a shadow property for the owned type. The value of the key of an instance of the owned type will be the same as the value of the key of the owner instance.
Collections of owned types
Note
This feature is new in EF Core 2.2.
To configure a collection of owned types OwnsMany should be used in OnModelCreating. However the primary key will not be configured by convention, so it needs to be specified explicitly. It is common to use a complex key for these type of entities incorporating the foreign key to the owner and an additional unique property that can also be in shadow state:
Mapping owned types with table splitting
When using relational databases, by convention reference owned types are mapped to the same table as the owner. This requires splitting the table in two: some columns will be used to store the data of the owner, and some columns will be used to store data of the owned entity. This is a common feature known as table splitting.
Tip
Owned types stored with table splitting can be used similarly to how complex types are used in EF6.
By convention, EF Core will name the database columns for the properties of the owned entity type following the pattern Navigation_OwnedEntityProperty. Therefore the StreetAddress properties will appear in the 'Orders' table with the names 'ShippingAddress_Street' and 'ShippingAddress_City'.
Sharing the same .NET type among multiple owned types
An owned entity type can be of the same .NET type as another owned entity type, therefore the .NET type may not be enough to identify an owned type.
In those cases, the property pointing from the owner to the owned entity becomes the defining navigation of the owned entity type. From the perspective of EF Core, the defining navigation is part of the type's identity alongside the .NET type.
For example, in the following class ShippingAddress and BillingAddress are both of the same .NET type, StreetAddress:
In order to understand how EF Core will distinguish tracked instances of these objects, it may be useful to think that the defining navigation has become part of the key of the instance alongside the value of the key of the owner and the .NET type of the owned type.
Nested owned types
In this example OrderDetails owns BillingAddress and ShippingAddress, which are both StreetAddress types. Then OrderDetails is owned by the DetailedOrder type.
In addition to nested owned types, an owned type can reference a regular entity, it can be either the owner or a different entity as long as the owned entity is on the dependent side. This capability sets owned entity types apart from complex types in EF6.
It is also possible to achieve the same thing using OwnedAttribute on both OrderDetails and StreetAdress.
Storing owned types in separate tables
Also unlike EF6 complex types, owned types can be stored in a separate table from the owner. In order to override the convention that maps an owned type to the same table as the owner, you can simply call ToTable and provide a different table name. The following example will map OrderDetails and its two addresses to a separate table from DetailedOrder:
Querying owned types
When querying the owner the owned types will be included by default. It is not necessary to use the Include method, even if the owned types are stored in a separate table. Based on the model described before, the following query will get Order, OrderDetails and the two owned StreetAddresses from the database:
Limitations
Some of these limitations are fundamental to how owned entity types work, but some others are restrictions that we may be able to remove in future releases:
By-design restrictions
You cannot create a DbSet<T> for an owned type
You cannot call Entity<T>() with an owned type on ModelBuilder
Current shortcomings
Inheritance hierarchies that include owned entity types are not supported
Reference navigations to owned entity types cannot be null unless they are explicitly mapped to a separate table from the owner
Instances of owned entity types cannot be shared by multiple owners (this is a well-known scenario for value objects that cannot be implemented using owned entity types)
Shortcomings in previous versions
In EF Core 2.0, navigations to owned entity types cannot be declared in derived entity types unless the owned entities are explicitly mapped to a separate table from the owner hierarchy. This limitation has been removed in EF Core 2.1
In EF Core 2.0 and 2.1 only reference navigations to owned types were supported. This limitation has been removed in EF Core 2.2
Feedback
We'd love to hear your thoughts. Choose the type you'd like to provide:
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GitHub is having one hell of a week, with four outages in five weekdays. The social coding site is currently being hit by a Distributed Denial of Service (DDoS) attack, the second one in two days. The site’s engineers are working on fixing the issue, which thankfully has not brought the whole site to its knees (unlike yesterday), but has only resulted in a “partial service outage.”
Here’s what we know so far (we’ll update this list as we learn more courtesy of status.github.com):
12:15 PM PST: We are investigating issues with pages serving.
12:41 PM PST: Pages is currently being hit with a DoS attack. We’re working to mitigate the attack.
01:22 PM PST: We’re still working to mitigate the attack on Pages.
02:11 PM PST: GitHub Pages service has returned to normal. All systems go!
02:16 PM PST: A small percentage of git repos will be unvailable while we recover a fileserver pair. Updates as we have them.
02:25 PM PST: All git repositories are now accessible.
On Tuesday, GitHub experienced a 26-minute partial service outage (a network issue at one of its upstream providers), and on Wednesday the service experienced a 24-minute partial service outage (errors with its search service). On Thursday (yesterday), the site experienced a major service outage that lasted 99 minutes and that GitHub attributed to a DDoS attack. On Friday (today), we’re back to a partial service outage, but this one is again due to a DDoS attack.
For reference, here’s what happened yesterday:
01:05 PM PST: We are investigating issues with GitHub.com
01:17 PM PST: We’re experiencing some connectivity issues at the moment. GitHub.com is currently unavailable while we resolve this.
01:33 PM PST: We are experiencing issues due to a DDOS attack, working hard to restore service
01:41 PM PST: Performance is still substantially sub par as we fend of the connection flood. We’re on it.
01:41 PM PST: We’ve temporarily disabled service on port 80 while we investigate the source of a connection flood. HTTPS, GIT, and SSH service are unaffected.
02:44 PM PST: Performance is stabilizing. We’re investigating additional mitigation strategies to harden ourselves against future attacks, similar or otherwise.
We’ll keep you posted to see if this devolves to a major service outage or if GitHub manages to fix the problem. As you can see in the log above, GitHub yesterday started looking into mitigation strategies to fight DDoS attacks, though it’s not yet clear if the company has implemented these yet.
We have contacted GitHub for more information. We will update this article if and when we hear back.
Update at 5:25PM EST: GitHub is back for all users. Today’s outage lasted a total of 130 minutes, the longest yet this week.
Image credit: Adam Fast
Read next: TNW's Daily Dose: Chromebooks, invites, and Microsoft hearts China
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{
"pile_set_name": "OpenWebText2"
}
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---
abstract: |
We prove the analogue for continuous space-time of the quenched LDP derived in Birkner, Greven and den Hollander [@BiGrdHo10] for discrete space-time. In particular, we consider a random environment given by Brownian increments, cut into pieces according to an independent continuous-time renewal process. We look at the empirical process obtained by recording both the length of and the increments in the successive pieces. For the case where the renewal time distribution has a Lebesgue density with a polynomial tail, we derive the quenched LDP for the empirical process, i.e., the LDP conditional on a typical environment. The rate function is a sum of two specific relative entropies, one for the pieces and one for the concatenation of the pieces. We also obtain a quenched LDP when the tail decays faster than algebraic. The proof uses coarse-graining and truncation arguments, involving various approximations of specific relative entropies that are not quite standard.
In a companion paper we show how the quenched LDP and the techniques developed in the present paper can be applied to obtain a variational characterisation of the free energy and the phase transition line for the Brownian copolymer near a selective interface.
*MSC2010:* 60F10, 60G10, 60J65, 60K37.\
*Keywords:* Brownian environment, renewal process, annealed vs. quenched, empirical process, large deviation principle, specific relative entropy.\
*Acknowledgment:* The research in this paper is supported by ERC Advanced Grant 267356 VARIS of FdH. MB is grateful for hospitality at the Mathematical Institute in Leiden during a sabbatical leave from September 2012 until February 2013, supported by ERC.
author:
- |
M. Birkner\
F. den Hollander
date: 9th December 2013
title: A quenched large deviation principle in a continuous scenario
---
Introduction and main result {#intro}
============================
When we cut an i.i.d. sequence of letters into words according to an independent integer-valued renewal process, we obtain an i.i.d. sequence of words. In the *annealed* LDP for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. Birkner, Greven and den Hollander [@BiGrdHo10] considered the *quenched* LDP, i.e., conditional on a typical letter sequence. The rate function of the quenched LDP turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal time distribution.
The goal of the present paper is to derive the analogue of the quenched LDP for the case where the i.i.d. sequence of letters is replaced by the process of Brownian increments, and the renewal process has a length distribution with a Lebesgue density that has a polynomial tail.
In Section \[setting\] we define the continuous space-time setting, in Section \[LDPs\] we state both the annealed and the quenched LDP, while in Section \[disc\] we discuss these LDPs and indicate some further extensions. In Section \[proof\] we prove the quenched LDP subject to three propositions. In Sections \[props\]–\[removeass\] we give the proof of these propositions. In Section \[proofalpha1infty\] we prove the extensions. Appendix \[metrics\] recalls a few basic facts about metrics on path space, while Appendices \[entropy\]–\[contrelentr\] prove a few basic facts about specific relative entropy that are needed in the proof and that are not quite standard.
Continuous space-time {#setting}
---------------------
Let $X=(X_t)_{t \geq 0}$ be the standard one-dimensional Brownian motion starting from $X_0=0$. Let ${\mathscr{W}}$ denote its law on path space: the Wiener measure on $C([0,\infty))$, equipped with the $\sigma$-algebra generated by the coordinate projections. Let $T=(T_i)_{i \in {\mathbb{N}}_0}$ ($T_0=0$) be an independent continuous-time renewal process, with interarrival times $\tau_i=T_i-T_{i-1}$, $i\in{\mathbb{N}}$, whose common law $\rho=\mathscr{L}(\tau_1)$ is absolutely continuous with respect to the Lebesgue measure on $(0,\infty)$, with density $\bar{\rho}$ satisfying $$\label{ass:rhodensdecay}
\lim_{x\to\infty} \frac{\log\bar{\rho}(x)}{\log x} = - \alpha,
\qquad \alpha \in (1,\infty).$$ In addition, assume that $$\label{ass:rhobar.reg0}
\begin{minipage}{0.85\textwidth}
$\mathrm{supp}(\rho) = [s_*,\infty)$ with $0 \leq s_* < \infty$, and $\bar{\rho}$ is
continuous and strictly positive on $(s_*,\infty)$, and varies regularly near $s_*$.
\end{minipage}$$
(0,0) rectangle (361.35,361.35);
( 2.40, 2.40) rectangle (358.95,358.95);
( 15.61,162.33) – (345.74,162.33);
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(106.39,198.78) – (106.72,202.74) – (107.05,205.89) – (107.38,200.59) – (107.71,200.39) – (108.04,197.96) – (108.37,203.79) – (108.70,209.82) – (109.03,216.18) – (109.36,218.39) – (109.70,219.87) – (110.03,212.15) – (110.36,214.22) – (110.69,207.46) – (111.02,208.96) – (111.35,207.20) – (111.68,205.55) – (112.01,201.79) – (112.34,197.57) – (112.67,197.28) – (113.00,199.80) – (113.33,202.19) – (113.66,199.62) – (113.99,201.48) – (114.32,198.95) – (114.65,198.88) – (114.98,195.00) – (115.31,198.39) – (115.64,194.21) – (115.97,196.46) – (116.30,195.87) – (116.63,198.92) – (116.96,205.32) – (117.29,202.39) – (117.62,205.91) – (117.95,201.42) – (118.28,201.44) – (118.61,197.01) – (118.94,192.33) – (119.27,196.29) – (119.60,197.02) – (119.93,200.61) – (120.26,199.05) – (120.59,195.86) – (120.92,197.09) – (121.25,192.65) – (121.58,193.14) – (121.91,199.03) – (122.24,201.72) – (122.57,196.94) – (122.90,199.67) – (123.23,196.29) – (123.56,196.94) – (123.89,204.30) – (124.22,213.13) – (124.55,213.25) – (124.88,212.46) – (125.21,216.19) – (125.54,221.37) – (125.87,213.17) – (126.20,213.06) – (126.53,213.21) – (126.86,214.20) – (127.19,222.84) – (127.52,216.51) – (127.85,223.10) – (128.18,221.10) – (128.51,221.54) – (128.84,220.01) – (129.17,221.11) – (129.50,223.96) – (129.83,228.91) – (130.16,227.96) – (130.49,234.23) – (130.82,234.61) – (131.15,231.28) – (131.48,228.32) – (131.81,228.57) – (132.14,229.23) – (132.47,239.02) – (132.80,239.74) – (133.13,241.87) – (133.47,234.42) – (133.80,232.14) – (134.13,237.75) – (134.46,237.07) – (134.79,235.75) – (135.12,233.85) – (135.45,231.74) – (135.78,233.66) – (136.11,225.93) – (136.44,226.73) – (136.77,227.52) – (137.10,230.76) – (137.43,227.00) – (137.76,234.79) – (138.09,232.12) – (138.42,234.66) – (138.75,235.80) – (139.08,234.71) – (139.41,232.44) – (139.74,234.44) – (140.07,234.81) – (140.40,224.35) – (140.73,232.53) – (141.06,238.96) – (141.39,237.36) – (141.72,230.46) – (142.05,230.84) – (142.38,231.28) – (142.71,234.70) – (143.04,234.30) – (143.37,232.08) – (143.70,232.70) – (144.03,235.58) – (144.36,235.07) – (144.69,233.43) – (145.02,228.12) – (145.35,228.37) – (145.68,232.98) – (146.01,230.77) – (146.34,233.58) – (146.67,236.81) – (147.00,241.11) – (147.33,234.45) – (147.66,235.57) – (147.99,236.56) – (148.32,235.01) – (148.65,232.37) – (148.98,237.98) – (149.31,241.02) – (149.64,246.15) – (149.97,242.29) – (150.30,229.52) – (150.63,229.61) – (150.96,228.39) – (151.29,232.02) – (151.62,227.27) – (151.95,221.02) – (152.28,213.36) – (152.61,211.49) – (152.94,210.32) – (153.27,210.94) – (153.60,206.97) – (153.93,209.02) – (154.26,210.39) – (154.59,207.60) – (154.92,207.49) – (155.25,207.10) – (155.58,208.32) – (155.91,205.47) – (156.24,208.75) – (156.57,213.58) – (156.90,212.61) – (157.24,212.47) – (157.57,217.98) – (157.90,214.83) – (158.23,212.02) – (158.56,211.23) – (158.89,217.78) – (159.22,211.94) – (159.55,214.47) – (159.88,216.76) – (160.21,220.74) – (160.54,215.35) – (160.87,217.00) – (161.20,220.68) – (161.53,226.07) – (161.86,227.25) – (162.19,224.58) – (162.52,226.61) – (162.85,225.82) – (163.18,227.91) – (163.51,224.23) – (163.84,221.02) – (164.17,223.97) – (164.50,224.40) – (164.83,228.81) – (165.16,231.43) – (165.49,237.76) – (165.82,235.92) – (166.15,230.46) – (166.48,229.64) – (166.81,235.66) – (167.14,234.52) – (167.47,234.84) – (167.80,236.71) – (168.13,237.52) – (168.46,236.74) – (168.79,231.86) – (169.12,237.77) – (169.45,235.00) – (169.78,233.38) – (170.11,230.54) – (170.44,227.22) – (170.77,229.26) – (171.10,229.10) – (171.43,229.16) – (171.76,236.46) – (172.09,233.20) – (172.42,236.14) – (172.75,239.23) – (173.08,236.66) – (173.41,244.17) – (173.74,248.64) – (174.07,239.55) – (174.40,237.44) – (174.73,238.78) – (175.06,236.14) – (175.39,240.74) – (175.72,240.68) – (176.05,241.19) – (176.38,237.89) – (176.71,242.98) – (177.04,247.72) – (177.37,246.98) – (177.70,247.28) – (178.03,248.02) – (178.36,248.41) – (178.69,245.31) – (179.02,252.07) – (179.35,253.23) – (179.68,252.12) – (180.01,248.55) – (180.34,244.15) – (180.67,246.85) – (181.01,251.95) – (181.34,254.35) – (181.67,253.16) – (182.00,245.95) – (182.33,239.46) – (182.66,241.70) – (182.99,236.71) – (183.32,234.62) – (183.65,236.87) – (183.98,230.99) – (184.31,232.04) – (184.64,223.22) – (184.97,223.69) – (185.30,219.03) – (185.63,214.00) – (185.96,212.67) – (186.29,213.98) – (186.62,214.49) – (186.95,221.15) – (187.28,222.66) – (187.61,225.91) – (187.94,227.62) – (188.27,226.36) – (188.60,224.53) – (188.93,224.52) – (189.26,223.79) – (189.59,224.46) – (189.92,225.43) – (190.25,225.91) – (190.58,226.08) – (190.91,225.43) – (191.24,218.70) – (191.57,211.95) – (191.90,213.77) – (192.23,218.90) – (192.56,223.07) – (192.89,220.63) – (193.22,222.21) – (193.55,219.58) – (193.88,222.59) – (194.21,221.76) – (194.54,221.55) – (194.87,226.57) – (195.20,220.80) – (195.53,222.11) – (195.86,221.04) – (196.19,219.93) – (196.52,215.40) – (196.85,207.54) – (197.18,209.61) – (197.51,206.69) – (197.84,205.55) – (198.17,206.15) – (198.50,202.37) – (198.83,207.16) – (199.16,204.30) – (199.49,206.52) – (199.82,199.35) – (200.15,194.79) – (200.48,200.97) – (200.81,202.15) – (201.14,201.79) – (201.47,203.54) – (201.80,199.50) – (202.13,205.55) – (202.46,206.47) – (202.79,205.18) – (203.12,202.96) – (203.45,206.30) – (203.78,207.63) – (204.11,202.62) – (204.44,200.70) – (204.78,196.79) – (205.11,204.41) – (205.44,201.80) – (205.77,200.74) – (206.10,205.37) – (206.43,204.00) – (206.76,197.62) – (207.09,194.13) – (207.42,194.06) – (207.75,198.72) – (208.08,196.40) – (208.41,193.98) – (208.74,192.00) – (209.07,190.51) – (209.40,192.45) – (209.73,201.69) – (210.06,201.17) – (210.39,205.64) – (210.72,208.75) – (211.05,207.28) – (211.38,206.20) – (211.71,205.38) – (212.04,201.88) – (212.37,200.89) – (212.70,201.29) – (213.03,194.23) – (213.36,195.33) – (213.69,197.48) – (214.02,194.52) – (214.35,199.85) – (214.68,198.18) – (215.01,195.67) – (215.34,201.15) – (215.67,199.51) – (216.00,207.69) – (216.33,205.32) – (216.66,210.90) – (216.99,213.68) – (217.32,212.47) – (217.65,211.42) – (217.98,209.79) – (218.31,211.95) – (218.64,216.32) – (218.97,218.96) – (219.30,216.24) – (219.63,222.50) – (219.96,217.14) – (220.29,211.39) – (220.62,214.94) – (220.95,216.84) – (221.28,211.99) – (221.61,212.00) – (221.94,213.57) – (222.27,222.54) – (222.60,224.81) – (222.93,228.58) – (223.26,224.40) – (223.59,230.01) – (223.92,232.62) – (224.25,238.72) – (224.58,245.31) – (224.91,245.21) – (225.24,249.23) – (225.57,250.78) – (225.90,246.38) – (226.23,249.82) – (226.56,253.98) – (226.89,251.08) – (227.22,253.96) – (227.55,252.86) – (227.88,247.13) – (228.21,245.41) – (228.55,243.95) – (228.88,248.27) – (229.21,244.80) – (229.54,248.41) – (229.87,250.08) – (230.20,246.72) – (230.53,245.55) – (230.86,248.37) – (231.19,248.10) – (231.52,247.01) – (231.85,248.99) – (232.18,251.64) – (232.51,250.12) – (232.84,250.16) – (233.17,244.70) – (233.50,244.72) – (233.83,236.17) – (234.16,233.66) – (234.49,234.23) – (234.82,239.14) – (235.15,250.48) – (235.48,245.34) – (235.81,240.36) – (236.14,240.69) – (236.47,243.92) – (236.80,246.40) – (237.13,243.28) – (237.46,244.01) – (237.79,251.20) – (238.12,254.25) – (238.45,255.04) – (238.78,255.00) – (239.11,259.04) – (239.44,259.14) – (239.77,266.13) – (240.10,264.21) – (240.43,269.21) – (240.76,266.62) – (241.09,263.77) – (241.42,266.82) – (241.75,264.14) – (242.08,264.15) – (242.41,254.93) – (242.74,251.71) – (243.07,246.16) – (243.40,242.13) – (243.73,241.97) – (244.06,240.62) – (244.39,241.02) – (244.72,249.78) – (245.05,256.52) – (245.38,258.86) – (245.71,262.44) – (246.04,260.04) – (246.37,262.31) – (246.70,257.04) – (247.03,256.15) – (247.36,250.51) – (247.69,253.00) – (248.02,258.47) – (248.35,255.29) – (248.68,259.00) – (249.01,261.94) – (249.34,263.23) – (249.67,262.69) – (250.00,267.16) – (250.33,267.94) – (250.66,271.27) – (250.99,278.01) – (251.32,273.25) – (251.65,270.81) – (251.98,270.29) – (252.32,270.85) – (252.65,269.61) – (252.98,267.66) – (253.31,271.13) – (253.64,269.60) – (253.97,263.28) – (254.30,263.28) – (254.63,260.89) – (254.96,266.78) – (255.29,265.33) – (255.62,275.53) – (255.95,277.96) – (256.28,272.13) – (256.61,275.83) – (256.94,279.30) – (257.27,278.67) – (257.60,275.19) – (257.93,270.61) – (258.26,273.76) – (258.59,272.35) – (258.92,265.35) – (259.25,261.65) – (259.58,258.02) – (259.91,262.44) – (260.24,254.63) – (260.57,252.54) – (260.90,250.93) – (261.23,251.18) – (261.56,250.71) – (261.89,246.91) – (262.22,251.58) – (262.55,254.17) – (262.88,256.25) – (263.21,249.34) – (263.54,246.51) – (263.87,247.81) – (264.20,247.92) – (264.53,251.01) – (264.86,252.13) – (265.19,248.78) – (265.52,251.45) – (265.85,251.79) – (266.18,249.59) – (266.51,248.59) – (266.84,244.71) – (267.17,240.02) – (267.50,239.30) – (267.83,239.88) – (268.16,239.07) – (268.49,234.20) – (268.82,238.51) – (269.15,233.07) – (269.48,227.14) – (269.81,229.75) – (270.14,232.43) – (270.47,238.94) – (270.80,233.41) – (271.13,231.07) – (271.46,235.33) – (271.79,234.62) – (272.12,234.92) – (272.45,240.07) – (272.78,242.76) – (273.11,244.46) – (273.44,251.92) – (273.77,249.04) – (274.10,250.86) – (274.43,246.15) – (274.76,244.27) – (275.09,238.48) – (275.42,240.93) – (275.75,241.16) – (276.09,240.75) – (276.42,247.29) – (276.75,245.28) – (277.08,247.95) – (277.41,251.00) – (277.74,251.67) – (278.07,250.16) – (278.40,249.70) – (278.73,250.12) – (279.06,258.25) – (279.39,252.94) – (279.72,240.96) – (280.05,243.59) – (280.38,242.00) – (280.71,242.91) – (281.04,237.02) – (281.37,235.68) – (281.70,235.73) – (282.03,237.13) – (282.36,234.99) – (282.69,231.86) – (283.02,234.34) – (283.35,233.88) – (283.68,235.64) – (284.01,235.89) – (284.34,231.60) – (284.67,227.84) – (285.00,230.97) – (285.33,233.05) – (285.66,228.66) – (285.99,230.28) – (286.32,233.51) – (286.65,237.46) – (286.98,236.76) – (287.31,240.35) – (287.64,240.32) – (287.97,247.11) – (288.30,250.31) – (288.63,249.51) – (288.96,250.87) – (289.29,252.69) – (289.62,249.74) – (289.95,248.08) – (290.28,253.85) – (290.61,252.54) – (290.94,250.68) – (291.27,252.45) – (291.60,247.94) – (291.93,250.95) – (292.26,251.98) – (292.59,249.74) – (292.92,257.42) – (293.25,251.20) – (293.58,257.37) – (293.91,260.03) – (294.24,265.07) – (294.57,269.85) – (294.90,267.27) – (295.23,264.21) – (295.56,266.82) – (295.89,262.48) – (296.22,257.93) – (296.55,247.13) – (296.88,248.21) – (297.21,244.84) – (297.54,250.42) – (297.87,251.20) – (298.20,254.59) – (298.53,258.18) – (298.86,262.72) – (299.19,257.03) – (299.52,253.77) – (299.86,255.90) – (300.19,254.51) – (300.52,251.41) – (300.85,256.59) – (301.18,254.11) – (301.51,249.85) – (301.84,251.42) – (302.17,247.50) – (302.50,243.68) – (302.83,243.08) – (303.16,246.73) – (303.49,250.11) – (303.82,255.08) – (304.15,256.28) – (304.48,252.91) – (304.81,254.17) – (305.14,257.11) – (305.47,259.03) – (305.80,259.48) – (306.13,262.14) – (306.46,264.52) – (306.79,261.40) – (307.12,263.11) – (307.45,267.93) – (307.78,265.37) – (308.11,275.41) – (308.44,276.54) – (308.77,273.85) – (309.10,271.35) – (309.43,272.37) – (309.76,274.79) – (310.09,271.90) – (310.42,275.56) – (310.75,276.66) – (311.08,276.26) – (311.41,283.76) – (311.74,283.90) – (312.07,280.50) – (312.40,275.85) – (312.73,279.26) – (313.06,285.42) – (313.39,289.36) – (313.72,291.51) – (314.05,291.02) – (314.38,287.60) – (314.71,284.52) – (315.04,288.90) – (315.37,289.94) – (315.70,294.69) – (316.03,286.10) – (316.36,290.07) – (316.69,290.24) – (317.02,284.59) – (317.35,286.91) – (317.68,284.17) – (318.01,288.19) – (318.34,294.16) – (318.67,298.83) – (319.00,298.05) – (319.33,303.31) – (319.66,304.55) – (319.99,304.07) – (320.32,309.41) – (320.65,313.13) – (320.98,313.67) – (321.31,316.29) – (321.64,317.33) – (321.97,314.70) – (322.30,316.53) – (322.63,308.27) – (322.96,302.30) – (323.29,297.26) – (323.63,296.35) – (323.96,297.05) – (324.29,304.16) – (324.62,302.97) – (324.95,302.08) – (325.28,308.07) – (325.61,311.21) – (325.94,310.19) – (326.27,310.64) – (326.60,309.98) – (326.93,310.49) – (327.26,309.83) – (327.59,307.09) – (327.92,302.60) – (328.25,302.22) – (328.58,308.29) – (328.91,312.81) – (329.24,317.89) – (329.57,317.26) – (329.90,319.15) – (330.23,320.71) – (330.56,326.20) – (330.89,323.12) – (331.22,326.01) – (331.55,330.71) – (331.88,331.80) – (332.21,331.79) – (332.54,332.55) – (332.87,329.31) – (333.20,336.26) – (333.53,336.09) – (333.86,336.15) – (334.19,343.27) – (334.52,342.85) – (334.85,346.95) – (335.18,346.22) – (335.51,345.34) – (335.84,345.01) – (336.17,342.60) – (336.50,347.99) – (336.83,341.90) – (337.16,344.08) – (337.49,343.88) – (337.82,338.46) – (338.15,340.81) – (338.48,338.70) – (338.81,336.82) – (339.14,338.28) – (339.47,338.43) – (339.80,341.46) – (340.13,346.47) – (340.46,349.57) – (340.79,350.21) – (341.12,346.49) – (341.45,347.69) – (341.78,337.64) – (342.11,333.52) – (342.44,329.43) – (342.77,328.51) – (343.10,327.76) – (343.43,326.43) – (343.76,328.38) – (344.09,325.85) – (344.42,323.55) – (344.75,319.87) – (345.08,320.35) – (345.41,320.71) – (345.74,322.50);
( 15.61,156.83) – ( 15.61,167.84);
at ( 21.55,148.83) [$\scriptstyle T_0$]{};
(131.15,156.83) – (131.15,167.84);
at (137.10,148.83) [$\scriptstyle T_1$]{};
(170.77,156.83) – (170.77,167.84);
at (176.71,148.83) [$\scriptstyle T_2$]{};
(213.69,156.83) – (213.69,167.84);
at (219.63,148.83) [$\scriptstyle T_3$]{};
(299.52,156.83) – (299.52,167.84);
at (305.47,148.83) [$\scriptstyle T_4$]{};
( 15.61, 2.40) – ( 15.61,358.95);
(131.15, 2.40) – (131.15,358.95);
(170.77, 2.40) – (170.77,358.95);
(213.69, 2.40) – (213.69,358.95);
(299.52, 2.40) – (299.52,358.95);
( 15.61, 55.96) – ( 15.94, 57.40) – ( 16.27, 55.21) – ( 16.60, 53.16) – ( 16.93, 54.46) – ( 17.26, 52.52) – ( 17.59, 53.22) – ( 17.92, 51.53) – ( 18.25, 56.61) – ( 18.58, 62.99) – ( 18.91, 65.15) – ( 19.24, 66.58) – ( 19.57, 66.11) – ( 19.90, 67.65) – ( 20.23, 75.06) – ( 20.56, 76.02) – ( 20.89, 76.48) – ( 21.22, 76.49) – ( 21.55, 70.58) – ( 21.88, 75.59) – ( 22.21, 84.06) – ( 22.54, 80.22) – ( 22.87, 81.52) – ( 23.20, 82.00) – ( 23.53, 84.09) – ( 23.86, 83.31) – ( 24.19, 85.04) – ( 24.52, 82.34) – ( 24.85, 83.26) – ( 25.18, 86.04) – ( 25.51, 87.06) – ( 25.84, 81.11) – ( 26.17, 80.70) – ( 26.50, 79.30) – ( 26.83, 78.55) – ( 27.16, 69.65) – ( 27.49, 71.31) – ( 27.82, 66.05) – ( 28.15, 72.26) – ( 28.48, 66.84) – ( 28.81, 58.15) – ( 29.14, 63.21) – ( 29.47, 65.26) – ( 29.80, 64.72) – ( 30.13, 59.88) – ( 30.46, 61.74) – ( 30.79, 63.97) – ( 31.12, 66.02) – ( 31.45, 64.26) – ( 31.78, 63.78) – ( 32.11, 62.12) – ( 32.44, 61.83) – ( 32.77, 65.59) – ( 33.10, 64.43) – ( 33.43, 60.69) – ( 33.76, 58.66) – ( 34.09, 61.45) – ( 34.42, 65.99) – ( 34.75, 68.72) – ( 35.08, 62.61) – ( 35.41, 64.70) – ( 35.74, 67.43) – ( 36.07, 69.22) – ( 36.40, 61.70) – ( 36.73, 66.47) – ( 37.06, 68.99) – ( 37.39, 64.67) – ( 37.72, 64.98) – ( 38.05, 62.89) – ( 38.39, 64.10) – ( 38.72, 66.43) – ( 39.05, 63.83) – ( 39.38, 61.89) – ( 39.71, 58.76) – ( 40.04, 55.85) – ( 40.37, 53.80) – ( 40.70, 53.60) – ( 41.03, 53.31) – ( 41.36, 52.88) – ( 41.69, 51.43) – ( 42.02, 56.66) – ( 42.35, 55.05) – ( 42.68, 50.24) – ( 43.01, 42.59) – ( 43.34, 44.28) – ( 43.67, 45.20) – ( 44.00, 44.98) – ( 44.33, 44.12) – ( 44.66, 33.77) – ( 44.99, 36.60) – ( 45.32, 32.95) – ( 45.65, 38.52) – ( 45.98, 30.53) – ( 46.31, 32.61) – ( 46.64, 25.92) – ( 46.97, 25.70) – ( 47.30, 29.21) – ( 47.63, 29.76) – ( 47.96, 25.95) – ( 48.29, 27.42) – ( 48.62, 30.47) – ( 48.95, 29.26) – ( 49.28, 31.01) – ( 49.61, 27.05) – ( 49.94, 22.79) – ( 50.27, 27.55) – ( 50.60, 28.95) – ( 50.93, 31.24) – ( 51.26, 30.10) – ( 51.59, 29.70) – ( 51.92, 28.52) – ( 52.25, 29.41) – ( 52.58, 24.45) – ( 52.91, 23.73) – ( 53.24, 26.35) – ( 53.57, 26.98) – ( 53.90, 36.55) – ( 54.23, 35.57) – ( 54.56, 35.26) – ( 54.89, 36.87) – ( 55.22, 43.03) – ( 55.55, 45.37) – ( 55.88, 46.68) – ( 56.21, 56.28) – ( 56.54, 62.26) – ( 56.87, 60.34) – ( 57.20, 58.00) – ( 57.53, 49.53) – ( 57.86, 45.25) – ( 58.19, 54.29) – ( 58.52, 55.62) – ( 58.85, 53.32) – ( 59.18, 51.34) – ( 59.51, 52.77) – ( 59.84, 55.18) – ( 60.17, 53.53) – ( 60.50, 49.74) – ( 60.83, 47.23) – ( 61.16, 48.94) – ( 61.49, 49.74) – ( 61.82, 51.65) – ( 62.16, 54.47) – ( 62.49, 54.73) – ( 62.82, 53.35) – ( 63.15, 54.26) – ( 63.48, 50.97) – ( 63.81, 54.22) – ( 64.14, 54.47) – ( 64.47, 53.97) – ( 64.80, 54.52) – ( 65.13, 62.32) – ( 65.46, 59.00) – ( 65.79, 60.46) – ( 66.12, 62.95) – ( 66.45, 60.88) – ( 66.78, 64.38) – ( 67.11, 62.44) – ( 67.44, 58.20) – ( 67.77, 63.75) – ( 68.10, 70.99) – ( 68.43, 69.77) – ( 68.76, 73.24) – ( 69.09, 73.34) – ( 69.42, 73.81) – ( 69.75, 72.54) – ( 70.08, 79.94) – ( 70.41, 77.46) – ( 70.74, 73.50) – ( 71.07, 73.62) – ( 71.40, 82.34) – ( 71.73, 75.73) – ( 72.06, 75.19) – ( 72.39, 73.74) – ( 72.72, 70.48) – ( 73.05, 70.92) – ( 73.38, 73.24) – ( 73.71, 76.17) – ( 74.04, 76.33) – ( 74.37, 76.93) – ( 74.70, 75.89) – ( 75.03, 75.84) – ( 75.36, 75.12) – ( 75.69, 77.84) – ( 76.02, 77.27) – ( 76.35, 72.20) – ( 76.68, 76.31) – ( 77.01, 82.56) – ( 77.34, 82.84) – ( 77.67, 80.41) – ( 78.00, 78.82) – ( 78.33, 76.25) – ( 78.66, 74.42) – ( 78.99, 70.78) – ( 79.32, 68.68) – ( 79.65, 63.24) – ( 79.98, 66.37) – ( 80.31, 68.98) – ( 80.64, 68.78) – ( 80.97, 69.18) – ( 81.30, 70.67) – ( 81.63, 71.11) – ( 81.96, 68.87) – ( 82.29, 67.70) – ( 82.62, 66.06) – ( 82.95, 65.90) – ( 83.28, 63.80) – ( 83.61, 67.42) – ( 83.94, 64.59) – ( 84.27, 65.46) – ( 84.60, 61.64) – ( 84.93, 64.97) – ( 85.26, 67.75) – ( 85.59, 65.75) – ( 85.93, 65.77) – ( 86.26, 70.67) – ( 86.59, 66.31) – ( 86.92, 63.75) – ( 87.25, 57.79) – ( 87.58, 62.60) – ( 87.91, 59.99) – ( 88.24, 61.93) – ( 88.57, 63.66) – ( 88.90, 66.59) – ( 89.23, 76.39) – ( 89.56, 77.24) – ( 89.89, 80.11) – ( 90.22, 81.25) – ( 90.55, 83.79) – ( 90.88, 82.33) – ( 91.21, 75.87) – ( 91.54, 74.49) – ( 91.87, 74.39) – ( 92.20, 74.22) – ( 92.53, 74.65) – ( 92.86, 73.22) – ( 93.19, 76.54) – ( 93.52, 70.83) – ( 93.85, 72.07) – ( 94.18, 72.83) – ( 94.51, 73.59) – ( 94.84, 74.23) – ( 95.17, 73.74) – ( 95.50, 74.15) – ( 95.83, 73.96) – ( 96.16, 84.20) – ( 96.49, 88.10) – ( 96.82, 84.84) – ( 97.15, 83.96) – ( 97.48, 78.22) – ( 97.81, 80.58) – ( 98.14, 75.42) – ( 98.47, 76.84) – ( 98.80, 75.45) – ( 99.13, 76.93) – ( 99.46, 75.20) – ( 99.79, 77.43) – (100.12, 77.53) – (100.45, 80.92) – (100.78, 85.38) – (101.11, 89.04) – (101.44, 89.26) – (101.77, 90.06) – (102.10, 88.35) – (102.43, 86.86) – (102.76, 85.22) – (103.09, 91.74) – (103.42, 90.14) – (103.75, 94.31) – (104.08, 95.16) – (104.41, 95.52) – (104.74, 95.89) – (105.07, 92.98) – (105.40, 89.00) – (105.73, 91.64) – (106.06, 89.94) – (106.39, 92.40) – (106.72, 96.36) – (107.05, 99.52) – (107.38, 94.21) – (107.71, 94.02) – (108.04, 91.58) – (108.37, 97.41) – (108.70,103.44) – (109.03,109.80) – (109.36,112.02) – (109.70,113.49) – (110.03,105.77) – (110.36,107.84) – (110.69,101.08) – (111.02,102.59) – (111.35,100.82) – (111.68, 99.17) – (112.01, 95.41) – (112.34, 91.19) – (112.67, 90.90) – (113.00, 93.42) – (113.33, 95.81) – (113.66, 93.24) – (113.99, 95.10) – (114.32, 92.57) – (114.65, 92.50) – (114.98, 88.63) – (115.31, 92.01) – (115.64, 87.83) – (115.97, 90.08) – (116.30, 89.49) – (116.63, 92.54) – (116.96, 98.94) – (117.29, 96.01) – (117.62, 99.53) – (117.95, 95.04) – (118.28, 95.06) – (118.61, 90.63) – (118.94, 85.95) – (119.27, 89.91) – (119.60, 90.64) – (119.93, 94.23) – (120.26, 92.67) – (120.59, 89.48) – (120.92, 90.71) – (121.25, 86.27) – (121.58, 86.76) – (121.91, 92.65) – (122.24, 95.34) – (122.57, 90.56) – (122.90, 93.29) – (123.23, 89.91) – (123.56, 90.56) – (123.89, 97.92) – (124.22,106.75) – (124.55,106.87) – (124.88,106.09) – (125.21,109.81) – (125.54,114.99) – (125.87,106.79) – (126.20,106.68) – (126.53,106.83) – (126.86,107.82) – (127.19,116.46) – (127.52,110.13) – (127.85,116.72) – (128.18,114.72) – (128.51,115.16) – (128.84,113.63) – (129.17,114.73) – (129.50,117.59) – (129.83,122.53) – (130.16,121.58) – (130.49,127.85) – (130.82,128.23) – (131.15,124.90);
at ( 73.38, 5.77) [$\scriptstyle Y^{(1)}$]{};
(131.15, 55.96) – (131.48, 53.00) – (131.81, 53.25) – (132.14, 53.91) – (132.47, 63.70) – (132.80, 64.42) – (133.13, 66.55) – (133.47, 59.10) – (133.80, 56.82) – (134.13, 62.43) – (134.46, 61.75) – (134.79, 60.43) – (135.12, 58.53) – (135.45, 56.42) – (135.78, 58.34) – (136.11, 50.61) – (136.44, 51.41) – (136.77, 52.20) – (137.10, 55.44) – (137.43, 51.68) – (137.76, 59.47) – (138.09, 56.80) – (138.42, 59.34) – (138.75, 60.48) – (139.08, 59.39) – (139.41, 57.12) – (139.74, 59.12) – (140.07, 59.49) – (140.40, 49.03) – (140.73, 57.21) – (141.06, 63.64) – (141.39, 62.04) – (141.72, 55.14) – (142.05, 55.52) – (142.38, 55.96) – (142.71, 59.38) – (143.04, 58.98) – (143.37, 56.76) – (143.70, 57.38) – (144.03, 60.26) – (144.36, 59.75) – (144.69, 58.11) – (145.02, 52.80) – (145.35, 53.05) – (145.68, 57.66) – (146.01, 55.45) – (146.34, 58.26) – (146.67, 61.49) – (147.00, 65.79) – (147.33, 59.13) – (147.66, 60.25) – (147.99, 61.24) – (148.32, 59.69) – (148.65, 57.05) – (148.98, 62.66) – (149.31, 65.70) – (149.64, 70.83) – (149.97, 66.97) – (150.30, 54.20) – (150.63, 54.29) – (150.96, 53.07) – (151.29, 56.70) – (151.62, 51.95) – (151.95, 45.70) – (152.28, 38.04) – (152.61, 36.17) – (152.94, 35.00) – (153.27, 35.62) – (153.60, 31.65) – (153.93, 33.70) – (154.26, 35.07) – (154.59, 32.28) – (154.92, 32.17) – (155.25, 31.78) – (155.58, 33.00) – (155.91, 30.15) – (156.24, 33.43) – (156.57, 38.26) – (156.90, 37.29) – (157.24, 37.15) – (157.57, 42.66) – (157.90, 39.51) – (158.23, 36.70) – (158.56, 35.91) – (158.89, 42.46) – (159.22, 36.62) – (159.55, 39.15) – (159.88, 41.44) – (160.21, 45.42) – (160.54, 40.03) – (160.87, 41.68) – (161.20, 45.36) – (161.53, 50.75) – (161.86, 51.93) – (162.19, 49.26) – (162.52, 51.29) – (162.85, 50.50) – (163.18, 52.59) – (163.51, 48.91) – (163.84, 45.70) – (164.17, 48.65) – (164.50, 49.08) – (164.83, 53.49) – (165.16, 56.11) – (165.49, 62.44) – (165.82, 60.60) – (166.15, 55.14) – (166.48, 54.32) – (166.81, 60.34) – (167.14, 59.20) – (167.47, 59.52) – (167.80, 61.39) – (168.13, 62.20) – (168.46, 61.42) – (168.79, 56.54) – (169.12, 62.45) – (169.45, 59.68) – (169.78, 58.06) – (170.11, 55.22) – (170.44, 51.90) – (170.77, 53.94);
at (150.96, 5.77) [$\scriptstyle Y^{(2)}$]{};
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at (192.23, 5.77) [$\scriptstyle Y^{(3)}$]{};
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Define the *word sequence* $Y = (Y^{(i)})_{i\in{\mathbb{N}}}$ by putting (see Fig. \[fig-wordsequence\]) $$\qquad Y^{(i)} = \Big(T_i-T_{i-1}, \big(X_{(s+T_{i-1}) \wedge T_i}
-X_{T_{i-1}}\big)_{s\geq 0}\Big),$$ which takes values in the *word space* $$\label{eq:defF}
F = \bigcup_{t>0} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\,
f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ equipped with a Skorohod-type metric (see Appendix \[metrics\]). Let $$Y^{N\text{-}\mathrm{per}} = \big(\,\underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\,
\underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\,\dots\big)$$ denote the $N$-periodisation of $Y$, and let $$\label{RNdef}
R_N = \frac1N \sum_{i=0}^{N-1}
\delta_{\widetilde{\theta}^i Y^{N\text{-}\mathrm{per}}}$$ be the *empirical process of words*, where $\widetilde{\theta}$ is the left-shift acting on $F^{\mathbb{N}}$. Note that $R_N$ takes values in $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, the set of shift-invariant probability measures on $F^{\mathbb{N}}$. Endow $F^{\mathbb{N}}$ with the product topology and $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with the corresponding weak topology. When averaged over $X$ and $T$, the law of $Y$ is ($\mathscr{L}$ denotes law) $$\label{qrhoWdef}
Q_{\rho,{\mathscr{W}}}=(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}} \quad \text{with} \quad
q_{\rho,{\mathscr{W}}} = \int_{(0,\infty)} \rho(dt)\,
\mathscr{L}\big((t, (X_{s \wedge t})_{s \geq 0})\big).$$ By the ergodic theorem, ${\mathop{\text{\rm w-lim}}}_{N\to\infty} R_N = Q_{\rho,{\mathscr{W}}}$ a.s., where ${\mathop{\text{\rm w-lim}}}$ denotes the weak limit.
Large deviation principles {#LDPs}
--------------------------
For definitions and properties of specific relative entropy, we refer the reader to Appendix \[entropy\].
The following theorem is standard (see e.g. Dembo and Zeitouni [@DeZe98 Section 6.5.3]).
\[thm0:contaLDP\] [[**\[Annealed LDP\]**]{}]{}\
The family $\mathscr{L}(R_N)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}
(F^{\mathbb{N}})$ with rate $N$ and with rate function $$\label{eq:Iann}
I^{\mathrm{ann}}(Q)= H(Q \mid Q_{\rho,{\mathscr{W}}}),$$ the specific relative entropy of $Q$ w.r.t. $Q_{\rho,{\mathscr{W}}}$. This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$.
To state the quenched LDP, we need to look at the reverse of cutting out words, namely, glueing words together. Let ${y}=(y^{(i)})_{i\in{\mathbb{N}}}=((t_i,f_i))_{i\in{\mathbb{N}}}
\in F^{\mathbb{N}}$. Then the *concatenation* of ${y}$, written $\kappa({y}) \in C([0,\infty))$, is defined by $$\begin{aligned}
&\kappa({y})(s) = f_1(t_1)+\dots+f_{i-1}(t_{i-1})
+f_i\big(s-(t_1+\cdots+t_{i-1})\big),\\
&t_1+\cdots+t_{i-1} \leq s < t_1+\cdots+t_{i}, \;\; i \in {\mathbb{N}}.
\end{aligned}$$ Write $\tau_i({y})=t_i$ to denote the length of the $i$-th word. For $Q
\in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with finite mean word length $m_Q =
{\mathbb{E}}_Q[\tau_1] ={\mathbb{E}}_Q[\tau_1(Y)]$, put $$\label{eq:PsiQcont}
\Psi_Q(A) =
\frac{1}{m_Q} {\mathbb{E}}_Q\left[ \int_0^{\tau_1} {1}_A(\theta^s \kappa(Y)) \, ds\right],
\quad A \subset C([0,\infty))\;\;\mbox{measurable},$$ where $\theta^s$ is the shift acting on $f \in C([0,\infty))$ as $\theta^s f(t)
= f(s+t)-f(s)$, $t \geq 0$. Note that $\Psi_Q$ is a probability measure on $C([0,\infty))$ with stationary increments, i.e., $\Psi_Q = \Psi_Q \circ (\theta^s)^{-1}$ for all $s \ge 0$. We can think of $\Psi_Q$ as the “stationarised” version of $\kappa(Q)$. In fact, if $m_Q<\infty$, then $$\label{eq:PsiQ}
\Psi_Q = {\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T}
\int_0^T \kappa(Q) \circ (\theta^s)^{-1}\,ds,$$ and $\kappa(Q)$ is asymptotically mean stationary (AMS) with stationary mean $\Psi_Q$. In fact, the convergence in also holds in total variation norm (see Lemma \[lemma:PsiQ:TVlim\] in Appendix \[entropy\]). Note that $\Psi_{Q_{\rho,{\mathscr{W}}}}={\mathscr{W}}$.
To state the quenched LDP, we also need to define word *truncation*. For $(t,f) \in F$ and ${{\rm tr}}> 0$, let $$[(t,f)]_{{\rm tr}}= \big(t \wedge {{\rm tr}}, (f(s \wedge {{\rm tr}})_{s\ge 0}\big)$$ be the word $(t,f)$ truncated at length ${{\rm tr}}$. Analogously, for ${y}=(y^{(i)})_{i\in{\mathbb{N}}}
\in F^{\mathbb{N}}$ set $[{y}]_{{\rm tr}}=([y^{(i)}]_{{{\rm tr}}})_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$, and denote by $[Q]_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}) \subset \mathcal{P}^{\mathrm{inv}}
(F^{\mathbb{N}})$ with $F_{0,{{\rm tr}}}=[F]_{{\rm tr}}$ the image measure of $Q \in \mathcal{P}^{\mathrm{inv}}
(F^{\mathbb{N}})$ under the map ${y} \mapsto [{y}]_{{\rm tr}}$.
\[thm0:contqLDP\] [[**\[Quenched LDP\]**]{}]{}\
Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:Iquelimitform}
I^{\mathrm{que}}(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),$$ where $$\label{def:Ique.tr}
I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) = H\big([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}\big)
+ (\alpha-1) m_{[Q]_{{\rm tr}}} H\big(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}\big).$$ This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$.
Theorem \[thm0:contqLDP\] is proved in Sections \[proof\]–\[removeass\]. Let $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})=\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})
\colon\,m_Q<\infty\}$. We will show that the limit in exists for all $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, and that $$\label{eq:Ique}
I^{\mathrm{que}}(Q) = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}),
\qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}}).$$ We will also see that $I^{\mathrm{que}}(Q)$ is the lower semi-continuous extension to $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ of its restriction to $\mathcal{P}^{\mathrm{inv,fin}}
(F^{\mathbb{N}})$.
Discussion {#disc}
----------
[**0.**]{} A *heuristic* behind Theorem \[thm0:contqLDP\] is as follows. Let $$\label{RNdefind}
R^N_{t_1,\dots,t_N}(X), \qquad 0<t_1<\dots<t_N<\infty,$$ denote the empirical process of $N$-tuples of words when $X$ is cut at the points $t_1,
\dots,t_N$ (i.e., when $T_i=t_i$ for $i=1,\dots,N$). Fix $Q \in \mathcal{P}^{\mathrm{inv,fin}}
(F^{\mathbb{N}})$ and suppose that $Q$ is shift-ergodic. The probability ${\mathbb{P}}(R_N \approx Q \mid X)$ is an integral over all $N$-tuples $t_1,\dots,t_N$ such that $R^N_{t_1,\dots,t_N}(X)
\approx Q$, weighted by $\prod_{i=1}^N \bar{\rho}(t_i-t_{i-1})$ (with $t_0=0$). The fact that $R^N_{t_1,\dots,t_N}(X) \approx Q$ has three consequences:
1. The $t_1,\dots,t_N$ must cut $\approx N$ substrings out of $X$ of total length $\approx N m_Q$ that look like the concatenation of words that are $Q$-typical, i.e., that look as if generated by $\Psi_Q$ (possibly with gaps in between). This means that most of the cut-points must hit atypical pieces of $X$. We expect to have to shift $X$ by $\approx\exp[N m_Q H(\Psi_Q \mid {\mathscr{W}})]$ in order to find the first contiguous substring of length $N m_Q$ whose empirical shifts lie in a small neighbourhood of $\Psi_Q$. By (\[ass:rhodensdecay\]), the probability for the single increment $t_1-t_0$ to have the size of this shift is $\approx
\exp[-N\alpha\,m_Q H(\Psi_Q \mid {\mathscr{W}})]$.
2. The “number of local perturbations” of $t_1,\dots,t_N$ preserving the property $R^N_{t_1,\dots,t_N}(X)\approx Q$ is $\approx \exp[NH_{\tau|K}(Q)]$, where $H_{\tau|K}$ stands for the *conditional specific entropy (density) of word lengths under the law $Q$*.
3. The statistics of the increments $t_1-t_0,\dots,t_N-t_{N-1}$ must be close to the distribution of word lengths under $Q$. Hence, the weight factor $\prod_{i=1}^N
\bar{\rho}(t_i-t_{i-1})$ must be $\approx \exp[N {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)]]$ (at least, for $Q$-typical pieces).
Since $$\label{eqnsre1}
m_Q H(\Psi_Q \mid {\mathscr{W}}) - H_{\tau|K}(Q) - {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)]
= H(Q \mid q_{\rho,{\mathscr{W}}}),$$ the observations made in (1)–(3) combine to explain the shape of the quenched rate function in . For further details, see [@BiGrdHo10 Section 1.5].
*Note:* We have not defined $H_{\tau|K}(Q)$ rigorously here, nor do we prove . Our proof of Theorem \[thm0:contqLDP\] uses the above heuristic only very implicitly. Rather, it starts from the discrete-time quenched LDP derived in [@BiGrdHo10] and draws out Theorem \[thm0:contqLDP\] via control of exponential functionals through a coarse-graining approximation. [**1.**]{} We can include the cases $\alpha=1$ and $\alpha=\infty$ in .
\[mainthmboundarycases\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}.\
[(a)]{} If $\alpha=1$, then the quenched LDP holds with $I^\mathrm{que}=I^\mathrm{ann}$ given by .\
[(b)]{} If $\alpha=\infty$, then the quenched LDP holds with rate function $$\label{eq:ratefctalphainfty}
I^\mathrm{que}(Q) =
\begin{cases}
H(Q \mid Q_{\rho,{\mathscr{W}}}) & \mbox{if} \;\;
\lim\limits_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0, \\
\infty & \mbox{otherwise}.
\end{cases}$$
Theorem \[mainthmboundarycases\] is the continuous analogue of Birkner, Greven and den Hollander [@BiGrdHo10 Theorem 1.4] and is proved in Section \[proofalpha1infty\].
[**2.**]{} We can also include the case where $\bar{\rho}$ has an exponentially bounded tail: $$\label{ass:rhoexp}
\bar{\rho}(t) \leq e^{-\lambda t} \mbox{ for some } \lambda >0 \mbox{ and }
t \mbox{ large enough}.$$
\[thmexp\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{} and . Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:ratefctexptail}
I^\mathrm{que}(Q) = \left\{\begin{array}{ll}
H(Q \mid Q_{\rho,{\mathscr{W}}}) &\mbox{if } Q \in {{\mathcal R}}_{\mathscr{W}},\\
\infty &\mbox{otherwise},
\end{array}
\right.$$ where $${{\mathcal R}}_{\mathscr{W}}= \left\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\,
{\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T} \int_0^T \delta_{\kappa(Y)}
\circ (\theta^s)^{-1}\,ds = {\mathscr{W}}\;\; \text{for $Q$-a.e.\ $Y$}\right\}.$$
Theorem \[thmexp\] is the continuous analogue of Birkner [@Bi08 Theorem 1] and is proved in Section \[proofalpha1infty\]. On the set $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ the following holds: $$\label{eq:calRchar}
\Psi_Q = {\mathscr{W}}\quad \mbox{ if and only if } \quad Q \in {{\mathcal R}}_{\mathscr{W}}.$$ The equivalence in is the continuous analogue of [@Bi08 Lemma 2] (and can be proved analogously).
[**3.**]{} By applying the contraction principle we obtain the quenched LDP for single words. Let $\pi_1\colon\,F^{\mathbb{N}}\to F$ be the projection onto the first word, and let $\pi_1R_N
= R_N \circ (\pi_1)^{-1}$.
\[cor:marginal\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}(\pi_1 R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}(F)$ with rate $N$ and with deterministic rate function $I^\mathrm{que}_1$ given by $$\label{eq:contractedratefct}
I^\mathrm{que}_1(q)
= \inf\big\{ I^\mathrm{que}(Q)\colon\,
Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}),\,\pi_1 Q = q \big\}.$$ This rate function is lower semi-continuous, has compact levels sets, is convex, and has a unique zero at $q=q_{\rho,{\mathscr{W}}}$.
For general $q$ it is not possible to evaluate the infimum in (\[eq:contractedratefct\]) explicitly. For $q$ with $m_q={\mathbb{E}}_q[\tau]={\mathbb{E}}_{q^{\otimes{\mathbb{N}}}}[\tau_1]=m_{q^{\otimes{\mathbb{N}}}}<\infty$ and $\Psi_{q^{\otimes{\mathbb{N}}}}={\mathscr{W}}$, we have $I^\mathrm{que}_1(q)=h(q \mid q_{\rho,{\mathscr{W}}})$, the relative entropy of $q$ w.r.t. $q_{\rho,{\mathscr{W}}}$.
[**4.**]{} We expect assumption to be redundant. In any case, it can be relaxed to (see Section \[prop1\]): $$\label{ass:rhobar.reg-mg}
\begin{minipage}{0.85\textwidth}
$\mathrm{supp}(\rho)= \cup_{i=1}^M [a_i,b_i] \cup [a_{M+1},\infty)$ with $M \in {\mathbb{N}}$
and $0 \leq a_1 < b_1 \leq a_2 < \cdots < b_M \leq a_{M+1}<\infty$, and $\bar{\rho}$
is continuous and strictly positive on $\cup_{i=1}^M (a_i,b_i) \cup (a_{M+1},\infty)$
and varies regularly near each of the finite endpoints of these intervals.
\end{minipage}$$
[**5.**]{} It is possible to extend Theorem \[thm0:contqLDP\] to other classes of random environments, as stated in the following theorem whose proof will not be spelled out in the present paper.
Theorems [\[thm0:contqLDP\]–\[thmexp\]]{} and Corollary [\[cor:marginal\]]{} carry over verbatim when the Brownian motion $X$ is replaced by a $d$-dimensional Lévy process $\bar{X}$ with the property that ${\mathbb{E}}[e^{\langle \lambda, \bar{X}_1 \rangle}] < \infty$ for all $\lambda \in {\mathbb{R}}^d$ (where $\langle\cdot\rangle$ denotes the standard inner product), ${\mathscr{W}}$ is replaced by the law of $\bar{X}$, and in the definition of $F$ in continuous paths are replaced by càdlàg paths.
[**6.**]{} In the companion paper [@BidHo13b] we apply Theorem \[thm0:contqLDP\] and the techniques developed in the present paper to the Brownian copolymer. In this model a càdlàg path, representing the configuration of the polymer, is rewarded or penalised for staying above or below a linear interface, separating oil from water, according to Brownian increments representing the degrees of hydrophobicity or hydrophilicity along the polymer. The reference measure for the path can be either the Wiener measure or the law of a more general Lévy process. We derive a variational formula for the quenched free energy, from which we deduce a variational formula for the slope of the quenched critical line. This critical line separates a *localized phase* (where the copolymer stays close to the interface) from a *delocalized phase* (where the copolymer wanders away from the interface). This slope has been the object of much debate in recent years. The Brownian copolymer is the unique attractor in the limit of weak interaction for a whole universality class of discrete copolymer models. See Bolthausen and den Hollander [@BodHo97], Caravenna and Giacomin [@CaGi10], Caravenna, Giacomin and Toninelli [@CaGiTo12] for details.
Proof of Theorem \[thm0:contqLDP\] {#proof}
==================================
The proof proceeds via a *coarse-graining* and *truncation* argument. In Section \[cogrtrun\] we set up the coarse-graining and the truncation, and state a quenched LDP for this setting that follows from the quenched LDP in [@BiGrdHo10] and serves as the starting point of our analysis (Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] below). In Section \[3prop\] we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below), involving expectations of exponential functionals of the coarse-grained truncated empirical process as well as approximation properties of the associated rate function, and we use these propositions to complete the proof of Theorem \[thm0:contqLDP\] with the help of Bryc’s inverse of Varadhan’s lemma. In Section \[3lem\] we state and prove two lemmas that are used in Section \[3prop\], involving approximation estimates under the coarse-graining. The proof of the three propositions is deferred to Sections \[props\]–\[removeass\].
Preparation: coarse-graining and truncation {#cogrtrun}
-------------------------------------------
### Coarse-graining {#cogr}
Suppose that, instead of the absolutely continuous $\rho$ introduced in Section \[setting\], we are given a discrete $\hat{\rho}$ with $\mathrm{supp}(\hat\rho) \subset h {\mathbb{N}}$ for some $h>0$. Let $$\label{def:Eh}
E_h = \{ f \in C([0,h])\colon\,f(0)=0 \}.$$ Path pieces of length $h$ in a continuous-time scenario can act as “letters” in a discrete-time scenario, and therefore we can use the results from [@BiGrdHo10]. Note that $(E_h)^{\mathbb{N}}$ as a metric space is isomorphic to $\{f \in C([0,\infty))\colon\,f(0)=0\}$ via the obvious glueing together of path pieces into a single path, provided the latter is given a suitable metric that metrises locally uniform convergence. Similarly, we can identify $\mathcal{P}^{\mathrm{inv}}(E_h^{\mathbb{N}})$ with $$\mathcal{P}^{h\text{-}\mathrm{inv}}(C([0,\infty)))
= \big\{ Q \in \mathcal{P}(C([0,\infty))) \colon\, Q = Q \circ (\theta^{h})^{-1} \big\},$$ which is the set of laws on continuous paths that are invariant under a time shift by $h$. Note that the set $$\label{def:Fh}
F_h = \bigcup_{t \in h{\mathbb{N}}} \Big(\{t\} \times
\big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ is isomorphic to $\widetilde{E_h} = \cup_{n \in {\mathbb{N}}} \left(E_h\right)^n$ via the map $\iota_h\colon\, F_h \to \widetilde{E_h}$ defined by $$\label{iotahdef}
\iota_h\big( (nh, f)\big) = \Big( \big(f\big((\,\cdot+(i-1)h) \wedge ih\big)
-f((i-1)h)\big)\Big)_{i=1,\dots,n}, \qquad (nh, f) \in F_h.$$
For $Q \in \mathcal{P}^{\mathrm{inv, fin}}(F_h^{\mathbb{N}})$, define $$\label{eq:definitionPsiQh}
\Psi_{Q,h}(A) = \frac1{m_Q} {\mathbb{E}}_Q\left[ \sum_{i=0}^{\tau_1-1}
{1}_A\big(\theta^i \iota_h \kappa(Y)\big)\right]
= \frac1{h \, m_Q} {\mathbb{E}}_Q\left[ \int_0^{h \tau_1}
{1}_A\big( \kappa(Y)(h \lfloor u/h\rfloor +s))_ {s \geq 0} \big) \, du\right]$$ for $A \subset C([0,\infty))$ measurable, where $\tau_1$ is the length of the first word (counted in letters, so that the length of the first word viewed as an element of $F_h$ is $h\tau_1$) and $\theta$ is the left-shift acting on $(E_h)^{\mathbb{N}}$. The right-most expression in can be viewed as a coarse-grained version of (\[eq:PsiQcont\]). The following coarse-grained version of the quenched LDP serves as our starting point.
\[thm00:contqLDP\] Fix $h>0$. Suppose that $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ and $\lim_{n\to\infty}
\log \hat\rho(\{nh\})/\log n = -\alpha$ with $\alpha \in (1,\infty)$. Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:ratefctfixedh}
I^{\mathrm{que}}_h(Q) = H(Q \mid Q_{\hat\rho,{\mathscr{W}}})
+ (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}}),
\qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ and $$I^{\mathrm{que}}_h(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_h([Q]_{{\rm tr}}),
\qquad Q \notin \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ where $Q_{\hat\rho,{\mathscr{W}}}=(q_{\hat\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$ with $q_{\hat\rho,{\mathscr{W}}}$ defined as in , and $\Psi_{Q,h}$ defined via .
The claim follows from [@BiGrdHo10 Corollary 1.6] by using $E_h$ as letter space and observing that $\widetilde{E_h}=\iota_h(F_h)$. Note that $F_h^{\mathbb{N}}$ is a closed subspace of $F^{\mathbb{N}}$. Since $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ by assumption, we have $I^{\mathrm{que}}_h(Q) \geq H(Q \mid Q_{\hat\rho,{\mathscr{W}}}) = \infty$ for any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $Q\big(F^{\mathbb{N}}\setminus F_h^{\mathbb{N}}\big)>0$. Therefore we can consider the random variable $R_N$ as taking values in $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$, $\mathcal{P}^{\mathrm{inv}}
(F_h^{\mathbb{N}})$ or $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, without changing the statement of Proposition \[thm00:contqLDP\]. Note that $I^{\mathrm{que}}_h$ is finite only on $\mathcal{P}^{\mathrm{inv}}(F_h^{\mathbb{N}})\subset\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$.
We want to pass to the limit $h \downarrow 0$ and deduce Theorem \[thm0:contqLDP\] from Proposition \[thm00:contqLDP\]. However, an immediate application of a projective limit at the level of letters appears to be impossible. Indeed, when we replace $h$ by $h/2$, each “$h$-letter” turns into two “$(h/2)$-letters”, so the word length changes, and even diverges as $h \downarrow 0$. This does not fit well with the way the projective limit was set up in [@BiGrdHo10 Section 8], where the internal structure of the letters was allowed to become increasingly richer, but the word length had to remain the same. In some sense, the problem is that we have finite words but only infinitesimal letters (i.e., there is no fixed letter space). To remedy this, we proceed as follows. For fixed discretisation length $h>0$ we have a fixed letter space, and so Proposition \[thm00:contqLDP\] applies. We will handle the limit $h \downarrow 0$ via Bryc’s inverse of Varadhan’s lemma. This will require several intermediate steps.
### Truncation {#trun}
It will be expedient to work with a *truncated* version of Proposition \[thm00:contqLDP\]. For $h>0$, let $\lceil t \rceil_h =h\lceil t/h \rceil$ for $t \in (0,\infty)$ and put $\lceil\rho\rceil_h
=\rho\,\circ (\lceil\cdot\rceil_h)^{-1}$, i.e., $$\label{def:rho.h.trunc}
\lceil \rho \rceil_h = \sum_{i\in{\mathbb{N}}} w_{h,i} \delta_{ih} \, \in \mathcal{P}(h{\mathbb{N}})
\subset \mathcal{P}((0,\infty)),$$ where $$w_{h,i} = \rho\big(((i-1)h,ih]\big) = \int_{(i-1)h}^{ih} \bar{\rho}(x)dx$$ is the coarse-grained version of $\rho$ from Section \[setting\]. It is easily checked that implies $$\lim_{n\to\infty} \frac{\log \lceil \rho\rceil_h(\{nh\})}{\log n} = -\alpha.$$ Write$ \mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$ for the law of the truncated empirical process $[R_N]_{{\rm tr}}$ conditional on $X$ when the $\tau_i$’s are drawn according to $\lceil \rho \rceil_h$.
\[prop:qLDPhtr\] For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^\mathrm{inv}(F_h^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:Ique.h.tr}
I^{\mathrm{que}}_{h,{{\rm tr}}}(Q)
= H\bigl(Q \mid Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}}\bigr)
+ (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}})$$ with $Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}} = ([q_{\lceil \rho \rceil_h, {\mathscr{W}}}]_{{\rm tr}})^{\otimes {\mathbb{N}}}$.
This follows from Proposition \[thm00:contqLDP\] and the contraction principle. Alternatively, it follows from the proofs of [@BiGrdHo10 Theorem 1.2 and Corollary 1.6].
Note that $I^{\mathrm{que}}_{h,{{\rm tr}}}(Q)=\infty$ when under $Q$ the word lengths are not supported on $h{\mathbb{N}}\cap (0,{{\rm tr}}]$.
Application of Bryc’s inverse of Varadhan’s lemma {#3prop}
-------------------------------------------------
In this section we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below) and show that these imply Theorem \[thm0:contqLDP\]. The proof of these propositions is deferred to Sections \[props\]–\[removeass\].
### Notations {#subsect:notations}
In what follows we obtain the quenched LDP for the truncated empirical process $[R_N]_{{{\rm tr}}}$ by letting $h \downarrow 0$ in the coarse-grained and truncated empirical process $[R_{N,h}]_{{\rm tr}}$ with ${{\rm tr}}\in {\mathbb{N}}$ fixed (for a precise definition, see in Section \[prop1\]) and afterwards letting ${{\rm tr}}\to\infty$. (We assume that ${{\rm tr}}\in {\mathbb{N}}$ and $h=2^{-M}$ for some $M \in {\mathbb{N}}$, in particular, ${{\rm tr}}$ is an integer multiple of $h$.)
In the coarse-graining procedure, it may happen that a very short continuous word $y=(t,f) \in F$ disappears, namely, when $0<t<h$. We remedy this by formally allowing “empty” words, i.e., by using $$\begin{aligned}
\label{def:Fhat}
\widehat{F} = F \cup \big\{ (0,0) \big\}
= \bigcup_{t \geq 0} \Big(\{t\} \times
\big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)\end{aligned}$$ as word space instead of $F$. The metric on $F$ defined in Appendix \[metrics\] extends in the obvious way to $\widehat{F}$.
Before we proceed, we impose *additional regularity assumptions* on $\bar{\rho}$ that will be required in the proof of Proposition \[prop:LambdaPhilimit1tr\]. Recall from that $\mathrm{supp}(\rho) = [s_*,\infty)$. Let $$\begin{aligned}
\label{eq:Vbarrhodef}
V_{\bar{\rho}}(t,h) =
\sup_{v \in (0,2h)} \left| \log
\frac{\int_t^{t+h} \bar{\rho}(s)\,ds}{\int_{t+v}^{t+h+v} \bar{\rho}(s)\,ds} \right|,
\qquad t,h>0.\end{aligned}$$ We assume that there exist monotone sequences $(\eta_n)_{n \in {\mathbb{N}}}$ and $(A_n)_{n\in{\mathbb{N}}}$, with $\eta_n \in (0,1)$ and $A_n \subset (s_*,\infty)$ satisfying $\lim_{n\to\infty} \eta_n = 0$ and $\lim_{n\to\infty} A_n = (s_*,\infty)$, such that $(s_*,\infty) \setminus A_n$ is a (possibly empty) union of finitely many bounded intervals whose endpoints lie in $2^{-n} {\mathbb{N}}_0$, and $$\begin{aligned}
\label{ass:rhobar.reg2}
\sup_{t \in A_n} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_n \qquad \forall\,n \in{\mathbb{N}}. \end{aligned}$$ In addition, we assume that there exists an $\eta_0 < \infty$ such that $$\begin{aligned}
\label{ass:rhobar.reg1}
\sup_{n \in {\mathbb{N}}} \sup_{t \in (s_*,\infty)} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_0. \end{aligned}$$ These assumptions will be removed only in Section \[removeass\]. Note that – are satisfied when $\bar{\rho}$ is continuous and strictly positive on $(s_*,\infty)$ and varies regularly near $s_*$ and at $\infty$.
### Proof of Theorem \[thm0:contqLDP\] subject to (\[ass:rhobar.reg2\]–\[ass:rhobar.reg1\]) and three propositions
A function $g$ on $\widehat{F}^\ell$ is Lipschitz when it satisfies $$\begin{aligned}
\label{eq:g_Lipschitz}
\big| g(y^{(1)},\dots,y^{(\ell)}) - g(y^{(1)}{}',\dots,y^{(\ell)}{}') \big|
\leq C_g \sum_{j=1}^\ell d_F(y^{(j)},y^{(j)}{}')
\quad \mbox{ for some } C_g < \infty. \end{aligned}$$ Consider the class $\mathscr{C}$ of functions $\Phi\colon\,\mathcal{P}(\widehat{F}^{\mathbb{N}}) \to {\mathbb{R}}$ of the form $$\label{eq:Phiform1}
\Phi(Q) = \int_{\widehat{F}^{\ell_1}} g_1 \, d\pi_{\ell_1} Q \wedge \cdots \wedge
\int_{\widehat{F}^{\ell_m}} g_m \, d\pi_{\ell_m}Q,
\quad Q \in \mathcal{P}^{\mathrm{inv}}(\widehat{F}^{\mathbb{N}}),$$ where $m \in N$, $\ell_1,\dots, \ell_m \in {\mathbb{N}}$, and $g_i$ is a bounded Lipschitz function on $\widehat{F}^{\ell_i}$ for $i=1,\dots,m$. This class is well-separating and thus is sufficient for the application of Bryc’s lemma (see Dembo and Zeitouni [@DeZe98 Section 4.4].
Our first proposition identifies the exponential moments of $[R_N]_{{\rm tr}}$.
\[prop:LambdaPhilimit1tr\] The families $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, and $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, ${{\rm tr}}\in {\mathbb{N}}$, are exponentially tight $X$-a.s. Moreover, for $\Phi \in \mathscr{C}$, $$\label{eq:LambdaPhilimit1tr}
\Lambda_{0,{{\rm tr}}}(\Phi)
= \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi([R_N]_{{{\rm tr}}})\big) ~\Big|~ X \Big]
= \lim_{h \downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) \quad
\text{exists $X$-a.s.},$$ where $\Lambda_{h,{{\rm tr}}}$ is the generalised convex transform of $I^{\mathrm{que}}_{h,{{\rm tr}}}$ given by $$\label{eq:Phihtrform}
\Lambda_{h,{{\rm tr}}}(\Phi)
= \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.$$ Furthermore, for $\Phi\in \mathscr{C}$, $$\label{eq:LambdaPhilimit3}
\Lambda(\Phi)
= \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi(R_N)\big) ~\Big|~ X \Big]
= \lim_{{{\rm tr}}\to\infty} \Lambda_{0,{{\rm tr}}}(\Phi) \quad
\text{exists $X$-a.s.}$$
Our second proposition identifies the limit in as the generalised convex transform of $I^{\mathrm{que}}_{{{\rm tr}}}$ defined in , $$I^{\mathrm{que}}_{{{\rm tr}}}(Q)
= \begin{cases} H\bigl( Q \mid Q_{\rho,{\mathscr{W}},{{\rm tr}}} \bigr)
+ (\alpha-1) m_Q H\left( \Psi_Q \mid {\mathscr{W}}\right) &
\text{if} \; Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}), \\[1ex]
\infty & \text{otherwise},
\end{cases}$$ and implies that the latter is the rate function for the truncated empirical process $[R_N]_{{\rm tr}}$.
\[prop:qLDPtrunc1\] For $\Phi\in \mathscr{C}$, $$\label{eq:Lambda0tr}
\Lambda_{0,{{\rm tr}}}(\Phi) =
\sup_{Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{{{\rm tr}}}(Q) \big\}.$$ Furthermore, for ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ with deterministic rate function $I^{\mathrm{que}}_{{{\rm tr}}}$.
Note that the family of truncation operators $[\cdot]_{{{\rm tr}}}$ forms a projective system as the truncation level ${{\rm tr}}$ increases. Hence we immediately get from Proposition \[prop:qLDPtrunc1\] and the Dawson-Gärtner projective limit LDP (see [@DeZe98 Theorem 4.6.1]) that the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP with rate function $Q \mapsto
\sup_{{{\rm tr}}\in {\mathbb{N}}} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$. Furthermore, since the projection can start at any initial level of truncation, we also know that the rate function is given by $Q \mapsto
\sup_{{{\rm tr}}\geq n} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$ for any $n\in{\mathbb{N}}$. Thus, Proposition \[prop:qLDPtrunc1\] in fact implies that the rate function is given by $$\begin{aligned}
\label{eq:Ique-DGform}
\tilde{I}^{\mathrm{que}}(Q) = \limsup_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}}).\end{aligned}$$ At this point, it remains to prove that $\tilde{I}^{\mathrm{que}}$ from actually equals $I^{\mathrm{que}}$ from and has the form claimed in .
This is achieved via the following proposition, note that is the continuous analogue of [@BiGrdHo10 Lemma A.1].
\[prop:Ique.tr.cont\] [(1)]{} For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$, $$\begin{aligned}
\label{eq:lemma:Ique.tr.cont1}
\lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}})
= H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ [(2)]{} For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$ there exists a sequence $(\widetilde{Q}_{{\rm tr}})_{{{\rm tr}}\in {\mathbb{N}}}$ in $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ such that ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and $$\begin{aligned}
\label{eq:lemma:Ique.tr.approx2}
\tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) \leq I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}})
+ o(1), \qquad {{\rm tr}}\to \infty.\end{aligned}$$
Proposition \[prop:Ique.tr.cont\] (1) implies that for $Q \in \mathcal{P}^{\mathrm{inv,fin}}
(F^{{\mathbb{N}}})$ the $\limsup$ in is a limit, i.e., it implies on $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ and also .
To prove for $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$, consider $\widetilde{Q}_{{\rm tr}}$ as in Proposition \[prop:Ique.tr.cont\] (2). Then $$\begin{aligned}
\tilde{I}^{\mathrm{que}}(Q) \leq \liminf_{{{\rm tr}}\to\infty} \tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}})
\leq \liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),\end{aligned}$$ where the first inequality uses that $\tilde{I}^{\mathrm{que}}$ is lower semi-continuous (being a rate function by the Dawson-Gärtner projective limit LDP), and the second inequality is a consequence of . For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $H(Q \mid Q_{\rho, {\mathscr{W}}})=\infty$ we have $$\begin{aligned}
\liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}})
\geq \liminf_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}})
= H(Q \mid Q_{\rho, {\mathscr{W}}}) = \infty,\end{aligned}$$ i.e., also in this case the $\limsup$ in is a limit and holds.
It remains to prove the properties of $I^{\mathrm{que}}$ claimed in Theorem \[thm0:contqLDP\]: lower semi-continuity of $I^{\mathrm{que}}=\tilde{I}^{\mathrm{que}}$ follows from the representation via the Dawson-Gärtner projective limit LDP in ; compactness of the level sets of $I^{\mathrm{que}}$ and the fact that $Q_{\rho,{\mathscr{W}}}$ is the unique zero of $Q \mapsto
I^{\mathrm{que}}(Q)$ are inherited from the corresponding properties of $I^{\mathrm{ann}}$ because $I^{\mathrm{que}} \leq I^{\mathrm{ann}}$; affineness of $Q \mapsto I^{\mathrm{que}}(Q)$ can be checked as in [@BiGrdHo10 Proof of Theorem 1.3].
[**Remark.**]{} Theorem \[thm0:contqLDP\] together with Varadhan’s lemma implies that $$\begin{aligned}
\label{eq:LambdaPhilimit2}
\Lambda(\Phi)
= \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(F^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}(Q) \big\}, \qquad \Phi\in \mathscr{C}, \end{aligned}$$ and identifies $I^{\mathrm{que}}(Q)$ as the generalised convex transform $$\label{eq:Iquetrafo}
I^{\mathrm{que}}(Q) = \sup_{\Phi \in \mathscr{C}}
\big\{ \Phi(Q) - \Lambda(\Phi)\big\}, \qquad Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$$ (see [@DeZe98 Theorems 4.4.2 and 4.4.10]). The supremum in can also be taken over all continuous bounded functions on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$.
Continuity of the empirical process under coarse-graining {#3lem}
---------------------------------------------------------
Before embarking on the proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] in Section \[props\], we state and prove two approximation lemmas (Lemmas \[obs:dSclose1\]–\[obs:Rdiscdiff\] below) that will be needed along the way.
For $N \in {\mathbb{N}}$, $0=t_0 < t_1 < \cdots < t_N$ and $\varphi \in C([0,\infty))$, let $y_\varphi = (y_\varphi^{(i)})_{i\in{\mathbb{N}}}$ with $$\label{def:yphii}
y_\varphi^{(i)} = \Big(t_i-t_{i-1}, \big(\varphi((t_{i-1}+s) \wedge t_i)
-\varphi(t_{i-1})\big)_{s\geq 0}\Big) \in F, \qquad i=1,\dots,N,$$ and define $$\label{eq:defRNphi}
R_{N;t_1,\dots,t_N}(\varphi)
= \frac1N \sum_{i=0}^{N-1}
\delta_{\widetilde{\theta}^i y_\varphi^{N\text{-}\mathrm{per}}}
\in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}).$$ We need a Skorohod-type distance $d_S$ on paths, which is defined in Appendix \[metrics\].
\[obs:dSclose1\] Let $i, j \in {\mathbb{N}}$, $i \leq j$, and $t, t' \in (0,\infty)$, $t<t'$, be such that $(i-1)h < t \leq ih$, $(j-1)h < t' \leq jh$. Then, for any $\varphi \in C([0,\infty))$ and $k \in {\mathbb{N}}$, $$\begin{aligned}
\label{eq:dS_wishful}
& d_S\big( \varphi((ih+\cdot) \wedge jh), \varphi((t+\cdot) \wedge t')\big) \notag \\
& \leq \log\tfrac{k+1}{k} + 2 \sup_{(i-1)h \leq s \leq (i+k)h}
|\varphi(s)-\varphi((i-1)h)| + 2 \sup_{(j-1)h \leq s \leq jh}
|\varphi(s)-\varphi((j-1)h)|.\end{aligned}$$ The same bound holds for $d_S([\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}},
[\varphi((t+\cdot) \wedge t')]_{{\rm tr}})$ for any truncation length ${{\rm tr}}> 0$.
Without loss of generality we may assume that $j \geq i+k$ (otherwise, employ the trivial time transform $\lambda(s)=s$ and estimate the left-hand side of by the second term in the right-hand side of ), and use the time transformation $$\begin{aligned}
\lambda(s) =
\begin{cases}
s \, \frac{(i+k)h-t}{kh} & \text{if} \; s < kh, \\
s + ih -t & \text{if} \; s \geq kh.
\end{cases}\end{aligned}$$ In that case $\lambda(s)+t=s+ih$ for $s \geq kh$ and $\gamma(\lambda)=|\log[((i+k)h-t)/kh]|
\leq \log\frac{k+1}{k}$. The same argument applies to the truncated paths $[\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}}$ and $[\varphi((t+\cdot) \wedge t')]_{{\rm tr}}$ (in fact, we can drop the third term in the right-hand side of when $(j-1)h>{{\rm tr}}$).
\[obs:Rdiscdiff\] Let $\varphi \in C([0,\infty))$, $h>0$, $N\in{\mathbb{N}}$ and $t_0=0<t_1<\cdots<t_N$. Let $\ell \in {\mathbb{N}}$, and let $g\colon\,\widehat{F}^\ell \to {\mathbb{R}}$ be bounded Lipschitz with Lipschitz constant $C_g$. Then, for $k \in {\mathbb{N}}$ with $k \geq \ell$, $$\begin{aligned}
&N \Big| \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;t_1,\dots,t_N}(\varphi)
- \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) \Big|\\
&\qquad \leq
4\ell \|g\|_\infty + C_g \ell N \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr)
+ 4 C_g \ell \sum_{i=1}^N \sup_{\lceil t_i \rceil_h-h \leq s \leq \lceil t_i \rceil_h+kh}
\big| \varphi(s) -\varphi({\lceil t_i \rceil_h-h}) \big|,
\end{aligned}$$ where $\pi_\ell\colon\,\widehat{F}^{\mathbb{N}}\to \widehat{F}^\ell$ denotes the projection onto the first $\ell$ coordinates. The same bound holds for the truncated versions $[R_{N;t_1,\dots,t_N}
(\varphi)]_{{\rm tr}}$ and $[R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) ]_{{\rm tr}}$ for any truncation length ${{\rm tr}}>0$.
For $i=1,\dots,N$, recall $y_\varphi^{(i)}$ from , i.e., $y_\varphi^{(i)}$ is the $i$-th word obtained by cutting the continuous path $\varphi$ along the time points $t_1,\ldots,t_n$, and let $$\begin{aligned}
\tilde{y}_\varphi^{(i,h)}
& = \Bigl(\lceil t_i \rceil_h-\lceil t_{i-1} \rceil_h,
\bigl(\varphi((\lceil t_{i-1} \rceil_h+s) \wedge \lceil t_i \rceil_h)
-\varphi(\lceil t_{i-1} \rceil_h)\bigr)_{s\geq 0}\Bigr), \end{aligned}$$ be the analogous quantity when the $h$-discretised time points $\lceil t_1 \rceil_h,\ldots,\lceil t_N \rceil$ are used. By Lemma \[obs:dSclose1\] we have $$\begin{aligned}
d_F\bigl(y_\varphi^{(i)}, \tilde{y}_\varphi^{(i,h)}\bigr)
&\leq \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr)
+ 2 \sup_{\lceil t_{i-1} \rceil_h-h \leq s \leq \lceil t_{i-1} \rceil_h+kh}
\big| \varphi(s) -\varphi({\lceil t_{i-1} \rceil_h}-h) \big|\\
&\qquad\qquad + 2 \sup_{\lceil t_{i} \rceil_h-h \leq s \leq \lceil t_{i} \rceil_h}
\big| \varphi(s) -\varphi({\lceil t_{i} \rceil_h}-h) \big|.
\end{aligned}$$ Writing $\tilde{y}^{(h)}=(\tilde{y}^{(i,h)})_{i\in{\mathbb{N}}}$ and putting, similarly as in , $$\label{eq:defRNphi_hdisc}
R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi)
= \frac1N \sum_{i=0}^{N-1}
\delta_{\widetilde{\theta}^i(\tilde{y}^{(h)})^{N\text{-}\mathrm{per}}},$$ we see that the claim follows from in combination with Lemma \[obs:dSclose1\]. Note that possible boundary effects due to the periodisation are estimated by the term $4\ell\|g\|_\infty$. The observation about the truncated versions of the empirical process follow analogously from Lemma \[obs:dSclose1\].
Proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] {#props}
======================================================================
Proof of Proposition \[prop:LambdaPhilimit1tr\] {#prop1}
-----------------------------------------------
The proof comes in 3 Steps.
#### Step 1.
A.s. exponential tightness of the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, is standard, because the family of unconditional distributions $\mathscr{L}(R_N)$ satisfies the LDP with a rate function that has compact level sets. Indeed, let $M > 0$, and pick a compact set $K \subset \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ such that $\limsup_{N\to\infty} \tfrac1N \log {\mathbb{P}}(R_N \not\in K) \leq -2M$. Then ${\mathbb{P}}({\mathbb{P}}(R_N \not\in K \mid X) > e^{-MN}) \leq e^{MN} {\mathbb{E}}[{\mathbb{P}}(R_N \not\in K\mid X)]
\leq \exp(MN -2MN +o(N))$, which is summable in $N$. Hence we have $\limsup_{N\to\infty}
\tfrac1N \log {\mathbb{P}}(R_N \not\in K \mid X) \leq -M$ a.s. by the Borel-Cantelli lemma. The same argument applies to $[R_N]_{{{\rm tr}}}$ (alternatively, use the fact that $[\cdot]_{{\rm tr}}$ is a continuous map).
#### Step 2a.
We next verify that the limits in exist. In Step 2a we consider the case $\mathrm{supp}(\rho)=[0,\infty)$, in Step 2b the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$.
Let ${{\rm tr}}\in {\mathbb{N}}$ and $h = 2^{-n}$. Let $Y^{(i,h)}=(\lceil T_i \rceil_h -\lceil T_{i-1} \rceil_h,
(X_{(s+\lceil T_{i-1} \rceil_h) \wedge \lceil T_i \rceil_h}-X_{\lceil T_{i-1} \rceil_h})_{s\geq 0}) \in
\widehat{F}$ be the $h$-discretised $i$-th word, and let $$\label{def:RNh}
R_{N,h} = \frac1N \sum_{i=0}^{N-1}
\delta_{\widetilde{\theta}^i (Y^{(h)})^{N\text{-}\mathrm{per}}}$$ be the $h$-discretised empirical process, where $Y^{(h)}=(Y^{(i,h)})_{i\in{\mathbb{N}}}$. Put $\ell = \ell_1 \vee \cdots \vee \ell_m$, $C_g=C_{g_1} \vee \cdots \vee C_{g_m}$. Let $$\label{eq:defDjh}
D_{j,h} = \sup_{(j-1)h \leq s \leq j h} |X_s-X_{j h}|,
\qquad
A_{\varepsilon,k,h}(N) = \left\{ \sum_{i=1}^N \sum_{j=0}^k
D_{\lceil T_i/h \rceil+j,h} \leq N \varepsilon \right\}.$$ By Lemma \[obs:Rdiscdiff\], on the event $A_{\varepsilon,k,h}(N)$ we have $$N \big| \Phi([R_N]_{{\rm tr}}) - \Phi([R_{N,h}]_{{\rm tr}}) \big|
\leq 4\ell \|\Phi\|_\infty + N C_g \ell m \Big(2h+ \log{\textstyle\frac{k+1}{k}}
+ 4\varepsilon \Big),$$ and hence $$\begin{aligned}
\label{eq:EeNPhiRNtr.ub1}
{\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big] \leq & \,
\exp\big[N C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}}
+ 4\varepsilon \big) + 4\ell \|\Phi\|_\infty \big]\,
{\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \big| X \big] \notag \\
& \, {} + e^{N \|\Phi\|_\infty} {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big), \end{aligned}$$ For $\lambda >0$, estimate $$\begin{aligned}
{\mathbb{P}}([A_{\varepsilon,k,h}(N)]^c | X)
\leq e^{-N \lambda \varepsilon} \,
{\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k
D_{\lceil T_i/h \rceil+m,h} \Big] \,\, \Big| \, X \right],\end{aligned}$$ so that, by Lemma \[lem:expmomentsDsum\] in Step 4 below, $$\begin{aligned}
\label{eq:limsuplogPrAN}
\limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big)
\leq -\varepsilon \lambda + \frac12 \log \chi\big(2 k \lambda \sqrt{h}\big). \end{aligned}$$ Since $\lim_{u\downarrow 0} \chi(u)= 1$, we have, for all $\varepsilon > 0$ and $k\in{\mathbb{N}}$, $$\label{eq_Aepskh.unlikely}
\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N
\log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big) = - \infty \quad \text{a.s.}$$ (pick $\lambda=\lambda(h)$ in (\[eq:limsuplogPrAN\]) in such a way that $\lambda\to\infty$ and $\lambda\sqrt{h}\to 0$).
