text: ....SE has said that they will have 5k enter per few days, close servers, do a maintenance/patch work, have another 5k enter the next time and so on and so on, then allow 10k to play per week or something along the lines of that, can't remember were I read the interview. But I'll find it if I have too. I think people are respecting the NDA and just not risking their account from being locked. But that is just me. I'm just thinking 1-2 weeks ago SE gave invites to the 1st wave of 5k, and let say for example, I wouldn't see mine till another 2 weeks, as they want to load the servers at a steady pace. Its just been too quite for me. I thinks its already started.


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Steak With Blue Cheese Butter
Recipe by The Flying Chef
A quick and easy BBQ meal.. Any leftover butter can be frozen and used at a later date. Prep time does not include the refrigeration time of the butter.
Top Review by breezermom
Very good! I used my own seasonings on the steak, but used the butter mix. I grilled my steaks over a charcoal grill. I had a 2 inch filet mignon, and that worked perfectly. I did have lots of butter left over, so I'll save that for another day. Delicious! Made for Best of 2011 Tag game. Thanks for sharing!
See Discussion:3 Reviews
Ingredients Nutrition
- 6 steak fillets, trimmed of excess fat
- 2 tablespoons olive oil
- 1 tablespoon fresh ground black pepper
Blue Cheese Butter
- 150 g butter, softened
- 65 g blue cheese
- 1 tablespoon parsley, finely chopped
- 1 teaspoon paprika
Directions See How It's Made
- Make butter.
- Combine butter, blue cheese, parsley and paprika in a bowl.
- Place butter on a plastic food wrap and roll into a log shape.
- Refrigerate 1 hour or until firm.
- Steaks.
- Preheat BBQ to a high heat.
- Put peppercorns and oil in a bowl.
- brush steaks with oil mixture.
- grill steaks 3-5 min's each side or until cooked to your desired doneness.
- Cut butter into approx 2cm thick slices and top each steak with 1 or 2 slices of butter.
- Serve dish immediately could serve with a nice garden salad.


text:
Latest content added for Digital Library Collection: World War One Collectionhttps://digital.library.unt.edu/explore/collections/WWI/browse/?sort=added_a&fq=untl_institution:UNTGD2006-06-12T21:34:08-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Collection: World War One CollectionHelp yourself : get up in the world by steady saving.2006-06-12T20:16:47-05:00https://digital.library.unt.edu/ark:/67531/metadc470/

All lettered poster with text in black, encouraging the public to save by buying War Savings Stamps, Treasury Savings Certificates, and Liberty Bonds.
[Wire to the Axis]2006-06-12T20:43:58-05:00https://digital.library.unt.edu/ark:/67531/metadc587/

A serpent made of metal wears a helmet with a swastika on it. The serpent has long tail, sharp teeth, and a long snout. Smoke billows in background. Metal beams in a "V" formation appear to approach the serpent from the upper left area of the poster.
La grande offensiva Italiana : la disfatta Austriaca sul Montello.2006-06-12T20:55:34-05:00https://digital.library.unt.edu/ark:/67531/metadc438/

Detailed color illustration of a battle scene. Italian soldiers in blue cross a river and charge against Austrian soldiers in green. Several Austrian soldiers and a horse are falling or dead. A toppled weapon is seen in the foreground. Several explosions can be seen on the river, its banks, the mountains, and a town in the distance. Airplanes are flying, cannons are firing, and multitudes of Italian troops are marching in the far distance.
V : invest.2006-06-12T20:56:54-05:00https://digital.library.unt.edu/ark:/67531/metadc437/

Red poster with blue and white border. In the center is a large blue "V" and below that is the word "INVEST" in white.
Marne, Yser, Somme, Verdun : l'aurore.2006-06-12T20:58:40-05:00https://digital.library.unt.edu/ark:/67531/metadc436/

At right, a French soldier in uniform and helmet holds a French flag. To the left and slightly behind him are two women, one wearing a traditional dress and bonnet of the Lorraine region and the other wearing the traditional dress and headdress of Alsace. The woman from Lorraine brings the flag to her lips as if to kiss it. The top section of the poster contains a portion of the title printed in red, which is partly obscured by laurel branches and a portion of the flag from the lower section.
Share in the victory : save for your country, save for yourself : buy war savings stamps.2006-06-12T20:59:21-05:00https://digital.library.unt.edu/ark:/67531/metadc435/

Color poster with turquoise blue background. A female figure dressed in a flowing ivory gown holds a sword in one hand and a palm branch in the other. She has wings and wears a laurel wreath around her head.