Next, observe that $$\begin{aligned}
{\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big]
&= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2
\times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N \notag \\
&\qquad \times \exp\big[ {N\Phi\big([R_{N;t_1,\dots,t_N}(X)]_{{\rm tr}}\big)} \big],\\[0.5ex]
\label{eq:expPhiRNhtr}
{\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big]
&= \sum_{1 \leq j_1 \leq \cdots \leq j_N} w_h(j_1,\dots,j_N)
\exp\big[ {N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big)} \big], \end{aligned}$$ where $$\begin{aligned}
\label{eq:wh.weights}
w_h(j_1,\dots,j_N)
&= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2
\times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N\\
&\qquad\qquad \times \prod_{k=1}^N {1}_{(h(j_k-1), h j_k]}(t_k).
\end{aligned}$$ The idea is to replace the right-hand side of by $\prod_{k=1}^N
\lceil \rho \rceil_h(h(j_k-j_{k-1}))$, which is the corresponding weight for a discrete-time renewal process with waiting time distribution $\lceil \rho \rceil_h$. The rigorous implementation of this idea requires some care, since the coarse graining can produce “empty” words.
For $\underline{j}=(j_1,\dots,j_N)$ appearing in the sum in , let $R(\underline{j}) = \# \{ 1 \leq i \leq N \colon j_i = j_{i-1}\}$ be the total number of repeated values and $\underline{\hat{\jmath}}=(\hat \jmath_1,\dots,\hat \jmath_M)$ with $M=M(\underline{j})
=N-R(\underline{j})$, $1 \leq \hat \jmath_1 < \cdots < \hat \jmath_M$, the unique elements of $\underline{j}$. Note that any given $\underline{\hat{\jmath}}$ with $M=\lceil (1-\varepsilon)
N \rceil$ can be obtained in this way from at most ${N \choose \lceil \varepsilon N \rceil}$ different $\underline{j}$’s.
In the following, we write $\eta(h)=\eta_n$ and $A(h) = A_n$ with $\eta_n$ and $A_n$ from when $h=2^{-n}$. Let us parse through the right-hand side of successively for $k=N,N-1,\dots,1$. When $j_k=j_{k-1}$, we integrate $t_k$ out over $(h(j_k-1), h j_k]$ and estimate the (multiplicative) contribution of this integral from above by $1$. When $j_k>j_{k-1}$, we replace $\bar{\rho}(t_k-t_{k-1})$ by $\bar{\rho}(t_k- h j_{k-1})$ and integrate $t_k$ out over $(h(j_k-1), h j_k]$. For $h(j_k-j_{k-1})
\in A(h)$ we can estimate the contribution of this integral from above by $e^{\eta(h)} \lceil \rho
\rceil_h(h(j_k-j_{k-1}))$) by using , while for $h(j_k-j_{k-1}) \not \in A(h)$ we can estimate it by $e^{\eta_0} \lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by using with $s_*=0$. Thus, for $\underline{j}$ with $R(\underline{j}) \leq \varepsilon N$ and $\#\{ 1 \leq i < N \colon h (j_i - j_{i-1}) \not\in A(h) \} \leq \varepsilon N$, we have $$\begin{aligned}
\label{eq:estwhbyrhohgr}
w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N}
\prod_{i=1}^{M} \lceil \rho \rceil_h \big(h(\hat \jmath_i- \hat \jmath_{i-1})\big) \end{aligned}$$ with $M=N-R(\underline{j})$. Furthermore, $$\begin{aligned}
\label{eq:estlast}
\Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big)
- M\Phi\big([R_{M;h \hat \jmath_1,\dots,h \hat \jmath_M}(X)]_{{\rm tr}}\big) \Big|
\le (N-M) \ell \| \Phi \|_\infty \le \varepsilon N \ell \| \Phi \|_\infty.\end{aligned}$$ Combining (\[eq:expPhiRNhtr\]–\[eq:estlast\]), we find $$\begin{aligned}
\label{eq:EeNPhiRNhtr.ub2}
&{\mathbb{E}}\big[e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big] \notag \\
&\leq e^{N \| \Phi \|_\infty} \Big\{ {\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h)
\geq \varepsilon N \, \Big| \, X \Big) \notag \\
&\hspace{5em} {} + {\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h
\not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) \Big\} \notag \\
&\quad + e^{[\varepsilon \eta_0 + \eta(h)]N} {N \choose \varepsilon N}
\sum_{M=\lceil (1-\varepsilon) N \rceil}^N \sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M}
e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)}
\prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1})) . \end{aligned}$$ But $$\begin{aligned}
\sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M}
e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)}
\prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1}))
= {\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big],\end{aligned}$$ where ${\mathbb{E}}_{\lceil \rho \rceil_h}$ denotes expectation w.r.t. the reference measure $Q_{\lceil \rho \rceil_h, {\mathscr{W}}}$, and so we can apply Corollary \[prop:qLDPhtr\] and Varadhan’s lemma to obtain $$\begin{aligned}
\label{eq:limEeMPhi}
\lim_{M\to\infty} \frac1M \log
{\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big]
= \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$
By elementary large deviation estimates for binomials we have, for any $\varepsilon>0$, $$\begin{aligned}
\label{eq:toomanywrongloops1}
&\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log
{\mathbb{P}}\big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h)
\geq \varepsilon N \, \big| \, X \big) = - \infty,\\
\label{eq:toomanywrongloops2}
&\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log
{\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon\, \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h
\not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) = - \infty. \end{aligned}$$ (Note that the events in (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\]) are independent of $X$.) Combining , and , and noting that $\lim_{N\to\infty} \frac1N \log {N \choose \varepsilon N}
= -\varepsilon\log\varepsilon - (1-\varepsilon)\log(1-\varepsilon)$, we find $$\label{eq:eNPhiRNasympt_upper0}
\begin{aligned}
&\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] \\
&\qquad \leq \bigg\{ \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\} \\
&\qquad \qquad + C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}}
+ 4\varepsilon \big) + \varepsilon \eta_0 + \eta(h)
+ \varepsilon\log\tfrac {1}{\varepsilon} + (1-\varepsilon)\log\tfrac{1}{1-\varepsilon} \bigg\}\\
&\qquad \vee \bigg( \|\Phi\|_\infty
+ \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big)\bigg\} \\
&\qquad \vee \bigg\{ \|\Phi\|_\infty
+ \limsup_{N\to\infty} \frac1N \log
{\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h)
\geq \varepsilon N \, \Big| \, X \Big) \bigg\} \\
&\qquad \vee \bigg\{ \|\Phi\|_\infty
+ \limsup_{N\to\infty} \frac1N \log
{\mathbb{P}}\Big( \#\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h
\not\in A(h) \} \geq \varepsilon N \, \Big| \, X \Big) \bigg\},
\end{aligned}$$ and hence $$\begin{aligned}
\label{eq:eNPhiRNasympt_upper}
\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big]
\leq \liminf_{h\downarrow 0}
\sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}\end{aligned}$$ (let $h\downarrow 0$ along a suitable subsequence, followed by $\varepsilon
\downarrow 0$ and $k\to\infty$, and use and (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\])).
Analogous arguments yield $$\begin{aligned}
\label{eq:eNPhiRNasympt_lower}
\liminf_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big]
\geq \limsup_{h\downarrow 0}
\sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})}
\big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$ Indeed, we can simply restrict the sum in to $\underline{j}$’s with $j_1 < \cdots < j_N$, so that the approximation argument is in fact a little easier because we need not pass to the $\underline{\hat\jmath}$’s.
Finally, combine (\[eq:eNPhiRNasympt\_upper\]–\[eq:eNPhiRNasympt\_lower\]) to obtain .
#### Step 2b.
Next we consider the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$ and indicate the changes compared to Step 2a. To some extent this case is easier than the case $s_*=0$, since for coarse-graining level $h<s_*$ no “empty” word can appear in the coarse-graining scheme. On the other hand, when implementing a replacement similar to , it can happen that an integral $\int \bar{\rho}(t_k-t_{k-1}) {1}_{(h(j_k-1),h j_k]}(t) \,dt_k$ gets mapped to $\lceil \rho \rceil_h(h(j_k-j_{k-1}))=0$ even though the true contribution of that integral to is strictly positive (namely, when $h (j_k-j_{k-1}) \leq s_* \leq h(j_k-j_{k-1}+1)$). The idea to remedy this problem is to replace $\lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by a sum of “neighbouring” weights of $\lceil \rho \rceil_h$ and to suitably control the overcounting incurred by this replacement. The details are as follows.
Fix $h>0$ and $s_{*,h} = \lceil s_* \rceil_h$. For $N\in N$, consider $\underline{j}=(j_1,\dots,j_N)$ as appearing in the sum in . We say that $k \in \{1,\dots,N\}$ is “problematic” when $h(j_k-j_{k-1}) \in \{ s_{*,h}-1, s_{*,h}, s_{*,h}+1\}$, and “relaxable” when $j_k-j_{k-1} \geq 2$ and $$\max_{m=-1,0,1}
\left| \log \frac{\lceil \rho \rceil_h(h(j_k-j_{k-1}+m))}{\lceil \rho \rceil_h(h(j_k-j_{k-1}))} \right| \leq 2.$$ Write $K_{\text{pro}}(\underline{j}) = \{ 1 \leq k \leq N \colon\, k\; \text{problematic}\}$ and $K_{\text{rel}}
(\underline{j}) = \{ 1 \leq k \leq N \colon\,k\; \text{relaxable}\}$. Try to construct an injection $f_{\text{rel},
\underline{j}}\colon\,K_{\text{pro}} \to K_{\text{rel}}$ with the property $f_{\text{rel},\underline{j}}(k) > k$ as follows:
- Start with an empty “stack” ${\sf s}$. For $k=1,\dots,N$ successively: when $k$ is problematic, push $k$ on ${\sf s}$; when $k$ is relaxable and ${\sf s}$ is not empty, pop the top element, say $k'$, from ${\sf s}$ and put $f_{\text{rel},\underline{j}}(k')=k$; when $k$ is neither problematic nor relaxable, proceed with the next $k$.
We say that $\underline{j}$ is “good” when the above procedure terminates with an empty stack (in particular, $f_{\text{rel},\underline{j}}(k')$ is defined for all $k' \in K_{\text{pro}}$) and $$\sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big) \leq \varepsilon N$$ (in particular, $\# K_{\text{pro}}(\underline{j}) \leq \varepsilon N$), and also $\# \{ 1 \leq k \leq N
\colon\, j_k-j_{k-1} \not\in A(h)\} \leq \varepsilon N$. For a given good $\underline{j}$, consider the set of all $\underline{\tilde\jmath}=(\tilde\jmath_1,\dots,\tilde\jmath_N)$ obtainable by setting $$\begin{aligned}
\tilde\jmath_k=j_k+\Delta_k, \; \tilde\jmath_{f_{\text{rel},\underline{j}}(k)}
= j_{f_{\text{rel},\underline{j}}(k)}-\Delta_k \quad
\text{with}\;\: \Delta_k \in \{-1,0,1\} \quad \text{for}\;\:k \in K_{\text{pro}},\end{aligned}$$ and $\tilde\jmath_k=j_k$ for $k \not\in(K_{\text{pro}} \cup f_{\text{rel},\underline{j}}(K_{\text{pro}}))$. Note that a given good $\underline{j}$ corresponds to at most $3^{\varepsilon N}$ different $\underline{\tilde\jmath}$’s and that, for any such $\underline{\tilde\jmath}$, $$\begin{aligned}
&\Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big)
- N\Phi\big([R_{N;h \tilde \jmath_1,\dots,h \tilde \jmath_M}(X)]_{{\rm tr}}\big) \Big| \notag \\
&\qquad \leq \ell \| \Phi \|_\infty
\sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big)
\leq \varepsilon N \ell \| \Phi \|_\infty .\end{aligned}$$
With $w_h(j_1,\dots,j_N)$ defined in , we now see that (analogously to the argument prior to ) for any good $\underline{j}$, $$\begin{aligned}
\label{eq:estwhbyrhohgr.c2}
w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N}
2^{\varepsilon N} \sum_{\underline{\tilde\jmath} \; \text{corresp.\ to}\; \underline{j}}
\; \prod_{i=1}^{N} \lceil \rho \rceil_h \big(h(\tilde \jmath_i -
\tilde \jmath_{i-1})\big). \end{aligned}$$ Moreover, we have $$\begin{aligned}
\label{eq:toomanywrongloops3}
\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log
{\mathbb{P}}\big( (\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h)
\; \text{not good} \, \big| \, X \big) = - \infty. \end{aligned}$$ To check , let $S_k$ be the size of the stack ${\sf s}$ in the $k$-th step of the above construction when we use $j_k=\lceil T_k \rceil_h$, and note that $(\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h)$ is good when $\sum_{k=1}^N S_k
< \varepsilon N$. A comparison of $(S_k)_{k\in{\mathbb{N}}}$ with a (reflected) random walk on ${\mathbb{N}}_0$ that draws its steps from $\{0, \pm 1 \}$, where $(+1)$-steps have a very small probability ($\leq \int_{s_*}^{s_*+2h} \bar{\rho}(t)\, dt$) and $(-1)$-steps have a very large probability ($\rho(A_h)$) when not from $0$, shows that $\limsup_{h \downarrow h} \frac1N
\log {\mathbb{P}}(\sum_{k=1}^N S_k \geq \varepsilon N) = -\infty$ for every $\varepsilon >0$. We can then estimate similarly as in , to obtain for the case $s_*>0$ as well.
Analogous arguments also yield the lower bound in .
#### Step 3.
We next verify that the limits in exist. Note that $$| \Phi(R_N) - \Phi([R_N]_{{\rm tr}})| \leq \|\Phi\|_\infty
\frac1N \#\big\{ \text{loops among the first $N$ loops that are longer than ${{\rm tr}}$} \big\},$$ which can be made arbitrarily small (also on the exponential scale, via a suitable annealing argument that uses that loop lengths are i.i.d.). A similar estimate holds for $| \Phi([R_N]_{{{\rm tr}}}) - \Phi([R_N]_{{{\rm tr}}'})|$ with ${{\rm tr}}< {{\rm tr}}'$. This shows that $\Lambda_{0,{{\rm tr}}}(\Phi)$ forms a Cauchy sequence as ${{\rm tr}}\to\infty$.
The arguments in Steps [2a]{} and [2b]{} can be combined to yield the same results when assumption is relaxed to assumption . Indeed, for a given coarse-graining level $h$, gives rise to finitely many types of “problematic points” that can be handled similarly as in Step [2b]{} (combined with arguments from Step [2a]{} when $a_1=0$).
#### Step 4.
We close by deriving the estimate on Brownian increments over randomly drawn short time intervals that was used in in Step 2. The intuitive idea is that even though there are arbitrarily large increments over short time intervals somewhere on the Brownian path, it is extremely unlikely to hit these when sampling along an independent renewal process. The proof employs a suitable annealing argument.
Recall $D_{j,h}$ from (\[eq:defDjh\]). For $h>0$ fixed, the $D_{j,h}$’s are i.i.d. and equal in law to $\sqrt{h}D_{1,1} = \sqrt{h} \sup_{0\leq s\leq 1} |X_s|$ by Brownian scaling.
\[lem:expmomentsDsum\] Let $T=(T_i)_{i\in{\mathbb{N}}}$ be a continuous-time renewal process with interarrival law $\rho$ satisfying $\mathrm{supp}(\rho) \subset [h,\infty)$. For $\lambda \geq 0$ and $k \in {\mathbb{N}}_0$, define $$\begin{aligned}
\xi(\lambda,h) = \limsup_{N\to\infty} \frac1N \log
{\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k D_{\lceil T_i/h \rceil+m,h} \Big] \,
\Big| \, \sigma(D_{j,h}, j \in {\mathbb{N}})\right],\end{aligned}$$ which is $\geq 0$ and a.s. constant by Kolmogorov’s $0$-$1$-law. Then $$\begin{aligned}
\lim_{h \downarrow 0} \xi(\lambda,h) = 0 \qquad \forall\,\lambda \geq 0. \end{aligned}$$
We consider only the case $k=0$, the proof for $k\in{\mathbb{N}}$ being analogous. Abbreviate $\mathscr{G}_h=\sigma(D_{j,h}, j \in {\mathbb{N}})$, and let $$\chi(u) = {\mathbb{E}}\Big[ \exp \big[ u \, {\textstyle\sup_{\,0 \leq t \leq 1} |X_t|} \big] \Big],
\quad u \in {\mathbb{R}}.$$ Note that $\chi(\cdot)$ is finite and satisfies $\lim_{u\to 0} \chi(u) = 1$. We have $$\begin{aligned}
{\mathbb{E}}\bigg[ {\mathbb{E}}\bigg[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h}
\Big] \, \Big| \, \mathscr{G}_h \bigg]^2 \bigg]
&\leq {\mathbb{E}}\bigg[ \exp\Big[ 2\lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h}
\Big] \bigg]\\
&= {\mathbb{E}}\big[ \exp[2\lambda D_{1,h}] \big]^N = \chi\big(2\lambda \sqrt{h}\big)^N.
\end{aligned}$$ Thus, for any $\epsilon>0$, $$\begin{aligned}
& {\mathbb{P}}\left( {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h}
\Big] \, \Big| \, \mathscr{G}_h \right]^2
\geq \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^N \right) \\
& \leq \, \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^{-N}
{\mathbb{E}}\left[ {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h}
\Big] \, \Big| \, \mathscr{G}_h \right]^2 \right]
\leq \left( \frac{\chi\big(2\lambda \sqrt{h}\big)}{\chi\big(2\lambda \sqrt{h}\big) + \epsilon} \right)^N,
\end{aligned}$$ which is summable in $N$. The Borel-Cantelli lemma therefore yields $$\limsup_{N\to\infty} \frac1N \log
{\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h}
\Big] \, \Big| \, \mathscr{G}_h \right]
\leq \frac12 \log \chi\big(2\lambda \sqrt{h}\big).$$
Proof of Proposition \[prop:qLDPtrunc1\] {#ss:prop2}
----------------------------------------
\[lemma:Iquetrregularised\] For ${{\rm tr}}\in {\mathbb{N}}$ and $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, $$\begin{aligned}
\label{eq:Iquetrregularised}
I^{\mathrm{que}}_{{\rm tr}}(Q)
= \lim_{\varepsilon \downarrow 0} \, \limsup_{h \downarrow 0} \,
\inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q)
\cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h,{{\rm tr}}})^{\mathbb{N}}) \Big\},\end{aligned}$$ where $h \downarrow 0$ along $2^{-m}$, $m\in {\mathbb{N}}$.
Note that after $\widetilde{E}_{h,{{\rm tr}}}$ is identified with a subset of $F_{0,{{\rm tr}}}$ (see ), states that $I^{\mathrm{que}}_{h,{{\rm tr}}}$ converges to $I^{\mathrm{que}}_{{\rm tr}}$ as $h \downarrow 0$ in the sense of Gamma-convergence.
Note that, when restricted to $\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{\otimes{\mathbb{N}}})$, $$\label{eq:PsiQcont1}
\text{both } Q \mapsto m_Q \text{ and } Q \mapsto \Psi_Q \text{ are continuous}$$ (by dominated convergence), while this is not true when $Q$ is allowed to vary over the whole of $\mathcal{P}^\mathrm{inv}(F^{\otimes{\mathbb{N}}})$. A more general statement is the following: if ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q_n = Q$ and $\{ \mathscr{L}_{Q_n}(\tau_1)\colon\,
n \in {\mathbb{N}}\}$ are uniformly integrable, then $\lim_{n\to\infty} m_{Q_n} = m_Q$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \Psi_{Q_n} = \Psi_Q$.
In the proof we use several properties of specific relative entropy derived in Appendix \[entropy\]. Let $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, and abbreviate the right-hand side of by $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q)$. Note that, by and the lower semi-continuity of $\Psi \mapsto H(\Psi \mid {\mathscr{W}})$, the map $$\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{{\mathbb{N}}}) \ni Q' \mapsto
m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}})$$ is lower semi-continuous. Hence, for any $\delta>0$, we have $m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}}) \geq
m_Q H(\Psi_Q \mid {\mathscr{W}}) - \delta$ for all $Q' \in B_\varepsilon(Q) \cap \mathcal{P}^{\mathrm{inv}}
(\widetilde{E}_{h,{{\rm tr}}}^{\mathbb{N}})$ when $\varepsilon$ is sufficiently small (depending on $\delta$). Combine this with in Lemma \[lemma:hregularised1\] in Appendix \[entropy\], and note that ${\mathop{\text{\rm w-lim}}}Q_{h,{{\rm tr}}} = Q_{{\rm tr}}$ as $h\downarrow0$, to obtain $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q)
\geq I^{\mathrm{que}}_{{{\rm tr}}}(Q)$.
For the reverse direction, we need to find $h_n>0$ with $\lim_{n\to\infty} h_n = 0$ and $Q'_n \in \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h_n,{{\rm tr}}})^{\mathbb{N}})$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\liminf_{n\to\infty} I^{\mathrm{que}}_{h_n,{{\rm tr}}}
(Q'_n) \le I^{\mathrm{que}}_{{\rm tr}}(Q)$. Here a complication stems from the fact that we must ensure that both parts of $I^{\mathrm{que}}_{h_n,{{\rm tr}}}(Q'_n)$, namely, $H(Q'_n \mid
Q_{\lceil \rho \rceil_{h_n},{\mathscr{W}},{{\rm tr}}})$ and $H(\Psi_{Q'_n,h_n} \mid {\mathscr{W}})$, converge simultaneously. The proof is deferred to Lemma \[lem:cg.2lev.blockapprox\] in Appendix \[entropy\].
We are now ready to give the proof of Proposition \[prop:qLDPtrunc1\].
Fix ${{\rm tr}}\in{\mathbb{N}}$. Denote the right-hand side of by $\tilde{\Lambda}_{{\rm tr}}(\Phi)$. Let $\Phi\colon\,\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}) \to {\mathbb{R}}$ be of the form (\[eq:Phiform1\]). For every $\delta > 0$ we can find a $Q^* \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ such that $\Phi(Q^*) - I_{{\rm tr}}^{\mathrm{que}}(Q^*) \geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - \delta$. For $\varepsilon>0$ sufficiently small (depending on $\delta$) we have $\big| \Phi(Q') - \Phi(Q^*)\big| \leq \delta$ for all $Q' \in B_\varepsilon(Q^*)$ and, by Lemma \[lemma:Iquetrregularised\], $$\begin{aligned}
\liminf_{h\downarrow 0} \, \inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\,
Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\}
\leq I^{\mathrm{que}}_{{\rm tr}}(Q^*) + \delta.\end{aligned}$$ Thus $$\begin{aligned}
\liminf_{h\downarrow 0} \,
\sup \Big\{ \Phi(Q') - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\,
Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\}
\geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - 3\delta. \end{aligned}$$ Let $\delta\downarrow 0$ to obtain $\liminf_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \Lambda_{0,{{\rm tr}}}(\Phi)
\geq \tilde{\Lambda}_{{\rm tr}}(\Phi)$.
For the reverse direction, pick for $h \in (0,1)$ a maximiser $Q^*_h \in \mathcal{P}^{\mathrm{inv}}
((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}})$ of the variational expression appearing in the right-hand side of , i.e., $\Phi(Q^*_h) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q^*_h) = \Lambda_{h,{{\rm tr}}}(\Phi)$. This is possible because $\Phi-I^{\mathrm{que}}_{h,{{\rm tr}}}$ is upper semi-continuous and bounded from above, and $I^{\mathrm{que}}_{h,{{\rm tr}}}$ has compact level sets. We claim that $$\label{claim:Q*htight}
\text{the family} \; \big\{ Q^*_h : h \in (0,1) \big\} \subset
\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\; \text{is tight}.$$ Assuming (\[claim:Q\*htight\]), we can choose a sequence $h(n) \downarrow 0$ such that $$\begin{aligned}
&\lim_{n\to\infty} \Big[ \Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big]
= \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi),\\
&{\mathop{\text{\rm w-lim}}}_{n\to\infty} Q^*_{h(n)} = \widetilde{Q}
\;\; \text{for some} \; \widetilde{Q} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}).
\end{aligned}$$ Then $\lim_{n\to\infty}\Phi(Q^*_{h(n)})=\Phi(\widetilde{Q})$ because $\Phi$ is continuous, and $\liminf_{n\to\infty} I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \geq I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q})$ by Lemma \[lemma:Iquetrregularised\]. Hence $$\begin{aligned}
\Lambda_{0,{{\rm tr}}}(\Phi) = \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \lim_{n\to\infty} \Big[
\Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big]
\leq \Phi(\widetilde{Q}) - I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q})
\leq \tilde{\Lambda}_{{\rm tr}}(\Phi).\end{aligned}$$
It remains to prove , which follows once we show that for each $N \in {\mathbb{N}}$ the family of projections $\pi_N(Q^*_h) \in \mathcal{P}^{\mathrm{inv}}(F^N)$, $h\in(0,1)$, is tight (because $F^{\mathbb{N}}$ carries the product topology; see Ethier and Kurtz [@EK86 Chapter 3, Proposition 2.4]). Let $M= \|\Phi\|_\infty+1$. Then necessarily $H( Q^*_h \mid [Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq M$, and hence $h(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq N M$ for all $h \in (0,1)$. Since $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ converges weakly to $\pi_N ([Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ as $h \downarrow 0$, the family $\{\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\colon\,h \in (0,1)\}$ is tight, and so for any $\varepsilon > 0$ we can find a compact $\mathcal{C} \subset F^N$ such that $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c) \leq \exp[-(NM + \log 2)/\varepsilon]$ uniformly in $h\in(0,1)$. By a standard entropy inequality (see in Appendix \[entropy\]), for all $h\in (0,1)$ we have $$\begin{aligned}
\pi_N(Q^*_h)(\mathcal{C}^c) \leq
\frac{\log 2 + h\big(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\big)}
{\log\Big(1+\big(\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c)\big)^{-1} \Big)}
\leq \frac{\log 2 + MN}{\log\big(1+\exp[(N M + \log 2)/\varepsilon]\big)}
\leq \varepsilon. \end{aligned}$$ This proves the representation of the limit $\Lambda_{0,{{\rm tr}}}(\Phi)$ from . From and , plus the exponential tightness in Proposition \[prop:LambdaPhilimit1tr\], we obtain the LDP via Bryc’s inverse of Varadhan’s lemma.
Proof of Proposition \[prop:Ique.tr.cont\] {#prop3}
------------------------------------------
### Proof of part (1)
We first verify , i.e., for $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned}
\label{eq:relentrsumlim}
\lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}})
& = \lim_{{{\rm tr}}\to\infty} \Big[
H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) +
(\alpha-1) m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \Big] \notag \\
& = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ The proof comes in 5 Steps.
[**Step 1.**]{} Note that $\lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = H(Q \mid Q_{\rho,{\mathscr{W}}})$ by the projective property of word truncations, $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} = m_Q<\infty$ by dominated convergence, and $$\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \geq H(\Psi_Q \mid {\mathscr{W}})$$ by the lower semi-continuity of specific relative entropy together with ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty}
\Psi_{[Q]_{{\rm tr}}} = \Psi_Q$. Hence, to obtain it remains to prove that $$\begin{aligned}
\label{ineq:HPsiQtr.upper}
\limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \leq H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$
[**Step 2.**]{} \[prop:Ique.tr.cont.part1step2\] To prove , we use coarse-graining. For every $h>0$ we can identify $\widetilde{E_h}$ with $F_h \subset F$ (recall ). In order to represent $Q\in\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ by a shift-invariant law on $(F_h)^{\mathbb{N}}$, we discretise the cut-points onto a *uniformly shifted* grid of width $h$, as follows. For $t \in {\mathbb{R}}$, $h>0$ and $u\in [0,1)$, define (compare with Section \[trun\]) $$\begin{aligned}
\label{eq:t.hu}
\lceil t \rceil_{h,u} = \min\big\{ (k+u)h \colon k \in {\mathbb{Z}}, (k+u)h \geq t \big\}
\quad \big(= \lceil t -uh \rceil_h + uh \big).\end{aligned}$$ Draw $Y=(Y^{(i)})_{i\in{\mathbb{N}}}=((\tau_i, f_i))_{i\in{\mathbb{N}}}$ from law $Q$, and let $U$ be an independent random variable with uniform distribution on $[0,1]$. Put $T_0=0$, $T_n=\tau_1+\cdots+\tau_n$, $n \in {\mathbb{N}}$, $$\tilde{T}_i = \lceil T_i \rceil_{h,U}, \quad i \in {\mathbb{N}}_0,
\qquad \tilde{\tau}_i = \tilde{T}_i - \tilde{T}_{i-1}, \;
\tilde{f}_i = \big(\theta^{\tilde{T}_{i-1}} \kappa(Y)\big)(\, \cdot \wedge \tilde{\tau}_i), \quad i\in{\mathbb{N}}.$$ (Note that it may happen that $\tilde{\tau}_i=0$. We can remedy this by allowing “empty words”, i.e., by formally passing to $\widehat{F}$ as in Section \[subsect:notations\].) Write $\lceil Q
\rceil_h$ for the distribution of $\tilde{Y}=(\tilde{Y}^{(i)})_{i \in {\mathbb{N}}}=((\tilde{\tau}_i,
\tilde{f}_i))_{i \in {\mathbb{N}}}$ obtained in this way. We view $\lceil Q \rceil_h$ as an element of $\mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$. To check the shift-invariance of $\lceil Q \rceil_h$, note that by construction an initial part of length $S_1=\tilde{T}_0-T_0 = U h$ of the content of the first word is removed (in a two-sided situation, this part would be added at the end of the zero-th word). The corresponding quantity for the second word is $S_2=\tilde{T}_1-T_1 = \lceil
T_1 \rceil_{h,U} - T_1$. Observe that, for measurable $A \subset [0,h)$ and $B \subset [0,\infty)$, $${\mathbb{P}}(S_2 \in A, T_1 \in B) = \int_B {\mathbb{P}}(T_1 \in dt) \int_{[0,1]} du\,
{1}_A\big( \lceil t -uh \rceil_h - (t-uh) \big) = \frac1h\, {\mathbb{P}}(T_1 \in B)\, \lambda(A),$$ i.e., $S_2$ is distributed as $U h$ and independent of $Y$, and so $(\tilde{Y}^{(i+1)})_{i\in{\mathbb{N}}}$ again has law $\lceil Q \rceil_h$. This settles the shift-invariance. The key feature of the construction of $\lceil Q \rceil_h$ is that $\kappa(\tilde{Y}) = (\theta^{U h} \kappa)(Y)$, so that $$\label{Psihrel}
\Psi_{\lceil Q \rceil_h, h} = \Psi_Q,$$ and therefore $$\begin{aligned}
\label{HhHrel}
H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) = H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ Thus, gives us a coarse-grained version of the right-hand of .
[**Step 3.**]{} If ${{\rm tr}}$ is an integer multiple of $h$, then the coarse-graining $\lceil Q \rceil_h \in
\mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ of $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ defined in Step 2 commutes with the word length truncation $[ \cdot ]_{{\rm tr}}$, i.e., $[ \lceil Q
\rceil_h ]_{{\rm tr}}= \lceil [ Q ]_{{\rm tr}}\rceil_h$. This is a deterministic property of the construction in . Indeed, fix $u \in [0,1)$ and $h$ with ${{\rm tr}}= Mh$ for some $M\in {\mathbb{N}}$, consider $t_{i-1}<t_i$ with $t_i-t_{i-1} > {{\rm tr}}$ (so that in the un-coarse-grained truncation procedure the $i$-th loop length would be replaced by ${{\rm tr}}$), let $k_{i-1}, k_i \in {\mathbb{N}}$ be such that $\lceil t_{i-1} \rceil_{h,u} = (k_{i-1}+u)h$ and $\lceil t_i \rceil_{h,u} = (k_i+u)h$. When we first truncate and then coarse-grain, the $i$-th point becomes $\lceil t_{i-1}
+ {{\rm tr}}\rceil_{h,u} = (k_{i-1}+M+u)h$. When we first coarse-grain and then truncate, the $i$-th point becomes $\lceil t_{i-1} \rceil_{h,u} + \big( (\lceil t_i \rceil_{h,u}
- \lceil t_{i-1} \rceil_{h,u}) \wedge M h \big) = (k_{i-1}+u)h + M h$, which is the same.
[**Step 4.**]{} Let $h=2^{-M}$, define $\lceil Q \rceil_h \in \mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ as in Step 2, and write $Q'_h = \lceil Q \rceil_h \circ \iota_h^{-1}$ for the same object considered as an element of $\mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}})$ (recall (\[def:Eh\]–\[iotahdef\])). Write $\nu_h=\mathscr{L}\big((X_{\cdot \wedge h})\big)$ for the Wiener measure on $E_h$. Then $m_{Q'_h} = m_{\lceil Q \rceil_h}/h$ (the mean word length counted in $h$-letters), while $$\begin{aligned}
\label{HPsihrel}
H(\Psi_{Q'_h} \mid \nu_h^{\otimes {\mathbb{N}}}) = H(\Psi_{\lceil Q \rceil_h,h} \mid {\mathscr{W}}),\end{aligned}$$ by construction, and $$\begin{aligned}
\label{Qtrhrel}
\lceil [ Q ]_{{\rm tr}}\rceil_h = [ \lceil Q \rceil_h ]_{{\rm tr}}= [Q'_h]_{({{\rm tr}}/h)} \circ \iota_h,\end{aligned}$$ where the first equality follows from the commutation property in Step 3 and the second equality is a truncation of the words from $Q'_h$ as elements of $\widetilde{E_h}$.
[**Step 5.**]{} Fix $\varepsilon>0$ and let ${{\rm tr}}_0 = {{\rm tr}}_0(Q,\varepsilon)$ be so large that $$\begin{aligned}
{\mathbb{E}}_Q\big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \big] < \tfrac13 \varepsilon m_Q,
\qquad {{\rm tr}}\geq {{\rm tr}}_0.\end{aligned}$$ Then, for $0<h<\tfrac{1}{24} \varepsilon m_Q$, we have $$\begin{aligned}
\label{eq:justso}
{\mathbb{E}}_{\lceil Q \rceil_h}\big[ h\big( \tfrac{|Y^{(1)}|}{h}-\tfrac{{{\rm tr}}}{h} \big)_+ \big]
< \tfrac13 \varepsilon m_Q + 2h < \tfrac12 \varepsilon m_{\lceil Q \rceil_h}. \end{aligned}$$ Divide both sides of by $h$, and observe that the continuum word of length $|Y^{(1)}|$ under $\lceil Q \rceil_h$ corresponds to the discrete word of $|Y^{(1)}|/h$ $h$-letters under $Q_h'$, to obtain $$\begin{aligned}
{\mathbb{E}}_{Q_h'}\big[ \big( |Y^{(1)}|-\tfrac{{{\rm tr}}}{h} \big)_+ \big]
< \tfrac12 \varepsilon m_{Q_h'}.\end{aligned}$$ This estimate allows us to use Lemma \[lem:trcontinuous\] in Appendix \[entropy\], which says that for every $0<\varepsilon<\tfrac12$, $$\begin{aligned}
\label{bepsest}
(1-\varepsilon) \Big[ H(\Psi_{[Q_h']_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}})
+ b(\varepsilon) \Big] \leq H(\Psi_{Q_h'} \mid \nu_h^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon)= - 2\varepsilon + [\varepsilon \log \varepsilon + (1-\varepsilon)
\log (1-\varepsilon)]/(1-\varepsilon)$. However, by (\[HPsihrel\]–\[Qtrhrel\]) we have $$H(\Psi_{[Q'_h]_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}})
= H(\Psi_{\lceil [Q]_{{\rm tr}}\rceil_h,h} \mid {\mathscr{W}}) = H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}).$$ Substitute this relation into and use (\[HhHrel\]–\[HPsihrel\]), to obtain $$(1-\varepsilon) \Big[ H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}})
+ b(\varepsilon) \Big] \leq H(\Psi_Q \mid {\mathscr{W}}).$$ Now let $\varepsilon \downarrow 0$ and use that $\lim_{\varepsilon\downarrow 0}
b(\varepsilon)=0$, to obtain .
### Proof of part (2) {#subsect:prop:Ique.tr.cont.part2}
Fix $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $m_Q= \infty$ and $H(Q \mid Q_{\rho,{\mathscr{W}}})
< \infty$. We construct $\widetilde{Q}_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, ${{\rm tr}}\in{\mathbb{N}}$, satisfying via a “smoothed truncation” that has the same concatenated word content as its “hard truncation” equivalent. The proof comes in 5 Steps.
[**Step 1.**]{} It will we be convenient to consider the two-sided scenario, i.e., we regard $Q$ as a shift-invariant probability measure on $F^{\mathbb{Z}}$. Define $$\chi_{{\rm tr}}\colon\,F_{0,{{\rm tr}}}^{\mathbb{Z}}\times [0,1]^{\mathbb{Z}}\to F^{\mathbb{Z}},
\qquad
\chi_{{\rm tr}}\colon\,\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}, (u_i)_{i\in{\mathbb{Z}}} \big) \mapsto (\tilde f_i, \tilde \tau_i)_{i\in{\mathbb{Z}}},$$ as follows. Put $t_0=0$, $t_i=t_{i-1}+\tau_i$, $t_{-i}=t_{-i+1}-\tau_{-i+1}$ for $i\in{\mathbb{N}}$, and $\varphi
= \kappa\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}\big)$, set $$\begin{aligned}
\tilde{t}_i =
\begin{cases} t_i-u_i & \text{if} \; \tau_i={{\rm tr}},\\
t_i & \text{if} \; \tau_i <{{\rm tr}},
\end{cases}\end{aligned}$$ $\tilde\tau_i=t_i-t_{i-1}$ and $\tilde f_i(\cdot)=\varphi( (\,\cdot \wedge \tilde\tau_i)+t_{i-1})$ for $i\in{\mathbb{Z}}$. In words, the total concatenated word content remains unchanged, and if the length of the $i$-th word $\tau_i$ equals ${{\rm tr}}$, then its end-point $t_i$ is moved $u_i$ to the left. Put $\widetilde{Q}_{{\rm tr}}= ([Q]_{{\rm tr}}\otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1} \in
\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{Z}})$. By construction, $\Psi_{\widetilde{Q}_{{\rm tr}}} = \Psi_{[Q]_{{\rm tr}}}$ and $m_{\widetilde{Q}_{{\rm tr}}} = m_{[Q]_{{\rm tr}}}$. In particular, $$\begin{aligned}
m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}})
= m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) .\end{aligned}$$
[**Step 2.**]{} Write $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif}
[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1}$ for the result of the analogous operation on the reference measure $(q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$. We have ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ
\chi_{{\rm tr}}^{-1} = (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$, and hence $$\begin{aligned}
\label{eq:string}
\liminf_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} )
& \geq \sup_{\varepsilon > 0} \liminf_{{{\rm tr}}\to\infty}
\inf_{Q' \in B_\varepsilon(Q)} H( Q' \mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \notag \\
& \geq H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}})
= \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}),\end{aligned}$$ where we use Lemma \[lemma:hregularised1\] (2) in the second inequality. (Note: Inspection of the proof of Lemma \[lemma:hregularised1\] (2) shows that the inequality “$\leq$” in also holds for $Q$’s that are not product.) The last equality in holds because the truncations $[\,\cdot\,]_{{\rm tr}}$ form a projective family (see [@BiGrdHo10 Lemma 8.1]). As specific relative entropy can only decrease under the operation of taking image measures, we have $H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H([Q]_{{\rm tr}}\mid [q_{\rho,
{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$, so $\limsup_{{{\rm tr}}\to\infty}
H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$ and, indeed, $$\begin{aligned}
\lim_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} )
= \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}})
= H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}).\end{aligned}$$ The proof of is complete once we show that $$\begin{aligned}
\label{eq:HtildeQtr.bd}
H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) \leq
H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) + o(1),\end{aligned}$$ since, by part (1), $$\begin{aligned}
\widetilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) =
H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) +
m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}}).\end{aligned}$$
[**Step 3.**]{} It remains to verify . Note that $$\begin{aligned}
H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})
- H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} )
& = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}\bigg[
\log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}}
- \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}} \bigg]
\notag \\
& = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}
\bigg[ \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}} \bigg],\end{aligned}$$ and that, by construction, ${d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}/{dq_{\rho,{\mathscr{W}}}^{\otimes N}}$ is a function of the word lengths $\tilde{\tau}_1,\dots,\tilde{\tau}_N$ only (indeed, because of the i.i.d. property of Brownian increments it easy to see that under both laws the word contents given their lengths are the same, namely, independent pieces of Brownian paths). Write $\widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ for the law of the sequence of word lengths under $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}$. Then we must show that $$\begin{aligned}
\label{eq:ElogdRtildedrho.bd}
\limsup_{{{\rm tr}}\to\infty}
\lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}
\bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}}
(\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq 0. \end{aligned}$$
[**Step 4.**]{} Denote the density of $\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ with respect to Lebesgue measure on ${\mathbb{R}}_+^N$ by $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$. Consider fixed $\tilde\tau_1,\dots,\tilde\tau_N$, and decompose into maximal stretches of $\tilde\tau_i$’s with values in $({{\rm tr}}-1,{{\rm tr}}+1)$ (note that under $\chi_{{\rm tr}}$ no word can become longer than ${{\rm tr}}+1$, while when $\tilde\tau_i < {{\rm tr}}-1$ the corresponding word is not truncated, i.e., $\tilde{t}_i=t_i$). Thus, there are $0 \leq M < N$, $i'_1 \leq j'_2 < i'_2 \leq j'_2
< \cdots < i'_M \leq j'_M \leq N$ such that $\{ 1 \leq i \leq N\colon\,\tilde\tau_i
\in ({{\rm tr}}-1,{{\rm tr}}+1) \} = \cup_{k=1}^M [i'_k, j'_k] \cap {\mathbb{N}}$. Observe that, by construction, $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)$ can be decomposed into a product of $\prod_{j \colon \tilde\tau_j\leq {{\rm tr}}-1} \bar{\rho}
(\tilde\tau_j)$ and $M$ further factors involving the $\tilde\tau_i$’s from these stretches, where the $k$-th factor depends only on $(\tilde\tau_i\colon\,
i'_k \leq i \leq j'_k)$. We claim that $$\begin{aligned}
\label{eq:ftrN.ref.bd}
\frac{\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)}{
\prod_{j=1}^N \bar{\rho}(\tilde\tau_j)}
\leq \prod_{k=1}^M \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{j'_k-i'_k+1}
= \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{\# \{ 1 \leq i \leq N\colon\, \tilde\tau_i > {{\rm tr}}-1 \}}\end{aligned}$$ for some $C_1=C_1(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. To see why holds, consider for example the first stretch and assume for simplicity that $i'_1=1<j'_1$ and that we know that the $0$-th word is not truncated (i.e., $\tilde{t}_0=t_0=0$). Let $\ell
\leq j'_1+1$, and pretend we know that the first $\ell-1$ words are truncated (i.e., $\tau_1=\cdots=\tau_{\ell-1}={{\rm tr}}$), while the $\ell$-th word is not ($\tau_\ell<{{\rm tr}}$). Then $\tilde\tau_1={{\rm tr}}-u_1$ and $\tilde\tau_i={{\rm tr}}-u_i+u_{i-1}$ for $2 \leq i \leq \ell-1$, and so $u_i=\sum_{j=1}^i ({{\rm tr}}-\tilde\tau_j)$ for $1 \leq i \leq \ell-1$ and $\tau_\ell
=\tilde\tau_\ell-u_{\ell-1} = \tilde\tau_\ell-\sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)$. This case contributes to $\widetilde{f}_{{{\rm tr}},\ell}^{\mathrm{ref}}$ the term $$\begin{aligned}
\label{eq:termf.ell.ref}
\rho([{{\rm tr}},\infty))^{\ell-1} \bar\rho\Big(\tilde\tau_\ell-
{\textstyle \sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)}\Big) \prod_{i=1}^{\ell-1}
{1}_{[0,1]}\Big( {\textstyle \sum_{j=1}^{i} ({{\rm tr}}-\tilde\tau_j)}\Big).\end{aligned}$$ Note that, by , we have $\eqref{eq:termf.ell.ref}/\prod_{j=1}^\ell
\bar{\rho}(\tilde\tau_j) \le C_2 (C_3 {{\rm tr}}^{1+\epsilon})^{\ell-1}$ for some $C_2=C_2(\rho),
C_3=C_3(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. The contribution of any given stretch of length $j'_k-i'_k+1$ can be written as a sum of at most $2^{j'_k-i'_k+1}$ cases where the indices of the truncated words are specified. Each such case can be estimated by a suitable product of terms as in . Furthermore, outside the stretches the words are necessarily untruncated and thus contribute $\bar{\rho}(\tilde\tau_i)$ to $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$, which cancels with the corresponding term in $\rho^{\otimes N}$.
[**Step 5.**]{} From and the shift-invariance of $\widetilde{Q}_{{\rm tr}}$ we obtain that $$\begin{aligned}
\label{eq:ElogdRtildedrho.bd2}
\lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}
\bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}}
(\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq
C(1+\log {{\rm tr}}) Q(\tau_1>{{\rm tr}}-1). \end{aligned}$$ Now, $h(\mathscr{L}_Q(\tau_1) \mid \rho) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\mathbb{N}}) < \infty$ by assumption. Because of , this implies that ${\mathbb{E}}_Q[\log(\tau_1)]
< \infty$, and hence that $Q(\tau_1>{{\rm tr}}) = o(1/\log {{\rm tr}})$. Therefore implies .
$\qed$
Removal of Assumptions – {#removeass}
========================
We give a brief sketch of the proof only, leaving the details to the reader. Assumptions – are satisfied when $\bar{\rho}$ satisfies and varies regularly at $\infty$ with index $\alpha$. The latter condition is stronger than . To prove the claim under alone, note that for every $\delta>0$ and $\alpha'<\alpha$ there exists a probability density $\bar{\rho}'=\bar{\rho}'
(\delta,\alpha')$ such that $\bar{\rho} \leq (1+\delta)\bar{\rho}'$, $\bar{\rho}'$ varies regularly at $\infty$ with index $\alpha'$, and $\bar{\rho}'(t)dt$ converges weakly to $\bar{\rho}(t)dt$ as $\delta
\downarrow 0$ and $\alpha' \uparrow \alpha$. Since the quenched LDP holds for $\bar{\rho}'$, we can proceed similarly as in [@BiGrdHo10 Sections 3.6 and 5] to get the quenched LDP for $\bar{\rho}$.
More precisely, for $B \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ we may write (recall and ) $$\begin{aligned}
P(R_N \in B \mid X) &= \int_{0 \leq t_1 < \cdots < t_N < \infty} dt_1 \cdots dt_N\,
\bar{\rho}(t_1)\,\bar{\rho}(t_2-t_1) \cdots \bar{\rho}(t_N-t_{N-1})\\[-2ex]
&\hspace{18em} \times 1_B\big(R_{N;t_1,\ldots,t_N}(X)\big), \notag\end{aligned}$$ and estimate $\bar{\rho}(t_1) \leq (1+\delta)\bar{\rho}'(t_1)$, etc., to get $P(R_N \in B \mid X)
\leq (1+\delta)^N\,P'(R_N \in B \mid X)$, where $P,P'$ have $\bar{\rho},\bar{\rho}'$ as excursion length distributions. Let $\mathcal{C} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ be a closed set, and let $\mathcal{C}^{(\varepsilon)}$ be its $\varepsilon$-blow-up. Then the LDP upper bound for $\bar{\rho}'$ gives $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X)
\leq \log (1+\delta) - \inf_{Q \in \mathcal{C}^{(\varepsilon)}} I^\mathrm{que}_{\bar{\rho}'}(Q)
\qquad X\text{-a.s.},$$ where the lower index $\bar{\rho}'$ indicates the excursion length distribution. Let $\delta \downarrow 0$ and $\alpha' \uparrow \alpha$, and use Lemma \[lemma:hregularised1\] (2), to get $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X)
\leq - \inf_{Q \in \mathcal{C}^{(2\varepsilon)}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$ Finally, let $\varepsilon \downarrow 0$ and use the lower semi-continuity of $I^\mathrm{que}_{\bar{\rho}}$ to get the LDP upper bound for $\bar{\rho}$.
An analogous argument works for the LDP lower bound: Now we pick $\alpha' > \alpha$, $\delta > 0$ and a probability density $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$, and $\bar{\rho}'$ satisfies the same conditions as above. Arguing as before, we obtain for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X)
\geq - \inf_{Q \in \mathcal{O}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$
Proof of Theorems \[mainthmboundarycases\]–\[thmexp\] {#proofalpha1infty}
=====================================================
We again give a brief sketch of the proofs only, leaving many details to the reader.
Theorem \[mainthmboundarycases\](a), which says that for $\alpha=1$ the quenched rate function coincides with the annealed rate function, can be proved as follows: Since the claimed LDP upper bound holds automatically by the annealed LDP, it suffices to verify the matching lower bound. For this we can argue as in the proof of the lower bound in Section \[removeass\]. For any $\alpha'>1$ and $\delta>0$ we can approximate $\bar{\rho}$ by a suitable $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$. Then, using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\delta \downarrow 0$, $\alpha' \downarrow 1$, we see that for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X)
\geq - \inf_{Q \in \mathcal{O} \cap \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})}
I^\mathrm{ann}(Q) \qquad X\text{-a.s.}$$ (recall ). Finally note that any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$ can be approximated by a sequence $(Q_n) \subset \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ in such a way that $H(Q_n \mid Q_{\rho,{\mathscr{W}}}) \to H(Q \mid Q_{\rho,{\mathscr{W}}})$ to obtain the claim (using for example a “smoothed truncation” operation similar to Section \[subsect:prop:Ique.tr.cont.part2\]).
Theorem \[mainthmboundarycases\](b), which says that for $\alpha=\infty$ the quenched rate function coincides with the annealed rate function on the set $\{Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\,\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}}
H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0\}$ and is infinite elsewhere, follows from arguments analogous to [@BiGrdHo10 Section 7, Part (b)]: For the upper bound, we can pick arbitrarily large $\alpha'>1$ and approximate $\bar{\rho} \leq (1+\delta) \bar{\rho}'$ with the help of a suitable probability density $\bar{\rho}'$ which has decay exponent $\alpha'$. Using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\alpha' \uparrow \infty$, $\delta \downarrow 0$, we see that the upper bound holds with the claimed form of the rate function. For the matching lower bound we can trace through the proof of the lower bound contained in Theorem \[thm0:contqLDP\] but replacing our “coarse-graining work horses” Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] (which rely on [@BiGrdHo10 Cor. 1.6]) by versions that are suitable for $\alpha=\infty$ (which rely on [@BiGrdHo10 Thm. 1.4 (b)] instead), still using a suitable truncation approximation of the quenched rate function analogous to the one proven in Proposition \[prop:Ique.tr.cont\]. This constitutes a way of rigorously implementing the “first long string strategy” from [@BiGrdHo10 Section 4], as explained in the heuristic given in item 0 of Section \[disc\], through the coarse-graining approximation.
Theorem \[thmexp\] follows from Theorem \[mainthmboundarycases\](b) via an observation that is the analogue of [@Bi08 Lemma 6]: subject to the exponential tail condition in , any $Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) <\infty$ necessarily has $m_Q<\infty$. Because of this observation we can argue as follows. If $m_Q<\infty$, then $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}}
= m_Q$ and $\lim_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$ by dominated convergence (recall ), which in turn imply that $\liminf_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}}
H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = m_Q H(\Psi_Q \mid {\mathscr{W}})$, as shown in Lemma \[lem:trcontinuous\] in Appendix \[entropy\]. The limit is zero if and only if $\Psi_Q = {\mathscr{W}}$, which by holds if and only if $Q \in {{\mathcal R}}_{\mathscr{W}}$. This explains the link between and .
Basic facts about metrics on path space {#metrics}
=======================================
We metrise $F$, defined in (and $F_h \subset F$ defined in ) as follows. Let $d_S(\phi_1, \phi_2)$ be a metric on $C([0,\infty))$ that generates Skorohod’s $J_1$-topology on $D([0,\infty)) \supset C([0,\infty))$, allowing for a certain amount of “rubber time” (see e.g. Ethier and Kurtz [@EK86 Section 3.5 and Eqs. (5.1–5.3)]) $$\label{def:dS}
d_S(\phi_1, \phi_2) = \inf_{\lambda \in \Lambda}
\bigg\{ \gamma(\lambda) \vee \int\nolimits_0^\infty
e^{-u} \sup_{t\geq 0} \big| \phi_1(t \wedge u) - \phi_2(\lambda(t) \wedge u)
\big| \, du \bigg\},$$ where $\Lambda$ is the set of Lipschitz-continuous bijections from $[0,\infty)$ into itself and $$\gamma(\lambda) = \sup_{0 \leq s < t < \infty}
\Big| \log \frac{\lambda(t)-\lambda(s)}{t-s} \Big|.$$ With $$\label{eq:metriconF}
d_F(y_1,y_2) = |t_1-t_2| + d_S(\phi_1,\phi_2)$$ for $y_i=(t_i,\phi_i) \in F$, $(F,d_F)$ becomes complete and separable, and the same holds for $(F_h,d_F)$ for any $h>0$.
[**Remark. **]{} We might at first be inclined to metrise $F$ in a more straightforward way than (\[eq:metriconF\]), e.g. via $$\label{eq:firstmetriconF}
d^{\mathrm{first}}_F(y_1,y_2) = |t_1-t_2| + \|\phi_1-\phi_2\|_\infty,
\quad y_i=(t_i,\phi_i) \in F, \: i=1,2.$$ However, if we would use Lipschitz functions with $d_F$ replaced by $d^{\mathrm{first}}_F$ in (\[eq:g\_Lipschitz\]), then in the analogue of Lemma \[obs:Rdiscdiff\] we would be forced to use terms of the form $\sup_{s \geq 0} |\varphi(s+t \wedge t') - \varphi(s+ih
\wedge jh)|$ in the right-hand side. When used for $\varphi=X$ (a realisation of Brownian motion as in Proposition \[prop:LambdaPhilimit1tr\]), this would in turn force us to control the increments of the Brownian motion not only locally near the beginning and the end of each loop, but uniformly inside loops. In fact, it seems plausible that an analogue of Proposition \[prop:LambdaPhilimit1tr\] where $d_F$ is replaced by $d^{\mathrm{first}}_F$ actually fails. Furthermore, note that we cannot arrange $d_S$ in such a way that, for $\phi \in C([0,\infty))$, $h>0$, $t_1 \leq t'_1 < t_2 \leq t_2'$ with $|t'_1-t_1| \leq h$, $|t'_2-t_2| \leq h$, $$\begin{aligned}
\label{eq:dS_wishful1}
d_S\big( \phi((t_1+\cdot) \wedge t_2),
\phi((t'_1+\cdot) \wedge t'_2)\big) \leq 2h
+ \sup_{t_1 \leq s \leq t'_1} |\phi(s)-\phi(t'_1)|
+ \sup_{t_2 \leq s \leq t'_2} |\phi(s)-\phi(t'_2)|.\end{aligned}$$ This is why in Lemma \[obs:Rdiscdiff\] we need the freedom to use an extra $k$ and to “look in a neighbourhood of the cut-points of size $kh$”.
Basic facts about specific relative entropy {#entropy}
===========================================
In Section \[ss:definitions\] we recall the definition of (specific) relative entropy of two probability measures. In Section \[ss:approximations\] we prove various approximation results for (specific) relative entropy, which were used heavily in Sections \[props\]. Especially the parts with $\Psi_Q$ require care because of the delicate nature of the word concatenation map $Q \mapsto \Psi_Q$. The latter is looked at in closer detail in Section \[subs:towards.Ique.tr.cont\].
Definitions {#ss:definitions}
-----------
For $\mu,\nu$ probability measures on a measurable space $(S,\mathscr{S})$, $$h(\mu \mid \nu)
=
\begin{cases}
\int_S (\log\frac{d\mu}{d\nu})\,d\mu,
&\text{if} \; \mu \ll \nu, \\
\infty,
& \text{otherwise,}
\end{cases}$$ is the relative entropy of $\mu$ w.r.t. $\nu$. When the measurable space is a Polish space $E$ equipped with its Borel-$\sigma$-algebra, we also have the representation (see e.g. [@DeZe98 Lemma 6.2.13]) $$\begin{aligned}
\label{eq:relentraslegendretransf}
h(\mu \mid \nu)
= \sup_{f \in C_b(E)} \Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\}
= \sup_{\scriptstyle f\colon\, E \to {\mathbb{R}}\; \atop \scriptstyle \text{bounded measurable}}
\Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\} \end{aligned}$$ (and if $\mu \ll \nu$ with a bounded and uniformly positive density, then the supremum in the right-hand side is achieved by $f=\log d\mu/d\nu$).
Equation implies the entropy inequality $$\label{ineq:entropy}
\mu(A)\leq \frac{\log 2 + h(\mu \mid \nu)}{\log[1+1/\nu(A)]}$$ by choosing $f=\alpha {1}_A$ and $\alpha=\log[1+1/\nu(A)]$ (see e.g. Kipnis and Landim [@KiLa99 Appendix 1, Proposition 8.2]).
For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, $$\begin{aligned}
\label{eq:SREwrtProd}
H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}})
= \lim_{N\to\infty} \frac1N h\big( \pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big)
= \sup_{N\in{\mathbb{N}}} \frac1N h\big(\pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big)\end{aligned}$$ with $\pi_N$ the projection onto the first $N$ words, is the specific relative entropy of $Q$ w.r.t. $(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$. Similarly, using the canonical filtration $(\mathscr{F}^C_t)_{t \ge 0}$ on $C([0,\infty))$, for a probability measure $\Psi$ on $C([0,\infty))$ with stationary increments we denote by $$\begin{aligned}
\label{eq:SREwrtWM}
H(\Psi \mid {\mathscr{W}}) = \lim_{t\to\infty}
\frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big)
= \sup_{t>0} \frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big)\end{aligned}$$ the specific relative entropy w.r.t. Wiener measure. See Appendix \[contrelentr\] for a proof of .
Approximations {#ss:approximations}
--------------
Let $E$ be a Polish space. Equip $\mathcal{P}(E)$ with the weak topology (suitably metrised). $E^{\mathbb{N}}$ carries the product topology, and the set of shift-invariant probability measures $\mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ carries the weak topology (also suitably metrised).
### Blocks
For $M\in{\mathbb{N}}$ and $r \in \mathcal{P}(E^M)$, denote by $r^{\otimes {\mathbb{N}}} \in \mathcal{P}(E^{\mathbb{N}})$ the law of an infinite sequence obtained by concatenating $M$-blocks drawn independently from $r$ (i.e., we identify $(E^M)^{\mathbb{N}}$ and $E^{\mathbb{N}}$ in the obvious way), and write $$\label{def:blockmeas}
{\mathsf{sblock}}_M(r) = \frac1M \sum_{j=0}^{M-1} r^{\otimes {\mathbb{N}}} \circ (\theta^j)^{-1}
\; \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$$ for its stationary mean.
\[lemma:HblockmeasQ\] For $Q = q^{\otimes {\mathbb{N}}} \in\mathcal{P}^\mathrm{inv}(E)$ and $r \in \mathcal{P}(E^M)$, $$\label{eq:HblockmeasQ}
H\big({\mathsf{sblock}}_M(r) \mid Q \big) = \frac1M h\big(r \mid \pi_M Q \big).$$ Moreover, for any $R \in \mathcal{P}^\mathrm{inv}(E)$, $$\label{eq:reconstrfromblocks}
{\mathop{\text{\rm w-lim}}}_{M\to\infty} {\mathsf{sblock}}_M\big(\pi_M R\big) = R.$$
This proof is standard. Equation follows from the results in Gray [@Gr09b Section 8.4, see Theorem 8.4.1] by observing that ${\mathsf{sblock}}_M(r)$ is the asymptotically mean stationary measure of $r^{\otimes{\mathbb{N}}}$. It is also contained in Föllmer[@Foe88 Lemma 4.8], or can be proved with “bare hands” by explicitly spelling out $d\pi_N {\mathsf{sblock}}_M(r)/dq^{\otimes N}$ for $N \gg M$ and using suitable concentration arguments under $q^{\otimes N}$ as $N\to\infty$. Equation is obvious from the definition of weak convergence.
### Change of reference measure
\[lemma:hregularised1\] [(1)]{} Let $\nu, \nu_1,\nu_2,\ldots \in \mathcal{P}(E)$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \nu_n = \nu$. Then $$\begin{aligned}
\label{eq:hregularised1}
h(\mu \mid \nu) = \lim_{\varepsilon \downarrow 0}
\limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n),
\quad \mu \in \mathcal{P}(E). \end{aligned}$$ [(2)]{} Let $Q=q^{\otimes {\mathbb{N}}}, Q_1=q_1^{\otimes {\mathbb{N}}},Q_2=q_2^{\otimes {\mathbb{N}}},\ldots \in
\mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ be product measures with ${\mathop{\text{\rm w-lim}}}_{n\to\infty}
Q_n$ $= Q$. Then $$\begin{aligned}
\label{eq:Hregularised1}
H(R \mid Q) = \lim_{\varepsilon \downarrow 0}
\limsup_{n \to \infty} \inf_{R' \in B_\varepsilon(R)} H(R' \mid Q_n),
\quad R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}}). \end{aligned}$$
\(1) Denote the term in the right-hand side of (\[eq:hregularised1\]) by $\tilde{h}(\mu)$. Let $f \in C_b(E)$, $\delta > 0$. We can find $\varepsilon_0 > 0$ and $n_0 \in {\mathbb{N}}$ such that $$\begin{aligned}
\forall \, 0 < \varepsilon \leq \varepsilon_0, \,
\mu' \in B_\varepsilon(\mu)\colon\,\,
&\int_E f\, d\mu' \geq \int_E f\, d\mu - \frac{\delta}{2}, \\
\forall \, n \geq n_0\colon\,\,
&\log \int_E e^f \, d\nu_n \leq \log \int_E e^f \, d\nu + \frac{\delta}{2}. \end{aligned}$$ Therefore, for $0 < \varepsilon \leq \varepsilon_0$ and $n \geq n_0$, $$\begin{aligned}
\inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n)
\geq \int_E f\, d\mu' - \log \int_E e^f \, d\nu_n
\geq \int_E f\, d\mu \smallskip - \log \int_E e^f \, d\nu - \delta.\end{aligned}$$ Now optimise over $f$ and take $\delta \downarrow 0$, to obtain $\tilde{h}(\mu)
\geq h(\mu \mid \nu)$ via .
For the reverse inequality, we may without loss of generality assume that $h(\mu \mid \nu)
= \int_E \varphi \log \varphi\, d\nu < \infty$, where $\varphi=d\mu/d\nu \geq 0$ is in $L^1(\nu)$. Then for any $\delta > 0$ we can find a $\widetilde{\varphi} \geq 0$ in $C_b(E) \cap L^1(\nu)$ such that $\int_E \widetilde{\varphi}\,d\nu = 1$ and $$\begin{aligned}
\int_E \big| \widetilde{\varphi} - \varphi \big| \, d\nu < \delta, \;\;
\int_E \big| \widetilde{\varphi}\log\widetilde{\varphi} -
\varphi\log\varphi \big| \, d\nu < \delta.\end{aligned}$$ Note that $\lim_{n\to\infty} \int_E \widetilde{\varphi}\,d\nu_n = 1$, and let $\widetilde{\varphi}_n = \widetilde{\varphi}/\int \widetilde{\varphi}\,d\nu_n$ and $\mu_n = \widetilde{\varphi}_n \nu_n$. Then, for $g \in C_b(E)$, $$\begin{aligned}
\Big| \int_E g \, d\mu_n - \int_E g \,d\mu \Big|
&= \Big| \frac1{\int_E \widetilde{\varphi} \, d\nu_n}
\int_E g \widetilde{\varphi}\, d\nu_n - \int_E g\varphi \,d\nu \Big| \\
&\leq \Big| \frac{1}{\int_E \widetilde{\varphi}\, d\nu_n} - 1 \Big|
\, \| g \widetilde{\varphi} \|_\infty
+ \Big| \int_E g \widetilde{\varphi}\, d\nu_n - \int_E g \widetilde{\varphi}\, d\nu \Big|
+ \Big| \int_E g (\widetilde{\varphi} - \varphi) \,d\nu\Big|,
\end{aligned}$$ which can be made arbitrarily small by choosing $\delta$ small enough and $n$ large enough. In particular, for any $\varepsilon > 0$ we can choose $\delta$, $\widetilde{\varphi}$ and $n_0$ such that $\mu_n \in B_\varepsilon(\mu)$ for $n \geq n_0$. Hence $$\begin{aligned}
\limsup_{n\to\infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n)
&\leq \limsup_{n\to\infty} h(\mu_n \mid \nu_n)\\
&= \limsup_{n\to\infty} \int_E \widetilde{\varphi}_n \log \widetilde{\varphi}_n \, d\nu_n
= \int_E \widetilde{\varphi} \log \widetilde{\varphi} \, d\nu \leq h(\mu \mid \nu) + \delta,
\end{aligned}$$ and letting $\delta \downarrow 0$ we $\tilde{h}(\mu) \leq h(\mu \mid \nu)$.
\(2) Recall that for $R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ and $Q$ a product measure, $$\lim_{N\to\infty} \frac1N h\left( \pi_N R \mid \pi_N Q\right)
= H(R \mid Q) = \sup_{N\in{\mathbb{N}}} \frac1N h\left( \pi_N R \mid \pi_N Q\right).$$ Denote the expression in the right-hand side of (\[eq:Hregularised1\]) by $\tilde{H}(R)$. Fix $N\in{\mathbb{N}}$. Since for each $\varepsilon>0$, we have $B_{\varepsilon'}(R) \subset
\{ R'\colon\, \pi_N R' \in B_{\varepsilon}(\pi_N R) \}$ for $\varepsilon'$ sufficiently small we also have $$\begin{aligned}
\lim_{\varepsilon' \downarrow 0} \limsup_{n\to\infty}
\inf_{R' \in B_{\varepsilon'}(R)} H(R' \mid Q_n)
\ge \limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\pi_N R)}
\frac1N h(\mu' \mid \pi_N Q_n). \end{aligned}$$ Let $\varepsilon \downarrow 0$ and use Part (1), to see that $\tilde{H}(R) \ge \frac1N
h(\pi_N R \mid \pi_N Q)$ for any $N$. Hence also $\tilde{H}(R) \geq H(R \mid Q)$.
For the reverse inequality, we may w.l.o.g. assume that $H(R \mid Q) < \infty$. Fix $\varepsilon>0$ and $\delta>0$. There is an $N \in {\mathbb{N}}$ such that $H(R \mid Q) \leq \frac1N h \big( \pi_N R \mid \pi_N Q \big) + \delta$, and since $\pi_N R \ll \pi_N Q=q^ {\otimes N}$ we can find a continuous, bounded and uniformly positive function $f_N\colon\,E^N \to [0,\infty)$ such that $\int_E f_N \,
dq^{\otimes N} = 1$, $\int_E f_N \log f_N \, dq^{\otimes N} \leq h \big( \pi_N R \mid
\pi_N Q \big) + N\delta$ and $\tilde{R}_N \in B_{\varepsilon/2}(R)$, where $\tilde{R}_N
= {\mathsf{sblock}}_N\big(f_N\,q^{\otimes N}\big) \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ (see Lemma \[lemma:HblockmeasQ\]). By , we have $$H(\tilde{R}_N \mid Q)
= \frac1N \int_E f_N \log f_N \, dq^{\otimes N}.$$ Now put $f_{N,n} = f_N/\int_E f_N\,q_n^{\otimes N}$, and define $\tilde{R}_{N,n} = {\mathsf{sblock}}_N
\big(f_{N,n} \, q^{\otimes N}\big)$ as the “stationary version” of $(f_{N,n}\,q_n^{\otimes
N})^{\otimes {\mathbb{N}}}$. In particular, $H(\tilde{R}_{N,n} \mid Q_n) = \frac1N \int f_{N,n} \log
f_{N,n} \, dq_n^{\otimes N}$. Since $f_N$ is continuous, we have $\tilde{R}_{N,n} \in
B_{\varepsilon}(R)$ and $\int_E f_{N,n} \log f_{N,n}\,dq_n^{\otimes N} \le H(R \mid Q)
+ 3\delta$ for $n$ large enough. Hence $$\limsup_{n\to\infty}
\inf_{R' \in B_{\varepsilon}(R)} H(R' \mid Q_n)
\le \limsup_{n\to\infty} H(\tilde{R}_{N,n} \mid Q_n)
\le H(R \mid Q) + 4\delta.$$ Now let $\delta \downarrow 0$ followed by $\lim_{\varepsilon\downarrow 0}$ to conclude the proof.
### Existence of sharp coarse-graining approximations to the quenched rate function
The following lemma was used in the proof of Lemma \[lemma:Iquetrregularised\].
\[lem:cg.2lev.blockapprox\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(F^{\mathbb{N}})$ with $H( Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$. There exist a sequence $(h_n)_{n\in{\mathbb{N}}}$ with $h_n>0$ and $\lim_{n\to\infty} h_n = 0$ and a sequence $(Q'_n)_{n\in{\mathbb{N}}}$ with $Q'_n \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}_{h_n}^{\mathbb{N}})$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\limsup_{n\to\infty} I^{\mathrm{que}}_{h_n}(Q'_n)
\leq I^{\mathrm{que}}(Q)$. The same holds with $F$ replaced by $F_{0,{{\rm tr}}}$ and $\widetilde{E}_{h_n}$ replaced by $\widetilde{E}_{h_n,{{\rm tr}}}$.
Recall the definition of $\lceil Q \rceil_h$ in Step 2 of the proof of part (1) of Proposition \[prop:Ique.tr.cont\] (see page ). For any $N\in{\mathbb{N}}$, we have $$\begin{aligned}
\label{eq:anyway1}
h(\pi_N \lceil Q \rceil_h \mid \pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)
\leq h(\pi_N Q \mid \pi_N Q_{\rho,{\mathscr{W}}}) \leq N \, H( Q \mid Q_{\rho,{\mathscr{W}}}).\end{aligned}$$ The second inequality follows from . For the first inequality, use the fact that the construction of $\lceil Q \rceil_h$ can be implemented as a deterministic function of the pair of random variables $(Y,U)$, together with the fact that relative entropy can only decrease when image measures are taken. Write $\hat{\tau}_i = (\tilde{T}_i-\tilde{T}_{i-1})/h$, $i \in {\mathbb{N}}$. Since letters both under $\lceil Q_{\rho,{\mathscr{W}}} \rceil_h$ and under $Q_{\lceil \rho \rceil_h,{\mathscr{W}}}$ are constructed from a Brownian path that is independent of the word lengths, we have (recall \[def:rho.h.trunc\]) $$\begin{aligned}
{1}(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N) \,
\frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}}
= \frac{(\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big)}
{\prod_{\ell=1}^N \lceil \rho \rceil_h(h k_\ell)}\end{aligned}$$ with $$\begin{aligned}
& (\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big) \notag \\
& = \int_{[0,1]} du \,
\int_0^\infty \bar{\rho}(t_1) dt_1
\int_{t_1}^\infty \bar{\rho}(t_2-t_1) d(t_2-t_1) \cdots
\int_{t_{N-1}}^\infty \bar{\rho}((t_N-t_{N-1})) d(t_N-t_{N-1}) \notag\\
&\qquad\qquad \times \prod_{\ell=1}^N {1}_{(h(\bar{k}_\ell-1+u), h(\bar{k}_\ell+u)]}(t_\ell),\end{aligned}$$ where $\bar{k}_\ell = k_1+\cdots+k_\ell$. Thus, by (\[eq:Vbarrhodef\]–\[ass:rhobar.reg1\]), $$\begin{aligned}
\label{eq:anyway2}
\sup_{N \in {\mathbb{N}}} \frac1N E_{\lceil Q \rceil_h}\left[ \Big| \log
\frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}}\Big|\right] \leq r_Q(h)\end{aligned}$$ with $$\begin{aligned}
r_Q(h) = \eta_n \lceil Q \rceil_h(\hat\tau_1 \in \bar{A}_n)
+ \eta_0 \lceil Q \rceil_h(\hat\tau_1 \not\in \bar{A}_n), \qquad h=2^{-n},\end{aligned}$$ where $\bar{A}_n \subset (s_*,\infty)$ is the set obtained from $A_n$ by removing pieces of length $2^{-n}$ from its edges (i.e., $\bar{A}_n$ is the $2^{-n}$-interior of $A_n$). But $\lim_{n\to\infty} \lceil Q \rceil_{2^{-n}}(\hat\tau_1 \not\in \bar{A}_n)=0$ because $A_n$ fills up $(s_*,\infty)$ as $n\to\infty$. Since $\lim_{n\to\infty} \eta_n=0$, we get $\lim_{h \downarrow 0}
r_Q(h) = 0$. Combining (\[eq:anyway1\]–\[eq:anyway2\]), we obtain that $$\begin{aligned}
H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}})
= \sup_{N \in {\mathbb{N}}}
\frac1N h(\pi_N \lceil Q \rceil_h \mid \pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}})
\leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + r_Q(h)\end{aligned}$$ and, finally, $$\begin{aligned}
&\limsup_{h \downarrow 0}
H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}})
+ (\alpha-1) m_{\lceil Q \rceil_h} H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) \notag\\
&\qquad \leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$
The truncated case, where $F$ is replaced by $F_{0,{{\rm tr}}}$, etc., can be handled analogously.
### Approximation of $\Psi_Q$
The approximation in is stronger than just weak convergence.
\[lemma:PsiQ:TVlim\] For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned}
\label{eq:PsiQTVlimit}
\lim_{T\to\infty} \sup_{A \subset C[0,\infty)\; \text{measurable}}
\bigg| \Psi_Q(A) - \frac{1}{T}
\int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| = 0,\end{aligned}$$ i.e., the convergence in holds in total variation.
Note that, by shift-invariance, $$\begin{aligned}
\Psi_Q(A) = \frac{1}{N m_Q}
{\mathbb{E}}_Q \left[ \int_0^{\tau_N} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right], \qquad N\in{\mathbb{N}}.\end{aligned}$$ Suppose that $Q$ is also ergodic. Then $\lim_{N\to\infty} \tau_N/N = m_Q$ $Q$-a.s. and in $L^1(Q)$. Hence, for given $\varepsilon > 0$ we can find a $T_0(\varepsilon)$ such that, for $T \geq T_0(\varepsilon)$, $$\begin{aligned}
\label{eq:Qerg.cons1}
{\mathbb{E}}_Q\Big[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \Big]
+ \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ where $N(T) = \lceil T/m_Q \rceil$. Thus, for $T \geq T_0(\varepsilon)$ and any measurable $A \subset C[0,\infty)$, we have $$\begin{aligned}
& \bigg| \Psi_Q(A) - \frac{1}{T}
\int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| \notag \\
& \leq \frac{1}{m_Q N(T)} \bigg|
{\mathbb{E}}_Q\left[ \int_0^{\tau_{N(T)}} {1}_A\big( \theta^s \kappa(Y) \big) \, ds
- \int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right] \bigg| \notag \\
& \qquad + \bigg| \Big(\frac{1}{m_Q N(T)}-\frac1T \Big)
\int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \bigg|
\leq {\mathbb{E}}_Q\left[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \right]
+ \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ i.e., holds.
If $Q$ is not ergodic, then use the ergodic decomposition $$Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} Q' \, W_Q(dQ')$$ and note that $$m_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} m_{Q'} \, W_Q(dQ'),
\quad \Psi_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} \frac{m_{Q'}}{m_Q}
\, \Psi_{Q'} \, W_Q(dQ')$$ (see also [@BiGrdHo10 Section 6]). We can choose $N(T)$ so large that the set of $Q'$s for which holds (with $Q$ replaced by $Q'$) has $W_Q$-measure arbitrarily close to $1$.
Continuity of the “letter part” of the rate function under truncation: discrete-time {#subs:towards.Ique.tr.cont}
------------------------------------------------------------------------------------
In this section we consider a discrete-time scenario as in [@BiGrdHo10]: $\rho \in \mathcal{P}({\mathbb{N}})$, $E$ is a Polish space, $\nu \in \mathcal{P}(E)$, the sequence of words $(Y^{(i)})_{i\in{\mathbb{N}}}$ with discrete lengths has reference law $q_{\rho,\nu}^{\otimes{\mathbb{N}}}$ with $q_{\rho,\nu}$ as in [@BiGrdHo10 Eq. (1.4)]. The following lemma extends [@BiGrdHo10 Lemma A.1] to Polish spaces (in [@BiGrdHo10] it was only proved and used for finite $E$, and without explicit control of the error term). Via coarse-graining, this lemma was used in the proof of Proposition \[prop:Ique.tr.cont\].
\[lem:trcontinuous\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}^{\mathbb{N}})$ and $0 < \varepsilon < \tfrac12$. Let ${{\rm tr}}\in {\mathbb{N}}$ be so large that $$\begin{aligned}
\label{eq:mQtr.qb}
{\mathbb{E}}_Q\Big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \Big] < \frac{\varepsilon}{2} m_Q.\end{aligned}$$ Then $$\begin{aligned}
\label{eq:HPsiQtr.qb}
(1-\varepsilon) \big( H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})
+ b(\varepsilon) \big) \leq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon) = -2\varepsilon + [\varepsilon \log\varepsilon
+ (1-\varepsilon) \log (1-\varepsilon)]/(1-\varepsilon) $, satisfying $\lim_{\varepsilon \downarrow 0} b(\varepsilon)=0$. In particular, $$\begin{aligned}
\label{eq:HPsiQtrlim.disc}
\lim_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})
= H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}}). \end{aligned}$$
We can assume w.l.o.g. that $H(\Psi_Q \mid \nu^{\otimes{\mathbb{N}}}) < \infty$ for otherwise is trivial and follows from lower-semicontinuity of specific relative entropy.
First, assume that $Q$ is ergodic, then $\Psi_Q$ is ergodic as well (see [@Bi08 Remark 5]). For $\Psi \in \mathcal{P}^{\mathrm{erg}}(E^{\mathbb{N}})$ and $\delta \in (0,1)$, $$\begin{aligned}
\label{eq:HPsinu.typset0}
H(\Psi \mid \nu^{\otimes {\mathbb{N}}})
& = \lim_{L\to\infty} -\frac{1}{L} \log \Big( \inf\big\{ \nu^{\otimes L}(B)
\colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \big\} \Big), \\
\label{eq:HPsinu.typset1}
& = \lim_{L\to\infty} \sup\Big\{ -\frac{1}{L} \log \nu^{\otimes L}(B) \,
\colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \Big\}.\end{aligned}$$ This replaces the asymptotics of the covering number and its relation to specific entropy for ergodic measures on discrete shift spaces that was employed in the proof of [@BiGrdHo10 Lemma A.1], and can be deduced with bare hands from the Shannon-McMillan-Breiman theorem. Indeed, asymptotically optimal $B$’s are of the form $\{ \frac1L \log \frac{d\pi_L\Psi}{d\nu^{\otimes L}} \in H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) \pm \epsilon\}$: Put $f_L = \frac{d\pi_L \Psi}{d\nu^{\otimes L}}$ and set $B_L = \{ \frac1L \log f_L > H(\Psi \mid
\nu^{\otimes {\mathbb{N}}}) - \epsilon \}$. Then $(\pi_L \Psi)(B_L) \to 1$ by the Shannon-McMillan-Breiman, and $\nu^{\otimes L}(B_L) = \int_{B_L} \frac1{f_L} d\pi_L\Psi \leq \exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}})
- \epsilon)]$, i.e., the right-hand side of is $\geq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$. For the reverse inequality, consider any $B \subset E^L$ with $(\pi_L\Psi)(B) \geq \tfrac12$, say. Set $B'=B \cap \{ \frac1L \log f_L < H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon\}$. Then $\pi_L\Psi(B') \geq
\tfrac13$ for $L$ large enough and $\nu^{\otimes L}(B) \geq \nu^{\otimes L}(B') \geq
\exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon)] \pi_L\Psi(B')$. Hence the right-hand side of is also $\leq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$.
To check , fix $\varepsilon>0$. For $L$ sufficiently large, we construct a set $B_L \subset E^L$ such that $\pi_L \Psi_Q(B_L) \geq \tfrac12$ and $\nu^{\otimes L}(B_L) \leq
\exp[ - L(1-\varepsilon)(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}))]$, i.e., $$\begin{aligned}
-\frac1L \log \nu^{\otimes L}(B_L) \geq
(1-\varepsilon) \big[ H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) + b_L(\varepsilon) \big], \end{aligned}$$ where $\lim_{L\to\infty} b_L(\varepsilon) = b(\varepsilon)$. Via applied to $\Psi=\Psi_Q$, this yields .
To construct the sets $B_L$, we proceed as follows. Put $N= \lceil (1+2\varepsilon) L/m_Q \rceil$. By the ergodicity of $Q$ (see [@BiGrdHo10 Section 3.1] for analogous arguments), we can find a set $A \subset \widetilde{E}^N$ such that $$\begin{aligned}
&\forall\, (y^{(1)},\dots,y^{(N)}) \in A \colon\, \notag \\
\label{eq:BL.prop.1}
& \hspace{2em} | \kappa(y^{(1)},\dots,y^{(N)}) | \geq L(1+\varepsilon), \;\;
|y^{(1)}| \leq {{\rm tr}}, \;\; \sum_{i=1}^N (|y^{(i)}|-{{\rm tr}})_+ < \varepsilon L, \\
\label{eq:BL.prop.2}
& \hspace{2em} {\mathbb{E}}_Q\Big[ |Y^{(1)}| {1}_A(Y^{(1)},\dots,Y^{(N)}) \big]
\geq (1-\varepsilon) m_Q, \end{aligned}$$ and the set $$\begin{aligned}
B'_L = B'_L(A) &= \Big\{ \pi_L \big( \theta^i \kappa([y^{(1)}]_{{\rm tr}},\dots,[y^{(N)}]_{{\rm tr}})\big)\colon \,\notag\\
&\qquad (y^{(1)},\dots,y^{(N)}) \in A, i=0,1,\dots,|y^{(1)}|-1 \Big\} \subset E^L \end{aligned}$$ satisfies $$\begin{aligned}
\pi_L\Psi_{[Q]_{{\rm tr}}}(B'_L) \geq \frac12, \quad
\nu^{\otimes \lceil L(1-\varepsilon) \rceil}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L)
\leq \exp\big[ -L(1-\varepsilon)
\big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big].\end{aligned}$$ Here, use in (\[eq:BL.prop.1\]–\[eq:BL.prop.2\]), and note that $N\big(1-\tfrac{\varepsilon}{2}\big)m_Q \sim (1+2\varepsilon)\big(1-\tfrac{\varepsilon}{2}\big)L
\geq (1+\varepsilon) L$ and $N \tfrac{\varepsilon}{2} m_Q \sim (1+2\varepsilon)\tfrac{\varepsilon}{2}
L < \varepsilon L$ as $L\to\infty$.
For $I \subset \{1,\dots,L\}$, $x \in E^L$ and $y \in E^{|I|}$, write $\mathsf{ins}_I(x; y) \in E^{L+|I|}$ for the word of length $L+|I|$ consisting of the letters from $y$ at index positions in $I$ and the letters from $x$ at index positions not in $I$, with the order of letters preserved within $x$ and within $y$ (the word $y$ is inserted in $x$ at the positions in $I$). Put $$\begin{aligned}
B_L = \pi_L\Big( \big\{ \mathsf{ins}_I(x; y) \colon \,
x \in B'_L, I\subset \{1,\dots,L\}, |I| \leq \varepsilon L, y \in E^{|I|} \big\} \Big).\end{aligned}$$ Then $\pi_L\Psi_Q(B_L) \geq \frac12$ by construction. Furthermore, for fixed $I\subset \{1,\dots,L\}$ with $|I|=k \le \varepsilon L$, $$\begin{aligned}
\nu^{\otimes L} \Big( \pi_L\big( \big\{ \mathsf{ins}_I(x; y)\colon \, x \in B'_L, y \in E^k \big\} \big)\Big)
= \nu^{\otimes L} \big( \pi_{L-k}(B'_L) \big)
\leq \nu^{\otimes \lceil L(1-\varepsilon \rceil]}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L),\end{aligned}$$ and hence $$\begin{aligned}
\nu^{\otimes L}(B_L) & \leq [\varepsilon L] {L \choose [\varepsilon L]}
\exp\big[ -L(1-\varepsilon) \big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big] \notag \\
& = \exp\big[ - L(1-\varepsilon) \big(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})\big) \big]\end{aligned}$$ with $b_L(\varepsilon) = - \frac{1}{(1-\varepsilon) L}(\log [\varepsilon L] + \log {L \choose [\varepsilon L]})
- 2\varepsilon$, which satisfies $\lim_{\varepsilon \downarrow 0} b_L(\varepsilon)= b(\varepsilon)$.
It remains to prove . Since ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$, we have $\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \geq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$, while the reverse inequality $ \limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \leq
H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$ follows from (\[eq:mQtr.qb\]–\[eq:HPsiQtr.qb\]) and the fact that $\lim_{{{\rm tr}}\to\infty} {\mathbb{E}}_Q[( |Y^{(1)}|-{{\rm tr}})_+] = m_Q$ by dominated convergence.
For non-ergodic $Q$, decompose as in [@BiGrdHo10 Eqs.(6.1)–(6.3)], use the above argument on each of the ergodic components, and use the fact that specific relative entropy is affine.
Existence of specific relative entropy {#contrelentr}
======================================
In this section we prove . For technical reasons, we consider the two-sided scenario. The argument is standard, but the fact that time is continuous requires us to take care.
Let $\Omega = \tilde{C}({\mathbb{R}})$ be the set of continuous functions $\omega\colon\, {\mathbb{R}}\to {\mathbb{R}}$ with $\omega(0)=0$, which is a Polish space e.g. via the metric $d(\omega, \omega') =
\int_{\mathbb{R}}e^{-|t|} \big(|\omega(t)-\omega'(t)| \wedge 1\big) dt$. The shifts on $\Omega$ are $\theta^t \omega(\cdot) = \omega(\cdot+t)-\omega(t)$. A probability measure $\Psi$ on $\Omega$ has stationary increments when $\Psi = \Psi \circ (\theta^t)^{-1}$ for all $t \in {\mathbb{R}}$. For an interval $I \subset {\mathbb{R}}$ denote $\mathcal{F}_I = \sigma(\omega(t)-\omega(s)\colon\,
s,t \in I)$. $\Psi_I$ denotes $\Psi$ restricted to $\mathcal{F}_I$. Write ${\mathscr{W}}$ for the Wiener measure on $\Omega$, i.e., the law of a (two-sided) Brownian motion.
Let $\Psi \in \mathcal{P}(\Omega)$ with stationary increments be given and assume that $h(\Psi_{[0,T]} \mid {\mathscr{W}}_{[0,T]}) < \infty$ for all $T>0$. To verify , we imitate well-known arguments from the discrete-time setup (see e.g. Ellis [@El85 Section IX.2]).
For $I_1$, $I_2$ disjoint intervals in ${\mathbb{R}}$, denote by $\kappa^\Psi_{I_1, I_2}\colon\,
\Omega \times \mathcal{F}_{I_2} \to [0,1]$ a regular version of the conditional law of (the increments of) $\Psi$ on $I_2$, given the increments in $I_1$, i.e., for fixed $\omega$, $\kappa^\Psi_{I_1, I_2}(\omega, \cdot)$ is a probability measure on $\mathcal{F}_{I_2}$, for fixed $A \in \mathcal{F}_{I_2}$, $\kappa^\Psi_{I_1, I_2}(\cdot, A)$ is an $\mathcal{F}_{I_1}$-measurable function, and $\kappa^\Psi_{I_1, I_2}(\omega, A)$ is a version of ${\mathbb{E}}_{\Psi}[{1}_A | \mathcal{F}_{I_1}]$. When $I_1=\emptyset$, $\kappa^\Psi_{\emptyset, I_2}(\omega, A) = \Psi_{I_2}(A)$. Similarly, define $\kappa^{\mathscr{W}}_{I_1, I_2}$ (which is simply $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, A) = {\mathscr{W}}_{I_2}(A)$ by the independence of the Brownian increments).
Put $$\begin{aligned}
a_{I_1,I_2} = \int_\Omega \Psi(d\omega_1)
\int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \,
\log\left[ \frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)}
{d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right], \end{aligned}$$ the expected relative entropy of the conditional distribution under $\Psi$ on $\mathcal{F}_{I_2}$ given $\mathcal{F}_{I_1}$ w.r.t. Wiener measure on $\mathcal{F}_{I_2}$). We have $a_{I_1,I_2}
< \infty$ for bounded intervals, because of the assumption of finite relative entropy of $\Psi$ w.r.t. ${\mathscr{W}}$ on compact time intervals. By stationarity, $a_{I_1,I_2}=a_{t+I_1,t+I_2}$ for any $t$.