I summon you to the comradeship : answer the Red Cross Christmas Roll Call for Universal Membership.2006-06-12T21:01:09-05:00https://digital.library.unt.edu/ark:/67531/metadc434/

Portrait drawing of U.S. President Woodrow Wilson in black and white. Large Red Cross symbol at lower left area. Signature of Woodrow Wilson appears in lower right area between the text captions.
Halt the Hun! : buy U.S. government bonds, Third Liberty Loan.2006-06-12T21:02:50-05:00https://digital.library.unt.edu/ark:/67531/metadc433/

Drawing of an Allied Forces soldier pushing a German soldier away from a frightened woman who is holding a small child in her lap. The German soldier has blood on his hands. Red flames rise from the ground in the background.
Buy Liberty Bonds.2006-06-12T21:04:06-05:00https://digital.library.unt.edu/ark:/67531/metadc432/

Color poster with picture frame drawn as border. Profile of a bust of Abraham Lincoln is in center.
Defeat the Kaiser and his U-boats : victory depends on which fails first, food or frightfulness : eat less wheat.2006-06-12T21:34:08-05:00https://digital.library.unt.edu/ark:/67531/metadc418/

In the foreground is the profile of Kaiser Wilhelm II wearing a German helmet. His face is colored green. In the background is a u-boat in the water; farther in the distance is a sinking ship with smoke rising from it. The sky is colored orange and the orange sky is reflected on the surface of the water. Centered at the bottom of the poster is a circular seal: a shield with a U.S. flag motif, surrounded by wheat stalks. (This is a seal of the U.S. Food Administration.)


text:
Patagonia Capilene Midweight baselayer hits the sweet spot of weight, breathability and wicking that, worn alone or with additional layers, is the essence of versatility. This baselayer keeps you dry and comfortable during high-exertion activities in cool-to-moderate temperatures and insulates beneath a shell in cold weather.
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Mount Snow Team Junior Accolade Jacket by Spyder
SMS Men's Team Jacket Winter 2017
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text:
Psych 2017 Chapter 7: Summarizing and interpreting data using statistics
- Morgan Watson
- 2 years ago
- Views:
1 Psych 2017 Chapter 7: Summarizing and interpreting data using statistics Descriptive statistics - The gal f descriptive statistics is t describe sample data - Can be cntrasted with inferential statistics where the gal is t make inferences abut ppulatins frm sample data Frequency distributins - A listing f values in a data set alng with their frequency Graphic representatins f data - Bar graph f frequency distributin Used with categrical variables Height f bar represents frequency f categry Bars shuld nt tuch - Histgram Used with cntinuus quantitative variables N space between bars - Line graphs Als used with quantitative variables Particularly useful fr interpreting interactins - Scatterplts Depicts relatinship between tw quantitative variables Crrelatins Measures f central tendency - Prvides a single value that is typical f the distributin f scres Mde Mst frequently ccurring value Least useful measure f central tendency Median Middle scre when numbers are in ascending r descending rder Mean Arithmetic average Mst cmmnly used measure f central tendency Measures f variability - Prvides a numerical value indicating the amunt f variatin in a grup f scres Range Highest scre minus lwest scre Rarely used as a measure f variability Variance Average deviatin f the data values frm their mean in squared units Standard deviatin Square rt f variance Rughly the average amunt that individual scres deviate frm the mean Standard deviatin and the nrmal curve Z-scres Standardized values transfrmed frm raw scres Mean f z-distributin is always zer; standard deviatin always ne
2 Indicates hw far abve r belw a raw scre is frm its mean in standard deviatin units; e.g., a z-scre f indicates a raw scre that is ne standard deviatin unit abve the mean In a nrmal distributin, the prprtin f scres ccurring between any tw pints can be determined Examining relatinships amng variables - Unstandardized difference between mean A cmparisn f mean differences between levels f an independent variable - Standardized difference between means Inferential statistics - Hyptheses - Significance testing - Errrs Hyptheses - Null hypthesis Effect size Chen s d is a cmmn measure f effect size Mean difference is divided by standard deviatin Small, medium, and large effect sizes are indicated by values f at least.2,.5, and.8 respectively Predicts n effect r relatinship in the ppulatin - Scientific/ Alternative hypthesis Predicts an effect r relatinship in the ppulatin Significance testing - Determines the likelihd f btaining sample data when null hypthesis is true - A decisin making prcess- determining if yu are dealing with ne r tw ppulatins Hw d yu decide t reject the null hypthesis? - Must base this n analysis f data cllected - Cmparisn f grup means Tw ppulatins if yu reject the null hypthesis One ppulatin if yu fail t reject the null hypthesis Mean- arithmetic average Example: peple wh sleep lnger will scre higher n a cgnitive skills test Randm assignment t tw levels f the IV 4 hurs r 8 hurs SCAT (Sam s Cgnitive Ability Test) Errr- grups will differ by chance alne - Standard deviatin and variance Bth measure the variability f scres arund the mean Standard deviatin is the square rt f variance If there is verlap in the scres f the varius grups the mean difference in the scres may be due t chance - Inferential statistical technique These techniques enable yu t estimate the prbability f the grup mean differences ccurring by chance By judging the difference between their means relative t the spread r variability f their scres
3 Significance testing - Alpha level Chsen by researcher- highest prbably value taken as evidence against the null hypthesis (i.e., sample value is t extreme fr null hypthesis) - Critical regin Prtin f distributin f sample means cntaining extreme scres- size is equal t alpha level - p value prbability f btaining sample values when the null hypthesis is true- cmpared with alpha t determine if null hypthesis can be rejected (i.e., test is significant) Errrs - Type I errr: reject the null hypthesis when it is true - Type II errr: fail t reject the null hypthesis when it is false Type I and Type II errrs are threats t validity Fail t Reject the H Reject the H
4 Chapter 14: The nuts and blts f using statistics Selectin f a statistical test - The statistical test selected is based n the design f the study Is it a within r between participants design Hw many independent variables are there- ne r mre than ne Hw many levels f the IV are there
5 Inferential statistical test guides - Independent samples t-test Between participants design Tw levels f ne independent variable - One- way analysis f variance Between participants design Mre than tw levels f ne independent variable - Tw- way analysis f variance Between participants design Tw r mre independent variables - Paired samples t-test Within-participants design Tw levels f ne independent variable - Repeated measures analysis f variance Within- participants design
6 Mre than tw levels f the independent variable Independent samples t-test - Used with a between participants design with ne IV and tw levels f the IV T-value rati f the between grup mean difference t the average within grup variability Hw large must the rati be t reject the null? Significance level - Definitin- prbability that an bserved difference is due t chance Reject the null if there is less than a 5% (alpha=.05) r 1% (alpha=.01) chance f the between grup difference being due t chance variability - Hw t determine if the significance level is reached Cmpare the t-value btained with the critical value f t Critical value f t identified by degrees f freedm and alpha Degrees f freedm= N-2 fr independent samples t-test Making a decin based n the cmputed and critical values - Reject the null if Cmputed t-value critical t-value P-value <.05 - Fail t reject the null if Cmputed t-value < critical t-value P-value >.05 One- way analysis f variance - Invlves partitining ttal variance int tw independent surces f variance - Between- grups variatin: measure f tw surces f variability Any treatment effect that exists Chance variatin - Within- grups variatin: Chance variatin - Ttal variatin= between+ within grups variatin Mean square - Mean square (MS) is a variance estimate used in Anvas - Between and within-grups variatin is divided by their degrees f freedm MS = grups variatin/ degrees f freedm - Between degrees f freedm number f grups minus 1 - Within degrees f freedm- number f research participants minus the number f grups F-rati - F-rati= MS (between)/ms (within)=(treatment + chance)/chance