Let $I_1' \subset I_1$, note that $\kappa^\Psi_{I_1, I_2}(\omega, \cdot) \ll \kappa^\Psi_{I_1', I_2}
(\omega, \cdot)$ for $\Psi$-a.e. $\omega$, and $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, \cdot) =
\kappa^{\mathscr{W}}_{I_1', I_2}(\omega, \cdot) = {\mathscr{W}}_{I_2}(\cdot)$. By the consistency property of conditional distributions, we have $$\begin{aligned}
a_{I_1',I_2} = \int_\Omega \Psi(d\omega_1) \int_\Omega
\kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2)
\log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)}
{d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right].\end{aligned}$$ Indeed, $$\int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2)
f(\omega_1,\omega_2)
= \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I'_1, I_2}(\omega_1, d\omega_2)
f(\omega_1,\omega_2)$$ for any function $f(\omega_1,\omega_2)$ that is $\mathcal{F}_{I_1'} \otimes
\mathcal{F}_{{\mathbb{R}}}$-measurable. Hence $$\begin{aligned}
&a_{I_1,I_2} - a_{I_1',I_2} \\
& = \int_\Omega \Psi(d\omega_1) \int_\Omega
\kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2)
\bigg( \log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)}
{d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right]
- \log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)}
{d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \bigg) \notag \\
\label{eq:hdeccont1}
& = \int_\Omega \Psi(d\omega_1)
\int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \,
\log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)}
{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \geq 0\end{aligned}$$ because the inner integral is $h( \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot) \mid \kappa^\Psi_{I_1', I_2}
(\omega_1, \cdot)) \geq 0$. Choosing $I_1'=\emptyset$, , we get $a_{I_1, I_2}
\geq a_{\emptyset, I_2} = h( \Psi_{I_2} \mid {\mathscr{W}}_{I_2})$.
Observe $$\begin{aligned}
\frac{d \Psi_{(0,s+t]}}{d {\mathscr{W}}_{(0,s+t]}}(\omega)
= \frac{d \Psi_{(0,t]}}{d {\mathscr{W}}_{(0,t]}}(\omega)
\, \frac{d \kappa^\Psi_{(0,t],(t,s+t]}(\omega, \cdot)}
{d \kappa^{\mathscr{W}}_{(0,t],(t,s+t]}(\omega, \cdot)}(\omega) \quad \Psi_{(0,s+t]}-\text{a.s.},\end{aligned}$$ take logarithms and integrate w.r.t. $\Psi$ (using consistency of conditional expectation on the right-hand side), to obtain $$\begin{aligned}
h\big( \Psi_{(0,s+t]} \mid {\mathscr{W}}_{(0,s+t]} \big) = h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big)
+ a_{(0,t], (t,s+t]}
\geq h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big) + h\big( \Psi_{(0,s]} \mid {\mathscr{W}}_{(0,s]} \big). \end{aligned}$$ Thus, the function $(0,\infty) \ni t \mapsto h( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]})$ is super-additive, and follows from Fekete’s lemma.
Under $\kappa^\Psi_{(-\infty,0], (0,h]}$, the coordinate process will be a Brownian motion with a (possibly complicated) drift process $U_t = \int_0^t u_s\, ds$, where $(u_t)_{t \geq 0}$ can be chosen adapted, and $${\mathbb{E}}_\Psi\big[ h(\kappa^\Psi_{(-\infty,0], (0,h]} \mid {\mathscr{W}}_{(0,h]}) \big] = {\mathbb{E}}_\Psi\big[{\textstyle \int_0^h} u_s^2 \,ds \big]$$ (see Föllmer [@Foe86]).
[AAA]{}
M. Birkner, Conditional large deviations for a sequence of words, Stoch. Proc. Appl. 118 (2008), 703–729.
M. Birkner, A. Greven and F. den Hollander, Quenched large deviation principle for words in a letter sequence, Probab. Theory Relat. Fields 148 (2010), 403–456.
M. Birkner and F. den Hollander, A variational view on the Brownian copolymer, manuscript in preparation.
E. Bolthausen and F. den Hollander, Localization transition for a polymer near an interface, Ann. Probab. 25 (1997), 1334–1366.
F. Caravenna and G. Giacomin, The weak coupling limit of disordered copolymer models, Ann. Probab. 38 (2010), 2322–2378.
F. Caravenna, G. Giacomin and F.L. Toninelli, Copolymers at selective interfaces: settled issues and open problems, In: *Probability in Complex Physical Systems. In honour of Erwin Bolthausen and Jürgen Gärtner* (eds. J.-D. Deuschel, B. Gentz, W. König, M. von Renesse, M. Scheutzow, U. Schmock), Springer Proceedings in Mathematics 11, 2012, pp. 289–312.
A. Dembo and O. Zeitouni, *Large Deviations Techniques and Applications* (2nd. Ed.), Springer, 1998.
R. Ellis, *Entropy, Large Deviations, and Statistical Mechanics*, Springer, 1985.
S. Ethier and T. Kurtz, *Markov Processes: Characterization and Convergence*, Wiley, 1986.
H. Föllmer, Time reversal on Wiener space, Lecture Notes in Math. 1158, Springer, 1986, pp. 119–129.
H. Föllmer, Random fields and diffusion processes, in: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203, Lecture Notes in Math. 1362, Springer, 1988.
R.M. Gray, *Entropy and Information Theory*, Springer, 1991.\
<http://ee.stanford.edu/~gray/it.html>
C. Kipnis and C. Landim, *Scaling Limits of Interacting Particle Systems*, Springer, 1999.
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Q:
SQL Server 2000: search through out database
Some how some records in my table are getting updated with value of xyz in a certain column. Out of hundred of stored procedures, functions, triggers, how can I determine which code is doing this action. Is there a way to search through the database some how through each and every script of the code?
Please help.
A:
One approach is to check syscomments
Contains entries for each view, rule,
default, trigger, CHECK constraint,
DEFAULT constraint, and stored
procedure within the database. The
text column contains the original SQL
definition statements..
e.g. select text from syscomments
If you are having trouble finding that literal string, the values could be coming from a table, or they could be being concatenated within a routine.
Try this
Select text from syscomments
where CharIndex('x', text) > 0
and CharIndex('y', text) > 0
and CharIndex('z', text) > 0
That might help you either find the right routine, or further indicate that the values are coming from a table.
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Akihisa Nagashima
is a member of the House of Representatives of Japan representing the Tokyo's 21st district, as well as a visiting professor at Chuo University's Graduate School of Public Studies. He served as the Parliamentary Vice Minister of Defense in the Kan Cabinet.
From 1993 to 1995, he was a visiting scholar at Vanderbilt University, Nashville, Tennessee, before becoming a research associate in Asian Security Studies in 1997, and an Adjunct Senior Fellow in Asia Studies at the Council on Foreign Relations, Washington, D.C., in 1999. From 2000 to 2001, he was a visiting scholar at the Edwin O. Reischauer Center for East Asian Studies at the Johns Hopkins University School of Advanced International Studies (SAIS), Washington, D.C. After coming back to Japan, he taught as a lecturer at Keio University's Graduate School of Law from 2003 to 2007.
Nagashima received his B.A. in Law in 1984, his B.A. in Government in 1986, and his Master of Laws (LL.M) from Keio University in 1988. He received his M.A. from Johns Hopkins SAIS in 1997. He was born on February 17, 1962, in Yokohama-City, Kanagawa Prefecture, Japan.
Political career
He started his political career with the Democratic Party of Japan (DPJ). During his time as an opposition legislator at the National Diet of Japan, he has served as the Senior Director of the Committee on National Security, Director of the Committee on Foreign Affairs, Special Committee on North Korean Abductions and Other Issues, as well as a member of the Committee on Education, Sports, Science and Technology, the Special Committee on Iraq and Terrorism and the Special Committee on Responses of Armed Attacks. From 2003 to 2004, he served as the Deputy Director-General of the Cultural and Organizations Department of the DPJ, as well as the Next Vice-Minister of Defense before becoming the Next Minister of Defense from 2005 to 2006. Later he has served as the Vice-Chair of the Diet Affairs Committee, the Policy Research Committee, and Deputy Secretary General of the DPJ. He left the DP in April 2017 due to a disagreement with the party's cooperation with the JCP. Prior to the 2017 general election, he participated in the foundation of the Party of Hope. When Hope merged with the Democratic Party in May 2018 to form the Democratic Party for the People, Nagashima decided not to join the new party and became an independent member instead.
Formerly affiliated to the openly revisionist lobby Nippon Kaigi, Nagashima contributed, with Yoshiko Sakurai, Eriko Sanya, and Masahiro Akiyama, to a forum on the Constitution about security, independence, and the article 9 in their journal in July 2009. In September 2015, Nagashima announced his withdrawal from Nippon Kaigi.
References
Category:1962 births
Category:Living people
Category:People from Yokohama
Category:Keio University alumni
Category:Members of Nippon Kaigi
Category:Johns Hopkins University alumni
Category:Members of the House of Representatives (Japan)
Category:Democratic Party of Japan politicians
Category:21st-century Japanese politicians
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Tundra and canyons in Tuktut Nogait National Park. (Photos by Charla Jones for The Globe and Mail)
Mysterious North
Few places in the world can push and pull you from agony to ecstasy
Looking at a map of Canada, most of us live huddled around the border of the United States; above us, there’s all this space, this land – sometimes referred to as Canada’s Great North. It remains a mystery to most of us; a romanticized concept of what it means to be Canadian. I don’t think I’m alone in my longing or curiosity to visit.
Glenn Gould put in this way in The Idea of North, a 1967 radio documentary. “I’ve read about it, written about it, and even pulled up my parka once and gone there. Yet like all but a very few Canadians, I’ve had no real experience of the North. I’ve remained, of necessity, an outsider. And the North has remained for me, a convenient place to dream about, spin tall tales about, and, in the end, avoid.”
Last summer I had the chance to visit Tuktut Nogait National Park, which is located 170 kilometres north of the Arctic Circle in the Northwest Territories. It is the country’s least visited national park, and in early August of 2014, I was the eighth and last visitor of the year.
A glimpse at Canada’s least-visited national park
Brightcove player
In preparation for my trip, Eric Baron, the park’s visitor experience manager, said: “Be prepared. It’s cold up here.” I was soon to learn just how cold August could be.
Tuktut Nogait hugs the border of Nunavut to the east and was created in 1998 to protect the calving grounds of the Bluenose-West caribou. “Tuktut nogait” means “young caribou” in Inuvialuktun. Estimates of the herd number vary but approximately 20,000 Bluenose-West caribou travel north to their calving grounds in the park each June.
The park is also a calving ground for the Bluenose-West caribou. 'Tuktut nogait' means 'young caribou' in Inuvialuktun. (Photo courtesy of Parks Canada)
To reach Inuvik, I flew two and half hours from Yellowknife. To reach the park, we then flew two more hours in a Twin Otter float plane from Inuvik. The plane was loaded with several barrels worth of gear. There are no facilities in the park, so everything must be brought in. Food and drink, tents for sleeping, tents for cooking, a tent for the toilet and, since nothing foreign can be left on the land, bags for human waste.
To get to the park: Some direct flights to Inuvik are available from Edmonton, Vancouver, Whitehorse and Yellowknife. Travellers can also drive the Dempster Highway to Inuvik. From Inuvik, charter a flight to the park from North-Wright Airways or purchase the six-day, self-catered base-camp trips from Parks Canada for $3,975 a person. Price includes return flight into Tuktut Nogait National Park, Parks Canada guides and the services of two Inuvialuit cultural hosts, backcountry fee and airport shuttle to and from Inuvik airport. This year’s trip runs Aug. 3 to Aug. 8. - Charla Jones
Two hours later, our floatplane landed on One Island Lake, which would be our campsite. Standing on the shore stood Ruben Green and Jonah Nakimayak, our two cultural hosts from nearby Paulatuk, wearing fur-lined coats and rain boots. They helped secure the plane to shore and greeted us warmly as we jumped off. Both had high cheekbones and brown skin weathered by a lifetime of exposure to the elements. I looked forward to hearing their stories.
We set up our tents and wrapped a bear-exclusion fence around the camp (grizzlies and black bears are resident in the park), then settled in for the night, or what the clock told us was the night since we were in a period of 24-hour daylight.
Eric was right. It was cold. I wore two hats, a sweater, long johns and two pairs of wool socks and slept in two sleeping bags, one tightened to cover the maximum amount of my face but still let me breathe. In my cocoon, in the light of night, I listened to the wind rolling across the tundra and thought of dinosaurs roaming the Arctic 240 million years ago.
The next morning we set out for a hike. I liked the people from Parks Canada, and we bantered like a close-knit family. But Jonah and Ruben were the real treasures. They sauntered across the tundra as though they were strolling through their backyard. Reuben, 58, was often quiet, but Jonah, a few months shy of 70, made jokes and rolled cigarettes. Both of them seemed peaceful. I wanted to know them better, be around them, listen to them. It was obvious they belonged here; they were a part of the ecosystem, at one with it.
Visitors sit at the edge of a canyon cliff overlooking the Hornaday river in Tuktut Nogait National Park.
There are more than 500 cultural sites in Tuktut Nogait. Only a handful have been excavated, but of those identified, the dates range from 100 to 40,000 years old. “Many Caches” is a picturesque spot overlooking the Hornaday River. As the name suggests, there were lots of 200-or-so-year-old underground caches once used by the Copper Inuit to store caribou meat. Old caribou bones lay close by and further down the site, a grave, thought to hold the body of a Copper Inuit woman. “They would bury all their belongings, even their seeds would be buried on top of them in the grave to take with them to the afterlife,” Ruben explained as he sprinkled some tobacco (a sacred plant connecting Earth to the heavens) in the air above the remains.
The following morning, I awoke sweating. The heat of the Arctic sun streamed through my orange tent as I unravelled my elaborate bedding and layers of clothing. That day T-shirts replaced parkas and clouds of mosquitoes replaced gusts of gale-force winds.
Warm or freezing, this landscape was always an intense experience. There were times when the park was more than a mere disconnection from the modern world. With more than 16,000 square kilometres of the natural world, untouched by the trappings of man, it felt like a different planet.
La Roncière Falls is a great example of natural beauty found in Tuktut Nogait National Park.
With a helicopter, the landscape opened up even more. I spent hours each day photographing Arctic beaches and waterfalls, flying through canyons, watching Bluenose-West caribou heading south along the tundra, a herd of muskox at a watering hole, a grizzly bear walking with her cub or looking at a nest of peregrine falcons in the cliffs of a canyon. I felt blessed to witness such beauty and stunned that this was my country. How could I not have known about this?
At the end of five days, I was filthy and exhausted. My face hurt from the extreme weather and my body ached. But I was elated. There are few places in the world that can push and pull you from agony to ecstasy: This is one of them.
My visit didn’t answer any existential questions about what it means to be Canadian. If anything, the North was even more mysterious, but I did feel liberated by the vastness and the full force of nature.
And I understood what Ruben meant when he said, “When we’re out here, everything is so real.”
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From the Archives
Voters wait in line to vote early Monday. Lines streched long as record numbers of people turned out to vote early.
Know where this slice of life in Arkansas is? Send along the answer to Times photographer Brian Chilson and win a prize. Once a month in this space, he'll post a shot from a relatively obscure spot in Arkansas for Times readers to identify. We also invite photographers to contribute submissions of both mystery and other pictures to our eyeonarkansas Flickr group. Write to brianchilson@arktimes.com to guess this week's photo or for more information. Last month's photo was taken on Hwy. 65 at St. Joe. The winner was Sandra Jackson.
Blood splatter could be seen in a parking lot across from Power Ultra Lounge at 220 W. Sixth St. after a Saturday morning's shooting.
Most Shared
One of the booths at this week's Ark-La-Tex Medical Cannabis Expo was hosted by the Arkansas Hemp Association, a trade group founded to promote and expand non-intoxicating industrial hemp as an agricultural crop in the state. AHA Vice President Jeremy Fisher said the first licenses to grow experimental plots of hemp in the state should be issued by the Arkansas State Plant Board next spring.
On the Buffalo National River, from Brian Cormack of the Arkansas Times' Flickr group.
Leaves cover the sidewalk along President Clinton Avenue in the
River Market district.
Chris Kingsby, president of the Arkansas State Conference of the NAACP Youth & College Division, led a rally Sunday at the state Capitol as a show of unity and solidarity with victims of the Charlottesville violence.
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---
abstract: 'We propose a method to electrically control electron spins in donor-based qubits in silicon. By taking advantage of the hyperfine coupling difference between a single-donor and a two-donor quantum dot, spin rotation can be driven by inducing an electric dipole between them and applying an alternating electric field generated by in-plane gates. These qubits can be coupled with exchange interaction controlled by top detuning gates. The qubit device can be fabricated deep in the silicon lattice with atomic precision by scanning tunneling probe technique. We have combined a large-scale full band atomistic tight-binding modeling approach with a time-dependent effective Hamiltonian description, providing a design with quantitative guidelines.'
author:
- 'Yu Wang\*'
- 'Chin-Yi Chen'
- Gerhard Klimeck
- 'Michelle Y. Simmons'
- 'Rajib Rahman\*'
title: 'All-electrical control of donor-bound electron spin qubits in silicon'
---
****
Introduction
============
Donor bound electrons in silicon have been demonstrated to have long spin coherence times [@Tyryshkin; @Buch; @Hsueh], making them promising candidates for solid-state qubits. Few-donor quantum dots [@Buch; @Weber_DQD] have been patterned by scanning tunneling microscopy (STM) based lithography in silicon with atomic precision [@Schofield], an excellent fabrication technique for building a scalable quantum computer. Due to different quantum confinement and hyperfine interaction compared to single donors, few-donor quantum dots provide more flexibility in addressing [@Buch; @YW_hyperfine] and engineering the exchange coupling [@YW_exchange], which are favorable attributes for multi-qubit operations.
Controlling individual electron spins is of great importance for donor-based quantum computation. Manipulation of electron spins with integrated microwave antenna has been demonstrated in both donor and gate-defined quantum dot qubits with long coherence and high gate fidelity [@Muhonen; @Menno1]. However, it is challenging to use an ac magnetic field to realize local spin control for many qubits, a crucial requirement for multi-qubit operations in a scalable quantum computer architecture. We also know that a microwave antenna can introduce deleterious noise to the coherence of a qubit [@Muhonen]. An alternative way to spin control is to utilize an oscillating electric field, which has been demonstrated in quantum dot systems [@Kato; @Laird; @Vandersypen]. Here, the qubit is modulated periodically by the difference in Zeeman energy caused by either non-uniform electron g-factor, external magnetic field or hyperfine couplings [@Pingenot; @Tokura; @Petta]. To date, all-electrical control of spins without a microwave magnetic field in donor systems in silicon has not been demonstrated.
=3.4in
\[fig\_devices\]
In this work, we propose all-electrical control of donor-based spin qubits taking advantage of the hyperfine coupling difference between single donor and few-donor quantum dots by introducing ancillary dots (The device schematic is shown in Fig. 1(a) and (b) with “P” denoting phosphorus donors). The electron and the 3 nuclear spins (of 2P + 1P atoms) in the dashed box in Fig. 1(b) define the single-qubit operation space, and the information is encoded in the electron spin. Here the ancillary dot (1P in green) creates a difference in the local hyperfine coupling between the 1P and 2P dots due to the different number of nuclei and asymmetric quantum confinement in the dashed box [@YW_hyperfine]. The (1,0)$\leftrightarrow$(0,1) [@charge_config_note] charge transition between them can be controlled by the pair of in-plane gates G1 and G2. An electric dipole can thereby be induced by biasing the system near the (1,0)–(0,1) charge degeneracy point, where the hyperfine couplings can be modulated with an ac electric field on top of the dc electric field from the in-plane gates G1 and G2 to drive the electron spin transition.
In this proposed approach, the donor dots can be placed with atomic precision far from interfaces or surfaces using an STM based lithography technique, making them less prone to noise sources close to the interface [@Shamim; @Shamim2; @Schenkel; @Paik]. Using Coulomb confined states leads to higher valley and orbital states that are not accessible as they are typically at least 10 meV above the ground state. This results in well-isolated states of operation within which the qubit coherence can be boosted. Moreover, this scheme could be realized with existing circuitry in STM-patterned devices [@Weber_DQD; @Tom_readout] without introducing extra control components. The donor dots can be coupled via the exchange interaction (e.g. the (1,1)$\leftrightarrow$(2,0) transition in Fig. 1(b)) controlled by the surface detuning gates GT1 and GT2 (Fig. 1(a)), retaining the highly tunable exchange coupling [@YW_exchange] and allowing fully electrical two-qubit operations. This design can be potentially extended to a scalable quantum computer architecture by repeating the fundamental structure (Fig. 1(a) and (b)) according to the requirements of the large-scale architecture [@Fowler; @Hill].
Methods
=======
In the following, we describe the spin and charge evolution in the 2P-1P system by an effective Hamiltonian with quantitative details. The effective Hamiltonian can be expressed as: $$H = H_e + H_T + H_{Z_e} + H_{Z_n} + H_{HF} , \label{eq1}$$ with the basis $\vert s,i_1i_2i_3,d\rangle$ including spin and charge information, where we denote $s=\uparrow,\downarrow$ as electron spin, $i_j=\Uparrow,\Downarrow$ (j = 1,2,3. See Fig. 2(a)) as the three nuclear spins, and $d=2P,1P$ as the specific dot site (e.g. $\vert \downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle$ is the lowest energy configuration). $H_e$ reflects both the on-site energy detuning and the applied ac electric field, which is expressed as: $$H_e = \sum_{d=1P,2P}(\epsilon_d +e \vec{E_{ac}}\cdot\vec{R})\vert d\rangle\langle d\vert, \label{eq1-1}$$ where $\epsilon_d$ is the on-site detuning energy, and $d$ is the dot index (1P or 2P). $e$ is the elementary charge, $\vec{E_{ac}}$ is the ac electric field, and $\vec{R}$ is the separation between the 2P and the 1P dots. The second term represents the tunnel coupling between the (1,0) and (0,1) charge states of the 2P and the 1P dots: $$H_T = \sum_{d\neq d'} t_c\vert d\rangle\langle d'\vert, \label{eq1-2}$$ where $t_c$ denotes the tunnel coupling between the two donor dots. $d$ and $d'$ denote different dot indices. The third and the fourth term denote the Zeeman energy of the electron spin and the nuclear spins: $$H_{Z_e} = g_e\mu_B\vec{B}\cdot\vec{S}, \label{eq1-3}$$ $$H_{Z_n} = \sum_jg_n\mu_B\vec{B}\cdot\vec{I_j}, \label{eq1-4}$$ where $g_e$ and $g_n$ are the electron and nuclear g-factors respectively. $\mu_B$ and $\mu_n$ are the Bohr and the nuclear magnetons respectively. $\vec{S}$ denotes the electron spin operator, and $\vec{I_j}$ denotes the spin operator of the $j$th nucleus.The fifth term gives the hyperfine coupling between the qubit electron and the donor nuclei: $$H_{HF} = \sum_j A_j\vec{I_j}\cdot\vec{S}, \label{eq1-5}$$ where $A_j$ is the Fermi-contact hyperfine coupling between the qubit electron and the $j$th nucleus.
To provide quantitative guidelines for exploiting this proposal, it is important to obtain the parameters in $H$, the tunnel coupling $t_c$, Fermi contact hyperfine couplings $A_j$, with sufficient accuracy. Here we model the system using an atomistic tight-binding approach to obtain the Stark-shifted electron wavefunctions, from which tunnel couplings $t_c$, hyperfine couplings $A_j$ and their electric field dependency can be extracted [@YW_hyperfine; @Rahman_prl; @Park]. In the tight-binding approach, the atoms are represented by $sp^3d^5s^*$ atomic orbitals with spin-orbit coupling and nearest-neighbor interactions. Each donor is represented by a Coulomb potential screened by the dielectric constant of silicon and an on-site constant potential for the Coulomb singularity, calibrated with the P donor energy spectrum [@Ahmed]. The donor ground-state wavefunction obtained from this approach agrees well with the recent STM imaging experiments [@Salfi]. This physics-based approach automatically includes silicon conduction band valley degrees of freedom, and captures valley-orbit interaction [@Rahman_gfactor] and Stark effect in donor orbitals [@Rahman_prl]. The experimentally spin relaxation times of a single P donor and few-donor dots can also be reproduced with our approach [@Hsueh]. This provides us confidence in accurately extracting the parameters $t_c$ and $A_j$ under realistic electric fields.
Results and discussions
=======================
=3.4in
\[fig\_hyperfine\]
Fig. 2 shows the modulation of the hyperfine couplings ($A_j$) between the qubit electron spin and the nuclear spins of the P donors (2P + 1P) with electric field, with j=1,2 labeling the nuclei of the 2P dots respectively and j=3 labeling the nucleus of the 1P dot. The detail of the system in the dashed box in Fig. 1(b) is depicted as Fig. 2(a), where an atom configuration of the 2P dot is shown at the bottom. An in-plane external electric field ($E_x$) between G1 and G2 applied along the direction between the 2P dot and its ancillary 1P dot can detune the system and redistribute the electron wavefunction between them, thereby controlling the hyperfine couplings of the 2P and the 1P dots, as shown in Fig. 2(b). As can be seen, $A_3\approx 0$ at $E_x-E_0 = 5kV/m$, and $A_1, A_2\approx 0$ at $E_x-E_0 = -5kV/m$, indicating that the charge states (0,1) and (1,0) can be accessed with a small electric field range ($\sim$ 10 kV/m) with an inter-dot separation R$\approx$15 nm. At $E_x \approx E_0$, the (1,0) and (0,1) charge states are degenerate, forming hybridized bonding and anti-bonding states. As observed, the hyperfine couplings ($A_j$) have a nearly linear dependence on the electric field near the (1,0)–(0,1) charge degeneracy point ($-1 kV/m <E_x - E_0 < 1 kV/m$), indicating that the hyperfine coupling difference between the 2P and the 1P dot has a linear response to the external electric field. Near the charge degeneracy point ($E_x \approx E_0$), an electric dipole transition can be driven by an ac electric field $\varepsilon_0 sin(\omega t)$ applied from G1 and G2, thereby causing the modulations of $A_j$ with time. When the ac electric field is in resonance with the qubit energy splitting (solved from the static part of $H$ in eq. (\[eq1\])), the electron spin transition can be induced by the overall local hyperfine coupling difference [@Laird; @Tosi] between the 2P and the 1P dot. The emulation of this process in a 2P-1P system is now described by solving the time evolution based on the Hamiltonian in eq. (\[eq1\]).
=3.4in
\[fig\_rabi\]
We assume the system is initially in its lowest energy configuration and the electron is located at the 2P dot (at (1,0)), i.e. $\vert\downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle$, and assume the 2P dot configuration as shown in Fig. 2(a) with the 2P in one dot $\leq$1.5 nm apart. A static magnetic field $B_0$ is applied along the separation direction $x$. One of the possible ways to prepare the qubit in this initial state is to utilize the dynamic nuclear polarization technique [[@Petta_dynamic]]{} by repeatedly and selectively loading up-spin electrons to the donor dots [[@Morello_readout]]{} and emptying the dots to dynamically drive the nuclear spins to up orientations, then depleting the dots and loading a down-spin electron onto the dots in the end.
To achieve universal quantum gates, two-axis control of single qubits, i.e. Z-gate and X-gate, is needed. A Z-gate can be simply realized by applying the external static B-field. We will focus on the X-gate in the following. As shown in Fig. 3(a), at the beginning of the control manipulation, an adiabatic in-plane gate bias between G1 and G2 is applied to ramp the static dc electric field up to and to keep it at $\sim$$E_0$, which is followed by a continuous ac electric field between the same gates. The X-gate rotation (the manipulation of $\downarrow\rightarrow\uparrow$ or $\uparrow\rightarrow\downarrow$) is then driven by the ac electric field if $h\gamma = \Delta E$, where $\Delta E$ is the energy difference between the two qubit states and $\gamma$ is the frequency of the ac electric field. To be explicit, the qubit ground state is $\frac{1}{\sqrt{2}}(\vert\downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle - \vert\downarrow,\Uparrow\Uparrow\Uparrow,1P\rangle)$, and the qubit excited state is a dressed state involving both electron and nuclear spins, which can be expressed as $\alpha(\vert\uparrow,\Downarrow\Uparrow\Uparrow,2P\rangle - \vert\uparrow,\Downarrow\Uparrow\Uparrow,1P\rangle) +\beta(\vert\uparrow,\Uparrow\Downarrow\Uparrow,2P\rangle$ $- \vert\uparrow,\Uparrow\Downarrow\Uparrow,1P\rangle) + \zeta(\vert\uparrow,\Uparrow\Uparrow\Downarrow,2P\rangle - \vert\uparrow,\Uparrow\Uparrow\Downarrow,1P\rangle)$. A second adiabatic gate bias is then applied after the X-gate control is finished to bring the electron back to the 2P dot for spin storage. Here, we choose 2P over 1P because the electron spin relaxation time is longer in a 2P donor cluster than a single donor dot [@Hsueh].
To achieve high-fidelity X-gate operation and make the system less prone to charge noise and relaxation, we need to make appropriate choices of the external B-field and the tunnel coupling. On the one hand, with regard to the external B-field, the qubit energy splitting $\Delta$E can be expressed as $E_{Z_e} + \delta$, where $E_{Z_e}$ is the electron Zeeman splitting, and $\delta$ includes the effects of nuclear spin Zeeman energies and hyperfine couplings, which contributes to an effective magnetic field in the order of mT. To form well-defined qubit states and suppress nuclear spin flip-flop, we need $E_{Z_e} >> \delta$ to preserve the qubit state when no ac field is applied. As a result, the external B-field is required to be in the order of 0.1 T.
On the other hand, regarding the tunnel coupling, we need $\Delta$E significantly smaller than $2t_c$ in order to make the higher anti-bonding states well separated from the lower qubit states, preventing state hybridization or excitation due to environmental noise. As an example, Fig. 3(b) shows the Rabi oscillations of the qubit electron spin under the driving ac electric-field under $\Delta E<2t_c$ ($B_0$ = 0.5 T) and $\Delta E\approx 2t_c$ ($B_0$ = 1.45 T) for $R\approx$ 11.4 nm, where the magnitude of the driving ac electric field is 15 kV/m, and its frequency is $\gamma\approx$ 14 GHz which satisfies $\Delta E = h\gamma$. The ac electric field is assumed to be a single-frequency sinusoid. As shown, a full X-gate spin rotation can be achieved for $\Delta E<2t_c$, while the X-gate fidelity (defined as max($\sum\vert\langle\Psi\vert\uparrow,i_1i_2i_3,d\rangle\vert^2$)) is diminished when $\Delta E\approx 2t_c$ due to the qubit spin-up state (solid blue curve in Fig. 3(a)) is hybridized with the upper anti-bonding spin-down state (dashed green curve). As a result, $t_c$ needs to be engineered large enough. Fig. 3(c) shows $t_c$ as a function of the inter-dot separation $R$. As can be seen, $t_c$ decreases exponentially as a function of $R$, because the wavefunction overlap of the 2P and the 1P dots decreases exponentially as $R$ increases. To achieve $\Delta E<2t_c$, if we choose B = 0.1 T, $R$ needs to be larger than 15.6 nm approximately according to Fig. 3(c). If B = 0.5 T is chosen, $R$ needs to be at least 13 nm. In the following, we investigate the effect of $R$ (or $t_c$) on the qubit coherence time. Both magnetic and charge noise can lead to qubit decoherence [@Kuhlmann]. In the proposed design, magnetic noise can be suppressed to a large extent if the substrate is made of enriched Si-28. In addition, the microwave antenna that introduces magnetic noise [@Muhonen] in the traditional magnetic qubit manipulation is excluded here. As a consequence, we mainly consider the effect of the charge noise from the charge fluctuations in the nearby gates on qubit coherence. We investigate the decoherence time $T_{2}^{*}$ possibly due to different types of charge noise from a single nearby in-plane gate, e.g. G1 in Fig. 1(b). $T_{2}^{*}$ can be obtained using [@Chirolli]: $$\begin{split}
\frac{1}{T_{2}^{*}} = \frac{e^2}{\hbar^2}\vert\sum_{r_i=x,y,z}\langle\Psi_{\uparrow}\vert r_i\vert \Psi_{\uparrow}\rangle-\langle\Psi_{\downarrow}\vert r_i\vert \Psi_{\downarrow}\rangle\vert^2\\\cdot\frac{S_E(\omega)}{\omega}\bigg|_{\omega\rightarrow 0}\frac{2k_BT}{\hbar},
\end{split}$$ where $e$ is the elementary charge, $\Psi_{\uparrow}$ and $\Psi_{\downarrow}$ are the electron spin-up and spin-down molecular wavefunctions solved by the atomistic tight-binding method respectively, $\omega$ is the noise frequency and $S_E(\omega)$ is the noise field spectrum. For $S_E(\omega)$, we study $1/f^{\alpha}$ noise, Johnson noise and evanescent wave Johnson noise (EWJN) [@PHuang], assuming the noise source is 65 nm (the distance between G1 and the two dot center) away from the qubit system to be consistent with Ref. [@Tom_readout]. The expressions of the noise field spectra and the parameter estimations based on experiments [@Tom_readout; @Bent_prl; @Buch_prb] are also included in the Supplementary.
=3.4in
\[fig\_noise\]
Fig. 4(a) demonstrates the decoherence rate $1/T_{2}^{*}$ as a function of the applied dc electric field ($E_x$) due to Johnson noise from a single noise source for the case $R \approx$ 13 nm, assuming B = 0.5 T. Using the estimated parameters in the Supplementary, we find that $T_{2}^{*}$ is limited by Johnson noise. Based on our calculations, the effect of EWJN is at least 2 orders of magnitude lower than Johnson noise, and $1/f^{\alpha}$ noise is negligible (see Supplementary), thus they are not shown here. As shown, the decoherence rate $1/T_{2}^{*}$ reaches a maximum at $E_x=E_0$, where the bonding and anti-bonding states are formed and the system is most sensitive to charge noise. The left y-axis of Fig. 4(b) shows the maximum decoherence rate $1/T_{2}^{*}$ due to Johnson noise as a function of inter-dot separation $R$. As shown, $T_{2}^{*}$ can be improved by shrinking the inter-dot separation. This can be explained by the curvature of the qubit energy curve (e.g. the solid green or blue curve in Fig. 3(a))), which serves as a metric of how the qubit is prone to charge noise near $E_x = E_0$. On the right y-axis of Fig. 4(b), we plot this curvature ($a$ in its absolute value) by fitting the energy curves with a quadratic function of $E_x$ for different $R$. The curvature term $|a|$ increases with $R$ because the tunnel coupling $t_c$ decreases with $R$ (Fig. 3(c)), causing more abrupt charge transition. As can be seen, $1/T_2^{*}$ agrees with the trend of the curvature term $|a|$. Consequently, larger tunnel coupling/smaller 2P-1P separation is preferred to enhance the qubit coherence time. So far, we have investigated the decoherence time due to a single charge noise source. In a real device, there could be multiple charge noise sources (other in-plane gates, top gates, etc.), resulting in $T_2^{*}$ being degraded by 1-2 orders of magnitude eventually. Even then, using a 2P-1P spin qubit with electrical control is likely to yield devices comparable to single electron spin qubit based on single donor ($T_2^*$ = 268 $\mu$s [@Muhonen]) and single quantum dot qubit ($T_2^*$ = 120 $\mu s$ [@Menno1]) based on magnetic control, and outperform the 1P-1P charge qubit ($T_2^*$ = 0.72 $\mu$s [@Hollenberg]) in terms of qubit coherence time.
Summary
=======
In summary, we propose a novel approach for all-electrical control of donor-based spin qubits in silicon using full-band atomistic tight-binding modeling and time-dependent simulations based on effective spin Hamiltonian. In this design, ancillary dots are introduced to form an asymmetric 2P-1P system to create a hyperfine coupling difference between 2P and 1P, utilized to realize electron spin control with an ac electric field. We perform a quantitative analysis to optimize this design in terms of X-gate fidelity and decoherence time through external static B-field and tunnel coupling determined by inter-dot separation. We show that a long qubit coherence time can be potentially achieved. This work can serve as an alternative design to those that exploit the hyperfine difference between the donor and the interface states [@Tosi], where the qubit coherence could be affected by the proximity of the oxide interface [@Schenkel; @Paik]. To further reduce possible sources of deleterious noise in the proposed design, we would further pursue all-in-plane electrostatic and qubit control without the top surface gates in the future.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project No. CE110001027), the US National Security Agency and the US Army Research Office under contract No. W911NF-08-1-0527. Computational resources on nanoHUB.org, funded by the NSF grant EEC-0228390, were used. M.Y.S. acknowledges a Laureate Fellowship.
Supplementary {#supplementary .unnumbered}
=============
[*Spectrum functions of charge noise fields.*]{} Following Ref. [@PHuang], we investigate three types of charge noise. The $1/f^{\alpha}$ noise field spectrum is expressed as: $$S_E(\omega) = \frac{N}{\omega^{\alpha}},$$ where $N$ is the noise field strength in $(V/m)^2$. In this work, we estimate $N$ based on Ref. [[@Kuhlmann]]{}, where the root-mean-square electric field noise is $F_{r.m.s}$ = 46 V/m. Then $N = (\sqrt{2}F_{r.m.s})^2$ = 4232 $(V/m)^2$. We assume the bandwidth starts at 0.1 Hz also in line with Ref. [[@Kuhlmann]]{}, and $\alpha$ = 1 in the calculations.
The Johnson noise field spectrum is: $$S_E(\omega) = \frac{2\xi\omega\hbar^2}{1+(\omega/\omega_R)^2}/(el_0)^2,$$ where $\xi = R/R_k$, $R_k$ is the fundamental quantum resistance $h/e^2$, $R$ is the circuit resistance, and $\omega_R = 1/RC$ is the cutoff frequency. R is estimated based on Ref. [@Bent_prl]. In this work, we assume the gate length ($l_g$) is 100 nm and the gate width is 6 nm ($w$) which leads to 18 conducting modes ($M$, number of modes) [@Bent_prl]. As estimated in Ref. [@Bent_prl], the mean-free-path ($\lambda$) of such a wire is $\sim$6 nm. Hence, R in this work is calculated by $1/R = e^2/h\cdot M\cdot \lambda/(\lambda+l_g)$. $l_0$ is the distance between the qubit and the noise source, and we assume $l_0$ = 65 nm based on experimental devices [@Tom_readout]. $C$ is estimated based on Ref. [@Buch_prb], where the donor-gate capacitance is 0.6 aF for a separation $\sim$35 nm. Therefore, $C$ in this work is evaluated as $35nm/65nm\cdot 0.36 aF = 0.17 aF$.
The evanescent wave Johnson noise (EWJN) field spectrum can be expressed as: $$S_E(\omega) = \frac{\hbar\omega}{8z^3\sigma},$$ where $\sigma$ is the conductivity of the gates. We extract $\sigma$ from Ref. [@Bent_prl] where the wire length ($l_w$) is 47 nm. We assume the thickness of the wire ($t_w$) to be 2 nm considering donor diffusion and segregation, then the conductivity can be expressed as $4.8\frac{e^2}{h}\frac{l_w+\lambda}{w\cdot t_w}$.
=3.4in
In Fig. [\[fis\]]{}, we compare the effects of the three types of charge noise stated above on the decoherence rate [$1/T_2^{*}$]{}. As can be seen and stated in the main text, [$T_{2}^{*}$]{} is limited by Johnson noise. EWJN is at least 2 orders of magnitude lower than Johnson noise. [$1/f^{\alpha}$]{} noise is negligible, which is consistent with Ref. [[@PHuang]]{}.
Electronic address: wang1613@purdue.edu; rrahman@purdue.edu
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Accessible Wearable Technology Hacks at evoHaX SE
Accessible Wearable Technology Hacks at evoHaX SE
evoHaX SE was two weeks ago and people are still reeling from the awesome technology and the innovative ideas produced by all five of the teams. The weekend long Hackathon, co-organized by EvoXLabs and Code for Philly to celebrate the Disability Awareness Month was a rousing success.
Benjamin's Desk provided their incredible space for the entire weekend, October 23-25, and Friday kicked off with an amazing panel discussion. Featured on the panel were artist and maker/blogger from Adafruit Industries Leslie Birch, accessibility consultant for Philly Touch Tours Austin Seraphin, Senior Vice President and CIO for Independence Blue Cross Michael R. Vennera, Founder and CEO of Glass-U Daniel Fine, and Founder of CreativeTechWorks Design Studio Jamie Bracey with Juliana Reyes, lead reporter from Technical.ly Philly, as the moderator. The entire panel was full of energy and there were excellent discussions about the practice and promotion of accessible design - basically the perfect way to start the weekend.
On Saturday, after a brief introduction and an explanation of the rules from EvoXLabs' Founder Ather Sharif, and a Hacking 101 talk by Code for Philly's Executive Director Dawn McDougall, the five teams were assigned their subject matter and industry experts. The subject matter experts each had a disability of some kind, blindness, deafness, cognitive disabilities, mobility disabilities, or advanced age, and served to mentor their team in designing a wearable technology solution to solve one of the challenges they face. The industry experts were people who have had professional interactions with people with disabilities, and ranged from doctors to therapists.
Sunday the teams continued working on their projects, putting on the final touches on their designs until the stop coding call was given at 1 pm. All the teams worked extremely hard and everyone was extremely excited to see what they had worked on. The judges included Leslie Birch and Austin Seraphin from the Friday night panel as well as CEO and Co-founder of ROAR for Good, Yasmine Mustafa and Neil McDevitt, who is the Director of the Deaf Hearing Communication Centre.
Here are the solutions developed at evoHaX SE:
Team Blue Steel
The team Blue Steel was paired with subject matter expert Andrew Rosenstien who has Dyslexia and ADHD, and created an app for both mobile and Pebble Smart watches to track a user's progress as he does different tasks so he can better estimate how long it might take him to do something similar in the future. The app claimed to improve study skills through prioritization and time management.
Team Haptic High Five
Team Haptic High Five was paired with Patrick Kilgallon, an expert who is deaf, and created a sensor to be placed in his back pocket that will vibrate whenever something gets too close or if there is a sudden spike in the noise level behind him.
Drexel Dragons
The Drexel Dragons were put with eight-year-old Helena Roberts who has cerebral palsy and is wheelchair bound, and uses an electronic, tablet-like device called a Tobii to communicate. They created a canopy to be used on her wheelchair to cut down on the glare the sun creates on her Tobii screen, which makes it difficult to use, and to provide cool shade on those hot sunny days. See more details about Drexel Dragons' hack.
Red Team
The Red Team worked with senior citizen Howard Bilofsky and developed an imaging device so users can tell how far apart their hands are to make doing tasks require a good sense of depth perception, like threading a needle, easier. The device was implemented using Google Glass and a Leap Motion.
West Chester University
The West Chester University team partnered with Walei Sabry, who is visually impaired, and developed a beacon that can be worn by a user's friend, sending signals to the user's phone, allowing him to find his friend in a crowd with significant ease. The purpose of the device is to allow blind individuals greater independence when attending large events with friends.
The judges were absolutely blown away by all five projects and how hard the teams worked to produce such amazing results in just a short time. After the presentations were over the judges deliberated and then Tim Wisniewski, Philadelphia's Chief Data Officer, talked about the importance of civic hacking and the value it can bring to the city. Then he revealed the winners... The Red Team took second place and with it a set of Arduino Unos. Taking home first place and Pebble Time watches, donated by Pebble, was the Drexel Dragons with their canopy design.
The event could not have been pulled off without the amazing team of volunteers. The volunteers were a mix of professional tech enthusiasts and students from local universities. A fleet of extremely generous sponsors also supported the event including AccessComputing, smart watch company Pebble, and Independence Blue Cross, among others.
Join us to bridge the gap between technology and people with disabilities. It matters.
About EvoXLabs
EvoXLabs is an initiative dedicated to bridging the gap between technology and people with disabilities. We research and develop universally designed state of the art tools to improve Web Accessibility, and run projects such as evoHaX, FAWN and SCI Video Blog to improve lives.
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The first episode of Netflix’s new horror series The Order, which premieres on March 7th, reads like a pretty straightforward supernatural drama with a touch of dark humor. The second episode starts to get a little weird, and by the third installment, the show transitions into a flat-out hilarious mashup of college and horror comedy. Like the werewolves that are featured prominently in its plot, The Order doesn’t make a particularly smooth transformation, but it’s a lot more impressive once it embraces the change.
The Order is being marketed as a drama, and the initial trailer doesn’t even hint at the humor that makes the show stand out. That’s bound to produce a mismatch between expectations and reality on the scale of the marketing for Drew Goddard and Joss Whedon’s Cabin in the Woods. Unfortunately, that mismatch also afflicts the show itself. The Order would be better off if creator Dennis Heaton (Ghost Wars) and his co-writer Shelley Eriksen (Continuum) ditched all pretensions of seriousness and just fully committed to their silly spin on dark fantasy.
The Order follows Jack Morton (Jake Manley), a freshman at Belgrave University and initiate into the Hermetic Order of the Blue Rose. Jack and his grandfather, played by Matt Frewer with the same gusto for spouting ridiculous lines he showed in Orphan Black and Eureka, have plotted for years to get Jack into the secret society as a way of avenging his mother and punishing his father Edward Coventry (Max Martini). That plan involves the prerequisite conspiracy theorist cork board covered with strings, newspaper clippings, and paranoid theories that’s pinned up in their garage. It comes with a whiteboard detailing a five-year plan, starting with Jack’s college admission. Jack is meant to keep all of this under wraps, but he’s terrible at secrecy. Along with periodically mentioning his “mission,” then having to backpedal, he interrupts a campus tour led by sophomore, Order member, and immediate love interest Alyssa Drake (Sarah Grey) to show off his encyclopedic knowledge about the school. She somehow finds this charming, rather than deeply weird.
The twist is that while Jack thinks he’s signing on for the plot of The Skulls, he’s actually landed inside The Craft. The Order has amassed power not just through conventional means, but with magic spells that sound like they’ve been cribbed from Harry Potter, and Edward is the organization’s Grand Magus. The Order doesn’t realize they’re being hunted by the Knights of St. Christopher, a sacred brotherhood of werewolves that fight dark magic. (Or “gender-neutral collective” of werewolves, according to the fiery Lilith Bathory, played by Devery Jacobs.) As with What We Do in the Shadows, the lycanthropes are a real highlight in The Order. The best of the bunch is Jack’s lackadaisical RA Randall Carpio (Adam DiMarco) whose job includes handing out “a rape whistle and a how-not-to-rape pamphlet.” When the rest of Randall’s pack decides they should kill Jack for being a member of the Order, Randall demands they settle the dispute in the traditional fashion: via a game of beer pong.
It starts as ‘The Skulls,’ then turns into ‘The Craft’
The show is filled with gags like this, playing with the concept of irresponsible college students who have the power to break the rules of physics. Much of the show’s third episode is devoted to the shenanigans of a trio of Order initiates who learn a single illusion spell, culminating in a Pinky and the Brain-style “Are you thinking what I’m thinking?” moment where they alternately suggest stopping the magic, trying to learn more powerful magic they can abuse, and having a threesome. The goofy humor is interwoven into the plot to produce stories that are not just surprisingly funny, but genuinely surprising. Characters who seem destined to be recurring buddies, rivals, or mentors are killed off with shocking alacrity, with magic providing an excuse for why Belgrave’s campus isn’t swarmed with reporters responding to those deaths.
The problems come when the laughs stop, and the show grinds to a halt as it falls into boilerplate fantasy horror clichés. That weight is really felt in the scenes with Alyssa, Edward, and Vera Stone (Katharine Isabelle), the leader of Belgrave’s Order chapter. While Edward manages at least a bit of gravitas during the power jockeying and magical experimentation, their intrigues play out more like a Vampire: The Masquerade live-action role-playing game than a conflict with real stakes. The Order would do better if it just let its villains embrace full over-the-top campiness rather than trying to present them as mysterious and cool by keeping them straight-faced and grim. Hopefully, Alyssa’s mysterious past, Vera’s ambitions, or Edward’s efforts to assemble a powerful magical item will pay off as the season goes on.
The comedy is strong, but the bland protagonist doesn’t help the drama
Jack is a particularly mediocre protagonist, with the combination of a bland performance and inconsistent script making him unconvincing as an underdog townie surrounded by more privileged students at Belgrave, or even as a straight man struggling to keep up with the pace of mystical reveals. His flirtation with Alyssa is downright painful, particularly when they attempt to engage in a battle of wits with a speed match on the school’s giant chess board. If the rest of the humor wasn’t so sharp, this familiar chess metaphor would feel like self-parody. The same can be said for the scripts, which lean too heavily on buzzy phrases like “fake news” or caricatures of self-centered, entitled young people.
Visually, The Order sometimes seems taxed by the constraints of its presumably low budget. The first scene where Jack and the other hopeful pledges meet the members of the Order would be a lot creepier if it didn’t take place in broad daylight and if the mages weren’t demonstrating their power with sleight-of-hand coin tricks that looked better when they were done in the first episode of American Gods. The werewolf special effects are decent enough, and a goofy color-shifting technique meant to show that a scene is being viewed by something inhuman is tolerable because it sets up a pretty clever bait-and-switch. Three episodes into the series, the magic is rarely particularly showy. But that works out fine, given the theme that suggests how just a little power can go a long way. Similarly, the clear limits on available sets work well within the constrained world of a small college campus, where it makes sense if there’s really only one bar where teens can hang out and drink.
It’s hard to guess where this show is heading, but it works best when it sticks to humor
Considering the breakneck speed of the plot and tonal shifts in The Order’s first three episodes, it’s hard to really guess where the show is heading. There’s clearly room within its world to add even more magical creatures into the mix to serve as allies and threats to members of both factions, while expanding on the feel that basically everyone at Belgrave has had at least some brush with the supernatural. The Order could continue to combine the resigned way that the characters in Buffy treated life on the Sunnydale Hellmouth with zaniness, which it touched on when Jack asked a professor for an extension on a paper because his roommate had gone missing, and the professor and one of his colleagues pulled out custom Bingo sheets to see if they had “missing roommate” or “dead roommate” listed as a predicted excuse.
A lot more could happen on The Order. The machinations of the veteran members of the Order could finally spill beyond the boundaries of their occult temple, causing their powers to be stretched too thin as they cover up murders and grotesque student injuries. The new pledges could get into increasingly dangerous hijinks. But whatever direction the writers take will be strongest if they scale back their halfhearted adult intrigue and embrace their well-realized juvenile humor. That could let The Order provide real competition for the What We Do in the Shadows series FX plans to premiere later in March.
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A. Field of the Invention
This invention is related to a method an a machine for the production of glassware articles and more specifically to a method and a machine, as an individual forming section including single or multiple cavities, which can be grouped to constitute a glassware forming machine of the type including multiple individual forming sections, normally including from six to eight individual forming sections, for the production of glass bottles, jars, tumblers and other glassware articles by the blow-and-blow, press-and-blow, press-and-blow paste mold or direct-press processes.
B. Description of the Related Art
Glassware articles such as narrow neck glass bottles are normally produced in glassware forming machines of the type which may include multiple similar forming sections, by the blow-and-blow process, while wide neck glass jars, tumblers and other glassware articles are produced in the so named “E” and “F” Series forming machines by the press-and-blow process, in both, the so named hot molds and paste mold.
Glass bottles known as narrow neck glass containers, can also be produced by the well-known press-and-blow process, in the above mentioned E and F machines.
Nowadays the production velocity or forming cycles of the machines including multiple-sections and E and F machines, have reached to an optimum status and the maximum number of glassware articles has been achieved by providing multiple cavities (usually two to four) in each individual forming section of the machine.
Looking for an increasing in the number of glassware articles per forming cycle on each section of the machine, attempts have been made to introduce additional forming stations in each section, for example an additional article forming apparatus (blow mold, blow head) which could carry out a forming task (receiving a just formed parison from a single parison forming apparatus and beginning the forming blown), while another similar equipment is carrying out a following forming task on the forming cycle (opening the blow mold for transferring a just formed article to a cooling dead plate and being prepared to receive another following parison from the parison forming apparatus).
Representative of such forming machines, are the so named “one-two station machines”, disclosed in U.S. Pat. Nos. 4,094,656; 4,137,061 and 4,162,911 of Mallory, including a single stationary parison forming station and two article finishing stations, one finishing station at each side of the parison forming station in the same line known as the “cold-side” of the machine, eliminating the so named hot-side, and in U.S. Pat. Nos. 4,244,756 and 4,293,327 of Northup, disclosing a single parison forming station placed in the hot-side of the machine, and two article finishing stations, mounted one above the other on a lifting and lowering mechanism, alternatively rising and lowering each forming station for forming the articles.
However, by increasing the number of forming stations, the number of forming molds and surrounding equipment (either for single or multiple cavities) are consequently increased, increasing in turn the operation cost of the machine.
Other attempts to increase the velocity of production and the quality of the glassware articles in the multiple-section machines and E and F machines, has been focused on providing three consecutive forming stations, comprising a first parison forming station, an intermediate station for re-heating and/or stretching of the parison, and a third station for finishing the glassware article.
Representative of these “three station” forming machines are the U.S. Pat. Nos. 3,914,120; 4.009,016; 4,010,021 of Foster; U.S. Pat. No. 4,255,177 of Fenton; U.S. Pat. No. 4,255,178 of Braithwite; U.S. Pat. No. 4,255,179 of Foster; U.S. Pat. No. 4,276,076 of Fenton; U.S. Pat. No. 4,325,725 of Fujimoto and U.S. Pat. No. 4,507,136 of Northup.
The differences between each of these tree step forming processes disclosed by the above U.S. patents, can be firstly determined by the parison forming orientation in an upright orientation, as disclosed in U.S. Pat. Nos. 3,914,1.20; 4,009,016; 4,010,021, all of them of Foster, and U.S. Pat. No. 4,255,178 of Braithwait, and in an inverted orientation, as disclosed in U.S. Pat. No. 2,555,177 of Fenton, U.S. Pat. No. 4,255,179 of Foster, U.S. Pat. No. 4,325,725 of Fujimoto, and U.S. Pat. No. 4,507,136 of Northrup.
Further differences between the above disclosed three step forming processes, are determined by the apparatuses to transfer the parison and finished article through the parison forming step, the intermediate step and the finishing and take out steps.
For example, U.S. Pat. Nos. 3,914,120; 4,009,016; 4,010,021, and 4,255,178 disclose a linear transference of the parison in an upright position from the parison forming station, to the intermediate station, then a linearly transference of the parison from the intermediate station to a blow molding station, and finally, a linearly transference of the finished article, to a cooling dead plate.
Unlikely to the above disclosed glassware forming machines and apparatuses, U.S. Pat. Nos. 4,255,177; 4,255,179; 4,325,725, and 4,507,136, disclose a first transference step including inverting of the parison from an inverted position at the parison forming station, to an upright position at the intermediate station; a second linear transference step from the intermediate station to a final forming (blowing) station; and a third linear transference step from the final forming station to the cooling dead plate. The second and third linear transference steps being carried out by a generally similar transference apparatus.
Other differences between the apparatuses disclosed in the above-referred patents can be found in connection with the very specific apparatuses to carry out the transference of the parison and the final glassware article.
The main objective sought by the introduction of the intermediate station in these glassware forming machines, has been to release the task of a previous mechanism to be in an conditions to repeat a new forming cycle, without having to wait that a following mechanism performs its respective task, to turn back at its original position to begin a new forming cycle.
However, the above objectives have been difficult to be achieved because of the configuration of the mechanisms constituting the machine, which have been the same as the conventional and well-known ones.
Applicants, looking for a win-to-win machine, i.e. seeking to obtain the objectives of increasing the velocity of the machine and a reduction of the forming cycle time, the efficiency of its performance and an increasing in the quality of the articles to be produced, as well as seeking to make standard some mechanisms which perform similar tasks, and equipping them only with their specific instruments to perform their specific function, reducing as much as possible the cost of equipping a machine, the number of mechanisms in storage, and simplicity of mounting the specific instruments on common mechanisms and apparatuses, applicants reached to the following concept of a new glassware forming machine comprising a combination of some new apparatuses, and a new method for the production of hollow glassware articles.
In the first place, applicants visualized that an intermediate station is conveniently necessary so that the re-heating of the glass surface of a just formed parison be continued outside the blank mold in order to immediately release the task of the blank mold, enabling it to carry out another forming cycle, and permitting to carry out a stretching of the parison, all of which also results in an increase in the velocity of production and in a better quality of the article.
Additionally, applicants recognized that the inverting arm including a neck ring mold, of a typical inverting mechanism, had to be in a standing position during a parison forming cycle and to wait for the opening of the blank mold, to initiate the inverting cycle, release the parison at the intermediate station and turn back at the parison forming position, to begin another forming cycle.
To overcome the former disadvantage, applicants introduced a new inventive concept for the inverting apparatus, consisting in providing two diametrically opposed and stepped inverting arms, each holding a transferable and open-able neck ring mold (either single or multiple-cavity), so that a first one of said arms, after a parison has been formed at a first parison forming cycle, can firstly rotate 180° clockwise (moving the parison upwardly constricting it) or counterclockwise (moving the parison downwardly stretching it) to release the parison held by a first transferable and open-able neck ring mold, at the intermediate station, while the second arm with a second transferable and open-able neck ring mold is simultaneously placed under the blank mold to perform a second parison forming cycle, and then the first arm with an empty transferable and open-able neck ring mold which has been turned back to said first arm, rotates additional 180° completing a 360° turn, to be placed under the blank mold for a third parison forming cycle, while the second arm is releasing the parison held by the corresponding transferable and open-able neck ring mold, at the intermediate station. In this way, the blank mold do not have to wait that the first arm release the parison at the intermediate station and turn back, to initiate a new parison forming cycle.
New first and second transferable and open-able neck ring molds (either single or multiple-cavity) are provided to be held and handled with absolute independence by each of the arms of the inverting apparatus, by the longitudinal transference apparatus and by the take out apparatus, have also been provided in order to improve the quality of the final product by handling the parison by the neck ring at a uniform temperature, thus avoiding that the formed parison had to be handled by other components at different temperatures which may cause checks, efforts or deformations in the parison, which result in a poor quality of the finished articles.
The independence and transference ability of these transferable and open-able neck ring molds of the present invention, is possible in the machine of the present invention because of the existence of the unidirectional indexing-rotary inverting apparatus including the first and second stepped and diametrically opposed arms, which are able to hold a transferable and open-able neck ring mold, so that, while a first transferable and open-able neck ring mold is transferred from the first arm at the intermediate station to the blown molding station for forming a finished article, the second arm with a second transferable and open-able neck ring mold is placed at the parison forming station, in a parison forming cycle and once the parison is formed and able to be inverted at the intermediate station, the first arm has received back the first transferable and open-able neck ring mold and rotated other 180° completing a 360° turn, to be placed again at the parison forming station.
Also, although a typical baffle apparatus could be included in the machine, mainly for the blow-and-blow forming process, this apparatus can be configured in accordance with the machine of the present invention, by including a new oscillating apparatus named “rotolinear apparatus”, which may also be useful for operating a glass gob guide channel, the blank mold apparatus, the final blow apparatus and any other apparatus, for firstly rotate, then place an actuating mechanism to their respective active positions, and then retire them to an initial inactive position, which includes a new configuration of cams and cam followers to impart reliable oscillation and lowering and lifting movements, overcoming any backlash which could cause misalignment of the baffle apparatus or any other apparatuses, with the following apparatuses of the forming sequence.
A new equalizing apparatus has also been provided at the baffle apparatus and at the final blow apparatus, for multiple-cavity, for mounting bottom blank mold heads and uniformly place them on the blank molds or the blow molds, effectively adjusting whatever misarrange which may exist both, in the baffle or blow heads, or in the blank mold or blow molds.
In this way, this new glassware forming machine overcomes a number of difficulties of the known glassware forming machines, affording a win in the forming cycle time, which is estimated at a 32.6%, and allows an increase in the production and an improvement in the quality of the hollow glassware articles, as will be specifically disclosed in the following detailed description of the invention.
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Ahead of his summit with Russian President Vladimir Putin on Monday, President Trump used his Twitter account to promote blatant lies and Kremlin-favored talking points.
First, Trump posted a tweet blaming former President Obama for Russia’s attack on American democracy, claiming Obama “did NOTHING” to counter Russia.
“President Obama thought that Crooked Hillary was going to win the election, so when he was informed by the FBI about Russian Meddling, he said it couldn’t happen, was no big deal, & did NOTHING about it,” Trump tweeted. “When I won it became a big deal and the Rigged Witch Hunt headed by Strzok!”
President Obama thought that Crooked Hillary was going to win the election, so when he was informed by the FBI about Russian Meddling, he said it couldn’t happen, was no big deal, & did NOTHING about it. When I won it became a big deal and the Rigged Witch Hunt headed by Strzok! — Donald J. Trump (@realDonaldTrump) July 16, 2018
Trump’s claim is a blatant lie. In the summer of 2016, Obama wanted to issue a bipartisan statement detailing what the intelligence community knew about Russian meddling and offering federal help to states. But his effort was stymied by Senate Majority Leader Mitch McConnell (R-KY), who expressed “skepticism that the underlying intelligence truly supported the White House’s claims” about Russia interference, as the Washington Post reported.
Special counsel Robert Mueller’s latest indictment, which was filed on Friday and which charges 12 Russian military intelligence officials for interfering in the 2016 presidential election, makes it clear that Obama was right about the intelligence all along.
After blaming Obama, Trump posted another tweet blaming the United States and Mueller’s investigation for ruining U.S. relations with Russia — a talking point that was quickly echoed by the Kremlin.
Our relationship with Russia has NEVER been worse thanks to many years of U.S. foolishness and stupidity and now, the Rigged Witch Hunt! — Donald J. Trump (@realDonaldTrump) July 16, 2018
It’s not the first time Trump’s rhetoric has mirrored Russia’s. Trump and Putin have also used the same talking points to dismiss concerns about Russia’s election interference. Both world leaders have suggested Russia has been unfairly blamed because the hacks could’ve originated from anywhere in the world. Mueller’s latest indictment indicates Trump and Putin are mistaken.
During remarks made to the press pool on Monday at the beginning of the summit — just ahead of a 90-minute meeting with Putin in which no notetaker would be present — Trump didn’t so much as mention Russia’s attack on American democracy or the country’s illegal invasion and occupation of Ukraine.
Instead, Trump appeared to wink at Putin — and told him, “I think we will end up having an extraordinary relationship.”
Before the summit even starts, Trump tells Putin, "I think we will end up having an extraordinary relationship." pic.twitter.com/JskjUNOYc3 — Julia Davis (@JuliaDavisNews) July 16, 2018
Trump received a briefing in August 2016 in which he was personally warned about Russia’s efforts to infiltrate his campaign.
Shortly before that, Trump publicly encouraged Russian hackers to go after Hillary Clinton. Mueller’s latest indictment indicates that on that same day — July 27, 2016 — Russian hackers launched unprecedented cyberattacks against Clinton’s team.
Then, in October, emails stolen by Russian hackers that were ultimately published by WikiLeaks became the centerpiece of Trump’s closing message.
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The cumulative risk of AIDS as the CD4 lymphocyte count declines.
A method is proposed for assessing the cumulative risk of various AIDS-defining conditions as the CD4 lymphocyte count declines in HIV-infected individuals. The method is analogous to survival analysis but is based on the CD4 lymphocyte count rather than on time. Thus, the level to which the CD4 lymphocyte count has declined, rather than the length of time since seroconversion, is considered as an individual's survival interval. The survival interval may be censored (due to lack of follow-up) or treated as an interval to failure (if the individual develops AIDS). The Kaplan-Meier (product-limit) estimates, of the proportion of individuals developing AIDS before reaching a given low CD4 lymphocyte count, may be useful for determining when prophylactic treatment should begin.
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{
"pile_set_name": "PubMed Abstracts"
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UNPUBLISHED
UNITED STATES COURT OF APPEALS
FOR THE FOURTH CIRCUIT
No. 03-7185
UNITED STATES OF AMERICA,
Plaintiff - Appellee,
versus
SHAUN A. BROOKS,
Defendant - Appellant.
Appeal from the United States District Court for the Northern
District of West Virginia, at Elkins. Frederick P. Stamp, Jr.,
District Judge. (CR-00-7-2, CA-02-22-2)
Submitted: October 9, 2003 Decided: October 21, 2003
Before LUTTIG, KING, and DUNCAN, Circuit Judges.
Dismissed by unpublished per curiam opinion.
Shaun A. Brooks, Appellant Pro Se. Sherry L. Muncy, OFFICE OF THE
UNITED STATES ATTORNEY, Clarksburg, West Virginia; Paul Thomas
Camilletti, OFFICE OF THE UNITED STATES ATTORNEY, Martinsburg, West
Virginia, for Appellee.
Unpublished opinions are not binding precedent in this circuit.
See Local Rule 36(c).
PER CURIAM:
Shaun A. Brooks seeks to appeal the district court’s order
dismissing his 28 U.S.C. § 2255 (2000) motion. Brooks cannot appeal
this order unless a circuit judge or justice issues a certificate
of appealability, and a certificate of appealability will not issue
absent a “substantial showing of the denial of a constitutional
right.” 28 U.S.C. § 2253(c)(2) (2000). A habeas appellant meets
this standard by demonstrating that reasonable jurists would find
that his constitutional claims are debatable and that any
dispositive procedural rulings by the district court are also
debatable or wrong. See Miller-El v. Cockrell, 537 U.S. 322, ,
123 S. Ct. 1029, 1039 (2003); Slack v. McDaniel, 529 U.S. 473, 484
(2000); Rose v. Lee, 252 F.3d 676, 683 (4th Cir.), cert. denied,
534 U.S. 941 (2001). We have independently reviewed the record and
conclude Brooks has not made the requisite showing. Accordingly,
we deny a certificate of appealability and dismiss the appeal. We
dispense with oral argument because the facts and legal contentions
are adequately presented in the materials before the court and
argument would not aid the decisional process.
DISMISSED
2
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{
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Walter Jacobi
Walter Jacobi (January 13, 1918 – August 19, 2009) was a rocket scientist and member of the "von Braun rocket group", at Peenemünde (1939–1945) working on the V-2 rockets in World War II. He was among the scientists to surrender and travel to the United States to provide rocketry expertise via Operation Paperclip. He came to the United States on the first boat, November 16, 1945 with Operation Paperclip and Fort Bliss, Texas (1945–1949). He continued his work with the team when they moved to Redstone Arsenal, and he joined Marshall Space Flight Center to work for NASA. Jacobi worked on rocket "structure and components."
He continued to support the space program and appear at public events until his death.
References
Category:German aerospace engineers
Category:German rocket scientists
Category:1918 births
Category:2009 deaths
Category:American aerospace engineers
Category:Early spaceflight scientists
Category:German emigrants to the United States
Category:German inventors
Category:V-weapons people
Category:Marshall Space Flight Center
Category:NASA people
Category:Operation Paperclip
Category:20th-century American engineers
Category:20th-century inventors
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{
"pile_set_name": "Wikipedia (en)"
}
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Q:
Does `git commit` commit changes that were not added
I'm trying to learn git using the 'Pro Git' book by Scott Chacon. When explaining how to Stage modified files (page 18), i understand with git add the files are scheduled for commit and then commited with git commit. It mentions that a commit is done, only the changes that were added will be actually be commited, and if i change a file again, i'd have to add it againg before commit so that all changes are comitted. The text says:
It turns out that Git stages a file exactly as it is when you run the git add
command.If you commit now, the version of the file as it was when you last ran the git add
command is how it will go into the commit, not the version of the file as it looks in your working directory when you run git commit. If you modify a file after you run git add
, you have to run git add again to stage the latest version of the file.
However, i'm seeing a different behaviour when trying it myself:
$ git status #start clean
#On branch master
nothing to commit (working directory clean)
$ echo "hello" >> README.TXT
git-question> git add README.TXT #added change to README
$ git status
# On branch master
# Changes to be committed:
# (use "git reset HEAD <file>..." to unstage)
#
# modified: README.TXT
#
$ echo "good bye" >> README.TXT #change README after adding
$ git status #now 'hello' is added to be committed but not 'good bye'
# On branch master
# Changes to be committed:
# (use "git reset HEAD <file>..." to unstage)
#
# modified: README.TXT
#
# Changes not staged for commit:
# (use "git add <file>..." to update what will be committed)
# (use "git checkout -- <file>..." to discard changes in working directory)
#
# modified: README.TXT
#
$ git commit -m "only hello" README.TXT #commit, i would expect just 'hello' gets commited
[master 86e65eb] only hello
1 file changed, 2 insertions(+)
$ git status #QUESTION: How come there's nothing to commit?!
# On branch master
nothing to commit (working directory clean)
So the question is: Shouldn't git commit just commit the changes that were added with git add? And if so, why does it commit the second change even if i didn't add it?
A:
git commit will commit what is currently in the index and therefore what you explicitly added.
However, in your example, you are doing git commit README.TXT which will commit the files you specify, that is the current file README.TXT. Just do git commit -m "only hello" to commit the index.
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"pile_set_name": "StackExchange"
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Reviews by category
Defrag 10 Professional
Review controls
As your hard drive is probably the slowest component of your computer system, it certainly makes sense to ensure you are getting the maximum possible response from this device.
Buy Now...
One way to help is to ensure that the drive is regularly defragmented so that files are stored contiguously rather than being scattered willy-nilly in different locations. The situation of files being broken up and scattered to various locations on your drive occurs through the normal use of the operating system. Rather than spend time looking for a space large enough to accommodate a file, the operating system will fill up spaces with portions of a file. When next required the operating system will locate the scattered segments and recreated the file ready for use but this process does take milliseconds and slows everything down.
While Windows has its own defrag option, this is rather limited, slow and with a propensity to keep restarting for various reasons. One third-party solution, developed by the Berlin-based O&O Software Company, is that of O&O Defrag 10 Professional Edition. Available on a 30-day trial basis before you need to make a decision regarding purchase, O&O Defrag opens with a wizard designed to help you define the appropriate settings to optimise system performance. You can opt for an automatic or manual defrag; state whether a desktop/laptop or server will be involved; and state whether the computer falls into the category of work, home or gaming machine.
O&O Defrag offers a choice of defragmentation methods. These include Stealth (optimised for large data drives); space (designed for heavy defragmentation); and Complete with files being sorted according to criteria based on name, modification or last access date. Let loose on my laptop, O&O Defrag displayed an interface that splits the screen into two horizontal areas designated as Tasks and Feedback with tabs allowing you to quickly switch between different options.
The top pane has tabs for selecting Defrag; Jobs & Reports; View; and Help. There is also a tool bar whose content changes to reflect whichever tab is selected. The lower pane lets you switch between views showing the clusters on your hard disk; jobs; reports; and the drive's status with regards to the current defrag operation.
Running a series of defrag operations using various modes on a 60GB hard drive divided into two partitions, the O&O software performed its designated task but could hardly be considered a speed king. In fact during one test, progress was so slow that I though the software had stopped responding. There appeared to be no disk activity and nothing was changing on the screen display. However I was wrong as the process was eventually completed. To give you some indication as to the speed, or lack of it, of operation, O&O Defrag took over 2½ hours to defrag less than 3GB of data made up of a mixture of small, medium and large files. You do get plenty of feedback about completed processes but this hardly makes up for the slowness of the program's performance.
O&O Defrag has support for both 32 and 64-bit systems. You will need to be running Windows 2000/XP/Vista. O&O Defrag 10 Professional carries a £29.99 price tag but does nothing to persuade me to change from my preferred defrag software.
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{
"pile_set_name": "Pile-CC"
}
|
Q:
Does $A=\{x\in\mathbb{Q}: x>0\}$ have any isolated points?
Let $A=\{x\in\mathbb{Q}: x>0\}$ and $B=\{x\in\mathbb{R}: x>0\}$. I think that it is easy to understand and to prove that the set $B$ does not contain any isolated points, but can we say the same about set $A$? Are the point's in $A$ all isolated?
A:
If $x<y \in A$ then $\frac{x+y}{2} \in A$ and $x<\frac{x+y}{2}<y$.Since $A$ contains at least two points, no point in $A$ can be an isolated point.
|
{
"pile_set_name": "StackExchange"
}
|
{
"pile_set_name": "PubMed Central"
}
|
|
require File.expand_path(File.dirname(__FILE__) + '/spec_helper')
module GovKit::FollowTheMoney
describe GovKit::FollowTheMoney do
before(:all) do
unless FakeWeb.allow_net_connect?
base_uri = GovKit::FollowTheMoneyResource.base_uri.gsub(/\./, '\.')
urls = [
['/base_level\.industries\.list\.php\?.*page=0', 'business-page0.response'],
['/base_level\.industries\.list\.php\?.*page=1', 'business-page1.response'],
['/candidates\.contributions\.php\?imsp_candidate_id=111933', 'contribution.response'],
['/candidates\.contributions\.php\?imsp_candidate_id=0', 'unauthorized.response'],
]
urls.each do |u|
FakeWeb.register_uri(:get, %r|#{base_uri}#{u[0]}|, :response => File.join(FIXTURES_DIR, 'follow_the_money', u[1]))
end
end
end
it "should have the base uri set properly" do
[Business, Contribution].each do |klass|
klass.base_uri.should == "http://api.followthemoney.org"
end
end
it "should raise NotAuthorized if the api key is not valid" do
api_key = GovKit.configuration.ftm_apikey
GovKit.configuration.ftm_apikey = nil
lambda do
@contribution = Contribution.find(0)
end.should raise_error(GovKit::NotAuthorized)
@contribution.should be_nil
GovKit.configuration.ftm_apikey = api_key
end
describe Business do
it "should get a list of industries" do
@businesses = Business.list
@businesses.should be_an_instance_of(Array)
@businesses.each do |b|
b.should be_an_instance_of(Business)
end
end
end
describe Contribution do
it "should get a list of campaign contributions for a given person" do
pending 'This API call is restricted'
@contributions = Contribution.find(111933)
@contributions.should be_an_instance_of(Array)
@contributions.each do |c|
c.should be_an_instance_of(Contribution)
end
end
end
end
end
|
{
"pile_set_name": "Github"
}
|
27 Cal.App.2d 240 (1938)
NELLA MARIE WIRES, Appellant,
v.
Dr. ELMER W. LITLE, Respondent.
Civ. No. 11793.
California Court of Appeals. Second Appellate District, Division Two.--
June 23, 1938.
Morris Lavine for Appellant.
W. I. Gilbert for Respondent.
WOOD, J.
The plaintiff seeks by this action to recover damages which she alleges were caused by the malpractice of defendant. The action was tried with a jury and upon the termination of plaintiff's evidence the trial court granted a motion for a nonsuit. Plaintiff appeals from the judgment thereafter entered.
According to the evidence presented by plaintiff, she was working on Saturday, March 2, 1935, as a seamstress and accidentally ran a needle into her right ring finger through the second joint, where part of it became imbedded. She was taken to the office of defendant, a physician, who was asked to remove the needle. It was shortly after noon and defendant was about to leave his office, but he stated that he was familiar with that type of case and would have no trouble in removing the needle. At the time plaintiff went to defendant's office her hand was discolored by a dark stain from the goods upon which she had been working. Defendant did not wash the finger or hand but swabbed the finger with a liquid which he stated was iodine. Defendant tried to locate the needle with a Zio-Lite, but was not able to do so. He sent plaintiff to Dr. Ekes, a dentist in the building, who had X-ray equipment for taking pictures of teeth. An X-ray picture was taken, but it was not clear or distinct. Dr. Ekes stated that the picture was not clear because it was wet and that it should have been allowed to stand for some time after being washed, but that this was not done because of the shortness of the time. He also stated that "it was not *242 very clear and distinct, because I am not in the habit of taking X-rays, only of matters of dentistry". Although the needle could not be seen from the picture that was taken, no effort was made to secure another picture. Defendant made an incision in the finger and probed for the needle for from four to five minutes without locating it. He put some gauze in the wound to establish drainage, stitched up the side of the incision and bandaged the finger and hand. He gave plaintiff a box of pills to relieve pain and sent her home with instructions to come back the following Monday morning in order to get X-ray pictures. He gave her no further advice or instructions.
Saturday evening the hand became very painful and on Sunday efforts were made to telephone to defendant, but he could not be located. The finger and hand became swollen and in order to relieve pain plaintiff cut the bandages. She became feverish and ill. Plaintiff was taken on Monday morning to Dr. Henderson, who found evidence of infection. An X-ray picture was taken and plaintiff was sent to a hospital, where her hand was put under a fluoroscope and the needle was extracted. Her arm became swollen up to the shoulder and she was treated by Dr. Henderson for two or three weeks, after which time she was sent to the General Hospital, where her finger was amputated.
Dr. Henderson testified that plaintiff was suffering from blood poisoning and that although the infection was present when the needle entered the finger, it could not be controlled otherwise than by subsequent treatment and that the removal of the needle accompanied by free bleeding and washing reduces the possibility of infection. He further testified that plaintiff's general physical condition was not good in that her resistance was low, that she "appeared to be tired and exhausted and worn out from probably a little too continuous hard work".
Dr. Burnight qualified as a medical expert. He testified that he was acquainted with the usual and customary practice in this type of case in Los Angeles; that the effect of leaving a needle in the finger of a person in a run-down condition would be to cause trauma or injury to the tissue and that if allowed to remain it would undoubtedly set up infection; that the leaving of the needle in the tissue was the primary cause of the infection and that as it became more advanced it would develop gangrene. He further testified *243 that it is the usual and ordinary custom in this community to take an X-ray of a finger before it is incised for the purpose of removing a needle; that where an X-ray is faded and the needle is entirely incased in the finger and not visible to the eye, it would not be the customary and usual practice in this community for a doctor to cut and incise the finger relying upon that type of an X-ray; that due to the infection, and pressure from the swelling, all the tissues and blood vessels in the immediate vicinity would become infected and deteriorated so as to stop the natural flow of the blood stream, resulting in gangrene; that when gangrene sets in amputation is necessary to save the life of the patient.
[1] The rules to be applied in the case of a motion for a nonsuit are too well established to require the citation of authority. The trial court must assume that all the evidence in plaintiff's favor is true and every favorable inference fairly deducible from the evidence and every favorable presumption fairly arising from the evidence must be drawn in plaintiff's favor. Measured by these rules it must be held that the issues should have been submitted to the jury. We are not unmindful of the general rule, which is relied upon by defendant, that in cases of malpractice it must be shown by expert testimony that the damages suffered by plaintiff resulted from the failure of the defendant to possess and use that degree of care and skill ordinarily possessed and used by physicians and surgeons in good standing practicing in the locality. The testimony of the experts in the case now before us was sufficient to meet the requirements of the general rule, considered in connection with the facts established by the lay witnesses. Even in actions for malpractice the jury can consider without the testimony of experts matters which are within the common knowledge of mankind. Although it is necessary that experts be called to establish matters peculiarly within the knowledge of experts there are also "facts which may be ascertained by the ordinary use of the senses of a nonexpert". (Barham v. Widing, 210 Cal. 206 [291 P. 173].) If the defendant had undertaken to remove a needle from the ring finger but had made an incision in the thumb it could not be successfully argued that an expert witness would be necessary to establish negligence. In Rankin v. Mills, 207 Cal. 438, 441 [278 P. 1044], the court stated: "Expert testimony appears in the record to the effect that, *244 under such conditions, and in view of the fact that no improvement occurred in the limb under the treatment for dislocation of the hip, the next step, logically and scientifically, was to investigate further the nature of the injury by taking X-ray pictures. Indeed, this would seem so obvious as to be a permissible inference without expert testimony on the subject, as would be the conclusion that a failure to do this was negligence." (See, also, Thomsen v. Burgeson, 26 Cal.App.2d 235 [79 PaCal.2d 136]; Evans v. Roberts, 172 Iowa, 653 [154 N.W. 923].) It might be well argued that it is a matter of common knowledge that a needle accidentally embedded in the finger of a seamstress might cause infection if not promptly extracted; that a clear X-ray picture would be of assistance to the surgeon; and that if an indistinct picture be secured from a dental office an effort should be made to secure a clear picture from those equipped with apparatus for that purpose.
Plaintiff complains that the trial court unduly curtailed the examination of the expert witnesses. In several instances the court erroneously sustained objections to plaintiff's questions. It would prolong this opinion unnecessarily to set forth the various questions, a number of which, being hypothetical, were quite lengthy. Upon a new trial it is to be assumed that the rulings on the admission of evidence will be correctly made. For the purposes of this appeal it is sufficient to state that if a verdict had been rendered in plaintiff's favor it would have found ample support in the evidence admitted.
The judgment is reversed. The attempted appeal from the order denying a motion for a new trial is dismissed.
Crail, P. J., and McComb, J., concurred.
A petition by respondent to have the cause heard in the Supreme Court was denied on August 22, 1938, and the following opinion was then rendered:
THE COURT.
The petition for hearing in this court after decision by the District Court of Appeal of the Second Appellate District, Division Two, is denied. However, such denial is not to be taken as indicating approval by this court of the discussion concerning matters of common knowledge *245 which, it is said, a jury may properly consider in an action for malpractice.
HOUSER, J.,
Concurring.
I concur in the general order by which a hearing of the cause by this court is denied; but I am not in accord with that part of the order which is to the effect that the denial of the petition for hearing in this court "is not to be taken as indicating approval by this court of the discussion concerning matters of common knowledge", etc.
The language which occurs in the opinion of the District Court of Appeal to which the order has reference is as follows: "It might be well argued that it is a matter of common knowledge that a needle accidentally embedded in the finger of a seamstress might cause infection if not promptly extracted; that a clear X-ray picture would be of assistance to the surgeon; and that if an indistinct picture be secured from a dental office an effort should be made to secure a clear picture from those equipped with apparatus for that purpose." (Emphasis added.)
It is obvious that an order of denial of hearing which contains express language that "such denial is not to be taken as indicating approval by this court" of specified discussion which appears in the opinion of the District Court of Appeal, is the equivalent, and is but another way of stating, that the criticized portion of the opinion is disapproved.
At the outset, it may be observed that it is questionable whether this court is constitutionally empowered or authorized as an incident to the instant proceeding, either specifically to approve or to "disapprove" of any declaration of law that has been made by the District Court of Appeal.
Of course, it must be conceded that the constitutional provisions are the source of power of the Supreme Court. Apparently sections 4 and 4c of article VI of the Constitution contain the only provisions that have a bearing upon such assumed authority. In the former section, it is provided that "the said court shall have appellate jurisdiction in all cases, matters and proceedings pending before a District Court of appeal, which shall be ordered by the Supreme Court to be transferred to itself for hearing and decision, as hereinafter provided"; and the pertinent provision of the latter section is that "the Supreme Court shall have power ... to *246 order any cause pending before a District Court of Appeal to be heard and determined by the Supreme Court". (Emphasis added.)
With respect to the instant inquiry, the jurisdiction of the Supreme Court is thus established, and is as fixed as is the jurisdiction of any other court. The constitutional provisions in that regard contemplate only that either a hearing of a cause shall be granted, or that it shall be denied. No middle course is available. If a hearing be granted, manifestly the cause is then before the court for determination; and in that event such a decision and opinion may be rendered as may accord with the views of a majority of the members of the court. But should the petition for a hearing be denied, by no constitutional provision is it implied or even suggested that the court possesses any power other than to make its unqualified order to that effect. In that regard, neither direct nor implied authority is conferred upon the Supreme Court either to modify an opinion theretofore rendered by a District Court of Appeal, or to exercise any sort of supervisory control with reference thereto. It is obvious that without and in the absence of a regular hearing of a cause, if the Supreme Court has authority to modify such a decision or opinion in any particular, it should follow that the court may so modify the decision or the opinion as to render it of no effect;--in other words, it might "modify" an opinion of the District Court of Appeal out of existence; and at the same time, in the place of such "modified" opinion, either expressly or impliedly, substitute an opinion and decision which might be to the same effect as that theretofore rendered by the District Court of Appeal, or, at its pleasure or option, expressly reach an opposite conclusion, accompanied by an appropriate opinion,--all without any hearing having been granted.
But even assuming the existence of the constitutional power in this court which herein has been exercised, it is not clear that the "discussion" to which the order herein has made reference, and regarding which "approval by this court" is withheld, does not constitute a fair and correct expression of the law in the matter to which it relates.
Directing attention to the opening words of the "discussion" which apply to each of the several statements thereafter following, it will be noted, not that a positive declaration *247 is made with reference thereto, but only that "it might be well argued that it is a matter of common knowledge", etc. Otherwise stated, the opinion does not purport to declare that either in fact or in law it is a matter of common knowledge, but only that whether such conditions exist may be debatable. In that light, I fail to discern any ground for "disapproval" by this court. Although not admitted, even should it be conceded that by the criticized language of the District Court of Appeal, an impression was sought to be conveyed that the specified statements actually constituted and were "matters of common knowledge", again I must confess my inability to detect any error therein. Is it not a fact that practically every one knows that either a needle or any other foreign substance which may become "embedded (either) in the finger", or in any other part of the body "might (may) cause infection if not promptly extracted"? And in these more advanced days, is there any responsible person who does not know that in treating a patient who has been so unfortunate as to have a broken needle embedded in a finger joint, "a clear X-ray picture (of such joint) would be of assistance to the surgeon"? Furthermore, would not common sense indicate that if in the first instance a poor or "indistinct picture be secured", either from a dental office or from any other source, "an effort should be made to secure a clear picture from those equipped with apparatus for that purpose"? A denial of the existence of such knowledge on the part of the average layman amounts to a refusal to accredit to him any observation whatsoever relating to the most common and ordinary conditions that surround transactions and happenings of our daily life. (Emphasis added.)
Nor in my judgment may it properly be declared that either of such assumed statements regarding that which may constitute "judicial notice" was erroneous. In 23 C.J., pages 58, 59, 61, it is declared that "The term 'judicial notice' means no more than that the court will bring to its aid and consider, without proof of the facts, its knowledge of those matters of public concern which are known by all well-informed persons. ... Courts may properly take judicial notice of facts that may be regarded as forming part of the common knowledge of every person of ordinary understanding and intelligence." And as affecting a situation such *248 as here is in question, in 15 Ruling Case Law, pages 1101 and 1130 (where many judicial illustrations are cited) it is respectively stated that "Courts will take judicial notice of those facts relating to human life, health, habits and acts known to men of ordinary understanding, ...; the general rule is that it is the duty of a court to take judicial cognizance of all matters affecting the public health which are of certainty to general or scientific knowledge. ..." Within cited cases where it has been held that the court may "take judicial notice" of facts which may include and affect the physical condition of persons, the following several rulings appear: The size of an ordinary man is a matter of common knowledge, which extends not only to height and thickness of the body as a whole, but also to the measurement of the various parts; a man could not accidentally fall through a hole of a certain size; the destruction of the sight of one eye impairs the general power of vision; the instinct of self-preservation will be judicially noticed; also the effect of fright or exposure on the nervous system; the habits and qualities of the more common animals; certain strains and breeds of animals of the same species are more valuable than others; certain objects or events are or are not such as to frighten horses of ordinary gentleness; Texas cattle have some contagious or infectious disease, communicable to native cattle outside that state; certain animals are the natural enemies of others; epilepsy tends to weaken mental force, and often descends from parent to child, or entails upon the offspring of the sufferer some other grave form of nervous malady; man's susceptibility to certain diseases; means or method by which disease spreads from one victim to another; diseases of the skin may be spread in barbershops; reclamation of swamp and overflowed lands may concern the public health; considerations of public health necessitates that streets be kept clean of refuse; hogs kept within thickly populated cities tend to create a condition hazardous to the public health; a high, rank growth of weeds in a thickly populated district tends to injuriously affect the health of the inhabitants; the manufacture of wearing apparel in unsanitary and overcrowded working quarters may promote or spread disease; and that long hours in certain employments are injurious to health. *249
As a conclusion, it is clear to my mind that that part of the order to which reference hereinbefore has been had, is erroneous.
|
{
"pile_set_name": "FreeLaw"
}
|
Microcytic anemia. Differential diagnosis and management of iron deficiency anemia.
Microcytic anemia is defined as the presence of small, often hypochromic, red blood cells in a peripheral blood smear and is usually characterized by a low MCV (less than 83 micron 3). Iron deficiency is the most common cause of microcytic anemia. The absence of iron stores in the bone marrow remains the most definitive test for differentiating iron deficiency from the other microcytic states, ie, anemia of chronic disease, thalassemia, and sideroblastic anemia. However, measurement of serum ferritin, iron concentration, transferrin saturation and iron-binding capacity, and, more recently, serum transferrin receptors may obviate proceeding to bone marrow evaluation. The human body maintains iron homeostasis by recycling the majority of its stores. Disruptions in this balance are commonly seen during menstruation, pregnancy, and gastrointestinal bleeding. Although the iron-absorptive capacity is able to increase upon feedback regarding total body iron stores or erythropoietic activity, this physiologic response is minimal. Significant iron loss requires replacement with iron supplements. The vast majority of patients respond effectively to inexpensive and usually well-tolerated oral iron preparations. In the rare circumstances of malabsorption, losses exceeding maximal oral replacement, or true intolerance, parenteral iron dextran is effective. In either form of treatment, it is necessary to replete iron stores in addition to correcting the anemia.
|
{
"pile_set_name": "PubMed Abstracts"
}
|
Related Content
Jerry Seib: Battling ISIS After the Fall of Ramadi
5/21/2015 6:00AM
After the fall of the Iraqi city of Ramadi to the Islamic State, the United States is faced with several difficult choices on how to move forward in combating the extremist group. WSJ’s Jerry Seib explains. Photo: AP
This transcript has been automatically generated and may not be 100% accurate.
... U S military and civilian officials are perplexed by the fall of Ramadi in Iraq this week to Islamic State fighters in to clear they're wondering why the Iraqi army abandoned from IV ... my colleague Julian Barnes had an interview this week with General Martin Dempsey the Chairman of Joint Chiefs of staff ... who said what happened there was an Iraqi forces decided unilaterally the Iraqi commander on the scene decided unilaterally ... to pull out move to safer ... positions because they're there was a sandstorm underway and they concluded ... because of the system they couldn't get U S airstrikes they needed to support them ... and to quote the general Dempsey provided Julie was the rocky army was not driven out of her body they grow value for money ... this is as a separate box of U S officials are raises the question where things go now in Iraq and I think the US faces some tough decisions ... a couple things need to happen first of all U S officials are already talking that they need ... to increase training of Sunni militias these are ... delicious to operate in the area around for a body which is the Sunni part of Iraq ... the government in Baghdad Shiite dominated so they're two different sides of this the Sunni militias who are willing to defend their their territory needs more help in the form of training secondly they also need more equipment provided to them by the central government in Baghdad one of these us been introduced puts more pressure on the Iraqi government ... to start providing equipment to the Sunni militias so they can find a better fight in the area around the Mahdi in the on bar province ... and thirty of those things don't work the US is been have to confront a question it is uncomfortable with but which is unavoidable if things ... go badly should the USB providing arms directly ... to Sunni militias and even the Kurdish militias in northern Iraq that would undermine the authority of the Iraqi central government ... Uniphase's not going there now but it's a question to remember down the road ...
|
{
"pile_set_name": "Pile-CC"
}
|
94 U.S. 741 (1876)
CORCORAN
v.
CHESAPEAKE AND OHIO CANAL COMPANY.
Supreme Court of United States.
Mr. J.D. McPherson, Mr. Conway Robinson, and Mr. Joseph Bryan, for the appellant.
Mr. John P. Poe and Mr. Bernard Carter, for the appellee
MR. JUSTICE MILLER delivered the opinion of the court.
The Chesapeake and Ohio Canal Company, from the date of its organization in 1824-25, issued several series of bonds, secured by as many mortgages on its property. The largest of these mortgages was the earliest, and was given to the State of Maryland for several millions of dollars; another was made to the State of Virginia; both of which States contributed largely, by the use of their credit, to the construction of this important work. In the last stages of the struggle to extend the canal to Cumberland, where it reached the coal-beds, which alone have made it of any value, the company issued another series of bonds to the amount of $1,700,000, for the payment of which it *742 pledged, by way of mortgage, the revenues and tolls of the canal, after deducting the necessary costs of running the canal and its repairs, and perhaps some other defined outlays.
In this mortgage, Corcoran, the complainant and appellant in the present suit, was one of several trustees for the benefit of the bondholders. He also became, and according to the statements of the present bill is now, a larger holder of these bonds, or of the coupons for interest on them.
The purpose of this bill, which was filed by him on behalf of himself and all others in like condition as holders of this class of bonds, is to enforce the payment of the coupons of interest due and unpaid for many years past.
The defendants to the bill are the Chesapeake and Ohio Canal Company, the State of Maryland, and the remaining trustees of the mortgage bonds on which the suit is founded. They have all answered, except the State of Maryland.
The answer of the trustees is unimportant. The canal company admit the indebtedness and the failure to pay, but deny that, under the reservations of the mortgage of the tolls and revenues in plaintiff's mortgage, there is now or has been in their hands any part of the said revenues which they could lawfully appropriate to the payment of said coupons, except so far as they have already done so. After several amendments of the pleadings and stipulations as to facts, the issue was finally narrowed to two questions; namely, the jurisdiction of the Supreme Court of the District, and the right of the holders of the interest coupons to exact out of the net revenues of the company payment of interest on those coupons from the respective dates when they fell due.
The first of these questions is raised by the proposition of the defendants, the canal company, that the State of Maryland is a necessary party to this suit; and, as she has not voluntarily appeared, and cannot be made amenable to any process to compel an appearance, the bill must be dismissed on that ground.
In the view which this court takes of the other question, and as the court has jurisdiction as to the canal company, it is unnecessary to consider or decide this one.
In reference to the question of interest upon the interest coupons, the canal company, in its answer to complainant's bill, *743 alleges that, in a suit brought by the State of Virginia in the Circuit Court of Baltimore City, to which suit the present complainant and his co-trustees, the State of Maryland, the canal company and others, representing all the various classes of bondholders, were parties, "the issue raised in this case, that the coupons upon said preferred bonds are entitled to bear interest from their maturity, which is to be allowed payment out of the revenues of this respondent in preference to the claims of the State of Maryland, was distinctly presented, was argued, amongst others, by the solicitors of complainant in this case, and was decided by the court in opposition to the claims of said complainant as then asserted and as reiterated in the bill in this case." The record of that suit, including the opinion of the Court of Appeals and the brief of the counsel of the present appellant, are made exhibits.
The bill of the State of Virginia distinctly claims interest upon the coupons which she held, standing in the same relation as those of the appellant here. The right to that interest as a preference to the debt of the State of Maryland is denied by the answers of the canal company and of the State of Maryland. Corcoran and his co-trustees submit all those matters to the decision of the court. It was, therefore, properly in issue. Indeed, the whole subject of priority of lien as to the revenues and tolls of the canal was before the court and was the very matter to be decided, and necessarily included the question whether the State of Maryland in the statute by which she waived her prior lien, so far as the revenues of the company were concerned, in favor of the class of bonds and coupons held by the State of Virginia, and those represented by Corcoran, as trustee, included interest upon interest, or only principal and current interest.
The opinion of the Court of Appeals of Maryland, found in the record as an exhibit, and reported in 32 Md. 501, while conceding the general rule, that where the annual or semiannual interest on a bond is represented by a distinct coupon, capable of separation and removal from the main instrument, it bears interest from its maturity, if unpaid; holds that, under the special statute of Maryland authorizing the pledge by the canal company of its revenues for the payment of these preferred *744 bonds and interest, and waiving her own existing priority of claim on those revenues, simple interest only was meant, and that as to the lien on those revenues and tolls, the interest on the coupons was not included in the lien.
The opinion, undoubtedly, decides the very point in controversy here.
It is said, however, that this is only an opinion, and that unless a judgment or decree is produced there can be no estoppel; and the principle asserted is undoubtedly correct. But, in a stipulation signed by the parties to the present suit, it is agreed "that a decree has been passed by the Circuit Court of Baltimore City making distribution of the net revenues of said canal company, and ordering their payment from time to time as the same accrue, in conformity with the said opinion."
The opinion of the court, then, by virtue of that decree, has become, by the well-settled principles of jurisprudence, the law of the case as to the parties who are bound by that decree.
In avoidance of the application of this doctrine to the present case several objections are urged, some of which are answered sufficiently by the foregoing statement of the record of that suit. We will notice one or two others.
It is said that Corcoran and his co-trustees, the canal company, and the State of Maryland, were all defendants to that suit, and that as between them no issue was raised by the pleadings on this question, and no adversary proceedings were had.
The answer is, that in chancery suits, where parties are often made defendants because they will not join as plaintiffs, who are yet necessary parties, it has long been settled that adverse interests as between co-defendants may be passed upon and decided, and if the parties have had a hearing and an opportunity of asserting their rights, they are concluded by the decree as far as it affects rights presented to the court and passed upon by its decree. It is to be observed, also, that the very object of that suit was to determine the order of distribution of the net revenue of the canal company, and that the Corcoran trustees were made defendants for no other purpose than that they might be bound by that decree. And, lastly, as the decree did undoubtedly dispose of that question, its conclusiveness cannot now be *745 assailed collaterally on a question of pleading, when it is clear that the issue was fairly made and was argued by Corcoran's counsel, as is shown by the third head of their brief, made a part of this record by stipulation.
It is also argued that in that suit Corcoran was only a party in his representative capacity of trustee, and he here sues in his individual character as owner of the bonds and coupons, and in this latter capacity is not bound by that decree.
But why is he not bound? It was his duty as trustee to represent and protect the holders of these bonds; and for that reason he was made a party, and he faithfully discharged that duty. It would be a new and very dangerous doctrine in the equity practice to hold that the cestui que trust is not bound by the decree against his trustee in the very matter of the trust for which he was appointed. If Corcoran owned any of these bonds and coupons then, he is bound, because he was representing himself. If he has bought them since, he is bound as privy to the person who was represented. Kerrison, Assignee, v. Stewart et al., 93 U.S. 155, and the authorities there collected.
It seems to us very clear that the question we are now called on to decide has been already decided by a court of competent jurisdiction, which had before it the parties to the present suit; that it was decided on an issue properly raised, to which issue both complainant and defendant here were parties, and in which the appellant here was actually heard by his own counsel; and that it therefore falls within the statutory rule of law which makes such a decision final and conclusive between the parties, and that none of the exceptions to that rule exists in this case.
Decree affirmed.
MR. JUSTICE CLIFFORD dissented.
|
{
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Originally published by Haïti Liberté as “Five years post-quake: Haiti’s promised rebuilding is unfulfilled as Haitians challenge authoritarian rule.”
Five years after the Jan. 12, 2010 earthquake that struck Haiti’s capital region, the loudly-trumpeted reconstruction of the country is still an unrealized dream.
2015 finds Haitians fighting tooth and nail in renewed political mobilizations to create the nation-building project that big governments and aid agencies pledged but then cruelly betrayed.
North American and European powers rushed planeloads and shiploads of soldiers and bottled water to Haiti in the days and weeks following the disaster, saying they would help Haiti “build back better.” The world was aghast at the rare glimpse of Haiti’s poverty provided by earthquake coverage. Leaders like Bill Clinton even acknowledged that the failed economic policies they had imposed over decades had impoverished Haiti and, indeed, are the source of its economic underdevelopment.
But the promises of the multi-billion dollar international relief effort and aid which will reach the grassroots have proven largely illusory.
A key admission in the months following the earthquake was that democratic governance and national sovereignty were essential tools for building Haiti on a new and progressive foundation. Today, the lack of democracy and sovereignty is at the epicenter of the political firestorm sweeping the country.
For many months, the Haitian people have carried out a sustained political mobilization demanding President Michel Martelly’s resignation. They want elections, now postponed for over three years, to bring a new government and parliament that is not afraid to take up the unfinished tasks of post-earthquake reconstruction.
The protest movement calls itself “Operation Burkina Faso,” inspired by events in that west African nation. In October, the people of Burkina Faso overthrew an unpopular president, Blaise Compaoré, and his government. Haitians draw inspiration from that event and, crucially, are aware that it is inspired by the socialist, egalitarian and anti-imperialist ideas of former president Thomas Sankara, killed and overthrown by Compaoré’s forces in 1987.
Haiti’s movement scored an important victory on Dec. 13 when Martelly’s prime minister, Laurent Lamothe, resigned. But Oxygène David, a leader of the Dessalines Coordination (KOD), one of the parties leading the protests, told Haiti Liberté weekly, “Lamothe was just the smallest part of a trinity holding Haiti down. The other two elements are Martelly and MINUSTAH. They also must go for Haiti to have democracy and sovereignty.”
MINUSTAH is the UN Security Council military occupation regime that deployed in Haiti in June 2004 to consolidate the Feb. 29, 2004 coup against Haiti’s progressive and elected President Jean-Bertrand Aristide.
The next wave of large protests is planned to take place in cities across Haiti on Jan. 12, the earthquake’s fifth anniversary.
Foreign occupation and the slide to authoritarian rule
Three factors are driving the protest movement — Martelly’s march towards authoritarian rule since coming to power in March 2011, the ongoing MINUSTAH occupation, and the failed record of earthquake reconstruction.
Although two presidential elections have been held in the years following the 2004 coup against Aristide, both Presidents René Préval and Michel Martelly have been dominated by and essentially subservient to imperialist powers. This weak state was dramatically symbolized by the partial collapse of Haiti’s iconic century-old presidential palace in the earthquake. It could not be salvaged and has been razed.
Right after the quake, the U.S., Canada, and Europe rushed Haiti into a election which they brazenly meddled in to establish even stronger neo-colonial rule. A two-round presidential election in November 2010 and March 2011 brought Martelly to the presidency, but only after the Organization of American States (OAS) intervened and illegally changed the outcome. The largest political party in the country — Aristide’s Fanmi Lavalas — was excluded, producing the lowest voter turnout of any polling in the Western Hemisphere’s history. The election was entirely financed from abroad.
President Michel Martelly finally managed to get his long-time business partner Laurent Lamothe named as prime minister, and the two declared Haiti “open for business,” meaning that foreign, sweatshop factory investment was to be Haiti’s economic salvation, complemented by foreign aid and charity. Public sector intervention to tackle housing, healthcare, education, and other emergency needs was eschewed.
Martelly was a close ally of the extreme right-wing that twice overthrew Aristide in 1991 and 2004. He honored former tyrant Jean-Claude Duvalier, who was driven out of Haiti by a popular uprising in 1986 and was content (and permitted) to live in France until his embezzled funds ran out and he returned to Haiti in January 2011.
Martelly’s family faces widespread allegations of corruption, including abuse of authority, money laundering, and the squandering public funds. But the Haitian people are also alarmed by Martelly’s steady march toward authoritarian rule. Martelly and Lamothe found excuses not to hold parliamentary and municipal elections, allowing electoral mandates expire. Rather than bargain in good faith with his political opposition to create a provisional electoral commission (CEP) to oversee democratic elections, Martelly sought to create a “permanent” CEP, stacking it with his partisans.
On Jan. 12, 2015, the mandates of most Parliamentarians expire, effectively dissolving the legislative branch. Martelly says he is then prepared to rule by decree.
In the past week, Martelly has nominated a controversial prime minister and concocted a political accord that would extend parliamentary terms and guarantee his own survival until May 14, 2016, but as we go to press, six vanguard senators have refused to vote, saying the prime minister and political map forward should come from the opposition and parliament, not Martelly’s back rooms.
Haiti Liberté‘s Thomas Péralte reported on Dec. 31 that large political protests (for Martelly’s resignation) took place for the first time ever in Haiti on Christmas Eve. Protesters said there is nothing to negotiate with the doomed regime, some saying they would prefer “civil war.”
Cholera and public health care
Tens of thousands of people died in the earthquake, and half the houses in Port-au-Prince, with a population of nearly three million, were destroyed or seriously damaged. Acute needs were intensified – for health care, sanitation, housing, public education, and economic development (including agriculture).
Early gains in earthquake relief were achieved with the public health initiatives taken by Haiti’s Public Health Ministry in cooperation with international missions, particularly those of Cuba (working in Haiti since 1999), Partners In Health (present since the 1980s) and many smaller, vital health care projects.
Cuban personnel and hundreds of students and graduates from other countries of the Latin American School of Medicine in Haiti fanned out into some of Haiti’s remotest parts to meet new and existing medical needs.[1] Other Latin American countries made substantial contributions to the Cuban-led health care effort. Cuba proposed a plan to the UN to create a comprehensive, public health care program for the country.
The Boston-based Partners In Health (PIH) expanded its work substantially, including building a second training hospital, opened in Mirebalais in 2012. PIH, too, voiced support and hope for a public health plan.
Tragically, the advances in building medical infrastructure suffered a huge blow in the autumn of 2010. The culprit was Haiti’s familiar old nemesis — foreign political intervention. MINUSTAH soldiers recklessly and criminally introduced cholera into the country when a Nepalese contingent allowed their cholera-infected sewage to flow into Haiti’s largest river system in October 2010. Over four years later, cholera has killed 8,500 people and sickened nearly 800,000, the world’s worst epidemic. The number of reported cases monthly was averaging 2,000 in 2014 but jumped in the latter months of the year.
Although UN Secretary General Ban Ki-Moon has promised money and resources to combat and eventually eradicate cholera, a report one year ago by the Washington DC-based Center for Economic Policy Alternatives noted, “The UN itself has pledged just one per cent of the funding needed for cholera treatment [estimated $2.2 billion], even as the UN’s mostly military and police mission in Haiti costs over $572 million a year”.
A recent report by Doctors Without Borders (MSF) blames those in authority in Haiti for persistent “shortages of funding, human resources, and drugs” in Haiti’s health care system, including for cholera. The UN as well as the major governments participating in MINUSTAH are denying any culpability for introducing cholera to Haiti and then failing to assist in its prevention. Cholera is easy to treat and prevent if there is the will and funds. It just requires potable water delivery and sanitary sewage disposal. That’s why people in New York or Toronto don’t get or die from cholera.
The cholera disaster only deepened the festering wound on Haiti’s body politic known as MINUSTAH. The continued presence of the force is an affront to the dignity and sovereignty of the Haitian people. [2]
The Housing Crisis
Housing was another of the most immediate needs in Haiti following the earthquake. International aid provided short-term shelters to protect from the elements. A reported 110,000 plywood shelters and tens of thousands of tent shelters were provided. Beginning in 2011, one-year rental subsidies were provided to families as an incentive for them to leave tent camps. The camps were an eyesore as well as visible testimony to the absence of substantial programs to build housing.
After mountains of studies highlighting the need for a massive home-building program in Haiti, the gains are few. According to a recent fact sheet on housing prepared by Church World Service and the Mennonite Central Committee (drawing on figures reported to UN agencies), some 85,000 earthquake victims still live in 123 camps of internally displaced persons within Port-au-Prince’s city limits.
Many tens of thousands more live in the new, sprawling informal suburban shantytowns of Canaan, Onaville, and Jerusalem, located beyond the pre-earthquake northern limits of the city. By a stroke of a pen, these communities are not considered as earthquake survivor settlements. That also means they don’t qualify for formal assistance.
Thirty four per cent of the families that left survivor camps were forced out by people claiming land ownership or by government officials. Twenty two of the remaining camps face eviction.
The aforementioned fact sheet reports that in the past five years, 27,353 houses have been repaired and 9,053 have been built, at a cost of $215 million. That amount compares to $500 million spent on the plywood shelters, most of which have long since deteriorated in the tropical weather or have been dismantled to build more permanent structures.
The UN-sponsored housing coordination body said in 2013: “Haiti needs to meet the challenge of constructing 500,000 new homes in order to meet the current housing deficit between now and 2020.”
The key instrument of Martelly’s housing “policy,” in keeping with the “Open for business” mantra, has been promises of financing for house construction. No housing agency of the government was created. But Haiti does not have networks of personal banking where people could obtain loans, and in any event, the proposal was laughable because most Haitians don’t have incomes to speak of. According to the updated country report on Haiti by the World Bank, more than six million out of Haiti’s population of 10.4 million live under the national poverty line of $2.44 per day. Over 2.5 million Haitians live under the national extreme poverty line of $1.24 per day. How are they to obtain loans to build houses?
In reality, the most active area of housing policy has been the clearing of survivor camps by force or by short-term economic lures. The latter has been facilitated by the Canada-funded, $20 million program of providing one-year rental subsidies.
Education
Public education was another key social need identified after the earthquake. Before the disaster, half of Haitian children did not attend school. The number reaching secondary school was much less. In 2011, the Martelly regime created a national education fund whose goal was said to get every Haitian child into school. It was to be financed by taxes on international phone calls and money transfers, which were never ratified or overseen by Parliament as constitutionally dictated. The plan has been plagued by a lack of transparency, and its achievements are very slim.
School administrators say that promised funding under the plan does not get delivered. Or it arrives months late. This year, the opening of the school year in September was delayed by a month because parents said they couldn’t afford to buy the textbooks and other supplies that schools were not supplying.
One of the outcomes of the fund, according to a lengthy investigation by Haiti Grassroots Watch published (in French) last July is that private schools have been favored over public schools. About 80% of Haiti’s primary and secondary schools are private, typically operated by churches and other charities from abroad.
Teacher unions in Haiti opposed the fund because it had no legislative authority and therefore operates outside of public oversight. Teachers have battled for years to establish a public education system and to pay teachers living wages. Last spring, strike action won salary increases of 30% to 60%, but salaries are still woefully inadequate.
Misguided economic development
Economic development was cited as key to Haiti’s future following the earthquake, including for agriculture. Most Haitians still live in the countryside, and those forced to move to the cities by economic circumstance have not done so freely. But international aid and governments never came close to fundamental change in this sphere. They rehabilitated the failed dogma that posits Haiti’s low-wage, factory labor force as an economic asset to be built upon. And they perpetuated the neglect of Haiti’s all–important agricultural production, including environmental decline prompted by deforestation.
A centrepiece of the sweatshop labor strategy promoted by former U.S. President Bill Clinton and current presidential aspirant Hillary Clinton is the Caracol Industrial Park, located far from the earthquake zone in Haiti’s north. It was touted to create tens of thousands of jobs when its idea was launched in 2010. But a 2013 investigation by reporter Jonathan Katz revealed that “fewer than 1,500 jobs have been created — paying too little, the locals say, and offering no job security.”
Katz reports, “Hundreds of smallholder farmers were coaxed into giving up more than 600 acres of land for the [industrial park] complex, yet nearly 95% of that land remains unused. A much-needed power plant was completed on the site, supplying the town with more electricity than ever, but locals say surges of wastewater have caused floods and spoiled crops.”
Assembly factories in the new park routinely pay below the meager US$4.76 average daily minimum wage. A report by the International Labor Organization (ILO) and the International Finance Cooperation (IFC) in 2013, which monitor and enforce factories’ compliance with national and international standards, found that all 24 of the factories it monitored in Haiti were “non-compliant”. All violate occupational safety and health standards. All violate minimum wage laws, and 11 violate overtime standards. None provide adequate health and first aid services, and 22 were in violation of worker protection standards.
And what has become of the billions of dollars of aid promised to Haiti” A report by CEPR in 2013 said that much of the aid earmarked for Haiti was not spent in Haiti at all; it went to foreign contractors. “67.1% of USAID contracts has gone to Beltway-based firms, while just 1.3% has gone to Haitian companies”, it wrote. And “of the $6.43 billion do-gooders by bilateral and multilateral donors to Haiti from 2010-2012, just nine percent went through the Haitian government.”
Writing in July of 2014, the CEPR reported that of the $1.38 billion awarded by USAID to projects in Haiti, just $12.36 million has gone to Haitian organizations. Of the Haitian amount, 57% went to Cemex Haiti, a local cement mixing outlet and subsidiary of the Mexican Cemex, the Mexican company that is one of the largest cement producers in the world. (Cemex purchased the former state-owned cement producer in Haiti some 15 years ago.)
A lot of celebrities and other prominent people have come and gone from Haiti over the past five years. Careers have been created or polished up by charitable works. The Clintons come to mind. Many Hollywood actors. Canada’s former governor-general (titular head of state), Michaëlle Jean, was a mouthpiece for the 2004 coup while she was governor general, then she became a Special Ambassador to Haiti for UNESCO following the earthquake. Recently, she rode rough over the objections of African countries to become the head of the Francophonie organization of French-speaking countries. What all these people as well as many other foreign do-gooders shared in common was their support for the political project keeping MINUSTAH and local clients (Martelly or some other derivative of him) in charge of the country, at the expense of the Haitian people.
CEPR Director Mark Weisbrot wrote one year ago that the lasting legacy of the earthquake “is the international community’s profound failure to set aside its own interests and respond to the most pressing needs of the Haitian people.”
But then there is the Haitian people – their mounting political actions and their unrelenting determination to build a country based on sovereignty and social justice. And their true and faithful international allies. Like the countries and healthcare projects mentioned earlier in this article. Like the lawyers of Bureau des avocats internationaux (BAI) and Institute for Justice and Democracy in Haiti (IJDH) who are suing the UN on behalf of the victims of cholera. Like the SOIL sanitation project and the organizations of peasants and farmers of Latin America who are working in the Haitian countryside. Like many school support projects which are an important form of the struggle for public education in Haiti.
These are the organizations who are working together with the Haitian people to help shape Haiti’s future.
Notes:
[1] For an early 2010 report of these efforts see ‘Field Notes from Haiti: After the Earthquake’, by MEDICC (Medical Education Cooperation with Cuba). [2] Read an eight page essay on the history of foreign intervention in Haiti: ‘ Haiti’s humanitarian crisis: Rooted in history of military coups and occupations ‘, by Roger Annis and Kim Ives, May 2011.
Here are places to go for information:
CHAN website, Haiti Liberté, IJDH
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I am a recent college graduate, and due to the bad economy have so far been unable to find employment, despite aggressively looking. I agree with Obama for what he's done for gay rights, but not the economy and jobs. I have been on Atripla since 2010 and thankfully Ryan White grant has enabled me to get my meds free of charge. So here's my question...exactly how does the Supreme Court's upholding of the Obama healthcare law affect those of us with HIV and who are on Ryan White? I also have insurance, but it hardly would make a dent in the cost of my HIV medication without Ryan White...theres a lot of bullshit and spin comng out of the mouths of politicians. So how does the healthcare law affect people like me who have HIV, and are on meds?
As fate would have it, I am due to go down to my clinic today for an appointment and some pills, so I am planning to ask about this directly today. I am also on RW and ADAP. I am wondering how this might change things up through my IDP.
I imagine others will have some feedback for you, but I'll try to post what I'm told later today or tomorrow.
RW/ADAP was renewed up until 2014, after that things may/probably will change
the American Academy of HIV Medicine explains it well on their site
Quote
Starting in 2014 thousands of Ryan White patients who previously received care and treatment through the Ryan White program will begin to receive coverage through other programs created under the ACA. A majority will become part of the new Medicaid expansion; another portion are likely to gain coverage through the State Insurance Exchanges and the new state Pre-Existing Condition Plans (PCIPs).
Because Ryan White is a “payer of last resort” program, those patients who will have access to the new programs must use them first.However, since the states will have significant flexibility in establishing the new programs, it is possible that some former Ryan White patients may end up with insufficient benefits, inadequate access to medications, or possibly even less than adequate access to care. They may also see new or increased co-pays, deductibles, or other cost-sharing. It is possible that the Ryan White Program may be able to aid in supplementing some of these areas.
Ryan White providers and clinics will likely help transitioning into these new systems, and ensuring inclusion in the new networks of care. Providers need to pay attention to the requirements and opportunities in their state to be included in the billing networks, referral networks, and care networks for their state’s Exchanges and state Medicaid program.
Most of the provisions of the ACA do not take effect until 2014, and their overall effects of ACA implementation will not become clear until sometime after that. The Ryan White Program as a whole will likely see many changes. Some of these may include the types of patients accessing services, to what services are provided, how those services wrap around the new systems. However, it seems as though there will ultimately be a significant need for the Ryan White safety net after 2014.
The Ryan White Program was reauthorized in 2009 through September 30, 2013. One positive change in the 2009 reauthorization was the removal of the “sunset” provision that had existed under previous reauthorizations.
My pharmacy bill is just over $3600 each month which is 3X my SSDI income. It is paid by Medicare Part D, and other insurances. The other insurances are Medi-Cal and ADAP.
A few years ago the Medi-Cal budget was cut so hard our ASO had to terminate Nurse visits. The ASO was also hit with the loss of RWCA Title I funding and struggle to serve a growing list of clients. In addition, Medi-Cal cut alternative medical practices, dental, vision and chiropractic. Our ASO had to close it's Russian River office and their main office which was once thriving with program, support services and staff, now looks like ghost town parade. I have been self case managed for about 5 years. That was Sonoma, they didn't put up much of a fight for their funding.
In San Francisco, which remains a Title I EMA, 6 million was taken from the budget. Mayor Ed Lee has replaced the Title I funds for AIDS services in San Francisco for 2 years.
Being already more than 133% below the poverty level, a 1/2 price reduction in the cost of medications are still not affordable. I will be really old in 2020 and wonder, who is going to pay for my medications then? Have the best dayMichael
In San Francisco, which remains a Title I EMA, 6 million was taken from the budget. Mayor Ed Lee has replaced the Title I funds for AIDS services in San Francisco for 2 years.
Being already more than 133% below the poverty level, a 1/2 price reduction in the cost of medications are still not affordable. I will be really old in 2020 and wonder, who is going to pay for my medications then? Have the best dayMichael
Micheal honey, by the yr. 2020 I'll probably be DEAD ( I'm 55+ now) so I'm not worried about whats happened then, I can't even think past next-week, the thought of it makes my head-swell
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"it's so nice to be insane, cause no-one ask you to explain" Helen Reddy cc 1974
Being already more than 133% below the poverty level, a 1/2 price reduction in the cost of medications are still not affordable. I will be really old in 2020 and wonder, who is going to pay for my medications then? Have the best dayMichael
Its a sobering thought isn't it . While I'm very happy for all the people the health care law will impact in a positive way Im aware that for some of us on medicare who still face the same problems shouldn't count on much more happening for a long time in regards to any new reforms in heath care , its a great start but we sure have a long way to go before we feel the relief . No pun intended but we must stay positive .
Its a sobering thought isn't it . While I'm very happy for all the people the health care law will impact in a positive way Im aware that for some of us on medicare who still face the same problems shouldn't count on much more happening for a long time in regards to any new reforms in heath care , its a great start but we sure have a long way to go before we feel the relief . No pun intended but we must stay positive .
Hey Jeff,
He has the tenacity, I read the President's books last year and was impressed with his Senate campaign. He walked neighborhoods, knocked on doors, stopped people on the street and spoke at every opportunity to find out what the American people want and need. Access to healthcare for all! The President has my vote and I am just looking at the possible details and how they concern me. Have the best dayMichael
I am not sure how any of this will work out. My new case workers is so green behind the ears she actually believes we have laws that protect people. lol....Anyway, she told me they are under great pressure to get people off RWhite and onto the high risk pool for the state. I looked it up and you can not get a straight answer from their own web site. It will be anywhere from 199 to 600 per month for limited coverage and a one million dollar cap. There will be co-pays for everything.
She informed me that under the risk pool that R.White would pay the premiums and co-pays and still save money over the old method. This made no sense to me, how can they save money by paying the premiums and co-pays. They also will allow a total income of 32 thousand gross and remain in the pool.
Under R White I am currently allowed two doctor visits per year and they pay for my meds.
I am going to give it one more try possibly with someone who is older and has more experience in the system but this may prove difficult as the case workers tend to hang around for a year or two at the max and then you get the nice letter saying they have moved up in the organization and I would be getting a new case worker.
She informed me that under the risk pool that R.White would pay the premiums and co-pays and still save money over the old method. This made no sense to me, how can they save money by paying the premiums and co-pays.
Right now, RW often pays the premium and co-pays for people who are financially eligible for ADAP. Since many HAART regimens cost around $2500 to $3000 per month, covering someone's insurance premium and co-pays is a lot less expensive than having to purchase the meds outright.
I am going to give it one more try possibly with someone who is older and has more experience in the system but this may prove difficult as the case workers tend to hang around for a year or two at the max and then you get the nice letter saying they have moved up in the organization and I would be getting a new case worker.
you may not get many answers right now. AHA won't be going into full effect, especially in regards to RW, until the RW act comes up for renewal in 2014. By then many things will have changed as some states will have created high risk pool, expanded their medicaid programs and created health care exchanges; while other states won't have any of that and won't be accepting any federal funds.
if you re-read the large quote I posted from the American Academy of HIV Medicine website, and realize that it was just this last week the Supreme Court finally upheld the AHA so that it can now aggressively move forward, you'll understand that many of the details you're looking to find are probably just now starting to be worked out for a hopeful transition in 2014. I hope you can find out more about your situation; but in the middle of an transition from RW to AHA there may not be many answers for you quite yet.
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leatherman (aka mIkIE)
All the stars are flashing high above the seaand the party is on fire around you and meWe're gonna burn this disco down before the morning comes- Pet Shop Boys chart from 1992-2015Isentress/Prezcobix
BC/BS offers 3 plans Basic, Enhanced, and Premium. I can not afford any of them today. I called the toll free number for a quote from an Agent. She said: Nothing is in the works now because the election is close and Obamacare will be replaced with whatever Mitt Romney has planned.
The advantage to being on Medicare/Medi-Cal for prescription coverage, over the person who already has private insurance is... The Medi/Medi client can change coverage any time of the year, while the private insurance client can only change vendors at the beginning of the year.
The Formulary, plan pricing, premiums and deductibles will be available to the Sales Staff on October 1st for the year 2013. On October 1st, 2013, the plan with pricing for 2014 will be available. We know nothing on this for 2 years. Have the best dayMichael
As a person who uses Medicare, Medi-Cal and ADAP for prescription services
Micheal.......what is the current-cut-off level for Medical, I had it for about 2 yrs. when I 1st got my SSDI back in 98...is it still 20k a yr. last i recall it was 17K or 18K a yr. but that was almost 12 yrs ago when I had it in Nor-Cal
EDITED TO ADD: I did have ADAP also back then, but in Cal, Adap has no income level cut-off, as far as I know they don't, and they have a very nice Formulary, way better than in any other State except maybe NY-State..........duuno
« Last Edit: July 07, 2012, 07:41:26 PM by denb45 »
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"it's so nice to be insane, cause no-one ask you to explain" Helen Reddy cc 1974
Micheal.......what is the current-cut-off level for Medical, I had it for about 2 yrs. when I 1st got my SSDI back in 98...is it still 20k a yr. last i recall it was 17K or 18K a yr. but that was almost 12 yrs ago when I had it in Nor-Cal
EDITED TO ADD: I did have ADAP also back then, but in Cal, Adap has no income level cut-off, as far as I know they don't, and they have a very nice Formulary, way better than in any other State except maybe NY-State..........duuno
-----------
As I know of things, ADAP in CA has an income level cut-off of one's Federal Adjusted Gross Income of $50,000.
Gentlemen, I have saved files for years and at this point can not find the income guidelines for Medi-Cal or ADAP. I have found a scale w/ 1-10 member families, where a 1 person family has to make $903.00 per month (last year). That same 1 person family may not have over $3000 in property but... I can not find a maximum income amount to qualify in the documents I have filed. Have the best dayMichael
Once I got medicare A, B in 2000, medical told me, that you are on your own, I only got that when I didn't have any health Ins. after my Cobra & Obra run out, due to having a per-exstsing condition...Full-Blown-AIDS
My guess ( If I wanted to move back to Nor-Cal) I would still get ADAP, but NOT medical, and I would probably have to sign all of my medicare over to an HMO, County Hospital, or something else to get Health care, like Labs, and Doctors Visits, I had Kaiser Permanente back then with co-pays for these such things....
If things go really south & downhill for me, I can always use the VA
dunno, I feeling very home-sick for Nor-Cal, and kinda done with ABQ, I don't wanna die here, as I'm not a real native of New Mexico, Nor-Cal is my home and I miss my family
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"it's so nice to be insane, cause no-one ask you to explain" Helen Reddy cc 1974
Today, I was at the Pharmacy picking up my monthly refills. The Pharmacist meets me once each month for less than 5 minutes and always remembers my name. Not bad in a city of 722,000 people... She does not know how much will be paid by whom in 2014 but will contact all of the client/patients when they do know. Not much more can be done until October 1, 2013. Have the best dayMichael
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“The phenomena of ‘paid news’ is one of increasing concern for the EC, especially during polls run-up.”
—Navin Chawla, CEC
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“The rot has set in in the system...language papers emulated the leading ones, which started the trend.”
—Justice G.N. Ray, Press Council of India
“Ultimately, the superintendence of elections rests with the EC and it should look into ‘paid news’.”
—Rajdeep Sardesai, Editors Guild
***
It was April last year and the general elections had just been announced when marketing whiz kids in the media saw a golden opportunity springing up. Poll campaigns had begun to roll out and they ran the whole gamut, from asking politicians to pay a premium for favourable reports to flattering interviews on TV to running down opponents who refused to oblige. In essence, media marketers worked overtime to milk netas and political parties, raking in the moolah even at a time when the general economy was reeling under a severe recession.
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While it is generally believed that the malaise of publishing news in return for money had kicked in way back in General Elections 2004, it was the scale of the phenomenon of paid news in 2009 that made many wake up and take notice. This has now forced statutory bodies like the Press Council of India (PCI) and the Election Commission of India to take note and try to set in place “guidelines” to rein in the practice. Chief election commissioner Navin Chawla says that “the EC proposes to convene a meeting of recognised national political parties shortly to discuss the issue”. The PCI has called for a meeting with editors of leading national dailies on January 12. The Editors Guild will also be seeking an appointment with the Election Commission to discuss the issue.
There will be many questions on the agenda when the EC and the PCI hold discussions on the ‘paid news’ syndrome. Sale of editorial space is lucrative business even though it compromises on objectivity. How, then, is one to ensure that the news in the day’s papers is not paid for. More importantly, how can editors and managements be persuaded to keep a distinction between news, advertising and marketing supplements?
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But for things to even get off the ground, the body entrusted with the mandate of watching over the media has to have some teeth. It is widely agreed that the Press Council has very little powers, apart from issuing strictures against publications. In effect, its writ has no legal standing even if it identifies an unhealthy trend in the media.
Justice G.N. Ray, who heads the council, had even earlier expressed concern at the malaise of paid news. “The rot has set in in the system and, unfortunately, language papers emulated the leading papers, which had started the paid-news trend...this reached a peak during the recent elections. But the Press Council is trying to assert itself,” he says. Now the council is in consultations with the Election Commission and will be examining the issue. The Press Council is, however, still in the process of collecting and examining “paid news items”, for which it has set up a committee.
Meanwhile, what is the Election Commission doing? In its note to the PCI on December 16, it has acknowledged that it received “informal” complaints from politicians but also admitted that some of its proposed recommendations have not been acted upon by the government of India. The commission cites an instance in which a “national daily”, on being asked to furnish details of a surrogate advertisement, refused to oblige. “On the face of it, such advertisements give an impression of a genuine news report covering the election campaign of a particular candidate. But when such reports repeatedly appear in a newspaper, more or less on a regular basis, the matter does give rise to doubts about whether it was honest coverage,” the note from the commission said. Incidentally, a bjp delegation met the CEC recently and submitted a memorandum asking the commission to probe the Ashok Chavan/Maharashtra elections imbroglio, which sparked off the whole paid-news imbroglio.
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Again, while the rules governing the Representation of People’s Act (RPA), 1951, clearly state that no person shall print or cause to be printed any election pamphlet or poster unless a declaration as to the identity of the publisher thereof, signed by him and attested by two persons to whom he is personally known, is delivered by him to the printer in duplicate and handed to the chief electoral officer or the district magistrate—the note states that rarely are copies of posters etc furnished to the district magistrate. Quite clearly, it is the EC that needs to be reminded of its own responsibility at the time of elections, because its writ runs supreme.
Interestingly, the commission has directed the PCI to determine what is paid news, so that the expenditure on such news becomes accountable. Sources in the EC admitted that they did not wish to be seen as overstepping their limits. “Our role only comes into play once the election dates are announced and the model code of conduct kicks in,” a senior official in the commission told Outlook.
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But as was noted during the assembly elections in Andhra Pradesh, politicians complained bitterly that the Election Commission did not give adequate importance to the issue. CEC Navin Chawla admitted to receiving complaints from political parties. “The phenomena of ‘paid news’ is one of increasing concern for the Election Commission, especially during the run-up to elections. We are awaiting the proceedings of the Press Council of India’s recent meeting in this regard. The Editors Guild is coming to see us on this subject shortly. If readers lose faith in the printed word, especially at election time, then an important pillar of democracy will stand compromised,” Chawla says.
But Paranjoy Guha Thakurta, member of the two-man inquiry committee set up by the Press Council to examine paid news, asks, “How does one prove that a monetary transaction has taken place? All such transactions are clandestine and one can only go by circumstantial evidence.” The council has invited editors of leading publications on January 11 to put in place a mechanism of check and balances. Rajdeep Sardesai, president of the Editors Guild, says, “Ultimately, the superintendence of the elections rests with the Election Commission and it should look into ‘paid news’.”
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Ironically, the EC had recommended a crucial change in the RPA, which could have gone some way to ensure that advertisements and news are distinct from each other. “In the case of advertisements/election matter for or against any political party or candidate in the print media during the election period, the name and address of the publisher should be given along with the material/advertisement,” the Commission had suggested. Till date, the government has not responded.
So, will the current exercise in reining in paid news pay any dividends? Most in the industry feel that unless changes are made in the RPA and organisations like the PCI are given powers, there will be no significant change. Perhaps newspapers and TV channels may become more careful in selling editorial space. But business, many say, will go on.
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I don't know why such a hue and cry NOW only, and about "paid news" only. Another one of the many ways to corrupt the press is to give them very expensive , genuine advertisements: e.g., on the inaguration of a railway station in some remote area (say, in the North East), by some cOnGRESS person. The full page ad will appear in some national newspaper, in its Delhi edition, on the morning of the same day as the scheduled inauguration. "All are welcome" to attend the inauguration.
When our MPs and MLAs are paid for their vote even on nationally important issues, what more hell is going to happen if news are paid for? For sure our freedom has been vandalised by feudal family interests who are perpetually in power. Corruption is tolerated at all levels and the system allows the corrupt to continue in power even trial based on prima facie evidence goes on endlessly in courts. See the present scenario despite food prices sky rocketing, the voters can hardly do anything democratically to force our legislatures (many of them having vast business interests) to arrest it. Congress need not fear till next general elections and by that time will come out with another great 'waiver' thamasha. Till such time we get the right to call back our worthless reps and we limit their service to two terms, this tragedy will continue.
It has been a well acknowledged fact of selection of 'KHAS' reporters and correspondents to help out and report flattering news of the contesting candidate as seen in all recent elections of the country. Ask Mr. Vinod Mehta any candidate on 'moral oath' whether he be it a winner or a loser did not go for the press on such counts? T. Vinod
The media lacks objectivity in the first place.It usually favours one political party or other in pursuance of its own political agenda.So,when the media prostitutes itself in this fashion it is no big deal if it does such things for money.
>There are limitations to what governmental bodies can do to exclude paid news from the news-space.
There are umpteen numbers of ways government can do to control the wayward media & its excesses. Question is whether govrnments have any intention to do so. Absolutely no. Media on the other hand serves as an impotant adjunct to government misdeeds.
For example never before overall governance standards fell to such appalling low as it is now during the current regime & its vegitating head who seems have taken a hands off policy in all matters of governmance. This old gentleman with his antedeluvial economic sense & with no faith in welfare state, only becomes hyperactive only in when matters which serve the interest of money bags & a superpower. 90% percent of his cabinetis packed with representatives of special interest groups - some mining, some aviation, some energy, some real estate, some telecom, some sugar & the list is endless.It was alwyas so, but has never been so blatant , all pervasive & so corruptive.
By diverting & refocussing on perepherial issues such sexual escapades of politicians or a particular criminal police officer , media takes off the heat off politicians from common man, the beast of burden which is reeling under the pressure of 30% increase in prices of essentials.You will not hear or see a whip of mention in media about this unprecedented engineered price rise of essentials .
So media & degenerating politicians are in cahoots. Thinking that government at all wants to control media is simplistic.
What is the point discussing about smaller language print/tv channels? Just look at so called national dailies/tv channels specially English language? There are enough media took interview of Priyanka Vadra just before the general election talking about her father’s killer? What was that? I am sure everybody know it is method of publicity to garner sympathy vote? What about CNNIBN? Their whole profiling of politicians?
Why gun down smaller mediums? What about VM himself? Sometimes it is paid by Indian politicians and some other times by some global vested interest parties!
There are limitations to what governmental bodies can do to exclude paid news from the news-space. There are all kinds of ways to avoid being caught. It is up to the press itself to monitor and regulate itself. It is also up to an intelligent readership to shun corrupt outlets. The power of money is overwhelming. Look what it has done to the U.S. Congress!
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Q:
Can you give an example why the active player wants to have priority in Beginning of Combat first?
I've read this article published on blogs.magicjudges.org explaining the combat shortcut which is about (contrary to the normal proceeding) the Not Active Player (NAP) receiving priority prior to the Active Player (AP) in the Beginning of Combat step (BoC).
This is also stated in the Magic Tournament Rules:
If the active player passes priority with an empty stack during their
first main phase, the non-activeplayer is assumed to be acting in
beginning of combat unless they are affecting whether a beginning of
combat ability triggers.Then, after those actions resolve or no
actions took place, the active player receives priority at the
beginning of combat. Beginning of combat triggered abilities (even ones that target) may be announced at this time.
At the end of the article the author mentions that normally this is not a problem since most of the time the AP does not want to act first anyway. However, one scenario in which the AP wants to be the first one to receive priorioty in the BoC step is mentioned:
AP Acting First in Combat
The new structure makes it look like the active player can’t be the
first person to act in the beginning of combat step. That’s not true,
but it does reflect the fact that the active player needing to act
first is unlikely. The only scenario I’m aware of is holding a split
second spell while your opponent is floating mana, which is not
something that’s going to come up every day! In that situation, the
protocol is the same as ever – you ask your opponent if they want to
do something with that mana in the main phase. If they do, you’re
still in main phase, since they used mana they couldn’t use in
beginning of combat, nullifying the default.
Otherwise, there is a way to do it, but it does give the opponent some
information. While in your main phase, simply say “I do this thing in
Beginning of Combat”. Done! Of course, the non-active player has the
ability to interrupt and do something in your main phase. That’s not
really any different than it was under the previous shortcut.
However, I can't reconstruct an example for such a scenario from the information given in these two paragraphs. The following information is given as stated in the paragraphs:
NAP has mana floating in the first main phase
AP has a split second spell
Question: Can you give an example for this scenario in which in order to reach his or hear goal, AP needs to be the first one to to receive priority in BoC to then cast the split second spell?
To illustrate the difficulties I have with this, here is an example which doesn't work:
AP is in his main phase, controls a Grizzly Bears enchanted with a Rancor. He has a Sudden Spoiling in his hand and enough mana to cast it.
NAP has 2 life, controls an Endbringer and has one mana floating.
AP wants to have priority in BoC to then cast Sudden Spoiling. He does not want to cast Sudden Spoiling in the main phase because NAP has floating mana he or she can use to respond. If everything goes well, AP then can attack with his bear (4/2, trample) and win the game (NAP now has a 0/2 creature).
This example doesn't work because of several aspects:
Why does AP need to have priority in BoC to cast Sudden Spoiling before NAP receives priority? Why does AP not let NAP have priority first in BoC, NAP then passes priority, then AP uses his priority to cast Sudden Spoiling?
Why doesn't AP cast Sudden Spoiling in his main phase? Sure, NAP has mana floating, but he can't use it to activate Endbringer's cant-attack-ability in response anyway because Sudden Spoiling has split second.
A:
I think the two different pieces here are in some sense independent. The opponent's floating mana makes you want to act in the beginning of combat step instead of the main phase, and holding a split second spell makes you want to act first in the beginning of combat step instead of waiting for your opponent to act.
Holding up mana here isn't necessarily about responding to the split second spell; it's more generally about having more options for actions to take, perhaps after the spell resolves. If you force the mana to empty from their mana pool, you cut off those options.
So, here is a scenario in which it matters that you specifically act first during the beginning of combat step: The active player has some mana available and Extirpate in hand, and two attackers with at least two toughness each. The non-active player has one floating red mana, two islands, Electrolyze in hand, and Feeling of Dread in their graveyard. The non-active player is better off casting Feeling of Dread, but they don't want to cast it during their main phase because then the active player could follow up with another creature, possibly with haste.
If the active player plays Extirpate during their main phase, the non-active player can use their floating mana to cast Electrolyze. If the active player waits until the non-active player acts in the beginning of combat step, the Feeling of Dread is no longer in the graveyard for the Extirpate to target. They get the best result if they act first in the beginning of combat step.
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Bob Kravitz
USA TODAY Sports
INDIANAPOLIS — Would it be impertinent to suggest that Indiana Pacers coach Frank Vogel might want to rest his starters once again? Like, in Game 2 of this first-round playoff series against the Atlanta Hawks?
Although the case could be made that the Pacers' collapsing, disintegrating starting unit — especially Roy Hibbert and George Hill — already took the night off Saturday during an embarrassing and thoroughly dispiriting 101-93 Game 1 loss to the Atlanta Hawks.
Somewhere, Tom Petty is crooning Free Falling.
Indiana has become the most confounding team in the NBA, a No. 1 seed that has completely lost its way this second half of the regular season. Suddenly, the Pacers seem poised for a very hard and inexplicable fall. It's too early to write the epitaph on a once-great season gone terribly wrong, but the time is getting close.
"Now we'll find out what we're made of," Hibbert said in a quiet postgame locker room.
Here's the crazy thing: This wasn't a surprise. If you've watched the Pacers closely the second half of this season, this wasn't completely unexpected. This team has lost 13 of its past 23, and two of those victories came with Vogel playing his reserves in an effort to provide the starters with rest.
They said they had to keep Hawks point guard Jeff Teague out of the paint. They didn't. Not once. Hill consistently got worked over and didn't get much help from his teammates.
They said they needed Hibbert to make Atlanta's Pero Antic pay on the defensive end. They didn't.
They said they needed to play with force and passion, yet they didn't corral a single loose ball and somehow gave up 14 offensive rebounds to an undersized team that is near the bottom of the league in offensive rebounding.
They said, as they've been saying all season, that they had to guard without fouling. They didn't, as the Hawks went 24-for-29 from the free-throw line, with 11 attempts by Paul Millsap and 10 by Teague.
They said they had to keep Kyle Korver under control from behind the three-point arc, yet there was Korver, time and time again, getting wide-open looks.
This team is in a funk, a fog, and it gives so indication it's inclined to break out of it any time soon.
By now, Vogel has tried everything to wake his team up during this second-half origami act.
He's whispered sweet nothings into the players' ears.
He's brought down the hammer.
He benched Hibbert against Atlanta the last time the teams met, and he benched the entire starting five in a game against the Milwaukee Bucks.
Then he tried to use the placebo effect, giving them several days off from practice and games in the hope that they would somehow believe that fatigue was their foe, and that time off would be the tonic for the troops.
Nothing has worked.
This was nothing less than a carryover from the last two or three months of a season that is rapidly slipping away. Somewhere along the way, they lost it, lost their cohesion, lost their mojo, lost their implicit trust in one another. And they can't seem to get it back.
What does Vogel do now? Desperate times, desperate measures.
It's time to think about fighting Atlanta's small lineup with a small lineup of their own. It's time to think about sitting Hibbert and starting Ian Mahinmi or playing a lineup that features David West and Luis Scola at power forward and center. It's time to give C.J. Watson more minutes; Hill is in his own sort of funk, and failed repeatedly to contain Teague. And, what the heck, why not find minutes for the forgotten one, Chris Copeland?
This is no time to get bull-headed, or their season will be over before you know it.
"We're going to stick with what we have, but in the playoffs, you've got to contemplate everything," Vogel said. "We've got a difficult matchup with a team that has a unique offensive attack, so you consider everything."
West was asked if it's time to make some seismic changes, either in the starting lineup or in the rotation.
He shook his head.
"Right now, no," he said. "We can't change who we've been all year."
All this time, all this season, they've been talking about earning the No. 1 seed and maintaining home-court advantage, and yet, here they are, locked in a series where they've already ceded home-court advantage to a team that finished 38-44.
It wasn't like the Hawks came into Bankers Life Fieldhouse and stole Game 1. No, they took Game 1, and grabbed the Pacers' lunch money in the process. This was a pretty thorough thrashing, and the Pacers' lack of fight through the second half was especially galling.
"I think once we eliminate them from spreading us out like they did (Saturday night), we'll be all right,'' Paul George said. "…(The Hawks) played as good a game as they can possibly play."
Empty words. It's all the Pacers have to offer right now. If they weren't ready for Game 1 — especially after getting embarrassed the last time the Hawks came to town — what makes anybody think they'll get this thing figured out in time to save this season?
The collapse isn't complete, not yet, but you can see it from here.
Bob Kravitz writes for The Indianapolis Star.
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---
abstract: 'We discuss the isospin symmetry breaking (ISB) of the valence- and sea-quark distributions between the proton and the neutron in the framework of the chiral quark model. We assume that isospin symmetry breaking is the result of mass differences between isospin multiplets and then analyze the effects of isospin symmetry breaking on the Gottfried sum rule and the NuTeV anomaly. We show that, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model. We also find that the correction to the NuTeV anomaly is in an opposite direction, so the NuTeV anomaly cannot be removed by isospin symmetry breaking in the chiral quark model. It is remarkable that our results of ISB for both valence- and sea-quark distributions are consistent with the Martin-Roberts-Stirling-Thorne parametrization of quark distributions.'
author:
- Huiying Song
- Xinyu Zhang
- 'Bo-Qiang Ma[^1]'
title: Isospin Symmetry Breaking in the Chiral Quark Model
---
Introduction
============
Isospin symmetry was originally introduced to describe almost identical properties of strong interaction of the proton and the neutron by turning off their electromagnetic interaction, i.e., their charge information. This symmetry is commonly expected to be a precise symmetry [@Henley:1979ig; @Miller:1990iz], and its breaking is assumed to be negligible in the phenomenological or experimental analysis. This is, in general, true, since electromagnetic interactions are weak compared with strong interactions. However, it is possible for isospin symmetry breaking (ISB) to have important influence on some experiments, especially its effects on the parton distributions. Therefore, it is necessary to analyze it carefully.
The isospin symmetry between the proton and the neutron originates from the SU(2) symmetry between $u$ and $d$ quarks, which are isospin doublets with isospin $I= 1/2$ and isospin three-components ($I_3$) 1/2 and -1/2, respectively. The isospin symmetry at parton level indicates that the $u~(d,~\bar{u},~\bar{d})$-quark distribution in the proton is equal to the $d~(u,~\bar{d},~\bar{u})$-quark distribution in the neutron. Accordingly, the ISBs of both valance-quark and sea-quark distributions are defined, respectively, as $$\begin{aligned}
\delta u_{\mathrm{V}}(x)&=&u_{\mathrm{V}}^{\mathrm{p}}(x)-d_{\mathrm{V}}^{\mathrm{n}}(x),\nonumber\\
\delta
d_{\mathrm{V}}(x)&=&d_{\mathrm{V}}^{\mathrm{p}}(x)-u_{\mathrm{V}}^{\mathrm{n}}(x),\nonumber\\
\delta \bar{u}(x)&=&\bar{u}^{\mathrm{p}}(x)-\bar{d}^{\mathrm{n}}(x),\nonumber\\
\delta \bar{d}(x)&=&\bar{d}^{\mathrm{p}}(x)-\bar{u}^{\mathrm{n}}(x),\end{aligned}$$ where $q_{\mathrm{V}}^{\mathrm{N}}(x)=q^{\mathrm{N}}(x)-\bar{q}^{\mathrm{N}}(x)~(q=u,~d,~\mathrm{N}=\mathrm{p},~\mathrm{n}).$
ISB at the parton level and its possible consequences for several processes were first investigated by one of us [@Ma:1991ac]. It was pointed out that both flavor asymmetry in the nucleon sea and isospin symmetry breaking between the proton and the neutron can lead to the violation of the Gottfried sum rule reported by the New Muon Collaboration [@Amaudruz:1991at; @Arneodo:1994sh]. The possibility of distinguishing these two effects was also discussed in detail [@Ma:1992gp].
In 2002, the NuTeV Collaboration [@Zeller:2001hh] extracted $\sin^{2}\theta_{\mathrm{W}}$ by measuring the ratios of neutral current to charged current $\nu$ and $\bar{\nu}$ cross sections on iron targets. The reported $\sin^{2}\theta_{\mathrm{W}}=0.2277\pm0.0013\left(\mathrm{stat}\right)\pm0.0009\left(\mathrm{syst}\right)$ has approximately 3 standard deviations above the world average value $\sin^{2}\theta_{\mathrm{W}}=0.2227\pm0.0004$ measured in other electroweak processes. This remarkable deviation is called the NuTeV anomaly and was discussed in a number of papers from various aspects, including new physics beyond the standard model [@Davidson:2001ji], the nuclear effect [@Kovalenko:2002xe], nonisoscalar targets [@Kumano:2002ra], and strange-antistrange asymmetry [@Cao:2003ny; @Ding:2004ht; @Ding:2004dv]. Moreover, the possible influence of ISB on this measurement was also studied in a series of papers [@Sather:1991je; @Rodionov:1994cg; @Davidson:1997mb; @Londergan:2003ij; @Cao:2000dj; @Gluck:2005xh; @Ding:2006ud]. However, the correction from ISB to the NuTeV anomaly is still not conclusive.
The Martin-Roberts-Stirling-Thorne (MRST) group [@Martin:2003sk] provided some evidence to support the ISB effects on parton distributions of both valance and sea quarks and included ISB in the parametrization based on experimental data. They obtained the ISB of valance quarks as $$\begin{aligned}
\delta u_{\mathrm{V}}=-\delta
d_{\mathrm{V}}=\kappa(1-x)^{4}x^{-0.5}(x-0.0909),\end{aligned}$$ where $-0.8\leq\kappa\leq+0.65$ with a $90\%$ confidence level, and the best fit value is $\kappa=-0.2$. They also obtained the ISB of sea quarks, as can be deduced from Eqs. (28) and (29) in Ref. [@Martin:2003sk], $$\begin{aligned}
\delta\bar{u}(x)=k\bar{u}^{\mathrm{p}}(x),~~~~\delta\bar{d}(x)=k\bar{d}^{\mathrm{p}}(x),\end{aligned}$$ with the best fit value $k=0.08$.
In this paper, we calculate the ISB of the valance- and sea-quark distributions between the proton and the neutron in the chiral quark model and discuss some possible effects of ISB. We assume that the ISB between the proton and the neutron is entirely from the mass difference between isospin multiplets at both hadron and parton levels.[^2] In Sec. \[section2\], we compute ISB in the chiral quark model, with the constituent-quark-model results as the bare constituent-quark-distribution inputs. Then, we calculate the ISB effect on the violation of the Gottfried sum rule. In Sec. \[section3\], we discuss the ISB correction to the measurement of the weak angle and point out the significant influence on the NuTeV anomaly. In Sec. \[section4\], we provide summaries of the paper.
isospin symmetry breaking in the chiral quark model {#section2}
===================================================
The chiral quark model, established by Weinberg [@Weinberg:1978kz] and developed by Manohar and Georgi [@Manohar:1983md], has an apt description of its important degrees of freedom in terms of quarks, gluons, and Goldstone (GS) bosons at momentum scales relating to hadron structure. This model is successful in explaining numerous problems, including the violation of the Gottfried sum rule from the aspect of flavor asymmetry in the nucleon sea [@Eichten:1991mt; @Wakamatsu:1991tj], the proton spin crisis [@Ashman:1987hv; @Cheng:1994zn; @Song:1997bp], and the NuTeV anomaly resulting from the strange-antistrange asymmetry [@Ding:2004dv], and has been widely recognized as an effective theory of QCD at the low-energy scale.
In the chiral quark model, the minor effects of the internal gluons are negligible. The valence quarks contained in the nucleon fluctuate into quarks plus GS bosons, which spontaneously break chiral symmetry. Then, the effective interaction Lagrangian is $$L=\bar{\psi}\left(iD_{\mu}+V_{\mu}\right)\gamma^{\mu}\psi+ig_{\mathrm{A}}\bar{\psi}A_{\mu}\gamma^{\mu}\gamma_{5}\psi+\cdots,$$ where $$\psi=\left(\begin{array}{c}
u \\
d \\
s \\
\end{array}\right)$$ is the quark field and $D_{\mu}=\partial_{\mu}+igG_{\mu}$ is the gauge-covariant derivative of QCD. $G_{\mu}$ stands for the gluon field, $g$ stands for the strong coupling constant, and $g_{\mathrm{A}}$ stands for the axial-vector coupling constant. $V_{\mu}$ and $A_{\mu}$ are the vector and the axial-vector currents, which are defined as $$\left(\begin{array}{c}
V_{\mu} \\
A_{\mu} \\
\end{array}\right)=\frac{1}{2}\left(\xi^{+}\partial_{\mu}\xi\pm\xi\partial_{\mu}\xi^{+}\right),$$ where $\xi=\mathrm{exp}(i\Pi/f)$, and $\Pi$ has the form: $$\Pi\equiv\frac{1}{\sqrt{2}}\left(
\begin{array}{ccc}
\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & \pi^{+} & K^{+} \\
\pi^{-} & -\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & K^{0} \\
K^{-} & \overline{K^{0}} & \frac{-2\eta}{\sqrt{6}} \\
\end{array}
\right).$$ Expanding $V_{\mu}$ and $A_{\mu}$ in powers of $\Pi/f$, one gets $V_{\mu}=0+O(\Pi/f)^{2}$ and $A_{\mu}=i\partial_{\mu}\Pi/f+O(\Pi/f)^{2}$. The pseudoscalar decay constant is $f\simeq93$ MeV. Thus, the effective interaction Lagrangian between GS bosons and quarks in the leading order becomes [@Eichten:1991mt] $$L_{\Pi
q}=-\frac{g_{\mathrm{A}}}{f}\bar{\psi}\partial_{\mu}\Pi\gamma^{\mu}\gamma_{5}\psi.$$ Based on the time-ordered perturbative theory in the infinite momentum frame, all particles are on-mass-shell, and the factorization of the subprocess is automatic, so we can express the quark distributions inside a nucleon as a convolution of a constituent-quark distribution in a nucleon and the structure functions of a constituent quark. Since the $\eta$ is relatively heavy, we neglect the minor contribution from its suppressed fluctuation in this paper. Then, the light-front Fock decompositions of constituent-quark wave functions are $$|U\rangle=\sqrt{Z_{u}}|u_{0}\rangle+a_{\pi^{+}}|d\pi^{+}\rangle+\frac{a_{u\pi^{0}}}{\sqrt{2}}|u\pi^{0}\rangle+a_{K^{+}}|sK^{+}\rangle,\label{u}$$ $$|D\rangle=\sqrt{Z_{d}}|d_{0}\rangle+a_{\pi^{-}}|u\pi^{-}\rangle+\frac{a_{d\pi^{0}}}{\sqrt{2}}|d\pi^{0}\rangle+a_{K^{0}}|sK^{0}\rangle,\label{d}$$ where $Z_{u}$ and $Z_{d}$ are the renormalization constants for the bare constituent $u$ quark $|u_{0}\rangle$ and $d$ quark $|d_{0}\rangle$, respectively, and $|a_{\alpha}|^{2}$ ($\alpha=\pi,
K$) are the probabilities to find GS bosons in the dressed constituent-quark states $|U\rangle$ and $|D\rangle$. In the chiral quark model, the fluctuation of a bare constituent quark into a GS boson and a recoil bare constituent quark is given as [@Suzuki:1997wv] $$q_{j}(x)=\int^{1}_{x}\frac{\textmd{d}y}{y}P_{j\alpha/i}(y)q_{i}\left(\frac{x}{y}\right),\label{q}$$ where $P_{j\alpha/i}(y)$ is the splitting function, which gives the probability of finding a constituent quark $j$ carrying the light-cone momentum fraction $y$ together with a spectator GS boson $\alpha$, $$\begin{aligned}
P_{j\alpha/i}(y)=\frac{1}{8\pi^{2}}\left(\frac{g_{\mathrm{A}}\overline{m}}{f}\right)^{2}\int
\textmd{d}k^{2}_{T}\frac{(m_{j}-m_{i}y)^{2}+k^{2}_{T}}{y^{2}(1-y)[m_{i}^{2}-M^{2}_{j\alpha}]^{2}}.
\label{splitting}\end{aligned}$$ $m_{i}$, $m_{j}$, and $m_{\alpha}$ are the masses of the $i$- and $
j$-constituent quarks and the pseudoscalar meson $\alpha$, respectively, and $\overline{m}=(m_{i}+m_{j})/2$ is the average mass of the constituent quarks. $M^{2}_{j\alpha}=\left(m^{2}_{j}+k^{2}_{T}\right)/y+\left(m^{2}_{\alpha}+k^{2}_{T}\right)/\left(1-y\right)$ is the square of the invariant mass of the final state. We can also write the internal structure of GS bosons in the following form $$q_{k}(x)=\int\frac{\textmd{d}y_{1}}{y_{1}}\frac{\textmd{d}
y_{2}}{y_{2}}V_{k/\alpha}\left(\frac{x}{y_{1}}\right)P_{\alpha
j/i}\left(\frac{y_{1}}{y_{2}}\right)q_{i}\left(y_{2}\right),$$ where $V_{k/\alpha}(x)$ is the quark $k$ distribution function in $\alpha$ and satisfies the normalization $\int_{0}^{1}V_{k/\alpha}(x)dx=1$.
When we take ISB into consideration, the renormalization constant $Z$ should take the form $$\begin{aligned}
Z_{u}=1-\langle P_{\pi^{+}}\rangle-\frac{1}{2}\langle
P_{u\pi^{0}}\rangle-\langle P_{K^{+}}\rangle,\nonumber\\
Z_{d}=1-\langle P_{\pi^{-}}\rangle-\frac{1}{2}\langle
P_{d\pi^{0}}\rangle-\langle P_{K^{0}}\rangle,\end{aligned}$$ where $\langle P_{\alpha}\rangle\equiv \langle
P_{j\alpha/i}\rangle=\langle P_{\alpha
j/i}\rangle=\int^{1}_{0}x^{n-1}P_{j\alpha/i}(x)\mathrm{d}x$ [@Suzuki:1997wv]. It is conventional to specify the momentum cutoff function at the quark-GS-boson vertex as $$g_{\mathrm{A}}\rightarrow
g_{\mathrm{A}}^{\prime}\textmd{exp}\bigg{[}\frac{m^{2}_{i}-M^{2}_{j\alpha}}{4\Lambda^{2}}\bigg{]},$$ where $g_{\mathrm{A}}^{\prime}=1$, following the large $N_{c}$ argument [@Weinberg:1990xm], and $\Lambda$ is the cutoff parameter, which is determined by the experimental data of the Gottfried sum and the constituent-quark-mass inputs for the pion. Such a form factor has the correct $t$- and $u$-channel symmetry, $P_{j \alpha /i} (y) = P_{\alpha j/i} (1-y)$. Then, one can obtain the quark-distribution functions in the proton [@Suzuki:1997wv], $$\begin{aligned}
u(x)&=&Z_{u}u_{0}(x)+P_{u\pi^{-}/d}\otimes
d_{0}(x)+V_{u/\pi^{+}}\otimes P_{\pi^{+}d/u}\otimes
u_{0}(x)+\frac{1}{2}P_{u\pi^{0}/u}\otimes
u_{0}(x)\nonumber\\&+&V_{u/K^{+}}\otimes P_{K^{+}s/u}\otimes
u_{0}(x)+
\frac{1}{2}V_{u/\pi^{0}}\otimes \left[P_{\pi^{0}u/u}\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\
d(x)&=&Z_{d}d_{0}(x)+P_{d\pi^{+}/u}\otimes
u_{0}(x)+V_{d/\pi^{-}}\otimes P_{\pi^{-}u/d }\otimes d_{0}(x)+
\frac{1}{2}P_{d\pi^{0}/d}\otimes
d_{0}(x)\nonumber\\&+&V_{d/K^{0}}\otimes P_{K^{0}s/d}\otimes
d_{0}(x) +\frac{1}{2}V_{d/\pi^{0}}\otimes \left[P_{\pi^{0}u/u }\otimes
u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\
\bar{u}(x)&=&V_{\bar{u}/\pi^{-}}\otimes P_{\pi^{-}u/d}\otimes
d_{0}(x)+\frac{1}{2}V_{\bar{u}/\pi^{0}}\otimes
\left[P_{\pi^{0}u/u}\otimes
u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\
\bar{d}(x)&=&V_{\bar{d}/\pi^{+}}\otimes P_{\pi^{+}d/u}\otimes
u_{0}(x)+\frac{1}{2}V_{\bar{d}/\pi^{0}}\otimes
\left[P_{\pi^{0}u/u}\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\end{aligned}$$ where the constituent quark-distributions $u_{0}$ and $d_{0}$ are normalized to two and one, respectively. Convolution integrals are defined as $$\begin{aligned}
P_{j \alpha / i}\otimes q_i &=&\int_{x}^{1}\frac{\textmd{d}y}{y}P_{j
\alpha / i}\left(y\right)q_i\left(\frac{x}{y}\right),\nonumber\\
V_{k/ \alpha}\otimes P_{\alpha j/i}\otimes
q_i&=&\int_{x}^{1}\frac{\textmd{d}y_{1}}{y_{1}}\int_{y_{1}}^{1}\frac{\textmd{d}y_{2}}{y_{2}}V_{k/
\alpha}\left( \frac{x}{y_{1}}\right)P_{\alpha
j/i}\left(\frac{y_{1}}{y_{2}}\right)q_{i}\left(y_{2}\right).\end{aligned}$$ In addition, $V_{k/\alpha}(x)$ follows the relationship $$\begin{aligned}
&&V_{u/\pi^{+}}=V_{\bar{d}/\pi^{+}}=V_{d/\pi^{-}}=V_{\bar{u}/\pi^{-}}=2V_{u/\pi^{0}}
=2V_{\bar{u}/\pi^{0}}=2V_{d/\pi^{0}}=2V_{\bar{d}/\pi^{0}}
=\frac{1}{2}V_{\pi},\nonumber\\
&&V_{u/K^{+}}=V_{d/K^{0}}.\end{aligned}$$
We postulate that the bare-quark distributions are isospin-symmetric between the proton and the neutron, so we can obtain the quark distributions of the neutron by interchanging $u_{0}$ and $d_{0}$. Employing the quark distributions of the chiral quark model, we get the Gottfried sum determined by the difference between the proton and the neutron structure functions, $$\begin{aligned}
S_{\mathrm{G}}&=&\int^{1}_{0}\frac{\mathrm{d}x}{x}\left[F^{\mathrm{p}}_{2}(x)-F^{\mathrm{n}}_{2}(x)\right]\nonumber\\
&=&\frac{1}{9}\int_{0}^{1}\mathrm{d}x\left[4u^{\mathrm{p}}(x)+4\bar{u}^{\mathrm{p}}(x)-4u^{\mathrm{n}}(x)-4\bar{u}^{\mathrm{n}}(x)+d^{\mathrm{p}}(x)+\bar{d}^{\mathrm{p}}(x)-d^{\mathrm{n}}(x)-\bar{d}^{\mathrm{n}}(x)\right]\nonumber\\&
=&\frac{1}{3}+\int^{1}_{0}\mathrm{d}x\left\{\frac{8}{9}\left[\bar{u}^{\mathrm{p}}(x)-\bar{u}^{\mathrm{n}}(x)\right]+\frac{2}{9}\left[\bar{d}^{\mathrm{p}}(x)-\bar{d}^{\mathrm{n}}(x)\right]\right\}\nonumber\\
&=&\frac{1}{3}-\frac{8}{9}\left<P_{\pi^{-}}\right>+\frac{2}{9}\left<P_{\pi^{+}}\right>+\frac{5}{18}
\left(\left<P_{u\pi^{0}}\right>-\left<P_{d\pi^{0}}\right>\right).\label{gottfried}\end{aligned}$$
We assume that the ISB is entirely from the mass difference between isospin multiplets. In this paper, we adopt $(m_{u}+m_{d})/2=330$ MeV, $m_{\pi^{\pm}}=139.6$ MeV, $m_{\pi^{0}}=135$ MeV, $m_{K^{\pm}}=493.7$ MeV, and $m_{K^{0}}=497.6$ MeV. We choose two sets of the mass difference between $u$ and $d$ quarks, namely $\delta m=4$ MeV and $\delta
m=8$ MeV, respectively, in order to show the dependence on this important parameter. Based on Eq. (\[gottfried\]) and the experimental data of the Gottfried sum [@Arneodo:1994sh], one can find that the appropriate value for $\Lambda_{\pi}$ is $1500$ MeV. However, one cannot determine $\Lambda_{K}$ in the same method, because $\langle P_{K}\rangle$ in the Gottfried sum is canceled out. Usually, it is assumed that $\Lambda_{K}=\Lambda_{\pi}=1500~\mathrm{MeV}$[@Suzuki:1997wv; @Szczurek:1996tp]. However, it is implied by the SU$\left(3\right)_{f}$ symmetry breaking that $\left<P_{K}\right>$ should be smaller, and, accordingly, one should adopt a smaller $\Lambda_{K}$. In this paper, we adopt a wide range of $\Lambda_{K}$ from $900$ to $1500$ MeV. In addition, the parton distributions of mesons are the parametrization GRS98 given by Gluck-Reya-Stratmann [@Gluck:1997ww], since the parametrization is more approximate to the actual value, $$\begin{aligned}
V_{\pi}(x)=0.942x^{-0.501}(1+0.632\sqrt{x})(1-x)^{0.367},\nonumber\\
V_{u/K^{+}}(x)=V_{d/K^{0}}(x)=0.541(1-x)^{0.17}V_{\pi}(x).\end{aligned}$$ We should point out that, in principle, it is possible that the parton distributions of different mesons in the same multiplet are different, and this can contribute to ISB simultaneously. However, in this paper, we simply neglect this possibility, and calculations in future can be improved if we have a better understanding of the quark structure of mesons. Moreover, we have to specify constituent-quark distributions $u_{0}$ and $d_{0}$, but there is no proper parametrization of them because they are not directly related to observable quantities in experiments. In this paper, we adopt the constituent-quark-model distributions [@Hwa:1980mv] as inputs for constituent-quark distributions. For the proton, we have $$\begin{aligned}
u_{0}(x)&=&\frac{2x^{c_{1}}(1-x)^{c_{1}+c_{2}+1}}{\textmd{B}[c_{1}+1,c_{1}+c_{2}+2]},\nonumber\\
d_{0}(x)&=&\frac{x^{c_{2}}(1-x)^{2c_{1}+1}}{\textmd{B}[c_{2}+1,2c_{1}+2]},\end{aligned}$$ where $\textmd{B}[i,j]$ is the Euler beta function. Such distributions satisfy the number and the momentum sum rules $$\begin{aligned}
\int^{1}_{0}u_{0}(x)\textmd{d}x=2, ~~~~
\int^{1}_{0}d_{0}(x)\textmd{d}x=1,\nonumber\\
\int_{0}^{1}xu_{0}(x)\textmd{d}x+\int_{0}^{1}xd_{0}(x)\textmd{d}x=1.\end{aligned}$$ $c_{1}=0.65$ and $c_{2}=0.35$ are adopted in the calculation, following the original choice [@Hwa:1980mv; @Kua:1999yf].
We display the ISB of the valance- and sea-quark distributions in Figs. \[uvalance\], \[dvalance\], and \[sea\], respectively. It is shown that in most regions, $x\delta u_\mathrm{V}(x)>0$ and $x\delta \bar{u}(x)>0$, and on the contrary that $x\delta
d_\mathrm{V}(x)<0$ and $x\delta \bar{d}(x)<0$. Our predictions that $x\delta \bar{u}(x)>0$ and $x\delta \bar{d}(x)<0$ are consistent with the MRST parametrization [@Martin:2003sk], and, moreover, the shapes of $x\delta \bar{u}(x)$ and $x\delta \bar{d}(x)$ are similar to the best phenomenological fitting results given by the MRST group. We should point out that our results are analogous to the results calculated in the framework of the meson cloudy model by Cao and Signal [@Cao:2000dj], and the shapes and magnitudes of $x\delta \bar{u}(x)$ and $x\delta \bar{d}(x)$ are similar to the results given in the framework of the radiatively generated ISB [@Gluck:2005xh], but with different signs. It can also be found that the difference between various choices of $\Lambda_K$ is minor, but the different choices of $\delta m$ can have remarkable influence on the distributions. Especially, larger $\delta m$ can lead to larger ISB, and this is concordant with our principle that ISB results from the mass difference between isospin multiplets at both hadron and parton levels. From the figures, we can see that $\delta u_\mathrm{V}(x)$ reaches a maximum value at $x\approx 0.5$, and $\delta d_\mathrm{V}(x)$ has a minimum value at $x\approx 0.4$. It should also be noted that $\delta q_\mathrm{V}(x)$ $(q=u,d)$ must have at least one zero point due to the valance-quark-normalization conditions. We should also point out that at large $x$, $\delta
u_\mathrm{V}/u_\mathrm{V} \approx -\delta
d_\mathrm{V}/d_\mathrm{V}$, and this implies that the magnitudes of the ISB for $u_{\mathrm{V}}$ and $d_{\mathrm{V}}$ are almost the same, but with opposite signs. Moreover, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model.
![The ISB of the $u_{\mathrm{V}}$-quark distribution $x\delta u_{\mathrm{V}}(x)$ versus $x$ in the chiral quark model with different inputs. The red solid line is the result with $\delta
m=4$ MeV and $\Lambda_K=1500$ MeV as inputs. The blue dashed line is the result with $\delta m=8$ MeV and $\Lambda_K=1500$ MeV as inputs. The green dotted line is the result with $\delta m=4$ MeV and $\Lambda_K=900$ MeV as inputs.[]{data-label="uvalance"}](uv.eps){width="95.00000%"}
![The ISB of the $d_{\mathrm{V}}$-quark distribution $x\delta d_{\mathrm{V}}(x)$ versus $x$ in the chiral quark model with different inputs. The red solid line is the result with $\delta
m=4$ MeV and $\Lambda_K=1500$ MeV as inputs. The blue dashed line is the result with $\delta m=8$ MeV and $\Lambda_K=1500$ MeV as inputs. The green dotted line is the result with $\delta m=4$ MeV and $\Lambda_K=900$ MeV as inputs.[]{data-label="dvalance"}](dv.eps){width="95.00000%"}
![The ISB of the sea-quark distributions $x\delta
\bar{q}(x)$ versus $x$ in the chiral quark model. The red solid line and the blue dashed line are the behaviors of $x\delta \bar{u}(x)$, with $\delta m=4$ MeV and $\delta m=8$ MeV, respectively. The green dotted line and the orange dash-dotted line are the behaviors of $x\delta \bar{d}(x)$, with $\delta m=4$ MeV and $\delta m=8$ MeV, respectively.[]{data-label="sea"}](sea.eps){width="95.00000%"}
The contribution from isospin symmetry breaking to the NuTeV anomaly {#section3}
====================================================================
The measured $\sin^2\theta_\mathrm{W}$ by the NuTeV Collaboration is closely related to the Paschos-Wolfenstein (PW) ratio [@Paschos:1972kj] $$\begin{aligned}
R^{-}=\frac{\left<\sigma_{\mathrm{NC}}^{\nu
\mathrm{N}}\right>-\left<\sigma_{\mathrm{NC}}^{\overline{\nu}\mathrm{N}}\right>}{\left<\sigma_{\mathrm{CC}}^{\nu
\mathrm{N}}\right>-\left<\sigma_{\mathrm{CC}}^{\overline{\nu}\mathrm{N}}\right>}=\frac{1}{2}-\sin^{2}\theta_{\mathrm{W}},\label{pw}\end{aligned}$$ where $\left<\sigma_{\mathrm{NC}}^{\nu \mathrm{N}}\right>$ is the neutral-current-inclusive cross section for a neutrino on an isoscalar target. If we take the ISB between the proton and the neutron into account, we obtain $$\begin{aligned}
R^{-}_{\mathrm{N}}=\frac{\left<\sigma_{\mathrm{NC}}^{\nu
\mathrm{N}}\right>-\left<\sigma_{\mathrm{NC}}^{\overline{\nu}\mathrm{N}}\right>}{\left<\sigma_{\mathrm{CC}}^{\nu
\mathrm{N}}\right>-\left<\sigma_{\mathrm{CC}}^{\overline{\nu}\mathrm{N}}\right>}
=R^{-}+\delta R^{\mathrm{ISB}}_{\mathrm{PW}},\label{mpw}\end{aligned}$$ where $\delta R^{\mathrm{ISB}}_{\mathrm{PW}}$ is the correction from the ISB to the PW ratio and takes the form $$\begin{aligned}
\delta
R^{\mathrm{ISB}}_{\mathrm{PW}}=\bigg{(}\frac{1}{2}-\frac{7}{6}\sin^{2}\theta_{\mathrm{W}}\bigg{)}\frac{\int^{1}_{0}x\bigg{[}\delta
u_{\mathrm{V}}(x)-\delta d_{\mathrm{V}}(x)\bigg{]}\mathrm{d}x}
{\int^{1}_{0}x\bigg{[}u_{\mathrm{V}}(x)+d_{\mathrm{V}}(x)\bigg{]}\mathrm{d}x},\label{rs}\end{aligned}$$ with $u_{\mathrm{V}}(x)$ and $d_{\mathrm{V}}(x)$ standing for valance-quark distributions of the proton. We show the renormalization constant $Z$, the total momentum fraction of valance quarks $Q_{\mathrm{V}}=\int^{1}_{0}x\left[u_{\mathrm{V}}(x)+d_{\mathrm{V}}(x)\right]\mathrm{d}x$, and the correction of the ISB to the NuTeV anomaly $\Delta
R^{\mathrm{ISB}}_{\mathrm{PW}}$, with different $\delta m$ and $\Lambda_{K}$ as inputs in Table \[ISB\]. It can be found that the ISB correction is of the order of magnitude of $10^{-3}$ and is more significant with a larger $\delta m$ or $\Lambda_K$. Our result is consistent with the range $-0.009\leq\Delta
R_{\mathrm{PW}}^{\mathrm{ISB}}\leq+0.007$, which is derived based on the parametrization given by the MRST group [@Martin:2003sk]. We should stress that the correction is remarkable, since the NuTeV anomaly can be totally removed if $\Delta R_{\mathrm{PW}}=-0.005$, and, consequently, we should pay special attention to ISB in such problem. It is also worthwhile to point out that the correction is in an opposite direction to remove the NuTeV anomaly in the chiral quark model. Such a conclusion is the same as that given in the baryon-meson fluctuation model [@Ding:2006ud], but the value is one or 2 orders of magnitude larger. Our result of the ISB correction to the NuTeV anomaly differs from the results in Refs. [@Londergan:2003ij; @Gluck:2005xh].
------------------ --------------------- ---------- ---------- ------------------ -----------------------------------------
$\delta m$ (MeV) $\Lambda_{K}$ (MeV) $Z_{u}$ $Z_{d}$ $Q_{\mathrm{V}}$ $\Delta R^{\mathrm{ISB}}_{\mathrm{PW}}$
$4$ $900$ $0.7497$ $0.7463$ $0.8451$ $0.0008$
$4$ $1200$ $0.7220$ $0.7185$ $0.8222$ $0.0008$
$4$ $1500$ $0.6932$ $0.6896$ $0.7985$ $0.0009$
$8$ $900$ $0.7515$ $0.7444$ $0.8455$ $0.0016$
$8$ $1200$ $0.7239$ $0.7165$ $0.8227$ $0.0017$
$8$ $1500$ $0.6953$ $0.6874$ $0.7990$ $0.0019$
------------------ --------------------- ---------- ---------- ------------------ -----------------------------------------
: The renormalization constant, the total momentum fraction of valance quarks, and the correction of the ISB to the NuTeV anomaly in the chiral quark model.[]{data-label="ISB"}
summary {#section4}
=======
In this paper, we discuss the ISB of the valance-quark and the sea-quark distributions between the proton and the neutron in the framework of the chiral quark model. We assume that isospin symmetry breaking is the result of mass differences between isospin multiplets. Then, we analyze the effects of isospin symmetry breaking on the Gottfried sum rule and the NuTeV anomaly. We show that, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model. It is remarkable that our results of ISB for both the valence-quark and sea-quark distributions are consistent with the MRST parametrization of the ISB of valance- and sea-quark distributions. Moreover, we find that the correction to the NuTeV anomaly is in an opposite direction, so the NuTeV anomaly cannot be removed by isospin symmetry breaking in the chiral quark model. However, its influence is remarkable and should be taken into careful consideration. Therefore, it is important to do more precision experiments and careful theoretical studies on isospin symmetry breaking.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by the National Natural Science Foundation of China (grant numbers. 10721063, 10975003, 11035003), and National Fund for Fostering Talents of Basic Science (grant numbers. J0630311, J0730316). It is also supported by Principal Fund for Undergraduate Research at Peking University.
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[^1]: Corresponding author. Email address: `mabq@pku.edu.cn`
[^2]: As mass difference between isospin multiplets, especially that between $u$ and $d$ quarks, is not entirely due to charge difference, we refer such effect as Isospin Symmetry Breaking (ISB) instead of Charge Symmetry Breaking (CSB) as called in some papers.
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---
abstract: 'Waveform design is a pivotal component of [*the fully adaptive radar*]{} construct. In this paper we consider waveform design for radar space time adaptive processing (STAP), accounting for the waveform dependence of the clutter correlation matrix. Due to this dependence, in general, the joint problem of receiver filter optimization and radar waveform design becomes an intractable, non-convex optimization problem, Nevertheless, it is however shown to be individually convex either in the filter or in the waveform variables. We derive constrained versions of: a) the alternating minimization algorithm, b) proximal alternating minimization, and c) the constant modulus alternating minimization, which, at each step, iteratively optimizes either the STAP filter or the waveform independently. A fast and slow time model permits waveform design in radar STAP but the primary bottleneck is the computational complexity of the algorithms.'
author:
- 'Pawan Setlur, *Member, IEEE*, Muralidhar Rangaswamy, *Fellow, IEEE*[^1] [^2] [^3]'
bibliography:
- 'refs.bib'
title: Joint Filter and Waveform Design for Radar STAP in Signal Dependent Interference
---
[AFRL, Sensors Directorate Tech. Report, 2014: Approved for Public release]{}
Waveform design, waveform scheduling, space time adaptive radar, Capon beamformer, constant modulus, convex optimization, alternating minimization, regularization, proximal algorithms.
Introduction
============
objective of this report is to address waveform design in radar space time adaptive processing (STAP) [@klemm2002; @ward1994; @guerci2003; @Brennan1973]. An air-borne radar is assumed with an array of sensor elements observing a moving target on the ground. We will assume that the waveform design and scheduling are performed over one CPI rather than on an individual pulse repetition interval (PRI).To facilitate waveform design, we develop a STAP model considering the fast time samples along with the slow time processing. This is different from traditional STAP which generally considers the data after matched filtering [@klemm2002; @ward1994]. Nonetheless STAP research efforts have been proposed which consider inclusion of fast time samples in space time processing, see for example [@klemm2002; @madurasinghe2006; @seliktar2006] and references therein.
In line with traditional STAP, we formulate the waveform design, as an minimum variance distortion-less response (MVDR) type optimization [@capon1969]. As we will see in the sequel, inclusion of the waveform increases the dimensionality of the correlation matrix. Classical Radar STAP is computationally expensive but the waveform adaptive STAP increases the complexity by several orders of magnitude. Therefore, benefits of waveform design in STAP come at the expense of increased computational complexity. The noise, clutter, and interference are modeled stochastically and are assumed to be mutually uncorrelated. Endemic to airborne STAP, clutter is persistent in most range gates resulting from ground reflections. The clutter correlation matrix is a function of the waveform causing the joint reliever filter and waveform optimization to be non-convex with no closed form solution. However, it is analytically shown here that the STAP MVDR objective is convex with respect to (w.r.t.) the receiver filter for a fixed but arbitrary waveform, and vice versa. Therefore, alternating minimization approaches arise as natural candidate solutions. As such, alternating minimization itself has a rich history in the optimization literature, possibly motivated directly from the works in [@Powell1964; @Powell1973; @Zangwill1967; @Ortega1970], with some not so recent seminal contributions [@Luo1992; @Auslender1992; @Bertsekas1999; @Grippo2000] and recent contributions (not exhaustive) [@Attouch2010; @Beck2013]. Other celebrated algorithms such as the Arimoto-Blahut algorithm to calculate channel capacity, and the expectation-maximization (EM) algorithm are all examples of the alternating minimization.
Here we address the joint optimization problem via a constrained alternating minimization approach, which has the favorable property of monotonicity in successive objective evaluations. Convergence, performance guarantees and other properties pertinent to this algorithm are further addressed. Full rank correlation matrices are required in implementing the constrained alternating minimization approach. In practice, radar STAP contends with rank deficient correlation matrices due to lack of homogeneous training data. In this case, the constrained alternating minimization approach is not implementable. To addresses this issue, we consider regularization of the STAP objective via strongly convex functions resulting in the constrained proximal alternating minimization [@Setlurasilomar2014]. Proximal algorithms, originally proposed by [@Martinet1970; @Rockafeller1973] are well suited candidate techniques for constrained, large scale optimization [@Attouch2010; @Bertsekas1994; @Rockafeller1976; @Combettes2011; @Parikh2013], applicable readily to our waveform adaptive STAP problem. In fact, as we will see subsequently the constrained proximal alternating minimization results in diagonal loading solutions, and for optimization-specific interpretations, the load factors may be related to the Lipschitz constants (w.r.t. the gradient).
[**Signal dependent interference: Chicken or the Egg?**]{} The fundamental problem in practical radar waveform design is analogous to the chicken or the egg problem. Signal dependent interference, i.e., clutter, can only be perfectly characterized by transmitting a signal. Herein lies the central problem. The estimated clutter properties could therefore be dependent on what was transmitted in the first place. This is especially true for frequency selective and dispersive clutter responses frequently encountered in radar operations, for example, urban terrain. Therefore, any claim of optimality is myopic. Sadly the same problem would also persist when the target impulse responses are used to shape the waveform. Unfortunately, and as famously stated by Woodward [@woodward1952information; @woodward1953theory], “…what to transmit remains substantially unanswered” [@woodward1953probability; @Benedetto2009].