7 - N treatment effect, then F-rati=1 - Fr a treatment effect t exist F-rati must be greater than 1 Interpreting results f analysis f variance - First identify the critical value f F crrespnding t the selected significance level--.05 r.01 significance level Identified by lking in the table f critical values f F Enter tables using the between and within degrees f freedm - Reject null f F-value critical value f F - Fail t reject the null if F- value < critical value f F - If yu reject the null, then yu must interpret the results Calculate the mean f each grup t see which grup perfrmed best D pst hc tests t see which grups were significantly different Tw-way analysis f variance - Statistical test applied t data cllected frm factrial designs - Prvides a test f main and interactin effects - Lgic is the same as with ne-way ANOVA Partitin ttal variatin fr each independent variable Interactin variatin Measure f chance r errr variatin - Steps in calculating a tw-way ANOVA Cmpute each surce f variatin Cmpute the MS fr each surce f variatin- divide the variatin by its df Calculate an F-rati fr each main effect and interactin surce f variatin by dividing each by the MS (errr) Interpreting tw-way ANOVA - Cmpare each F-value with the critical value f F Reject null if F-value critical value f F Fail t reject the null if F-value < critical value f F - If yu reject the null then must interpret the results Main effects Interactins Rejecting r failing t reject the null hypthesis - Reject the null if the t r F-value yu calculated meets r exceeds the critical value btained frm the t r F tables - Fail t reject if it des nt meet the critical value Because it is lgically impssible t accept the null Smetimes may need t prvisinally accept it When n data refute a null cnclusin When the prbability f the null being false is large When the design is pwerful in cntrlling fr extraneus variable Ptential errrs in statistical decisin- making - Type I errr- falsely reject the null Cntrlled by the significance level - Type II errr- fail t reject a false null - Tends t be an inverse relatinship between Type I and II errrs - Assume Type I errrs are the wrst s fcus n cntrlling these
8 Chapter 8: Reprting Research Guidelines fr effective writing - Knw yur audience - Identify yur purpse Expsitry writing discurse designed t cnvey infrmatin r explain what is difficult t understand - Write clearly Ecnmy f expressin: using as few wrds as pssible t cnvey an idea Precisin Adherence t grammatical rules - Write an interesting reprt Structure f a research reprt - Title page - Abstract - Intrductin - Methds - Results - Discussin
9 - References - Appendixes - Ftntes The APA frmat - Page 1- title page Running head Title Authr(s) name(s) and institutinal affiliatins Authr nte Department affiliatins f each authr Identify grants r ther supprt and acknwledge assistance in the cmpletin f the prject Identify persn f cntact - Page 2- abstract On every page put a header and the page number in the upper right hand curner f the manuscript Abstract in nly ne paragraph 150 wrds State prblem under investigatin Describe experimental methd State the main findings State cnclusin and implicatins - Next series f cntinuus pages Intrductin Intrduce the general tpic/research questin Review backgrund studies in terms f relevance t research questin and hypthesis develpment Briefly describe current study State hyptheses and justificatin fr hyptheses Overall, argue that current study is imprtant and will add new knwledge t the field Methd Purpse f the methds sectin Allws reader t assess validity and reliability f the study Allws researchers t replicate yur study Cmpnents f the methd sectin Participants Design Materials/apparatus Prcedure Can cmbine sectins- Participants and prcedure are mst imprtant Participants Number Imprtant demgraphics Where btained Cmpensatin
10 Design Type f research design Variables (dependent, independent) Hw variables were measured/manipulated Materials/ apparatus Descriptin f stimuli r scales used in the study Where stimuli/scales were btained r hw they were develped Descriptin f any apparatus used in the study Prcedure Chrnlgical descriptin f what the participants/ subjects experienced in the study Descriptin f instructins given t participants Order f tasks Results The results sectin summarizes the data cllected and the statistical treatment f them. First, briefly state the main results f findings. Discussing the implicatins f the results is nt apprpriate here. Mentin all relevant results, including thse that run cunter t the hypthesis. D nt include raw scres r raw data with the exceptin, fr example, f single case designs r illustrative samples. Dependent measure analyzed Statistical tests used Alpha level used fr tests Descriptive statistics- figures/tables Statistical values frm tests p values frm tests Statements abut significant effects r relatinships Discussin Begin with a clear and cncise statement f the essential findings What have I cntributed here? Hw has my study helped t reslve the riginal prblem? Cmpare current results with thse frm past studies Prvide explanatins f unexpected results What cnclusins and theretical implicatins can I draw frm my study? Discuss pssible flaws in the study Discuss future research - References -- starts a new page The reference sectin prvides a cmplete listing f all the surces cited in the article The references are listed in alphabetical rder by the last name f the first authr - Ftntes- n a separate page - Tables - Figure captins APA 6 th - n same page as figures - Figures - Appendixes- used t prvide a verbatim cpy f questinnaires, respnse sheets, and/r instructins - Multiple study articles Intrductin sectin that prvides backgrund fr all yur studies