We will assume like other works in the signal dependent interference waveform design [@Delong1967; @Delong1969; @Delong1970; @Kay2007; @vaidyanathan2009; @Pillai2000; @Demaio2012; @Demaio2013; @Hongbin2014], that the clutter response is known [*a priori*]{}. To a certain extent, this may be obtained via a combination of, either previous radar transmission [@cochran2009waveform], or assuming that the topography is known from ground elevation maps, synthetic aperture radar imagery [@SetlurJSTSP2014], or access to knowledge aided databases as in the DARPA’s KASSPER program [@Guerci2006].
[**Literature:**]{} The signal dependent interference waveform design problem has had a rich history [@middleton1996; @van2004detection]. Iterative approaches but not limited alternating minimization type techniques have been the subject of work in [@Delong1967; @Delong1969; @Delong1970; @Kay2007; @vaidyanathan2009; @Pillai2000; @Stoica2012; @Demaio2012; @Demaio2013; @Hongbin2014] for SISO, MIMO radars but never in radar STAP. Waveform design for STAP without considering the signal dependent interference clutter was addressed in [@pattonstap2012], where the authors premise is that the degrees of freedom from the waveform could be used in suppressing the interference and noise, while the degrees of freedom from the filter could be used exclusively for suppressing the clutter. A joint STAP waveform and STAP filter design was never considered. Further, their premise is erroneous for the following several reasons. For any radar application, but especially in STAP, obtaining range cells which are interference free or clutter free is impossible. Nonetheless assuming this was possible, then, the weight vector for exclusive clutter suppression uses [*the inverse of the clutter interference correlation matrix*]{} only, and not, as stated in [@pattonstap2012], the inverse of the (clutter+noise+interference) correlation matrix. Furthermore such a detector may have disastrous consequences, because control in the false alarm rates becomes impossible due to the self induced coloring on other range cells which are contaminated by the clutter plus interference plus noise.
Other contributions in waveform design and waveform scheduling for extended targets in radar using information theoretic measures, tracking etc can be seen in [@setlurspie2012; @setlurssp2012; @bell1993information; @bell1988thesis; @Lesham2007; @romero2008information], [@paper:AussiesSensorScheduleRadar; @sira2006waveform; @sira2009waveform; @Kershaw:2004; @kershaw1997waveform; @Li2008], and the references therein.
We outline some of the contributions for the signal dependent interference problem which have thus far appeared in the literature.
[*Approaches different from Alternating minimization*]{}: In [@Delong1967; @Kay2007; @Pillai2000; @Stoica2012] a single sensor radar was assumed. In [@Delong1967], the authors used the symmetry property of the cross-ambiguity function to design an iterative algorithm for the signal dependent interference problem. Their algorithm cannot be modified easily for the multi-sensor framework and when noise is in general colored. The problem was addressed from a detection perspective in [@Kay2007], and lead to a waterfilling [@bell1993information] type solution. A similar waterfilling type metric albeit in the discrete time domain was obtained in [@Stoica2012], where the authors also imposed constant modulus and peak to average power ratio (PAPR) waveform constraints. An iterative algorithm was derived in [@Pillai2000], where monotonic increase in SINR was not guaranteed, and was shown that waveform could always be chosen as minimum phase.
[*Alternating minimization type approaches*]{}: In [@vaidyanathan2009], a MIMO sensor framework was employed, convergence was not addressed, convexity was not proven, and no practical waveform constraints were imposed on the design. See also in this report, Section III, paragraph following Rem. \[limitpointremk\] where some of the conclusions drawn in [@vaidyanathan2009] are further discussed. Alternating minimization was used in [@Demaio2012; @Demaio2013; @Hongbin2014] but for reasons unknown, was called as sequential optimization. In [@Demaio2012; @Demaio2013], a SISO model advocating joint filter and radar code design (after matched filtering) was employed. Analysis of the convexity of the objective in the individual filter or radar code was never shown. Convergence in iterates was not proven formally, neither was it shown via simulations. The constant modulus constraint was not invoked directly but through a similarity constraint. In [@Hongbin2014], the authors used a MIMO radar framework, and relaxation techniques were employed in their iterative algorithm. Neither convergence nor convexity was demonstrated analytically. Constant modulus constraint and similarity constraints were enforced separately in the waveform design.
[**Notation:**]{} The variable $N$ is used interchangeably with the number of the fast time samples, as well as, the conventional dimension of arbitrary real or complex (sub)spaces. Its meaning is readily interpreted from context. The symbol $|| \cdot||$ always denotes the $l_2$ norm. Vectors are always lowercase bold, matrices are bold uppercase, $\lambda$ is typically reserved for eigenvalues (with $\lambda_o$ being an exception it used for the spatial frequency, defined later) and $\gamma$ is strictly reserved for the Lagrange multipliers ($\gamma_{pq}$ is an exception used for the radar cross section of the $p$-th scatterer in the $q$-th clutter patch). Solutions to the optimization are denoted as $(\cdot)_o$, i.e. the subscript $o$. the complex conjugate is denoted with $(\cdot)^{\ast}$. The set of reals, complex numbers, and natural numbers are denoted as $\mathbbm{R},\mathbbm{C},\mathbbm{N}$, respectively. Other symbols are defined upon first use and are standard in the literature.
[**Organization:**]{} The STAP fast time-slow time model is delineated in Section II, and in Section III, the filter and waveform optimization is derived. Some preliminary simulations are presented in section IV and the resulting conclusions are drawn in Section V.
STAP Model
==========
The radar consists of a calibrated air-borne linear array, comprising $M$ sensor elements, each having an identical antenna pattern. Without loss of generality, assume that the first sensor in the array is the phase center, and acts as both a transmitter and receiver, the rest of the elements are purely receivers. The first sensor is located at $\mathbf{x_r}\in\mathbbm{R}^3$ and the ground based point target at $\mathbf{x_t}\in\mathbbm{R}^3$. The radar transmits the burst of pulses: $$\label{eq1}
u(t)=\sum\limits_{l=1}^{L}s(t-lT_p)\exp(j2\pi f_o(t-lT_p)),t\in[0,T)$$ where, $f_o$ is the carrier frequency, and $T_p=1/f_p$ is the inverse of the pulse repetition frequency, $f_p$. The pulse width and bandwidth are denoted as $T$, $B$, respectively. The coherent processing interval (CPI) consists of $L$ pulses, each of width equal to $T$. The geometry of the scene is shown in Fig. 1, where $\theta_t$ and $\phi_t$ denote the azimuth and elevation. The radar and target are both assumed to be moving. For the time being, we ignore the noise, clutter and interference and assume a non-fluctuating target. Then the desired target’s received signal for the $l$-th pulse, and at the $m$-th sensor element is given by $$\label{eq2}
s_{ml}(t)=\rho_t s(t-lT_p-\tau_m)e^{(j2\pi (f_o+f_{dm}) (t-lT_p-\tau_m))}$$ where the target’s observed Doppler shift is denoted as $f_{dm}$, and its complex back-scattering coefficient as $\rho_t$. Assume that the array is along the local $x$ axis as shown in Fig. 1. Then, the coordinates of the $m$-th element is given by $\mathbf{x_t}+m\mathbf{d},\mathbf{d}:=[d,0,0]^T,m=0,1,2\ldots,M-1$, where $d$ is the inter-element spacing. The delay $\tau_m$ could be re-written as $$\begin{aligned}
&\tau_m=||\mathbf{x_r}-\mathbf{x_t}||/c+||\mathbf{x_r}+m\mathbf{d}-\mathbf{x_t}||/c \nonumber \\
&=\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c} \sqrt{1+\frac{||m\mathbf{d}||^2}{||\mathbf{x_r}-\mathbf{x_t}||^2}+\frac{2m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{||\mathbf{x_r}-\mathbf{x_t}||^2}} \nonumber \\
&\overset{(a)}{\equiv}\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c} \left(1+\frac{m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{||\mathbf{x_r}-\mathbf{x_t}||^2} \right) \label{eq3} \\
&=2\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\frac{m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||}, \label{eq4}\end{aligned}$$ where in approximation (a), the term $\propto||m\mathbf{d}||^2$ was ignored, i.e. it is assumed that $d/|| \mathbf{x_r}-\mathbf{x_t}||<<1$, and then a binomial approximation was employed. From geometric manipulations, we also have: $$\frac{\mathbf{x_r}-\mathbf{x_t}}{||\mathbf{x_r}-\mathbf{x_t}||}=[\sin(\phi_t)\sin(\theta_t),\sin(\phi_t)\cos(\theta_t),\cos(\phi_t)]^T.$$ Using the above equation in , the delay $\tau_m,m=0,1,\ldots,M-1$ can be rewritten as $$\label{eq5}
\tau_m=2\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{md\sin(\phi_t)\sin(\theta_t)}{c}.$$ The Doppler shift, i.e. $f_{dm}$ is computed as $$\begin{aligned}
\label{eq6}
&f_{dm}=2f_o\dfrac{(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||} \\
&+ f_o\dfrac{m\mathbf{d}^T}{c}\left[\dfrac{\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t}}{||\mathbf{x_r}-\mathbf{x_t}||^2} -\dfrac{(\mathbf{x_r}-\mathbf{x_t})(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{\|| \mathbf{x_r}-\mathbf{x_t}||^3} \right] \nonumber\end{aligned}$$ where $\mathbf{\dot{x}_{(\cdot)}}$ is the vector differential of $\mathbf{x_{(\cdot)}}$ w.r.t. time. In practice $d$ is a fraction of the wavelength, and assuming that $d/|| \mathbf{x_r}-\mathbf{x_t}||<<1$ we approximate the second term in as $0$. The Doppler shift is no longer a function of the sensor index, $m$, and is rewritten as $$\label{eq7}
f_{dm}=f_d=2f_o\dfrac{(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||}$$
Vector signal model
-------------------
Let $s(t)$ be sampled discretely resulting in $N$ discrete time samples. Consider for now the single range gate corresponding to the time delay $\tau_t$. After a suitable alignment to a common local time (or range) reference, and invoking some standard assumptions, see also , the radar returns in $l$-th PRI written as a vector defined as $\mathbf{y}_{\mathbi{l}}\in\mathbbm{C}^{NM}$, is given by $$\begin{aligned}
&\mathbf{y}_{\mathbi{l}}=\rho_t\mathbf{s}\otimes \mathbf{a}(\theta_t,\phi_t)\exp(-j2\pi f_d(l-1)T_p) \label{eq9} \\
&\mathbf{a}(\theta_t,\phi_t):=[1,e^{-j2\pi\vartheta},\ldots,e^{-j2\pi (M-1)\vartheta}]^T \in\mathbbm{C}^M \nonumber\end{aligned}$$ where $\mathbf{s}:=[s_1,s_2,\ldots,s_N]^T \in\mathbbm{C}^N $ and $\vartheta:=d\sin(\theta_t)\sin(\phi_t)/\lambda_o$ is defined as the spatial frequency. Further it is noted that in , the constant phase terms have been absorbed into $\rho_t$. Considering the $L$ pulses together, i.e. concatenating the desired target’s response for the entire CPI in a tall vector $\mathbf{y}$, is defined as $$\begin{aligned}
&\mathbf{y}\in\mathbbm{C}^{NML}=[\mathbf{y_0}^T,\mathbf{y_1}^T,\ldots,\mathbf{y}_{\mathbi{L-1}}^T]^T =\rho_t \mathbf{v}(f_d) \otimes\mathbf{s}\otimes \mathbf{a}(\theta_t,\phi_t)\nonumber \\
&\mathbf{v}(f_d):=[1,e^{-j2\pi f_dT_p},\ldots,e^{-j2\pi f_d(L-1)T_p}]^T \label{eq10}.\end{aligned}$$ The vector $\mathbf{y}$ consists of both the spatial and the temporal steering vectors as in classical STAP, as well as the waveform dependency, via waveform vector $\mathbf{s}$. Due to inclusion of the fast time samples in the waveform $\mathbf{s}$, the STAP data cube is modified to reflect this change, and is depicted in Fig. \[fig2\].
At the considered range gate, the measured snapshot vector consists of the target returns and the undesired returns, i.e. clutter returns, interference and noise. The contaminated snapshot at the considered range gate is then given by $$\begin{aligned}
\mathbf{\tilde{y}}=\mathbf{y}+\mathbf{y_i}+\mathbf{y_c}+\mathbf{y_n}
=\mathbf{y}+\mathbf{y_{u}} \label{eq.11} \end{aligned}$$ where $\mathbf{y_i,y_c,y_n}$ are the contributions from the interference, clutter and noise, respectively, and are assumed to be statistically uncorrelated with one another. The contribution of the undesired returns are treated in detail, starting with the noise as it is the simplest.
[**Noise**]{}: The noise is assumed to be zero mean, identically distributed across the sensors, across pulses, and in the fast time samples. The correlation matrix of $\mathbf{y_n}$ is denoted as $ \mathbf{R_n}\in\mathbbm{C}^{NML\times NML}$. [**Interference**]{}: The interference consists of jammers and other intentional / un-intentional sources which may be ground based, air-borne or both. Let us assume that there are $K$ interference sources. Further, since nothing is known about the jammers waveform characteristics, the waveform itself is assumed to be a stationary zero mean random process. Consider the $k$-th interference source in the $l$-th PRI, and at spatial co-ordinates $(\theta_k,\phi_k)$. Its corresponding snapshot contribution is modeled as, $$\mathbf{y}_{kl}=\boldsymbol{\alpha_{kl}}\otimes\mathbf{a}(\theta_k,\phi_k),k=1,2,\ldots,K,l=0,1,\ldots,L-1$$ where $\boldsymbol{\alpha_{kl}}=[\alpha_{kl}(0),\alpha_{kl}(1),\ldots,\alpha_{kl}(N-1)]^T\in\mathbbm{C}^{N}$ is the random discrete segment of the jammer waveform, as seen by the radar in the $l$-th PRI. Stacking $\mathbf{y}_{kl}$ for a fixed $k$ as a tall vector, we have $$\begin{aligned}
\label{eq12}
\mathbf{y_k}&=\boldsymbol{\alpha_k}\otimes\mathbf{a}(\theta_k,\phi_k)
=[\mathbf{y}_{ko}^T,\mathbf{y}_{k1}^T,\ldots,\mathbf{y}_{kL-1}^T]^T \in\mathbbm{C}^{NML} \nonumber\\
\boldsymbol{\alpha_k}:&=[\boldsymbol{\alpha_{k0}}^T,\boldsymbol{\alpha_{k1}}^T,\ldots,\boldsymbol{\alpha_{kL-1}}^T]^T \in\mathbbm{C}^{NL} \end{aligned}$$ Using the Kronecker mixed product property, (see for e.g. [@horn1994]), the correlation matrix of $\mathbf{y}_k$ is expressed as $\mathbbm{E}\{\mathbf{y}_k\mathbf{y}_k^H \}=\mathbf{R}_{\boldsymbol{\alpha}}^k\otimes \mathbf{a}(\theta_k,\phi_k)\mathbf{a}(\theta_k,\phi_k)^H$ where, $\mathbbm{E}\{ \boldsymbol{\alpha_k} \boldsymbol{\alpha_k}^H\}:=\mathbf{R}_{\boldsymbol{\alpha}}^k$. For $K$ mutually uncorrelated interferers, the correlation matrix is $\mathbf{R_i}=\sum\limits_{k=1}^K\mathbbm{E}\{\mathbf{y}_k\mathbf{y}_k^H \}=\sum\limits_{k=1}^K \mathbf{R}_{\boldsymbol{\alpha}}^k\otimes \mathbf{a}(\theta_k,\phi_k)\mathbf{a}(\theta_k,\phi_k)^H=\sum\limits_{k=1}^K (\mathbf{I}_{NL}\otimes\mathbf{a}(\theta_k,\phi_k))\mathbf{R}_{\boldsymbol{\alpha}}^k(\mathbf{I}_{NL}\otimes\mathbf{a}(\theta_k,\phi_k)^H) $, and is simplified as $$\begin{aligned}
\label{eq13}
\mathbf{R_i}&= \mathbf{A}(\theta,\phi)\mathbf{R}_{\boldsymbol{\alpha}}\mathbf{A}(\theta,\phi)^H\end{aligned}$$ where $\mathbf{R}_{\boldsymbol{\alpha}}:=\mbox{Diag}\{\mathbf{R}_{\boldsymbol{\alpha}}^1,\mathbf{R}_{\boldsymbol{\alpha}}^2,\ldots,\mathbf{R}_{\boldsymbol{\alpha}}^K\} \in\mathbbm{C}^{NMLK\times NMLK}$ and $\mathbf{A}(\theta,\phi)\in\mathbbm{C}^{NML\times NMLK}\\
=[\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_1,\phi_1),\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_2,\phi_2),\ldots,\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_K,\phi_K)]$, here $\mathbf{I}_{NL}$ the identity matrix of size $NL\times NL$, and $\mbox{Diag}\{ \cdot,\cdot,\ldots,\cdot\}$ the matrix diagonal operator which converts the matrix arguments into a bigger diagonal matrix. For example, $\mbox{Diag}\{\mathbf{A,B,C}\}=\left[ \begin{smallmatrix} \mathbf{A}&\mathbf{0}&\mathbf{0}\\
\mathbf{0}&\mathbf{B}&\mathbf{0} \\ \mathbf{0}&\mathbf{0}&\mathbf{C} \end{smallmatrix} \right].$
[**Clutter**]{}: The ground is a major source of clutter in air-borne radar applications and is persistent in all range gates upto the gate corresponding to the platform horizon. Other sources of clutter surely exist, such as buildings, trees, as well as other un-interesting targets, which are ignored. We therefore consider only ground clutter and treat it stochastically.
Let us assume that there are $Q$ clutter patches indexed by parameter $q$. Each of these clutter patches are comprised of say $P$ scatterers. The radar return from the $p$-th scatterer in the $q$-th clutter patch is given by $$\gamma_{pq}\mathbf{v}(fc_{pq})\otimes\mathbf{s} \otimes a(\theta_{pq},\phi_{pq})$$ where $\gamma_{pq}$ is its random complex reflectivity, $fc_{pq}$ is the Doppler shift observed from the $p$-th scatterer in the $q$-th clutter patch, and $\theta_{pq},\phi_{pq}$ are the azimuth and elevation angles of this scatterer.The Doppler $fc_{pq}$ is given by, $$\label{eq14}
fc_{pq}:=\dfrac{2f_o\mathbf{\dot{x}_r}^T (\mathbf{x_r}-\mathbf{x_{pq} })}{c||\mathbf{x_r}-\mathbf{x_{pq} }||}.$$ where $\mathbf{x_{pq}}$ is the location of the $p$-th scatter in the $q$-th clutter patch. Since the clutter patch is stationary, the Doppler is purely from the motion of the aircraft as seen in . The contribution from the $q$-th clutter patch to the received signal is given by $$\label{eq15}
\mathbf{y}_q=\sum \limits_{p=1}^P \gamma_{pq} \mathbf{v}(fc_{pq} )\otimes\mathbf{s} \otimes a(\theta_{pq},\phi_{pq} ),$$ with corresponding correlation matrix $$\begin{aligned}
\label{cluteq}
&\mathbf{R}_{\boldsymbol{\gamma}}^q:=\mathbf{B_q}\mathbf{R}_{\boldsymbol{\gamma}}^{pq} \mathbf{B_q}^H\end{aligned}$$ where, $\mathbf{B_q}
=[\mathbf{v}(fc_{1q})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{1q},\phi_{1q}), \mathbf{v}(fc_{2q})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{2q},\phi_{2q})\ldots,\mathbf{v}(fc_{Pq})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{Pq},\phi_{Pq})] \in\mathbbm{C}^{NML\times P}$ and $\mathbf{R}_{\boldsymbol{\gamma}}^{pq}$ is the correlation matrix of the random vector, $[\gamma_{1q},\gamma_{2q},\ldots,\gamma_{Pq}]^T$. It is readily shown that the matrix $\mathbf{B_q}$ could be simplified as, $\mathbf{B_q}:=\mathbf{\breve{B}_q}(\mathbf{I}_P\otimes\mathbf{s})$, where $\mathbf{\breve{B}_q}:=[\mathbf{v}(fc_{1q})\otimes\mathbf{A_{1q}},\mathbf{v}(fc_{2q})\otimes\mathbf{A_{2q}},\ldots, \mathbf{v}(fc_{Pq})\otimes\mathbf{A_{Pq}}]\in \mathbbm{C}^{NML\times PN}$, and the structure of the matrix $\mathbf{A_{pq}}\in\mathbbm{C}^{NM\times N}$ (straightforward but not shown here) is defined such that $\mathbf{s}\otimes\mathbf{a}(\theta_{pq},\phi_{pq} )=\mathbf{A_{pq}}\mathbf{s},p=1,\ldots,P$. Assuming that a particular scatterer from one clutter patch is uncorrelated to any other scatterer belonging to any other clutter patch, we have the net contribution of clutter $\mathbf{y_c}=\sum\limits_{q=1}^Q \mathbf{y}_q$, with corresponding correlation matrix given by $$\label{eq16}
\mathbf{R_c}=\sum\limits_{q=1}^Q \mathbf{R}_{\boldsymbol{\gamma}}^q.$$ The clutter model could further be simplified by the following arguments. Assuming a large range resolution which is typically the case for radar STAP [@ward1994] the scatterers in a particular clutter patch are in the same range gate and hence are assumed to possess approximately identical Doppler shifts, i.e. $fc_{pq}\approx fc_q=\tfrac{2f_o\mathbf{\dot{x}_r}^T (\mathbf{x_r}-\mathbf{x_{q} })}{c||\mathbf{x_r}-\mathbf{x_{q} }||}$. Similarly for the far field operation, and considering scatterers in the same azimuth resolution cell, and from the large range resolution argument, we may assume $\theta_{pq}\approx \theta_q$ and $\phi_{pq}\approx \phi_{q}$, i.e. their nominal angular centers. These assumptions can now be incorporated in matrix $\mathbf{B_q}$ to simplify the clutter model, see also [@Setlurradar2013].
Waveform Design
===============
The radar return at the considered range gate is processed by a filter characterized by a weight vector, $\mathbf{w}$, whose output is given by $\mathbf{w}^H\mathbf{\tilde{y}}$. Since the vector $\mathbf{s} \in\mathbbm{C}^{N}$ prominently figures in the steering vectors, the objective is to jointly obtain the desired weight vector, $\mathbf{w}$ and waveform vector, $\mathbf{s}$. It is desired that the weight vector will minimize the output power, $\mathbbm{E}\{|\mathbf{w}^H\mathbf{y_u}|^2\}=\mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}$. Mathematically, we may formulate this problem as: $$\begin{aligned}
\min_{\mathbf{w},\mathbf{s}}\;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}\label{eq17} \\
\mbox{ s. t } \;\;\;\;\;&\mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\
\;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o \nonumber \nonumber\end{aligned}$$ In , the first constraint is the renowned,well known Capon constraint with $\kappa\in\mathbbm{R}$, typically $\kappa=1$. An energy constraint enforced via the second constraint is to addresses hardware limitation. Before we derive the solutions to the optimization problem, it is useful to recall Lem. \[crlemma\], which is well-known, used throughout this report but not stated explicitly. This fundamental result discusses the technique to compute stationary points of a real valued function w.r.t. its complex valued argument and its conjugate.
\[crlemma\] Let $f(\mathbf{x},\mathbf{x}^{\ast}):\mathbbm{C}^N\rightarrow\mathbbm{R}$. The stationary point of $f(\mathbf{x},\mathbf{x}^{\ast})=\bar{f}(\mathbf{x_r},\mathbf{x_i})$ is found from the three equivalent conditions, 1. $\nabla_{\bf x_r}\bar{f}(\mathbf{x_r},\mathbf{x_i})=\mathbf{0}$ and $\nabla_{\bf x_i}\bar{f}(\mathbf{x_r},\mathbf{x_i})=\mathbf{0}$, or 2. $\nabla_{\bf x}f(\mathbf{x},\mathbf{x}^{\ast})=\mathbf{0}$, or 3. $\nabla_{\bf x^{\ast}} f(\mathbf{x},\mathbf{x}^{\ast})=\mathbf{0}$. Here $\bar{f}:\mathbbm{R}^{N}\times\mathbbm{R}^{N}\rightarrow\mathbbm{R}$ is the real equivalent of $f(\cdot,\cdot)$, $\mathbf{x_r}=\mathrm{Re}\{\mathbf{x}\},\mathbf{x_i}=\mathrm{Im}\{\mathbf{x}\}$, where we define the gradient $\nabla_{\bf x} f(\mathbf{x},\mathbf{x}^{\ast}):=[\tfrac{\partial f( \cdot,\cdot)}{\partial x_1},\tfrac{\partial f( \cdot,\cdot)}{\partial x_2},\cdots,\tfrac{\partial f( \cdot,\cdot)}{\partial x_N} ]^T$ with $x_i$ as the $i$-th element of ${\bf x},i=1,2,\ldots N$, and $\mathbf{0}$ is a column vector of all zeros of dimension $N$.
This arises from the Wirtinger calculus see [@Gesbert2007] [^4] for a recent formal proof.
Optimizing w.r.t. $\mathbf{w}$ first, the solution to is well known, and expressed as $$\label{weightcomp}
\mathbf{w}_{o}=\frac{\kappa \mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))}{(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1} (\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))}$$ where $\mathbf{R_u}(\mathbf{s})=\mathbf{R_i}+\mathbf{R_c}(\mathbf{s})+\mathbf{R_n}$. We further emphasize that the weight vector is an explicit function of the waveform. Now substituting $\mathbf{w}_{o}$ back into the cost function in , the minimization is purely w.r.t. $\mathbf{s}$, and cast as, $$\begin{aligned}
\label{eq18}
\min_{\mathbf{s}} &\frac{\kappa^2}{(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))} \nonumber \\
\mbox{ s. t. } & \mathbf{s}^H \mathbf{s}\leq P_o \end{aligned}$$ A solution to is not immediate, given the dependence of $\mathbf{R_u}$ on the waveform vector $\mathbf{s}$. We consider first, the case when the clutter dependence on the waveform is ignored. Solutions to the design when clutter is considered are treated subsequently.
Rayleigh-Ritz: Minimum eigenvector solution
-------------------------------------------
Ignoring the dependency of $\mathbf{R_u}$ on $\mathbf{s}$, we readily see that the can be recast as a Rayleigh-Ritz optimization, whose solution is given by $$\begin{aligned}
\label{eq21}
\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)=\boldsymbol{\mu}_{\min}(\mathbf{R_u})\end{aligned}$$ where $\boldsymbol{\mu}_{\min}(\mathbf{R_u})$ is the eigenvector corresponding to the minimum eigenvalue of $\mathbf{R_u}$. This tensor equation implicitly defines the optimal $\mathbf{s}$. It is readily seen that, $\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)=\mathbf{G}\mathbf{s}$, where $\mathbf{G}=\mathbf{v}(f_d)\otimes\mathbf{A_t}$, and $$\begin{aligned}
\mathbf{A_t}=\begin{bmatrix}
\mathbf{a}(\theta_t,\phi_t) &\mathbf{0} &\mathbf{0} &\cdots &\mathbf{0} \\
\mathbf{0}& \mathbf{a}(\theta_t,\phi_t) & \mathbf{0} &\cdots &\mathbf{0} \\
\mathbf{0}& \mathbf{0} &\mathbf{a}(\theta_t,\phi_t) &\vdots &\vdots \\
\vdots & \vdots &\vdots &\vdots &\vdots
\end{bmatrix} \in\mathbbm{C}^{MN\times N}.\end{aligned}$$ In general, the system is over-determined, and we solve this equation approximately via least squares (LS), $$\begin{aligned}
\label{lsubopt}
\mathbf{\hat{s}}=(\mathbf{G}^H\mathbf{G})^{-1}\mathbf{G}^H\mu_{\min}(\mathbf{R_u}).\end{aligned}$$ Moreover from and the structure of the temporal and spatial steering vectors, as well as the orthonormality of the eigenvectors, it is readily seen that, $$\begin{aligned}
\label{eqrayleigh}
||\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)||^2&=
|| \mathbf{v}(f_d)||^2 || \mathbf{s}||^2 ||\mathbf{a}(\theta_t,\phi_t) ||^2=|| \mathbf{s}||^2 \nonumber \\
||\boldsymbol{\mu}_{\min}(\mathbf{R_u})||^2&=1.\end{aligned}$$ Hence the LS solution in must be scaled to satisfy the desired energy requirements of the radar system.
[**Decoupling LS:**]{} The LS solution in can be further simplified due to the following linear relation between elements of $\mathbf{v}(f_d),\mathbf{a}(\theta_t,\phi_t),\mathbf{s}$ and elements of $\boldsymbol{\mu}_{\min}(\mathbf{R_u})$, expressed as $$\begin{aligned}
\label{linearrel}
v_la_ms_n=\mu_h,\; l&=1,2,\ldots,L,\;m=1,2,\ldots,M,\;n=1,2,\ldots,N \nonumber \\
h&=(l-1)MN+(n-1)M+m.\end{aligned}$$ where $v_l$, $a_m$, $s_n$ are the $l$-th, $m$-th, $n$-th elements of $\mathbf{v}(f_d)$, $\mathbf{a}(\theta_t,\phi_t)$, $\mathbf{s}$, and $\mu_h$ is the $h$-th element of $\boldsymbol{\mu}_{min}(\mathbf{R_u})$, respectively. Therefore, the LS solution in decouples as $$\begin{aligned}
\label{lsfinal}
s_n=\frac{(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )^H\boldsymbol{\mu}_{\bf n} }{(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )^H(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )},\; n=1,2,\ldots,N\end{aligned}$$ where the vector $\boldsymbol{\mu}_{\bf n}\in\mathbbm{C}^{ML}$ for a [*particular $n$*]{} consists of the $ML$ appropriate elements, $\mu_h,\; h=(l-1)MN+(n-1)M+m,\; m=1,2,\ldots,M, \;l=1,2,\ldots,L$, as highlighted in .
The min. eigenvector solution is most relevant when noise and interference are considered and clutter is ignored in the waveform design [@guerci2003]. it has some nice spectral properties similar (but not identical) to water-filling [@guerci2003; @bell1993information]. Therefore this solution, although suboptimal, is a good initial waveform to interrogate the radar scene, but is unfortunately well known to suffer from poor modulus and sidelobe properties. Nonetheless, in certain exceptional cases and in the presence of clutter, this suboptimal solution is shown to be optimal, and is discussed at a later stage. The ensuing definitions and lemma proves useful subsequently.
\[lemma1\] (a) If vectors $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ consist of the eigenvalues of the square but not necessarily Hermitian matrices, $\mathbf{X}\in\mathbbm{C}^{N\times N}$, $\mathbf{Y}\in\mathbbm{C}^{M\times M}$ and $\mathbf{X}\otimes\mathbf{Y}$, respectively. Then $\boldsymbol{\gamma}=\boldsymbol{\alpha}\otimes\boldsymbol{\beta}$. (b) Also, $\mathrm{rank}(\mathbf{X}\otimes\mathbf{Y})=\mathrm{rank}(\mathbf{X})\otimes\mathrm{rank}(\mathbf{Y})$.
For (a), let $\mathbf{x}_i,i=1,2,\ldots,N$ and $\mathbf{y}_j,j=1,2,\ldots,M$ are the eigenvectors corresponding to $\alpha_i,\beta_j$ i.e. the $i$-th and $j$-th eigenvalues, of $\mathbf{X,Y}$, respectively. Then, from the mixed property of the Kronecker product, $\mathbf{X}\mathbf{x}_i\otimes\mathbf{Y}\mathbf{y}_j=(\mathbf{X}\otimes\mathbf{Y})(\mathbf{x}_i\otimes\mathbf{y}_j)$ but the eigenvector relations imply that $\mathbf{X}\mathbf{x}_i=\alpha_i\mathbf{x}_i,\mathbf{Y}\mathbf{y}_j=\beta_j\mathbf{y}_j$. This implies that the $ij$-th eigenvalue of of $\mathbf{X}\otimes\mathbf{Y}$ is $\gamma_{ij}=\alpha_i\beta_j$ with associated eigenvector $\mathbf{x}_i\otimes\mathbf{y}_j$. Since the rank is equal to the number of non-zero eigenvalues for square matrices, the second follows directly from (a). Hence proved.
\[mydef1\] ([*Convexity*]{}) A function $f(\mathbf{x}):\mathbbm{R}^N\rightarrow\mathbbm{R}$ is convex if : (a) $f(t \mathbf{x}_1+(1-t)\mathbf{x}_2)\leq t f(\mathbf{x}_1)+(1-t)f(\mathbf{x}_2)$ for any $t\in[0,1]$ (b) If $f(\mathbf{x})$ is first order differentiable, then it is convex if $f(\mathbf{x}_j)\geq f(\mathbf{x}_i)+ \nabla_{\mathbf{x}_i} f(\mathbf{x}_i)^T( f(\mathbf{x}_j)-f(\mathbf{x}_i) )$where in (a)(b) $\mathbf{x}_i\in\mathbbm{R}^N,i=1,2, j=1,2,j\neq i$.
From our extensive simulations, we noticed that the original cost function in is [*not jointly convex*]{} in $\mathbf{w}$ and $\mathbf{s}$. Nevertheless, it is not straightforward to prove / disprove joint convexity w.r.t. both $\mathbf{w}$ and $\mathbf{s}$ analytically. Consider, then, the following propositions:
\[propos1\] The objective function in is individually convex w.r.t. $\mathbf{s}$, for any fixed but arbitrary $\mathbf{w}$
Definition \[mydef1\] cannot be directly invoked as the objective $g(\mathbf{s})= \mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}:\mathbbm{C}^N\rightarrow \mathbbm{R}$ depends on the waveform $\mathbf{s}$, which is complex. Consider the following transformation[^5], $\mathbf{s}=\mathbf{D}\bar{\mathbf{s}}$ where $\bar{\mathbf{s}}\in\mathbbm{R}^{2N}=[\mbox{Re}\{\mathbf{s}\}^T,\mbox{Im}\{\mathbf{s}\}^T] ^T$ and $\mathbf{D}=[\mathbf{I}_N, j\mathbf{I}_N]\in\mathbbm{C}^{N\times2N}$. Now, we may define an equivalent $g(\bar{\mathbf{s}}) :\mathbbm{R}^{2N}\rightarrow \mathbbm{R}$ to invoke the definition of convexity. We have to prove that, $$\begin{aligned}
\label{eq19}
&\mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\
&+\sum\limits_{q=1}^Q \mathbf{\breve{B}_q}\begin{aligned} &(\mathbf{I}_P\otimes \mathbf{D}(t\bar{\mathbf{s}}_1+(1-t)\bar{\mathbf{s}}_2 ) )\mathbf{R}_{\gamma}^{pq} \\
&( \mathbf{I}_P\otimes (t\bar{\mathbf{s}}_1+(1-t)\bar{\mathbf{s}}_2 )^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H \end{aligned} \end{aligned} \right) \mathbf{w} \nonumber \\
&\leq t \mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\
&+\sum\limits_{q=1}^Q \mathbf{\breve{B}_q}
(\mathbf{I}_P\otimes \mathbf{D}\bar{\mathbf{s}}_1 )\mathbf{R}_{\gamma}^{pq} ( \mathbf{I}_P\otimes \bar{\mathbf{s}}_1^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H
\end{aligned} \right) \mathbf{w} \nonumber \\
&+(1-t)\mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\
&+\sum\limits_{q=1}^Q
\mathbf{\breve{B}_q}(\mathbf{I}_P\otimes \mathbf{D}\bar{\mathbf{s}}_2 )\mathbf{R}_{\gamma}^{pq} ( \mathbf{I}_P\otimes \bar{\mathbf{s}}_2^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H
\end{aligned} \right) \mathbf{w} \end{aligned}$$ where $t\in[0,1]$ and $\mathbf{\bar{s}_i}\in \mbox{dom}\{ g(\bar{\mathbf{s}})\},i=1,2$. After elementary algebra, the convexity requirement in transforms to:
$$\begin{aligned}
\label{eq20}
\sum\limits_{q=1}^{Q} \mathbf{x}^H_{\bf q} \left(\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H\right)\mathbf{x_q} \geq 0\end{aligned}$$
where $\mathbf{x_q}\in\mathbbm{C}^{NP}:=\mathbf{\breve{B}_q}^H\mathbf{w}$. In other words, it is sufficient to show that iff is true then is also true and therefore convex. We notice immediately that is a sum of Hermitian quadratic forms. Consider the matrix $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H$, we know that $\mathbf{R}_{\gamma}^{pq}\succeq \mathbf{0}$[^6], since it is a covariance matrix and by definition atleast positive semi-definite (PSD). The other matrix, i.e. $\mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H$ is of course rank-1 Hermitian, and is clearly PSD. From Lem. \[lemma1\], it is straightforward to show that $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T \mathbf{D}^H\succeq \mathbf{0},\forall q$. Then from the definition of positive semi-definiteness, each of the $Q$ Hermitian quadratic forms in is greater than zero, hence their sum is also greater than zero.
\[propos2\] The objective function in is individually convex w.r.t. $\mathbf{w}$, for any fixed but arbitrary $\mathbf{s}$.
Given the guaranteed positive semi-definiteness of $\mathbf{R_u}(\mathbf{s})$, the proof is straightforward to demonstrate by invoking the convexity definition on the vector consisting of the real and imaginary parts of $\mathbf{w}$.
In fact, Prop. 1, Prop. 2 may be sharpened to include strong convexity, which, as we will show subsequently is desired for the solutions to exist, see the note immediately after $\eqref{eq32}$. For now, however, individual convexity is sufficient to proceed with our analysis.
\[remarkstapobj\] ([*Characteristic of STAP objective*]{}) The STAP objective in has [*at most one*]{} minima for a fixed but arbitrary $\mathbf{w}\in\mathbbm{C}^{NML}$ but $\forall \mathbf{s} \in\mathbbm{C}^{N}$. Likewise, it has [*at most one*]{} minima for a fixed but arbitrary $\mathbf{s}\in\mathbbm{C}^{N}$ but $\forall \mathbf{w} \in\mathbbm{C}^{NML}$
This is concluded readily from Prop. \[propos1\], Prop. \[propos2\], i.e. the individual convexity. An illustrative example is provided in Fig. \[figlocalmin\].
\[htbp!\] ![An [*illustrative*]{} non-convex example with multiple local minima. Contours in black are characteristic of the objective. Contours in blue violate convexity in the $\mathbf{w}$, and $\mathbf{s}$ dimension individually, and are therefore [*not characteristic*]{} of the objective function.[]{data-label="figlocalmin"}](Localminima "fig:")
Constrained alternating minimization
------------------------------------
Motivated from Prop. \[propos1\], and Prop. \[propos2\], we propose a constrained alternating minimization technique which is iterative. Before we present details on this technique, consider the following minimization problem, which optimizes $\mathbf{s}$, but for a fixed and arbitrary $\mathbf{w}$: $$\begin{aligned}
\min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\
\mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq22} \\
\;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o. \nonumber \nonumber\end{aligned}$$ In , the objective function could be rewritten as, $$\begin{aligned}
\label{eq23}
\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w}=&\mathbf{w}^H(\mathbf{R_n}+\mathbf{R_i})\mathbf{w} \\
+&\sum\limits_{q=1}^Q \mbox{Tr}\{ \mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H (\mathbf{I}_P\otimes\mathbf{s})\} \nonumber.\end{aligned}$$ In , the trace operation is further simplified as: $$\begin{aligned}
\label{eq24}
\mbox{Tr}\{ &\mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H (\mathbf{I}_P\otimes\mathbf{s})\} \nonumber \\
&=\mbox{vec}\left( \left(\mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H \right)^T \right)^T \mbox{vec}( \mathbf{I}_P\otimes\mathbf{s}) \nonumber
\\
&=\mathbf{s}^H\mathbf{H}^T ( \mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H)\mathbf{H}\mathbf{s} \nonumber\\
&=\mathbf{s}^H\mathbf{Z_q}(\mathbf{w})\mathbf{s}\end{aligned}$$ where $\mbox{vec}(\mathbf{I}_P\otimes\mathbf{s})=\mathbf{H}\mathbf{s}$, with $\mathbf{H}\in \mathbbm{R}^{P^2N \times N}=[\mathbf{H_1}^T,\mathbf{H_2}^T,\ldots,\mathbf{H_P}^T]^T$. The matrix $\mathbf{H_k}\in\mathbbm{R}^{PN \times N}, k=1,2,\ldots,P$ is further decomposed into $P$, $N\times N$ matrices, and is defined such that the $k$-th $N \times N$ matrix is $\mathbf{I}_N$ and the other $(N-1)$, $N\times N$ matrices are all equal to zero matrices.
\[PSDremark\] (a) At the very least, $\sum\limits_{q=1}^Q\mathbf{Z_q}\succeq \mathbf{0}$. (b) The matrix $\mathbf{Z_q}\succeq \mathbf{0}$ for $P<N$, always. (c) However, it may be positive definite, i.e. $\mathbf{Z_q}\succ\mathbf{0}$ and hence $\sum\limits_{q=1}^Q\mathbf{Z_q}\succ\mathbf{0}$ for $P\geq N$ and for $\mathbf{R}_\gamma^{pq}\succ \mathbf{0}$.
We note that (a) is readily implied from Prop. \[propos1\] since a Hermitian quadratic form $\mathbf{x}^H\mathbf{B}\mathbf{x}$ is convex (strictly convex) iff $\mathbf{B}\succeq\mathbf{0}$ ( $\mathbf{B}\succ\mathbf{0}$). Since $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H \succeq \mathbf{0}$ always ($\leq P$ non-zero eigenvalues and the rest are zeros) and that $P<N$, in other words, the transformation $\mathbf{H}^T(\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H)\mathbf{H}:\mathbbm{C}^{P^2 N\times N} \times \mathbbm{C}^{P^2 N\times N} \rightarrow \mathbbm{C}^{N\times N}$ and from the structure of $\mathbf{H}$, the result (b) is obvious. For (c), we know that $\mathrm{rank}(\mathbf{H})=N$, hence it could be shown after some tedious algebra that $\mathbf{Z_q}$ may be PD only when $P\geq N$ and that $\mathbf{R}_{\gamma}^{pq}$ is PD in the first place, also see for example [@horn1994 pg. 399]. Using and , the Lagrangian of is readily cast as, $$\begin{aligned}
\label{eq25}
\mathcal{L}(\mathbf{s},\gamma_1,\gamma_2)&=\mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w}+\sum\limits_{q=1}^Q\mathbf{s}^H\mathbf{Z_q}(\mathbf{w})\mathbf{s} \\
&+\mbox{Re}\{ \gamma_1^{\ast} (\mathbf{w}^H\mathbf{G}\mathbf{s}-\kappa)\}+\gamma_2\mathbf{s}^H\mathbf{I}_N\mathbf{s}-\gamma_2P_o \nonumber\end{aligned}$$ where $\gamma_1\in\mathbbm{C}$ and $\gamma_2\in\mathbbm{R}^{+}$ are the complex and real Lagrange parameters.
[**Lagrange Dual:**]{} The Lagrange dual, denoted as $\mathcal{H}(\gamma_1,\gamma_2)=\inf\limits_{\mathbf{s}} \mathcal{L}( \mathbf{s},\gamma_1,\gamma_2)$. Since consists of Hermitian quadratic forms and other linear terms of $\mathbf{s}$, we have $\mathcal{H}(\gamma_1,\gamma_2)=\mathcal{L}(\mathbf{s_o}(\gamma_1,\gamma_2),\gamma_1,\gamma_2)$, where $\mathbf{s_o}(\gamma_1,\gamma_2)$ is obtained by solving the first order optimality conditions, i.e. $$\begin{aligned}
\label{eq26}
\frac{\partial \mathcal{L}(\mathbf{s},\gamma_1,\gamma_2)}{\partial \mathbf{s}} ={\bf 0}\end{aligned}$$ where, $\mathbf{0}$ is a column vector of size $N$ and consists of all zeros. Further, in , while taking the derivative the usual rules of complex vector differentiation apply, i.e. treat $\mathbf{s}^H$ independent of $\mathbf{s}$. The solution to is readily obtained by differentiating , and expressed as: $$\begin{aligned}
\label{eq27}
\mathbf{s_o}(\gamma_1,\gamma_2)=-\frac{\gamma_1}{2}\Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}.\end{aligned}$$ Using , the dual $\mathcal{H}(\gamma_1,\gamma_2)$ is given by: $$\begin{aligned}
\label{eq28}
&\mathcal{H}( \gamma_1,\gamma_2)= \mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w} -\kappa\mbox{Re}\{ \gamma_1^\ast \}-\gamma_2P_o \nonumber \\
-&\frac{|\gamma_1|^2}{4} \mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}.\end{aligned}$$ Equation is further simplified by decomposing, $\gamma_1=\gamma_{1r}+j\gamma_{1i}$. In which case, we notice that is quadratic in $\gamma_{1r},\gamma_{1i}$,and purely linear in $\lambda_2$. The Lagrange dual optimization is therefore, $$\begin{aligned}
\label{eq29}
\max \limits_{\gamma_{1r},\gamma_{1i},\gamma_2} \;\;\;\; &\mathcal{H}(\gamma_{1r},\gamma_{1i},\gamma_2) \nonumber \\
\mbox{s. t } \;\;\;\; &\gamma_2\geq0.\end{aligned}$$ Maximizing first w.r.t. $\gamma_{1r},\gamma_{1i}$, we have the solutions, $$\begin{aligned}
\bar{\gamma}_{1r}=\frac{-2\kappa}{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}},\;\bar{\gamma}_{1i}=0.\end{aligned}$$ Substituting the above solutions into , the Lagrange dual optimization problem and after ignoring an unnecessary additive constant, takes the form, $$\begin{aligned}
\label{eq30}
\max\limits_{\gamma_2} \;\; &\kappa^2 \Bigl( \mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}\Bigr)^{-1}-\gamma_2P_o \nonumber \\
\mbox{s. t. } \;\;&\gamma_2\geq0\end{aligned}$$ The associated Lagrangian for is $$\begin{aligned}
\label{lagra1}
\mathcal{D}(\gamma_2,\gamma)=\frac{\kappa^2} {\mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w}} -\gamma_2P_o-\gamma_3\gamma_2\end{aligned}$$ where $\mathbf{F}:=\sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N$. The first order optimality condition for the optimization is given by: $$\begin{aligned}
&\frac{\partial}{\partial \gamma_2} \bigl( \frac{\kappa^2} {\mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w}} \bigr)-P_o-\gamma_3 =0 \nonumber \\
\mbox{or } &\frac{-\kappa^2}{ ( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w})^{2}} \mathbf{w}^H\mathbf{G}\frac{\partial\mathbf{F}^{-1}}{\partial\gamma_2}\mathbf{G}^H\mathbf{w}-P_o -\gamma_3 =0 \nonumber \\
\mbox{or } &\frac{\kappa^2}{ ( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w})^{2}} \mathbf{w}^H\mathbf{G}\bigl( \mathbf{F}^{-1}\frac{\partial\mathbf{F}}{\partial\gamma_2} \mathbf{F}^{-1} \bigr)\mathbf{G}^H\mathbf{w}-P_o -\gamma_3 =0 \nonumber \end{aligned}$$ where $\gamma_3$ is the Lagrange multiplier associated with the Lagrangian , and we also have $\tfrac{\partial \mathbf{F}}{\partial \gamma_2}=\mathbf{I}_N$. The complementary slackness and constraint qualifier for i.e. $\gamma_3\gamma_2=0$ and $\gamma_2\geq0$ form the rest of the equations comprising the KKT conditions. It is now readily shown that the solution to is given by $$\begin{aligned}
\label{eq31}
&\bar{\gamma}_2=\max [0,\gamma_2] \\
&\gamma_2 \mbox{ solves } \gamma_2\left( \kappa^2\mathbf{w}^H\mathbf{G}\mathbf{F}^{-2}\mathbf{G}^H\mathbf{w}-P_o( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1}\mathbf{G}^H\mathbf{w})^2\right)=0 \nonumber.\end{aligned}$$
\[propos3\] The parameter $\bar{\gamma}_2=0$ solves .
The spectral theorem for Hermitian matrices, allows for a decomposition, $\mathbf{F}=\mathbf{E}(\boldsymbol{\Lambda}+\gamma_2\mathbf{I}_N)\mathbf{E}^H$. The matrix $\boldsymbol{\Lambda}$ is a diagonal matrix comprising eigenvalues in descending order, whereas, $\mathbf{E}$ is unitary and whose columns are the corresponding eigenvectors of $\mathbf{F}$. For ease of exposition, denote $\mathbf{z}\in\mathbbm{C}^{N}:=\mathbf{E}^H\mathbf{G}^H\mathbf{w}$, then assume a function $f(\gamma_2):\mathbbm{R}^{+}\rightarrow\mathbbm{R}$, expressed as $$\begin{aligned}
\label{lagra2}
f(\gamma_2)&:=\kappa^2\mathbf{w}^H\mathbf{G}\mathbf{F}^{-2}\mathbf{G}^H\mathbf{w}-P_o( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1}\mathbf{G}^H\mathbf{w})^2 \nonumber \\
&=\sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{(d_n+\gamma_2)^2} -P_o\left( \sum\limits_{n=1}^N \frac{|z_n|^2}{d_n+\gamma_2}\right)^2\end{aligned}$$ where $z_n,d_n$ are the $n$-th elements of $\mathbf{z}$, and the $n$-th eigenvalue in $\boldsymbol{\Lambda}$. We analyze $f(\gamma_2)$ and $\gamma_2f(\gamma_2)$ in detail. The following (behavior at $0$ and $\infty$) are readily observed
\[lagra3\] $$\begin{aligned}
\lim_{\gamma_2\rightarrow \infty}f(\gamma_2)&=f(\infty) =0 \label{lagra32}\\
\lim_{\gamma_2\rightarrow 0}f(\gamma_2)&=\sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{d_n^2} -P_o\left( \sum\limits_{n=1}^N\frac{|z_n|^2}{d_n}\right)^2=f(0) \label{lagra31}\end{aligned}$$
Furthermore, it is seen that $$\label{lagra4}
\begin{aligned}
&\lim_{\gamma_2\rightarrow\infty}\gamma_2f(\gamma_2)=\lim_{\gamma_2\rightarrow\infty}\frac{f(\gamma_2)}{1/\gamma_2}=\lim_{\gamma_2\rightarrow\infty}\frac{\frac{\mathrm{d}f(\gamma_2)}{\mathrm{d}\gamma_2}}{(-1/\gamma_2^2)} =0
\end{aligned}$$ Moreover, consider $f(\gamma_2)=h_1(\gamma_2)-h_2(\gamma_2)=0$, where $h_1(\gamma_2)=\kappa^2\sum\limits_{n=1}^N\frac{f_n^2(\gamma_2)}{|z_n|^2},h_2(\gamma_2)=P_o (\sum\limits_{n=1}^N f_n(\gamma_2))^2$, where $f_n(\gamma_2)=\frac{|z_n|^2}{d_n+\gamma_2}$. Note that $fn(\gamma_2)\downarrow, n=1,2,\ldots,N$ and that $h_i(\gamma_2)\downarrow,i=1,2$, i.e. decreasing functions w.r.t. $\gamma_2 \in[0,\infty)$. Then equation$f(\gamma_2)=0$ implies that $$\label{lagra41}
\begin{aligned}
\sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{(d_n+\gamma_2)^2} -P_o\left( \sum\limits_{n=1}^N \frac{|z_n|^2}{d_n+\gamma_2}\right)^2=0 \\
\mbox{or } \sum\limits_{n=1}^N (\frac{\kappa^2}{|z_n|^2}-P_o )f_n^2(\gamma_2)=2\sum\limits_{n_1}\sum\limits_{\substack{n_2\\ n_2\neq n_1}} f_{n_1}(\gamma_2)f_{n_2}(\gamma_2)
\end{aligned}$$ where $(n_1,n_2)\in (1,2,\ldots,N)$. Recall that $d_n\neq 0 \forall n$, $d_n\geq d_{n+1}, n=1,2,\ldots, N$, and $ |z_n|\neq 0 \forall n$. A solution to for $\gamma_2 \in[0,\infty)$ is readily derived in the trivial case, for example when $f_{n_1}(\gamma_2)=f_{n_2}(\gamma_2)$, $P_o\neq \kappa^2$, and for $|z_n|$ to be some arbitrary constant for all $n$. For $P_o\geq \kappa$ it may now be shown numerically that a solution to for $\gamma_2 \in [0,\infty)$ does not exist.
In fact, our extensive numerical simulations reveal that in general and assuming $P_o\geq \kappa$ and for $\gamma_{21}\leq \gamma_{22}$ $$\begin{aligned}
\label{lagra51}
\begin{cases}
f(\gamma_{21})\geq f(\gamma_{22}) \mbox{ if } f(0)>0 \\
f(\gamma_{21})\leq f(\gamma_{22}) \mbox{ if } f(0)<0
\end{cases} \gamma_{21} \mbox{ and } \gamma_{22} \in[0,\infty).\end{aligned}$$ That is, $f(\gamma_2)$ is monotonic. From the above arguments, therefore, $\gamma_2f(\gamma_2)=0$ implies that $\gamma_2=0$. Alternatively nevertheless, a solution to may be found numerically and is computationally cheap.
[**Note:**]{} [*(Inactive power constraint)*]{} It is noted that trivially $\bar{\gamma}_2=0$ [*may always*]{} be chosen as a solution with suitable choices of the free parameter $P_o$. This implies that the power constraint is always satisfied and hence is an inactive constraint in the corresponding Lagrangian. A graphical behavior of $h_i(\gamma_2),i=1,2$ and thus the behavior of $f(\gamma_2)$ is seen from Fig. \[figlagrange\]. Using Prop. \[propos3\], the waveform design solution is unique, a function of $\mathbf{w}$ and expressed as, $$\begin{aligned}
\label{eq32}
\mathbf{s}_o(\mathbf{w})=\frac{\kappa \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w} }{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w}}.\end{aligned}$$
[**Note:**]{} ([*Strong convexity*]{}) To compute the constrained alternating minimization solutions, the respective matrices in , must be invertible, implying strong convexity individually w.r.t. $\mathbf{w}$, $\mathbf{s}$, respectively. This directly necessitates, $\lambda_{\min}\Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)\neq 0$ and $\lambda_{\min} (\mathbf{R_u}(\mathbf{s}) )\neq 0$, and hence also, positive definiteness of these matrices.
The alternating minimization algorithm is now succinctly stated in Table \[table1\].
\[tbp!\] ![Two cases are presented assuming $P_o\geq \kappa$. (a) Blue: $h_1(\gamma_2)$, Red: $h_2(\gamma_2)$ and therefore $f(\gamma_2)$ is decreasing, (b) Blue: $h_2(\gamma_2)$, Red: $h_1(\gamma_2)$ and therefore $f(\gamma_2)$ is increasing. The blue and red curves intersect at $\infty$.[]{data-label="figlagrange"}](lagrange_param "fig:")
\[dualprop\]
([*Strong duality*]{}) The optimal value of the lagrange dual problem is given by $$\mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w} +\frac{\kappa^2}{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w}}.$$ It is therefore trivial to show that the duality gap between and is zero. In other words, strong duality holds between the primal in and the dual in . From Slaters condition [@Boyd2004] the sufficient condition to ensure strong duality is the existence of , i.e. the inverse of $\sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})$ exists (see note below), and that the solution in satisfies the power constraint.
[**Note:**]{} ([*Lower bound on $Q$*]{}) Since $\mathrm{rank}( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w}))\leq \sum\limits_{q=1}^Q\mathrm{rank}( \mathbf{Z_q}( \mathbf{w}))$, [*assume the worst case*]{} $P=1$, then we have that $\mathrm{rank}(\mathbf{Z_q})=1$. Therefore for $Q$ distinct (different spatial signature and Doppler) clutter patches, $Q\geq N$ ensures invertibility of $\sum\limits_q \mathbf{Z_q}$.
[|p[3.3in]{}|]{}
1. [*Initialize*]{}: Start with an initial waveform design, defined as $\mathbf{s}_o^{(0)}$, set counter $k=1$
2. [ *Filter design*]{}: Design the optimal filter weight vector, $\mathbf{w}_{o}^{(k)}=\mathbf{w}_o(\mathbf{s}_o^{(k-1)})$, where is used to compute $\mathbf{w}_o( \cdot)$.
3. [ *Waveform design*]{}: Design the updated waveform $\mathbf{s}_o^{(k)}=\mathbf{s}_o(\mathbf{w}_o^{(k)})$, where is used to compute $\mathbf{s}_o(\cdot)$.
4. [*Check:*]{} If convergence is achieved, exit, else $k=k+1$, go back to step-2.
\
### Convergence, performance guarantees, and other properties
Denote $(\mathbf{w}_k,\mathbf{s}_k)$ as the sequence of iterates of the algorithm in Table \[table1\] and define $g(\mathbf{w}_k,\mathbf{s}_k):=\mathbf{w}_k\mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_k^H$, then for $k=1,2,\ldots$ $$\begin{aligned}
\label{eq33}
\cdots g(\mathbf{w}_k,\mathbf{s}_{k-1} )\geq g(\mathbf{w}_k,\mathbf{s}_k ) \geq g( \mathbf{w}_{k+1},\mathbf{s}_k)\cdots.\end{aligned}$$ Moreover, since at least $\mathbf{R_u}(\mathbf{s})\succeq \mathbf{0}$, i.e. PSD $\forall\mathbf{s}$, we have that $g(\mathbf{w},\mathbf{s})\geq0,\forall\mathbf{w}$. Therefore each of the individual terms in are [*lower bounded*]{} by zero, in other words $g(\mathbf{w}_{k_1},\mathbf{s}_{k_2})\geq0,k_1=k,\mbox{ or }k+1$ and $k_2=k,\mbox{ or }k+1$, for $k=1,2,\ldots$ .
\[propos4\] Iff the iterates $(\mathbf{w}_k,\mathbf{s}_k)$ of the constrained alternating minimization exist, then $\lim\limits_{k\rightarrow\infty}\mathbf{g ( \mathbf{w}_k,\mathbf{s}_k)}$ is finite.
The non-increasing property in , and since each term in is lower bounded, straightforward application of the monotone convergence theorem to the sequence, $\{\mathbf{g ( \mathbf{w}_k,\mathbf{s}_k)}\}$,completes the proof.
We note that convergence to a finite limit as evidenced from Prop. \[propos4\] is indeed dependent on the constraints via the existence of the iterates $(\mathbf{w}_k,\mathbf{s}_k)$. This however does not imply convergence of the sequence $\{(\mathbf{w}_k,\mathbf{s}_k) \}$, for which, consider the following.
\[remark1\] The alternating minimization is a special case of the block Gauss-Siedel and block co-ordinate descent (BCD) algorithm with block size equal to two [@Grippo2000; @Luo1992].
\[deflimit\] ([*Convergence in $\mathbbm{R}^{N}$*]{}) A sequence $\{ \mathbf{x}_k\} \in\mathbbm{R}^{N} ,k=1,2,\ldots$ is said to converge to $\tilde{\mathbf{x}}$, a limit point, if, $\forall \epsilon>0,\;\; \exists K\in{\mathbbm{N}}:\;||\mathbf{x}_k-\tilde{\mathbf{x}}|| \leq\epsilon,\; k>K$.
\[altminlemma\] (Constrained alternating minimization lemma) Assume that a function $g(\mathbf{z}):\mathbbm{R}^{2N}\rightarrow\mathbbm{R},\mathbf{z}=[\mathbf{x}^T,\mathbf{y}^T]^T$ is continuously differentiable over a closed nonempty convex set, $\mathcal{A}=\mathcal{A}_1\times\mathcal{A}_2$. Also, suppose the solution to the constrained optimization problems, $\min \limits_{\mathbf{x}\in\mathcal{A}_1} g(\mathbf{x},\mathbf{y} )$ and $\min \limits_{\mathbf{y}\in\mathcal{A}_2} g(\mathbf{x},\mathbf{y} )$ are uniquely attained. Let $\{ \mathbf{z}_k \}$ be the sequence generated by this algorithm, then every limit point of this sequence is also a stationary point.
The proof in [@Bertsekas1999 Prop. 2.7.1] follows immediately to the alternating minimization assuming two blocks. Also see [@Grippo2000], where the convergence of the two block BCD was analyzed.
The above Lem. \[altminlemma\] discusses convergence of the constrained alternating minimization.This lemma can be applied by decomposing our problem into its real equivalent along-with real and imaginary decomposition of $\mathbf{w},\mathbf{s}$, and assuming the our constraint set $\mathcal{A}=\mathcal{A}_1\times\mathcal{A}_2$ is closed convex and the minimizers are unique. The necessary condition of a unique minimizer [@Zangwill1967] at each step is not obvious, but [@Powell1973] showed that in the absence of this assumption the algorithm cycles endlessly around a particular objective value [@Bertsekas1999]. Further the algorithm provides limit points which are not stationary points [@Grippo2000]. To discuss the characteristics of the limits points at convergence, consider the remark, presented next.
\[limitpointremk\] ([*Characterizing the solutions at convergence*]{}) If $(\mathbf{w}_\star,\mathbf{s}_\star)$ are the limit points of the sequence $\{(\mathbf{w}_k,\mathbf{s}_k)\}$. Then, $(\mathbf{w}_\star,\mathbf{s}_\star)$ is a local minima, i.e. by definition $g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w},\mathbf{s})$,$\exists \epsilon>0$ with $(\mathbf{w},\mathbf{s}):\,\sqrt{|| \mathbf{w}-\mathbf{w}_\star||^2+|| \mathbf{s}-\mathbf{s}_\star||^2}\leq \epsilon$. Further, $(\mathbf{w}_\star,\mathbf{s}_\star):g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w}_\star,\mathbf{s}),\,\forall \mathbf{s}\in\mathcal{A}_2\mbox{ and } g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w},\mathbf{s}_\star),\,\forall \mathbf{w}\in\mathcal{A}_{1}$.
The first statement in Rem. \[limitpointremk\] directly results from from the stationarity condition as given in Lem. \[altminlemma\] and also since the objective is non-convex. The second statement in Rem. \[limitpointremk\] arises from the individual convexity in $\mathbf{w}$ and $\mathbf{s}$ as shown in Prop. \[propos1\], Prop. \[propos2\]. We note readily from Rem. \[remarkstapobj\], that unfortunately there is nothing special or strong about $(\mathbf{w}_\star,\mathbf{s}_\star)$ except the fact that they are local minima. It is well known that global extrema (minima or maxima) are attained only when the objective is either convex or concave. For a problem similar to ours and where the alternating minimization was applied, see [@vaidyanathan2009 pg.3537] the authors state that their algorithm produces limit points which are stronger than local maxima, in our opinion this conclusion is suspect. They further claim that their algorithm produces global extrema in their filter design and waveform dimensions individually, which leads us to believe that their objective is concave, although this was never proved in [@vaidyanathan2009]. In our opinion, Rem. \[remarkstapobj\] is also relevant to their objective by replacing minima by maxima, and hence we do not believe that the limit points produced by their algorithm are stronger than local extrema.
To derive the upper and lower bounds on $g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)$, the following well known lemmas are useful.
\[lemma2\] For any Hermitian matrix, $\mathbf{A}\in\mathbbm{C}^{N\times N}$ and any arbitrary vector $\mathbf{x}\in\mathbbm{C}^{N\times N}$ , we always have $\lambda_{\min}(\mathbf{A})||\mathbf{x}||^2\leq \mathbf{x}^H\mathbf{Ax}\leq\lambda_{\max}(\mathbf{A})||\mathbf{x}||^2$, where $\lambda_{\min}(\mathbf{A})$ and $\lambda_{\max}(\mathbf{A})$ are the min. and max. eigenvalues of matrix $\mathbf{A}$, respectively.
The proof can be seen in [@horn1994], and is in fact fundamental to the Rayleigh-Ritz theorem.
\[lemma3\] For any two Hermitian matrices, $\mathbf{A,B}$, both in $\mathbbm{C}^{N\times N}$, $$\begin{aligned}
\sum\limits_{i=1}^N \lambda_i(\mathbf{A}) \lambda_{N-i+1}(\mathbf{B})\leq \mathrm{Tr}\{ \mathbf{AB}\}\leq\sum\limits_{i=1}^N \lambda_i(\mathbf{A})\lambda_i(\mathbf{B})\end{aligned}$$ where $\lambda_i(\cdot)\geq \lambda_{i+1}(\cdot)$, $i=1,2,\ldots,N$.
See [@Lasserre1995 Lemma. II. I] for a proof.
Consider $g(\mathbf{w}_k,\mathbf{s}_k)$, we have $$\begin{aligned}
\label{eq34}
g(\mathbf{w}_k,\mathbf{s}_k)&=\mathbf{w}_k^H\mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_k \nonumber \\
&=(\mathbf{w}_k-\mathbf{w}_{k+1} +\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}+\mathbf{w}_{k+1} ) \nonumber\\
&=(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}) \\
&+ \mathbf{w}_{k+1} ^H \mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_{k+1}+\textrm{Re}\{ (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}( \mathbf{s}_k) \mathbf{w}_{k+1} \} \nonumber\end{aligned}$$ Moreover since the square root decomposition exists i.e., $\mathbf{R_u}( \cdot)=\mathbf{R}_{\bf u}^{1/2}( \cdot) \mathbf{R}_{\bf u}^{1/2}(\cdot)$, then application of the Cauchy-Schwartz inequality produces, $$\begin{aligned}
\label{eq35}
&\textrm{Re}\{ (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}( \mathbf{s}_k) \mathbf{w}_{k+1} \} \leq \\
&\sqrt{(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}) } \sqrt{\mathbf{w}_{k+1} ^H \mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_{k+1}} \nonumber\end{aligned}$$ Using in and since $\mathbf{R_u}(\cdot)$ is PSD, we can show that $
g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)\leq (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1})
$. Further using , we have the following upper and lower bounds $$\begin{aligned}
\label{eq36}
0&\leq g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \nonumber \\
&\leq(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1})\end{aligned}$$ We notice immediately, that at convergence $(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1})\rightarrow 0$ since $\mathbf{w}_k\rightarrow\mathbf{w}_{k+1}$. Other bounds as in can be readily derived. From Lem. \[lemma2\], we can show that $$\begin{aligned}
\label{eq37}
&\leq \lambda_{\min}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_k||^2-\lambda_{\max}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_{k+1}||^2\nonumber \\
&g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \\
&\leq \lambda_{\max}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_k||^2-\lambda_{\min}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_{k+1}||^2 \nonumber.\end{aligned}$$
Consider the following.
\[lemma4\] If $\mathbf{x}$, $\mathbf{y}$ are arbitrary but distinct complex vectors of size $N$ and let $\mathbf{A}:=\mathbf{xx}^H-\mathbf{yy}^H$, then, (a) matrix $\mathbf{A}$ has exactly two real non-zero eigenvalues, the rest $N-2$ eigenvalues are all zeros, (b) of the two real and non-zero eigenvalues one is always positive and the other is always negative, and (c) if the $\mathbf{x}$, $\mathbf{y}$ are not distinct, i.e. $\mathbf{y}=\beta \mathbf{x}$, $\beta\in\mathbbm{C}$, then there exists only one non-zero eigenvalue, $(|1-|\beta|^2|) ||\mathbf{x}||^2$and the rest $N-1$ eigenvalues are purely zeroes.
First of all we notice $\mathbf{A}$ is Hermitian and hence its eigenvalues are real. The proof for (a) is obvious given the fact that $\mathbf{A}$ is a sum of two distinct outer products. In other words, $\mathrm{rank}(\mathbf{A})=2$, for all $\mathbf{y}\neq\beta\mathbf{x}$ .
Now we know that $$\begin{aligned}
\mathrm{Tr}\{ \mathbf{A}\}&=\lambda_1+\lambda_2=\mathbf{x}^H\mathbf{x}-\mathbf{y}^H\mathbf{y} \\
\mathrm{Tr}\{\mathbf{AA}^H\}&=\lambda_1^2+\lambda_2^2=||\mathbf{x}||^4+||\mathbf{y}||^4-2|\mathbf{x}^H\mathbf{y}|^2\end{aligned}$$ where $\lambda_i,i=1,2$ are the two non zero eigenvalues of $\mathbf{A}$. The above set of equations can be reduced to a quadratic in any one eigenvalue. It can be shown that the only two possible solutions are then $$\label{eq38}
\begin{aligned} \lambda_1&=\frac{||\mathbf{x}||^2-||\mathbf{y}||^2}{2}
\left( 1+ \sqrt{ 1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} } \right) \\
\lambda_2&=\frac{||\mathbf{x}||^2-||\mathbf{y}||^2}{2}
\left( 1- \sqrt{ 1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} } \right) \end{aligned}$$ Since $\lambda_i,i=1,2$ are purely real we have, $1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} \geq0$ and from Cauchy Schwarz inequality, we also have that $|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2\leq0 $. Using these two facts, consider two specific cases, both of which are shown easily from elementary algebra, $$\label{eq39}
\begin{cases}
\lambda_1>0,\lambda_2 <0, & \mbox{ if } ||\mathbf{x}||^2-||\mathbf{y}||^2 \geq0 \\
\lambda_1<0,\lambda_2>0, & \mbox{ if } ||\mathbf{x}||^2-||\mathbf{y}||^2 <0
\end{cases}.$$ When $||\mathbf{x}||^2-||\mathbf{y}||^2=0$, it is easily seen that $\lambda_1=\sqrt{||\mathbf{x}||^2||\mathbf{y}||^2-|\mathbf{x}^H\mathbf{y}|^2 } >0$, $\lambda_2=-\lambda_1<0$. We also note immediately from that when, $\mathbf{y}=\beta \mathbf{x}$, $\lambda_1 =(1-|\beta|^2)||\mathbf{x}||^2$, $\lambda_2=0$. This completes the proof.
It is readily shown that $g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)=\mathrm{Tr}\{\mathbf{R_u}(\mathbf{s}_k) ( \mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \}$. Therefore, from Lem. \[lemma3\], and Lem. \[lemma4\], we have, $$\begin{aligned}
\label{eq40}
&\leq \lambda_{\max}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{-}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\\
&+\lambda_{\min}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{+}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber \nonumber \\
&g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \\
&\leq \lambda_{\max}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{+}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\\
&+\lambda_{\min}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{-}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\end{aligned}$$ It is not immediately evident from the analysis which set of bounds in , , are tight, hence combining them we have $$\begin{aligned}
&\max\left\{ \begin{aligned} g_{lb}^1( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \;\; &g_{lb}^2( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \\
&g_{lb}^3( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}) \end{aligned}\right\} \\
&\leq g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \leq \\
&\min\left\{ \begin{aligned} g_{ub}^1( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \;\; &g_{ub}^2( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \\
&g_{ub}^3( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}) \end{aligned}\right\} \\\end{aligned}$$ where $g_{lb}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $g_{ub}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $i=1,2,3$ are the lower and upper bounds as given in -, , for $i=1,2,3$, respectively.
Similar upper and lower bounds can be readily derived for the other corresponding terms, $g(\mathbf{w}_{k+1},\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_{k+1})$ using analysis presented thus far, and is not the focus now. Let us however denote these corresponding lower and upper bounds to be $h_{lb}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $ h_{ub}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $i=1,2,3$.
Constrained proximal alternating minimization
---------------------------------------------
The proximal version of the constrained alternating minimization is iterative, and for the filter design step, optimizes at the $k$-th iteration, $$\begin{aligned}
\label{eq41}
\min_{\mathbf{w} }\;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}( \mathbf{s}_{k-1})\mathbf{w}+\frac{\alpha_{k-1}}{2} || \mathbf{w}-\mathbf{w}_{k-1} ||^2\\
\mbox{ s. t } \;\;\;\;\;&\mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber
$$ where $\alpha_{k-1} \in \mathbbm{R}^{+}$ can be seen as a weight attached to the regularizer / penalizer $|| \mathbf{w}-\mathbf{w}_{k-1} ||^2$. This parameter can be interpreted as follows, if it is small, it encourages the optimizer to look for viable solutions in the vicinity of $\mathbf{w}_{k-1}$. However, if large, it penalizes the optimizer heavily for focusing even slightly in the immediate vicinity of $\mathbf{w}_{k-1}$.
In a similar spirit, the proximal version of the constrained alternating minimization for the waveform design step at the $k$-th iteration optimizes, $$\begin{aligned}
\label{eq42}
\min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H_{k}\mathbf{R_u}(\mathbf{s})\mathbf{w}_{k} +\frac{\beta_{k-1}}{2} ||\mathbf{s}-\mathbf{s}_{k-1} ||^2\nonumber \\
\mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H_{k}(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \\
\;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o \nonumber \nonumber\end{aligned}$$ where $\beta_{k-1}\in\mathbbm{R}^{+}$ is the weight attached to the regularizer $ ||\mathbf{s}-\mathbf{s}_{k-1} ||^2$. Bounds on $\alpha_{k-1},\beta_{k-1}$ relating it to the Lipschitz constants are deferred to forthcoming analysis. A graphical example comparing the constrained alternating minimization and the proximal constrained alternating minimization is shown in Fig. \[amcamfig\].
\[remark2\] The objective functions in , are still individually convex in $\mathbf{w}$, $\mathbf{s}$, respectively. The regularizer terms $|| \mathbf{w}-\mathbf{w}_{k-1} ||^2$ and $ ||\mathbf{s}-\mathbf{s}_{k-1} ||^2$ are strongly convex, and $\nabla^2_{\mathbf{w}} ( || \mathbf{w}-\mathbf{w}_{k-1} ||^2)=\mathbf{I}\succ \mathbf{0}$, $\nabla^2_{\mathbf{s}} ( || \mathbf{s}-\mathbf{s}_{k-1} ||^2)=\mathbf{I} \succ \mathbf{0}$, and therefore do not alter the individual convexity of $\mathbf{w}^H\mathbf{R_u}( \mathbf{s}_{k-1})\mathbf{w}$ and $\mathbf{w}^H_{k}\mathbf{R_u}(\mathbf{s})\mathbf{w}_{k}$, w.r.t. $\mathbf{w}$, $\mathbf{s}$, respectively.
The solutions to , can be cast in terms of the proximal operator as $$\begin{aligned}
\mathbf{w}_{k}=&\mathrm{prox}_{(\alpha_{k-1},\mathbf{w} )} \big( g(\mathbf{w},\mathbf{s}_{k-1}) ;\mathbf{w}_{k-1} \big) \label{eq43} \\
&\mbox{ s. t } \; \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\
\nonumber \\
\mathbf{s}_{k}=&\mathrm{prox}_{(\beta_{k-1},\mathbf{s} )} \big( g(\mathbf{w}_{k},\mathbf{s}) ;\mathbf{s}_{k-1} \big) \label{eq44} \\
&\mbox{s. t. } \; \mathbf{w}^H_{k}(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\
& \;\;\;\;\;\;\;\;\mathbf{s}^H \mathbf{s}\leq P_o \nonumber
\end{aligned}$$ where, for a general $f(\mathbf{x}):\mathbbm{C}^N\rightarrow \mathbbm{R}$, the proximal operator is defined as $$\begin{aligned}
\label{eq45}
\mathrm{prox}_{(\alpha,\mathbf{x})} \big(f(\mathbf{x});\mathbf{y} \big):= \operatorname*{arg\,min}\limits_{\mathbf{x} } \; \mathbf{f} (\mathbf{x}) +\frac{\alpha}{2} ||\mathbf{x}-\mathbf{y} ||^2.\end{aligned}$$ The proximal operator has a rich history in the literature, and well documented properties, see for example [@Parikh2013; @Rockafeller1973; @Rockafeller1976; @Bertsekas1994]. A useful and interesting fact of this operator is that iff $\mathbf{x}_{o}$ minimizes $f(\mathbf{x}$) then $\mathbf{x}_{o}=\mathrm{prox}_{(\alpha,\mathbf{x})} (f(\mathbf{x});\mathbf{x}_{o})$, a proof is seen in [@Parikh2013].
[**Trust region interpretation**]{}. The objective now is to relate the unconstrained proximal minimization as in to a well known technique in numerical optimization. A generalized trust region subproblem can be formulated for $\mathbf{f}(\mathbf{x}):\mathbbm{C}^N\rightarrow \mathbbm{R}$ [@More93] $$\begin{aligned}
\label{trustreg}
\min \limits_{\mathbf{x}} \;\;\; &f( \mathbf{x}) \nonumber \\
\mbox{s. t. } \;\;\; & || \mathbf{U}\mathbf{x}-\mathbf{v}||^2 \leq \delta\end{aligned}$$ where $\mathbf{U}$, $\mathbf{v}$ are a general nonsingular matrix, and a vector, both characterizing the trust region. The positive scalar $\delta$ may be interpreted as a parameter which specifies the extent of the trust region. For $\mathbf{U}=\mathbf{I}$ and $\mathbf{v}=\mathbf{y}$, the proximal minimization as in and the trust region problem in are equivalent for specific values of $\alpha$ and $\delta$. In particular every solution of is a solution to for a particular $\delta$. In the same spirit, every solution to is an unconstrained minimizer to $f(\cdot)$ or a solution to for a particular $\alpha$, see also [@Rockafeller1976; @Parikh2013].
The proximal optimizations problems, , can be cast as equivalent constrained trust region subproblems, where for the $k$-th iteration, the trust region is characterized by the previous iteration, $\mathbf{w}_{k-1}$, $\mathbf{s}_{k-1}$, respectively.
[**Closed form:**]{} A closed form solution to is readily derived, expressed as in $$\label{eq46}
\begin{aligned} \mathbf{w}_{k}&=\big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\frac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1} \big( \frac{\alpha_{k-1}}{2}\mathbf{w}_{k-1}-\frac{\gamma_4^{\ast}}{2} \big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big) \big) \\
\gamma_4&=\frac{\alpha_{k-1} \mathbf{w}_{k-1}^H \big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\dfrac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1} \big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)-2\kappa }{\big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)^H
\big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\dfrac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1}
\big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)} \end{aligned}$$
where $\gamma_4$ is the Lagrange parameter associated with . The solution to is also in closed form and the procedure to obtain it is similar to that used in deriving . Assuming that the Lagrange parameters for are $\gamma_5=\gamma_{5r}+j\gamma_{5i},\,\gamma_6\in\mathbbm{R}^{+}$, the solution is expressed in , $$\label{eq47}
\mathbf{s}_{k}=\big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \big(\frac{\beta_{k-1}}{2}\mathbf{s}_{k-1} -\frac{\gamma_5}{2}\mathbf{G}^H\mathbf{w}_{k}\big)$$ where, $$\begin{aligned}
\gamma_{5r}&=2\frac{\frac{\beta_{k-1}} {2}\mathrm{Re} \left\{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\}-\kappa} {\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} } \\
\gamma_{5i}&=\frac{\beta_{k-1} \mathrm{Im} \left\{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\} }{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} }.
\end{aligned}$$ The Lagrange parameter $\gamma_6$ is obtained by solving, the following $$\begin{aligned}
\label{eq48}
\gamma_6 r(\gamma_6)=0,\;\gamma_6\geq0\end{aligned}$$ obtained from the complementary slackness constraint on the Lagrange dual and where, $$\begin{aligned} r(\gamma_6)&=(P_o-\frac{\beta_{k-1}^2}{4} a_k) \bigg( \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} \bigg)^2 \\
-&2\big( b_i\frac{db_i}{d\gamma_6} +(b_r-\kappa) \frac{db_r}{d\gamma_6}\big) \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} \\
-&(b_{i}^2+(b_r-\kappa)^2) \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{G}^H\mathbf{w}_{k}.
\end{aligned}$$ Where we also define $$\begin{aligned}
a_{k}=&\mathbf{s}_{k-1}^H\big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1} \\
b_{r}&=\frac{\beta_{k-1}}{2}\mathrm{Re}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\} \\
b_{i}&=\frac{\beta_{k-1}}{2}\mathrm{Im}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\}
\end{aligned}$$
Further, since the derivative, $\mathrm{Re}\{ \cdot\},\mathrm{Im}\{ \cdot\}$ are all linear we also have $$\begin{aligned}
\dfrac{db_r}{d\gamma_6}&=-\frac{\beta_{k-1}}{2} \mathrm{Re}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{s}_{k-1}\right\} \\
\dfrac{db_i}{d\gamma_6}&=-\frac{\beta_{k-1}}{2} \mathrm{Im}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{s}_{k-1}\right\}.
\end{aligned}$$
\[propos5\] In general $r(\gamma_6)$ is not monotone and there exist one or more zero crossings excluding $\gamma_6=\infty$. However in our extensive numerical simulations, and assuming $P_o>>\kappa^2, \gamma_6=0$ solves .
It is readily seen that $\lim \limits_{\gamma_6\rightarrow 0} r(\gamma_6)=r(0)\neq 0,\lim \limits_{\gamma_6\rightarrow \infty} r(\gamma_6)=0,\lim \limits_{\gamma_6\rightarrow \infty} \gamma_6 r(\gamma_6)=0$. Nevertheless unlike Prop. \[propos3\], Rem. \[propos5\] is not straightforward to demonstrate analytically, however can be shown numerically. See Section IV for some demonstrative examples not specific to the radar problem.
The value of $\gamma_6=0$ is substituted in to obtain the final waveform solution $\mathbf{s}_{k}(\cdot)$.
\[remstrongdual\] ([*Strong duality*]{}) The primal problem, and its associated dual have zero duality gap. This is straightforward but tedious to show. However we provide the optimal values attained by the primal as well as the dual, given below, $$\begin{aligned}
\label{dualopt2}
\mathbf{w}_k^H(\mathbf{R_i+R_n })\mathbf{w}_k&+\mathbf{s}_k^{\ast H} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}\big)^{-1} \mathbf{s}_k^{\ast} \nonumber \\
&+\frac{\beta_{k-1} || \mathbf{s}_{k-1}||^2}{2}
-\beta_{k-1}\mbox{Re}\{ \mathbf{s}_{k}^{\ast H} \mathbf{s}_{k-1}\}\end{aligned}$$ where using , Prop. \[propos5\], $$\begin{aligned}
\mathbf{s}_{k}^{\ast}&=(\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} ( \frac{\beta_{k-1}}{2}\mathbf{s}_{k-1} -\frac{\gamma_5}{2}\mathbf{G}^H\mathbf{w}_{k}) \\
\gamma_5&=\frac{\beta_{k-1}\mathbf{w}_{k}^H\mathbf{G} ( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} \mathbf{s}_{k-1}-2\kappa} {\mathbf{w}_{k}^H\mathbf{G} ( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} \mathbf{G}^H\mathbf{w}_{k} }.
\end{aligned}$$
This is not surprising since it is similar to Rem. \[dualprop\]. However, in this case the condition on the existence of the matrix is irrelevant, since the inverse in always exists. Hence Slater’s condition now is a simple constraint qualifier (the power constraint) which must be satisfied as in Rem. \[dualprop\].
[**Interpretation with specific ranges of $\alpha_{k-1},\,\beta_{k-1}$ and related to the Lipschitz constants**]{}. Some definitions and lemmas are useful for future discussions and are expressed below
\[mydef2\] ([*Lipschitz continuous gradient*]{}) A function $f(\mathbf{\bar{x}}) :\mathbbm{R}^N\rightarrow\mathbbm{R}$ has a Lipschitz constant (and trivially real positive), $\mathtt{L}$, when $||\nabla_{\mathbf{\bar{x}}}f(\mathbf{\bar{x}})-\nabla_{\mathbf{\bar{y}}}f(\mathbf{\bar{y}})|| \leq\mathtt{L} || \mathbf{\bar{x}-\bar{y}}||$, and $\forall \mathbf{\bar{x}}$, $\mathbf{\bar{y}}\in \mathbbm{R}^N$.
[**Note:**]{} ([*upper bound on Hessian*]{} ) If $f(\mathbf{\bar{x}})$ has a Lipschitz continuous gradient, with constant $\mathtt{L}$, then using Taylor’s theorem, it can be proved that $\nabla_{\mathbf{\bar{x}}}^2 f(\mathbf{\bar{x}})\preceq \mathtt{L}\mathbf{I}$.
\[remark3\] The Lipschitz constant for $f(\mathbf{\bar{x}})=\mathbf{\bar{x}}^T\mathbf{\bar{B}}\mathbf{\bar{x}}$ is the maximum eigenvalue of $\mathbf{\bar{B}}$, i.e. $\lambda_{\max}(\mathbf{\bar{B}})$, where $\mathbf{\bar{B}}\in\mathbbm{R}^{N\times N}, \; \mathbf{\bar{x}}\in\mathbbm{R}^N$.
This is readily seen since $\nabla_{\mathbf{\bar{x}}}\mathbf{\bar{x}}^T\mathbf{\bar{B}\bar{x}}=\mathbf{\bar{B}\bar{x}}$. Further since the induced (by an arbitrary $\mathbf{\bar{z}} \in \mathbbm{R}^N$) spectral norm (notation: $||| \cdot |||$) is defined as $$\begin{aligned}
|||\mathbf{\bar{B}}|||:=\sup\limits_{\mathbf{\bar{z}}} \{ \frac{||\bf \bar{B}\bar{z}||}{||\bf \bar{z}||} : \mathbf{\bar{z}}\in\mathbbm{R}^N,\mathbf{\bar{z}}\neq\mathbf{0} \},||{\bf \bar{B}\bar{z}}||=\sqrt{ \mathbf{\bar{z}}^T\mathbf{\bar{B}}^T\mathbf{\bar{B}\bar{z}}}\end{aligned}$$ but we know from Lem. \[lemma2\] that $\mathbf{\bar{z}}^T\mathbf{\bar{B}\bar{z}}\leq \lambda_{\max}( \mathbf{\bar{B}})|| \bf \bar{z}||^2$ and that eigenvalues of $\mathbf{\bar{B}}$ and $\mathbf{\bar{B}}^T$ are identical. This further implies that $\mathbf{\bar{z}}^T\mathbf{\bar{B}}^T\mathbf{\bar{B}\bar{z}}\leq \lambda_{\max}^2( \mathbf{\bar{B}})|| \bf \bar{z}||^2$. Therefore from Definition \[mydef2\], it is readily seen that the Lipschitz constant is the maximum eigenvalue of $\mathbf{\bar{B}}$.
\[lemma5\] ([*Descent lemma*]{}) If $f(\mathbf{\bar{x}}) :\mathbbm{R}^N\rightarrow\mathbbm{R}$ is continuously differentiable and has a Lipschitz continuous gradient described by constant $\mathtt{L}$, then $f(\mathbf{\bar{x}})\leq f(\mathbf{\bar{y}})+\nabla_{\mathbf{\bar{y}}}f(\mathbf{\bar{y}})^T(\mathbf{\bar{x}}-\mathbf{\bar{y}}) +\frac{\mathtt{L}}{2}||\mathbf{\bar{x}}-\mathbf{\bar{y}}||^2$.
See [@Bertsekas1999 Prop. A.24] and also [@Beck2013 Lem2.2] relevant in general for the BCD.
Consider an arbitrary $g(\mathbf{x}):= \mathbf{x}^H\mathbf{Bx}$, and $\mathbf{B}=\mathbf{B}^H$, $\mathbf{x}\in\mathbbm{C}^N$. Since $g(\mathbf{x}):\mathbbm{C}^N\rightarrow\mathbbm{R}$, a real equivalent of $g(\mathbf{x})$ could be defined as $\bar{g}(\bar{\mathbf{x}}):=\bar{\mathbf{x}}^T\bar{\mathbf{B} }\bar{\mathbf{x}}$ where $$\bar{\mathbf{B}}:= \left[ \begin{matrix} \mbox{Re}\{ \mathbf{B} \} & -\mbox{Im} \{\mathbf{B} \} \\
\mbox{Im} \{\mathbf{B} \} & \mbox{Re}\{ \mathbf{B} \} \end{matrix} \right] \in\mathbbm{R}^{2N\times 2N}, \;\; \bar{\mathbf{x}}=[\mbox{Re} \{ \mathbf{x} \}^T \mbox{Im} \{ \mathbf{x} \}^T ]^T \in \mathbbm{R}^{2N}.$$
\[lemma6\] The matrix $\bar{\mathbf{B}} :=\left[ \begin{smallmatrix} \mathrm{Re}\{ \mathbf{B} \} & -\mathrm{Im} \{\mathbf{B} \} \\
\mathrm{Im} \{\mathbf{B} \} & \mathrm{Re}\{ \mathbf{B} \} \end{smallmatrix} \right]\in\mathbbm{R}^{2N\times 2N}$ and $ \left[ \begin{smallmatrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{smallmatrix} \right] \in\mathbbm{C}^{2N\times 2N} $ have identical eigenvalues, $\tilde{\lambda}_i,i=1,2,\ldots,2N$. Moreover, if $\mathbf{B}$ is Hermitian, then $\tilde{\lambda}_i \in \mathbbm{R}^{+},i=1,2,\ldots,2N$ are equal to twice the multiplicity of the eigenvalues of $\mathbf{B}\in\mathbbm{C}^{N\times N}$.