11 Methd and results fr each study General discussin sectin that describes cnclusins fr all the studies cmbined Other APA frmatting issues t keep in mind - Cite all surces Past studies (Smith & James, 2009) have shwn r Smith and James (2009) have shwn Include a title page with a title, running head (ges n every page), page number, authrs names, and authrs affiliatins - 1 inch margins fr all pages - Fllw heading structure: centered in bld, left justified in bld - New page fr abstract, intrductin, references, tables, figures nly - Qutatins- fewer than 40 wrds, insert int text - Numbers- use wrds fr numbers less than 10 - Physical measurements- use metric - Tables- use nly when they can cnvey and summarize the data mre ecnmically and clearly than can a discussin - Figures- use when they cnvey a cncept mre effectively than can a table - Figure captins- a brief descriptin f the cntent Presentatin f statistical results - Mean and standard deviatin The sample as a whle was relatively yung (M = 19.22, SD = 3.45) The average age f students was years (SD = 3.45) - Percentages Nearly half (49%) f the sample was married - Chi-square - T-tests - ANOVAs The percentage f participants that were married did nt differ by gender, X^2(1, N = 90) = 0.89, p >.05. There was a significant effect fr gender, t(54) = 5.43, p <.001, with men receiving higher scres than wmen. There was a significant main effect fr treatment, F(1, 145) = 5.43, p <.01, and a significant interactin between treatment and time f day, F(2, 145) = 3.13, p < Crrelatins The treatment and depressin scres were strngly crrelated, r(55) =.49, p <. 01. Specifically, as amunt f treatment increased, depressin scres increased (see Figure 2). Submissin f the research reprt fr publicatin - Chse a jurnal t submit yur article t - Jurnal impact factr Based n the average frequency f citatins fr articles in a jurnal The higher the impact factr, the mre selective the jurnal and the harder it is t get papers published in the jurnal Psychlgical sciences Send t the editr f the selected jurnal Include a cver letter stating yu are submitting the manuscript Editr will send the manuscript ut fr review
12 In several mnths yu will have the manuscript returned with the reviewers cmments and the editrs decisin Reject Accept revise - Jurnals fr undergraduate research Jurnal f Undergraduate Studies Mdern Psychlgical Studies: A Jurnal f Undergraduate Research The Jurnal f Psychlgy and the Behaviral Sciences The Psi Chi Jurnal f Student Research Presenting research results at prfessinal cnferences - Oral presentatin Fllw general APA fr rder f infrmatin Briefer frm than paper Designed fr visual and ral acquisitin Clarity f visual aids (e.g., PPTs) imprtant - Pster presentatin Fllw general APA fr rder f infrmatin Visual acquisitin imprtant Use pictures and bullet pints fr clarity and space issues Make sure infrmatin flws clearly dwn clumns and frm left t right n pster Prepare shrt ral explanatin f what is n the pster t guide viewer thrugh - Cnferences where undergraduates can present Suthern Sciety fr Philsphy and Psychlgy MPA (Midwestern Psychlgy Assciatin) Psi Chi APA (American Psychlgical Assciatin) and APS (American Psychlgical Sciety). Cmmn pitfalls - Prblem: misuse f citatins and qutes- students tend t veruse qutes in research reprts and incrrectly cite surces - Prblem: incrrect use f verb tense- students ften use the wrng verb tense t describe their studies - Prblem: prviding t many details f backgrund studies- writing a full summary f a past study in the intrductin f a research reprt - Prblem: describing backgrund literature frm secndary surces0 students ften use secndary surces fr literature reviews leading t prblems with misrepresentatin f the primary surce - Prblem: fcus n statistics in results sectin- making statements abut the statistics instead f the effect r relatinship tested
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text:
Reflection Positivity of the Free Overlap Fermions
- Ira Hudson
- 2 years ago
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1 Yoshio Kikukawa Institute of Physics, the University of Tokyo, Tokyo , Japan Department of Physics, the University of Tokyo , Japan Institute for the Physics and Mathematics of the Universe (IPMU), the University of Tokyo, Chiba , Japan It is shown that free lattice Dirac fermions defined by overlap Dirac operator fulfill the Osterwalder-Schrader reflection positivity condition with respect to the link-reflection. The XXVIII International Symposium on Lattice Field Theory, Lattice2010 June 14-19, 2010 Villasimius, Italy Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
2 1. Introduction In this short article, we will examine the reflection positivity [1, 2, 3] of lattice fermions defined through overlap Dirac operator [4, 5], a gauge-covariant solution to the GW relation [4, 5, 6, 7, 8, 9], which has been derived in the five-dimensional domain wall approach [10, 11, 12]. The reflection positivity of the GW fermions is not fully understood yet [13, 14, 15], while that of Wilson fermions has been rigorously proved in various ways including the gauge interacting case [3, 16, 17]. In the following, It will be shown rigorously that free overlap Dirac fermion fulfills the reflection positivity with respect to the link-reflection. In ref [13], Lüscher discussed the unitarity property of free overlap Dirac fermion by investigating the positivity through the spectral representation of free propagator and concluded that free overlap Dirac fermion has a good unitarity property. Our direct proof of the reflection positivity given here is consistent with this observation. Our proof will be also extended to the non-gauge models with interactions such as chiral Yukawa models. For gauge models, however, a proof of reflection positivity, if any, seems to be more involved and we will leave it for future study. 2. Reflection positivity Reflection positivity is a sufficient condition for reconstructing a quantum theory in the canonical formalism, i.e. the Hilbert space of state vectors and the Hermitian Hamiltonian operator acting on the state vectors, from the lattice model defined in the Euclidean spacelet us formulate the reflection positivity condition for lattice Dirac fermions. We assume a finite lattice Λ = [ L + 1,L] 4 Z 4 in the lattice unit a = 1, and impose antiperiodic boundary condition in the time direction, and periodic boundary conditions in the space directions. The fermionic action is defined in the bilinear form A( ψ,ψ) = ψ(x)d L ψ(x), (2.1) x Λ with a lattice Dirac operator D L 1. The kernel of the Dirac operator should be written as D L (x,y) = n Z 4 ( 1) n 0 D(x + 2nL,y), x,y Λ, (2.2) where D(x,y) is the kernel of the Dirac operator in the infinite lattice Z 4. The quantum theory is then completely characterized by the expectational functional defined by the fermionic pathintegration: F := 1 Z D[ψ]D[ ψ]e A( ψ,ψ) F( ψ,ψ), (2.3) 1 The reader might prefer the sign convention where A = ψd L ψ in stead of (2.1). These two sign convention in the fermionic action are connected to each other by the transformation ψ = i ψ and ψ = iψ. We have chosen this sign convention for the sake of the proof of the reflection positivity following [3]. As we will see in section 3, this sign convention is suitable to prove the statement (iv) A P. 2
3 where the Grassmann integration for each field variable is specified as dψ α (x)ψ α (x) = 1, d ψ α (x) ψ α (x) = 1, (2.4) and the functional measure is defined by D[ψ]D[ ψ] := x Λ;α=1,2,3,4 {dψ α (x)d ψ α (x)}. (2.5) The reflection positivity condition a condition on this expectational functuonal is formulated as follows: let us define time reflection operator θ which acts on polynomials of the fermionic field variables by the relations θ(ψ(x)) = ( ψ(θx)γ 0 ) T, θ( ψ(x)) = (γ 0 ψ(θx)) T (2.6) θ(αf + βg) = α θ(f) + β θ(g), θ(fg) = θ(g)θ(f), (2.7) where we denote θ(t, x) = ( t + 1, x) and F,G are arbitrary polynomials of fermionic fields and * means complex conjugation. Let Λ ± Λ be the sets of sites with positive or non-positive time respectively. Let A ± be the algebra of all the polynomials of the fields on Λ ±, and A on Λ. Then one says the theory is reflection positive if its expectation : A C satisfies θ(f + )F + 0 for F + A +. (2.8) A popular choice of lattice Dirac operator is the Wilson Dirac operator, D w = Here we consider the overlap Dirac operator { 1 µ=0,1,2,3 2 γ µ( µ µ) + 1 } 2 µ µ., (2.9) ( ) D = X 2 X X, X = D w m, (2.10) for 0 < m 1. This lattice Dirac operator describes a single massless Dirac fermion and satisfies the GW relation, γ 5 D + Dγ 5 = 2Dγ 5 D. Although the action is necessarily non ultra-local [18], the free overlap Dirac fermion indeed satisfies the reflection positivity condition, as will be shown below. 3. Proof of Reflection Positivity of overlap Dirac fermion To prove the reflection positivity, we need some additional definitions and notations. First, let us denote F 0 := D[ψ]D[ ψ]f( ψ, ψ). (3.1) 3