Owing to the complex to real-real isomorphism, it can be shown after algebraic manipulations that $$\begin{aligned}
\label{eq49}
\left[ \begin{matrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{matrix} \right] =\mathbf{P}^{H}\bar{\mathbf{B}}\mathbf{P}, \;\; \mathbf{P}=\frac{1}{\sqrt{2}}\left[ \begin{matrix} j\mathbf{I} & \mathbf{I} \\ \mathbf{I} & j\mathbf{I} \end{matrix} \right], \;\; \mathbf{P}^H=\mathbf{P}^{-1}.\end{aligned}$$ That is indicates that $\bar{\mathbf{B}}$ and $\left[ \begin{smallmatrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{smallmatrix} \right]$ are unitary equivalent. Therefore they share the same eigenvalues. Furthermore if $\mathbf{B}$ is Hermitian its eigenvalues are purely real, and hence trivially, the eigenvalues of $\mathbf{B}$, $ \mathbf{B}^{\ast}$ are identical, and their eigenvectors are complex conjugates of one another. Hence the block diagonal matrix has identical eigenvalues as $\mathbf{B}$ but with multiplicity two.
Consider the objective in , . Define $\bar{g}( \mathbf{\bar{w}},\mathbf{\bar{s}}_{k-1}),\bar{g}(\mathbf{\bar{w}}_{k-1},\mathbf{\bar{s}}_{k-1})$ as the real equivalents of $g( \mathbf{w},\mathbf{s}_{k-1}),g (\mathbf{w}_{k-1},\mathbf{s}_{k-1})$, respectively for the filter design objective as in . In addition, denote $\mathtt{L}_{1k-1}$ as the Lipschitz constant associated with $\bar{g}(\bar{\mathbf{w}}_{k-1},\mathbf{\bar{s}}_{k-1})$. Similarly using the same notation and for the objective in the waveform design objective as in consider the real equivalents, $\bar{g}( \mathbf{\bar{s}},\mathbf{\bar{w}}_{k}),\bar{g}(\mathbf{\bar{s}}_{k-1},\mathbf{\bar{w}}_{k})$ and the Lipschitz constant denoted as $\mathtt{L}_{2k-1}$. Then the following inequalities can now be shown. $$\label{eq50}
\begin{aligned} &\bar{g}( \mathbf{\bar{w}})+\frac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}} ||^2
\geq \bar{g}(\mathbf{\bar{w}}_{k-1}) \\
+&\nabla \bar{g}(\mathbf{\bar{w}}_{k-1})^T(\mathbf{\bar{w}}-\mathbf{\bar{w}}_{k-1})
+\frac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}}||^2 \geq \bar{g}(\mathbf{\bar{w}}) \end{aligned}$$ $$\label{eq51}
\begin{aligned} &\bar{g}( \mathbf{\bar{s}})+\frac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}} ||^2
\geq \bar{g}(\mathbf{\bar{s}}_{k-1}) \\
+&\nabla \bar{g}(\mathbf{\bar{s}}_{k-1})^T(\mathbf{\bar{s}}-\mathbf{\bar{s}}_{k-1})
+\frac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}}||^2 \geq \bar{g}(\mathbf{\bar{s}}) \end{aligned}$$ where in , the known’s $\mathbf{\bar{s}}_{k-1}$ and in , the known’s $\mathbf{\bar{w}}_{k}$ are respectively treated as constants, therefore suppressed in notation for brevity. We further note that , are tight, i.e. for $\mathbf{\bar{w}}_{k}=\mathbf{\bar{w}}_{k-1}$, $\mathbf{\bar{s}}_{k}=\mathbf{\bar{s}}_{k-1}$ the inequalities are strict equality’s. The Lipschitz constants, $\mathtt{L}_{1k-1}$, $\mathtt{L}_{2k-1}$ are readily derived using Lem. \[lemma6\].
\[remark4\] It is readily seen that if $\alpha_{k-1}\geq \mathtt{L}_{1k-1}$ and $\beta_{2k-1}\geq \mathtt{L}_{2k-1}$ the inequalities in , are valid by replacing $\mathtt{L}_{1k-1},\mathtt{L}_{2k-1}$ with $\alpha_{k-1},\beta_{k-1}$, respectively.
The term in the first inequalities of , are the proximal minimization objectives with $\alpha_{k-1}=\mathtt{L}_{1k-1},\beta_{k-1}=\mathtt{L}_{2k-1}$. The inequalities of , are obtained from first applying the convexity Def. \[mydef1\](b) (first order definition) and then subsequently adding the respective terms $\tfrac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}} ||^2$, $ \tfrac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}}||^2$ and then using Lem. \[lemma5\], the descent lemma.
Additionally, it is recalled that the functions associated with the second inequalities of , are the (unconstrained) objectives which are minimized by the gradient descent with step size $\mathtt{L}_{1k-1}$, $\mathtt{L}_{2k-1}$, respectively. That is, the new iterations are then $\mathbf{\bar{w}}_{k}=\mathbf{\bar{w}}_{k-1}-\frac{1}{\mathtt{L}_{1k-1}} \nabla_{\mathbf{\bar{w}}}\bar{g}( \mathbf{\bar{w}})$, and $\mathbf{\bar{s}}_{k}=\mathbf{\bar{s}}_{k-1}-\frac{1}{\mathtt{L}_{2k-1}} \nabla_{\mathbf{\bar{s}}}\bar{g}( \mathbf{\bar{s}})$. Therefore from , and Rem. \[remark4\] [*we note that the proximal objective, the gradient descent objective are all surrogate albeit tight upper bounds on the true objective $\forall \alpha_{k-1}\geq\mathtt{L}_{1k-1}$ and $\forall \beta_{k-1}\geq\mathtt{L}_{2k-1}$*]{}. This interpretation is graphically depicted in Fig. \[fig3\] for the filter design objective as in but for $\alpha_{k-1}=\mathtt{L}_{1k-1}$. A similar graphic interpretation is obvious for the waveform design stage and is therefore not shown.
[**Tikhonov interpretation**]{} This interpretation is immediate from , . In fact from , , the quadratic regularizers $||\mathbf{w}-\mathbf{w}_{k-1} ||^2, || \mathbf{s}-\mathbf{s}_{k-1} ||^2$ are essentially Tikhonov regularization terms. Geometrically they are spheres centered at $\mathbf{w}_{k-1}$, $\mathbf{s}_{k-1}$ and encourage the current iterates to be in the vicinity of the previous iterates. Furthermore, since in the limit, the regularizer terms only decrease, this may be also seen as a vanishing Tikhonov regularization problem [@Parikh2013] for each iteration in both the waveform and the filter vectors.
[**Proximal minimization: A training data starved STAP solution**]{} The regularization in , leads to [*diagonally loaded*]{} solutions , when compared to the constrained alternating minimization solutions as in and . In particular, the diagonal loading serves two important purposes, [*firstly it offers a numerically stable solution by conditioning . Secondly and more importantly, it permits a weight vector solution when $\mathrm{rank}( \mathbf{R_u}(\mathbf{s}))\leq NML$*]{}.
Practical STAP contends with rank deficient correlation matrices due to lack of sufficient training data from neighboring range cells due to outlier contamination or heterogeneity in the data. The solution in ameliorates over the training data starved STAP scenarios.
So far, we have considered the algorithms for waveform design without enforcing constraints such as const. modulus or sidelobe constraints. The minimum eigenvector solution belongs to this class of unconstrained waveform design. We will revisit this design by considering and Lem. \[lemma2\].
\[remark5\] The min. eigenvector solution in is still optimal in the presence of clutter, provided $\mathbf{R_i}+\mathbf{R_n}$ and $\mathbf{R_c}(\mathbf{s})$ share the same eigenvector corresponding to their min.eigenvalues, but with $\lambda_{\min}(\mathbf{R_c}(\mathbf{s}))=0$, always.
This is readily seen since the optimization in , ignoring the constraint for now could be recast as $\max \limits_\mathbf{s} (\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))$. Now using Woodbury’s identity [@Kayest1998], we have $$\label{eqeigmin}
\begin{aligned}
&(\mathbf{R_i}+\mathbf{R_n}+\mathbf{R_u}(\mathbf{s}))^{-1} =(\mathbf{R_i}+\mathbf{R_n})^{-1} \\
-&(\mathbf{R_i}+\mathbf{R_n})^{-1}\mathbf{R_c}(\mathbf{s} ) \bigl( \mathbf{I}+ (\mathbf{R_i}+\mathbf{R_n})^{-1} \mathbf{R_c}(\mathbf{s})\bigr) ^{-1}(\mathbf{R_i}+\mathbf{R_n})^{-1}.
\end{aligned}$$ Further using the eigenvector relations, $(\mathbf{R_i}+\mathbf{R_n})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}(\mathbf{R_i}+\mathbf{R_n})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))$ and $\mathbf{R_c} (\mathbf{s}) (\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}(\mathbf{R_c}( \mathbf{s}))(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\mathbf{0}$ in , it is readily seen that $(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H(\mathbf{R_i}+\mathbf{R_n}+\mathbf{R_u}(\mathbf{s}))^{-1} )(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}^{-1}( \mathbf{R_i}+\mathbf{R_n})$.
The simplest example where Rem. \[remark5\] is satisfied is when the noise correlation matrix is scaled identity (may not be practical for narrowband radar), clutter correlation matrix is low rank. In STAP and for ideal scenarios, insights to the clutter rank are obtained by the Brennan’s rule [@guerci2003; @klemm2002; @ward1994]. A high clutter rank prevails due to the practical effects such as, the intrinsic clutter motion,velocity misalignment and crabbing, mutual coupling and antennae element mismatches as well as clutter ambiguities in Doppler resulting in aliasing [@ward1994].
\[tbp!\] ![Upper bounds on the objective for the proximal algorithm w.r.t. the filter design. A similar graphical interpretation for the waveform design but with $\mathtt{L}_{2k-1}$ is also easy depicted but not shown here.[]{data-label="fig3"}](Lipschitz_upper "fig:")
Constant modulus alternating minimization
-----------------------------------------
So far, the optimization problems had no specific constraints (except the power/energy constraint) on the waveform, constant modulus is a desirable property to have in a waveform [@Setlurradar2014]. The optimum weight vector is unchanged by introducing the const. modulus constraint, and is identical to for the constrained alternating minimization [^7].
Since the optimization w.r.t. weight vector is unchanged, we only treat the optimization for $\mathbf{s}$ but with the const. mod. constraint for a fixed but arbitrary $\mathbf{w}$, formulated below $$\begin{aligned}
\min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\
\mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq52} \\
\;\;\;\;\;\; & |s_i|=\rho,i=1,2,\ldots,N. \nonumber \nonumber\end{aligned}$$ where $s_i$ is the $i$-th component in $\mathbf{s}$. Unlike say , notice that in , constraining the power of the waveform is unnecessary since $\rho$ is fixed but could be chosen arbitrarily to scale up / down the waveforms energy to satisfy hardware limitations. Therefore, the last $N$ constraints in implicitly impose the power requirements, but more importantly also impose the constant modulus constraint.
The Lagrangian of is expressed as $$\begin{aligned}
\label{lagraconmod}
\mathcal{L}(\mathbf{s}, \gamma_7,\boldsymbol{\gamma}_5)&=\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} +\mbox{Re}\{ \gamma_7^{\ast} (\mathbf{w}^H\mathbf{Qs}-\kappa)\} \nonumber \\
&+\mathbf{s}^H\mathbf{D}_{\gamma}\mathbf{s} -\rho\mathbf{1}^T\boldsymbol{\gamma}_8\end{aligned}$$ where the Lagrange parameter, $\gamma_7\in\mathbbm{C}$, and the Lagrange parameter vector $\boldsymbol{\gamma}_8=[\gamma_{8_1},\gamma_{8_2},\ldots,\gamma_{8_N}]^T\in\mathbbm{R}^{N}$ are for the Capon constraint and the $N$ const. mod. constraints, respectively. Furthermore in , define $\mathbf{D}_{\gamma}= \left[ \begin{smallmatrix} \gamma_{8_1}& & \\
& \ddots & \\ & &\gamma_{8_N} \end{smallmatrix} \right]$, i.e. a diagonal matrix. The KKT conditions are expressed as
\[kkt1\] $$\begin{aligned}
\mathbf{s}_o(\mathbf{w})&=\frac{\kappa \bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\mathbf{D}_{\gamma}\bigr)^{-1}\mathbf{G}^H\mathbf{w} }{\mathbf{w}^H\mathbf{G} \bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w}) +\mathbf{D}_{\gamma}\bigr)^{-1}\mathbf{G}^H\mathbf{w}} \\
|s_{oi}(\mathbf{w})|&=\rho,i=1,2,\ldots,N.\end{aligned}$$
The waveform which simultaneously satisfies (a)(b) is the solution. Moreover, note that (a)(b) are $2N$ non-linear equations with $2N$ unknowns. The first $N$ unknowns are $s_{oi}(\mathbf{w}),i=1,2\ldots,N$ and the next $N$ unknowns are the Lagrange parameters $\gamma_{8_i}$. Unfortunately, is not in closed form but can be solved numerically for the $N$ parameters, $\gamma_{8_i},i=1,2,\ldots,N$ via a numerical root finder. Nonetheless we note that $\gamma_{8_i}\in (-\infty,\infty)$ and a reasonable initialization point is not forthcoming for the numerical root finding.
[**Eliminating the constant modulus constraints**]{} Instead of solving the $2N$ non-linear equations as in (a)(b), we take an alternative approach. One may reformulate the optimization by eliminating the last $N$ constraints, by imposing a structure on $\mathbf{s}$, namely, $s_i=\rho\exp(j\alpha_i)$. Other structures exists but from our experience, complex exponentials are the easiest to manipulate. The new optimization problem is now w.r.t. $\boldsymbol{\alpha}=[\alpha_1,\alpha_2,\ldots,\alpha_{N}]^T\in\mathbbm{R}^N$, expressed as $$\begin{aligned}
\min\limits_{\boldsymbol{\alpha}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\
\mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq53}\end{aligned}$$ where in, $\mathbf{s}=\rho[\exp(j\alpha_1),\exp(j\alpha_2),\ldots,\exp(j\alpha_{N})]^T$ and $\alpha_i \in [0,2\pi),i=1,2,\ldots, N$. The Lagrangian corresponding to is $$\begin{aligned}
\label{eq54}
\mathcal{L}(\boldsymbol{\alpha},\gamma_9)=\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} +\mbox{Re}\{ \gamma_9^{\ast}(\mathbf{w}^H\mathbf{Gs}-\kappa)\}.\end{aligned}$$ The KKT’s are expressed as, $\tfrac{\partial\mathcal{L}(\boldsymbol{\alpha},\gamma_9)}{\partial \boldsymbol{\alpha}}=\mathbf{0}$ and $\mathbf{w}^H\mathbf{Gs}=\kappa$. Noting that $\boldsymbol{\alpha}$ is purely real, we have $$\label{eq56}
\begin{aligned}
\frac{\partial\mathcal{L}(\boldsymbol{\alpha},\gamma_9)}{\partial \boldsymbol{\alpha}}=-&j\sum\limits_{q=1}^Q\mathbf{Z_q}\mathbf{s}\odot \mathbf{s}^{\ast}+j\sum\limits_{q=1}^Q\mathbf{Z}_{\bf q}^{\ast}\mathbf{s}^{\ast}\odot \mathbf{s} \\
+&\mbox{Im}\{ \gamma_9^{\ast}(\mathbf{w}^H\mathbf{G} )^T\odot \mathbf{s}\} =\mathbf{0}.
\end{aligned}$$ The above equation can be simplified as, $\mbox{Im}\{ \sum\limits_{q=1}^Q\mathbf{Z}_{\bf q}^{\ast}\mathbf{s}^{\ast}\odot \mathbf{s} -\frac{\gamma_9^{\ast}}{2}(\mathbf{w}^H\mathbf{G})^T\odot \mathbf{s}\}$. Using this in , and taking the complex conjugate, while absorbing the negative sign into the constant $\gamma_9$[^8], we have the KKTs in final form expressed as
\[eq57\] $$\begin{aligned}
\mbox{Im}\left\{ \left(\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}) \mathbf{s}_o+\frac{\gamma_9}{2} \mathbf{G}^H \mathbf{w} \right)\odot \mathbf{s}_o^* \right\}&=\mathbf{0} \\
\mathbf{w}^H\mathbf{G}\mathbf{s}_o&=\kappa\end{aligned}$$
where $\mathbf{0}$ is a column vector of all zeros and of dimension $N$. The optimal solution, $\mathbf{s}_o$, is a function of the optimal $\boldsymbol{\alpha}_o$. This relationship although evident from is not explicitly stressed in for notational succinctness. Define $\mathbf{Z}_{\bf Q}:=\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w})$ and let $z_{ij},i=1,2,\ldots,N,j=1,2\ldots,N$ be the $ij$-th element of $\mathbf{Z}_{\bf Q}$. Noting that $\mathbf{Z}_{\bf Q}$ is Hermitian, we also have $\mbox{Im}\{z_{ii}\}=0,\,\forall i$, $z_{ji}=z_{ij}^{\ast}$.
\[propos6\] The Lagrange parameter $\gamma_9=0$ solves .
For any $z\in\mathbbm{C}$, and any $\theta\in [0,2\pi]$, we have $\mbox{Im}\{ z\exp(j\theta)\}=\mbox{Re}\{z\}\sin(\theta)+\mbox{Im}\{z\}\cos(\theta)$. Using this and the fact that $\mathbf{Z}_{\bf Q}=\mathbf{Z}_{\bf Q}^H$, the $i$-th equation in (a) can be simplified as $$\label{eq58}
\begin{aligned}
&2\rho
\big( \sum\limits_{j=1, j\neq i}^{N} \mbox{Re}\{ z_{ij}\}\sin(\alpha_{j}^o-\alpha_{i}^o )+\mbox{Im}\{z_{ij} \} \cos( \alpha_{j}^o-\alpha_{i}^o )\big)\\
&=\mbox{Im}\{ \gamma_9u_i\exp(-j\alpha_{i}^o)\}, i=1,2,\ldots,N
\end{aligned}$$ where $u_i$ is the $i$-th element of $\mathbf{u}=\mathbf{G}^H\mathbf{w}$ Adding the $N$ equations in , it easily seen that $\sum \limits_{i=1}^N\mbox{Im}\{ \gamma_9u_i\exp(-j\alpha_i^o\}=0$ but we know from (b) that $\rho\sum \limits_{i=1}^N u_i\exp(-j\alpha_i^o)=\kappa$, where $\kappa\in\mathbbm{R}$. Therefore this implies that $\mbox{Im}\{\gamma_9\}=0$ or in other words, $\gamma_9 $ is purely real. Substituting this back into (a) and following the same arguments as before, this is possible if trivially $\rho=0$ or $\gamma_9=0$, the former is false since $\rho=0$ does not solve (b), therefore the latter must be true.
[**Interpretation of $\gamma_9=0$**]{}. With $\gamma_9=0$, from (a) we have that $$\label{finalkkts_cm}
\begin{aligned}
\mbox{Im}\left\{ \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}) \mathbf{s}_o \right\}&=\mathbf{0} \\
\mathbf{w}^H\mathbf{G}\mathbf{s}_o&=\kappa.
\end{aligned}$$ The first equation in does not depend on $\rho$, but the second does. Therefore $\gamma_9=0$ does not imply that the constraint in is inactive. Rather, this implies that the KKTs enforce the Capon constraint in for the constant modulus waveform by varying the [*unspecified*]{} modulus parameter $\rho$.
The result in Prop. \[propos6\] has some very interesting consequences. Using $\gamma_9=0$, the $N$ equations in and therefore (a), can be rewritten as a some linear matrix equation $\bar{\mathbf{Z}}_{\bf Q}\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}}=\mathbf{0}$, where $\bar{\mathbf{Z}}_{\bf Q}\in\mathbbm{R}^{N\times\binom{N}{2}}$ and the vector $\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}}=[\sin(\alpha_{2}^o-\alpha_{1}^o) ,\sin(\alpha_{3}^o-\alpha_{1}^o) \ldots, \sin(\alpha_{N}^o-\alpha_{N-1}^o), \cos(\alpha_{2}^o-\alpha_{1}^o),\ldots,\cos(\alpha_{N}^o-\alpha_{N-1}^o) ]^T$ i.e. has $\binom{N}{2}$ components consisting of sines and cosines of all possible differences of $\alpha_i^o-\alpha_j^o,\forall i, \forall j\neq i$. In other words, $\mathbf{p}_{\boldsymbol{\alpha}^{\bf o}}\in \mbox{null}\big( \bar{\mathbf{Z}}_{\bf Q}\big)$. The rank of $\bar{\mathbf{Z}}_{\bf Q}$ is not easy to calculate here but its maximum value is $N$. Therefore from the rank-nullity theorem, $\dim(\mbox{null}(\bar{\mathbf{Z}}_{\bf Q}))\geq N(N-2)$. Clearly there could exist multiple vectors which are in this null space but we are not certain if this translates to multiple solutions of $\boldsymbol{\alpha}_o$ from this linear equation alone. Nonetheless, if multiple solutions exist to this linear equation, they must also satisfy (b) to be considered as a solution to . In any case the optimal solution(s) are in, $\mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }\subset\mathbbm{R}^{N}$, with $$\begin{aligned}
\label{eq59}
\mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }=\{\boldsymbol{\alpha}^o:\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}} \in \mbox{null}( \bar{\mathbf{Z}}_{\bf Q}),\sum\limits_{i=1}^N u_i^{\ast}\exp(j \alpha_i^o)=\frac{\kappa}{\rho} \}. \end{aligned}$$ It remains to be seen if $\mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }$ is singleton, or comprises many elements, but we are optimistic that it would not turn out to be empty.
Practical Considerations: Classical STAP v.s Waveform adaptive STAP
-------------------------------------------------------------------
Here we addresses practical considerations on the fast time-slow time model in STAP which aids in the waveform design and compare this with the classical model in STAP (slow time).
[**Hardware**]{} The fast-time slow-time model in STAP does not necessitate newer hardware nor does it require any modifications to the existing hardware. It does however assume that the current state-of-art permits arbitrary waveform generation and adaptive transmitting capabilities [@cochran2009waveform].
[**Computational complexity**]{} The inclusion of the waveform causes the correlation matrices to have larger dimension. Inverting large matrices are computationally prohibitive. Classical STAP requires inverting a complex $ML\times ML$ matrix which has a complexity of $O((ML)^{2.373})$-$O((ML)^{3})$ [@VirginiaWilliams2012]. Waveform adaptive STAP requires inverting complex $NML\times NML $ complex matrices which has a computational complexity of $O((NML)^{2.373})$-$O((NML)^{3})$ [@VirginiaWilliams2012].
[**Training data**]{} Due to the larger dimensions of the correlation matrices by inclusion of the waveform, it suddenly appears, albeit deceivingly, that more training data (from more neighboring range cells) are needed to estimate the correlation matrices. This is not true since inclusion of waveform simply includes the fast time samples. Hence the fast-time slow-time model uses the raw data prior to pulse compression or matched filtering, hence the training data requirements is identical to that required in the classical STAP case. Note that we are not interested in resolving targets within the pulse duration but rather outside it.
\
\
\
![Constrained alternating minimization: objective costs vs. iterations for 3 random, independent waveform initializations (inset: for 25 random initializations).[]{data-label="fig4"}](AMmono)
![Convergence of non const. mod initial waveform to a con. mod Con. waveform:[]{data-label="fig6"}](FJ_CM_noCMinit)
Simulations {#sec:simulations}
===========
First we will addresses simulations not specific to radar.
Simulations supporting: Prop. \[propos3\] and Rem. \[propos5\]
--------------------------------------------------------------
We ran simulations with random $z_n$ and random $d_n$ to analyze $f(\gamma_2)$ and $\gamma_2f(\gamma_2)$ numerically. In our extensive simulations we chose $z_n$ from complex normal distributions with different means and different variances. Since $d_n>0$ for all $n$, we used uniform distributions with different supports on the positive real axis excluding zero. We show only two representative simulation results for the monotonically increasing and decreasing cases in Fig. \[lagrasupport1\](a)(c), respectively. The corresponding function $\gamma_2f(\gamma_2)$ are also shown in Fig. \[lagrasupport1\](b)(d) for the two cases.
Simulations for supporting Rem. \[propos5\] is presented next. Some parameters specifying the function $r(\gamma_6)$ were simulated randomly with the identical distributions used as in generating Fig. \[lagrasupport2\]. The parameter $\kappa=2,P_o=10$ was used in generating Fig. \[lagrasupport2\](a), the function $\gamma_6r(\gamma_6)$ is also shown in Fig. \[lagrasupport2\](b). As such, it is noted that $P_o=10$ is a a contrived example, typical radar applications will require $P_o$ to be in several hundred KW or several Hundred MW.
The zero crossing is the intersection of the dashed line (black) with the blue curve in Fig. \[lagrasupport2\](a). Now using $P_o=20$ and keeping the other parameters fixed we obtain Fig. \[lagrasupport2\](c) which shows that $r(\gamma_6)$ is monotonic decreasing whose limit at $\infty$ is 0.
[**Radar Specific simulations:**]{} Here onward, some parameters are common to all the simulation examples and are stated now. The simulation parameters are in SI units unless mentioned otherwise. To reduce computation complexity while inverting large matrices and computing their eigen-decompositions, we considered the number of, sensors, waveform transmissions, and fast time samples in the waveform as $M=5,L=32, N=5$, respectively. The carrier frequency was chosen to be 1GHz, and the radar bandwidth was 50MHz. The element spacing $d=\lambda_o/2$.
Constrained alternating minimization
------------------------------------
The noise correlation matrix was assumed to have a correlation function given by $\exp(-|0.005n|),\; n=0,1,\ldots,NML$. Two interference sources were considered at $(\theta=0.3941,\phi=0.3)$ and at $(-0.4941,0.3)$. Both these interference sources had identical discrete correlation functions given by $0.2^{|n|},\; n=\pm 0, \pm 1,\ldots$. To simulate clutter we considered two clutter patches, consisting of five scatters each. The clutter correlation functions corresponding to the two patches were $\exp(-0.2|p|) \mbox{ and } \exp(-0.1|p|), \;p=\pm 0,\pm 1,\ldots,\pm P$. The rest of the parameters are identical to those used in [@Setlurradar2013].
In Fig. \[fig4\], the STAP beamformer objective vs. iterations are shown for 3 independent, random waveform initializations but the [*inset*]{} shows 25 independent initializations or trials. The alternating minimization was initialized with waveforms whose fast time samples are chosen independently from a standard complex Gaussian distribution. The algorithm was terminated as soon as the current waveform iterate invalidated the set power constraint. From the figure and its inset it is clear that the STAP beamformer output is non-increasing thereby validating the monotonicity property of this algorithm. More importantly from Fig. \[fig4\], we see that the final objective value and the iterations to reach it for each trial are different from one another, attributed to the joint non-convexity of the objective w.r.t. $\mathbf{w}$ and $\mathbf{s}$. Sensitivity to the random initialization is therefore duly noted.
Constrained proximal alternating minimization
---------------------------------------------
All the simulations parameters are identical to the previous case. The constrained alternating minimization was initialized with random waveforms as in Fig. \[fig4\], immediately followed by its proximal counterpart. The termination of the former algorithm was identical to the previous case, then, the latter was run for 200 iterations. Three representative trials are shown in Fig. \[fig5\](a)(b), for the constrained alternating minimization and its proximal counterpart. In Fig. \[fig5\](b), the dashed black lines are the final objective values obtained from the min. eigenvector waveform having the same energy as its proximal counterpart. For the three trials and not surprisingly, the proximal objective value, for all practical purposes, is identical to that obtained from the waveform derived from as evidenced from the [*inset*]{}. Therefore validating the implementation of both the constrained as well as its proximal counterpart. From Fig. \[fig5\](b) and unlike Fig. \[fig4\], three accumulation points w.r.t. the objective are clearly visible for the three trials indicating [*strong convergence*]{}.
Constant modulus
----------------
The constant modulus algorithm was implemented numerically via the KKTs (i.e. ) and using the results from Prop. \[propos6\]. The simulation parameters are identical to the two previous scenarios. In Fig. \[fig6\], the modulus of the fast time waveform samples vs. iterations are shown for the constant modulus alternating minimization algorithm. As seen from this figure, the algorithm was initialized with a non-constant modulus waveform. For this random initialization, convergence to a constant modulus is achieved in three iterations or less. We have however encountered cases where the algorithm has not converged for several iterations. Nevertheless this problem was not encountered when the algorithm was initialized with a random constant modulus waveform. Thus in practice, it is advocated that this algorithm be initialized with an arbitrary constant modulus waveform, viz. a chirp, rectangular pulse, etc..
The ratio of the final objective for the constant modulus algorithm to the objective for the non-constant modulus waveform design using the constrained alternating minimization is seen in Fig. \[fig7\](a)(b) for 200 random waveform initializations. After convergence, not unexpectedly, the constant modulus objective is more than the non-constant modulus objective. This trend is readily observed from Fig. \[fig7\](a)(b) for the 200 trials. This is to be expected since constant modulus waveforms are a subset of their non-constant modulus counterparts. In particular, the amplitude is constrained temporally in the constant modulus design, while the phase is allowed to be optimized. Whereas, the phase and amplitude are both optimized the non-constant modulus design. From these figures we can see that on one end, this ratio is as much as 10dB, and on the other it is almost 0dB. Nonetheless on the average, the non-const. modulus waveforms have lower objective values than objective values derived from the const. modulus waveforms.
Oracle sample support requirements
----------------------------------
The ideal SINR is $\tfrac{\rho_t^2|\mathbf{w}^H_{o}(\mathbf{v}(f_d)\otimes\mathbf{s}_{o}\otimes\mathbf{a}(\theta_t,\phi_t))|^2}{\mathbf{w}^H_{o}\mathbf{R_u}(\mathbf{s}_{o})\mathbf{w}_{o}}$ where $\mathbf{w}_o,\mathbf{s}_o$ are obtained after optimization. Using the estimated covariance matrix, say the sample covariance matrix, the definition of the estimated SINR is $\tfrac{\rho_t^2|\mathbf{w}^H_{est}(\mathbf{v}(f_d)\otimes\mathbf{s}_{est}\otimes\mathbf{a}(\theta_t,\phi_t))|^2}{\mathbf{w}^H_{est}\hat{\mathbf{R}}_{\bf u}(\mathbf{s}_{est})\mathbf{w}_{est}}$, where $\hat{\mathbf{R}}_{\bf u}(\cdot)$ is the estimated sample covariance matrix, and $\mathbf{w}_{est},\mathbf{s}_{est}$ are the optimized weight and waveform vectors by using the estimated covariance in the optimization instead.
A true SINR loss can be computed by using the estimated i.e. $\hat{\mathbf{R}}_{\boldsymbol{\gamma}}^{pq}$ in and running the optimization algorithm for each Monte Carlo trial, resulting in an estimated $\mathbf{s}_{est}$. This is computationally heavy on our current resources, therefore not reported here. However, we will assume that an oracle has provided the optimal waveform to be transmitted. Then the oracle loss of SINR due to the estimated covariance is a random variable, captured by, $$\begin{aligned}
SINR_{\mathrm{loss}}=\frac{\mathbf{w}^H_{o}\mathbf{R_u}(\mathbf{s}_{o})\mathbf{w}_{o}}{\mathbf{w}^H_{est}\hat{\mathbf{R}}_{\bf u}(\mathbf{s}_{o})\mathbf{w}_{est}} .\end{aligned}$$ Random data is now generated from zero mean multivariate complex Gaussian distributions to compute the sample covariance matrices, i.e. $\hat{\mathbf{R}}_{\bf i}, \hat{\mathbf{R}}_{\bf n}$ and $\hat{\mathbf{R}}_{\boldsymbol{\gamma}}^{pq}$. Two hundred Monte Carlo trials were run with differing sample supports. The mean and standard deviation of the oracle $SINR_{\mathrm{loss}}$ are shown in Fig. \[fig8\](a)(b). Not surprisingly the RMB rule is followed perfectly. For the same sample support, the standard deviation is a few orders less than the mean.
Adapted patterns
----------------
The adapted pattern for the waveform dependent STAP objective function is expressed as $$\begin{aligned}
\mathcal{P}(f_d,\theta)=\lvert\mathbf{w}_o^H ( \mathbf{v}(f_d)\otimes\mathbf{s}_o \otimes \mathbf{a}(\theta,\phi) ) \rvert^2,\;\mbox{ for a fixed }\phi.
\label{eq60}\end{aligned}$$ The adapted pattern in is a function of angle, Doppler, the optimal weight and the waveform vectors, $\mathbf{w}_o,\mathbf{s}_o$, respectively. Two examples are shown in Fig. \[fig9\](a)(b). Two interferers at $(\theta=-0.2,\phi=\pi/3)$ and at $(-0.2,\pi/3)$ were chosen. We modeled the clutter discretely from all azimuth angles from $-\pi/2\mbox{ to } \pi/2$ in discrete increments of $-0.005\pi/2$ radians. The clutter patches were fixed at an elevation angle of $\pi/4$ radians. The target was assumed to be at $\theta_t=0.7,\phi_t=\pi/4$ with normalized Doppler equal to 0.31 and $\theta_t=0,\phi_t=\pi/4$ with normalized Doppler equal to -0.4 in Fig\[fig9\](a)(b), respectively. The adapted patterns in Fig. \[fig9\] are identical (upto a scaling) to those obtained from the classical STAP adapted pattern. This is not a surprise but is rather reassuring since the waveform in affects all the Doppler frequencies and the azimuths identically. Moreover, we can always consider $\mathbf{s}_o \otimes \mathbf{a}(\theta,\phi)$ as a new /modified spatial steering vector. Hence as expected the inclusion of the optimal waveform will not alter the shape of the classical STAP adapted pattern.
Detection
---------
Here, we investigate the impact of detection using the optimized waveforms and randomly selected waveforms. The detection test for the presence of a target at a particular range cell is cast as a binary hypothesis test, $$\label{eq61}
\begin{aligned}
&\mathcal{H}_0 \,\, : \mathbf{w}^H\bar{\mathbf{y}}=\mathbf{w}^H\mathbf{y_u} \,\,\,
&\mathcal{H}_1\,\, : \mathbf{w}^H\bar{\mathbf{y}}=\mathbf{w}^H\mathbf{y}+\mathbf{w}^H\mathbf{y_u}
\end{aligned}$$ where $\mathbf{y},$ $\mathbf{y_u}$ have been been defined in , . Assuming that $\mathbf{y_u}$ is complex normal distributed, the test in is readily evaluated. The weight vector is obtained after the optimization. The ROC curves for SINRs 0dB, 3dB and 6dB are shown in Fig. \[fig10\](a)(b) for the non const. modulus and const. modulus design, respectively. For generating Fig. \[fig10\](a), a random waveform was used having the same energy as that obtained after the alternating minimization algorithm. The waveform samples were drawn independently from a complex Gaussian distribution. In Fig. \[fig10\](b), a chirp waveform was used having the same bandwidth and energy as its optimized constant modulus counterpart. From these figures and as expected, from a detection standpoint, an optimized waveform performs much better than transmitting an un-optimized waveform.
Realistic STAP waveform design
------------------------------
We consider a scenario frequently encountered in STAP, the sample covariance matrix is rank deficient due to the paucity of training data. The simulation parameters are identical to those used as in Fig. \[fig4\], except that we considered ground clutter from all azimuths in $[-\pi/2,\pi.2]$, similar to those used in generating Fig. \[fig9\]. Furthermore, we constrained the rank of the resulting correlation matrices to be 30, equal to the numerical rank of the clutter correlation matrix for generating Fig. \[fig11\]. The alternating minimization is first used for 20 iterations assuming an arbitrary diagonal loading factor equal to 100. After termination of this algorithm, the proximal algorithm was employed for 50 iterations. The results are shown in Fig. \[fig11\](a)(b). It is noted that in practice the ‘true’ min. eigenvector cannot be computed due to the rank deficiency. Interestingly nonetheless, the designed waveforms after the proximal optimization result in a STAP objective value which is close to that obtained from the waveform estimated from the ’true’ min. eigenvector. However, extensive simulations for the rank deficient STAP are needed to verify if this behavior is seen for other classes of noise plus interference, and clutter correlation matrices.
Conclusions
===========
Waveform design in STAP was the focus of this report assuming the dependence of the clutter response on the transmitted waveform. Our preliminary simulations indicate that the objective function was jointly non-convex in the weight and waveform vectors. However, we showed analytically that the objective function is individually convex in the waveform and the weight vector. This motivated a constrained alternating minimization technique which iteratively optimizes one vector while keeping the other fixed. A constrained proximal alternating minimization technique was propose to handle rank deficient STAP correlation matrices. To addresses practical design constraints we incorporated constant modulus constraints in our alternating minimization formulation. Simulations were chosen to demonstrate the monotonic decrease of the MVDR objective function using this alternating minimization algorithm. Preliminary simulations were presented to validate the theory.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was sponsored by US AFOSR under project 13RY10COR. All views and opinions expressed here are the authors own and does not constitute endorsement from the Department of Defense or the USAF.
[^1]: P. Setlur is affiliated with the Wright State Research Inst., and as a research contractor with the US AFRL, WPAFB, OH, email:pawan.setlur.ctr@wpafb.af.mil.
[^2]: M. Rangaswamy is with Sensors Directorate, U.S. AFRL, WPAFB, OH, email:muralidhar.rangaswamy@wpafb.af.mil.
[^3]: [Approved for Public Release No.: 88ABW-2014-3392]{}.
[^4]: also see refs. Brandwood, and A. van den Bos in [@Gesbert2007]
[^5]: Ideally one must decompose the function into real and imaginary components (as accomplished subsequently), but due to Hermitian symmetry, real valued-ness e.t.c., we take this shortcut, here, instead
[^6]: Here $\succeq$ is the Löwner partial order [@horn1994]
[^7]: The analysis of the proximal constrained alternating minimization with the const. mod. constraint is omitted, but can be readily derived from the analysis of its non-proximal counterpart, presented here.
[^8]: new $\gamma_9$=old $-\gamma_9$.
|
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"pile_set_name": "ArXiv"
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|
Background
==========
Peripheral T-cell lymphomas (PTCLs) represent an heterogeneous group of non-Hodgkin\'s lymphomas (NHL), characterized by poor outcome, accounting approximately for 10%-15% of all non-Hodgkin lymphomas in the western countries, and with an higher prevalence in Asia \[[@B1],[@B2]\]. Peripheral T-cell lymphomas derive from lymphocytes at the post-thymic stage of maturation. According to recent WHO (World Health Organization) classification more than 20 biologically and clinically distinct entities of Peripheral T-cell lymphomas have been described, such as Peripheral T-cell lymphomas Not Otherwise Specified (NOS), angioimmunoblastic T-cell lymphoma (AITL), natural killer/T-cell lymphoma, adult T-cell leukemia/lymphoma (ATLL) and anaplastic large-cell lymphoma (ALCL), the most common ones \[[@B3]\]. Cutaneous lymphomas represent a distinct entity of T lymphomas according to WHO, because same of those show even an indolent course \[[@B4]\]. Unlike other non-Hodgkin\'s lymphomas, only two subtypes of Peripheral T-cell lymphomas are characterized by disease-defining genetic abnormalities, such as the t(2;5)(p23;q35) in anaplastic large-cell lymphoma and DNA integration of human T-lymphotropic virus 1 (HTLV1) in adult T-cell leukemia/lymphoma \[[@B5],[@B6]\]. Peripheral T-cell lymphomas- Not Otherwise Specified account approximatively for 60-70% of T-cell lymphomas and it cannot be furtherly classified on the basis of morphology, phenotype, and conventional molecular studies, representing often a diagnosis of exclusion with respect to other T cell lymphomas histotypes \[[@B7]\].
*Cyclin D1* is well-established human oncogene, frequently deregulated in cancer, playing a specific role in cancer phenotype characterization and disease progression \[[@B8]\]. Cyclin D1 over-expression is often due to chromosomal aberration. Among lymphomas, translocation (11;14)(q13;q32) is typically observed in mantle cell lymphoma. Thus the *cyclin D1* gene at chromosome 11q13 is juxtaposed to *IgH* gene on chromosome 14q32, resulting in overexpression of cyclin D1 \[[@B9],[@B10]\].
Moreover *cyclin D1* amplification and gain copies with consequent protein over-expression have been frequently described in multiple myeloma, T cutaneous lymphomas and in solid cancer, such as oral squamous cell carcinoma, lung cancer, melanoma, breast cancer \[[@B11],[@B12]\]. *Cyclin D1* gene abnormality has also described in cutaneous lymphoma where the *cyclin D1* gene copy gain is an infrequent event and it seems associated to malignant phenotype \[[@B12]\].
Here we report a case of Peripheral T-cell lymphomas Not Otherwise Specified with *cyclin D1* gene copy gain associated with protein overexpression.
Case presentation
=================
A 74 year old man was admitted to Hematology Unit of Moscati Hospital, Avellino, because of multiple superficial adenopathies, splenomegaly and bilateral lower limbs lymphedema. Laboratory data revealed elevated LDH levels, hyperuricemia and positivity for hepatitis B antibodies. Peripheral blood counts were normal, since leucocitosi was not found. Parametres are as follows: white blood cells count (WBC) 8500/mmc (Neutrophilis 73.5%; Leukocytes 20.8% Monocytes 5,5%); red blood cells count (RBC) 3.970000/mmc Haemoglobin (Hb) 13,4 g/dl; hematocrit **(**HCT) 38.3%; mean corpuscular volume (MCV) 96,4 Platelet count (PLT) 129.000/mmc. On the basis of this clinical presentation and histological findings was excluded a diagnosis of lymphoma with a leukemic presentation. CT (computed tomography) scan showed multiple deep and superficial lymph-nodes enlargement in the neck, thorax and abdomen. Focal hypoechoic lesions were detectable in the spleen. A latero-cervical/submandibular nodal biopsy was performed for diagnosis purpose. The specimen was fixed in 10% neutral buffered formalin and paraffin embedded. Five microns thick sections were stained with hematoxylin and eosin for histological examination (Figure[1](#F1){ref-type="fig"}).
{#F1}
Further sections were utilized for immunohistochemical study, performed with Ventana automatic stainer . Antibodies against CD2, CD3, CD4, CD8, CD5, CD43, bcl2, bcl6, CD10, CD56, CD57, CD1a, CD34, CD99, CD30, ALK1, CD23, CD20, CD79a, BSAP/Pax5, MIB1 and cyclin D1 were tested.
The histological examination showed an effacement of the normal lymphoid parenchyma, because of the diffuse relatively monotonous proliferation of atypical, small-medium size cells with rounded or irregularly cleaved nuclei, finely dispersed chromatin and inconspicuous nucleoli. The proliferation showed a predominantly paracortical pattern of growth, entrapping residual follicles. Occasional larger cells were interspersed. The mitotic activity was high.
Neoplastic cells show immunohistochemical positivity for T cell markers (CD2, CD3, CD5, CD43, CD4) and bcl2 (Figure[1](#F1){ref-type="fig"}); B cell markers (CD20, CD79a and BSAP/Pax5) were expressed in residual follicles, in which a CD23 positive dendritc cells meshwork was occasionally observed (Figure[1](#F1){ref-type="fig"}). The atypical cells were unreactive to CD10, CD8, CD56, CD57, CD1a, CD34, CD99 and ALK1. Only rare larger cells stained with CD30. The proliferation marker MIB1 was positive in almost 80% of cells. Unexpectedly cyclin D1 antibody was expressed by a cospicous part of the neoplastic T cells (Figure[2](#F2){ref-type="fig"}).
{#F2}
A peripheral T cell lymphoma, unspecified, was diagnosed, according to the morphology and immunophenotype, with unusual cyclin D1 expression.
Moreover molecular analysis of IgH and TCR rearrangement was done. In particular detection of B clonality was investigated by identification of VDJ segments amplification of the hypervariable region of immunoglobulin heavy chain (IgH) using multiple primers complementary to conserved regions in the involved gene (*Nanogen-Master Diagnòstica)*; the detection of T clonality was investigated by identification of VJ segments amplification of TCRgamma gene using primers complementary flanking regions of the V and the J segments (Nanogen-*Master Diagnòstica*). In electrophoresis study, clonal rearrangement of TCR gamma gene is shown by the presence of a single strong sharp band within the expected size range from clonal control ( Figure[3](#F3){ref-type="fig"}). Our PCR analysis definitively demonstrated neoplastic T cell proliferation, being clonally rearranged for TCRgamma gene, in particular our sample presents VJ-B rearrangement as shown from band approximately for 215bp (Figure[3](#F3){ref-type="fig"}).
{#F3}
Although morphological and immunoprofile excluded a mantle cell lymphoma, chromosomal translocation (11; 14) (q13; q32) involving cyclin D1/IGH genes has been searched. FISH analysis for the detection of *cyclin D1* status was performed using Vysis LSI IGH/CCND1 XT Dual Color Dual Fusion Probes (Vyses). This probe set uses the dual-color, dual fusion strategy and consists of a mixture of locus-specific fluorophore-labelled DNA probes containing sequences homologous to the IGH regions (Spectrum Green) and *cyclin D1* breakpoint region (Spectrum Orange). The *cyclin D1* contig is composed of three segments covering a region approximately of 942Kb locus 11q13 where are present different genes including *cyclin D1.* Green fluorescent spots represent *Igh* and red spots stand for *cyclin D1*. In a normal tissue we have two split signals of both colors while in a traslocated sample we have two or one fused signals (yellow). The cytogenetic analyses revealed a copy gain of the *cyclin D1* without evidence of translocation because the sample shows two split signals of both colours (Figure[3](#F3){ref-type="fig"}). Bone marrow biopsy showed a huge CD3+ T cell lymphoma infiltration (Figure[4](#F4){ref-type="fig"}).
{#F4}
The patient was placed on GEMOX chemotherapy regimen (gemcitabine and oxalyplatin). Only one cycle of therapy was administered because of hematological and systemic toxicity. The patient died of disease two months after the diagnosis.
In this short report we show overexpression of cyclin D1 in a peripheral T-cell lymphoma.
Worldwide, Peripheral T-cell lymphomas represent approximately 12% of all non-Hodgkin\'s lymphomas \[[@B13]\].
Although Peripheral T-cell lymphomas Not Otherwise Specified represent most of the T-cell lymphomas, the genetic features are only poorly characterized \[[@B14]\]. Gene profiling studies performed on small series of Peripheral T-cell lymphomas showed frequent aberrations, particularly over-expression of critical genes involved in a *proliferation signature*, also significantly associated with shorter survival. This *proliferation signature* included genes commonly involved in cell cycle progression, such as CCNA, CCNB, TOP2A, and PCNA \[[@B15]\].
Cyclins play a central role in cell cycle regulation and are involved in the pathogenesis of specific hematologic malignancies. D-cyclins (D1, D2, and D3) are structurally and functionally similar proteins that bind and activate cyclin-dependent kinases 4 and 6 during the G1 phase of the cell cycle as the cell prepares to initiate DNA synthesis \[[@B16]\]**.** In mammalian cells, deregulation of these proteins leads to significantly increased cell proliferation and turnover \[[@B17]\].
In the current literature, T-cell lymphoma subtypes could be characterized by overexpression of cyclin D2, D3, in particular when proliferation rate is greater than 50% \[[@B8]\]. In addition increased Cyclin D1 expression has been observed in 9 of 23 Mycosis Fungoides (39%), 7 of 10 C- primary cutaneous CD30+ anaplastic large-cell lymphoma (70%), and 6 of 30 Sezary Syndrome (20%) \[[@B12]\]. On the contrary cyclin D1 overexpression, to the best of our knowledge, has not been hitherto described in nodal Peripheral T-cell lymphomas, Not Otherwise Specified \[[@B17]\].
Cyclin D1 overexpression is described as a driving molecular event in various types of cancer, including mantle cell lymphoma (MCL), plasmacellular dyscrasia, a subset of cutaneous T cell lymphomas, ,non-small cell lung cancer, and carcinomas of breast, head and neck, and esophagus \[[@B12],[@B18]-[@B22]\]. In various studies, cyclin D1 immunohistochemical expression in several tumors seems to be related to other proliferation markers such as Ki-67, PCNA and other cell-cycle regulatory proteins such as CDK4, p21, E2F1 proapoptotic protein p53, and inversely correlated with expression of tumor suppressor pRb protein, and bcl-2 \[[@B23]-[@B26]\]. In the literature, there are conflicting reports about prognostic impact of cyclin D1 expression and clinical outcome of different cancers. Cyclin D1 overexpression is responsible for the cell cycle deregulation playing a significant role for a greater aggressiveness, tumour extension, regional lymph node metastases and advanced clinical stage in many cancer types, such as oral cancer, breast cancer and lung cancer \[[@B27]-[@B29]\]. Cyclin D1 may be a prognostic indicator for survival \[[@B30],[@B31]\]. Many studies confirm that cyclin D1 over expression is indicative of poor outcome in B cell lymphoma patients and as it might be used also as poor prognostic index as in our case \[[@B32]\].
Aberrant expression of Cyclin D1 can be due to chromosomal translocations, single nucleotide polymorphism and gene amplification or copy gains. Chromosomal translocation is a common genetic mechanism for the pathogenesis of B-cell lymphomas \[[@B18]\]. Indeed more than 90% of mantle cell lymphoma is characterized by t(11; 14) (q13; q32) \[[@B9]\]. In addition translocation involving *Cyclin D1*, i.e. t(11; 14) (q23; q32), is also observed in 15-25% of non-IgM MGUS (monoclonal gammopathy of undetermined significance) \[[@B33]\]. As a consequence of this translocation, *cyclin D1* is constitutively expressed under the control of an active *Ig* locus in B cells presenting traslocation. Elevated expression of cyclin D1 has also been demonstrated in other lymphoproliferative disorders as hairy cell leukemia, plasma cell dyscrasias, rare cases of B-cell chronic lymphocytic leukemia/ small lymphocytic lymphoma and epithelial malignancies. Copy number change at locus 11q13 *cyclin D1* has been described in melanomas and it is strictly related to prognosis. \[[@B34]\] Gene amplifications of *cyclin D1* with consequent overexpression has been reported in several tumor types such as head and neck cancer, pituitary tumors, esophageal squamous cell carcinoma, and breast cancer. \[[@B20],[@B35]-[@B37]\] In addition, also small genetic changes such as single nucleotide polymorphisms, producing specific *cyclin D1* splice variant, have been described as responsible of cyclin D1 overexpression \[[@B38]\]. Further cyclin D1 G/A870 polymorphism has been implicated as a modulator of cancer risk and/or poor prognosis in human disease. \[[@B39]\]
Conclusion
==========
In this paper we described a case of peripheral T cell lymphoma with atypical expression of cyclin D1. The monomorphic Cyclin D1 high proliferation observed in this case led to a diagnosis with other lymphomas, in particular with mantle cell lymphoma. In addition, we showed molecular alteration never described in the literature for this type of lymphoma which affects cyclin D1 expression through gene gain of function. This abnormality produces a dysregulation of the cell cycle and it could have contribute to a more aggressive behavior of this lymphoma. Since we consider the difficult that the pathologists encounter in the diagnosis of T cell lymphomas and we underline the importance of molecular biology integration tests as a diagnostic tool to escape the pitfall.
Consent
-------
Written informed consent was obtained from our patient for the treatment of biological material for diagnostic and researching pourpose. A copy of the written consent is available by the Hospital "S. G. Moscati" AV, Italy.
Competing interests
===================
The authors declare that they have no competing interests.
Authors\' contributions
=======================
LP, Fro and RF have been directly involved in Diagnosis and interpretation of patient\'s examinations. RFr and LP were responsible for the conception and design of the Case Report. LP and FRo were responsible for provision of case report biological sample. LR is responsible for the technical part concerning the processing of the biological material. GA, RFr have been involved in PCR and FISH analysis. ADC, AA and LP have been involved in clinic-patological elaboration. The manuscript was prepared by GA under the supervision of RFr and LP. All authors read and approved the final manuscript.
|
{
"pile_set_name": "PubMed Central"
}
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It's been two weeks since the first case of COVID-19 was identified in Manitoba. Schools have suspended classes, gyms are closed and casinos are shut down but video lottery terminals in lounges and hotels remain in operation.
On March 20, half of Manitoba's 5,000 VLTs were remotely shut off by Manitoba Liquor & Lotteries, to ensure physical distancing between gamblers, said Scott Jocelyn of the Manitoba Hotel Association. He estimates half of his member hotels decided to close their lounges at that point.
But the partial shutdown of VLTs doesn't go far enough for the general manager of one small Winnipeg hotel.
"People weren't sitting right next to one another, but there's still not the six feet between them," he said.
CBC is not naming the manager or the hotel because he is concerned speaking out could affect his relationship with Manitoba Liquor & Lotteries.
"I myself did not understand because there is social distancing right now and of course, you're supposed to stay home if you can. I don't think that was one of the essentials — to have VLTs going," said the manager.
Atlantic Lotteries, which runs gambling operations for the four Atlantic provinces, shut down its VLTs on March 15 in response to advice from public health officials to ensure the safety of gamblers, workers and the public at large.
Manitoba has nearly 500 locations that run VLTs, which generate $199 million per year or just under $17 million per month in net revenue for the province.
The unnamed hotel's restaurant and lounge shut down this week out of concerns for staff safety. The general manager says the loss of revenue is not as important as keeping his workers — who have been laid off — healthy.
"You can have all the money in the world, but if you're sick, it's not worth it. So it's important that we just shut down so that we can keep our staff at home, keep them healthy."
Jocelyn says members of his association have had mixed reactions to the partial shutdown on VLTs. Some members took it upon themselves to completely shut their bars and lounges while others are staying open in order to keep their small businesses alive.
"The fact that some of those people have wrestled with that decision and come up with that, I'm glad they had the opportunity to make that decision [to close], because a lot of times we don't have the opportunity to make decisions," said Jocelyn. He says his members are subject to heavy regulation by Manitoba Liquor & Lotteries.
As for lounges that stay open, "I respect whatever decisions those people have made," said Jocelyn.
Occupancy limits
A spokesperson for Crown Services Minister Jeff Wharton says the government is closely following the advice of Manitoba's chief provincial public health officer Dr. Brent Roussin, who has not recommended closing licensed hospitality premises.
"At this time licensed hospitality premises, some of which include VLTs, can remain open but are subject to occupancy limits," said a spokesperson for Crown Services Minister Jeff Wharton.
The occupancy limits are 50 people or 50 per cent capacity whichever is lesser. Measures are also in place to ensure people stand at least one metre apart from one another.
Less than 20 per cent of VLTs in Manitoba are currently operating, a Manitoba Liquor & Lotteries spokesperson told CBC News in an email.
"VLT sites are independent private businesses. Many of them have closed and we have disabled alternating VLTs at any sites that remain open to maintain customer spacing (social distancing)," the spokesperson said.
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Wednesday, November 9, 2011
New releases > Balblair 2001 Vintage
The 2001 version of Balblair is the latest in a line of Vintages released by the Highland distillery - a range of single malt whiskies which are specially selected on the basis that, "our whisky tells us when it's ready, not the other way round" according to Distillery Manager John MacDonald. This policy started in 2007 and differs from the philosophy of most distilleries who release their whisky at various, well established age points such as 12, 18 and 25 years old.
Balblair is located in the picturesque village of Edderton, near to the town of Tain. It lies close to the shores of the Dornoch Firth, one of Scotland’s largest estuaries, with the the Highlands rising up behind and the Inverness-Thurso railway track running next to it. The distillery is one of Scotland's oldest (it was founded by John Ross in 1790), although the current buildings were constructed in 1893. Balblair is currently owned by Inver House Distillers, which is a subsidiary of the larger Thai Beverages group, and they have owned it since 1996. Balblair is currently running at full capacity, which gives an annual production of 1.8 million litres.
The Balblair 2001 Vintage is a landmark whisky for the distillery and Inver House. It is the first of their single malts to be released with a combination of being non-chill filtered, at a higher strength of 46% ABV and with no artificial colouring. Going forwards, all Balblair releases are being planned as such. This whisky has been matured in ex-bourbon casks and bottled at 10 years of age. Balblair 2001 was launched last week at the
distillery and we were delighted to be invited to
be part of the festivities. It will be available from specialist whisky retailers and on-line retailers very shortly and has a recommended price of £32.99.
Our tasting notes
The colour of the new Vintage is pale lemon yellow and the nose is delicate and fresh with initial aromas of green orchard fruits (especially pears and apples), honey and vanilla. Underneath are a series of subtle aromas that are interwoven with the more obvious ones. These include malty cereals, coconut, something floral (think of honeysuckle), orange zest and hints of almond and nutmeg. A lovely and highly distinctive note of marzipan develops with significant time (15+ minutes) in the glass. On the palate, a delicious zesty tangy kicks things off (imagine lime and lemon) and it feels like this whisky has been dipped in honey. This is followed by a pleasant combination of sweet malty barley, juicy and crisp green fruit (think of the pear and apple again), vanilla and a pinch of drying baking spices, especially cinnamon and nutmeg. The finish again offers immediate and sustained freshness in the form of citrus zest, dry wood spices and the green fruits, although these are more reminiscent of dried pear and apple by this stage and this gives a good level of sweetness.
What's the verdict?
This Balblair 2001 Vintage is a fine example of a whisky in the lighter and fresher style. It shows excellent delicacy, subtlety and what can be achieved with the sympathetic use of quality casks during maturation. If you like your whiskies richer or smoky, then you may come away disappointed but this is a great whisky to sip on during warm weather or as an aperitif. While talking with John MacDonald at the launch, he commented that "this is my favourite of the younger Vintages" - it's hard to disagree with him.
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Welcome To Whisky For Everyone
Blogging about whisky since 2008.
Whisky For Everyone is a blog for all lovers of whisky - beginner, keen enthusiast or connoisseur. Whisky is one of the most popular spirit drinks in the world and its sales and number of drinkers are increasing each year. This blog includes regularly updated tasting notes of all manner of whisky from the popular, inexpensive ones to the limited edition, rare bottlings.
We also try to demystify the sometimes daunting world of the whisky industry by explaining in layman's terms about the language, processes, terminology and answering commonly asked questions. We also have a weekly whisky news round up feature called Inbox, which includes news on new whisky releases, events and other items of interest. There is a Whisky For Everyone and we will help you discover yours ...
Drammie Award 2013 Winner
All content (images, text and photography) on Whisky For Everyone is subject to copyright unless explicitly stated otherwise. The content of this blog may be linked to, printed, downloaded or quoted in an unaltered form (not distorted from its original format or taken out of context) with permission and copyright acknowledgment. Requests for permission to reproduce content for other purposes should be directed to whiskyforeveryone@gmail.com
|
{
"pile_set_name": "Pile-CC"
}
|
/*
Copyright 2014 The Kubernetes Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package cache
import (
"errors"
"sync"
"k8s.io/apimachinery/pkg/util/sets"
)
// PopProcessFunc is passed to Pop() method of Queue interface.
// It is supposed to process the element popped from the queue.
type PopProcessFunc func(interface{}) error
// ErrRequeue may be returned by a PopProcessFunc to safely requeue
// the current item. The value of Err will be returned from Pop.
type ErrRequeue struct {
// Err is returned by the Pop function
Err error
}
var FIFOClosedError error = errors.New("DeltaFIFO: manipulating with closed queue")
func (e ErrRequeue) Error() string {
if e.Err == nil {
return "the popped item should be requeued without returning an error"
}
return e.Err.Error()
}
// Queue is exactly like a Store, but has a Pop() method too.
type Queue interface {
Store
// Pop blocks until it has something to process.
// It returns the object that was process and the result of processing.
// The PopProcessFunc may return an ErrRequeue{...} to indicate the item
// should be requeued before releasing the lock on the queue.
Pop(PopProcessFunc) (interface{}, error)
// AddIfNotPresent adds a value previously
// returned by Pop back into the queue as long
// as nothing else (presumably more recent)
// has since been added.
AddIfNotPresent(interface{}) error
// HasSynced returns true if the first batch of items has been popped
HasSynced() bool
// Close queue
Close()
}
// Helper function for popping from Queue.
// WARNING: Do NOT use this function in non-test code to avoid races
// unless you really really really really know what you are doing.
func Pop(queue Queue) interface{} {
var result interface{}
queue.Pop(func(obj interface{}) error {
result = obj
return nil
})
return result
}
// FIFO receives adds and updates from a Reflector, and puts them in a queue for
// FIFO order processing. If multiple adds/updates of a single item happen while
// an item is in the queue before it has been processed, it will only be
// processed once, and when it is processed, the most recent version will be
// processed. This can't be done with a channel.
//
// FIFO solves this use case:
// * You want to process every object (exactly) once.
// * You want to process the most recent version of the object when you process it.
// * You do not want to process deleted objects, they should be removed from the queue.
// * You do not want to periodically reprocess objects.
// Compare with DeltaFIFO for other use cases.
type FIFO struct {
lock sync.RWMutex
cond sync.Cond
// We depend on the property that items in the set are in the queue and vice versa.
items map[string]interface{}
queue []string
// populated is true if the first batch of items inserted by Replace() has been populated
// or Delete/Add/Update was called first.
populated bool
// initialPopulationCount is the number of items inserted by the first call of Replace()
initialPopulationCount int
// keyFunc is used to make the key used for queued item insertion and retrieval, and
// should be deterministic.
keyFunc KeyFunc
// Indication the queue is closed.
// Used to indicate a queue is closed so a control loop can exit when a queue is empty.
// Currently, not used to gate any of CRED operations.
closed bool
closedLock sync.Mutex
}
var (
_ = Queue(&FIFO{}) // FIFO is a Queue
)
// Close the queue.
func (f *FIFO) Close() {
f.closedLock.Lock()
defer f.closedLock.Unlock()
f.closed = true
f.cond.Broadcast()
}
// Return true if an Add/Update/Delete/AddIfNotPresent are called first,
// or an Update called first but the first batch of items inserted by Replace() has been popped
func (f *FIFO) HasSynced() bool {
f.lock.Lock()
defer f.lock.Unlock()
return f.populated && f.initialPopulationCount == 0
}
// Add inserts an item, and puts it in the queue. The item is only enqueued
// if it doesn't already exist in the set.
func (f *FIFO) Add(obj interface{}) error {
id, err := f.keyFunc(obj)
if err != nil {
return KeyError{obj, err}
}
f.lock.Lock()
defer f.lock.Unlock()
f.populated = true
if _, exists := f.items[id]; !exists {
f.queue = append(f.queue, id)
}
f.items[id] = obj
f.cond.Broadcast()
return nil
}
// AddIfNotPresent inserts an item, and puts it in the queue. If the item is already
// present in the set, it is neither enqueued nor added to the set.
//
// This is useful in a single producer/consumer scenario so that the consumer can
// safely retry items without contending with the producer and potentially enqueueing
// stale items.
func (f *FIFO) AddIfNotPresent(obj interface{}) error {
id, err := f.keyFunc(obj)
if err != nil {
return KeyError{obj, err}
}
f.lock.Lock()
defer f.lock.Unlock()
f.addIfNotPresent(id, obj)
return nil
}
// addIfNotPresent assumes the fifo lock is already held and adds the provided
// item to the queue under id if it does not already exist.
func (f *FIFO) addIfNotPresent(id string, obj interface{}) {
f.populated = true
if _, exists := f.items[id]; exists {
return
}
f.queue = append(f.queue, id)
f.items[id] = obj
f.cond.Broadcast()
}
// Update is the same as Add in this implementation.
func (f *FIFO) Update(obj interface{}) error {
return f.Add(obj)
}
// Delete removes an item. It doesn't add it to the queue, because
// this implementation assumes the consumer only cares about the objects,
// not the order in which they were created/added.
func (f *FIFO) Delete(obj interface{}) error {
id, err := f.keyFunc(obj)
if err != nil {
return KeyError{obj, err}
}
f.lock.Lock()
defer f.lock.Unlock()
f.populated = true
delete(f.items, id)
return err
}
// List returns a list of all the items.
func (f *FIFO) List() []interface{} {
f.lock.RLock()
defer f.lock.RUnlock()
list := make([]interface{}, 0, len(f.items))
for _, item := range f.items {
list = append(list, item)
}
return list
}
// ListKeys returns a list of all the keys of the objects currently
// in the FIFO.
func (f *FIFO) ListKeys() []string {
f.lock.RLock()
defer f.lock.RUnlock()
list := make([]string, 0, len(f.items))
for key := range f.items {
list = append(list, key)
}
return list
}
// Get returns the requested item, or sets exists=false.
func (f *FIFO) Get(obj interface{}) (item interface{}, exists bool, err error) {
key, err := f.keyFunc(obj)
if err != nil {
return nil, false, KeyError{obj, err}
}
return f.GetByKey(key)
}
// GetByKey returns the requested item, or sets exists=false.
func (f *FIFO) GetByKey(key string) (item interface{}, exists bool, err error) {
f.lock.RLock()
defer f.lock.RUnlock()
item, exists = f.items[key]
return item, exists, nil
}
// Checks if the queue is closed
func (f *FIFO) IsClosed() bool {
f.closedLock.Lock()
defer f.closedLock.Unlock()
if f.closed {
return true
}
return false
}
// Pop waits until an item is ready and processes it. If multiple items are
// ready, they are returned in the order in which they were added/updated.
// The item is removed from the queue (and the store) before it is processed,
// so if you don't successfully process it, it should be added back with
// AddIfNotPresent(). process function is called under lock, so it is safe
// update data structures in it that need to be in sync with the queue.
func (f *FIFO) Pop(process PopProcessFunc) (interface{}, error) {
f.lock.Lock()
defer f.lock.Unlock()
for {
for len(f.queue) == 0 {
// When the queue is empty, invocation of Pop() is blocked until new item is enqueued.
// When Close() is called, the f.closed is set and the condition is broadcasted.
// Which causes this loop to continue and return from the Pop().
if f.IsClosed() {
return nil, FIFOClosedError
}
f.cond.Wait()
}
id := f.queue[0]
f.queue = f.queue[1:]
if f.initialPopulationCount > 0 {
f.initialPopulationCount--
}
item, ok := f.items[id]
if !ok {
// Item may have been deleted subsequently.
continue
}
delete(f.items, id)
err := process(item)
if e, ok := err.(ErrRequeue); ok {
f.addIfNotPresent(id, item)
err = e.Err
}
return item, err
}
}
// Replace will delete the contents of 'f', using instead the given map.
// 'f' takes ownership of the map, you should not reference the map again
// after calling this function. f's queue is reset, too; upon return, it
// will contain the items in the map, in no particular order.
func (f *FIFO) Replace(list []interface{}, resourceVersion string) error {
items := map[string]interface{}{}
for _, item := range list {
key, err := f.keyFunc(item)
if err != nil {
return KeyError{item, err}
}
items[key] = item
}
f.lock.Lock()
defer f.lock.Unlock()
if !f.populated {
f.populated = true
f.initialPopulationCount = len(items)
}
f.items = items
f.queue = f.queue[:0]
for id := range items {
f.queue = append(f.queue, id)
}
if len(f.queue) > 0 {
f.cond.Broadcast()
}
return nil
}
// Resync will touch all objects to put them into the processing queue
func (f *FIFO) Resync() error {
f.lock.Lock()
defer f.lock.Unlock()
inQueue := sets.NewString()
for _, id := range f.queue {
inQueue.Insert(id)
}
for id := range f.items {
if !inQueue.Has(id) {
f.queue = append(f.queue, id)
}
}
if len(f.queue) > 0 {
f.cond.Broadcast()
}
return nil
}
// NewFIFO returns a Store which can be used to queue up items to
// process.
func NewFIFO(keyFunc KeyFunc) *FIFO {
f := &FIFO{
items: map[string]interface{}{},
queue: []string{},
keyFunc: keyFunc,
}
f.cond.L = &f.lock
return f
}
|
{
"pile_set_name": "Github"
}
|
List of supermarket chains in Belgium
This is a list of supermarket chains in Belgium. As of 2011, in Belgium three major groups form more than two thirds of the market: Colruyt group 27%, Delhaize 22.5% and Carrefour 22%. Then there are Aldi 11%, Lidl 5.6% and Makro 4.5%.
Current supermarket chains
Defunct supermarket chains
Écomarché (owned by Les Mousquetaires, now rebranded to Intermarché Contact or Intermarché Super)
GB Supermarkets, Taken over by Carrefour. Before that, the stores belonged to the now defunct GIB Group, almost all GB stores were later rebranded to become Carrefour stores:
Maxi GB (now: Carrefour)
Super GB (now: Carrefour Market or Carrefour GB)
GB Express (now: Carrefour Express)
Bigg's Continent (now: Carrefour hypermarkets)
Jawa (was a supermarket chain, all its stores were taken over in 1995 to become Match supermarkets)
Profi (was a discount store owned by Louis Delhaize Group, rebranded to Smatch supermarket)
Unic (rebranded to Super GB and later Carrefour GB.)
References
Belgium
Supermarket
|
{
"pile_set_name": "Wikipedia (en)"
}
|
Coronavirus Pandemic Weekly Briefs
Thursday 4th June 2020
Briefing number 13 (unlucky for some – more on that later!) sees NI move to the next stage of the Executives Recovery Plan. As well as being able to go to garden and recycling centres, we can now also take part in certain outdoor activities, such as tennis, golf, and angling, and meet in groups of up to six people outdoors. Of course, progress will depend on controlling the rate of transmission of the virus – something in which we all have a part to play. At Brooke House we are starting to explore how we may open more for our staff and our clients safely and slowly, in a way that is fully compliant with the social distancing guidelines.
Volunteers Week
This week (1st to 7th June) is also Volunteers Week – a chance to celebrate and say thank you for the contribution millions of volunteers make across the UK. As well as helping others, volunteering has been shown to improve volunteers’ wellbeing, gain valuable new skills and experiences, and boost confidence. We’d like to take this opportunity to give our thanks to the volunteers who help to make the Colebrooke Garden as productive and beautiful as it is.
So, back to number 13 – lucky or unlucky? There is no scientific evidence to suggest that any number is inherently lucky or unlucky, but many people believe that 13 is unlucky. There is even a term for the specific fear of the number 13 – triskaidekaphobia. And it’s a fear believed to be rooted in ancient times. The Norse god Baldur, who stood for beauty and good in the world was murdered by Loki, the 13th guest at a dinner honouring Baldur’s memory. But maybe our view on the number 13 could be coloured by our level of optimism?
Optimism
Psychologists report that most of the population is largely optimistic. Research has shown that higher levels of optimism are associated with better mental health, more effective pain management, improved immune and cardiovascular function, and greater physical functioning overall. It is also associated with better health outcomes after physical illness as well as increased life expectancy. It has even been shown to be connected to increased success in sports and work. Optimism is designed to give us the energy, enthusiasm, determination, and positive attitude to move forward in our lives. It helps us to appreciate the glass as half-full. Optimists tend to apply better coping strategies when faced with adversity and look for meaning in that adversity, which can make them more resilient. The good news is that, while optimism may be part of our character, it can also be learned.
Take 10 minutes or more out of your day and imagine yourself in a future, say 5 years from now, that has turned out to be the rosiest that is possible (but also realistic). In this future, you have achieved all the things that you wanted to, whether that’s being at the height of your dream career, living with the love of your life, being in peak physical shape, having a small circle of close and supportive friends, etc. You get the picture. Visualise what such a future will be like and feel like to you in as much detail as possible (see: https://www.youtube.com/watch?v=G_jEsnDEIa0). The more vividly your imagination can conjure up your best possible self, the more successful this exercise is. Even if you perform the best possible self exercise just once, your optimism will get a temporary boost. And if you perform it repeatedly, say every night, or a few times a week, there will be a persistent spike in your optimism. What is more, your mood will also improve, and you will feel happier.
Last week we included a link to express interest in our Armed Forces Covenant grant funded online workshops allowing us to deliver existing and new material to support the health and wellbeing of our clients in a different way. We hope to offer online workshops in Nutrition, Gardening, Wellness and Horticultural Therapy. So far we have had a great response! There’s still time to register your interest using the following link: https://forms.gle/kzrH16VoHGrryYfh6
Here you can see all our previous posts, as well as new posts every week! This week we are sharing with you planting out brassicas and how the new market garden is developing.
It is our hope to make the blog site interactive so you can comment and ask questions and share your own thoughts. Watch this space.
If you have any queries or need additional advice/support you can call us on 07885 808550 or 07885 808546 during office hours or email us on info@brookehouse.co.uk . In a crisis please call Lifeline on 0808 808 8000 or Samaritans on 116 123.
In the meantime, please maintain all the precautionary measures advised by the Government to protect yourselves and others.
This week we are talking about your whole wellbeing and resilience. But first, some good news, Brooke House has been successful in obtaining grant funding from the Armed Forces Covenant to deliver online workshops during the Coronavirus Pandemic. This will allow us to deliver existing and new material to support the health and wellbeing of you, our clients, in a different way. If you would be interested in attending our online workshops, please register your interest using the following link:https://forms.gle/kzrH16VoHGrryYfh6
The 5 ways to wellbeing (Connect, Be Active, Take Notice, Keep Learning and Give: https://www.nhs.uk/conditions/stress-anxiety-depression/improve-mental-wellbeing/) are the foundations for resilience. The walls and the roof of our resilience ‘house’ are constructed from additional skills/behaviours. In briefing number 11 we covered ‘building and maintaining high quality connections’ – this week we’re talking about having meaning and purpose.
As humans we need to find purpose in what we do. Imagine being on holiday (one day soon!), relaxing with your favourite food and drink to hand, feeling the warmth of the sun on your face, listening to the waves breaking gently on the beach – you’re feeling pretty good right? How good would it feel after being sat there all day? Maybe still good. What about after a week, a month, or a year? As much as we might hate to admit it, this same pleasurable situation will gradually become less enjoyable over time and not have nearly the same effect on us as it did at the beginning. So, what’s missing? Purpose – beyond feeling good, we need to find purpose in what we do.
Resilient people have a strong sense of purpose – they know what they are doing and why. And here’s the science bit – having a sense of purpose in life has been shown to improve overall wellbeing and life satisfaction, to be good for our health and even to reduce the risk of death. What’s not to like?
We have been advised of an opportunity to help make a difference in the current situation that some of you may be interested in. PT Security Services Ltd (https://pt-ss.com/about-us) has been awarded the contract to establish and staff the Covid-19 Mobile Testing Centre in Belfast. There is a requirement for 20 members of staff to begin training immediately at the SSE Arena, Belfast. The email address for those interested in applying is info@pt-ss.com.
We all want to be part of something bigger. As an organisation we at Brooke House are looking to refresh our values and we’d be interested in hearing what you see as core to our way of doing things. Get in touch at info@brookehouse.co.uk with any ideas and please express an interest in our upcoming online workshops:https://forms.gle/kzrH16VoHGrryYfh6
Here you can see all our previous posts, as well as new posts every week! This week we are sharing with you how to support your runner beans using hazel rods, an update from the garden and sheep shearing, no lockdown haircuts here!
It is our hope to make the blog site interactive so you can comment and ask questions and share your own thoughts. Watch this space.
If you have any queries or need additional advice/support you can call us on 07885 808550 or 07885 808546 during office hours or email us on info@brookehouse.co.uk . In a crisis please call Lifeline on 0808 808 8000 or Samaritans on 116 123.
In the meantime, please maintain all the precautionary measures advised by the Government to protect yourselves and others.
Friday 22nd May 2020
Briefing number 11 comes towards the end of Mental Health Awareness week the theme of which is ‘kindness’.
Studies have found that people who carry out acts of kindness experience greater wellbeing, increased feelings of happiness, and improved life satisfaction. It does not seem to matter whether we show kindness to our friends and family, our communities or even ourselves – the effect is the same. Not only that but, just reflecting on and remembering kind things we have done in the past may also increase our wellbeing. ‘Give’ is one of the 5 ways to wellbeing that we have covered before in our previous briefings.
“Kindness is a gift everyone can afford to give”
The 5 ways to wellbeing are the foundations for resilience.
“Resilience enables us to function effectively, deal with the ups and downs of everyday life, adapt to change, bounce back from adversity and even grow as a result”.
Some researchers have termed resilience ‘ordinary magic’ because we all have it to varying degrees and we can all learn to strengthen it over time. Have a look at this video – the power of resilience with Dr Sam Goldstein, Neuropsychologist: https://www.youtube.com/watch?v=isfw8JJ-eWM
One of the other important building blocks of resilience is developing and maintaining high quality connections (‘Connect’). Now that in NI we are permitted to be outside at a social distance with up to 6 people, we have an increased opportunity to enhance those connections. Resilient people are resourceful, and friends and family are among their most important resources. Resilient people have strong social networks, close connections to family and friends, can self-disclose about their troubles to people close to them, and ask for help when they need it
We all need to actively build and maintain our own resilience; act early on any warning signs that our resilience is under strain; and, mobilise support when our resilience is threatened. The Action for Happiness ‘10 days of Happiness’ online coaching programme will help: https://10daysofhappiness.org/
Here you can see all our previous posts, as well as new posts every week! This week we are sharing with you how to grow the best carrots, an update from the garden and information about the Sika Deer who call Colebrooke Park home.
It is our hope to make the blog site interactive so you can comment and ask questions and share your own thoughts. Watch this space.
The ‘office’ will be closed for the public holiday on Monday 25th May. If you have any queries or need additional advice/support you can call us on 07885 808550 or 07885 808546 during office hours or email us on info@brookehouse.co.uk . In a crisis please call Lifeline on 0808 808 8000 or Samaritans on 116 123.
In the meantime, please maintain all the precautionary measures advised by the Government to protect yourselves and others.
The document sets out the approach the Executive will take when deciding how to ease coronavirus restrictions in the future. There are however no fixed dates for when any single restriction will be lifted leaving us all feeling a little uncertain.
Uncertainty
As human beings, we seek security. We all want to feel safe and have a sense of control over our lives and well-being. Research shows that fear and uncertainty can leave us feeling stressed and anxious. It can drain us emotionally and trap us in a downward spiral of negativity. But it doesn’t have to be like that – there are steps we can all take to better deal with the impact of uncontrollable circumstances and manage our anxiety.
Uncertainty is often centred on worries about the future and all the bad things we anticipate happening. It can make things feel worse than they are and even paralyse us from taking action to overcome a problem.
Worrying won’t give us more control over uncontrollable events – it just takes away our enjoyment in the present, drains our energy, and keeps us awake at night! But there are healthier ways to cope with uncertainty, for example creating a ‘safe space’ in your imagination.
A ‘safe space’ is somewhere we can go to regardless of what is happening around us. Imagine a place that has a lot of positive associations, where you feel safe, comfortable, peaceful, or calm. Give it a name such as ‘home’, ‘woods’, ‘beach’ etc. Really focus on the image and describe the positive emotions it creates and notice where in your body you feel any pleasant physical sensations. Notice the sounds, smells, textures associated with the image. Make the image as vivid in your mind as you can. Identify a single word or phrase that fits the picture, e.g. “relax”, “safe”, “in control”, “at peace”. When your mind wanders back to worrying or the feelings of uncertainty return, refocus your mind on the image and your own slow, steady, deep breathing. The more practice you put into this, the more effective it will be at helping you to manage anxiety and worry.
And, without some uncertainty and unpredictability, we would never have any surprises. As we have seen during this pandemic, good things do sometimes happen unexpectedly – the rise of volunteering and acts of kindness for instance. Facing uncertainty in life can also help us learn to adapt, overcome challenges, and increase our resiliency. It can help us to grow.
If you have any queries or need additional advice/support you can call us on 07885 808550 or 07885 808546 during office hours or email us on info@brookehouse.co.uk.
Out of hours, in a crisis please call Lifeline on 0808 808 8000 or Samaritans on 116 123.
In the meantime, please maintain all the precautionary measures advised by the Government to protect yourselves and others.
Thursday 7th May 2020
Briefing number 9 falls on an auspicious week. This year’s early May Bank Holiday has been moved back by four days to coincide with the 75th anniversary of VE (Victory in Europe) Day marking the day, towards the end of World War Two, when fighting came to an end in Europe. This holiday was to form part of a three-day weekend of commemorative events to remember and honour the heroes of the Second World War and reflect on the sacrifices of a generation. But all that was before the lockdown and the need for social distancing. But there is no reason why we can’t still mark the occasion.
On Wednesday evening of this week, four members from Maguiresbridge Silver Band, came to the Walled Garden to help us to create a short clip to help celebrate VE Day. We hope you enjoy it. It can be viewed here https://youtu.be/MxL4JQBxO_Q
A host of virtual and remote initiatives have been planned including a UK-wide two-minute silence at 11am, church bells being rung to mark the moment the guns of the Second World War fell silent, and a 9pm singalong to Vera Lynn’s We’ll Meet Again. Belfast City Hall will be illuminated and in Fermanagh there will be the projection of two beams of light in Enniskillen from Thursday to Saturday, to replicate the V for victory sign which lit up the skies in 1945. We’re all being encouraged to take part in the nation’s toast from the safety of our own homes at 3pm on Friday.
Food and mood
Weekly wartime food rations made creative cooks of everyone and there was little in the way of treats. You went to the butcher for weekly meat rations, visited the baker for your allocation of bread and so on. Some similarities with the lockdown you may think! But people now have more of an opportunity to adopt unhealthy behaviour patterns – overeating the wrong kinds of foods and drinking more. There are physiological reasons for some of these behaviours – when the body is stressed it produces too much cortisol, which makes us more likely to over-eat, particularly foods which are high in fat and sugar. The Association of UK Dietitians have written an article with hints and tips to help us eat for a healthy body and mind during the COVID-19 pandemic – https://www.bda.uk.com/resource/eating-well-during-coronavirus-covid-19.html.
But food also affects our mood directly for instance, protein (found in lean meat, fish, eggs, cheese, peas, beans, lentils etc) contains amino acids, which make up the chemicals our brain needs to regulate our thoughts and feelings. MIND, the mental health charity, has information exploring the relationship between what we eat and how we feel, including tips on how to incorporate healthy eating into our life. https://www.mind.org.uk/media-a/2929/food-and-mood-2017.pdf
Jane McClenaghan, a BANT Registered Nutritional Therapist, who ran a number of courses for us earlier this year has launched a Tuesday Teabreak where, every Tuesday morning at 10am, we can join her in her kitchen for an informal chat and catch up on all things nutrition. From planning healthy meals, to making sense of food labels and using simple ingredients in creative ways. We’re hoping to do some more work with Jane through lockdown so watch this space!
Garden Blog
We hope you are enjoying our Garden blog. This week we share with you an update from Dougal, how to look after your potatoes and carrots, and pictures of the Bluebells starting to appear in the Colebrooke woodland: https://brookehouse.co.uk/walled-garden-blog/
Veterans’ Gateway
A 24-hour point of contact for veterans’ support, has launched a new app enabling any ex-Service personnel who are in need to get help with issues such as finances, housing, employment, relationship, physical and mental health. The link to the Directory of Local Support is here: https://www.veteransgateway.org.uk/local-support/. There is also a new category within the Local Support tool entitled Covid-19 Local Support that lists local ‘pop-up’ volunteer services. These have a Covid-19 icon to help individuals find them easily. The App also has links to allow an individual to call, email or live chat with the team at the contact centre – that continues to be fully manned 24/7 by Connect Assist advisers. The Veterans’ Gateway app is available for free on the Apple App Store and Google Play: https://support.veteransgateway.org.uk/app/answers/detail/a_id/820 There is also a YouTube video of the app in use.
We will be closed on Friday 8th for the May Day holiday. If you have any queries or need additional advice/support you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk. In a crisis please call Lifeline on 0808 808 8000 or Samaritans on 116 123.
In the meantime, please maintain all the precautionary measures advised by the Government to protect yourselves and others.
Friday 1st May 2020
Welcome to the eighth weekly briefing from Brooke House. This is our way of keeping you up to speed with events and providing information and tools to help us all manage through these unusual and unsettling times.
Food parcels
As you know, we have been assisting those of you who don’t have local support with a food delivery – if you think you would now benefit from this scheme please get in touch and we will discuss your needs with you. We will also call those who received a delivery previously to see if your needs have changed.
To plant a garden is to believe in tomorrow
This week our focus is on Hope. The Rainbow of Hope has become a national symbol for supporting the NHS and getting through the coronavirus pandemic together.
Hope has been shown to be connected to mental health, happiness, satisfaction with life, and psychological well-being across all age groups. There is an association between high hope and lower levels of depression, while low hope is associated with a reduction in well-being. Research has also shown that hope acts as a protective factor during life crises. People who have high levels of hope view barriers as challenges to overcome and can more easily find another way to achieve their goals.
If we can value the positive things in life and be thankful for all the good, instead of worrying about what we don’t have or can’t change, our hope will grow. Like a virus, hope can also be transmitted to others but, unlike a virus, providing hope to others makes us feel happier and makes our lives feel more meaningful.
Action for Happiness have produced a ‘Coping Calendar’ with 30 suggested actions to look after ourselves and each other as we face this global crisis together. You can download it as a PDF for printing, or pass it on to others and help spread the word.
To sum up, hope can protect us against negativity and despair. It helps us keep in sight a better future in times of adversity, uncertainty, and crises.
Hope is like a road in the country; there was never a road, but when many people walk on it, the road comes into existence
May Day Holiday
We will be closed on Friday 8th for the May Day holiday. If you have any queries or need additional advice/support you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Friday 24th April 2020
Briefing number seven is here already and we have just had the ‘lockdown’ restrictions extended for another 3 weeks. You will recall the Government advice to:
As a result, all the Brooke house staff will be continuing to work from home until the advice changes.
Walled Garden Blog
We have developed in conjunction with Colebrooke Walled Garden, a blog to help keep members and friends of the Walled Garden in touch with what is happening during the Coronavirus lock down and beyond. This blog also contains information about the wider happenings in Colebrooke Park, check it out here: Walled Garden Blog
Food parcels
As you know, we have been assisting those of you who don’t have local support with a food delivery – if you think you would now benefit from this scheme please get in touch and we will discuss your needs with you.
Mental Health
As the lockdown continues there is increasing focus on mental health and wellbeing. We have already provided information and links to our website (https://brookehouse.co.uk/category/coronavirus/), the Health and Social Care NI leaflet – ‘Take 5 steps to wellbeing’, an NHS Scotland resource called ‘Tips on how to cope if you are worried about Coronavirus and in isolation’, and the Action for Happiness’ 3 good things exercise.
This week our focus is on self-soothing techniques. These are simple things that you can do wherever you are that can bring calm to your mind and body. The most effective techniques involve one or more of our five senses – touch, taste, smell, sight, and sound. Here are some examples:
Sound – listen to relaxing music, sing to yourself (or your neighbours!), try some positive statements out loud like I choose hope over fear, Positivity is a choice that I choose to make, I am grateful for the things I have,I can go with the flow.
When trying these techniques focus completely on the task. If you get distracted simply bring your focus back. The more you practice the more effective they will be.
Other Helplines
Specialist helplines providing a variety of vital support services including information, advice, counselling, a listening ear, and befriending are available through the Helplines Network NI.
Helplines are confidential, non-judgemental, and accessible sources of information, advice, and support. The Helplines Network NI website provides a single point of access to NI Helplines telephone numbers and websites. For further information visit www.helplinesnetworkni.com.
Lifeline is a free, confidential telephone helpline available anytime every day on 0808 808 8000. Counsellors answer all telephone calls. They listen, help, and support you in confidence. They do not judge you. They can deal with different concerns including depression, anxiety, self-harm, suicide, trauma, sexual violence, and abuse. Lifeline can arrange an appointment for face-to-face counselling or other therapies in your area within seven days. They can put you in touch with follow-up services, so you get the best possible response.
If you have any queries or need additonal advice/support you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Friday 17th April 2020
This is the Brooke House briefing number six and will be slightly ‘briefer’ than usual due to the Easter break. We hope you all had a restful one.
Reading
In these unusual times many of us are rediscovering the joy of the written word. Not only is reading pleasurable, but research has shown that it can help to reduce stress. Reading as little as six minutes a day can reduce stress levels by 60% by slowing your heart rate, easing muscle tension and altering your state of mind. It often provides an ‘escape’ from the worries of the day, has been shown to slow memory loss and is good for our mental health.
In case you are not aware, thousands of eBooks and Audiobooks are available to download for free from Libraries NI. You can borrow up to eighteen at a time for three weeks. To make use of this great free service you need to have a Libraries NI membership number. If you have a smartphone or tablet device download the Libby App or Overdrive App and create an account (Choose Libraries NI as your library). For PCs and laptops you need to install Adobe Digital Editions (ADE) software. So, get reading!
Additional support
So far we have made you aware of our website (https://brookehouse.co.uk/about/), the Health and Social Care NI leaflet – ‘Take 5 steps to wellbeing’, and an NHS Scotland resource called ‘Tips on how to cope if you are worried about Coronavirus and in isolation’.
This week we’re focussing on gratitude – a thankful appreciation for what we receive and an acknowledgement of the goodness in our lives. Gratitude helps us connect to something larger than ourselves, be that other people, nature, or a higher power. In positive psychology research, gratitude is strongly and consistently associated with greater happiness and higher overall wellbeing. Gratitude helps people feel more positive emotions, relish good experiences, improve their health, deal with adversity, and build strong relationships. We can all cultivate gratitude on a regular basis and, although it may feel contrived at first, this mental state grows stronger with use and practice. You could write a thank-you note, thank someone mentally, or keep a gratitude journal, reflecting on what you are grateful for. See Action for Happiness’ 3 good things exercise: https://www.actionforhappiness.org/take-action/find-three-good-things-each-day
If you have any queries or need additonal advice/support you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Thursday 9th April 2020
This the fifth of our Brooke House updates. We hope you have been receiving the previous ones and finding them useful?