4 This 0 defines a linear function from A into C. Second, we decompose the lattice action A into the following three parts : A = A + + A + A (3.2) where A + A +, A A, and A is the part of the action which contain both positive and negative time fields. Thirdly, let us call P the set of all polynomials of the form j θ(f + j )F + j in a finite summation, where F + j A +. Although the above definition of P works well for the proof of the Wilson fermion, it is not enough for the proof of the overlap fermion. In our case of the overlap fermion, one needs to consider not only finite summations of the form j θ(f + j )F + j, but also infinite summations or integrations like dsθ(f(s))f(s) = lim N N k=1 θ(f(s k ))F(s k ) s k, (3.3) where the integration is defined as a limit of a finite Riemanian summation (see also eq. (3.18)). To this end, we consider P, the closure of P. The closure P contains not only elements of the original P, but also all the limit points of conversing sequences in P. That is, F P {F n } n=1 P : lim F n = F. (3.4) n Here, the sequence {F n } n A is defined to be convergent to some F A, if any coefficient in F n converges to the corresponding coefficient in F as a complex number 2. Note that with respect to this definition of convergence, the linear operation, the product operation in A, and the linear mappings 0, : A C are all continuous functions, i.e. if F n F, G n G, then αf n + βg n αf + βg, F n G n FG, (3.5) F n F, F n 0 F. (3.6) Now, we note the fact that the following four statements (i)-(iv) imply the reflection positivity: (i) If F,G belong to P then FG also belongs to P. (ii) For all F P, F 0 0. (iii) θ(a + ) = A. (iv) A P. In fact, from these statements, it follows that e A θ(f + )F + 0 = e A ++A + A θ(f + )F + 0 = e A ++θ(a + )+ A θ(f + )F + = θ(e A + )e A + e A θ(f } {{ } + )F } {{ + } P (by (i),(iv)) P 0 0 (3.7) 0 2 This definition of convergence in A is equivalent to the norm convergence of the Grassmann algebra induced from the metric of the underlying vector space, where the fermionic fields form an orthonomal basis. 4
5 for arbitrary F + A +. Considering the special case where F + = 1 A +, we have e A 0 0. Hence, we obtain e A θ(f + )F + θ(f + )F + = 0 e A 0 0. (3.8) Therefore the proof is reduced to showing these four statements (i)-(iv). Next, we will give the proofs of the statements (i)-(iv). The statement (i) follows from the similar statement with P, which has been proved for the Wilson case [3]. To show the statement (ii), one should refer to the definition of fermionic integration measure. With the definition (2.5), it is sufficient to consider F + A + of the form for which one can see F + = x Λ + ;α=1,2,3,4 { ψ α (x)ψ α (x)} P, (3.9) D[ψ]D[ ψ]θ(f + )F + = {det(γ 2 0 )} 16L4 = 1 0. (3.10) Therefore, one concludes that for arbitrary F P, F 0 0. Take arbitrary F P. Then there exists a converging sequence {F n } n such that F n F. From the continuity of 0 (see (3.6)), we obtain F 0 = lim F n = lim F n n 0 n 0 0. (3.11) The statement (iii) can be shown by using the property of the overlap Dirac kernel: D L (x,y) = γ 0 D L (θx,θy)γ 0. To show the statement (iv) A P, we use a spectral representation of D L (x,y). To derive the spectral representation of D L, we first Fourier transform the overlap Dirac operator kernel D(x,y) in the infinite volume: D(x, y) = x0 y 0 d 4 p eip (x y) X(p 0, p) (2π) 4 2 X X(p 0, p), (3.12) where X(p 0, p) = µ iγ µ sin p µ + µ (1 cos p µ ) m. Then, we change the p 0 integration region, [ π,π], to the contours along the imaginary axis in the complex p 0 plane by Cauchy s integration theorem, as shown in FIG. 1. Depending whether x 0 y 0 > 0 or x 0 y 0 < 0, we choose the contours [ie 1,i ] or [ ie 1, i ], respectively, to obtain d 3 D(x, y) = p x0 y 0 >0 (2π) 3 d 3 p D(x, y) = x0 y 0 <0 (2π) 3 E 1 de 2π e E(x 0 y 0 ) e ip (x y) X(iE, p) X X(iE, p) E 1 (3.13) de 2π ee(x 0 y 0 ) e ip (x y) X( ie, p) X X(iE, p). (3.14) where E 1 is the edge of the cut coming from the square root, and is determined by the relations X X(iE 1, p) = 0, E 1 > 0. (3.15) 5
6 In this spectrum representaion of D, it is very crucial to notice the fact that γ 0 X(±iE, p) (E E 1 ) are positive definite matrices and there exist matrices Y ± (E, p) such that γ 0 X(±iE, p) = Y ±Y ± (E, p) (E E 1 ). (3.16) In fact, it is not difficult to check that Y ± (E, p) are given by Y ± (E, p) = 3 k=1 l(e, p)sin p k W(E, p) where W(E, p) = 3 k=1 (1 cos p k) + 1 coshe m and l(e, p) = 1 sinhe 2 3 k=1 sin2 p k /W(E, p) ( 1 + W(E, p) γ k i 2l(E, p) γ 0 + il(e, p), 1 3 k=1 sin2 p k +W(E, p) 2 sinh 2 E ) 1 2 From the equations (2.2), (3.13) and (3.14), we find the spectrum representation of D L (x,y) as follows: putting V = 1/(2L) 3, de 1 1 D L (x,y) = x0 y 0 E 1 2π 1 + e 2EL V e E x 0 y 0 e ip (x y) X(εiE, p) X X(iE, p) p + p E 1 de e 2EL 1 2π 1 + e 2EL V ee x 0 y 0 ip (x y) X( εie, p) e X X(iE, p), (3.17) where ε is defined as the sign of x 0 y 0, and the spacial momentum p k runs over p k = n k π/l, ( L n L) in the above summation. In (3.17), the first term becomes D(x,y) in the limit L, and the second term represents a finite lattice effect which vanishes in the limit L. The latter is the contribution of the wrong-sign-energy modes and the minus sign appearing in front of X( εie, p) comes from the anti-periodicity in the time direction, which is required for the positivity, as will be seen. From these observations, now we can show that A P: for the term with x 0 > 0, y 0 0 (in this case ε = 1), we obtain x Λ + y Λ [ ] de 1 ψ(x)d L (x,y)ψ(y) = C E,p θ(c E,p ) + D E,p θ(d E,p ), (3.18) p E 1 2π V π Im ie 1 0 π Re π Im 0 π Re ie 1. Figure 1: Complex integration contours 6
7 where C E,p and D E,p are defined by 1 C E,p = 1 + e 2EL x Λ + e D E,p = 2EL 1 + e 2EL x Λ + ψ(x)γ 0 Y + (E, p) e Ex 0 e ip x, (3.19) ψ(x)γ 0 Y (E, p) e Ex 0 e ip x, (3.20) with Y + (E, p) = Y + (E, p)/( X X(iE, p)) 4 1. The overall minus sign in the r.h.s. of (3.18) results from (3.17) by using (3.16). This minus sign is canceled after exchanging the order of the Grassmann products in (3.18), and we see that this term belongs to P. One can show similarly that the term with x 0 0, y 0 > 0 (in this case ε = 1) also belongs to P. Thus we obtain A P and complete the proof of the reflection positivity. Acknowledgements K.U. would like to thank Tsutomu T. Yanagida for valuable encouragements. K.U. is supported by Global COE Program the Physical Science Frontier", MEXT, Japan. This work was supported by World Premier International Center Initiative (WPI Program), MEXT, Japan. Y.K. would like to thank S. Hashimoto and H. Fukaya for discussions. Y.K. is supported in part by Grant-in-Aid for Scientific Research No , References [1] K. Osterwalder and R. Schrader, Commun. Math. Phys. 31, 83 (1973). [2] K. Osterwalder and R. Schrader, Commun. Math. Phys. 42, 281 (1975). [3] K. Osterwalder and E. Seiler, Annals Phys. 110, 440 (1978). [4] H. Neuberger, Phys. Lett. B 417, 141 (1998). [5] H. Neuberger, Phys. Lett. B 427, 353 (1998). [6] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982). [7] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B 427, 125 (1998). [8] P. Hasenfratz, Nucl. Phys. B 525, 401 (1998). [9] M. Luscher, Phys. Lett. B 428, 342 (1998). [10] D. B. Kaplan, Phys. Lett. B 288, 342 (1992). [11] Y. Shamir, Nucl. Phys. B 406, 90 (1993). [12] V. Furman and Y. Shamir, Nucl. Phys. B 439, 54 (1995). [13] M. Luscher, arxiv:hep-th/ [14] M. Creutz, Phys. Rev. D 70, (2004). [15] J. E. Mandula, Phys. Rev. D 80, (2009). [16] M. Luscher, Commun. Math. Phys. 54, 283 (1977). [17] P. Menotti and A. Pelissetto, Nucl. Phys. Proc. Suppl. 4, 644 (1988). [18] I. Horvath, Phys. Rev. Lett. 81, 4063 (1998). 7
1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
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The disaster information lists multiple types of environmental disaster including:
Drought: prolonged or abnormally dry weather or lack of rain, especially that which affects the growth of crops or deplete reserves of water for industrial or domestic use. A drought poses a risk of serious environmental damage through vegetation loss and soil erosion, and a significant threat to the agricultural sector.
Heatwave: according to the Bureau of Meteorology, a heatwave is defined by three or more days of unusually high maximum and minimum temperatures in any area. There are three categories: Areas of Heatwave can expect unusually hot conditions sustained for three days. The level of heat is unusual, but injury to people is not expected unless inappropriate activities are conducted or sensible precautions are not followed. In a Severe Heatwave, vulnerable people are at risk of injury. Areas of Extreme Heatwave are likely to impact area such as infrastructure, transport, energy, agriculture and both healthy and vulnerable people are at risk of injury.
Image: Wetlands in the Coorong National Park, Mundoo (flickr)