Telephone checks
It’s been great to hear how supported most of our clients are during these challenging times. Many of you have family and friends who have rallied round or local services/shops who have stepped up to deliver groceries and/or medicines. We have supplemented this with additional support through a food parcel to some of you and many of these have now been delivered or are on their way.
We also want to make you aware of the Department for Communities NI food parcel initiative recently announced: https://www.communities-ni.gov.uk/news/hargey-announces-thousands-benefit-food-parcels-service. If you are in receipt of a GP shielding letter or otherwise vulnerable you may be eligible for a weekly food parcel delivery. Ring 0808 802 0020, email covid19@adviceni.net or text ACTION to 81025. You will be assessed and, if suitable, you will receive deliveries through the registered voluntary/community groups operating in the area. If there are medical related supplies in the parcel, deliveries will be authorised by a community pharmacist.
Additional support
For those of you who prefer phone contact rather than emails we are setting up a periodic call system with our Health and Wellbeing Coordinators. We have also re-launched our website which, for the duration of the pandemic, will have regularly updated information and materials: https://brookehouse.co.uk/about/.
We are planning to take a break for Easter so the ‘office’ will be closed from Friday 10th to Tuesday 14th April inclusive. Otherwise you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk at any time if your circumstances change or you think we might be able to help with something.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Friday 3rd April 2020
This the fourth of our Brooke House updates. Home working has not been without its challenges for our staff what with issues around connectivity and online tools! However, we are working our way around these.
New referrals, initial screenings and reviews
Since we started working from home, we have only received 3 new referrals which is not unexpected given the current situation. We have started to carry out initial screenings by telephone.
Interventions
All face-to face interventions have had to cease which means complementary therapies, physiotherapy, groups etc are on hold for now. However, approximately 20 of our clients remain in talking therapy and a few have been newly referred for this – all delivered remotely.
Telephone checks
We have also managed to call over 200 of you individually to check in and see if you are supported locally or need our help with anything. This has led to us arranging a small number of food deliveries across Fermanagh, Tyrone and Armagh for those who cannot manage to access food and supplies themselves using an existing support network. We are continuing with this work across the rest of NI. Most people we talk to have asked for ongoing contact by email and one or two have asked for a periodic telephone call which we will set up with the Health and Wellbeing Coordinators.
Just a reminder to all of you that you can call us on 028 8953 1223 or email us on info@brookehouse.co.uk at any time if your circumstances change or you think we might be able to help with something.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Friday 27th March 2020
This the third of what we hope will be fairly regular updates as we all see our way through the coronavirus pandemic. Currently all our substantive staff have transferred to home working.
New referrals, initial screenings and reviews
We are still receiving new referrals although we anticipate that these will drop off significantly in the coming weeks as people reprioritise and as a result of a lack of active promotion on our part. We are also conducting initial screenings and reviews by telephone wherever possible. These will be carried out by 3 new staff members: Amanda Armstrong, Andrea Milligan and Christine Spence.
Interventions
We have provided our Associate Talking Therapists with some guidance on how to maintain therapy remotely either by telephone or using an online platform. For those of you involved in this as clients you will receive some guidance shortly to help you manage this new way of working.
Telephone checks
For us to fully understand your current circumstances and how we might be able to support you further, we will be calling you all individually over the next few weeks. We’d like to know who is around to provide you with any practical support you may need at this time. It would also be useful for us to know if you have any benefits and/or pension concerns as we may be able to connect you with advice on this. We would also like to check that you still want ongoing contact with us. Some people prefer to wait until things are more settled, others are happy enough with periodic emails where others appreciate the option of a telephone call from time to time.
Future support
Once we have bedded in our new processes, we will start to work up some materials to help us all manage our psychological, and physical, wellbeing in these challenging times. Please watch this space.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
Wednesday 18th March 2020
As you know the situation in relation to the coronavirus pandemic in the UK changes daily. In response to the most recent Government advice and in order to protect our clients, our employees and NHS capacity in the coming weeks, Brooke House is suspending normal services effective immediately. We will be supporting our employees to work from home wherever possible.
We are investigating the facilities for potentially delivering the following remotely, either by telephone or online:
Screening appointments following referral
Review appointments on completion of services
Psychological assessment appointments
Talking therapies interventions (Counselling, Clinical Psychology) – if your therapist has already made arrangements directly with you then please continue with that.
Please do not come to the Centre in person. Instead call 028 8953 1223 or email us on info@brookehouse.co.uk. We will be contacting you all in the coming days with further updates so please keep an eye on your emails or for those of you who do not have an email address we will try to call instead.
In the meantime, please maintain all the precautionary measures advised by the Government in order to protect yourselves and others.
These are:
Stay at home if you have coronavirus symptoms: If you experience recent onset of a new continuous cough and/or high temperature, however mild, stay at home and do not leave your house for 7 days from when your symptoms started. If you live with other people, they should stay at home for 14 days from the day the first person got symptoms. If you live with someone who is 70 or over, has a long-term condition, is pregnant or has a weakened immune system, try to find somewhere else for them to stay for 14 days. If you have to stay at home together, try to keep away from each other as much as possible. See: https://www.nhs.uk/conditions/coronavirus-covid-19/self-isolation-advice/
Only use the NHS 111 online or telephone service if needed: If your symptoms worsen during home isolation or are no better after 7 days, contact NHS 111 online or call NHS 111. For a medical emergency dial 999.
Wash your hands: Wash your hands with soap and water often – do this for at least 20 seconds. Always wash your hands when you get home or into work and/or after coughing, sneezing and blowing your nose, or after being in areas where other people are doing so. Use hand sanitiser gel if soap and water are not available. Clean and disinfect door handles, objects and surfaces using suitable cleaning products.
Cover your mouth and nose when you cough or sneeze: To reduce the spread of germs when you cough or sneeze, cover your mouth and nose with a tissue, or your sleeve (not your hands) if you don’t have a tissue, and throw the tissue away immediately. Then wash your hands or use a hand sanitising gel.
Keep more physically distant from other people: Do not use public transport unless you have to. Work from home, if you can. Avoid social activities, such as going to pubs, restaurants, theatres and cinemas. Avoid events with large groups of people. Use phone, online services, or apps to contact your GP surgery or other NHS services.
As things change so may this advice and we will attempt to keep you as up to date as possible. In the meantime, look after yourselves and others.
Monday 16th March 2020
As you know the situation in relation to the coronavirus pandemic in the UK changes daily. In response to the most recent Government advice and in order to protect our clients, our employees and NHS capacity in the coming weeks, Brooke House is suspending normal services effective immediately. We will be supporting our employees to work from home wherever possible.
In particular, we will no longer deliver the following at all until advised it is safe to do so:
We are investigating the facilities for potentially delivering the following remotely, either by telephone or online:
Screening appointments following referral
Review appointments on completion of services
Psychological assessment appointments
Talking therapies interventions (Counselling, Clinical Psychology) – if your therapist has already made arrangements directly with you then please continue with that.
Please do not come to the Centre in person. Instead call 028 8953 1223 or email us on info@brookehouse.co.uk. We will be contacting you all in the coming days with further updates so please keep an eye on your emails or for those of you who do not have an email address we will try to call instead.
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Heinrich and Julius Hart
The brothers Heinrich and Julius Hart were Jewish-German writers and literary critics who collaborated closely. They were among the pioneers of naturalism in German literature.
Heinrich was born 30 December 1855, in Wesel and died 11 June 1906, in Tecklenburg. Julius was born 9 April 1859, in Münster and died 7 July 1930, in Berlin.
The Hart brothers published works of literary criticism, notably Kritische Waffengänge (parts 1–6, 1882–1884), in which they opposed the light reading chosen by the bourgeoisie.
Works
Hart, J. Sansara (1879)
Hart, J. The Triumph of Life (1898)
Hart, H. Gesammelte Werke, vols. 1–4. Berlin (1907)
Hart, J. Revolution der Ästhetik. Berlin (1908)
Hart, H. Song of Humanity, an attempt to depict the panorama of man's development from ancient times. He finished only three "songs":
"Tul and Nahila" (1888)
"Nimrod" (1888)
"Moses" (1896)
References
Hart, Heinrich and Julius at the Great Soviet Encyclopedia. Accessed April 1013
Jürgen, I. Der Theaterkritiker Julius Hart. Berlin, 1956. (Dissertation.)
Secondary literature on Heinrich and Julius Hart
List of works by the Hart brothers
Modern poet-characters in Zeno.org
Category:1855 births
Category:1859 births
Category:1906 deaths
Category:1930 deaths
Category:German literary critics
Category:Sibling duos
Category:German male non-fiction writers
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Q:
Calculating area of sphere with constraint on zenith
Let $\mathbb S_R$ be the sphere of radius $R$ centered about the origin. Consider $$A_R= \left\{ (x,y,z)\in \mathbb S_R \mid x^2+y^2+(z-R)^2\leq 1 \right\}.$$
I want to calculate the area of this region of the sphere with radius $R$ about the origin. I already calculated the area of the region of $\mathbb S_R$ of vectors with zenith $\leq \alpha$ is given by $2\pi R^2(1-\cos \alpha)$.
For $A_R$, here's my idea. I want to find the zenith $\alpha$ which satisfies $R\sin \alpha=\sin \alpha+R$, since this is the zenith of points of both $\mathbb S_R$ and the unit sphere translated up $R$ units at points which are of the same height. Then, I just want to integrate the zenith $0\leq \phi\leq \alpha$.
Solving gives $\phi=\arcsin \frac R{R-1}$ and things get a little messy.
On the other hand, another student posted a solution which makes sense. He just writes $\alpha$ should satisfy $2R^2-2R^2\cos\alpha=1$ and then obtains the area of $A_R$ is $\pi$, independently of the radius.
Why is my method wrong?
Picture
A:
The problem seems to be what you mean by
"the zenith of points of both $\mathbb S_R$ and the unit sphere translated up $R$ units at points which are of the same height."
The quantity $R \sin \alpha$ is the distance from the $z$ axis of points whose spherical coordinates are $(R, \theta, \alpha)$ for any $\theta,$
that is, the points on the sphere $\mathbb S_R$ that are at a zenith angle of $\alpha$ relative to the center of that sphere.
So far, this makes sense.
The quantity $\sin \alpha$ is the distance from the $z$ axis of points on the unit sphere about the origin at a zenith angle of $\alpha.$
This does not change when you translate that sphere upward $R$ units,
so it's unclear why you would add $R.$
In fact, since $0 < \sin\alpha \leq 1$ whenver $0 < \alpha < \pi,$
for any such $\alpha$ we would find that
$R \sin\alpha \leq R < R + \sin\alpha,$
so it is impossible to have $R \sin\alpha = R + \sin\alpha.$
And that equation is just as impossible for $\alpha = 0$ or $\alpha = \pi$
as long as $R > 0.$
In short, the equation you proposed just does not make sense.
Now there may be some sense in equating the heights ($z$ coordinates)
of points at the intersection of $\mathbb S_R$ and the unit sphere around $(0,0,R).$ For that, we use the cosine of the azimuth angle relative to each sphere, and we note that the azimuth angle of these points relative to the sphere around $(0,0,R)$ is greater than their azimuth angle relative to $\mathbb S_R.$
There is a relationship between the two angles, but it's not what your equation says.
As for the claim that $2R^2-2R^2\cos\alpha=1,$ which the other student presumably did not explain, let $P$ be a point where the two spheres intersect. Then the points $(0,0),$ $(0,R),$ and $P$ form an isoceles triangle with two legs of length $R$ (each a radius of $\mathbb S_R$) and a base of $1$ (the segment from $(0,R)$ to $P,$ which is a radius of the unit sphere around $(0,R).$)
If we drop a perpendicular from either $P$ or $(0,R)$ to the opposite leg, we divide the isoceles triangle into two right triangles, one with hypotenuse $R$ and angle $\alpha$ and one with hypotenuse $1$ and angle $\frac\alpha2.$
If we drop a perpendicular from $(0,0)$ to the base of the isoceles triangle, we get two congruent right triangles with hypotenuse $R$ and angle $\frac\alpha2,$ with leg $\frac12$ opposite the angle $\frac\alpha2.$
In either case we end up having to deal with a right triangle with angle $\frac\alpha2.$
Let's try working with the second pair of right triangles.
The construction tells us that $R \sin\frac\alpha2 = \frac12.$
Square both sides and multiply by $2$:
$$2R^2 \sin^2\frac\alpha2 = 2\left(\frac14\right) = \frac12.$$
Use the trigonometry identity
$2 \sin^2\frac\alpha2 = 1 - \cos\alpha$
(which you can get from either the half-angle sine formula or the double-angle cosine formula)
to substitute $1 -\cos\alpha$ for $2 \sin^2\frac\alpha2$:
$$R^2(1 - \cos\alpha) = \frac12.$$
You can then distribute the $R^2$ over $1-\cos\alpha$ and multiply both sides by $2$ to get the other student's formula if you want, although I think the formula above is more convenient for just plugging into your known area formula.
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arcana75 wrote:I too find EWs terrible. Other than roleplaying, why bother investing in a weapon skill that becomes semi-good in LA, when I can invest in nearly any other skill and be useful throughout the game? I fought the Dugan robots using all ballistic weapons and didn't really have an issue, no one went down, and the same weapon skills took me through AZ.
I almost feel like the addition of Precision Strikes was a bad idea. I enjoy them and I love how much more tactical they make combat on SJ (it actually becomes pretty important to cripple legs and especially arms since there are many times where you're going to have to let them shoot back), but at the same time, it precludes things like, for example, instead of "conductive vs non-conductive", all energy weapons have a chance of simply shredding enemy armor with every successful hit, or anything like that.
I love the concept behind energy weapons, I just think it's awfully hard to find a way to implement it where it won't be either underpowered or overpowered. If EWs ever do get fixed, Gamma Ray Blaster is going to need some massive nerfs, though. That thing should not be nearly as good as it is.
I'll whine a little more about difficulty: why did you remove the whole fight against Leather Jerks in Rodia? Player is led to believe he/she's going to overthrow and kick Leather Jerks out of this nice town, enlists the help of locals & weakens minibosses (for no less than 5-10 hours of game time!) and in the end gets a 5 minute fight with 4 guys in a living room. All the scary Jerks (that you can even "lock up" to delay their vengeance!) instantly disappear without a single shot, or even a strongly worded retort.
C'mon. Again, you can even lock some of them up, and they... disappear through the chimney? Fade away into thin air? MELT? Surprising that the villain's castle doesn't crumble into dust =)
Nearly agree altho my energy weapons are nearly holding their own in combat, having two different energy weapons in the rangers hands gets two shots off, one higher costing AP and a lower AP for a second shot in a turn (high Dam + low Damage as opposed to one high Dam shot or two low Dam shots per turn), im at California so plenty of bots and cyborgs with conductive armour. My beef is the unique energy weapons, the one from Silo 7 and Tinkers weapons are completely and utterly RUBBISH! crap damage, the standard energy weapons are better than the unique weapons which shudnt be the case at all.
PizzaSHARK wrote:I feel like there need to be a LOT more conductive enemies across the board. Energy weapons are incredibly weak in Arizona because even the toughest guys you run across aren't wearing conductive armor, even on SJ... with the obvious exception of the robots, but aside from Damonta, they're pretty much entirely optional. If I'm fighting a dude in Highpool and he's got 5 armor and we're talking like lvl 3-6 characters... man, that dude's clearly wearing Goat Hide Armor (and probably has Hardened or some other passive giving him free armor), and that shit's conductive. Are we supposed to believe a bunch of pissant raiders can afford tier 5 light armor that isn't even available until you get to California? Not even the random roaming vendor sells better than, what, +3 light armor?
My thoughts exactly. Even at the start of California, you'll meet enemies with armor values of 8-9 that aren't conductive. How is that possible? Unless of course everyone is wearing Spectrum Assault Vests and have Hardened perk, which would still only come out to 8. Maybe they all have the Thick-Skinned quirk?
Sorry for resurrecting this old topic, but I'm currently finishing my first play-through of WL2:DC and Dugan and his robots are definitely not conductive. Their armor does display a blue shield with a lightning bolt across it, however there is no damage multiplier added when I take the shot. Another issue is that I cannot use Computer Science on the two Worker Robots that fight with Dugan, and neither on the synths in the final battle, too. Is this intended behavior?
I like Pizepi's combat performance more than Scotchmo's (char I used to roll with before). This is mostly because of her superior combat stats but also because energy weapon as a weapon class is much more flexible than shotgun class. (has longish range option, multiple low precision shots per turn with energy pistol option or heavy burst fire vs conductive armor using the gamma ray - only endgame weapon available in Arizona ..).
It is impossible to precisely measure/compare the amount of damage she inflicted so far since the game forces me to dismiss her every time I get a (1 time) chance to recruit new personnel and resets her total damage accumulator stat. Pizepi + energy weapons offers in most fights way more options that the Scotchmo + shotgun combo.
I think even a 7 man full energy party play-trough wouldn't be significantly to tedious even if there is only 1 gamma ray in the game ..
Energy weapons do seem redundant, unless you are running low on 5.56mm or 7.62mm ammo. The same can be said for the other ranged weapons that don't use rifle calibers. I kept some energy weapons for my my heavy weapons guy though, because he was constantly needing more ammo for a while.
Jozape wrote:Energy weapons do seem redundant, unless you are running low on 5.56mm or 7.62mm ammo. The same can be said for the other ranged weapons that don't use rifle calibers. I kept some energy weapons for my my heavy weapons guy though, because he was constantly needing more ammo for a while.
I think there's some issue on how much damage they do in the endgame on conductive opponents, when/if that gets fixed it might change things. At any rate, Final Assessment with Pizepi is a great debuffer, and debuffers are vital on SJ.
crimsoncorporation wrote:I think there's some issue on how much damage they do in the endgame on conductive opponents, when/if that gets fixed it might change things. At any rate, Final Assessment with Pizepi is a great debuffer, and debuffers are vital on SJ.
I didn't notice energy weapons being any less or more useful at the end of the game, but I played at the default difficulty which is one step from the easiest setting. At this setting energy weapons are under powered but thankfully not useless.
After a play through on Ranger difficulty putting points into Energy Weapons was by far one of the biggest wastes of skill points I made throughout the game.
I even got more value from the points in handguns I gave my snipers.
The whole conductive / non-conductive armor came across as an unrealistic and poorly implemented mechanic that mostly served to prevent me from using heavy armor on my team with zero value vs. enemies.
The only thing that they were good for was when I was struggling with ammo needs in the mid game and it let me use a different ammo type on one party member.
Playing on Ranger - just made it to Hollywood. I'm finding them pretty useful actually. Put Energy Weapons on my skill guy with Two-Pump Chump, and he's very reliably being able to throw 3 or 4 precision strikes out at the start of combat to cripple limbs and strip armor. I like energy weapons much more than handguns for debuffing because they have deeper clips, no deadzone, and the later ones have really decent range.
Fighting Conductive armor is a nice when it happens but he's really there to cripple limbs and finish enemies.
If you are any sort of good player you will already have your regular weapons AND energy weapons. That way when you do fight enemies that energy weapons can affect, you can quickly kill them with no sweat. Sure, 99% of the game you won't need them, but you'll level up a few weapon skills anyway so level up energy so when you do reach California you can blast down robots, etc super easy. What are you going to pick instead, pistols? SMGS? Heavy weapons? haha yea right. Get an assault/energy gunner and a sniper/energy gunner, and everyone else with the same, and your set. There isn't any 2ndary weapon that's useful. If your melee sharp/blunt your weapons can pen their armor excellent anyway you won't need energy. unarmed level 10, already know that. So with guns, there aren't many choices in the game. Assault and sniper are the only useful ones, and energy weapons the 3rd most useful. Any smart person leaves SMGS, pistols, heavy weapons left on the doorstep for the trash people.
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Q:
How does Sodium Valproate cause neural plasticity
I have been reading a fascinating paper: Valproate reopens critical-period learning of absolute pitch
18 individuals were given Sodium Valproate (VPA) for a fortnight during which they trained on a pitch-training game. Results suggest that VPA reopens the plasticity window that normally closes by adolescence.
However, the paper seems to suggest that the exact mechanism of action is unknown.
Valproic acid is believed to have multiple pharmacological actions,
including acute blockade of GABA transaminase to enhance inhibitory
function in epileptic seizures and enduring effects on gene
transcription as an histone deacetlyase (HDAC) inhibitor (Monti et
al., 2009). Of relevance here is the epigenetic actions of this drug,
as enhancing inhibition does not reactivate brain plasticity in
adulthood (Fagiolini and Hensch, 2000), but reopening chromatin
structure does (Putignano et al., 2007). While systemic drug
application is a rather coarse treatment, the effects may differ
dramatically by individual cell type (TK Hensch and P Carninci,
unpublished observations). VPA treatment mimics Nogo receptor deletion
to reopen plasticity for acoustic preference in mice (Yang et al.,
2012), suggesting a common pathway through the regulation of
myelin-related signaling which normally closes critical period
plasticity (McGee et al., 2005). Future work will address the cellular
actions of VPA treatment in the process of reactivating critical
periods. Future MRI studies will also be needed to establish whether
HDAC inhibition by VPA induces hyperconnectivity of myelinated,
long-range connections concurrent with renewed AP ability (Loui et
al., 2011).
So it is saying that the standard use of VPA is to increase GABA levels (which keeps firing rate down -- it is used as an antiepileptic), however it also acts as an HDAC inhibitor, which means it causes unwrapping of chromatin and consequently increased mRNA transcription, maybe even transcription of genes that would normally be entirely deactivated in an adult.
So my guess is that some protein is getting produced that messages neurons to generate new axon/dendrite growth and/or new synaptic connections.
Can anyone clarify how VPA might accomplish plasticity?
EDIT (one month later): I have more detail, but I still can't quite make the connection. Here goes:
Neurites get wrapped by myelin/oligodendrocyte, which produces and exudes some of the chemical messagers {Nogo, OMgp, MAG}. The membrane surface of the neurite contains nogo 66 receptors (NgR-s) that get triggered by these messagers and inform the neuron to inhibit axon-growth. Somehow the 'HDAC inhibition' property of VPA is unwinding DNA enough to alter transcription rates of certain proteins, and one of these must be disabling the NgR.
But how is this happening?
A:
As far as I can see this paper is being a little misleading, by saying "VPA mimics Nogo-66 receptor deletion".
The action of VPA doesn't seem to be related to this receptor.
It seems that blocking this receptor and applying VPA both increase plasticity, but it is like taking a car or taking a train -- entirely different modes of transport that achieve the same effect.
VPA seems to facilitate LTP through increased availability of relevant proteins.
That is unconnected with growing new neural structure.
The principal problem with growing new structure is, as hinted at the end of the question, that the adolescent/adult brain secretes a chemical that gets picked up by Nogo 66 receptor, which signals to collapse the axon growth cone.
It's why adult humans can't recover from spinal injuries. The axons won't reconnect.
It so happens that a small molecule Nogo Antagonist has recently been developed by Professor Strittmatter. He was kind to reply to my query, and I learned that this molecule is currently in the early stages of FDA approval. It's a very exciting discovery!
I would caution anyone against taking VPA -- I have been encountering chest pains since experimenting with it (even though I dropped the experiment after a week due to pain). It looks as though this pain is inflammation of the stomach lining and a common reaction to VPA (VPA is an acid!).
If someone is determined to take VPA, they should at least investigate taking it together with Omeprazole, which discourages the stomach from producing excess acid.
I am disappointed that I now need to take omeprazole quazi-regularly, it appears that my brief VPA experiment has caused some long-term upset in my chemical balance.
Also of note is that (again discovered by Strittmatter) Ibuprofen has been found to also interrupt the signalling pathway that leads to axon growth collapse. But it requires a high dosage, and ibuprofen also causes a similar imbalance.
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---
abstract: 'We construct integral bases for the $SO(3)$-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus $3$ and $p=5,$ we still give an explicit basis.'
address:
- |
Department of Mathematics\
Louisiana State University\
Baton Rouge, LA 70803\
USA
- |
Institut de Mathématiques de Jussieu (UMR 7586 du CNRS)\
Université Paris 7 (Denis Diderot)\
Case 7012\
2, place Jussieu\
75251 Paris Cedex 05\
FRANCE
- |
Department of Mathematics\
Louisiana State University\
Baton Rouge, LA 70803\
USA
author:
- 'Patrick M. Gilmer'
- Gregor Masbaum
- Paul van Wamelen
date: 'July 8, 2002'
title: Integral bases for TQFT Modules and unimodular representations of mapping class groups
---
Introduction
============
Integrality properties of Witten-Reshetikhin-Turaev quantum invariants of $3$-manifolds have been studied intensively in the last several years. H. Murakami [@Mu1; @Mu2] showed that the $SU(2)$- and $SO(3)$-invariants at a root of unity $q$ of prime order are algebraic integers. This was reproved in [@MR] and generalised to all classical Lie types in [@MW; @TY] and then to all Lie types in [@Le]. These integrality properties are crucial for establishing the relationship of the invariants with the Casson invariant [@Mu1; @Mu2] and with the perturbative invariants or Ohtsuki series [@OhCambridge; @Oh1; @Le].
Quantum invariants fit into Topological Quantum Field Theories (TQFT). This means in particular that there are representations of mapping class groups associated with them. (Actually the representations are usually only projective-linear; equivalently, one has to consider certain central extensions of mapping class groups here.) If a $3$-manifold $M$ is presented as a Heegaard splitting where two handlebodies are glued together by a diffeomorphism $\varphi$ along their boundary, the quantum invariant of $M$ can be recovered from the representation of $\varphi$ on the TQFT-vector space $V({{\Sigma}})$ associated to the boundary surface ${{\Sigma}}$.
The TQFT-representations are finite-dimensional and can be defined over a finite extension of the cyclotomic number field $\mathbb{Q}(q)$, where the quantum parameter $q$ is a root of unity. They also preserve a non-degenerate Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on $V({{\Sigma}})$ (which may or may not be unitary; this usually depends on the choice of the embedding of the cyclotomic field into $\mathbb{C}$).
A quite striking result was recently announced by Andersen [@An] who proved that in the $SU(n)$ case the representations are asymptotically faithful (here asymptotically means letting the order of $q$ go to infinity). At a fixed root of unity, they are certainly not faithful, as Dehn twists are always represented by matrices of finite order. Roberts [@R] showed that the representations are irreducible in the $SU(2)$-case if the order of $q$ is prime.
An interesting question is to determine the image of the mapping class group in the TQFT-representations. For the $SU(2)$ and $SO(3)$-theories, the first author proved that in the genus one case the image is a finite group [@G1]. However in higher genus, the image is not finite [@Fu]; in fact, it contains elements of infinite order [@Madeira]. One might hope that this image is equal to the linear transformations which are automorphisms of some (yet to be found) structure, just as a linear transformation of the homology group $H_1({{\Sigma}};\mathbb{Z})$ is represented by a mapping class if and only if it preserves the intersection form.
In this paper we are concerned with integrality properties of the TQFT representations. For simplicity, we restrict ourselves to the $SO(3)$ case; specifically, we use a variant of the $V_p$-theories of [@BHMV2] with $p$ an odd prime. Here, $p$ is the order of the root of unity $q$. Let ${{\mathcal{O}}}$ denote the ring of algebraic integers in the cyclotomic ground field. The main idea to obtain an integral structure on the TQFT already appears in [@G]. Namely, we define an ${{\mathcal{O}}}$-submodule ${{\mathcal{S}_p}}({{\Sigma}})$ of the TQFT-vector space ${{V_p}}({{\Sigma}})$ as the ${{\mathcal{O}}}$-span of vectors represented by connected $3$-manifolds with boundary ${{\Sigma}}$. The point of this definition is that the submodule ${{\mathcal{S}_p}}({{\Sigma}})$ is clearly preserved under the mapping class group.
It was shown in [@G] that ${{\mathcal{S}_p}}({{\Sigma}})$ is always a free finitely generated ${{\mathcal{O}}}$-module. One can also rescale the Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ to obtain a non-degenerate ${{\mathcal{O}}}$-valued form $(\ ,\ )_{{\Sigma}}$ on ${{\mathcal{S}_p}}({{\Sigma}})$. This relies on the integrality results for the invariants mentioned above.
The form $(\ ,\ )_{{\Sigma}}$ is again preserved by the mapping class group. In particular, the image of the mapping class group in the TQFT-representation $V_p({{\Sigma}})$ lies in the subgroup preserving a lattice defined over ${{\mathcal{O}}}$ and a non-degenerate Hermitian form on it.
In what sense is ${{\mathcal{S}_p}}$ a TQFT defined over ${{\mathcal{O}}}$? For instance one might hope that the form $(\ ,\ )_{{\Sigma}}$ was unimodular. Here, we show that this is indeed the case in genus one and two. This is a consequence of our main result which is to describe explicit bases of ${{\mathcal{S}_p}}({{\Sigma}})$ in genus one and two.
In fact, we will describe two quite different bases in genus one. The first basis is given in Theorem \[firstbasis\]. It is $$\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$$ where $\omega$ is the element appearing in the surgery axiom of the ${{V_p}}$-theory and $t$ is the twist map. It is easy to see that these elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$ and we use a Vandermonde matrix argument to show that they form a basis. A crucial step is to show that the Hermitian form $(\ ,\ )_{S^1\times S^1}$ is unimodular with respect to this basis.
The second basis given in Theorem \[secondbasis\] is of a quite different nature. It is $$\{ 1, v ,v^2 , \ldots , v^{d-1}\}$$ where $v=(z+2)/(1+A)$ (here $z$ is represented by the core of the solid torus). We call it the $v$-basis. This time, it is not even obvious [*a priori*]{} that its elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$. The proof involves two different arguments. One is to show that the ${{\mathcal{O}}}$-span of the $v$-basis is stable under the twist map. This is shown in Section \[Kauf\]. In fact, we prove it in the more general context where the skein variable $A$ is an indeterminate rather than a root of unity. The second ingredient is to express $\omega$ in the $v$-basis and thereby relate the $v$-basis to the first basis. This is done in Section \[2nd\].
The $v$-basis lends itself nicely to finding bases in higher genus. In Section \[g2\], we describe a basis of ${{\mathcal{S}_p}}(\Sigma)$ in genus two consisting of $v$-colored links in a genus two handlebody. These links are described by arrangements of curves in a twice punctured disk. Again, the unimodularity of the Hermitian form with respect to this basis is a crucial step in the argument.
In principle this method can be used to study ${{\mathcal{S}_p}}(\Sigma)$ in higher genus as well. It turns out, however, that the Hermitian form $(\ ,\ )_{{{\Sigma}}}$ is not always unimodular. For example, a simple argument given in Section \[non-uni\] shows that it cannot be unimodular for surfaces of genus 3 and 5, assuming $p \equiv 5 \pmod{8}.$
In this paper we will not attempt to deal with the higher genus case in general. We only give in Section \[g3\] a basis of ${{\mathcal{S}_p}}(\Sigma)$ for a surface of genus three when $p=5.$ Although in this case the Hermitian form is not unimodular, it is nearly so. This allows us to find a basis easily in this one case.
Note that our definition of ${{\mathcal{S}_p}}({{\Sigma}})$ is analogous to the construction of integral modular categories in [@MW]; in both cases one constructs integral structures by considering the span, over the subring of algebraic integers of the coefficient field, of the morphisms of the geometrically defined category (tangles in the case of [@MW], $3$-dimensional cobordisms in the case at hand). It might be that this is not always enough: It is conceivable that one might be able to enlarge ${{\mathcal{S}_p}}({{\Sigma}})$ in some way to make the form always unimodular; however this enlargement would not be generated by $3$-cobordisms anymore.
We conclude the paper by showing how the ${{\mathcal{S}_p}}$-theory defined over ${{\mathcal{O}}}$ can be used to prove a divisibility result for the Kauffman bracket of links in $S^3$. This generalizes a result of Cochran and Melvin [@CM] for zero framed links (see also [@OhCambridge; @KS]).
[*Notational conventions.*]{} Throughout the paper, $p\geq 3$ will be an odd integer, and we put $d=(p-1)/2$. From Section \[elem\] onwards, $p$ is supposed to be prime.
The twist map on the Kauffman Bracket module of a solid torus {#Kauf}
=============================================================
In this section we define a sequence of submodules $K(n)$ of the Kauffman Bracket skein module of the solid torus $S^1 \times D^2$ and show that they are preserved under the twist map. We use the notations of [@BHMV1].
Suppose $R$ is a commutative ring with identity and an invertible element $A.$ The universal example is $R={{\mathbb{Z}}}[A,A^{-1}]$ which we also denote by ${{\mathbb{Z}}}[A^{\pm}]$. Recall that the Kauffman bracket skein module $K(M,R)$ of a $3$-manifold $M$ is the free $R$ module generated by isotopy classes of banded links in $M$ modulo the submodule generated by the Kauffman relations.
We let $z$ denote the skein element of $K(S^1 \times D^2,R)$ given by the banded link $S^1\times J,$ where $J$ is a small arc in the interior of $D^2.$
As is well known, $K(S^1 \times D^2,R)$ is a free $R$-module on the nonnegative powers of $z,$ where $z^n$ means $n$ parallel copies of $z$. This also makes $K(S^1 \times D^2,R)$ into an $R$-algebra isomorphic to the polynomial ring $ R [z].$
Let $t:K (S^1 \times D^2,R) \rightarrow K (S^1 \times D^2,R)$ denote the twist map induced by a full right handed twist on the solid torus. It is well known (see [*e.g.*]{} [@BHMV1]) that there is a basis $ \{e_i\}_{i\ge0}$ of eigenvectors for the twist map. It is defined recursively by $$\label{ei}
e_0=1, \quad e_1=z, \quad e_i= z e_{i-1}- e_{i-2} ~.$$ The eigenvalues are given by $$\label{mui}t(e_i)=\mu_i e_i, \text{ where } \mu_i=(-1)^i A^{i^2+2i} ~.$$
[*Let ${K}(n)$ denote the ${{\mathbb{Z}}}[A^{\pm}]$-submodule of $K (S^1 \times D^2, {{\mathbb{Z}}}[A^{\pm},\frac 1 {1+A}])$ generated by $\{1,v,v^2,\ldots,v^n\}$, where $$v= \frac {z+2}{1+A}~.$$* ]{}
\[tK\] The twist map $t$ sends $K(n)$ to itself.
Consider the basis $ \{ (z+2)^i\}_{i\ge0}$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]).$ The following Lemma gives the change of basis formulas.
\[changeofbasis\] For each $n \ge 1,$ $$(z+2)^{n-1}= \sum_{k=1}^n \binom{2n}{n-k} \frac k n e_{k-1}$$ $$e_{n-1}= \sum_{i=1}^n (-1)^{n-i} \binom{n+i-1}{n-i} (z+2)^{i-1}$$
Prove each separately by induction on $n$ using the recursion formula (\[ei\]).
[*It follows that $\binom{2n}{n-k} \frac k n \in {{\mathbb{Z}}},$ which can also be seen directly:*]{} $$\binom{2n}{n-k} \frac k n=
\binom{2n}{n-k} (1-\frac {n-k} n) =
\binom{2n}{n-k}- 2\binom{2n-1}{n-k-1}~.$$
It is enough to show Theorem \[tK\] for the endomorphism $-At$ in place of $t$. Let us compute $-At$ in the basis $(z+2)^n.$ Note that $-At(e_{i-1})=(-A)^{i^2} e_{i-1}$. $$\begin{aligned}
-At\left((z+2)^{n-1}\right) & = \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A) t(e_{k-1}), \\
&= \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A)^{k^2} \sum_{i=1}^k (-1)^{k-i} \binom {k+i-1}{k-i} (z+2)^{i-1},\\
&= \sum_{i=1}^n (-1)^i
\left( \frac 1 n
\sum_{k=i}^n k \binom {2n}{n-k} \binom {k+i-1}{k-i} A^{k^2} \right)(z+2)^{i-1},\\
&= \sum_{i=1}^n (-1)^{i} S_{1,i,n}(A) (z+2)^{i-1}, \end{aligned}$$
Here, for $ m \ge 1,$ we define $$S_{m,i,n}(A)= \frac 1 n \sum_{k=i}^n k^m \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2}\in {{\mathbb{Z}}}[A].$$
\[at -1\] $$S_{1,i,n}(-1) =
\begin{cases}
(-1)^n, &i=n\\
0 &i \ne n
\end{cases}$$
If we put $A=-1,$ then all $\mu_i=1$ and hence $-At$ is the identity.
The following formula is a very special case of a transformation formula for terminating hypergeometric series due to Bailey [@B Formula 4.3.1]. This was pointed out to us by Krattenthaler’s [HYP]{} package, [@hyp].
\[recurs\] $$S_{m,i,n}= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}$$
$$\begin{aligned}
S_{m,i,n}
&=\frac 1 n \sum_{k=i}^n k^{m-2} \left( (k+i) (k-i) +i^2 \right) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\
&= i^2 S_{m-2,i,n} + \frac 1 n \sum_{k=i+1}^n k^{m-2} (k+i) (k-i) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\
& \text{ (the term with $k=i$ is zero) }\\
&= i^2 S_{m-2,i,n} + \frac 1 n 2 i (2i+1)\sum_{k=i+1}^n k^{m-2} \binom {2n} {n-k} \binom {k+i}{k-i-1} A^{k^2} \\
&= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}\end{aligned}$$
Here we use the simple identity: $$(k+i)(k-i) \binom {k+i-1}{k-i} = 2i (2i+1) \binom {k+i}{k-i-1}$$
\[eq2.7\] \[divis\] $S_{1,i,n}(A)$ is divisible by $(1+A)^{n-i}$ in ${{\mathbb{Z}}}[A]$ for $i \le n.$
It suffices to show: $$\label{eq2}
\left[ \left( \frac {d} {dA} \right)
^{k}
S_{1,i,n}(A) \right]_{A=-1} =0$$ for all $k=0,1,\ldots , n-i-1$. Note that $\frac {d} {dA} S_{m,i,n}= A^{-1} S_{m+2,i,n} \in {{\mathbb{Z}}}[A].$ Thus
$$\left( \frac {d} {dA} \right)
^k
S_{1,i,n} \in \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,|\, m \text{ odd}, \, 1 \le m \le 2n-2i-1\}$$ for all $k$ in the required range. But using Lemma \[recurs\] one may decrease $m$ at the cost of increasing $i,$ and see that $$\text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,| \, m \text{ odd},\, 1 \le m \le 2n-2i-1\} \subseteq \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{1,j,n} \,|\, j < n\}.$$
By Lemma \[at -1\], $S_{1,j,n}(-1) =0,$ for $ j < n,$ and (\[eq2\]) follows.
[*Proof of Theorem \[tK\].*]{} We have $$t(v^{n-1}) = - A^{-1} \sum_{i=1}^n s_{i,n}(A) v^{i-1},$$ where $s_{i,n}(A)= (-1)^i (1+A)^{i-n}S_{1,i,n}(A)$ lies in ${{\mathbb{Z}}}[A]$ by Proposition \[eq2.7\].
[*Theorem \[tK\] remains valid if we take $A$ to be a root of unity, other than $-1,$ rather than an indeterminant.* ]{}
[*Let $\tilde {K}(n)$ be defined as $K(n)$ but with $v=({z+2})/({1+A})$ replaced with $\tilde v= ({z+2})/({1-A^2})$. Then a similar argument shows that $\tilde {K}(n)$ is stable under $t^2$, the square of the twist map. (To see this, one should replace $-At$ with $A^2 t^2$ in the above and express everything in terms of $q=A^2$. This leads to polynomials $\tilde S_{m,i,n}(q)$ defined similarly as the $ S_{m,i,n}(A)$ except that $A$ is replaced with $q$ and an extra factor of $(-1)^k$ is inserted in the sum. The remainder of the argument is the same.)* ]{}
The $SO(3)$-TQFTs
=================
Let $p\geq 3$ be an odd integer. (In this section, $p$ need not be prime.) We consider a variation of the $2+1$ dimensional cobordism category considered in [@BHMV2] whose objects are closed oriented surfaces (with extra structure) with a (possibly empty) collection of banded points ($=$ small oriented arcs) colored by integers in the range $[0,p-2].$ The morphisms are (equivalence classes of) oriented 3-dimensional manifolds (with extra structure) with $p$-admissibly colored banded trivalent graphs. (Two morphisms are considered equivalent if they are related by a homeomorphism respecting the boundary identifications.) For the definition of $p$-admissibility in the $p$-odd case see [@BHMV2 Theorem 1.15]; see also Section \[g2\].
The variation consists of replacing the $p_1$-structures of [@BHMV2] with structures put forward by Walker [@W] and Turaev [@Tu]. Surfaces are equipped with a Lagrangian subspace of their first homology. We use homology with rational coefficients when considering Lagrangian subspaces. Cobordisms are equipped with integer weights, as well as Lagrangian subspaces for the target and source. This is also described in [@G]. We will denote this category by $\mathcal{C}.$ We call the objects of this category e-surfaces, and call the morphisms 3-e-manifolds.
The procedure of [@BHMV2] defines a TQFT-functor ${{V_p}}$ on $\mathcal{C}$ over a commutative ring $R$ containing $p^{-1}$, a primitive 2pth root of unity $A$ and a solution of $\kappa^2= A^{ -6- p(p+1)/2 }.$ The number $\kappa$ here plays the role of $\kappa^3$ in [@BHMV2]. Here we use the term TQFT slightly loosely as the tensor product axiom does not hold unless only even colors are used in the cobordism category. The even colors correspond to irreducible representations of $SU(2)$ which lift to $SO(3)$. Therefore the ${{V_p}}$-theory for odd $p$ is considered a $SO(3)$ variant of the Witten-Reshetikhin-Turaev $SU(2)$-TQFT.
For us it is convenient to use odd colors as well as even colors. However, if we insist that only even colors be used in coloring the banded points on the surfaces, then we do obtain an honest TQFT with the tensor product axiom, but we are still allowed us to use the language of odd colors to describe states. This will be useful in Sections \[g2\] and \[g3\].
If $M$ is a 3-e-manifold viewed as morphism from ${{\Sigma}}$ to ${{\Sigma}}'$ in $\mathcal{C}$, we denote the associated endomorphism from ${{V_p}}({{\Sigma}})$ to ${{V_p}}({{\Sigma}}')$ by ${{Z_p}}(M)$. (It is denoted by $(Z_p)_M$ in [@BHMV2]). If $M$ is a closed 3-e-manifold viewed as morphism from $\emptyset$ to $\emptyset,$ ${{Z_p}}(M)$ induces multiplication by a scalar from $R= V(\emptyset).$ This scalar is denoted by $\langle M\rangle.$ If $M$ is a 3-e-manifold viewed as morphism from $\emptyset$ to $\Sigma,$ let $[M]$ denote ${{Z_p}}(M)(1) \in V(\Sigma).$ ($[M]$ is denoted by $Z_p(M)$ in [@BHMV2]). We call such an element $[M]$ a [*vacuum state.*]{} If $M$ is connected, $[M]$ is called a connected vacuum state.
The modules ${{V_p}}({{\Sigma}})$ are always free over $R$. They also carry a nonsingular Hermitian form [@BHMV2] : $$\langle \ ,\ \rangle _{\Sigma}: {{V_p}}(\Sigma) \times {{V_p}}(\Sigma) \rightarrow R$$ given by $$\langle [N_1],[N_2]\rangle _{\Sigma}= \langle N_1 \cup_{\Sigma} -N_2\rangle~.$$ Here $-N_2$ is the 3-e-manifold obtained by reversing the orientation, multiplying the weight by $-1, $ and leaving the Lagrangian on the boundary alone.
If $\Sigma$ is an e-surface with no colored points, and $H$ is a handlebody (weighted zero) with boundary $\Sigma,$ then ${{V_p}}(\Sigma)$ has a specified isomorphism to a quotient of the skein module $K(H,R)$ [@BHMV2 p. 891]. In fact if $H$ is a subset of $S^3$ then two skeins represent the same element if and only if they are equal as “maps of outsides” in Lickorish’s phrase [@L].
Let $S^1 \times S^1$ denote an e-surface of genus one with no colored points. Let $d$ denote $(p-1)/ 2.$ It turns out that $d$ is the dimension or rank of ${{V_p}}(S^1 \times S^1).$ In fact, the module ${{V_p}}(S^1 \times S^1) $ is isomorphic as an $R$-module to the quotient of $K(S^1 \times D^2,R)=R[z]$ by the ideal generated by $e_d- e_{d-1}\in R[z].$ It follows [@BHMV1 p.696] that $e_{p-1}=0$ in ${{V_p}}(S^1 \times S^1) $ and $e_{d+i}= e_{d-1-i}.$ Thus the module ${{V_p}}(S^1 \times S^1) $ has indeed rank $d$ with the basis $\{e_0,e_1,\ldots ,e_{d-1}\}$. Note that this basis is the same, up to reordering, as the even basis $\{e_0,e_2,\ldots, e_{p-3}\}.$
Let $\Sigma_g$ denote an e-surface of genus $g$ with no colored points on the boundary. The rank of the free module ${{V_p}}(\Sigma_g)$ is given by the formula [@BHMV2 Cor. 1.16] $$\text{rank}\left({{V_p}}(\Sigma_g)\right) = \left( \frac p 4\right)^{g-1} \sum_{j=1}^{d}
\left( \sin \frac {2 \pi j}{p} \right)^{2-2g} .$$ This is the same as $2^{-g}$ times the dimension of $V_{2p}({{\Sigma}}_g)$ (this fact comes from a tensor product formula, see [@BHMV2 Thm. 1.5]). Note that $V_{2p}({{\Sigma}}_g)$ is an $SU(2)$-TQFT module, with dimension given by the $SU(2)$ Verlinde formula at level $p-2$ (where the colors are again the set of integers in the range $[0,p-2]$).
In genus 2, we have $$\text{rank}\left({{V_p}}(\Sigma_2)\right)= \frac {d (d+1)(2d+1)}{6}$$ as will be seen by an explicit counting argument in Section \[g2\].
Some facts from elementary number theory {#elem}
========================================
In the remainder of this paper, we assume $p$ is an odd prime. We continue to use the notation $d=(p-1)/2$.
In this section, we collect some notation and a few elementary number-theoretical facts. To be specific we pick particular values for $A$ and $\kappa$. We put $A= \zeta_{2p}$ where $\zeta_n = e^{2 \pi i/{n}},$ and also use the notation[^1] $$q=A^2.$$ We may then take $\kappa = A^{-3}(-i)^{({p+1})/{2}}.$ Note that $\zeta_{2p}\in {{\mathbb{Z}}}[\zeta_p]$. Thus the coefficient ring is $R ={{\mathbb{Z}}}[\zeta_p, \frac 1 p]$ if $p \equiv -1 \pmod{4},$ and $R ={{\mathbb{Z}}}[\zeta_{p},i, \frac 1 p]={{\mathbb{Z}}}[\zeta_{4p}, \frac 1 p]$ if $p \equiv 1 \pmod{4}.$ Of course, the coefficient ring remains unchanged if $A$ is replaced by another primitive $2p$-th root of unity, and $\kappa$ is changed accordingly.
We let $\eta$ denote $\langle S^3\rangle$, the invariant of $S^3$ with weight zero, and put $\mathcal{D}=
\eta^{-1}$. Then using equations on [@BHMV2 p.897] $$\label{D}
\mathcal{D} = \frac{i\ \sqrt{p} }{q-q^{-1}} = \frac{i^{\frac {p+1} 2} }{q-q^{-1}}
\left(\frac 1 2 \sum _{m=1}^{2p}(-1)^m A^{m^2}\right)$$ In particular $$\label{DD} \mathcal{D}^2= \frac {-p}{ ({q-q^{-1}})^2}~.$$
We denote by $\mathcal{O} $ the ring of integers in $R.$ Note that $\mathcal{O}={{\mathbb{Z}}}[\zeta_p]$ if $p \equiv -1 \pmod{4},$ and $\mathcal{O} ={{\mathbb{Z}}}[\zeta_{p},i]={{\mathbb{Z}}}[\zeta_{4p}]$ if $p \equiv 1 \pmod{4}.$
The following notation will be useful. If $x,y$ are elements of ${{\mathcal{O}}}$ (or, more generally, of its quotient field), we write $x\sim y$ if there exists a unit $u\in {{\mathcal{O}}}$ such that $x=uy$.
\[num\]
- $1-A$ is a unit in ${{\mathcal{O}}}$, and $1-q\sim 1+A$.
- One has $\mathcal{D}\in {{\mathcal{O}}}$. Moreover, $\mathcal{D}\sim (1-q)^{({p-3})/2}=(1-q)^{d-1}$.
- The quantum integers $[n]=(q^n-q^{-n})/(q-q^{-1})$ are units for $1\leq n\leq p-1$.
- If $0\leq i,j \leq d-1$ and $i\neq j$, then the twist coefficients $\mu_i$ (see (\[mui\])) satisfy $\mu_i-\mu_j\sim 1-q$.
- Put $\lambda_i=-q^{i+1}-q^{-i-1}$. If $0\leq i \leq d-1$, then $\lambda_0-\lambda_i\sim (1-q)^2$.
The fact that $1-A$ is a unit follows easily from the fact that $A$ is a zero of the $2p$-th cyclotomic polynomial $1-X+X^2-\ldots +X^{p-1}$. This proves (i). It is well-known that $p\sim (1-q)^{p-1}$ (see [*e.g.*]{} [@MR Lemma 3.1]). Together with Formulas (\[D\]) and (\[DD\]), this shows (ii). Observing that $[n]\sim
1+q^2+\ldots +q^{2n-2}$, (iii) is also shown in [@MR Lemma 3.1]. For (iv), observe that $\mu_i=\mu_{p-2-i}$ so that the set of $\mu_i$ in question is equal to the set of $\mu_{2i}=q^{
2 i^2+2i}$ for $i=0,1,\ldots d-1$. These powers of $q$ are all distinct, which implies $$\mu_i-\mu_j \sim 1-q^n \sim 1-q$$ for some $0<n<p$. This proves (iv). Similarly, (v) follows from $\lambda_0-\lambda_i\sim(1-q^{i+2})(1-q^i)$.
[*It is well-known that $1-q$ is a prime in ${{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$. But if $p \equiv 1 \pmod{4},$ then $1-q$ is not a prime in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ (it splits as a product of two conjugate prime ideals).* ]{}
Associated Integral Cobordism Functors
======================================
In [@G], a cobordism functor from a restricted cobordism category to the category of free finitely generated $\mathcal{O}$-modules is described. Let $\mathcal{C''}$ denote the subcategory of $\mathcal{C}$ defined by considering only nonempty connected surfaces and connected morphisms between such surfaces. This represents a further restriction of $\mathcal{C}$ than that considered in [@G], but it suffices for our purposes.
[*If $\Sigma$ is a connected e-surface, define ${{\mathcal{S}_p}}(\Sigma)$ to be the $\mathcal{O}$-submodule of ${{V_p}}(\Sigma)$ generated by connected vacuum states. If $N: \Sigma \rightarrow \Sigma'$ is a morphism of $\mathcal{C''}$ then ${{Z_p}}(N)$ sends $ [M] \in {{\mathcal{S}_p}}(\Sigma)$ to $[M \cup_{\Sigma}N] \in {{\mathcal{S}_p}}(\Sigma').$ In this way we get a functor from $\mathcal{C''}$ to the category of finitely generated $\mathcal{O}$-modules. We also rescale the Hermitian form on ${{V_p}}(\Sigma)$ to obtain an $\mathcal{O}$-valued Hermitian form $$(\ ,\ )_{\Sigma}: {{\mathcal{S}_p}}(\Sigma)
\otimes_{{\mathcal{O}}}{{\mathcal{S}_p}}(\Sigma) \rightarrow \mathcal{O},$$ defined by $$([N_1],[N_2])_{\Sigma}=
\mathcal{D} \langle [N_1],[N_2]\rangle_{\Sigma} =
\mathcal{D} \langle N_1 \cup_{\Sigma} -N_2\rangle
.$$* ]{}**
This form takes values in $\mathcal{O}$ by the integrality result for closed 3-e-manifolds [@Mu2; @MR]. These theorems are also used in proving that ${{\mathcal{S}_p}}(\Sigma)$ is finitely generated [@G].
[*Over a Dedekind domain such as ${{\mathcal{O}}}$, a finitely generated torsion-free module is always projective, but it need not be free. (The typical examples are non-principal ideals in ${{\mathcal{O}}}$.) Somewhat surprisingly, however, it turns out that the modules ${{\mathcal{S}_p}}(\Sigma)$ are always free. This is proved in [@G]. We will not actually make use of this fact in genus $1$ and $2$: freeness will follow from the construction of explicit bases.* ]{}
[*A Hermitian form on a projective ${{\mathcal{O}}}$-module $S$ is called [*non-degenerate*]{} (or [*non-singular*]{}) if its adjoint map $S\rightarrow S^*$ is injective. It is called [*unimodular*]{} if the adjoint map is an isomorphism.* ]{}
Note that if $S$ is free and $M$ is the matrix of the Hermitian form in some basis, then the form is non-degenerate (resp. unimodular) if $\det M$ is non-zero (resp. a unit in ${{\mathcal{O}}}$).
In our situation, the form $ (\ ,\ )_{\Sigma}$ is always non-degenerate (since the original form $\langle \ ,\ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ is). We will show that $ (\ ,\ )_{\Sigma}$ is unimodular in genus $1$ and $2$. There is a standard basis $\{u_\sigma\}$ of ${{V_p}}(\Sigma_g)$ given by p-admissible [ *even*]{} colorings $\sigma$ of the graph
(-.5,-.2)(5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5) (1.35,.5)(1.65,.5) (3.35,.5)(3.65,.5) (4,.5)[.35]{} (2,.45)[$\cdots$]{}
(where there are $g$ loops) embedded in a 3-e-handlebody $H_g$ of genus $g$ with boundary the e-surface ${{\Sigma}}_g$ (see [@BHMV2 4.11]). One may actually use any trivalent graph in $H_g$ to which $H_g$ deformation retracts. (In the case $g=1$, this is the same as the basis given by the elements $e_i$.) These basis elements lie in ${{\mathcal{S}_p}}(\Sigma_g)$ because the denominators appearing in the Jones-Wenzl idempotents needed to expand colored graphs into skein elements are invertible in ${{\mathcal{O}}}$ (see [@MR]). Warning: the $u_\sigma$ do [*not*]{} generate ${{\mathcal{S}_p}}(\Sigma_g)$ over ${{\mathcal{O}}}$.
\[detb\] The elements $u_\sigma$ are orthogonal for the form $(\ ,\ )_{\Sigma_g}.$ Moreover, one has $$(u_\sigma,u_\sigma)_{\Sigma_g}\sim {{\mathcal{D}}}^g\sim (1-q)^{(d-1)g}.$$
By [@BHMV2 Theorem 4.11] one has that $\langle u_\sigma,u_\sigma\rangle_{\Sigma_g}$ is equal to $\eta^{1-g}={{\mathcal{D}}}^{g-1}$ times a product of non-zero quantum integers or their inverses, which are units in ${{\mathbb{Z}}}[q]$ by Lemma \[num\]. Since the form $(\ ,\ )_{\Sigma_g} $ is just a rescaling of the form $\langle \ ,\ \rangle_{\Sigma_g} $, the result follows.
One of the reasons to study the form $(\ ,\ )_{\Sigma_g} $ is that it is preserved by the TQFT-action of the mapping class group. More precisely, let $\tilde{\Gamma}(\Sigma)$ denote the central extension of the mapping class group $\Gamma(\Sigma)$ of $\Sigma$ realized by the subcategory of $C''$ consisting of e-manifolds homeomorphic to $\Sigma \times I$ such that the colored graph is given by $I$ times the colored banded points of $\Sigma.$ This homeomorphism need not respect the boundary identification at $\Sigma \times \{1\},$ but should respect the boundary identification at $\Sigma \times \{0\}.$ In fact considering this boundary identification at $\Sigma \times \{1\},$ defines the quotient homomorphism from $\tilde{\Gamma}(\Sigma)$ to $\Gamma(\Sigma),$ which has kernel ${{\mathbb{Z}}}$ given by the integral weights on $\Sigma \times I$ with standard boundary identifications. The group $\tilde \Gamma(\Sigma)$ is isomorphic to the signature extension (see [*e.g.*]{} Atiyah [@At], Turaev [@Tu].) This extension can be described nicely using skein theory [@MR1].
The group $\tilde \Gamma(\Sigma)$ acts on ${{V_p}}(\Sigma)$ preserving the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma)$ and the ${{\mathcal{O}}}$-valued Hermitian form $ (\ ,\ )_{\Sigma}$.
This follows from the definition of ${{\mathcal{S}_p}}(\Sigma)$ and the fact that the group $\tilde \Gamma(\Sigma)$ preserves the original Hermitian form $ \langle\ ,\ \rangle_{\Sigma}$.
The module ${{\mathcal{S}_p}}(\Sigma)$ can be described using the notion of ‘mixed graph’. Recall the element $$\omega= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle e_i \in K(S^1 \times D^2,R).$$ Here $\langle e_i\rangle=(-1)^i [i+1]$. It plays an important role in the surgery axioms of the ${{V_p}}$-theory.
By a [*mixed graph*]{} in a weighted 3-manifold $M, $ we mean a trivalent banded graph in $M$ whose simple closed curve components may possibly be colored $\omega$ or by integer colors in the range $[ 0,p-2]$ and whose other edges are colored p-admissibly by integers in the range $[ 0,p-2]$. A mixed graph can be expanded multilinearly into a $R$-linear combination of colored graphs. The result should be thought of as a superposition of e-morphisms. If the graph is a link and every component is colored $\omega,$ we say the link is $\omega$-colored. A mixed graph in a handlebody $H$ specifies an element in ${{V_p}}(\partial H).$
\[ml\] A mixed graph in a connected 3-e-manifold $M$ with boundary $\Sigma$ represents an element of ${{\mathcal{S}_p}}(\Sigma).$ If $H$ is a 3-e-handlebody with boundary e-surface $\Sigma$ then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements specified by mixed graphs in $H.$
The first statement follows from the fact that ${{V_p}}$ satisfies the surgery axiom (S2) [@BHMV2 p 889]. The second statement follows from the fact that any connected 3-manifold with boundary $\Sigma$ can be obtained by a sequence of 2-surgeries to $H$ [@BHMV2 Proof of Lemma p. 891].
\[5.7\] [*Suppose that we know that some collection $T$ of elements of ${{V_p}}(\Sigma)$ lie in the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma).$ Then $\text{Span}_\mathcal{O}(T)$ is a $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma).$ This sublattice might not be invariant under $\tilde{\Gamma}(\Sigma)$. Let $G=\{g_i\} \in \tilde \Gamma(\Sigma)$ be a finite set of elements whose image in $\Gamma(\Sigma)$ generate. The sequence of submodules of ${{\mathcal{S}_p}}(\Sigma)$: $\text{Span}_\mathcal{O}(T),$ $\text{Span}_\mathcal{O}(T\cup G(T)),$ $\text{Span}_\mathcal{O}(T\cup G(T) \cup G(G(T))),$ , $ \ldots $ etc. must stabilize in an $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma)$ which is invariant under the mapping class group. This procedure is well suited to computer investigation. The basis given in Section \[2nd\] was originally found by this procedure. We used the computer program Kant [@D] starting with $T= \{e_0,e_1,\ldots,
e_{d-1}\}$ in ${{\mathcal{S}_p}}(S^1\times S^1).$* ]{}
First integral basis in genus 1 {#1st}
===============================
By a slight abuse of notation, we let $\omega$ denote the element in ${{\mathcal{S}_p}}(S^1 \times S^1)$ given by coloring the core of $S^1 \times D^2$ with $\omega.$ Let $t$ also denote the induced map on ${{V_p}}(S^1 \times S^1)$ given by giving $S^1 \times D^2$ a full right handed twist. Note that $t^n(\omega)\in {{\mathcal{S}_p}}(S^1 \times S^1)$ for all $n$.
\[firstbasis\] $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is a basis for the module ${{\mathcal{S}_p}}(S^1 \times S^1).$ The form $( \ ,\ )_{S^1 \times S^1}$ is unimodular.
Note that it follows in particular that the ${{\mathcal{O}}}$-span of $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is stable under the action of the mapping class group $\tilde{\Gamma}(
S^1 \times S^1)
$.
Recall that $\mu_i=(-1)^{i} A^{i^2+2i}$ denotes the eigenvalue of $e_i$ under the twist map $t$. We have that $$t^j (\omega)= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle \mu_i^j e_i~.$$ Note that $\langle e_i\rangle=(-1)^i [i+1]$ is a unit by Lemma \[num\](iii). The matrix $W$ which expresses $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$ has as determinant a unit (the product of the $\langle e_i\rangle$) times ${{\mathcal{D}}}^{-d}$ times the determinant of the Vandermonde matrix $[ \mu_i^j ]$ where $0 \le i, j\le d-1.$ Moreover by Lemma \[num\](iv) $$\det [ \mu_i^j ] = \pm \prod_{i<j} (\mu_i-\mu_j)\sim (1-q)^{{d(d-1)}/2}~.$$
As ${{\mathcal{D}}}\sim (1-q)^{d-1},$ we conclude that $$\label{detW} \det W\sim (1-q)^{- {d(d-1)}/2}~.$$
In particular, this determinant is non-zero, hence the $t^j(\omega)$ are linearly independent. Let $\mathcal {W}$ denote the $\mathcal{O}$-module spanned by the $t^j(\omega)$. Clearly $\mathcal W \subset {{\mathcal{S}_p}}(S^1 \times S^1)$. Now we know by Proposition \[detb\] that $(e_i,e_i)\sim (1-q)^{d-1}$ (here we simply write $(\ ,\ )$ for the Hermitian form $(\ ,\ )_{S^1 \times S^1} $). Therefore the matrix for $(\ ,\ )$ with respect to the orthogonal basis $\{e_0,e_1,\ldots, e_{d-1}\}$ has determinant $(1-q)^{d(d-1)}$. By (\[detW\]) it follows that the matrix for $(\ ,\ )$ with respect to $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ has unit determinant. (Here we use that $\overline{1-q}=1-q^{-1}\sim 1-q$.) In other words, the form $(\ ,\ )$ restricted to $\mathcal {W}$ is unimodular. But then $\mathcal {W}$ must be equal to ${{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof.
\[wc\] If $H$ is a 3-e-handlebody with boundary the e-surface of $\Sigma$ and $\Sigma$ has no colored points in the boundary, then $
{{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $\omega$-colored banded links in $H.$
By the above theorem, each $e_i$ (in particular $e_1=z$) can be expressed as an $\mathcal{O}$-linear combinations of the elements $t^j(\omega)$. Therefore every mixed graph can be written as an $\mathcal{O}$-linear combination of $\omega$-colored banded links in $H.$ The result now follows from Theorem \[ml\].
Second integral basis in genus 1 {#2nd}
================================
Consider $K(d-1)$ in the notation of Section \[Kauf\], now taking $A= \zeta_{2p}.$ Let $\mathcal{V}$ denote its image in ${{V_p}}(S^1 \times S^1).$ In other words $\mathcal{V}$ is the $\mathcal{O}$-submodule of ${{V_p}}(S^1 \times S^1)$ generated by $\{ 1, v ,v^2 , \ldots , v^{d-1}\},$ where $v=(z+2)/(1+A)$.
\[secondbasis\] One has $\mathcal{V}={{\mathcal{S}_p}}(S^1 \times S^1).$ In particular, $\{ 1, v ,v^2 , \ldots , v^{d-1}\}$ is a basis for the free module ${{\mathcal{S}_p}}(S^1 \times S^1).$
We refer to this basis as the $v$-basis of ${{\mathcal{S}_p}}(S^1 \times S^1).$ We originally found it by the procedure outlined in Remark \[5.7\]. Since $\tilde{\Gamma}(\Sigma)$ preserves ${{\mathcal{S}_p}}(S^1 \times S^1)$, we have the following Corollary.
$\mathcal{V}=\text{\em Span}_{{\mathcal{O}}}\{1,v,v^2,\ldots,v^{d-1}\}$ is stable under the action of the mapping class group $\tilde{\Gamma}(S^1 \times S^1)$.
[*The mapping class group $\tilde{\Gamma}(S^1 \times S^1)$ is a central extension of $SL(2,{{\mathbb{Z}}})$. Its image in $GL({{V_p}}(S^1 \times S^1))$ is generated by $\kappa $ times the identity matrix (the central generator acts as multiplication by $\kappa$), the twist map $t$, and the so-called $S$-matrix. The entries of the $S$-matrix in the $e_i$-basis are well-known. One can therefore write down its entries in the $v$-basis (using the change of basis formulas in Lemma \[changeofbasis\]). The fact that these entries lie in ${{\mathcal{O}}}$ is by no means obvious. We originally proved this fact using some identities involving binomial coefficients. The argument is similar to the proof that the $v$-basis is stable under the twist map $t$ given in Section \[Kauf\], but considerably more complicated. We found proofs of these identities using Zeilberger’s algorithm together with some identities from [@B] as above. In particular the [Gosper]{} command in the Mathematica package “Fast Zeilberger” (V 2.61) by Peter Paule and Markus Schorn, [@Zeil] was used. As the proof we give below is much simpler, we omit the details of this computation.* ]{}
One has $\omega \in {{\mathcal{V}}}~.$
Let $\lambda_i= -q^{i+1}- q^{-i-1}$. Recall [@BHMV1] that $e_i$ is an eigenvector with eigenvalue $\lambda_i$ for the endomorphism $c$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]) $ given by sending a skein in $S^1 \times D^2$ to the skein circled by a meridian.
Let $\langle\ ,\ \rangle_H$ be the Hopf pairing ([*i.e.*]{} the symmetric bilinear form on ${{V_p}}(S^1 \times S^1)$ which sends two elements $x,y$ to the bracket of the zero-framed Hopf link with one component cabled by $x$, and the other component cabled by $y$). Then $$\label{omip} \langle \omega,e_i\rangle_H = \begin{cases}
\langle\omega\rangle={{\mathcal{D}}}, &\text{if } i=0\\
\ \ \ 0 &\text{if } 1\leq i\leq d-1
\end{cases}$$ Note that $\langle z-\lambda_i,e_i\rangle_H=0$ for $1\leq i\leq d-1$, and $\langle z-\lambda_i,e_0\rangle_H=\lambda_0-\lambda_i$ (since $\langle z,e_0\rangle = \langle z\rangle =-q-q^{-1}=\lambda_0$). It follows that $$\label{omi} \omega={{\mathcal{D}}}\prod_{i=1}^{d-1} \frac {z-\lambda_i} {\lambda_0-\lambda_i}$$ since the pairing $\langle\ ,\ \rangle_H$ is non-degenerate. (Note the similarity with the polynomials $Q_n$ of [@BHMV1].)
Since ${{\mathcal{D}}}\sim (1-q)^{d-1}$ and $\lambda_0-\lambda_i\sim (1-q)^2$ by Lemma \[num\], it follows that $$\label{omi2} \omega \sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1-q}\sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1+A}$$ (where $\sim$ means equality up to multiplication by a unit). Now $$\begin{aligned}
z-\lambda_i &= (z+2)-(2 +\lambda_i)\\
&= (z+2)-(1-q^{i+1})(1-q^{-i-1})\\
&= (z+2)+ u_i (1+A)^2\end{aligned}$$ where $u_i \in {{\mathcal{O}}}.$ It follows that $$(z-\lambda_i)/(1+A) \in \text{Span}_{{\mathcal{O}}}\{1,v\}~,$$ and so (\[omi2\]) implies $\omega \in {{\mathcal{V}}}~,$ proving the lemma.
By Theorem \[tK\], $K(n)$ hence ${{\mathcal{V}}}$ is stable under the twist map $t$. It follows that $$\mathcal{W}=\text{Span}_{{\mathcal{O}}}\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\}
\subseteq \mathcal{V}~.$$ Now recall from the proof of Theorem \[firstbasis\] that the matrix $W$ which expresses $\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots,e_{d-1}\}$ has determinant $\det W \sim
(1-q)^{-d(d-1)/2}$. Remembering $v=(z+2)/(1+A)$ and $1+A\sim 1-q$, it is easy to see that the same is true for the matrix which expresses $\{ 1,v,\ldots,v^{d-1}\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$. Since $\mathcal{W}\subset\mathcal{V}$, it follows that actually $\mathcal{W}= \mathcal{V}$. By Theorem \[firstbasis\] we conclude $\mathcal {V}={{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof.
By a $v$-colored banded link in a 3-manifold, we mean a banded link whose components are colored $v.$ As before this should be interpreted as the linear combination (superposition) of the colored banded links that one obtains by expanding multilinearly. We note that $i$ parallel strands colored $v$ is the same as one strand colored $v^i.$
\[v\] If $H$ is a 3-e-handlebody with boundary the e-surface $\Sigma$ and $\Sigma$ has no colored points in the boundary, then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $v$-colored banded links in $H.$
*The matrix of the Hermitian form $(\ ,\ )_{S^1\times S^1}$ in the $v$-basis is easily computed. One has for $0\leq i,j\leq d-1$ $$\begin{aligned}
(v^i,v^j)_{S^1\times S^1}&={{\mathcal{D}}}\langle v^i,v^j\rangle_{S^1\times S^1} = \langle v^{i+j},\omega \rangle_H
={(1+A)^{-(i+j)}} \langle (z+2)^{i+j},\omega \rangle_H\\
&=\frac {1} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1} \langle e_0,\omega \rangle_H\\
&=\frac {{{\mathcal{D}}}} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1}\end{aligned}$$ Here we have used Lemma \[changeofbasis\] to express $(z+2)^{i+j}$ in terms of the $e_n$, and then retained only the $e_0$ term. Indeed, the others are annihilated by the Hopf pairing with $\omega$ since $0\leq i+j\leq 2d-2$ (see (\[omip\]) and remember that $e_{d+i}=e_{d-1-i}$ in ${{V_p}}(S^1\times S^1)$).*
It is instructive to check directly that the expression above lies in ${{\mathcal{O}}}$ (use that $p$ divides the binomial coefficient $\binom{2i+2j+2} {i+j}$ if $d\leq i+j\leq 2d-2$).
Integral basis in genus 2 {#g2}
=========================
Let ${{\Sigma}}_2$ be a closed surface of genus $2$. In this section, we describe a basis for the module ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and show that the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular.
Let $H_2$ be a regular neighborhood of the hand cuff graph ${
\psset{unit=.5cm}
\begin{pspicture}[.4](-.5,0)(1.5,1)
\pscircle(0,.5){.35}
\pscircle(1,.5){.35}
\psline(.35,.5)(.65,.5)
\end{pspicture}
}$ in $\mathbb{R}^3$. Then $H_2$ is a genus $2$ handlebody and by Corollary \[v\], ${{\mathcal{S}_p}}({{\Sigma}}_2)$ is spanned by $v$-colored banded links in $H_2$. We think of $H_2$ as $P_2 \times I$ where $P_2$ is a disk with two holes.
The skein module $K(H_2,R)$ is free on the set of isotopy classes of collections of nonintersecting essential simple closed curves in $P_2$. We refer to these isotopy classes as arrangements of curves. Such arrangements can be indexed by 3-tuples of nonnegative integers. Let $C_{\alpha,\beta,\gamma}$ denote the arrangement with $\gamma$ parallel curves going around both holes, and within them $\alpha$ parallel curves going around the left hole, and $\beta$ parallel curves going around the right hole. See Figure \[fig0\] for an example.
(-.6,-.8)(2.6,.8) (0,0)[.35]{}(2,0)[.35]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.45]{}(2,0)[.45]{} (2,0)[.55]{} (0,0)[.7]{}[45]{}[315]{}(2,0)[.7]{}[225]{}[135]{} (.5,.49)(.6,.43)(.8,.4)(1.2,.4)(1.4,.43)(1.5,.49) (.5,-.49)(.6,-.43)(.8,-.4)(1.2,-.4)(1.4,-.43)(1.5,-.49)
\[thg2\] Let $C_{\alpha,\beta,\gamma}(v)$ be the element of ${{\mathcal{S}_p}}({{\Sigma}}_2)$ obtained by coloring each curve of $C_{\alpha,\beta,\gamma}$ by $v=(z+2)/(1+A)$. Then the set $$\{ C_{\alpha,\beta,\gamma}(v)\,|\, 0\le \gamma \le d-1, \quad
0\le \alpha,\beta \le d-1-\gamma \}$$ is a basis of ${{\mathcal{S}_p}}({{\Sigma}}_2).$ Moreover, the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular.
Note that $C_{\alpha,\beta,\gamma}(v)$ lies in ${{\mathcal{S}_p}}({{\Sigma}}_2)$ because $v$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ by Theorem \[secondbasis\].
Let us first describe a basis of ${{V_p}}({{\Sigma}}_2)$ consisting of elements represented by colorings of the hand cuff graph
(-.5,0)(1.5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5)
. Let $G(i,j,k)$ be the element defined by the colored graph
(-.5,0)(2,1) (0,.5)[.35]{} (1.5,.5)[.35]{} (.35,.5)(1.15,.5) (-.7,.5)[$i$]{} (2,.5)[$j$]{} (.6,.7)[$k$]{}
For this element to exist, $k$ must be even. Then the coloring is $p$-admissible if and only if $\frac k 2 \leq i,j\leq p-2-\frac k 2$ (see [@BHMV2 Thm 1.15]). The standard basis of ${{V_p}}({{\Sigma}}_2)$ would be to take the $p$-admissible $G(i,j,k)$ with both $i$ and $j$ even. It is also possible to impose that one or both of $i,j$ be odd [@BHMV2 Thm 4.14]. We will need a different basis where $i,j$ are allowed to be both even and odd, but $\leq d-1$. This is given in the following Lemma.
\[lemG\] The $G(i,j,k)$ with $k$ even in the range $[0,p-3]$, and both $i$ and $j$ in the range $[\frac k 2, d-1]$ (but not necessarily even), form a basis of ${{V_p}}({{\Sigma}}_2)$.
[ *Let $ \mathcal{G}$ be the basis described in the above Lemma. Let $ \mathcal{G}_k$ be the subset of elements of $ \mathcal{G}$ with middle arc colored $k$. The cardinality of $ \mathcal{G}_k$ is $(d-\frac k 2)^2.$ Thus we see directly that the cardinality of this basis is $ \sum_{j=0}^{d-1} (d-j)^2= \sum_{j=1}^{d} j^2= d (d+1)(2d+1)/6 .$* ]{}
Lemma \[lemG\] could be proved using the methods of [@BHMV2]. Here we give a different, more direct proof.
For $i$ in the range $[0,p-2]$ we let $i'=p-2-i$. We claim that $$G(i,j,k)\sim G(i',j,k)\sim G(i,j',k)\sim G(i',j',k)$$ (where $\sim$ means equality up to multiplication by a unit in ${{\mathcal{O}}}$). It is enough to prove that $G(i,j,k)\sim G(i',j,k).$ This is done in Figure \[fig1\]. Note that if $i$ is even and $> d-1$ then $i'$ is odd and $\leq d-1$. Thus the basis of where all $i,j$ are even may be replaced by the basis of Lemma \[lemG\].
(-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{}
$=$
(-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{} (-.6,.25)[$\tilde p$]{} (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{}
\
$ = c_1$
(-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{}[270]{}[90]{} (.35,0)[$\tilde p$]{} (0,0)[1.5]{}[270]{}[90]{} (0,0)[1.25]{}[118]{}[242]{} (0,1.5)(-.3,1.4)(-.6,1.1) (0,1)(-.3,1.02)(-.6,1.1) (0,-1.5)(-.3,-1.4)(-.6,-1.1) (0,-1)(-.3,-1.02)(-.6,-1.1) (1.5,0)(3,0) (-1.8,.1)[$i'$]{} (2,.3)[$k$]{} (1.1,1.4)[$i$]{} (1.3,-1.3)[$i$]{}
$ = c_1 c_2$
(-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i'$]{} (2,.3)[$k$]{}
Let $ \mathcal{A}(v)$ be the set of the $v$-colored elements $C_{\alpha, \beta,\gamma}(v)$ claimed to be a basis in Theorem \[thg2\], and let $ \mathcal{A}$ be the set of the uncolored ([*i.e.*]{} colored by $z=e_1$) elements $C_{\alpha, \beta,\gamma}$ (in the same range for $\alpha, \beta,\gamma$).
The set $ \mathcal{A}$ is a basis of ${{V_p}}(\Sigma_2)$. Moreover, the basis change from $ \mathcal{G}$ to $\mathcal{A}$ has determinant $\pm 1$.
Using the Wenzl recursion formula for the idempotents of the Temperley-Lieb algebra, one can expand the elements of $\mathcal{A}$ as $\mathcal{O}$-linear combinations of elements of the graph basis $\mathcal{G}.$ In fact, in the expansion of $C_{\alpha,\beta,\gamma}$, only those $G(i,j,k)$ occur where $i\leq \alpha +\gamma$, $j\leq \beta +\gamma$, and $k\leq 2\gamma$; moreover, $G(\alpha +\gamma,\beta +\gamma,
2\gamma)$ occurs with coefficient one. We can find orderings of $\mathcal{A}$ and $\mathcal{G}$ so that the matrix which expresses $\mathcal{A}$ in terms of $\mathcal{G}$ is triangular with ones on the diagonal (use the lexicographical orderings where $\gamma$ resp. $k$ is counted first). This implies the Lemma.
Let $ r=d (d+1)(2d+1)/{6}$ be the rank of ${{V_p}}(\Sigma_2)$. By Proposition \[detb\], the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to the orthogonal basis $\mathcal{G}$ has determinant $\sim (1-q)^{2 (d-1) r}$. The preceding Lemma shows that the same holds true for the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}$.
Let $N$ denote the sum over $ \mathcal{A}$ of the number of curves appearing in each arrangement. The change of basis matrix for writing $\mathcal{A}(v)$ in terms of $\mathcal{A}$ is again triangular and has determinant $\sim (1-q)^{-N}.$ Thus the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}(v)$ has determinant $\sim (1-q)^{2 (d-1) r -2N}$. The following Lemma \[lemf\] shows that this determinant is a unit. As in the genus one case (see the proof of Theorem \[firstbasis\]), we conclude that $\mathcal{A}(v)$ is a basis for ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and that the form $(\ ,\ )_{\Sigma_2} $ is unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_2)$.
\[lemf\] $N=(d-1)r$.
To count $N, $ we write ${{\mathcal{A}}}= \cup_{ 0\le \gamma \le d-1} {{\mathcal{A}}}_\gamma,$ where $${{\mathcal{A}}}_\gamma= \{ C_{\alpha,\beta,\gamma} | 0 \le \alpha,\beta \le d-\gamma-1 \}~.$$
Note that $|{{\mathcal{A}}}_\gamma|= (d-\gamma)^2.$ The total number of curves appearing in ${{\mathcal{A}}}_\gamma$ is
$$\gamma (d-\gamma)^2 + \sum_{\alpha=0}^{d-\gamma-1}\sum_{\beta=0}^{d-\gamma-1} (\alpha +\beta) = (d-1) (d-\gamma)^2~.$$ Thus each ${{\mathcal{A}}}_\gamma$ contributes $d-1$ times its cardinality to the count. As $\sum_{\gamma =0}^{d-1} |{{\mathcal{A}}}_\gamma| =r,$ we see that $N=(d-1) r.$
This completes the proof of Theorem \[thg2\].
Non-Unimodularity {#non-uni}
==================
Even without knowing an explicit basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$, it is possible to see that the form $(\ ,\ )_{{{\Sigma}}_g}$ is sometimes not unimodular.
\[9.1\] If $p \equiv 1 \pmod{4}$ and both the genus $g$ and the rank of ${{V_p}}({{\Sigma}}_g)$ are odd, then the form $(\ ,\ )_{{{\Sigma}}_g}$ is not unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_g)$.
For example, if $g=3$ and $p=5$ then the rank is $15$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular on $\mathcal{S}_5({{\Sigma}}_3)$.
[*We used Mathematica [@Wo] to calculate the rank of $V_{p}(\Sigma_g)$ for small $g$ using the formula [@BHMV2 1.16(ii)]. We found that: $$\begin{aligned}
\text{rank} \left( V_{4k+1}(\Sigma_3)\right) &=
(1 /{45} )(3 \ k+32 \ k^2+120 \ k^3+200 \ k^4+192 \ k^5+128 \ k^6)
\\
\begin{split}
\text{rank} \left(V_{4k+1}(\Sigma_5)\right) &=
( 1/ {14175} ) (45 \ k + 864 \ k^2 + 6892 \ k^3 + 30184 \ k^4\\
& +
83760 \ k^5 + 172512 \ k^6 + 304896 \ k^7 + 458112 \ k^8\\
& + 542720 \ k^9 + 487424 \ k^{10} + 294912 \ k^{11} + 98304 \ k^{12})
\end{split} \end{aligned}$$ Thus the rank of $V_p({{\Sigma}}_3)$ is odd if $p \equiv 5 \pmod{8},$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular in this case. Similarly $(\ ,\ )_{{{\Sigma}}_5}$ is not unimodular if $p \equiv 5 \pmod{8}. $* ]{}
The argument relies on the following result of [@G]. Assume $p \equiv 1 \pmod{4}$ and recall that ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ in this case. Put ${{\mathcal{O}}}^+={{\mathbb{Z}}}[\zeta_{p}]\subset {{\mathcal{O}}}$. Let the Lagrangian assigned to $\Sigma_g$ be the kernel of the map on the first homology induced by the inclusion of $ \Sigma_g$ to $H_g$ and assign $H_g$ the weight zero. Then $H_g$ is an [ *even*]{} (in the sense of [@G]) morphism from $\emptyset$ to $\Sigma_g.$ Note that the quantum integers $[n]$ for $1\le n \le p-1$ are units in ${{\mathcal{O}}}^+.$
[@G] If $p \equiv 1 \pmod{4}$ then ${{\mathcal{S}_p}}({{\Sigma}}_g)\simeq {{\mathcal{S}^+_p}}({{\Sigma}}_g)\otimes {{\mathcal{O}}}$ where ${{\mathcal{S}^+_p}}({{\Sigma}}_g)\subset {{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}^+$-module. Moreover, one has $\mathcal {G}\subset{{\mathcal{S}^+_p}}({{\Sigma}}_g)$, where $\mathcal {G}$ is the colored graph basis of ${{V_p}}({{\Sigma}}_g)$ (see Proposition \[detb\]).
The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to $\mathcal {G}$ has determinant ${{\mathcal{D}}}^{gr}\sim (1-q)^{(d-1)gr}$ where $r$ denotes the rank of ${{V_p}}({{\Sigma}}_g)$. Let $\mathcal {B}$ be a basis of the free ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$, and let $D$ be the determinant of the matrix expressing $\mathcal {B}$ in terms of $\mathcal {G}$. The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to the basis $\mathcal {B}$ has determinant $\sim \Delta$, where $$\label{Delta}
\Delta = D \overline D
(1-q)^{(d-1)gr}~.$$ If the form is unimodular, $\Delta$ must be a unit in ${{\mathcal{O}}}$, and since $\Delta$ lies in ${{\mathcal{O}}}^+$, it must be a unit in ${{\mathcal{O}}}^+$. But $1-q$ is a self-conjugate prime in ${{\mathcal{O}}}^+={{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$, and since $D^{-1}$ lies in ${{\mathcal{O}}}^+$ as well, $\Delta$ can be a unit only if $(d-1)gr$ is even. Thus one of $g$ and $r$ must be even (since $d-1=(p-3)/2$ is odd in our situation). This completes the proof.
\[9.4\] [*The use of the ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$ can in general not be avoided in this argument. Here is why. Recall that $1-q$ splits in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ as the product of two conjugate prime ideals $\mathfrak{p}$ and $\overline{\mathfrak{p}}$. If $\mathfrak p$ is principal (this happens for example if $p=5$), then there exists $D\in {{\mathcal{O}}}$ such that the number $\Delta$ defined as in (\[Delta\]) is a unit even when $(d-1)gr$ is odd. Of course, such a $D$ does not exist in ${{\mathcal{O}}}^+$.*]{}
[*If we assign extra structure to ${{\Sigma}}_g$ and $H_g$ as described above in the proof of 9.1, then ${{\mathcal{O}}}^+$ linear combinations of banded links in $H$ represent elements in ${{\mathcal{S}^+_p}}({{\Sigma}}).$ Moreover the bases described in Sections \[1st\], \[2nd\], \[g2\] for ${{\mathcal{S}_p}}(S^1 \times S^1), $ and ${{\mathcal{S}_p}}({{\Sigma}}_2)$ are actually bases for ${{\mathcal{S}^+_p}}(S^1 \times S^1), $ and ${{\mathcal{S}^+_p}}({{\Sigma}}_2).$ There are also plus versions of Theorem \[ml\] and Corollaries \[wc\] and \[v\].*]{}
[*When restricted to ${{\mathcal{S}^+_p}}(\Sigma_g)$, the Hermitian form $(\ ,\ )_{\Sigma_g}$ does not take values in ${{\mathcal{O}}}^+,$ if $g$ is odd. This follows from the proof of Proposition \[detb\], since ${{\mathcal{D}}}\not\in {{\mathcal{O}}}^+$. In the next section, we will use the sesquilinear form $$(\ ,\ )^+_{\Sigma_g}: {{\mathcal{S}^+_p}}(\Sigma_g)
\times
{{\mathcal{S}^+_p}}(\Sigma_g) \rightarrow \mathcal{O}^+$$ obtained by multiplying the form $(\ ,\ )_{\Sigma_g}$ by $i^{\varepsilon (g)},$ where ${\varepsilon (g)}$ is zero or one accordingly as $g$ is even, or odd. This form takes values in ${{\mathcal{O}}}^+$ since $i {{\mathcal{D}}}\in {{\mathcal{O}}}^+$.* ]{}
Genus three at the prime five {#g3}
=============================
In genus $g\geq 3$, one can also try to find a set of banded links in a handlebody so that one obtains a basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$ by cabling each curve component with $v=(z+2)/(1+A)$. This is suggested by Corollary \[v\] and the fact that ${{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}$-module [@G]. In fact, we now find such a set of links giving a basis for $\mathcal{S}^+_5({{\Sigma}}_3)$ (and therefore also for $\mathcal{S}_5({{\Sigma}}_3)$) by adapting the above procedures. These links are described by arrangements of curves in a thrice punctured disk. Although the Hermitian form and the related ${{\mathcal{O}}}^+$-valued sesquilinear form are not unimodular, in this particular situation they are nearly so, and this is essential for our argument. It seems more difficult to find an explicit collection of banded links with this property for $\mathcal{S}_p({{\Sigma}}_3)$ for $p>5,$ and for $\mathcal{S}_p({{\Sigma}}_g)$ for $g>4.$ We plan to return to this question elsewhere.
We think of the handlebody $H_3$ as $P_3 \times I$ where $P_3$ is a disk with three holes. We give $H_3$ weight zero. We equip ${{\Sigma}}_3$ with the Lagrangian given by the kernel of the map induced on the first homology by the inclusion of ${{\Sigma}}_3$ in $P_3 \times I.$
Consider the set of 15 arrangements of curves in $P_3$ $$\begin{aligned}
{{\mathcal{A}}}=\{A_\emptyset,&A_1,A_2,A_3, A_1A_2,A_2A_3,A_3A_1,
A_1A_2A_3,\\
&A_{12},A_{23},A_{13}, A_{12}A_3, A_{23}A_1, A_{31}A_2, A_{123}\}~.\end{aligned}$$ Here, $A_\emptyset$ is the empty arrangement, $A_i$ (resp. $A_{ij}$, resp. $A_{123}$) is a curve of the shape pictured in Figure \[fig3\] around just the $i$-th hole (resp. around both the $i$-th and $j$-th hole, resp. around all three holes), and the multiplicative notation $A_\alpha A_\beta$ means disjoint union of $A_\alpha$ and $A_\beta$. See Figure \[fig3\] for two examples. Note that the total number of curves in ${{\mathcal{A}}}$ is $22$.
\[thg3\] The set ${{\mathcal{A}}}(v)=\{A_\emptyset(v)=A_\emptyset,A_1(v),A_2(v),\ldots\}$ consisting of the curve arrangements in ${{\mathcal{A}}}$ colored $v$ is a basis of $\mathcal{S}^+_5({{\Sigma}}_3), $ and thus also a basis for $\mathcal{S}_5({{\Sigma}}_3).$
Note that it follows in particular that the ${{\mathcal{O}}}^+$-span of ${{\mathcal{A}}}(v)$ is stable under the action of the index two subgroup of even morphisms in the mapping class group $\tilde{\Gamma}({{\Sigma}}_2)$.
(-.6,-2.8)(2.6,.8) (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{} (2,0)[.6]{} (1,-1.73)[.6]{} (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (1,-.577)(.51,-.29) (1,-.577)(1.49,-.29) (1,-.577)(1,-1.13) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{}
(-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{123}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{}[350]{}[310]{} (2,0)[.6]{}[230]{}[190]{} (1,-1.73)[.6]{}[110]{}[70]{} (.6,-.1)(1,-.33) (1.4,-.1)(1,-.33) (.385,-.475)(.8,-.7) (1.615,-.475)(1.2,-.7) (1.2,-.7)(1.2,-1.175) (.8,-.7)(.8,-1.175) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78)
(-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{3}$]{} (2.3,-.8)[$A_{12}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (0,0)[.6]{}[20]{}[340]{} (2,0)[.6]{}[200]{}[160]{} (.56,.2)(1.44,.2) (.56,-.2)(1.44,-.2) (1,-1.73)[.6]{}
The set ${{\mathcal{A}}}$ (where its elements are considered as planar banded links in $H_3$) is a basis of $V_5({{\Sigma}}_3)$.
By the proof of Lemma \[lemG\], we can find a graph basis $\mathcal{G}$ for $V_5({{\Sigma}}_3)$ by 5-admissible colorings of the graph $G$ in Figure \[fig3\], where the loops are colored zero or one and the non-loop edges are colored zero or two. If a non-loop edge is colored two, then the loop at the end of the edge must be colored one. Also, the number of non-loop edges colored two must be zero, two, or three. This summarises 5-admissibility in this case. There are 15 such colorings. Again using Wenzl’s recursion formula, there is a triangular change of basis with ones on the diagonal from the basis $\mathcal{G}$ to the set $\mathcal{A}$ which is therefore also a basis of $V_5({{\Sigma}}_3)$.
Recall that ${{\mathcal{A}}}(v)$ consists of the $15$ elements of ${{\mathcal{S}^+_5}}(\Sigma_3)$ obtained by replacing each of the 22 curves in ${{\mathcal{A}}}$ with $v=(z+2)/(1+A)$. Again there is a triangular change of basis matrix from ${{\mathcal{A}}}$ to ${{\mathcal{A}}}(v)$. Therefore the elements of ${{\mathcal{A}}}(v)$ span $V_5({{\Sigma}}_3)$ and hence are linearly independent over ${{\mathcal{O}}}^+$. Consider the inclusion $$\label{strict}
\text{Span}_{{{\mathcal{O}}}^+}{{\mathcal{A}}}(v) \subset {{\mathcal{S}^+_5}}(\Sigma_3)~.$$ The matrix for $(\ ,\ )^+_{\Sigma}$ with respect to $\mathcal{A }(v)$ has determinant $\sim (1-q)^{3\cdot 15-2 \cdot 22}=1-q.$ Since $1-q$ is a prime in ${{\mathcal{O}}}^+ $ and ${{\mathcal{S}^+_5}}(\Sigma_3)$ is also a free ${{\mathcal{O}}}^+$-module, we conclude that the inclusion (\[strict\]) cannot be strict. Thus $\mathcal{A }(v)$ is a basis for ${{\mathcal{S}^+_5}}(\Sigma_3).$
[*Theorem \[thg3\] remains true if we replace $v$ by $\omega$ throughout. The same proof works.*]{}
[*As in Remark \[9.4\], it is crucial for this argument to use ${{\mathcal{S}^+_5}}({{\Sigma}}_3)$ rather than ${{\mathcal{S}_5}}({{\Sigma}}_3)$, since for $p=5$ there exists $a\in {{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{20}]$ such that $1-q=a\overline{a}$.*]{}
[*Kerler has announced in [@Ke] a construction of integral bases for the Reshetikhin-Turaev $SO(3)$ TQFT at the prime $p=5$ for any genus.*]{}
A divisibility result for the Kauffman bracket
==============================================
In this final section, we let $A$ again be an indeterminant. The fact that $v=(z+2)/(1+
\zeta_{2p}
)$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ for all odd primes $p$ has the following application to the Kauffman bracket $\langle\ \rangle$ of banded links in $S^3$.
\[th11\] Let $L$ be a banded link in $S^3$ with $\mu$ components. Let $L(z+2)$ denote this link colored $z+2.$ Then the Kauffman Bracket $\langle L(z+2) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is divisible by $(1+A)^{\mu}$.
Here the Kauffman bracket is normalized so that the bracket of the empty link is $\langle \emptyset\rangle=1$. Note that $$\langle L(z+2)\rangle = \sum_{L'\subset L} 2^{\mu-\mu(L')}
\langle L'\rangle~,$$ where the sum is over all sublinks $L'$ of $L$, and $\mu(L')$ denotes the number of components of $L'$.
When we evaluate the Kauffman bracket $\langle J \rangle$ of a banded link $J$ in $S^3$ at $A=\zeta_{2p}$, we obtain the quantum invariant of the pair $(S^3,J)$ (where $S^3$ is given the weight zero) in the normalization $$\langle J \rangle\vert_{A=\zeta_{2p}}=I_p(S^3,J)=\frac {\langle (S^3,J) \rangle} {\langle S^3 \rangle}={{\mathcal{D}}}{\langle (S^3,J) \rangle} ~.$$ This normalization $I_p(M,J)$ of the quantum invariant is precisely the one which is always an algebraic integer [@Mu1; @MR] and which is at the basis of the integral cobordism functors ${{\mathcal{S}_p}}$.
Let $f(A)$ denote the Kauffman bracket $\langle L(z+2)\rangle \in {{\mathbb{Z}}}[A^\pm].$ Since $v= (z+2)/(1+
\zeta_{2p}
)\in S_p(S^1 \times S^1),$ we have $I_p(S^3,L(v)) \in {{\mathbb{Z}}}[\zeta_{2p}],$ for every odd prime $p.$ Thus $$\label{divisi}
f(\zeta_{2p}) = I_p(S^3,L(z+2)) \in (1+\zeta_{2p})^\mu {{\mathbb{Z}}}[\zeta_{2p}],$$ for every odd prime $p.$
Now recall the following elementary Lemma (see [@Mu1 Lemma 5.5] and note that $-\zeta_{2p}$ is a primitive $p$-th root).
Suppose $f(A) \in {{{\mathbb{Z}}}} [A^\pm].$ Let $f^{(k)}(A)$ denote the $k$-th derivative of $f(A)$. Assume $0\leq \mu<p$ where $p$ is prime. Then $f(\zeta_{2p})\in {{{\mathbb{Z}}}} [\zeta_{p} ]$ is divisible by $(1+\zeta_{2p})^\mu$ if and only if $f^{(k)}(-1)\equiv 0\pmod{p}$ for every $0\le k< \mu.$
By this lemma, (\[divisi\]) implies $f^{(k)}(-1)\equiv 0\pmod{p}$ for each $0\le k< \mu,$ provided $p$ is larger than $\mu.$ Since there are infinitely many such primes, it follows that $f^{(k)}(-1)=0$ for each $0\le k< \mu.$ But this means that $ (1+A)^{\mu}$ divides $f(A).$
\[Kcor\] If $L$ is as in the theorem, then the Kauffman Bracket $\langle L(z+[2]) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is also divisible by $(1+A)^{\mu}$.
This follows immediately from the fact that $2-[2]=2-A^2-A^{-2}=(1-A^2)(1-A^{-2})$ is divisible by $1+A.$
[*In a similar way, Theorem \[tK\] remains true if we replace $v$ with $\hat v= (z+ [2])/(1+A)$ in the definition of $K(n).$ Similarly Theorems \[secondbasis\], \[thg2\], \[thg3\] remain true if we replace $v$ by $\hat v.$* ]{}
[*Theorem \[th11\] can also be proved by computing the Kauffman bracket from the (framed) Kontsevich integral via an appropriate weight system. Actually this proof is an adaptation of an argument going back to Kricker and Spence [@KS Proof of Thm. 2], but they only considered algebraically split links. Previously Ohtsuki [@OhCambridge Prop. 3.4] had obtained a stronger divisiblity result for $\langle L(z+[2]) \rangle$ using quantum groups, for algebraically split links satisfying some extra conditions. Later, Cochran and Melvin generalised the Kontsevich integral argument, and their result [@CM Theorem 2.5] contains Corollary \[Kcor\] for zero-framed links. (The results of [@OhCambridge; @KS; @CM] are stated in terms of the Jones polynomial, but it is well-known that the Jones polynomial and the Kauffman bracket are equivalent.) However, the restriction to zero framing is not really necessary (although a small additional argument is needed). We will not give details of this alternative proof here, as the techniques are completely different from the ones in the present paper.* ]{}
[BHMV2]{}
Asymptotic Faithfulness of the quantum $SU(n)$ representations of the mapping class groups. arXiv:math.QA/0204084 On framings of $3$-manifolds. [*Topology*]{} [**29**]{} (1990) 1-7. . Cambridge University Press, Cambridge, 1935.
Three manifold invariants derived from the Kauffman bracket, [*Topology*]{} [**31**]{} (1992), 685-699 Topological quantum field theories derived from the Kauffman bracket, [*Topology*]{} [**34**]{} (1995), 883-927
Quantum cyclotomic orders of 3-manifolds. [*Topology*]{} [**40**]{} (2001), no. 1, 95-125.
, [*J. Symbolic Comp,*]{} [**24**]{} (1997), 267-283
On the TQFT representations of the mapping class groups. [*Pacific J. Math.*]{} [**188**]{} (1999), no. 2, 251–274. On the Witten-Reshetikhin-Turaev representations of mapping class groups. [*Proc. AMS*]{} [**127**]{} (1999) 2483-2488. , arXiv:math.QA/0105059.
HYP and HYPQ - Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, [ *J. Symbol. Comput.* ]{}[**20**]{} (1995), 737-744
, arXiv:math.GT/0110007.
Ohtsuki’s invariants are of finite type. [*J. Knot Theory Ramifications*]{} [**6**]{} (1997) 583-597.
Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. arXiv:math.QA/0004099. An element of infinite order in TQFT-representations of mapping class groups. [*Contemp. Math.*]{} [**233**]{} (1999) 137-139.
(1993), no. 2, 171–194.
On central extensions of mapping class groups. [*Math. Ann.*]{} [**302**]{}, 131-150 (1995)
, Math. Proc. Cambridge Philos. Soc. [**121**]{} (1997) no. 3, 443–454 $3$-valent graphs and the Kauffman bracket. [*Pacific J. Math.*]{} [**164**]{}, (1994) 361-381. Integral modular categories and integrality of quantum invariants at roots of unity of prime order. [*J. reine angew. Math. (Crelle’s Journal)*]{} [**505**]{} (1998) 209-235.
(1994), no. 2, 253–281
(1995), no. 2, 237–249
A polynomial invariant of integral homology $3$-spheres. [*Math. Proc. Camb. Phil. Soc.*]{} [**117**]{} (1995) 83-112.
A polynomial invariant of rational homology $3$-spheres. [*Invent. Math.*]{} [**123**]{} (1996) 241-257.
Irreducibility of some quantum representations of mapping class groups. [*J. Knot Theory Ramifications*]{} [**10**]{} (2001), no. 5, 763–767. The $PSU(n)$ invariants of $3$-manifolds are polynomials. [*J. Knot Theory Ramifications*]{} [**8**]{} (1999) 521-532
de Gruyter (1994)
A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. (1995) 673–698.
Preprint 1991 Wolfram Media, Inc., Champaign, IL; Cambridge University Press, Cambridge. (1999)
[^1]: Warning: In many places ([*e.g.*]{} in [@MR]), $q$ denotes $A^4$ rather than $A^2.$
|
{
"pile_set_name": "ArXiv"
}
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Depression and anxiety correlate with disease-related characteristics and quality of life in Chinese patients with gout: a case-control study.
This study aims to evaluate the prevalence of depression and anxiety and investigate the potential risk factors for depression and anxiety in Chinese gout patients. A self-report survey was administered to 226 gout patients and 232 age- and gender-matched healthy individuals. Patients were asked to complete a set of standardized self-report questionnaires. Univariate and mutiple regression were used to analyze the data. We found 15.0% of gout patients had depression, and 5.3% had anxiety. After adjusted demographic variables, the prevalence of depression was significantly higher than the healthy controls (6.0%). There were significant correlations among education, total pain, disease duration, stage of gout, functional disability, number of tophi, number of flares/last year, presence of tender joints, nephropathy comorbidity, health-related quality of life (HRQoL), and psychological status. Meanwhile, logistic regression analysis identified number of tophi, functional disability, and mental component summary (MCS) as predictors of depression in gout patients. Education and MCS were significantly accounted for anxiety. In summary, the prevalence of depressive symptoms among gout patients was higher than healthy individuals. Education, disability, tophi and HRQoL were important risk factors linked to depression/anxiety in Chinese gout population.
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{
"pile_set_name": "PubMed Abstracts"
}
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// Copyright 2015 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// +build arm64,darwin
package unix
import (
"syscall"
"unsafe"
)
func Getpagesize() int { return 16384 }
func TimespecToNsec(ts Timespec) int64 { return int64(ts.Sec)*1e9 + int64(ts.Nsec) }
func NsecToTimespec(nsec int64) (ts Timespec) {
ts.Sec = nsec / 1e9
ts.Nsec = nsec % 1e9
return
}
func NsecToTimeval(nsec int64) (tv Timeval) {
nsec += 999 // round up to microsecond
tv.Usec = int32(nsec % 1e9 / 1e3)
tv.Sec = int64(nsec / 1e9)
return
}
//sysnb gettimeofday(tp *Timeval) (sec int64, usec int32, err error)
func Gettimeofday(tv *Timeval) (err error) {
// The tv passed to gettimeofday must be non-nil
// but is otherwise unused. The answers come back
// in the two registers.
sec, usec, err := gettimeofday(tv)
tv.Sec = sec
tv.Usec = usec
return err
}
func SetKevent(k *Kevent_t, fd, mode, flags int) {
k.Ident = uint64(fd)
k.Filter = int16(mode)
k.Flags = uint16(flags)
}
func (iov *Iovec) SetLen(length int) {
iov.Len = uint64(length)
}
func (msghdr *Msghdr) SetControllen(length int) {
msghdr.Controllen = uint32(length)
}
func (cmsg *Cmsghdr) SetLen(length int) {
cmsg.Len = uint32(length)
}
func sendfile(outfd int, infd int, offset *int64, count int) (written int, err error) {
var length = uint64(count)
_, _, e1 := Syscall6(SYS_SENDFILE, uintptr(infd), uintptr(outfd), uintptr(*offset), uintptr(unsafe.Pointer(&length)), 0, 0)
written = int(length)
if e1 != 0 {
err = e1
}
return
}
func Syscall9(num, a1, a2, a3, a4, a5, a6, a7, a8, a9 uintptr) (r1, r2 uintptr, err syscall.Errno) // sic
// SYS___SYSCTL is used by syscall_bsd.go for all BSDs, but in modern versions
// of darwin/arm64 the syscall is called sysctl instead of __sysctl.
const SYS___SYSCTL = SYS_SYSCTL
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{
"pile_set_name": "Github"
}
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You've probably known for a while now that you quite like having sex – but what you may not have realised is just how good it is for you.
Yes, that's right: sex helps to keep you healthy. And we're not referring to that hoary old playground myth that a tumble in the hay three times a week is the same as running a marathon every year (it's much more). No, sex is beneficial for an array of scientifically proven reasons – and since today is World Sexual Health Day, now is a good time to learn them.
So, dim the lights, put some Barry White on, and snuggle up with our low-down on how more sex can improve your life ...
via GIPHY
Sex is a natural painkiller
According to research in the Bulletin of Experimental Biology and Medicine, orgasms half the body's sensitivity to pain, due to the flow of endorphins (read: natural painkillers) they create. What's more, those endorphins take effect in a matter of minutes – which is far quicker than most over-the-counter drugs available on the high street.
So, the next time your partner claims that he or she "has a headache," politely inform them that the best remedy is a spot of sexual healing. All together now ...
Sex combats illness
You know those people who say "I never get ill"? Well, it's possible that they're revealing more about themselves than they realise.
According to a study carried out in Pennsylvania and published in Psychology Report, people who have sex once or twice a week have, on average, 30pc more Immunoglobulin A (IgA), which is used to fight illness, than those who are not sexually active.
However, it's worth pointing out that this link between intercourse and immunity is not always positive. The same study found that the people with the lowest level of Immunoglobulin A were those who had sex more than twice a week.
Sex beats stress
Do you lie in bed awake at night worrying about tomorrow's meeting or that big DIY project on the horizon? Well, rather than turning the problems over in your head, the best course of action might be turning your partner over and making a play for a spell of stress-easing sex.
According to a study that was published in Biological Psychology, men who have recently had sex respond better to stressful situations.
To an extent, this should come as no shock: sex is largely considered to be an enjoyable pastime, so it's natural that it lowers your stress level. However, what is more surprising is the de-stressing affect that mere touch has. Another experiment, published in Behavioural Medicine, looked at couples who held each other's hands for 10 minutes, followed by a 20-second hug. They were shown to have healthier reactions to subsequent stress, such as public speaking, than couples who rested quietly without touching.
The huggers also had: lower heart rates, lower blood pressure, and smaller heart rate increases.
What he needs is a good ... Credit: Alamy
So, if your partner isn't interested when you turn him or her over, settle for a nice snuggle instead.
Sex helps the heart
Men who have sex at least twice a week can almost halve their risk of heart disease, according to research published in 2010 by scientists at the New England Research Institute in Massachusetts.
The study, of over 1,000 men, showed that sex has such a protective effect on the male that its authors went as far as calling for doctors to screen men for sexual activity when assessing their risk of heart disease.
They suggested that the benefits of sex could be due to both the physical and emotional effects on the body.
via GIPHY
Sex powers up the brain
Ok, so it's only been proven in rats to date – but one scientific study suggests that a rumble in the (pubic) jungle can boost brain power.
In 2010, research published in the journal PLoS ONE suggested that rats who mate regularly had a higher rate of cell proliferation in the hippocampus, which is the part of the brain linked to memory. The rats also experienced more brain cell growth and a rise in the number of connections between brain cells than those who did not.
So, forget about sudoku and instead consider more virile endeavours as your brain exercise.
Who said sex isn't a thinking man's game?
Sex makes you look younger
Regular sex could now be the key to looking up to seven years younger, according to Dr David Weeks, who is the former head of old age psychology at the Royal Edinburgh Hospital.
In 2013, Dr Weeks told the British Psychological Society that sex has a number of health benefits which can make men and women look between five and seven years younger. Partly, this is due to the health benefits summarised above (a healthy person tends to look younger than an unhealthy person), but he also pointed to the release of human growth hormone in the act of love making, which makes skin look more elastic.
Another piece of research, conducted by scientists at the Royal Edinburgh University, found that couples who merrily romp at least four times a week look a whole decade younger than less libidinous twosomes. The pleasure of penetration releases positive hormones such as adrenalin, dopamine and norepinephrine, which help preserve skin cells and relax muscles, therefore preventing wrinkles.
Which all suggests that the elixir of youth might spout from a very unexpected fountain ...
Good sex helps you last longer
Not only are men who have frequent sex more likely to last longer in the sack than those who don't; they're also more likely to last longer in the world.
As you've probably figured by now, sex is good for you – which is why a study conducted in a Welsh village in 1997 found that men who have two or more orgasms a week add up to eight years onto their lives than their less passionate brethren.
''Sexual activity seems to have a protective effect on men's health,'' wrote Dr. George Davey-Smith and his team of researchers from the University of Bristol and Queen's University of Belfast, pointing to the positive effects on the immune system, heart, and brain.
'My knight takes your castle': Kim Kardashian playing chess Credit: Melissa Whitworth
Indiana Jones wonders whether he really wants to try the elixir of youth Credit: Indiana Jones and the Last Crusade
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{
"pile_set_name": "OpenWebText2"
}
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'At least for Europe it is obvious: All roads lead to Rome! You can reach the eternal city on almost 500.000 routes from all across the continent. Which road would you take?
To approach one of the biggest unsolved quests of mobility, the first question we asked ourselves was: Where do you start, when you want to know every road to Rome? We aligned starting points in a 26.503.452 km² grid covering all of Europe. Every cell of this grid contains the starting point to one of our journeys to Rome.
Now that we have our 486.713 starting points we need to find out how we could reach Rome as our destination. For this we created a algorithm that calculates one route for every trip. The more often a single street segment is used, the stronger it is drawn on the map. The maps as outcome of this project is somewhere between information visualization and data art, unveiling mobility and a very large scale.'
'Heavily tinted blue paintings form space stations, spacesuits, and rockets just after blast. Michael Kagan paints these large-scale works to celebrate the man-made object—machinery that both protects and holds the possibility of instantly killing those that operate the equipment from the inside. To paint the large works, Kagan utilizes an impasto technique with thick strokes that are deliberate and unique, showing an aggression in his application of oil paint on linen. The New York-based artist focuses on iconic images in his practice, switching back and forth between abstract and representational styles. “The painting is finished when it can fall apart and come back together depending on how it is read and the closeness to the work,” said Kagan about his work. “Each painting is an image, a snapshot, a flash moment, a quick read that is locked into memory by the iconic silhouettes.”'
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{
"pile_set_name": "Pile-CC"
}
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Q:
DataGrid In Java Struts Web Application
After scouring the web I have edited my question from the one below to what it is now.
Ok I seem to understand that I don't need all the capabilities of excel right now. I think i am satisfied having a data grid to display data. Basically i am working on Struts 2 and I wat my jsp page to have an excel like feel and hence looks like even a datagrid is sufficient. I came across This Technology
I am not sure whether I must go ahead and use it. Any other suggestions, alternatives are welcome
The older version of the question
"I have a java web application running on windows currently. I may host it in future in a Linux Server.
My application allows people to upload data. I want to display the data they have uploaded in an excel file and render it in a portion of my webpage.
How do I go about this ?"
A:
Basically you would need to read the excel files, get the data in some kind of java objects, and then show it back to user as a normal HTML page with tables etc..
If you want to show the excel files in such a way that your users are also able to edit these then you need to look into javascript / ajax to make a UI as per your needs.
An easy and open source way of reading the uploaded excel files in java is via Apache POI. It is capable of reading .xls files as well as the newer OOXML .xlsx files.
http://poi.apache.org/spreadsheet/
They have very helpful examples which can get you started within 10 minutes..
http://poi.apache.org/spreadsheet/quick-guide.html
If you can allow data to go to another site, then you can use ZOHO. Their online Excel Editing is reasonably good and you don't really have to do anything much.
|
{
"pile_set_name": "StackExchange"
}
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Sea Containers
Sea Containers was a Bermudan registered company which operated two main business areas: transport and container leasing. It filed for bankruptcy on 16 October 2006. In 2009 its maritime container interests were transferred to a new company SeaCo Ltd, with the winding down and liquidation of the remainder of the group continuing.
History
Yale University graduate and retired United States Navy officer James Sherwood founded Sea Containers in 1965, with initial capital of $100,000. It was later listed on the New York Stock Exchange. In May 1989, Tiphook launched an unsuccessful takeover bid for the company.
Over 40 years, Sherwood expanded Sea Containers from a supplier of leased cargo containers, into various shipping companies, as well as expanding the company into luxury hotels and railway trains, including the Venice-Simplon Orient Express and the Great North Eastern Railway train operating company.
Although valued with a net worth of £60million in the 2004 Sunday Times Rich List, as Sea Containers hit financial troubles, he resigned from each of his companies in 2006. In 2005, Sea Containers sold its 25% share in Orient-Express Hotels.
Chapter 11
In March 2006, Sherwood resigned from all positions in the various Sea Containers Companies. Sherwood was replaced by company doctor Bob Mackenzie, while Ian Durant became senior vice-president of finance.
Despite selling various businesses and assets, Sea Containers announced in early October 2006 that it was unlikely to be able to pay a $115m (£62m) bond due up on 15 October. On 16 October, the company filed for Chapter 11 bankruptcy protection.
On 6 November 2006 the Department for Work & Pensions wrote to Sea Containers that it must pay £143m to its two UK pension schemes if it wants to wind them up.
On 11 February 2009, its maritime container interests were transferred to a new company SeaCo Ltd, with the wind down and liquidation of the remainder of the group continuing. The major shareholders in the new company were the former Sea Containers Ltd bondholders and two of the group's UK pension funds.
Transport
Ferry services
Isle of Man Steam Packet Company: fast and conventional services in the Irish Sea. Acquired in 1996, sold in 2003.
Silja Line: fast and conventional services in the Baltic Sea. In June 2006 Silja Line was purchased by Tallink, a ferry company from Estonia. The fast catamaran service SuperSeaCat was separated from Silja Line and operated until 2008 when it went bankrupt.
Orient-Express Hotels: (25% shareholding) sold in 2005
SeaStreak: following the Sea Containers bankruptcy of 2006, this operation was sold to New England Fast Ferry
SNAV-Hoverspeed: a joint venture with Italian ferry operator SNAV. Used the former Seacat Danmark as Zara Jet.
Aegean Speed Lines: a joint venture in Greece with the Eugenides Group. The service uses the former Hoverspeed Great Britain as Speedrunner 1, which operated in the English Channel and held the Hales Trophy and Blue Riband for the fastest crossing of the North Atlantic.
Hoverspeed English Channel services ceased in 2005
SeaCat: (Belfast& Troon).
Other
Related activities include:
Hart Fenton: a naval architecture and marine engineering company, sold to Houlder in 2006
Sea Containers Chartering
Rail
GNER: a train operating company that commenced operating the InterCity East Coast franchise in April 1996. After winning a further 10-year extension when re-tendered in 2005, GNER ran into financial difficulties with Sea Containers handing back the franchise in December 2007. It also bid for the South Western franchise in 2001 and South Eastern franchise in 2006.
Containers
Sea Containers container leasing business was conducted mainly through GE SeaCo, a joint venture with GE Capital formed in 1998. GE SeaCo was sold to the HNA Group for approximately $1 billion on 15 December 2011 and now operates as Seaco.
Other former activities
Sea Containers Property Services Ltd – property development, property asset management.
The Illustrated London News Group (ILNG) – publishing
Fruit farming – Sea Containers owned plantations in West Africa and South America.
Fairways & Swinford – UK-based business travel agency
Former internet property of Sea Containers Ltd
In March 2016 the domain of seacontainers.com was acquired by World Sea Containers
References
Category:Companies formerly listed on the New York Stock Exchange
Category:Defunct companies of Bermuda
Category:Shipping companies of Bermuda
Category:Container shipping companies
Category:Transport companies established in 1965
Category:1965 establishments in Bermuda
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{
"pile_set_name": "Wikipedia (en)"
}
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The influence of drug incorporation on the structure and release properties of solid dispersions in lipid matrices.
The effect of incorporating caffeine and paracetamol on the structure and behaviour of Gelucire 50/13 has been studied with a view to establishing whether the choice of drug influences the solid structure and release mechanism. Dispersions containing up to 30% w/w drug were prepared and studied using differential scanning calorimetry (DSC), hot stage differential interference contrast microscopy (HSM), dissolution studies and erosion, water uptake (WU) and diameter change measurements. Gelucire 50/13 alone showed a broad melting endotherm using DSC, with two dominant peaks at 36 and 44 degrees C. While incorporation of caffeine did not result in marked changes to the profile, the presence of paracetamol increased the proportion of material in the lower melting peak. HSM studies indicated that the Gelucire crystallised into two main spherulitic conformations; paracetamol appeared to act as a nucleation site for the lower melting fractions while caffeine particles changed into a needle-shaped morphology on cooling the system from the liquid state. Dissolution studies at 37 degrees C showed the caffeine to be released at a relatively faster rate than the paracetamol. Kinetic modeling and direct measurement of the erosion profile indicated that the caffeine systems showed a greater preponderance for erosion than did the corresponding paracetamol systems. It is suggested that the paracetamol promotes the generation of the lower melting form of Gelucire 50/13 which in turn influences the release rate and mechanism. The study therefore indicates that the influence of the drug should be carefully considered when studying Gelucire matrix systems.
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{
"pile_set_name": "PubMed Abstracts"
}
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No bodies found in capsized fishing boat off N.S.
The capsized fishing boat Miss Ally is shown in an undated Canadian Forces handout photo.
Share:
Text:
The Canadian Press
Published Saturday, February 23, 2013 10:57AM EST
Last Updated Saturday, February 23, 2013 9:08PM EST
WOODS HARBOUR, N.S. -- The father of one of five young Nova Scotia fishermen lost at sea says word that divers found no sign of their bodies beneath a capsized vessel is hard to accept, yet it gives him a sense of closure.
"There's no need for any more search," said George Hopkins, whose 27-year-old son Joel Hopkins was among the missing.
Nearby, a dozen cars were parked and a steady stream of family friends came and went, offering condolences as they entered his brightly lit house.
"It wasn't the result we wanted," said Hopkins.
"But for me there's closure knowing the search is over and there's no hope now of anybody being alive."
RCMP said Saturday they've been told no bodies were found in a search of an overturned boat off southwest Nova Scotia by divers on a private fishing vessel.
The Mounties said the captain of the Slave Driver told the Canadian Coast Guard Vessel Sir William Alexander around 6 p.m. Saturday that divers had visually confirmed that there were no bodies in the Miss Ally.
Supt. Sylvie Bourassa-Muise said according to the information police received, the divers also found that no wheelhouse or sleeping quarters were attached to the vessel's hull.
For Hopkins, the details of the missing wheelhouse was decisive and crushing news. He said that no wheelhouse meant that the Miss Ally's life-raft was also torn away.
"With the wheelhouse gone, I think things happened so fast, they didn't have a chance to get in the life-raft. It would be false hope to continue," he said.
Searchers spotted the hull of the boat Saturday morning, about 239 kilometres southeast of Halifax and 46 kilometres northwest of the boat's last known position, RCMP said.
Bourassa-Muise said the HMCS Glace Bay was expected to arrive at the site of the Miss Ally sometime overnight Saturday and would conduct an assessment with a remotely operated vehicle on Sunday morning.
"That is simply done to confirm what the report was from the private fishing boat," said Bourassa-Muise from Woods Harbour, N.S., on Saturday evening. "That will conclude the efforts."
Photos will also be taken during the assessment, RCMP said.
The Mounties said an aircraft would continue to maintain a visual sighting of the boat overnight.
The Miss Ally capsized in heavy seas last Sunday with the loss of five young fishermen. Police have not formally released the names of the fisherman, but a family member has identified one of those aboard as Cole Nickerson. The three other men were identified at a local prayer service last Tuesday evening as Katlin Nickerson, Billy Jack Hatfield and Tyson Townsend.
The 13 metre vessel, which was on an extended halibut fishing trip, was last spotted by the coast guard on Tuesday.
After the search for survivors was called off, the families of the fishermen asked federal authorities to recover the overturned vessel to determine if there were bodies inside.
Sandy Stoddard, a Woods Harbour fisherman who helped organize the continued search by local fishermen, said he pleaded with the local RCMP to continue the search for the bodies.
"They knew we weren't going to let this go until we were satisfied that nobody was on the boat," he said.
"We are men of little patience. We don't wait for protocol to do things. When you're a fishermen on the ocean you don't follow protocol, you follow knowledge."
The 57-year-old fisherman now praises the local RCMP officers for passing their message on to the Defence Department and the coast guard.
"They went to bat for us," he said.
Stoddard said he and many other fishermen in the community still recall a lost vessel from 39 years ago, when another seven men were lost at sea and never recovered.
"There was never any closure to that accident. There was pleading to the authorities this time. We can't live through this for another 30 or 40 years," he said.
Pastor Rod Guptill of the Wesleyan Church in Woods Harbour said people were gathered at a community centre Saturday to await word from searchers.
"It's not good news. But it's news that does help us accept us begin grieving," he said.
He said he will now turn his attention to a sermon for Sunday.
"We will mourn with those who mourn. We are there to express our support and sympathy and grief for those who are going through the grieving process," he said.
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{
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Flow cytometric identification of cancer cells in effusions with Ca1 monoclonal antibody.
The fluorescence of cells from 42 pleural and peritoneal effusions stained with Ca1 monoclonal antibody (Ca1MA) was studied by flow cytometry. In 14 of 17 malignant effusions a significantly higher intensity of fluorescence was observed in samples exposed to Ca1MA when compared with controls. There was no increase of fluorescence intensity in 25 benign effusions. The method failed in three malignant effusions: one due to endometrial carcinoma and two to malignant lymphoma. The sensitivity of the method was tested in experimental samples with a known percentage of malignant cells. The positive fluorescence with Ca1MA was detected in samples containing 0.1% of carcinoma cells. Flow cytometry with Ca1MA can be a relatively simple method of identification of malignant cells in effusions.
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From Dead Media Archive
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Originally known as ImageK, Silicon FIlm is best known for never really existing. The digital film attachment for 35 mm film cameras generated lots of interest and publicity only to die out before every reaching the market. The product (dubbed the EFS-1) looks like a reel of 35mm film, and replaces the normal film to record the images digitally. At the bottom of the camera, a receiver is attached that allows users to view their photos
The EFS-1 and its Limitations
The unit itself only had 64MB of space, leaving room for only 24 images. To upload images, there are two options. The EFS-1 can be inserted into an e-port, which as a PCMCIA connector which can then be inserted into a PCMCIA slot for download. For those who are not around a computer, the e-port can be inserted into an e-box, which then transfers the images onto compactFlash cards. The EFS-1 took two batteries that could last for several hundred shots.
The 1.3 megapixel CMOS sensor only utilized 30% of the center of the frame, meaning that when looking in the viewfinder, there is a very small field of view (marked by a supplied rub-on transfer). What results from this is a multiplication of focal length by 2.58x, making a 28mm lens become 72mm for example.
Only certain cameras were supported by the EFS-1: Nikon F5, F3, N60/N90 and Canon EOS-1N, EOS-A2, EOS-5.
Shooting an image writes an RAW file onto the EFS-1 which is then decoded by a Photoshop plugin which performs bayer interpolation, white balance, gamma and exposure compensation.
Brief History
June 1998 - The first model of what would become Silicon Film appeared as ImageK, a subsidiary of Irvine Sensors, a company in Irvine California. The prototype of this model was an insert that they said had a 1.3 mega-pixel sensor and a 2.58x crop factor.
September 1999 - Irvine Sensors renames the product "Silicon Film"
February 2001 - The first public demo of Silicon FIlm takes place. According to those in attendance, the pictures shown in the demonstration were different from the models, causing speculation that the prototype was entirely nonfunctional and the images had been created prior to the demo.
September 2001 - Irvine Sensors announces the suspension of Silicon Film operations. The reasons cited include a supposed failure to pass European product safety (EMC) tests. The company claims however to have passed the USS (FCC) tests, but the FCC denied this claim
Business Decisions
“Will digital photography do to traditional photography what CDs did to records, make them a collectors item? Not as far as (e)film creators Silicon Film Technologies is concerned”
When it was first announced, Silicon Films’ Technology was considered to have the potential to be “as revolutionary as the Polaroid cameras of the 1950’s.” With headquarters in Irvine, California, the company defined the scope of its operations to include R&D in micro-miniaturization of imager packaging and integrated circuit sensor implementation.
Originally known as Imagek, the majority-owned, independently operated subsidiary of parent company Irvine Sensors Corporation was renamed Silicon Film Technologies in 1998. Under CEO Robert Webber the company was plagued with missed release dates and continual disappointments. As a result, Kenneth P. Fay from Kodak was brought on as Chief Operating Officer in early 2000 and later elected to the position of President and Chief Executive Officer. The board hoped the change in leadership might turn the company around and bring the vaporware technology into being, but it only lead them further towards a set of turbulent strategic decisions. Also in 2000, SFT partnered with MeltroniX, who specialized in design, volume manufacturing, and testing capabilities for high growth industries, to help with engineering and volume manufacturing issues. With the ongoing struggle to produce a working prototype, the issues of scale never became practical issues.
At the end of 2000, Irvine Sensor Corporation’s stock was considered to have a potentially very high reward after factoring in the risk Silicon Film Technologies brought to the company as a whole. Just over 6 months later, it announced a plan to change its CEO once again. In recruiting William Patton, the parent company thought the veteran business man could succeed where previous appointment Kenneth Lay had not. In the midst of these management shifts on March 15th 2000, Silicon Films signed an exclusive distribution contract with Ringfoto for immediate access into all European Union countries, Scandinavia and Eastern Europe photo market. These contracts were made under the assumption there would be a product to ship.
After its first life trade show in February 2001, the timeline for release appeared to be back on track. The product shown at show was partially outsourced to Quest Manufacturing, a vendor of shaped metal parts. In May 2001, Silicon Film Technologies announced it would begin accepting orders. This was under the assumption that it had received the necessary testing for public release, which it in fact had not. When taking orders, the company priced the EFS-1 System at $699, compatible with Nikon F5, N90, and F3 and the Canon EOS 1N A2/5 models.
After failing certification tests, Silicon Film Technologies was forced to suspend operations on September 15, 2001. The company attributed the issues to FCC and European emissions standards, but its parent company and largest creditor refused to extend further lines of credit. In an attempt to cover outstanding debts, SFT was bought out of bankruptcy shortly there after by Quest Manufacturing, a previous supplier.
But this isn’t where the patent stopped changing hands. When trying to continue development, Quest hired Applied Color Science to help engineer. While a working prototype was developed, the modular adaptor was larger than the camera. When Quest ran out of funding before it could miniaturize the product, the patent was again used to cover debt obligations, finally ending with Applied Color Science.
Fraud
In 2001, Silicon Film attended the Photo Marketing Association (or PMA) International Convention and Trade Show to unveil its long awaited EFS-1 digital film product and its accompanying accessories. At the time of this “demo” the product could only hold 24 pictures, was only compatible with Nikon F5, F3, N60/N90 and Canon EOS-1N, EOS-A2, and EOS-5 models, and limited the viewing area of the image. These limitations were a small compromise for the final unveiling of the product, but some message board enthusiasts speculate in hindsight that the performance put on at the stand might have been a well-orchestrated fraud where pictures were produced with a digital camera prior to the event.
Failure
More than 18 months after the product was promised, parent company Irvine Sensors Corporation was sued because of allegations of misleading information in regard to the state of Silicon Film Technologies Inc. The claims of the case are reproduced below:
(1) the EFS-1 suffered from serious and insurmountable technical design flaws; (2) these design problems would prevent the unit from passing the required FCC and CE certifications necessary to publicly release the product; (3) the current design of the EFS-1 was extremely difficult to produce. Specifically, it took hundreds of engineering hours to produce one unit with a success rate of about one unit in three working; (4) an internal design review was conducted in May, 2001 with all the top officers of SFI, ISC and all of the suppliers for the EFS-1 that were owed millions of dollars. The results of the internal design review were that SFI had a design and parts to produce about 200 units. However, the biggest contract SFI had was for 100 units to a European distributor who would not accept the units since they would not pass CE certification. The web-site sale commitments for domestic sales was for only a few dozen units; (5) EFS-1 technology presented potential patent conflicts with those already registered by Kodak; (6) SFI and ISC had scrapped the initial design of the EFS-1 and were scrambling to develop a new prototype; (7) several key employees on the EFS-1 project left SFI further hampering the development process; and (8) William Patton never accepted the position of Chairman and Chief Executive Officer of SFI..
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The product did suffer from technical design flaws. In its attempt to build on existing technologies it was forced to adapt. In the case of internal inconsistency between camera models the company found that 6 difference versions would cover the majority of “popular high-end camera bodies”. SFT might have been able to achieve the miniaturization, but luck was never on their side and Applied Color Science developed the technology long after the grass had grown over the grave of Silicon’s bankruptcy.
While there were technical design flaws in Silicon’s ambitious plans to create sensor, processor, and storage components, this obstacle was overcome by competitors in a timely manner. The real problem for Silicon Film was casting the product as a transitory step. It was targeted with lines such as “if you just aren't ready to go totally digital, the EFS-1 System from Silicon Film Technologies Inc. is a good compromise”. While this easy adaptation did have the benefits of using the SLR’s superior optics and features beyond those of the digital cameras at the time to sell to an existing consumer base, time was limited. Even with this quicker learning curve, the product would have to be in the market to be adopted. A senior engineer was on the record saying, “EFS was only ever going to be a short term product. We knew that DSLRs would fall in price and eventually make the system obsolete”. Here he recognized the time pressure that was too much for SFT.
As a compromise between the digital camera and popular SLR’s, the EFS effort exposes the expanding gap between consumer camera products and professional equipment brought on with the invention and spread of digital.
Beyond the market pressures to get the product out there were a vast marketing effort. Some critize the focus on profitability over development, joking that car rental expenses exceeded those for R&D. Beyond the technological and strategic reasons opposing success, an ultimate lesson from the failure of Silicon Film Technologies is that interesting ideas do not automatically turn into practical products.
Legacy
“When Silicon Film Technologies filed for Chapter 7 bankruptcy in Sept of 2001 EFS-1 was probably 3 months (my estimate, our management was saying 2 months) from reaching the market place.” Jon Stern - Senior Engineer, Silicon Film Technologies
The people at Silicon Film Technologies may have dropped the EPS-1, but they didn’t stop working. After going through iterations of selling and reselling, the company was known as Voyager One, and then finally settled on Monarch Petroleum Products, with a focus on producing gasoline additives.
References
"PMA 2001 Show Report: Section Four: Digital Photography Review." Digital Photography Review, News, Reviews, Forums, FAQ. 16 Feb. 2001. Web. 16 Nov. 2010. <http://www.dpreview.com/news/0102/01021404pma04.asp#siliconfilm>. "Silicon Film Technologies Promotes COO to CEO; Former Kodak Executive to Lead Company into Next Phase of Business. - Free Online Library." Free Online Library. /PRNewswire, 5 May 2001. Web. 16 Nov. 2010. <http://www.thefreelibrary.com/Silicon Film Technologies Promotes COO to CEO; Former Kodak Executive...-a061895730>. "MeltroniX Inks $2.8M Deal for Volume Manufacturing With Silicon Film." Business Wire. Business Wire. 2000. HighBeam Research. 16 Nov. 2010 <http://www.highbeam.com>. Nemeth, Andrew. "Silicon Film - an inside Story by Jon Stern." Andrew Nemeth Φ Nemeng.com. Web. 16 Nov. 2010. <http://nemeng.com/leica/004fa.shtml>. B.K. GREEN, on Behalf of Himself and All Others Similarly Situated v IRVINE SENSORS CORPORATION, ROBERT RICHARDS, JOHN STUART JR., JAMES D. EVERT, JOHN CARSON and ROBERT WEBBER,. CENTRAL DISTRICT OF CALIFORNIA, Southern Division. 14 Feb. 2002. Print.
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A remote fluid reservoir blow-molded of polymeric material is provided with an antechamber for the reception of incoming high velocity fluid and impinging of same in a sinuous course on walls of the antechamber which are so configured as to smoothly reduce flow velocity and avoid undue turbulence upon...http://www.google.com/patents/US4431027?utm_source=gb-gplus-sharePatent US4431027 - Reservoir for remote fluid system
A remote fluid reservoir blow-molded of polymeric material is provided with an antechamber for the reception of incoming high velocity fluid and impinging of same in a sinuous course on walls of the antechamber which are so configured as to smoothly reduce flow velocity and avoid undue turbulence upon introduction of the fluid into a main storage chamber.
Images(2)
Claims(3)
The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:
1. A fluid reservoir molded of polymeric material and adapted for enhanced anti-turbulent storage of fluid entering at a high velocity from a fluid system, comprising, a main storage chamber portion, an antechamber portion separated from the main chamber portion, a fluid inlet opening formed adjacent the upper end of the antechamber portion for receiving high-velocity fluid from the fluid system, means adjacent the lower end of the antechamber portion defining an opening communicating the interiors of the antechamber and main chamber portions, the area of said communicating opening being a predetermined large multiple of the area of said inlet opening for passage of fluid from the antechamber portion to the main chamber portion at a velocity substantially reduced from that entering at said inlet opening, and the cross-sectional area across the direction of fluid flow of said antechamber between the ends thereof varying without discontinuity from a smallest area adjacent said inlet opening to a largest area adjacent said communicating opening for anti-turbulent transition of the velocity of fluid flow between said high and reduced values thereof.
2. A blow-molded fluid reservoir of polymeric material and adapted for enhanced anti-turbulent storage of fluid entering at a high velocity from a fluid system, comprising, a main storage chamber portion, an antechamber portion including walls separated from the walls of the main chamber portion, a fluid inlet opening formed adjacent the upper end of the antechamber portion for receiving high-velocity fluid from the fluid system, walls adjacent the lower end of the antechamber portion integrally associating said lower end with the main chamber portion and defining an opening communicating at said lower end the interiors of the antechamber and main chamber portions, said antechamber defining a sinuous course between said openings and the area of said communicating opening being a predetermined large multiple of the area of said inlet opening for passage of fluid from the antechamber portion to the main chamber portion at a velocity substantially reduced from that entering at said inlet opening, and the cross-sectional area of said antechamber across the direction of fluid flow between the ends thereof varying without discontinuity from a smallest area adjacent said inlet opening to a largest area adjacent the communicating opening for anti-turbulent transition of the velocity of fluid flow between said high and reduced values thereof.
3. A blow-molded fluid reservoir of polymeric material and adapted for enhanced anti-turbulent storage of fluid entering at a high velocity from a fluid system, comprising, a main storage chamber portion, an antechamber portion of generally L-shape in vertical section and including walls separated by a mold-closure gap from the walls of the main chamber portion, a fluid inlet tube formed adjacent the upper end of one leg of the antechamber portion and extending laterally of said leg for receiving high-velocity fluid from the fluid system and impinging the same on the opposed walls of the antechamber portion, walls within the other leg of the antechamber portion integrally associating said other leg with the main chamber portion and defining an opening communicating at said other leg the interiors of the antechamber and main chamber portions, the area of said communicating opening being a predetermined large multiple of the area of said inlet opening for passage of fluid from the antechamber portion to the main chamber portion at a velocity substantially reduced from that entering at said inlet opening, and the lateral cross-sectional area of said antechamber across the direction of fluid flow between the ends thereof smoothly varying from a smallest area adjacent said inlet opening to a largest area at the communicating opening for enhanced anti-turbulent transition of the velocity of fluid flow between said high and reduced values thereof.
Description
This invention relates to fluid systems and more particularly to reservoirs for such systems located remote from the system proper and connected thereto by conduits.
Recent designs of automotive power steering systems separate the fluid reservoir of the system from what formerly had been unitary association thereof with the engine-driven pump. Crowding of engine compartments has led to need for more efficient utilization of space through location of the reservoir remote from the engine and the pump and connection between the two and with the power steering gear via connecting hoses.
It is desirable to mold the remote reservoir of polymeric material for a shape suited to the available space at the selected location in the engine compartment. Additionally, it is necessary to ensure that as the reservoir receives relatively high velocity fluid flow returning from the power steering gear, it does so without undue turbulence within the reservoir and entrainment of air bubbles within the fluid. Such air entrainment affects the fluid viscosity and in turn degrades the efficiency of the fluid system.
By the present invention there is provided a remote fluid reservoir moldable from polymeric or like material to a desired shape and provided with a main reservoir portion and an antechamber portion especially designed to receive the high-velocity incoming fluid flow, traverse it in a sinuous course through smoothly widening cross-sectional areas of the antechamber to substantially reduce the flow velocity, and introduce the flow to a main storage portion of the reservoir without undue turbulence and air entrainment.
In a preferred embodiment, the reservoir is blow-molded and the main storage and the antechamber portions suitably defined as by the known use of mold-closure regions, the reservoir further featuring an antechamber of generally L-shape in vertical section adapted to reception of the incoming high-velocity fluid, as a first stage of the sinuous course, laterally at an upper narrow end of the antechamber and impingement upon opposed walls of the latter, thence a direction of the flow into smoothly widening areas of the two legs of the antechamber which achieve the desired velocity reduction.
The fluid flow is turned and introduced to the main storage portion through an opening communicating the separate cavities of the two reservoir portions. In the blow-molded embodiment the communicating opening is formed incidental to a merging of walls of the two otherwise separated portions below a mold-closure gap defined by mating generally mirror-image mold pieces and at the margins of the mold cavities for the antechamber and main chamber portions.
These and other objects, features and advantages of the invention will be readily apparent in the following specification and from the drawings wherein:
FIG. 1 is a partially broken away elevational view of a reservoir according to the invention;
FIG. 2 is a partially broken away elevational view taken generally along the plane indicated by lines 2--2 of FIG. 1;
FIG. 3 is a plan view in the direction of lines 3--3 of FIG. 1;
FIG. 4 is a sectional view taken along the plane indicated by lines 4-4 of FIG. 1;
FIG. 5 is a sectional view taken along the plane indicated by lines 5--5 of FIG. 1;
FIG. 6 is a sectional view taken generally along the planes indicated by lines 6--6 in FIG. 1; and
FIG. 7 is an enlarged perspective view.
The remote reservoir is indicated generally as 10 and is preferably fabricated by use of blow molding within mated mold halves. High ambient temperature-resistant nylon has been found to be a preferred material but other polymeric materials may be found to be equally acceptable. The illustrated shape of reservoir 10 has been found in one application to be well suited to the noted objective of efficient space utilization. It is mounted within the vehicle engine compartment generally in the upright or vertical condition illustrated. This design includes a main storage chamber portion 12 provided at its upper end with a filler neck 14 threaded for installation of a removable closure cap. Immediately therebelow, the main storage portion is molded with shoulder formations 16 and 18 defining a waist section adapted to cooperate with a clamp strap or band to mount the reservoir on a selected wall of the engine compartment. At the bottom of the main storage portion there is provided a depending well 20 containing a magnet disc 22 which attracts and holds any foreign metallic particles which may be introduced into the fluid circulating through reservoir 10.
Connection of the reservoir to the supply side of a remotely located power steering pump and associated fluid system is provided via an outlet hose nipple or tube 24 molded integrally with and extending laterally of the bottom of the main storage portion 12. Connection of the reservoir to incoming fluid exiting the remote power steering gear or other element of such system is provided by an inlet hose nipple or tube 26. As is typical with power steering gears and like fluid motors the exiting fluid thereof is carried in relatively small diameter conduits or hoses at relatively high velocity. Nipple 26 is molded or sized to conform to existing automotive specifications for such hose, as is nipple 24. It has been found that in the circumstances of such high velocity, special needs do arise, when providing a remote reservoir, to handle the incoming fluid prior to introduction into the main chamber portion 12 so that it does enter at much reduced velocity and free of turbulence tending toward undue entrainment of air that may be contained in the upper region of the main chamber 12.
To this end, reservoir 10 is molded with an antechamber portion 30 integrally associated with inlet nipple 26. Referring to FIGS. 1, 2 and 7, such antechamber portion is generally L-shaped in vertical section and structured to have a gradually varying cross sectional area taken in sections laterally across a sinuous path of fluid flow traversing the antechamber. Such fluid flow path is generally indicated by the arrows in FIG. 1 emanating inwardly from nipple 26. The fluid flow path leads through essentially a vertical leg of the antechamber ultimately to an opening 32 at the exit of the lower end or lateral leg region 34 of the antechamber. Such lower end or leg is generally triangular in lateral cross section as indicated best in FIG. 6 and has its walls 36 merging integrally with the walls of the main storage chamber 12 as seen best in FIG. 7. To minimize turbulence in the main chamber 12, the area of opening 32 is made many times larger than the area of the passage through nipple 26, thereby to substantially reduce fluid velocity. In one commercial embodiment, the area of opening 32 is in the order of 180 times larger than that of a nipple 26 having an I.D. of about 6 mm.
In the particular illustrated embodiment, the cross-sections continuously widen toward the lower end 34, FIGS. 4 through 6 showing the transition that occurs. Above such lower end the antechamber, by known blow-mold techniques, has its walls separated from the walls of the main chamber by a mold-closure gap 38, this term being employed to identify regions where the two halves of the mold pieces assume very close proximity or actual engagement and define the margin of major mold cavities for the two chamber portions. Such mold halves are most conveniently of mirror image construction. During molding, some separation between the mold halves may be used to result in filling of the gap by a molded stiffening web 40 generally the full length of the gap.
A similar mold-closure gap 42 exists just below the lower end 34 of the antechamber with a web 44 molded into such gap. During molding, the mold-closure represented by gap 42 defines the bottom wall of lower end 34 and the upper wall of a tapered outlet section 46 of the main storage chamber 12 leading to outlet nipple 24. As seen best in FIGS. 1 and 2 the outlet section 46 has generally the same shape in lateral cross-sections, i.e., triangular, as does the lower end 34 of the antechamber portion 30.
As set forth, the conformation of antechamber portion 30 as defined by the mold closure gaps 38 and 42 is thus generally of L-shape to define a like fluid flow path. Additionally, with the lateral orientation of nipple 26, the flow first impinges upon the opposed wall 48 of the antechamber portion 30 and is directed downwardly to traverse the gradually widening cross-sectional area of successive lateral cross-sections of the antechamber legs until the fluid approaches opening 32. There, the bottom wall of the lower end or leg 34 redirects the fluid laterally through the opening 32 to the lowest levels of the main chamber. The incoming fluid has at that point a low velocity and relatively little tendency toward disturbance of the quantity of fluid contained within the storage chamber 12.
While blow-molding is preferred, the reservoir may alternatively be fabricated otherwise, as by injection molding of halves and subsequent bonding thereof. Also, while the particular L-shaped antechamber with ever-widening sections has proven effective, it will be recognized that departure to similar shapes have and will yield beneficial results within the spirit of the invention.
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To meet the dual threats of emerging infectious diseases and engineered biowarfare/bioterror agents, there is a pressing need for more efficient systems for vaccine development. TRIAD, or the Translational Immunology Research and Accelerated [Vaccine] Development program, based in the Biotechnology Program at the University of Rhode Island, has pioneered the development and application of an integrated "gene to vaccine" in silico, in vitro and in vivo vaccine design program to address this need. TRIAD has selected Category A pathogens F. tularensis, Category B agents Burkholderia pseudomallei and Burkholderia mallei, and emerging infectious diseases (HCV, H. pylori, tick borne diseases) as the focal point of this proposal. Using the TRIAD immunoinfomatics Toolkit, TRIAD investigators will pursue the development of second generation epitopebased immunome-derived vaccines for these pathogens, while addressing the failings of prior generations of epitope based vaccines. We will maximize payload quantity using validated immunoinformatics tools that permit selection of optimal T cell epitopes that are highly conserved and immunogenic. We will ensure payload quality by choosing epitopes that demonstrate antigenicity in human PBMC as well as protection in established murine models of disease/infection. We will select a combination of promiscuous Class II epitopes and Class I supertype epitopes will provide >99% coverage of human populations. We will avoid cross-reactive epitopes and explore the role of regulatory T cells in the context of improving vaccine design. Where appropriate, we will combine our epitope-driven vaccines with broad-spectrum anti-LPS vaccines. We will optimize payload, delivery, formulation, and adjuvanting by exploring a range of delivery options [Dendritic cells, DEC205, DNA, electroporation, mucosal delivery). The TRIAD project aims to develop vaccines demonstrating broad spectrum activity include crossprotective and multiple component vaccines, and delivery technologies that have the potential to be effective against multiple emerging and re-emerging infectious diseases. Our efforst to merge rational design with rcentadvances in vaccine deliverywill manifest in a coordinated toolkit and a cadre of informed users, who will be ready and able to apply the tools to discover new treatments for emerging infectious disease and biodefense.
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Background {#Sec1}
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Colorectal surgery has traditionally been associated with significant morbidity and prolonged hospital stay \[[@CR1]--[@CR4]\]. Overall complication rates have been reported to be 26--35 % \[[@CR1], [@CR3], [@CR4]\]. Infectious complications, in particular, represent a major cause of morbidity and mortality after colorectal surgery \[[@CR4], [@CR5]\].
Albumin is considered a negative acute-phase protein because its concentration decreases during injury and sepsis. The rate of loss of albumin to the tissue spaces (measured as transcapillary escape rate) rises by more than 300 % in patients with septic shock \[[@CR6], [@CR7]\]. Hypoalbuminemia is a risk factor for mortality and postoperative complications \[[@CR8]--[@CR13]\]. Therefore, nutritional control has been an important focus of perioperative management \[[@CR14]\].
The magnitude of the systemic inflammatory response during the perioperative period, as indicated by the acute-phase proteins---C-reactive protein (CRP) in particular---may help to identify the risk of a postoperative infectious complication \[[@CR4], [@CR15]--[@CR21]\].
The correlation between serum albumin and CRP with gastrointestinal cancer has been reported \[[@CR22], [@CR23]\]. However, it is unclear whether antecedent CRP could be used to predict future hypoalbuminemia in the perioperative period of colorectal surgery.
The primary endpoint of this study was to reveal whether antecedent CRP could be used to predict future hypoalbuminemia in the perioperative period of colorectal surgery. The secondary endpoint was to clarify the relationship between CRP on postoperative day (POD) 3 and postoperative infectious complications.
Methods {#Sec2}
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Study design {#Sec3}
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This retrospective study included patients who had been admitted for elective open colorectal surgery from July 2011 to March 2013 at the Izumi Regional Medical Center. The following patient data were collected from medical charts: sex, age, albumin administration in the postoperative period, body mass index (BMI), type of surgery, tumor site, American Joint Committee on Cancer (AJCC) tumor-node-metastasis (TNM) staging, depth of tumor invasion, lymph node involvement, and postoperative oral intake. The tumors were staged according to the TNM criteria \[[@CR24]\].
The following laboratory data were determined preoperatively and on PODs 3 and 7: serum albumin, CRP, aspartate aminotransferase (AST), alanine aminotransferase (ALT), gamma-glutamyl transpeptidase (γ-GTP), lactate dehydrogenase (LDH), alkaline phosphatase (ALP), serum creatinine (Scr), blood urea nitrogen (BUN), hemoglobin (Hb), and white blood cell (WBC) count. Serum levels of albumin (normal range 4.0--5.0 g/dL) and CRP (normal range 0--0.3 mg/dL) were measured using the bromocresol green dye-binding method and turbidimetric assay with an autoanalyzer (Hitachi 7180; Hitachi, Tokyo, Japan).
Patients underwent mechanical bowel preparation with 2 L of polyethylene glycol electrolyte solution (Niflec; Ajinomoto Pharma, Tokyo, Japan). Prophylactic cefmetazole was administered from the day of the surgery (3 g/day) to POD 2 (2 g/day on POD 1 and POD 2). The study protocol was approved by the ethics committee of Izumi Regional Medical Center (approval number 20130812-1).
Examination of factors affecting perioperative serum albumin with colorectal surgery {#Sec4}
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Preoperative hypoalbuminemia is a risk factor for postoperative complications \[[@CR8], [@CR11], [@CR12]\]. Platt et al. provided data WBC, CRP, and albumin concentrations on preoperative and PODs 1--7 in 454 patients undergoing surgery for colorectal cancer, of whom 104 developed infectious complications. Results demonstrated that CRP measurements on POD 3 could accurately predict infectious complications, including anastomotic leak, after resection for colorectal cancer \[[@CR4]\]. The average time to development of an infectious complication, including an anastomotic leak, was between 6 and 8 days postoperatively \[[@CR4]\]. These results demonstrated the utility of factors affecting serum albumin on the preoperative day and PODs 3 and 7.
Moyes et al. reported that preoperative elevated modified Glasgow Prognostic Score predicts postoperative infectious complications in patients undergoing potentially curative resection for colorectal cancer \[[@CR25]\]. Therefore, the independent variables with a possible effect on serum albumin were chosen by referring to this report \[[@CR25]\]. The dependent variable was serum albumin and the independent variables were CRP, sex (male, 1; female, 0), age, albumin administration on the postoperative day (yes, 1; no, 0), tumor site (rectum, 1; colon, 0), AJCC TNM cancer stage (I, 1; II, 2; III, 3; IV, 4), depth of tumor invasion (T1, 1; T2, 2; T3, 3; T4, 4), lymph node involvement (N0, 0; N1, 1; N2, 2), BMI, postoperative oral intake (bad, 1; good, 0), AST, ALT, γ-GTP, LDH, ALP, Scr, BUN, Hb, and WBC count. Postoperative oral intake was used as an independent variable only on POD 7. Postoperative albumin administration was used as an independent variable on PODs 3 and 7.
Correlations between antecedent CRP and future serum albumin {#Sec5}
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We examined correlations between preoperative CRP and serum albumin on POD 3, between preoperative CRP and serum albumin on POD 7 and between CRP on POD 3 and serum albumin on POD 7.
Relationships between antecedent CRP and future hypoalbuminemia {#Sec6}
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By receiver operating characteristic (ROC) analysis \[[@CR4]\], we examined relationships between preoperative CRP and hypoalbuminemia on POD 3, between preoperative CRP and hypoalbuminemia on POD 7 and between CRP on POD 3 and hypoalbuminemia on POD 7. Hypoalbuminemia was defined as serum albumin ≤3.0 g/dL \[[@CR14], [@CR26]\].
Relationship between CRP on POD 3 and postoperative infectious complications {#Sec7}
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Patients were assessed for the following infectious complications: wound infection, intra-abdominal abscess, anastomotic leak, pneumonia and septicemia \[[@CR25]\]. The criteria used to define infectious complications were taken from the methods reported by Moyes et al. \[[@CR25]\]: (1) wound infection was defined as the presence of pus, either discharged spontaneously or requiring drainage. Wound infection included a subgroup of patients who developed perineal infection after abdominoperineal resection of the rectum. (2) Intra-abdominal abscess was verified by either surgical drainage or by ultrasonographically guided aspiration of pus. (3) Anastomotic leakage was defined as radiologically verified fistula to bowel anastomosis or diagnosed by repeat laparotomy. (4) Pneumonia was defined as a positive chest radiograph and requirement for antibiotic treatment. (5) Septicemia was defined by clinical symptoms combined with a positive blood culture. To reveal the relationship between CRP on POD 3 and postoperative infectious complications, the diagnostic accuracy of CRP was assessed by ROC analysis \[[@CR4]\].
Statistical analysis {#Sec8}
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Multiple regression analysis with stepwise variable selection was used to examine the factors affecting preoperative day and PODs 3 and 7 serum albumin with significance level of entering a selection at *p* \< 0.05 and of keeping a selection at *p* \< 0.10 \[[@CR27]\]. The significance level for keeping an independent variable in the final model was set at 0.01.
The relationships between antecedent CRP and future hypoalbuminemia were examined by ROC analysis \[[@CR4]\]. The relationships between CRP on POD 3 and postoperative infectious complications were performed using ROC analysis \[[@CR4]\]. The area under the ROC curve (AUC) results were considered excellent for AUC values between 0.9 and 1, good for AUC values between 0.8 and 0.9, fair for AUC values between 0.7 and 0.8, poor for AUC values between 0.6 and 0.7 and failed for AUC values between 0.5 and 0.6 \[[@CR28]\]. Statistical analysis was performed using Excel 2010 (Microsoft Corp., Redmond, WA, USA) with the add-in software Ekuseru-Toukei 2012 (Social Survey Research Information Co., Ltd., Tokyo, Japan). Additionally, EZR (Saitama Medical Center, Jichi Medical University, Japan), which is a graphical user interface for R \[[@CR29]\] (The R Foundation for Statistical Computing, Vienna, Austria) was used for ROC analysis only.
Results {#Sec9}
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Patient characteristics {#Sec10}
-----------------------
Patient characteristics are presented in Table [1](#Tab1){ref-type="table"}. Three-quarters of patients were older than 65 years of age. Laboratory values revealed no severe perioperative liver or kidney dysfunction.Table 1Characteristics of 37 patients who underwent colorectal surgeryPatients characteristics (n = 37)ValuesSex, no (%) Male/female19 (51):18 (49)Age (years)77 (38--86)BMI (kg/m^2^)22 (15.8--31)Tumor site, no. (%) Colon/rectum25 (68):12 (32)Type of surgery, no. (%) Colectomy25 (68) Anterior resection9 (24) Abdominoperineal resection of rectum2 (5) Hartmann procedure1 (3)TNM staging, no. (%) Stage I3 (8) Stage II14 (38) Stage III17 (46) Stage IV3 (8)Laboratory values in preoperative period Serum albumin (g/dL)4.1 (2.5--5.0) CRP (mg/dL)0.31 (0.03--16.67) AST (IU/L)21 (10--37) ALT (IU/L)17 (6--46) γ-GTP (IU/L)22 (8--123) LDH (IU/L)186 (124--500) ALP (IU/L)257 (122--679) Serum creatinine (mg/dL)0.64 (0.42--1.42) BUN (mg/dL)13.5 (5.8--40.8) WBC (/μL)6100 (2400--16500) Hemoglobin (g/dL)10.7 (5.4--16.6)Quantitative variables are expressed as medians (minimum--maximum). Qualitative variables are expressed as absolute numbers (percentages)*ALP* alkaline phosphatase, *ALT* alanine transaminase, *AST* aspartate transaminase, *BMI* body mass index, *BUN* blood urea nitrogen, *CRP* C-reactive protein, *γ-GTP* gamma glutamyl transpeptidase, *LDH* lactate dehydrogenase, *WBC* white blood cell
Postoperative infectious complications {#Sec11}
--------------------------------------
Postoperative complications are presented in Table [2](#Tab2){ref-type="table"}. Twelve (32 %) patients experienced postoperative complications, and nine (24 %) experienced only infectious complications. The most serious infectious complication was anastomotic leak. The median time to development of an infectious complication was 5 postoperative days.Table 2Postoperative complications after colorectal surgeryPostoperative complicationsNumberPercentageInfectious complications924Anastomotic leak13Wound infection25Intra-abdominal abscess25Pneumonia4^a^10Ileus25Cardiac complications13All complications1232Mortality00^a^2 patients with ileus
Factors affecting perioperative serum albumin with colorectal surgery {#Sec12}
---------------------------------------------------------------------
In the preoperative period, CRP and BUN were effective variables. CRP was significant (*p* \< 0.01), and the partial correlation coefficient was −0.497 (Table [3](#Tab3){ref-type="table"}).Table 3Variables identified as predicting serum albumin in the perioperative period of colorectal surgeryPointVariableRegression coefficientStandard errorStandardized regression coefficientPartial correlation coefficient*P*Preoperative periodConstant4.6760.254\<0.001CRP−0.0780.026−0.468−0.4970.005BUN−0.0370.015−0.393−0.4340.016POD 3Constant2.3790.376\<0.001Hb0.1190.0350.6170.5320.002CRP−0.0220.008−0.416−0.4390.012Albumin administration0.3050.1330.3740.3870.029Lymph node involvement0.1820.0850.3110.3660.040Scr−0.5290.279−0.262−0.3270.068Tumor site−0.2640.141−0.332−0.3240.070POD 7Constant1.9120.388\<0.001Hb0.1070.0320.5010.5060.002CRP−0.0240.008−0.429−0.4570.007γGTP−0.0060.002−0.399−0.4240.012Depth of tumor invasion0.1620.0850.2680.3190.066*BUN* blood urea nitrogen, *CRP* C-reactive protein, *γ-GTP* gamma glutamyl transpeptidase, *Hb* hemoglobin, *POD* postoperative day
On POD 3, Hb, CRP, albumin administration on the postoperative day, lymph node involvement, SCr, and tumor site were effective variables. Hb was significant (*p* \< 0.01), and the partial correlation coefficient was 0.532 (Table [3](#Tab3){ref-type="table"}).
On POD 7, Hb, CRP, γ-GTP, and depth of tumor invasion were effective variables. Hb and CRP were significant (*p* \< 0.01), and partial correlation coefficients were 0.506 and −0.457, respectively (Table [3](#Tab3){ref-type="table"}).
Correlations between antecedent CRP and future serum albumin {#Sec13}
------------------------------------------------------------
Significant correlations were observed between preoperative CRP and serum albumin on POD 3 (*p* = 0.023), between preoperative CRP and serum albumin on POD 7 (*p* = 0.023) and between CRP on POD 3 and serum albumin on POD 7 (*p* \< 0.001) (Table [4](#Tab4){ref-type="table"}).Table 4Correlations between antecedent CRP and future serum albumin in the perioperative period of colorectal surgeryVariableCorrelation coefficient*P*CRP in preoperative period and serum albumin on POD 3−0.37420.0225CRP in preoperative period and serum albumin on POD 7−0.37230.0233CRP on POD 3 and serum albumin on POD 7−0.54470.0005*CRP* C-reactive protein, *POD* postoperative day
Relationship between antecedent CRP and future hypoalbuminemia {#Sec14}
--------------------------------------------------------------
The AUC of CRP in the preoperative period to the development of hypoalbuminemia on POD 3 was 0.579 (95 % CI 0.392--0.766) with an optimal threshold of 0.86 mg/dL, sensitivity of 36.4 % and specificity of 93.3 % (Fig. [1](#Fig1){ref-type="fig"}), and the diagnostic accuracy resulted as failed. The AUC of CRP in preoperative period to the development of hypoalbuminemia on POD 7 was 0.683 (95 % CI 0.481--0.886) with an optimal threshold of 0.94 mg/dL, sensitivity of 50 % and specificity of 92 % (Fig. [1](#Fig1){ref-type="fig"}) and the diagnostic accuracy was poor. The AUC of CRP on POD 3 to development of hypoalbuminemia on POD 7 was 0.833 (95 % CI 0.679--0.987) with an optimal threshold of 12.43 mg/dL, sensitivity of 75 % and specificity of 80 % (Fig. [1](#Fig1){ref-type="fig"}), and the diagnostic accuracy was good.Fig. 1Diagnostic accuracy of antecedent CRP with regard to development of future hypoalbuminemia. **a** AUC of CRP in preoperative period to development of POD 3 hypoalbuminemia was 0.579 (95 % CI 0.392--0.766) with an optimal threshold of 0.86 mg/dL, sensitivity 36.4 % and specificity 93.3 %. **b** AUC of CRP in preoperative period to development of hypoalbuminemia on POD 7 was 0.683 (95 % CI 0.481--0.886) with an optimal threshold of 0.94 mg/dL, sensitivity 50 % and specificity 92 %. **c** AUC of CRP on POD 3 to development of hypoalbuminemia on POD 7 was 0.833 (95 % CI 0.679--0.987) with an optimal threshold of 12.43 mg/dL, sensitivity 75 % and specificity 80 %. *AUC* the area under the receiver operating characteristic curve, *CRP* C-reactive protein, *POD* postoperative day
Relationships between CRP on POD 3 and postoperative infectious complications {#Sec15}
-----------------------------------------------------------------------------
The AUC of CRP on POD 3 was 0.96 (95 % CI 0.902--1) with an optimal threshold of 13.8 mg/dL, sensitivity of 100 % and specificity 88 % (Fig. [2](#Fig2){ref-type="fig"}), and the diagnostic accuracy was excellent.Fig. 2Diagnostic accuracy of CRP on POD 3 with regard to development of infective complications after colorectal surgery. AUC of CRP on POD 3 was 0.96 (95 % CI 0.902--1) with an optimal threshold of 13.8 mg/dL, sensitivity 100 % and specificity 88 %. *AUC* the area under the receiver operating characteristic curve, *CRP* C-reactive protein, *POD* postoperative day
Discussion {#Sec16}
==========
In the present study, we examined whether antecedent CRP could be used to predict future hypoalbuminemia in the perioperative period of colorectal surgery. The main finding is that CRP on POD 3 may be of use in predicting the development of hypoalbuminemia on POD 7 (Fig. [1](#Fig1){ref-type="fig"}).
Three-quarters of patients were older than 65 years of age in the present study (Table [1](#Tab1){ref-type="table"}). We searched for similar studies that evaluated infectious complications of colorectal surgery and found that 67 % of the patients in the study by Moyes et al. \[[@CR25]\] were over 65 years of age, which is similar to the 67 % in Platt's report \[[@CR4]\], suggesting that the population in the present study is similar to the population in previous reports.
Twelve (32 %) patients experienced postoperative complications, and nine (24 %) experienced only infectious complications in the present study (Table [2](#Tab2){ref-type="table"}). Overall complication rates have been reported to be 26--35 % in colorectal surgery \[[@CR1], [@CR3], [@CR4]\]. Infectious complication rates have been reported to be 15--42 % in colorectal surgery \[[@CR4], [@CR25], [@CR30], [@CR31]\]. Therefore, the rates of all complications and infectious complications in the present study are similar to those in previous reports.
The correlation between serum albumin and CRP with gastrointestinal cancer has been reported previously \[[@CR22], [@CR23]\]. In present study, correlations were observed between serum albumin and CRP preoperatively (*p* \< 0.01) and between serum albumin on POD 3 and CRP on POD 3 (*p* = 0.012) and between serum albumin on POD 7 and CRP on POD 7 (*p* \< 0.01) (Table [3](#Tab3){ref-type="table"}) in stepwise multiple regression analysis. These findings suggest that CRP has the greatest association with serum albumin, and concur with the results of other related reports.
Hypoalbuminemia is a risk factor for mortality and postoperative complications \[[@CR8]--[@CR13]\]. Therefore, the identification of a predictor of hypoalbuminemia may be clinically significant. In present study, significant correlations were observed between CRP in preoperative period and serum albumin on POD 3 (*p* = 0.023), between CRP in the preoperative period and serum albumin on POD 7 (*p* = 0.023) and between CRP on POD 3 and serum albumin on POD 7 (*p* \< 0.001) (Table [4](#Tab4){ref-type="table"}). Additionally, the AUC of CRP on POD 3 to the development of hypoalbuminemia on POD 7 was 0.833 (95 % CI 0.679--0.987) with an optimal threshold of 12.43 mg/dL, sensitivity 75 % and specificity 80 % (Fig. [1](#Fig1){ref-type="fig"}), suggesting that CRP on POD 3 could be useful in predicting the development of hypoalbuminemia on POD 7. Therefore, CRP on POD 3 may be valuable for the indicator of early nutritional intervention.
We consider that hypoalbuminemia resulted from increased CRP, which can be explained by the following: inflammatory cytokines decrease the synthesis of constitutive proteins, such as serum albumin, and increase its degradation \[[@CR7]\]. They also promote capillary permeability and leakage of serum albumin into the extravascular space \[[@CR7]\]. Because CRP is affected by increased interleukin-6 during acute inflammation, a decrease in serum albumin occurs with increased CRP \[[@CR32]\].
The clinical utility of postoperative CRP has been reported \[[@CR4], [@CR33]\]. In particular, a large study (n = 454) by Platt et al. showed that CRP was a predictor of postoperative infectious complications after curative resection in patients with colorectal cancer and that postoperative measurement of CRP on POD 3 was clinically useful in predicting surgical site infectious complications, including anastomotic leak \[[@CR4]\]. In that study, the AUC of CRP on POD 3 was 0.8 (*p* \< 0.001) and the optimal cutoff value was 17 mg/dL, and the AUC of serum albumin on POD 3 was 0.68 (*p* \< 0.001) and the optimal cutoff value was 2.5 g/dL. The diagnostic accuracy for postoperative infectious complications of CRP on POD 3 was better than that of serum albumin on POD 3 \[[@CR4]\]. In the present study, the AUC of CRP on POD 3 with regard to development of infective complications after colorectal surgery was 0.96 (95 % CI 0.902--1) with an optimal threshold of 13.8 mg/dL, sensitivity 100 % and specificity 88 % (Fig. [2](#Fig2){ref-type="fig"}), suggesting that CRP on POD 3 could be useful to predict postoperative infective complications. Therefore, these results are consistent with those reported by Platt et al.
A limitation of this study is that retrospective data collection relied only on evaluation of clinical progress notes, laboratory test results, and other documentation. However, three-quarters of patients in this study were older than 65 years of age. Therefore, we believe our results apply to the elderly, in whom serum albumin is likely decreased. Prospective studies are needed to confirm whether our findings can be adapted to all colorectal surgery patients.
Conclusions {#Sec17}
===========
The present study revealed that CRP has the greatest association with serum albumin in the preoperative period and on PODs 3 and 7 and that antecedent CRP was associated with future serum albumin. Additionally, CRP on POD 3 could be useful in predicting hypoalbuminemia on POD 7. This result suggests that CRP on POD 3 may be valuable as an indicator of early nutritional intervention.
AJCC
: American Joint Committee on Cancer
ALP
: alkaline phosphatase
ALT
: alanine aminotransferase
AST
: aspartate aminotransferase
AUC
: the area under the receiver operating characteristic curve
BMI
: body mass index
BUN
: blood urea nitrogen
CRP
: C-reactive protein
γ-GTP
: gamma-glutamyl transpeptidase
Hb
: hemoglobin
LDH
: lactate dehydrogenase
POD
: postoperative day
ROC
: receiver operating characteristic
Scr
: serum creatinine
TNM
: tumor-node-metastasis
WBC
: white blood cell
AS, KI, and TI designed the study. AS, SO, YaI, SN, NH, and KI performed research. AS, YoI, KI, and TI analyzed the data. AS, SO, KI, YK, YoI, and TI drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements {#FPar1}
================
We thank the Department of Gastroenterology for its contribution to the study.
Competing interests {#FPar2}
===================
The authors declare that they have no competing interests.
|
{
"pile_set_name": "PubMed Central"
}
|
Post-hoc secondary analysis of data from our recent Edinburgh and Lothians Viral Intervention Study (ELVIS) pilot randomised controlled trial (RCT) indicates that hypertonic saline nasal irrigation and gargling (HSNIG) reduced the duration of coronavirus upper respiratory tract infection (URTI) by an average of two-and-a-half days. As such, it may offer a potentially safe, effective and scalable intervention in those with Coronavirus Disease-19 (COVID-19) following infection with the betacoronavirus Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) \[[@R1]\].
ELVIS was undertaken in 66 adults with URTI. Results have been reported in detail elsewhere \[[@R2]\]. Briefly, volunteers with URTI were within 48 hours of symptom onset randomised to intervention (n = 32) or control (n = 34) arms. The intervention arm made hypertonic saline at home and performed HSNIG as many times as needed (maximum of 12 times/day). Control arm participants dealt with their URTI as they normally did. Nose swabs collected at recruitment and first thing in the morning on four consecutive days were sent to the laboratory for testing. Both arms kept a diary (which included the Wisconsin Upper Respiratory Symptom Survey-21 questionnaire) for a maximum of 14 days or until they were well for two consecutive days. Follow-up data were available for 92% of individuals (intervention arm: n = 30; control arm: n = 31). HSNIG reduced the duration of URTI by 1.9 days (*P* = 0.01), over-the-counter medication use by 36% (*P* = 0.004), transmission within household contacts by 35% (*P* = 0.006) and viral shedding by ≥0.5 log~10~/d (*P* = 0.04) in the intervention arm when compared to controls \[[@R2]\].
We also recently reported that epithelial cells mount an antiviral effect by producing hypochlorous acid (HOCl) from chloride ions \[[@R3]\]. HOCl is the active ingredient in bleach. Epithelial cells have this innate antiviral immune mechanism to clear viral infections. Since bleach is effective against all virus types \[[@R4]\], we tested to see if a range of DNA, RNA, enveloped and non-enveloped viruses were inhibited in the presence of chloride ions supplied via salt (NaCl). All the viruses we tested were inhibited in the presence of NaCl. The human viruses we tested were: DNA/enveloped: herpes simplex virus; RNA/enveloped: human coronavirus 229E (HCoV-229E), respiratory syncytial virus, influenza A virus; and RNA/non-enveloped: coxsackievirus B3 \[[@R3]\].
In COVID-19, high titres of SARS-CoV-2 are detectable in the upper respiratory tract of asymptomatic and symptomatic individuals \[[@R5]\]. The titres are higher in the nose than the throat suggesting measures that control the infection and viral shedding will help reduce transmission \[[@R5]\]. In the context of the COVID-19 pandemic, we have undertaken a post-hoc re-analysis of the ELVIS data with a focus on those infected with coronaviruses. Coronaviruses were the second most common cause of URTI (after rhinoviruses). Fifteen individuals were infected by a coronavirus: 7 in the intervention arm, 8 in the control arm. In the intervention arm, four participants were infected by an alphacoronavirus (HCoV 229E = 3, HCoV NL63 = 1) and three by a betacoronavirus (HCoV HKU1 = 3). In the control arm, two were infected by an alphacoronavirus (HCoV NL63 = 2) and six by a betacoronavirus (HCoV OC43 = 1, HCoV HKU1 = 5). An individual in the control arm with HCoV HKU1 had dual infection with rhinovirus.
{#Fa}
The duration of illness was lower in the intervention arm compared to the control arm in the subset of patients infected with coronavirus (mean days (SD): 5.6 (1.4) vs 8.1 (2.9)). Using a two-sample *t* test, this was difference of -2.6 days (95% confidence interval (CI) = -5.2, 0.05; *P* = 0.054). The difference in the duration of blocked nose was -3.1 days (95% CI = -6.0, -0.2; *P* = 0.04), cough -3.3 days (95% CI = -5.9, -0.7; *P* = 0.02) and hoarseness of voice -2.9 days (95% CI = -5.6, -0.3; *P* = 0.03) in favour of HSNIG ([**Table 1**](#T1){ref-type="table"}). The severity of symptoms in individuals in the two arms can be seen in [**Figure 1**](#F1){ref-type="fig"}.
######
Number of days for self reported symptom improvement in the control and intervention arms infected by a coronavirus
Variable label Treatment N Mean SD Difference in mean (intervention -- control) (95% CI for difference) *P*-value
------------------ -------------- --- ------ ----- ---------------------------------------------------------------------- -----------
Blocked nose Intervention 7 4.0 2.2 -3.1 (-6.0, -0.2) 0.0362
Blocked nose Control 8 7.1 2.9
Chest congestion Intervention 7 1.9 1.2 -0.8 (-2.7, 1.2) 0.4056
Chest congestion Control 8 2.6 2.1
Cough Intervention 7 2.7 1.3 -3.3 (-5.9, -0.7) 0.0179
Cough Control 8 6.0 3.0
Head congestion Intervention 7 3.4 1.9 -1.9 (-5.0, 1.1) 0.1931
Head congestion Control 8 5.4 3.3
Hoarseness Intervention 7 2.4 1.6 -2.9 (-5.6, -0.3) 0.0325
Hoarseness Control 8 5.4 2.9
Scratchy throat Intervention 7 2.6 1.0 -2.1 (-5.1, 1.0) 0.1712
Scratchy throat Control 8 4.6 3.6
Sneezing Intervention 7 3.9 1.7 -1.0 (-3.8, 1.8) 0.4469
Sneezing Control 8 4.9 3.0
Sore throat Intervention 7 3.6 1.9 -1.1 (-4.4, 2.3) 0.5139
Sore throat Control 8 4.6 3.7
Runny nose Intervention 7 4.4 1.3 -1.6 (-4.1, 0.9) 0.1999
Runny nose Control 8 6.0 2.8
Feeling tired Intervention 7 3.6 1.8 -2.1 (-5.1, 1.0) 0.1671
Feeling tired Control 8 5.6 3.3
SD -- standard deviation, CI -- confidence interval
{#F1}
The individual in the control arm with a co-existing rhinovirus infection could have affected the results. Excluding this individual, the duration of illness in the control arm was a mean of 7.3 days (SD = 1.8). The impact on the intervention control comparison was to reduce the size of the difference to -1.7 days (95% CI = -3.6, 0.2; *P* = 0.07).
In the absence of a suitable antiviral agent or a vaccine, we need a safe and effective intervention that can be globally implemented. Our *in-vitro* data gives the evidence that NaCl has an antiviral effect that works across viral types. The findings from this *post-hoc* analysis of ELVIS need to be interpreted with caution. These data do however suggest that HSNIG may have a role to play in reducing symptoms and duration of illness in COVID-19.
**Funding:** The study was funded by Edinburgh and Lothians Health Foundation. The funder reviewed the grant application, but had no role in the study design, collection, analysis, interpretation of data, writing of the report or and in the decision to submit the paper for publication.
**Authorship contributions:** SR conceived the ELVIS trial and was PI on this leading it together with AS. SR, AS and CG planned this post-hoc subgroup analysis. CG was the trial statistician and undertook the secondary analysis. JD managed the virological testing, LM supported with project management expertise. SR and AS jointly drafted the manuscript, which was contributed to by LM and CG. All authors approved the final version of the manuscript.
**Competing interests:** The authors have completed the ICMJE uniform disclosure form (available upon request from the corresponding author) and declare no conflicts of interest.
|
{
"pile_set_name": "PubMed Central"
}
|
The purpose was to determine if whole body and skeletal muscle glutamine and leucine metabolism are altered in HIV-infected subjects. Six HIV-infected men with chronic stable opportunistic infections and 10% weight loss, 8 HIV-infected men and 1 woman without wasting and 6 HIV- negative age-and weight-matched men were studied. Constant intravenous infusions of stable isotopically labeled leucine and glutamine were used to assess plasma GLN and LEU rates of appearance by mass spectrometric measurements of stable isotope content. Fasting whole body protein breakdown and synthesis rates were increased above control in the asymptomatic HIV-infected subjects and further increased in symptomatic HIV-infected subjects. These findings suggest that the rate of muscle protein breakdown was increased while the rate of muscle protein synthesis was unchanged in symptomatic HIV-infected subjects. Since lymphocytes require GLN release may be increased to provide energy for proliferati ng lymphocytes.
|
{
"pile_set_name": "NIH ExPorter"
}
|
In certain architectures, service providers and/or enterprises may seek to offer sophisticated online conferencing services for their end users. The conferencing architecture can offer an “in-person” conference experience over a network. Conferencing architectures can also deliver real-time interactions between people using advanced visual, audio, and multimedia technologies. Virtual conferences and conferences have an appeal because they can be held without the associated travel inconveniences and costs. In addition, virtual conferences can provide a sense of community to participants who are dispersed geographically.
|
{
"pile_set_name": "USPTO Backgrounds"
}
|
1. Field of the Invention
The present invention relates to the on-demand production of laser ablation transfer ("LAT") imaging films, and, more especially, to the on-demand (or on-line) economical production of LAT imaging films presenting options of flexibility hitherto unknown in laser ablation transfer imaging science ("LATIS").
2. Description of the Prior Art
U. S. Pat. No. 5,156,938 to Diane M. Foley et al, assigned to the assignee hereof and hereby expressly incorporated by reference and relied upon, recounts the LATIS prior art and describes a unique method/system for simultaneously creating and transferring a contrasting pattern of intelligence on and from an ablation-transfer imaging medium to a receptor element in contiguous registration therewith that is not dependent upon contrast imaging materials that must absorb the imaging radiation, typically laser radiation, and is well adopted for such applications as, e.g., color proofing and printing, computer-to-plate, the security coding of various documents and the production of machine-readable or medical items, as well as for the production of masks for the graphic arts and printed circuit industries; the ablation-transfer imaging medium, per se, comprises a support substrate and an imaging radiation-, preferably a laser radiation-ablative topcoat essentially coextensive therewith, such ablative topcoat having a non-imaging ablation sensitizer and an imaging amount of a non-ablation sensitizing contrast imaging material ("CIM") contained therein.
Ellis et al copending application Ser. No. 07/707,039, filed May 29, 1991, also assigned to the assignee hereof and hereby expressly incorporated by reference and relied upon, describes improved ablation-transfer imaging media having greater sensitivity, requiring less sensitizer and threshold energy (thus permitting a greater range of mass to be transferred), and which additionally are kinetically more rapid and facilitate the ablative transfer to a receptor element of an imaging radiation-ablative topcoat containing virtually any type of contrast imaging material (whether sensitizing or non-sensitizing).
Such Ellis et al method/system for simultaneously creating and transferring a contrasting pattern of intelligence on and from a composite ablation-transfer imaging medium to a receptor element in contiguous registration therewith is improvedly radiation sensitive and versatile, is kinetically rapid and not dependent on a sensitized ablative topcoat, and is also very well adopted for such applications as, e.g., color proofing and printing, computer-to-plate, the security coding of various documents and the production of machine-readable or medical items, as well as for the production of masks for the graphic arts and printed circuit industries; the Ellis et al composite ablation-transfer imaging medium, per se, comprises a support substrate (i), at least one intermediate "dynamic release layer" (ii) essentially coextensive therewith and an imaging radiation-ablative carrier topcoat (iii) also essentially coextensive therewith, said imaging radiation-ablative carrier topcoat (iii) including an imaging amount of a contrast imaging material contained therein, whether or not itself including a laser absorber/sensitizer, and said at least one dynamic release layer (ii) absorbing such imaging radiation, typically laser radiation, at a rate sufficient to effect the imagewise ablation mass transfer of at least said carrier topcoat (iii).
By "dynamic release layer" is intended an intermediate layer that must interact with the imaging radiation to effect imagewise ablative transfer of at least the carrier topcoat onto a receptor element at an energy/fluence less than would be required in the absence thereof. The dynamic release layer ("DRL") is believed to release the carrier topcoat by effectively eliminating the adhesive forces that bond or consolidate the carrier topcoat with the support substrate. Preferably, under the same conditions additional propulsion is simultaneously provided by the interaction of the imaging radiation therewith, e.g., by ablation of the dynamic release layer itself, thus further facilitating the imagewise ablative transfer of the entire carrier topcoat to a receptor element. Representative DRLs per Ellis et al include metal, metal alloy, metal oxide and metal sulfide thin films, etc., and the organics.
Nonetheless, to data the LAT imaging films employed in, for example, the Foley et al and Ellis et al LATIS' described above have been limited to those "permanent" films available from inventory, namely, pre-manufactured or pre-coated, and, thus, which inventory is typically inadequate to supply the complete spectrum of LAT imaging films that may be required for a particular application, e.g., not all colors, not all color densities, not all film thicknesses, etc., are usually available from inventory.
|
{
"pile_set_name": "USPTO Backgrounds"
}
|
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